microsoft word seyran_article_new2.doc ìàòåìàòè÷åñêèå âîïðîñû êèáåðíåòèêè è âû÷èñëèòåëüíîé òåõíèêè 24, 2005, 144-146. 144 ðàñïîçíàâàíèå ïî ïåðåñå÷åíèÿì ìíîæåñòâ ñåéðàí ì. âàðäàíÿí èíñòèòóò ïðîáëåì èíôîðìàòèêè è àâòîìàòèçàöèè íàí ðà e-mail seyranv@ipia.sci.am àííîòàöèÿ â ñòàòüå ðàññìàòðèâà þòñÿ çàäà÷è ðàñïîçíàâàíèÿ ïî ïåðåñå÷åíèÿì ìíîæåñòâ è äîêàçûâàåòñÿ ìèíèìàëüíîñòü è òóïèêîâîñòü íåêîòîðûõ ñèñòåì. ëèòåðàòóðà [1] ï. ýðäåø, äæ. ñïåíñåð. âåðîÿòíîñòíûå ìåòîäû â êîìáèíàòîðèêå, ìîñêâà, ìèð, 1976. [2] ñ.ì.âàðäàíÿí. îá îäíîé çàäà÷å ðàñïîçíàâàíèÿ ìíîæåñòâ, äîêëàäû àêàäåìèè íàóê àðì. ññð, òîì. 72, ñ.141-143, 1981ã. [3] ñ.ì.âàðäàíÿí. àëãîðèòì ïîñòðîåíèÿ îáîáùåííûõ òåñòîâ. ìîäåëèðîâàíèå, îïòèìèçàöèÿ, óïðàâëåíèå. âûï.4, ñ.74-77, åðåâàí 2001ã. ö³ý³ãáõù áëï µ³½ùáõãûáõýý»ñç ñ³ïáõùý»ñç ê. ì³ñ¹³ýû³ý ²ù÷á÷áõù ðá¹í³íáõù ¹çï³ñïíáõù ¿ ׳ý³ãù³ý ëý¹çñ áëï µ³½ùáõãûáõýý»ñç ñ³ïáõùý»ñç: òáõûó ¿ ïñíáõù áñáß ×³ý³ãáõ ñ³ù³ï³ñ·»ñç ùçýçù³éáõãûáõýá ¨ ÷³ïáõõ³ûýáõãûáõýá: microsoft word 30.doc mathematical problems of computer science 38, 73, 2012. 73 on a neo-logicistic conception of natural number hakob madoyan phd in theoretical philosophy, logic and philosophy of science armenian state pedagogical university chair of philosophy and logic email: hasama@inbox.ru in the report we discuss a neo-logicistic conception of natural number formulated by british philosopher crispin wright in his book frege’s conception of numbers as objects (1983). for appropriate understanding of the differences between logicistic and neo-logicistic conceptions of natural numbers, we recall bertrand russell’s criticism of formal axiomatic systems containing peano axioms. russell argued that such systems with peano axioms could not serve as real foundation for arithmetic, for they define not exactly our ordinary arithmetic but a group of systems sharing the same structure. russell’s argument can be generalized into a principle according to which a foundation of arithmetic may be called logicistic only if in addition to well-known principles of logicism it also retains our ordinary intuition of numbers. we recall also gottlob frege’s informal definition of natural number in his grundlagen der arithmetik (1884) and its formalization in the grundgesetze der arithmetik (1893, 1903). in the light of russell’s argument we outline the differences between frege’s initial conception and wright’s neo-logicistic conception of natural number. microsoft word tpel.doc математические вопросы кибернетики и вычислительной техники 32, 116--119, 2009. 116 èññëåäîâàíèå óñòîé÷èâîñòè èíôîðìàöèîííîé áàçû äàííûõ êàäðîâ íà îñíîâàíèè àíàëèçà õàðàêòåðà ðîòàöèîííûõ ïîòîêîâ àðìåí ìóðàäÿí ìèíèñòåðñòâî îáîðîíû ðåñïóáëèêè àðìåíèÿ аннотация â ðàáîòå ïðåäëàãàåòñÿ ìåòîä èññëåäîâàíèÿ èíôîðìàöèîííîé áàçû äàííûõ êàäðîâ îðãàíèçàöèè, ïóòåì àíàëèçà âîçíèêàþùèõ â íåé êðèòè÷åñêèõ ðîòàöèîííûõ è çàñòîéíûõ ïîòîêîâ çà íåêîòîðûé ïåðèîä âðåìåíè. ëèòåðàòóðà [1]. ². º. ´³µáõñû³ý, ².ð. ²ãáû³ý, ². ². øáõñ³¹û³ý. “èçëïç ëùµ»ñç ñ³ßí³ñïáõùá ¨ ï³ýë³ï»ëáõùá áëï ïíû³éý»ñç ï»õ»ï³ïí³ï³ý µ³½³ûç”, §ð³ûï³ï³ý µ³ý³ï¦, n3, ¿ç. 63, 2006. [2]. º. ð³ñáõãûáõýû³ý, î. ô³½³ýãû³ý, ü. ø»ëñáåû³ý, ¸.²ë³ïñû³ý, ø. ð³ñáõãûáõýû³ý, ø. ê³ñ³ïû³ý, ð. þ³ñáõùû³ý. ð³í³ý³ï³ýáõãûáõý ¨ ïçñ³é³ï³ý íç׳ﳷñáõãûáõý, ðð ¶²² §¶çïáõãûáõý¦ ññ³ï³ñ³ïãáõãûáõý, ºñ¨³ý, 2000. [3]. à. à. ìóðàäÿí. “ñîâìåùåííûé ìíîãîâàðèàíòíûé àëãîðèòì êîìïàíîâêè è ðàçìåùåíèÿ”, èçâåñòèÿ àí àðì.ññð, ñåð. òåõ.íàóê., ò. xli, 3, ñòð. 70, 1988. î³¹ñ»ñç ï»õ»ï³ïí³ï³ý ïíû³éý»ñç µ³½³ûç ï³ûáõýáõãû³ý ñ»ï³½áïáõùá éáï³óçáý ñáëù»ñç µýáõã³·ñ»ñç í»ñéáõíáõãû³ý ñçù³ý íñ³ ². øáõñ³¹û³ý ²ù÷á÷áõù ²é³ç³ñïíáõù ¿ ï³½ù³ï»ñåáõãû³ý ï³¹ñ»ñç ï»õ»ï³ïí³ï³ý ïíû³éý»ñç µ³½³ûç í»ñéáõíáõãû³ý ù»ãá¹՝ í»ñ³ñëïíáõ å³ù³ý³ï³ñ³ïí³íáõù, ïñçïçï³ï³ý éáï³óçáý ¨ é׳óù³ý å³ï׳éý»ñç ñ»ï³½áïáõãû³ý ñçù³ý íñ³: ռçëïç ëùµç áñáßù³ý ñ³ù³ñ ï³éáõóí»é ¿ ñ³ù³å³ï³ëë³ý ù³ã»ù³ïçï³ï³ý ùá¹»é: d:\sbornik\...\hodvac2.dvi mathematical problems of computer science 24, 2005, 11{15. linear p r ogr amming with changing coe±cients of objective function a r m e n h . a la ve r d ya n institue for informatics and automation problems of nas of ra e-mail armen am@yahoo.com abstract the article studies the problem of linear programming, when the objective function's coe±cients change depending on the data °ow. the set of objective function coe±cients under which the vertex of the feasible set of linear programming is an optimal solution is described. refer ences [1] ï.ëàíêàñòåð, òåîðèÿ ìàòðèö, ì.: íàóêà. 1978. [2] õ.ïàïàäèìèòðèó, ê.ñòàéãëèö, êîìáèíàòîðíàÿ îïòèìèçàöèÿ: àëãîðèòìû è ñëîæíîñòü. ì.: ìèð. 1985. [3] à. ñõðåéâåð, òåîðèÿ ëèíåéíîãî è öåëî÷èñëåííîãî ïðîãðàììèðîâàíèÿ. ì.: ìèð. 1991. ¶í³ûçý íñ³·ñ³íáñáõùª ÷á÷áëíáõ ·áñí³ïçóý»ñáí ýå³ï³ï³ûçý ýáõýïóç³ûáí ². ð. ²é³í»ñ¹û³ý ²ù÷á÷áõù ðá¹í³íá ýíçñí³í ¿ ·í³ûçý íñ³·ñ³íáñù³ý ëý¹ñç áõëáõùý³ëçñù³ýá, »ñµ ñáëù³ûçý ïíû³éý»ñçó ï³ëí³í ÷á÷áëíáõù »ý ýå³ï³ï³ûçý ýáõýïóç³ûç ·áñí³ïçóý»ñá: üï³ñ³·ñí³í ¿ ·í³ûçý íñ³·ñ³íáñù³ý ãáõûé³ïñ»éç ï»ï»ñç µ³½ùáõãû³ý ·³·³ãçý ñ³ù³å³ï³ëë³ýáõ ýå³ï³ï³ûçý ýáõýïóç³ûç ·áñí³ïçóý»ñç ³ûý µ³½ùáõãûáõýá, áñç ¹»åùáõù ³û¹ ·³·³ãá ñ³ý¹çë³ýáõù ¿ ûåïçù³é éáõíáõù: 1 1 d:\sbornik\...\article.dvi mathematical problems of computer science 24, 2005, 82{85. on t r ees with a special p r oper p ar tial 0 ¡ 1 color ings v a h a n v . mkr t c h ya n department of informatics and applied mathematics, yerevan state university e-mail vahanmkrtchyan2002yahoo.com abstract it is proved that in a tree in which the distance between any two endpoints is even, there is a maximum proper partial 0 ¡1 coloring such that the edges colored by 0 form a maximum matching. refer ences [1 ] l o va s z l ., p lu m m e r m.d ., ma t c h in g th e o r y, a n n a ls o f d is c r e t e ma t h . 2 9 , n o r t h h o lla n d , 1 9 8 6 . [2 ] h a r a r y f., \ gr a p h th e o r y" , a d d is o n -w e s le y, r e a d in g , ma , 1 9 6 9 . ð³ïáõï ïçåç ×ß·ñçï ù³ëý³ïç 0-1 ý»ñïáõùý»ñ å³ñáõý³ïáõ í³é»ñç ù³ëçý ì.ì. øïñïãû³ý ²ù÷á÷áõù ²ßë³ï³ýùáõù óáõûó ¿ ïñí»é, áñ ³ûý í³é»ñáõù, áñáýóáõù ó³ýï³ó³í »ñïáõ ï³ëí³í ·³·³ãý»ñç ñ»é³íáñáõãûáõýá ½áõû· ãçí ¿, ·áûáõãûáõý áõýç ù³ùëçù³é ×ß·ñçï ù³ëý³ïç 0-1 ý»ñïáõù, áñ 0 ·áõûýáí ý»ñïí³í ïáõ»ñç µ³½ùáõãûáõýá ï³½ùáõù ¿ í³éç ù³ùëçù³é ½áõ·³ïóáõù: 8 2 microsoft word intensiv.doc îçµ»éý»ïçï³ûç ¨ ñ³ßíáõ³ï³ý ï»ëýçï³ûç ù³ã»ù³ïçï³ï³ý ñ³ñó»ñ 26, 2006, 76–81. 76 ºñ㨻ïáõãû³ý çýï»ýëçíáõãû³ý ¨ íý³ë³ï³ñ ³ñï³ý»ïáõùý»ñç ù³ý³ïç ù³ã»ù³ïçï³ï³ý ùá¹»é (ºñ¨³ý ù³õ³ùç ûñçý³ïç íñ³) ²ñù»ý³ï úáõ. ²ûí³½û³ý ðð ¶²² ¾ïáéá·³ ýááëý»ñ³ûçý ñ»ï³½áïáõãûáõýý»ñç ï»ýïñáý ²ù÷á÷áõù ²ßë³ï³ýùáõù ¹çï³ñïíáõù ¿ ºñ¨³ý ù³õ³ùç ï³ñ¨áñ³·áõûý ëý¹çñý»ñçó ù»ïá` ³íïáùáµçé³ûçý ïñ³ýëåáñïç »ñ㨻ïáõãû³ý ï³½ù³ï»ñåáõùá, ýå³ï³ï áõý»ý³éáí ÷áõáóý»ñç »ñ㨻ïáõãû³ý çýï»ýëçíáõãû³ý ¨ ¹ñ³ ñ»ï ï³åí³í ãáõý³íáñ ³ñï³ý»ïáõùý»ñç ù³ý³ïç ñ³ßí³ñïá ¨ ñëïáõùá: àñå»ë »é³ï»ï³ûçý ïíû³éý»ñ áý¹áõýí³í ¿ ºñ¨³ý ù³õ³ùç ï绽»ñ³ï³ý ýï³ñá, ùß³ïí³í ¿ ñ³ù³å³ï³ëë³ý ïíû³éý»ñç µ³½³ ¨ íñ³·ñ³ûçý ³å³ñáíáõù, áñç ñçù³ý íñ³ ï³ñáõ ¿ ï³½ùí»é ºñ¨³ý ù³õ³ùç ïñ³ýëåáñï³ûçý ñáëù»ñç í³ýñ³µ»éýí³íáõãû³ý ù³ëý³·çï³óí³í ù³ñ﻽á: ¶ñ³ï³ýáõãûáõý [1] ð.ì.ø³ùë³å»ïû³ý, ²íïáùáµçéç ï»ëýçï³ß³ñ³·áñí³ï³ý ñ³ïï³ýçßý»ñç í»ñéáõíáõãûáõý: ºñ¨³ý, §èáõûë¦, 1989ã. [2] ë.à.àôîíîñåâ, àâòîìîáèëüíûå ïåðåâîçêè è åäèíàÿ òðàíñïîðòíàÿ ñèñòåìà. ìîñêâà, “âûñøàÿ øêîëà”, 1991 [3] â.í.ëóêàíèí, þ.â.òðîôèìåíêî, ïðîìûøëåííî/òðàíñïîðòíàÿ ýêîëîãèÿ. ìîñêâà, “âûñøàÿ øêîëà”, 2001 ². úáõ. ²ûí³½û³ý 77 mathematical model of the traffic intensity and of the quantity of poisonous rubbish (on the example of the yerevan city) a. ayvazyan center for ecological-noosphere studies of nas ra abstract in this paper one of the most important problems of the yerevan city are considered that is organising automobile transport traffic with purpose of control and calculation of street’s traffic intensity and quantity of poisonous rubbish in the result of it. the space picture of the yerevan city is taken as initial data. the corresponding database is created and automatic system is developed in virtue of which the specialized map of the yerevan city transport streams loading can be formed. microsoft word tpel.doc mathematical problems of computer science 31, 16-27, 2008. 16 completely normal elements in iterated quadratic extensions of finite fields of odd characteristics melsik k. kyuregyan and ofelya a. manukyan institute for informatics and automation problems, armenian national academy of sciences e-mails: melsik@ipia.sci.am, manofa81@yahoo.com abstract in this paper computationally easy explicit constructions of sequences of irreducible and normal monic polynomials over finite fields of odd characteristic are presented. references [1] r. chapman, “completely normal elements in iterated quadratic extensions of finite fields”, finite fields appl., volume 3, pp. 3-10, 1997. [2] s. d. cohen, “the explicit construction of irreducible polynomials over finite fields”, design, codes and cryptography, volume 2, pp. 169-174, 1992. [3] m. k. kyuregyan, “recurrent methods for constructing irreducible polynomials over qf of odd characteristics”, finite fields appl., volume 9, pp. 39-58, 2003. [4] g. mcnay, topics in finite fields, ph.d. thesis, university of glasgow, 1995. [5] a. j. menezes, i. f. blake, x. gao, r. c. mullin, s. a. vanstone, t. yaghoobian, “applications of finite fields”, kluwer publishers, boston, dordrecht, lancaster, 1993. [6] h. meyn, “explicit n-polynomials of 2-power degree over finite fields”, i, designs, codes and cryptography, volume 6, pp. 107-116, 1995. î»ýï µýáõã³·ñçãáí í»ñç³íáñ ¹³ßï»ñç ù³é³ïáõë³ûçý çï»ñ³ïçí áý¹é³ûýáõùý»ñç íñ³ ³ùµáõç³ï³ý ýáñù³é ï³ññ»ñ ø. îûáõñ»õû³ý ¨ ú. ø³ýáõïû³ý ²ù÷á÷áõù ü»ñï³û³óí³í »ý ãµ»ñíáõ ¨ ýáñù³é µ³½ù³ý¹³ùý»ñç ñ³çáñ¹³ï³ýáõãûáõýý»ñç ï³éáõóù³ý ³é·áñçãùý»ñ, áñáýù ñ³ßíáõ³ï³ý ï»ë³ýïûáõýçó ñ»ßï »ý ¨ ï³éçë »ý µ³½ù³ý¹³ùý»ñç µ³ó³ñ³ûï ï»ëù»ñá: microsoft word finaly 2006 rus.doc ìàòåìàòè÷åñêèå âîïðîñû êèáåðíåòèêè è âû÷èñëèòåëüíîé òåõíèêè 25, 2006, 57–63. 57 программный пакет генерации целочисленных преобразований григор а.петросян институт проблем информатики и автоматизации нан ра e-mail spl@ipia.sci.am аннотация пакет дает возможность в зависимости от диапазона представления чисел данной системы, генерировать матрицы целочисленных преобразования фурье, dct-1-4, dst-1-4 и хаарa порядка 2n. литература [1] john miano, “compressed image file fоrmat”, addison-wesley 2003. [2] михаил гук, “процессоры pentium 4 athlon и duron”, питер 2001. [3] emmanual c. ifeachor, “digital signal processing, a practical aproach”, prentice hall, 2002 [4] flores i., “the logic of computer arithmetic”. prentice hall, 1963 [5] l. cheng, y. zeng. fast multiplierless approximations of dct. ieee trans. circuits and systems-ii. 2002. [6] ahmed,rao. orthogonal transforms for digital signal processing. new york. 1975 [7] s. oriantara, y. j. chen, truong nguyen,.”integer fast fourier transform”, ieee 2001. [8] k.r. rao, p.c. yip, “the transform and data compression handbook”, crc press 2001 [9] g.a. petrosyan, h.g. sarukhanyan, idct-ii and realization using mmx technology, information technologies and management. number 3, 38–43 pages. 2004. [10] k.komatsu and k.sezaki, “reversible subband coding of images”, proc. spie vcip, vol.2501, pp.676-684, may 1995. программный пакет генерации целых преобразований 58 ²ùµáõçãí³ûçý ó¨³÷áëáõãûáõýý»ñç ë»ñù³ý íñ³·ñ³ß³ñ ¶. ². ä»ïñáëû³ý ü»ñï³û³óí³í ¿ ³ùµáõç³ãí³ûçý ó¨³÷áëáõãûáõýý»ñç ë»ñù³ý (itg) íñ³·ñ³ûçý ñ³ù³ï³ñ·ç áý¹ñ³ýáõñ ýï³ñ³·ñáõãûáõýá: æýãå»ë ñ³ûïýç ¿ ³ùµáõç³ãí³ûçý ó¨³÷áëáõãûáõýý»ñá ³ãùç áýïý»éáí ³ñ³·áõãû³ùµ ¨ ñçßáõáõãû³ý ëý³ûáõáõãû³ùµ áñ³ï³ï³ý óáõó³ýçßý»ñáí ·ñ»ã» ã»ý ½ççáõù ýñ³ýó: ü»ñï³û³óí³í ÷³ã»ãá ñý³ñ³íáñáõãûáõý ¿ ï³éçë ëï³ý³é 2 m ï³ñ·ç üáõñû»ç, êçýáõëáç¹³ûçý ¨ ð³³ñç ³ùµáõç³ãí³ûçý ó¨³÷áëáõãûáõýý»ñç ù³ïñçóý»ñ᪠ï³ëí³í ïíû³é ñ³ù³ï³ñ·ç ãí»ñç ý»ñï³û³óù³ý ñý³ñ³íáñ ùçç³ï³ûùçó: êï³óí³í ³ùµáõç³ãí³ûçý ó¨³÷áëáõãûáõýý»ñá ï³ñáõ »ý éç³ñå»ùáñ»ý û·ï³·áñíí»é å³ïï»ñý»ñç ¨ ³½¹³ýß³ýý»ñç ùß³ïù³ý ëý¹çñý»ñáõù: microsoft word 03.doc ìàòåìàòè÷åñêèå âîïðîñû êèáåðíåòèêè è âû÷èñëèòåëüíîé òåõíèêè 26, 2006, 15-20. 15 об одном методе пороговой локальной сегментации изображения* * работа выполнена в рамках госбюджетного тематического финансирования армении по проекту 734 давид г. асатрян и григорий с. сажумян èíñòèòóò ïðîáëåì èíôîðìàòèêè è àâòîìàòèçàöèè íàí ðà e-mail dasat@ipia.sci.am àííîòàöèÿ рассмотрена задача локальной сегментации изображения методом его разбиения на непересекающиеся однородные связные области. предложен новый подход к оцениванию качества сегментации, основанный на анализе поведения суммарного квадратического отклонения интенсивностей пикселов от средних по сегментам, в зависимости от принятой шкалы интенсивностей. предложен алгоритм решения задачи. приведены модельные примеры сегментации изображения. литература [1] w.pratt. digital image processing. 3-rd ed., j.wiley $ sons, inc., n.y., 2001. [2] g.x. ritter; j.n. wilson. handbook of computer vision algorithms in image algebra. boca raton: crc press llc, fl.,1996. [3] i. pitas. digital image processing fundamentals. thessaloniki, 1998. [4] advanced signal processing handbook. editor: stergios stergiopoulos. boca raton: crc press llc, 2001. [5] nikhil r. pal and sankar k. pal. a review on image segmentation techniques. pattern recognition, 26(9):1277-1294, 1993. [6] robert m. haralick and linda g. shapiro. image segmentation techniques. computer vision, graphics, and image processing, 29:100ñ132, 1985. [7] p. hubert. the segmentation procedure as a tool for discrete modeling of hydrometeorogical regimes. stoch. env. res. and risk ass., vol. 14, pp.297-304, 2000. об одном методе пороговой локальной сегментации изображения 16 ä³ïï»ñç ß»ù³ûçý ï»õ³ûçý ñ³ïí³í³íáñù³ý ùç »õ³ý³ïç ù³ëçý ¸. ²ë³ïñû³ý ¨ ¶. ê³åáõùû³ý ²ù÷á÷áõù ¸çï³ñïí»é ¿ ãñ³ïíáõ ñ³ù³ë»é ï³å³ïóí³í ïçñáõûãý»ñç ïñáñù³ý »õ³ý³ïáí å³ïï»ñç ß»ù³ûçý ï»õ³ûçý ñ³ïí³í³íáñù³ý ëý¹çñ: ²é³ç³ñïí»é ¿ ñ³ïí³í³íáñù³ý áñ³ïç ·ý³ñ³ïù³ý ýáñ ùáï»óáõù` ñ»ýí³í ñ³ïí³íý»ñç ï³ññ»ñç` ùçççý í³éáõãûáõýçó ·áõù³ñ³ûçý ù³é³ïáõë³ûçý ß»õáõùý»ñç í³ñùç í»ñéáõíáõãû³ý íñ³, ï³ëí³í í³éáõãûáõýý»ñç ñ³ù³ñ áý¹áõýí³í ë³ý¹õ³ïçó: ²é³ç³ñïí»é ¿ ëý¹ñç éáõíù³ý ³é·áñçãù: ´»ñí»é »ý å³ïï»ñç ñ³ïí³í³íáñù³ý ùá¹»é³ûçý ûñçý³ïý»ñ: d:\sbornik\...\tigran.dvi mathematical problems of computer science 24, 2005, 107{119. analysis of case splitting in an ar ithmetical system tig r a n m. ga lo ya n institue for informatics and automation problems of nas of ra e-mail tiger galo@yahoo.com abstract investigations in this paper concern the analysis of case splitting [(: a ! b) ! (a ! b) ! b] in an arithmetical system. it is shown that the case splitting can be done according to a thicker class of formulas (quanti¯er-free formulas) instead of decidable formulas. the latter includes the class of quanti¯er-free formulas. some approaches to the case splitting, promoted by other authors, have been investigated, and some corrections concerning the selection of quanti¯er-free formulas instead of decidable formulas have been done. according to these corrections, a new derivation of case splitting is suggested. refer ences [1 ] s . c. k le e n e , in t r o d u c t io n t o me t a m a t h e m a t ic s . b ib lio t e c h a ma t h e m a t ic a , 1 9 5 7 . [2 ] w ilfr ie d b u c h h o lz , u lr ic h b e r g e r a n d h e lm u t s c h wic h t e n b e r g , r e ¯ n e d p r o g r a m e xt r a c t io n fr o m cla s s ic a l p r o o fs . in s t it u t e mitta g-l e ffl e r , t h e r o ya l s we d is h a c a d e m y o f s c ie n c e s , 2 0 0 0 . [3 ] u lr ic h b e r g e r a n d h e lm u t s c h wic h t e n b e r g . p r o g r a m d e ve lo p m e n t b y p r o o f tr a n s fo r m a t io n . in h . s c h wic h t e n b e r g , e d it o r , p roof and computation, vo lu m e 1 3 9 o f series f : computer and systems sciences, p a g e s 1 -4 5 . n a to a d va n c e d s t u d y in s t it u t e , in t e r n a t io n a l s u m m e r s c h o o l h e ld in ma r kt o b e r d o r f, ge r m a n y, ju ly 2 0 a u g u s t 1 , 1 9 9 3 , 1 9 9 5 . ¸»åù»ñç ïñáñù³ý í»ñéáõíáõãûáõýá ãí³µ³ý³ï³ý ñ³ù³ï³ñ·áõù î. ¶³éáû³ý ²ù÷á÷áõù àõëáõùý³ëçñáõãûáõýý»ñá ýíçñí³í »ý ¹»åù»ñç ïñáñù³ý [( : a ! b ) ! ( a ! b ) ! b] í»ñéáõíáõãû³ýá ãí³µ³ý³ï³ý ñ³ù³ï³ñ·áõù£ òáõûó ¿ ïñí»é, áñ ¹»åù»ñç ïñáñáõùá µ³í³ï³ý ¿ ï³ï³ñ»é ñ³ù³ó³ûý µ³ý³ó¨»ñç ³í»éç ý»õ ¹³ëçª ùí³ýïáñý»ñçó ³½³ï µ³ý³ó¨»ñç, áñáß»éç µ³ý³ó¨»ñç ¹³ëç ÷áë³ñ»ý£ ì»ñççýë ³í»éç é³ûý ¹³ë ¿ ¨ áý¹·ñïáõù 1 0 7 1 0 8 analysis of case splitting in an arithmetical system ¿ ùí³ýïáñý»ñçó ³½³ï µ³ý³ó¨»ñç ¹³ëᣠàõëáõùý³ëçñí»é »ý ý³¨ ³ûé ñ»õçý³ïý»ñç ùáï»óáõùý»ñá ¹»åù»ñç ïñáñù³ýá ¨ ï³ï³ñí»é »ý áñáß ×ß·ñïáõùý»ñ ï³åí³í áñáß»éç µ³ý³ó¨ç ÷áë³ñçýù³ýá ùí³ýïáñý»ñçó ³½³ï µ³ý³ó¨áí£ ð³ù³ó³ûý ³ûë ³ù»ýç ³é³ç³ñïí»é ¿ ¹»åù»ñç ïñáñù³ý ýáñ ³ñï³íáõù£ microsoft word tpel1.doc mathematical problems of computer science 32, 96--100, 2009. 96 robust audio watermarking algorithm david g. asatryan1 and sergej v. tairyan2 1institute for informatics and automation problems of nas ra 2russian-armenian (slavonic) university 1dasat@ipia.sci.am, 2sergej_tairoff@mail.ru abstract in this paper, we propose an adaptive digital audio watermarking algorithm. the algorithm uses the local properties of an audiosignal to satisfy the properties of human auditory system. the efficiency and robustness of proposed algorithm is illustrated by examples of random attacks with various intensities and compression attacks in the spectrum domain. references [1] a. fabien, p. petitcolas, r. j. anderson and m. g. kuhn, “information hiding a survey”, proceedings of the ieee, special issue on protection of multimedia content, 87(7), pp. 10621078, july 1999. [2] g. voyatzis and i. pitas, “the use of watermarks in the protection of digital multimedia products’’, proceedings of the ieee, special issue on protection of multimedia content, 87(7), pp. 1197-1207, 1999. [3] wu guo-min, zhuang yue-ting, wu fei, pan yun-he, “adaptive audio watermarking based on snr in localized regions”, journal of zhejiang university science, 2005 6a (suppl. 1), pp. 53-57, 2005. [4] http://www.publicknowledge.org/pdf/citizens_guide_to_drm.pdf [5] j. seok, j. hong, and j. kim. “a novel audio watermarking algorithm for copyright protection of digital audio”, etri journal, vol. 24, no. 3, pp. 181-189, 2002. [6] д.г.асатрян, н.с.ланина, “адаптивный алгоритм встраивания цифровых водных знаков в изображение”, труды годичной научной конференции рау, ереван, cc. 59-65, 2006. èë³³½¹³ý³ß³ýç çñ³ýßù³ý ï³ûáõý ³é·áñçãù ¸. ²ë³ïñû³ý, ê. â³çñû³ý ²ù÷á÷áõù ðá¹í³íáõù ³é³ç³ñïí»é ¿ éë³³½¹³ýß³ýç çñ³ýßù³ý ³¹³åïçí ãí³ûý³óí³í ³é·áñçãù: ²ûý ñ»ýí³í ¿ éë³³½¹³ýß³ýç ï»õ³ûçý ñ³ïïáõãûáõýý»ñç û·ï³·áñíù³ý íñ³ ¨ µ³í³ñ³ñáõù ¿ ù³ñ¹áõ éëáõ³ï³ý ñ³ù³ï³ñ·ç ñ³ïïáõãûáõýý»ñçý: ²é³ç³ñïí³í ³é·áñçãùç ³ñ¹ûáõý³í»ïáõãûáõýá ¨ ï³ûáõýáõãûáõýá óáõó³¹ñí»é ¿ ï³ñµ»ñ áõå·ýáõãû³ý ¨ å³ï³ñ³ï³ý ýáõûãç ñ³ñó³ïáõùý»ñç, çýãå»ë ý³¨ ³½¹³ýß³ýç ëå»ïïñç ïçñáõûãáõù ë»õùù³ý ûñçý³ïý»ñç íñ³: d:\sbornik\...\alex_chub1.dvi mathematical problems of computer science 26, 2006, 117{122. on some p r opor ties of fr ege p r oofs s o n a r . a le ks a n ya n a n d a n a h it a . ch u b a r ya n department of applied mathematics, state engineering university of armenia, department of informations and applied mathematics, yerevan state university, e-mail: sonush@rambler.ru, achubaryan@ysu.am abstract in [4] a measure s on propositional formula was de¯ned such that for every tautology ' "high" value of s(') requires the large size of proof in the "weak" propositional systems. in this paper it is shown, that there is a tautology ', the measure s(') of which has exponential dependence on the size of ', but its proof complexity in frege systems is polynomially bounded. refer ences [1 ] s . r . b u s s , p o lyn o m ia l s iz e p r o o fs o f t h e p r o p o s it io n a l p ig e o n h o le p r in c ip le , j ournal of symbolic l ogic, 5 2 , 1 9 8 7 , 9 1 6 { 9 2 7 . [2 ] a . a . ch u b a r ya n , on t h e p r o o f c o m p le xit y in s o m e s ys t e m o f c la s s ic a l p r o p o s it io n a l lo g ic , izvestija nan armenii, m athematika, v o l. 3 7 , n 5 , 1 9 9 9 , 1 6 { 2 6 . [3 ] a . a . ch u b a r ya n , on t h e c o m p le xit y o f p r o o fs in fr e g e s ys t e m s , csit conference, yerevan, 2 0 0 1 , 1 2 9 { 1 3 2 . [4 ] a . a . ch u b a r ya n , r e la t ive e ± c ie n c y o f a p r o o f s ys t e m in c la s s ic a l p r o p o s it io n a l lo g ic , izvestija nan armenii, m athematika, v o l. 3 7 , n 5 , 2 0 0 2 , 7 1 { 8 4 . [5 ] s . a . co o k, a . r . r e c kh o w, th e r e la t ive e ± c ie n c y o f p r o p o s it io n a l p r o o f s ys t e m s , j ournal of symbolic l ogic, 1 9 7 9 , 4 4 , 3 6 { 5 0 . [6 ] e . me n d e ls o n , in t r o d u c t io n t o ma t h e m a t ic a l l o g ic , d . van nostrand company, inc. üñ»·»ûç ³ñï³íáõùý»ñç ùç ñ³ïïáõãû³ý ù³ëçý ê. è. ²é»ùë³ýû³ý ¨ ²ý³ñçï ². âáõµ³ñû³ý ²ù÷á÷áõù úáõñ³ù³ýãûáõñ ' ýáõûý³µ³ýáõãû³ý ñ³ù³ñ [4]-áõù ý»ñùáõíí»é ¿ ³ûýåçëç s( ') ù»íáõãûáõý, áñ s( ') -ç ”µ³ñóñ” ³ñå»ùç ¹»åùáõù '-ç ³ñï³íù³ý »ñï³ñáõãûáõýá »ãáõûé» ñ³ù³ï³ñ·»ñáõù ýáõûýå»ë ù»í ¿: êáõûý ñá¹í³íáõù óáõûó ¿ ïñíáõù, áñ ·áûáõãûáõý áõýç ³ûýåçëç ' ýáõûý³µ³ýáõãûáõý, áñç ñ³ù³ñ s( ') -ý áõýç óáõóã³ûçý ï³ëí³íáõãûáõý '-ç »ñï³ñáõãûáõýçó, ë³ï³ûý ýñ³ ³ñï³íáõùá üñ»·»ûç ñ³ù³ï³ñ·»ñáõù áõýç µ³½ù³ý¹³ù³ûçý µ³ñ¹áõãûáõý: 1 1 7 d:\sbornik\...\ident2.dvi mathematical problems of computer science 24, 2005, 76{81. on logar ithmically asymptotically optimal h ypothesis t esting of t hr ee distr ibutions for p air of i ndependent objects e vg u e n i a . h a r o u t u n ia n a n d p a r a n d z e m m. h a ko b ya n institute for informatics and automation problems of nas of ra e-mail evhar@ipia.sci.am, par h@ipia.sci.am abstract the problem of hypotheses testing for a model consisting from two independent objects is considered. it is supposed that three probability distributions are known and objects independently from each other follow to one of them. the matrix of asymptotic interdependencies (reliability{reliability functions) of all possible pairs of the error probability exponents (reliabilities) in optimal testing for this model is studied. the case with two independent objects and two given probability distributions was elaborated by haroutunian and ahlswede. refer ences [1 ] h a r o u t u n ia n e . a . " l o g a r it h m ic a lly a s ym p t o t ic a lly o p t im a l t e s t in g o f m u lt ip le s t a t is t ic a l h yp o t h e s e s " , p roblems of control and information theory, vo l. 1 9 ( 5 -6 ) , p p . 4 1 3 { 4 2 1 , 1 9 9 0 . [2 ] a h ls we d e r . a n d w e g e n e r i., s e a r c h p r o b le m s . w ile y, n e w y o r k, 1 9 8 7 . [3 ] a h ls we d e r . f. a n d h a r o u t u n ia n e . a ." on s t a t is t ic a l h yp o t h e s e s op t im a l te s t in g a n d id e n t i¯ c a t io n " . m athematical p roblems of computer science 24, p p .1 6 { 3 3 , 2 0 0 5 . [4 ] co ve r t. m. a n d th o m a s j. a . " e le m e n t s o f in fo r m a t io n th e o r y" . w iley, new york, 1 9 9 1 . [5 ] i. cs is z ¶a r , " th e m e t h o d o f t yp e s " , ie e e trans. inform. theory, vo l. 4 4 , n o . 6 , p p . 2 5 0 5 { 2 5 2 3 , 1 9 9 8 . [6 ] i. cs is z ¶a r a n d j. k äo r n e r , information theory: coding theorems for d iscrete m emoryless systems, a c a d e m ic p r e s s , n e w y o r k, 1 9 8 1 , r u s s ia n t r a n s la t io n , mir , mu s c o w, 1 9 8 5 . 7 6 e. a. haroutunian and p. m. hakobyan 7 7 ºñïáõ ³ýï³ë ûµû»ïïý»ñç ½áõû·ç ýï³ïù³ùµ »ñ»ù í³ñï³íý»ñç éá·³ñãùáñ»ý ³ëçùåïáïáñ»ý ûåïçù³é ëïáõ·áõù º. ². ð³ñáõãûáõýû³ý ¨ ö. ø. ð³ïáµû³ý ²ù÷á÷áõù ¸çï³ñïí³í »ý »ñïáõ ³ýï³ë ûµû»ïïý»ñçó ï³½ùí³í ùá¹»éç ñ³ù³ñ í³ñï³íý»ñç ëïáõ·ù³ý ëý¹çñá: ð³ûïýç »ý »ñ»ù ñ³í³ý³ï³ý³ûçý µ³ßëáõù»ý»ñ, ¨ ûµû»ïïý»ñçó ûáõñ³ù³ýãûáõñá ³ýï³ëáñ»ý áý¹áõýáõù ¿ ¹ñ³ýóçó ù»ïá: ²ûë ùá¹»éç ñ³ù³ñ áõëáõùý³ëñçí»é ¿ ûåïçù³é ï»ëï³íáñù³ý ³ñ¹ûáõýùáõù µáéáñ ñý³ñ³íáñ ½áõû·»ñç ëë³éý»ñç ñ³í³ý³ï³ýáõãûáõýý»ñç óáõóçãý»ñç (ñáõë³éçáõãûáõýý»ñç) ÷áëï³ëí³íáõãûáõýá: ºñïáõ ñ³í³ý³ï³ý³ûçý µ³ßëáõùý»ñáí ¹»åùá áõëáõùý³ëçñí»é ¿ ð³ñáõãûáõýû³ýç ¨ ²éëí»¹»ç ïáõùçó [3]: microsoft word e-learning.doc mathematical problems of computer science 26, 2006, 40-47. 40 e-learning tool and statistical analysis of video tace files over the network gohar sargsyan institute for informatics and automation problems of nas of ra e-mail gohar_s@yahoo.com abstract the following paper offers an overview on the e-learning tool “dileco” system structure and the advancement of features of it. the main features of this tool aim to support operations, using unstable and limited internet connection, utilizing novel solutions outlined in this paper. this system would be one of the best tools applicable for e-learning in countries with poor internet. the paper pursues the objectives of showing some of the innovative features of the system and demonstrates the progress of the system starting from 2002. this paper aims to express technological aspects on building this system. it offers a snapshot of application areas of the system and provides suggestions for future expansion of application areas, and content creation, particularly, on topics of statistics and mathematics. it is provided with statistical analysis of h.263 video traces over the network. references [1] sargsyan g., researching and developing a distance learning and video conferencing (dileco) system on internet and its application in e-learning; may 2003. http://www.opensourcearmenia.com/projects/dileco/dileco [2] sargsyan g., development of distance learning and video conferencing (dileco) tool and its application in e-learning; csit 2003 international conference, armenia, september 2003 pp. 424-428. [3] sargsyan g., e-learning tools dileco and vls, new information technologies in education, collected scientific works 2005, pp. 140-144. [4] sargsyan g., dileco, open source armenia www.opensourcearmenia.com/projects/dileco, may 2004. [5] sargsyan g. and others, e-learning assessment and e-learning tools for armenia needs; , multidementional statistical analysis and econometrics, moscow 2004, pp. 95-96 [6] hakobyan h, e learning assessment , april 2003. http://www.gateway.am/index.jsp?sid=1&id=8764&pid=8020, [7] information technologies group center for international development at harvard university: readiness for the networked world: a guide for developing countries with support of ibm, 2002. [8] hakobyan h, towards knowledge economy. e-readiness assessment in armenia, sept 2003. http://www.gateway.am/index.jsp?sid=1&id=12114&pid=8056 g. sargsyan 41 [9] haroutunian e. and shahumyan h., armenian statistical web lab, http://www.stat.auckland.ac.nz/~iase/publications/3/shahumyancps04.pdf [10] haroutunian e. and others: probability and applied statistics, “gitutyun” publ. house, yerevan, 2000 (in armenian) [11] beaudoin m., university of new england, distance education leadership for the new century, journal of distance learning and administration, 2003, pp.5-7. [12] fitzek f. and others, video and audio trance files of pre-encoded video content for network performance mesasurements, universita di ferrara and arizona state university, jan 2004 [13] fitzek f. and reisslein m., mpeg-4 and h.263 video traces for network performance evaluation, ieee network, 15(6); 40-54, nov/dec 2001 [14] seeling p., fitzek f. and reisslein m., videometer, ieee network magazine, page 5, january 2003 [15] gereoffy a., mplayer tool. http://mplayerhq.hu, may 2003, version 0.90 ð»é³áõëáõóù³ý íñ³·ñ³ûçý ·áñíçù ¨ ó³ýó»ñç ùççáóáí ï»ë³ûçý ñ»ï³·í»ñç íç׳ﳷñ³ï³ý í»ñéáõíáõãûáõý ¶. ê³ñ·ëû³ý ²ù÷á÷áõù ðá¹í³íý ³é³ç³ñïáõù ¿ ¸çé»ïá ñ»é³áõëáõóù³ý íñ³·ñ³ûçý ÷³ã»ãç ùççáóý»ñç ï³ï³ñé³·áñíáõùá ó³íñ áñ³ïç çýï»ñý»ï³ûçý ï³åç ¹»åùáõù: êï»õí³í ñ³ù³ï³ñ·á ï³ñáõ ¿ éçý»é ñ»é³áõëáõóáõù çñ³ï³ý³óý»éáõ é³í³·áõûý ·áñíçùý»ñçó ù»ïá: ðá¹í³íá ýå³ï³ï³áõõõí³í ¿ ý»ñï³û³óý»éáõ ñ³ù³ï³ñ·ç áñáß ýáñ³ñ³ñ³ï³ý ñ³ïïáõãûáõýý»ñ ¨ ·áñíçùç ½³ñ·³óù³ý ³é³çáýã³óá ëïë³í 2002-çó: ð³ù³éáï ý»ñï³û³óí³í ¿ ñ³ù³ï³ñ·ç ïçñ³éáõãû³ý áéáñïý»ñá: ²é³ç³ñïí³í »ý ïçñ³éáõãû³ý ñ»ï³·³ íñ³·ñ»ñ ¨ µáí³ý¹³ïáõãû³ý ñ³·»óáõù, ù³ëý³íáñ³å»ë, íç׳ﳷñáõãû³ý ¨ ù³ã»ù³ïçï³ûç ã»ù³ý»ñáí: øýý³ñïí³í ¿ ó³ýó»ñç ùççáí h.263 ï»ë³ûçý ñ»ï³·í»ñç íç׳ﳷñ³ï³ý í»ñéáõíáõãûáõý: d:\sbornik\...\dpm_full.dvi mathematical problems of computer science 25, 2006, 18{26. dynamic p r ocess m anagement system ar chitectur e for computational cluster s tig r a n m. gr ig o r ya n , v la d im ir g. s a h a kya n institue for informatics and automation problems of nas of ra e-mals tigrangr@ipia.sci.am, svlad@sci.am abstract the problem of e±cient utilization of computational resources of clusters arises as its load and number of users grow. tasks like the fair use of resources and load balancing are common and should be solved by the operation environment of cluster. existing mechanisms that solve the mentioned problems work ¯ne as long as parallel programs are run on a ¯xed number of resources. allocating and freeing resources dynamically can highly improve the performance of a parallel program as well as the e±ciency of using the cluster. in the following paper the system architecture is described, which supports dynamic resource allocation and process spawning, which is alternate to mpi-2 standard's dynamic process spawning mechanism. it is also introduced, how dynamic task/process spawning can improve the performance of the parallel program. r e fe r e n c e s [1 ] me s s a g e p a s s in g in t e r fa c e fo r u m . mp i-2 : e xt e n s io n s t o t h e me s s a g e -p a s s in g in t e r fa c e ( h t t p :/ / www.m p i-fo r u m .o r g / d o c s / m p i-2 0 .p s ) [2 ] w . gr o p p , e . l u s k. d yn a m ic p r o c e s s ma n a g e m e n t in a n mp i s e t t in g ; ma t h e m a t ic s a n d co m p u t e r s c ie n c e d ivis io n a r g o n n e n a t io n a l l a b o r a t o r y, 1 9 9 5 [3 ] g. b u r n s , r . d a o u d , j. v a ig l. l a m: a n op e n clu s t e r e n vir o n m e n t fo r mp i; p r o c e e d in g s o f s u p e r c o m p u t in g s ym p o s iu m , p p . 3 7 9 { 3 8 6 ; 1 9 9 4 [4 ] j. m. s qu yr e s a n d a . l u m s d a in e . a co m p o n e n t a r c h it e c t u r e fo r l a m/ mp i; p r o c e e d in g s , 1 0 t h e u r o p e a n p v m/ mp i u s e r s ' gr o u p me e t in g , p p . 3 7 9 { 3 8 7 ; 2 0 0 3 [5 ] b . b a r r e t t , j. m. s qu yr e s a n d a n d r e w l u m s d a in e . in t e g r a t io n o f t h e l a m/ mp i e n vir o n m e n t a n d t h e p b s s c h e d u lin g s ys t e m ; p r o c e e d in g s o f t h e 1 7 t h in t e r n a t io n a l s ym p o s iu m o n h ig h p e r fo r m a n c e co m p u t in g s ys t e m s a n d a p p lic a t io n s a n d os ca r s ym p o s iu m , p p . 2 7 7 { 2 8 3 ; 2 0 0 3 [6 ] a . b a yu c a n , r . l . h e n d e r s o n , j. p . jo n e s , c. l e s ia k, b . ma n n , b . n it z b e r g , t. p r o e t t , j. u t le y. p o r t a b le b a t c h s ys t e m , op e n p b s r e le a s e 2 .3 , a d m in is t r a t o r gu id e ; v e r id ia n in fo r m a t io n s o lu t io n s , in c ., 2 0 0 0 1 8 t. m. grigoryan, v. g. sahakyan 1 9 [7 ] â.â. âîåâîäèí, âë.â. âîåâîäèí. ïàðàëëåëüíûå âû÷èñëåíèÿ; ”áõâ-ïåòåðáóðã”, ñàíêòïåòåðáóðã, 2004. [8 ] t. gr ig o r ya n , v . s a h a kya n . d yn a m ic r e s o u r c e ma n a g e r fo r clu s t e r s . p r o c e e d in g s o f cs it2 0 0 5 , p p . 4 3 9 { 4 4 2 ; y e r e va n , 2 0 0 5 ð³ßíáõ³ï³ý ïé³ëï»ñý»ñç ¹çý³ùçï áýã³óùý»ñç õ»ï³í³ñù³ý ñ³ù³ï³ñ·³ûçý ׳ñï³ñ³å»ïáõãûáõý î. ø. ¶ñç·áñû³ý ²ù÷á÷áõù îé³ëï»ñ³ûçý ñ³ù³ï³ñ·ç í³ýñ³µ»éýí³íáõãû³ý ¨ û·ï³·áñíáõý»ñç ù³ý³ïç ³×ç ñ»ï ù»ïï»õ ³é³ç ¿ ·³éçë ¹ñ³ ñ³ßíáõ³ï³ý é»ëáõñëý»ñç ¿ý»ïïçí û·ï³·áñíù³ý åñáµé»ùá: ²é³ç³ýáõù »ý é»ëáõñëý»ñç §³½ýçí¦ û·ï³·áñíù³ý ¨ µ»éýí³íáõãû³ý ñ³í³ë³ñ³ïßéù³ý ëý¹çñý»ñá, áñáýù å»ïù ¿ éáõíí»ý ïé³ëï»ñç ûå»ñ³óçáý ùçç³í³ûñç ïáõùçó: ¶áûáõãûáõý áõý»óáõ ù»ë³ýç½ùý»ñá ³å³ñáíáõù »ý ³û¹ ëý¹çñý»ñá éáõíáõùá ù³ýç ¹»é ½áõ·³ñ»é íñ³·ñ»ñá ³ßë³ïáõù »ý ýçùëí³í ù³ý³ïáõãû³ùµ åñáó»ëáñý»ñç íñ³: ð³ßíáõ³ï³ý é»ëáõñëý»ñç ¹çý³ùçï ½µ³õ»óáõùá ¨ ³½³ïáõùá ï³ñáõ ¿ ½·³éçáñ»ý µ³ñóñ³óý»é çýãå»ë ½áõ·³ñ»é íñ³·ñç ³ñï³¹ñáõ³ï³ýáõãûáõýá, ³ûýå»ë ¿é ïé³ëï»ñç û·ï³·áñíù³ý ¿ý»ïïçíáõãûáõýá: êáõûý ñá¹í³íáõù ýï³ñ³·ñí³í ¿ ñ³ßíáõ³ï³ý é»ëáõñëý»ñç ¹çý³ùçï ½µ³õ»óáõù ¨ ¹çý³ùçï áý¹³óùý»ñç ë»ñáõù ³å³ñáíáõ ïé³ëï»ñç ñ³ù³ï³ñ·³ûçý ׳ñï³ñ³å»ïáõãûáõý, áñá ï³ñáõ ¿ ³ûéáýïñ³ýù ñ³ý¹çë³ý³é mpi2 ëï³ý¹³ñïç ¹çý³ùçï áý¹³óùý»ñç ë»ñù³ý ù»ë³ýç½ùçý: òáõûó ¿ ïñí³í ý³¨, ã» çýãå»ë ¹çý³ùçï ë»ñíáõ áý¹³óùý»ñç ïçñ³éáõùá ï³ñáõ ¿ µ³ñóñ³óý»é ½áõ·³ñ»é íñ³·ñç ³ñï³¹ñáõ³ï³ýáõãûáõýá: d:\sbornik\...\termody+.dvi mathematical problems of computer science 26, 2006, 101{113. quantum p r ocesses and p ossibility of t heir contr ol a s h o t s . ge vo r kya n institue for informatics and automation problems of nas of ra e-mail g ashot@sci.am abstract the dissipation and decoherence (for example, the e®ects of noise in quantum computations), interaction with thermostat or in general with physical vacuum, measurement and many other complicated problems of open quantum systems are a consequence of interaction of quantum systems with the environment. these problems are described mathematically in terms of complex probabilistic process (cpp). particularly, treating the environment as a markovian process we derive an langevinschräodinger type stochastic di®erential equation (sde) for describing the quantum system interacting with environment. for the 1d randomly quantum harmonic oscillator (qho) l-sh equation has a solution in the form of orthogonal cpp. on the basis of orthogonal cpp the stochastic density matrix (sdm) method is developed and in its framework relaxation processes in the uncountable dimension closed system of "qho+environment" is investigated. with the help of sdm method the thermodynamical potentials, like nonequilibrium entropy and the energy of ground state are exactly constructed. the dispersion for di®erent operators is calculated. in particular, the expression for uncertain relations depending on parameter of interaction between qho and environment is obtained. the weyl transformation for stochastic operators is speci¯ed. ground state winger function is developed in detail. refer ences [1 ] p r o c e e d in g s o f a d r ia t ic o r e s e a r c h co n fe r e n c e a n d min iwo r ks h o p quantum chaos, 4 ju n e { 6 ju ly 1 9 9 0 , tr ie s t e , it a ly [2 ] c. p r e s illa , r . on o fr io , u . ta m b in i, a n n .p h ys ., v. 2 4 8 , p . 9 5 ( 1 9 9 6 ) [3 ] c.w . ga r d in e r , m.j.co lle t t , p h ys .r e v. a , v. 3 1 , p . 3 7 6 1 ( 1 9 8 5 ) [4 ] n . gis in , i.c. p e r c iva l, j.p h ys . a , v. 2 5 , p . 5 6 7 7 ( 1 9 9 2 ) [5 ] n . k n a u f, y .g. s in a i, e -p r in t n 2 3 2 h t t p :/ / www.m a t h .t u -b e r lin .d e [6 ] a .v . b o g d a n o v, a .s . ge vo r kya n , p r o c e e d in g s o f in t . w o r ks h o p o n qu a n t u m s ys t e m s , min s k, b e la r u s , p . 2 6 ( 1 9 9 6 ) [7 ] a .v . b o g d a n o v, a .s . ge vo r kya n , a .g. gr ig o r ya n , a ms / ip s t u d ie s in a d va n c e d ma t h e m a t ic s , v. 1 3 , p . 8 1 ( 1 9 9 9 ) [8 ] a .v . b o g d a n o v, a .s . ge vo r kya n , a .g. gr ig o r ya n , s .a . ma t ve e v, in t . jo u r n . b ifu r c a t io n a n d ch a o s , v. 9 , n . 1 2 , p . 9 ( 1 9 9 9 ) 1 0 1 1 0 2 quantum processes and possibility of their control [9 ] a .s . ge vo r kya n , e xa c t ly s o lva b le m o d e ls o f s t o c h a s t ic qu a n t u m m e c h a n ic s wit h in t h e fr a m e wo r k o f l a n g e vin -s c h r e o d in g e r t yp e e qu a t io n , a n a lys is a n d a p p lic a t io n s . p r o c e e d in g o f t h e n a to a d va n c e d r e s e r a c h wo r s ks h o p , y e r e va n 2 0 0 2 , e d s . b y b a r s e g ia n g. a n d b e g e h r h ., n a to s c ie n c e p u b lic a t io n s , p p . 4 1 5 -4 4 2 , k lu we r , ( 2 0 0 4 ) . [1 0 ] a . n . b a z ', y a . b . ze l'd o vic h a n d a . m. p e r e lo m o v, scattering reactions and d ecays in nonrelativistic quantum m echanics, ( in r u s s ia ) , " n a u ka " , mo s c o w, 1 9 7 1 . [1 1 ] d .n . zu b a r e v, nonequilibrium statistical thermodinamics, n a u ka , 1 9 7 1 ( in r u s s ia n ) . [1 2 ] i.m. l ifs h it z , s .a .gr e d e s ku l a n d l .p . p a s t u r , introduction to the theory of non-r egular systems, ( in r u s s ia ) , " n a u ka " , mo s c o w, 1 9 8 2 . [1 3 ] c.w . ga r d in e r , handbook of stochastic m ethods for p hysics, chemistry and natural sciences, s p r in g e r -v e r la g b e r lin n e w-y o r k to kyo , 1 9 8 5 [1 4 ] j.glim m , a .ja ®e , quantum p hysics. a f unctional integral p oint of view, s p r in g e r v e r la g , 1 9 8 1 . [1 5 ] w .m. it a n o , d .j. h e in z e n , j.j. b o llin g e r a n d d .j. w in e la n d , p hys.r ev. a, v. 4 1 , p . 2 2 9 5 ( 1 9 9 0 ) . [1 6 ] v . go r in i, a . k o s s a ko ws ki a n d e .c.g. s u d a r s h a n , j .m ath.p hys., v. 1 7 , p . 8 2 1 ( 1 9 7 6 ) . [1 7 ] g. l in d b la d , comm. m ath. phys., v. 4 8 , p . 1 1 9 ( 1 9 7 6 ) . ä³ï³ñ³ï³ý ùí³ýï³ûçý áýã³óùý»ñá ¨ ýñ³ýó õ»ï³í³ñù³ý ñý³ñ³íáñáõãûáõýý»ñá ². ¶¨áñ·û³ý ²ù÷á÷áõù ø³ñáõùý áõ ¹»ïáñ»ñ»ýïáõãûáõýá, ÷áë³½¹»óáõãûáõýá ã»ñùáëï³ïç ï³ù áý¹ñ³ýáõñ ¹»åùáõù ýç½çï³ï³ý í³ïáõùç ñ»ï, ã³÷áõùý»ñá ¨ µ³ó ùí³ýï³ûçý ñ³ù³ï³ñ·»ñç ³ûé µ³ñ¹ åñáµé»ùý»ñá, í»ñççý ñ³ßíáí ùí³ýï³ûçý ñ³ù³ï³ñ·»ñç ýñ³ýó ßñç³ï³ûùç ñ»ï ÷áë³½¹»óáõãû³ý ³ñ¹ûáõýù »ý: ²ûë åñáµé»ùý»ñá ù³ã»ù³ïçïáñ»ý ýï³ñ³·ñíáõù »ý ïáùåé»ùë ñ³í³ý³ï³ûçý áýã³óùý»ñç 黽íáí (îäà): ø³ëý³íáñ³å»ë ýï³ñ³·ñ»éáí ßñç³ï³ûùá, çýãå»ë ø³ñïáíû³ý áýã³óù, ùí³ýï³ûçý ñ³ù³ï³ñ·ç ßñç³ï³ûùç ñ»ï ÷áë³½¹»óáõãû³ý ýï³ñ³·ñáõãû³ý ñ³ù³ñ ³ñï³í»é »ýù è³ýå»í»ý-þñ»¹çý·»ñ ïçåç å³ï³ñ³ï³ý ¹çý»ñ»ýóç³é ñ³í³ë³ñáõù (ä¸ð). òáõûó ¿ ïñí³í, áñ å³ï³ñ³ï³ý 1d ùí³ýï³ûçý ý»ñ¹³ßý³ï ï³ï³ý³ïá (øüî) è-þñ»¹ ñ³í³ë³ñù³ý ßñç³ý³ïý»ñáõù áõýç éáõíáõùý»ñ‘ ûñãá·áý³é îðà ï»ëùáí: úñãá·áý³é îðà-ç ñ»ýùç íñ³ ½³ñ·³óí³í ¿ å³ï³ñ³ï³ý ëïáõãû³ý ù³ïñçó³ûç ù»ãá¹ (äêø), áñç ßñç³ý³ïý»ñáõù ï³ï³ñí³í ¿ ³ýñ³ßí»éç ã³÷áõ³ï³ýáõãû³ý ÷³ï ñ³ù³ï³ñ·ç »øüî+þñç³ï³ûù» ñ»ï³½áïáõãûáõýá: äêø ù»ãá¹ç û·ýáõãû³ùµ ã»ñùá¹çý³ùçï³ï³ý åáï»ýóç³éý»ñá‘ ýù³ý áãñ³í³ë³ñ³ïßçé ¿ýïñáåç³ûç ¨ ñçùý³ï³ý íç׳ïç ¿ý»ñ·ç³ûç, ×ß·ñçï ï³éáõóí³í »ý: ð³ßí³í »ý ûå»ñ³ïáñý»ñç ¹çëå»ñëç³ý»ñá: ø³ëý³íáñ³å»ë ³ýáñáßáõãûáõýý»ñç ³éýãáõãûáõýý»ñç ñ³ù³ñ, ï³ëí³í øüî-ç ßñç³ï³ûùç ñ»ï ÷áë³½¹»óáõãû³ý å³ñ³ù»ïñçó, ëï³óí³í »ý ³ñï³ñ³ûïáõãûáõýý»ñ: àñáßí³í ¿ ì»ûéç ó¨³÷áëáõãûáõýá å³ï³ñ³ï³ý ûå»ñ³ïáñý»ñç ñ³ù³ñ: ðçùý³ï³ý íç׳ïç ìç·ý»ñç ýáõýïóç³ý áõëáõùý³ëçñí³í ¿ ù³ýñ³ù³ëý: microsoft word 6_albert_saribekyan.doc mathematical problems of computer science 39, 48--53, 2013. 48 performance of ndvi index on hpc resources albert g. saribekyan institute for informatics and automation problems of nas of ra e-mail: albert_saribekyan@ipia.sci.am abstract geographic information systems (gis) [1] are crucial to enable the gathering, analysis, presentation and distribution of spatial and non-spatial data. in some specific gis applications, such as time-critical simulations or data mining, we need to deal with massive amount of geospatial data storage, retrieval, and processing. the main aim of this article is to analyze and benchmark the performance of normalized difference vegetation index (ndvi) [2] of satellite images (16 gb [3]) using highperformance computational (hpc) resources, as the developers need to solve the problem of both data and task distribution among serial or parallel environments. the geographical resources analysis support system (grass) [4] is used as the main instrument for the study. keywords: grass gis, high performance computing, remote sensing, ndvi. 1. introduction satellite imagery provides a large amount of useful information and plays a vital role for research developments in astronomy, remote sensing (rs), gis, agriculture, disaster management and many other fields of study. gis is a system designed to capture, store, manipulate, analyze, manage, and present all types of geographical data. gis is used for geographical information science or geospatial information studies to refer to the academic discipline or career of working with gis. in the simplest terms, gis is the merging of cartography, statistical analysis, and database technology. these geospatial technologies allow us to monitor natural biological systems. new advances in spatial ecology permit us to put these data in the context of our ecological understanding and to generalize these patterns to advance ecological theories and their applications. remotely-sensed data is one of the most important sources of data for gis. rs means acquiring data from a distance usually uses electromagnetic energy sunlight, radar, laser originally captured on photographic film. recent platforms utilize digital sensors. rs is the observation of an object from a distance. information about features on the earth's surface can be gathered from orbiting rs satellites or from a plane (e.g., aerial photography). geographic resources analysis support system, commonly referred to as grass gis, is a gis used for data management, image processing, graphics production, spatial modeling, and a. saribekyan 49 visualization of many types of data. it is a free software/open source released under gnu general public license >= v2. grass gis is an official project of the open source geospatial foundation. a wide ranging analysis capabilities make it an ideal tool to set up environmental models, as well as to support land planning and management. grass was originally developed in the beginning of the 1980s by the us army construction engineering research laboratories (usa-cerl), published as a public domain software. the usa-cerl withdrew from the grass development in the early 1990s. from 1999 it is developed by an international developer team, which published grass as a free software under the terms of the gnu general public license. grass is developed mostly in linux but binaries for other operative systems are available. main grass features are:  2d raster analysis and 3d voxel (volumes) management  2d/3d vector engine with sql-based dbms support  image processing modules  vector network analysis, linear referencing system  visualization of 2d, 3d maps and volumes  interoperable with standard raster and vector formats  works on gnu/linux, mac os x, ms-windows and other posix compliant platforms  modular architecture and scripting capabilities for batch processing grass gis is capable of handling raster, topological vector, image processing, and graphic data. it includes more than 350 modules [5] for management, processing, analysis and visualization of georeferenced data. additionally, processing those data requires a large amount of computation time due to its complex and large processing criteria. this seems a barrier for real time decision making. to switch the job faster, hpc can be a suitable solution. multicomputerbased distributed systems (clusters and grids) have a large processing capacity for a lower cost, naturally, choice turns towards developing hpc applications. developing grass module framework on hpc, three case studies, i.e. “i.vi” to process 13 vegetation indices, “i.lmf” to remove the atmospheric effects from rs images and “r.gaswap” to find out the crop parameters that are not directly visible from rs images, are discussed. the foremost case study (i.vi) discusses the interoperability framework designing issues between the grass tool and hpc. vegetation index (vi) is the major set of indicators for vegetation. the grass module i.vi, is used to process 13 different vegetation indexes for the satellite images. ndvi (normalized difference vegetation index) is one of them. the ndvi is calculated from these individual measurements as ndvi=(nir-red)/(nir+red), where red and nir stand for the spectral reflectance measurements acquired in the red and near-infrared regions. fig.1 shows a snapshot of a ndvi output of grass software with red and nir bands. performance of ndvi index on hpc resources 50 figure 1: ndvi calculations with grass grass module i.vi works with raster images (rows x columns). grass functions are used to extract row-wise data from the specific band images and store them in buffers. then, each column value is extracted sequentially from the buffers and sent for generating the specific vi values. thus, after completing the vi from the row buffers, the row wise vi values are put back into the output image and this process will continue for each row. fig.2 presents the structure of running i.vi module (for simplicity only two band images have been presented). a. saribekyan 51 figure 2: serially running i.vi module structure in this paper the same task has been compared on two different environments to justify and practically understand the behavior of the parallelized i.vi module of grass (which is i.vi.mpi). to achieve this purpose grass gis 7.0.svn [6] was installed on two different platforms. the both hpc have the same hardware but they differ from each other only with the gcc version and operating system. the parallelization was done using mpi standard [7]. parallelization of i.vi [8] module was illustrated on these platforms using two to eight cores. in fig.3 the computational time has been presented of the same job running on predefined number of cores on two different platforms. figure 3: the computational time of job per cores on two platforms performance of ndvi index on hpc resources 52 dark grey scale shows the result of machine in which gcc version 4.4.5 (debian 4.4.5-8) and operating system is debian gnu/linux 6.0 with linux kernel 2.6.32-5-amd64 and light grey scale shows the result of machine with gcc version 4.1.2 20080704 (red hat 4.1.2-50) and operating system is scientific linux sl release 5.5 (boron) with linux kernel 2.6.18-274.17.1.el5 x86_64. these two results obviously show that the i.vi module works best on six cores. conclusion this research is leading us to be sure that the parallelization of i.vi module is very efficient in terms of the calculation speed, because the same task is done much slower without parallelization. and the second important aspect is the architecture of hpc, especially the version of gcc compiler and the operating system, by which the task solution time has been changed dramatically. references [1] geographic information systems as an integrating technology: context, concepts and definitions. kenneth e. foote and margaret lynch. [2] ndvi index http://gis-lab.info/qa/ndvi.html [3] database of landsat satellite images http://glcfapp.glcf.umd.edu:8080/esdi/ [4] neteler, m. and mitasova, h., open source gis: a grass gis approach. second edition, 2003. kluwer academic publishers. [5] grass gis 7.0.svn reference manual http://grass.osgeo.org/grass70/manuals/full_index.html [6] grass gis official website http://grass.osgeo.org/ [7] the message passing interface (mpi) standard: http://www-unix.mcs.anl.gov/mpi/ [8] parallel grass jobs http://grasswiki.osgeo.org/wiki/parallel_grass_jobs submitted 10.11.2012, accepted 04.02.2013. բուսականության նորմավորված տարբերության ինդեքսի հաշվարկման արտադրողականությունը բարձր արտադրողականության հաշվողական ռեսուրսներում ա.սարիբեկյան ամփոփում երկրատեղեկատվական համակարգերը (ետհ) կարևորվում են տարածական և ոչ տարածական տվյալները խմբավորելու, վերլուծելու, ներկայացնելու և a. saribekyan 53 տարանջատելու համար: որոշ տեսակի ետհ ծրագրային համակարգերում, ինչպիսիք են կարճ ժամկետներում մոդելավորման կամ տվյալների ձեռքբերման համակարգերը, գործ ենք ունենում մեծ քանակի տարածական տվյալների պահպանման, որոնման և մշակման հետ: այս հոդվածի հիմնական նպատակն է հետազոտել և վերլուծել տիեզերական պատկերների բուսականության նորմավորված տարբերության ինդեքսի հաշվարկումը` օգտագործելով բարձր արտադրողականության հաշվողական համակարգեր, քանի որ հետազոտողներին անհրաժեշտ է լուծել առաջացող խնդիրները միաժամանակ զուգահեռ և ոչ զուգահեռ միջավայրերում: ուսումնասիրությունների համար որպես հիմնական գործիք ընտրված է աշխարհագրական ռեսուրսների վերլուծության ապահովման համակարգեր ծրագրային փաթեթը: производительность вычисления нормализованного относительного индекса растительности на высокопроизводительных вычислительных ресурсах а.сарибекян аннотация геоинформационные системы имеют решающее значение для сбора, анализа, представления и расспределения пространственных и атрибутивных данных. в некоторых гис приложениях, таких как срочное моделирование или анализ данных, мы имеем дело с хранением, поиском и обработкой огромного количество геопространственных данных. основной целью данной статьи является анализ и тестирование производительности вычисления нормализованного относительного индекса растительности из спутниковых изображений, используя высокопроизводительные вычислительные ресурсы, так как разработчики должны решать задачи в параллельных и не параллельных средах. в качестве основного инструмента для исследований использован програмный пакет системной поддержки анализа географических ресурсов. . начиная с начала 2000 года осуществляется внедрение ghis в здравоохранении, в рамках принятого проекта о реформирование информ mathematical problems of computer science 50, 56--60, 2018. notification mechanisms by initiative of e-mail receiver in systems based on e-mail/sms technologies andranik e. mkhitaryan, arthur s. petrosyan and aram s. nanassian institute for informatics and automation problems of nas ra e-mail: and.mkhitaryan@gmail.com, arthur@sci.am, ananas@sci.am abstract the feature of mechanisms for the exchange of messages in computer and cellular networks is considered, in order to send a notification to the receiver of email on his/her own initiative using ip and gsm technologies. the peculiarities of technical models for designing a separated system are compared. keywords: notification, mail2sms, e-mail, sms. 1. introduction nowadays, electronic mail is one of the widespread info-communicational systems, which provides the opportunity to exchange messages between computers by the internet [1]. however, e-mail is a typical “on demand” system. despite the instant delivery of messages, the addressees will see new letters only when they request them from the server. another well-known info-communicational system is sms. it operates on a completely different platform: global system for mobile (gsm). this system has some limitations, such as text size, content, etc. however, an undeniable advantage of this technology is the delivery of the message directly to “the pocket” of the mobile subscriber. many modern mobile devices allow you to get e-mail messages via mobile networks directly to the device. but still, the cost of internet traffic via cellular networks forces many users to keep the mobile internet switched off, which prevents receiving urgent messages, because they are being delivered according to the client request. in case of sms the message is delivered to the device by the mobile operator’s (server) decision, thus it remains a more flexible solution than e-mail. the use of e-mail and sms mechanisms in one unified system could enhance the features of both as they complement each other. it’s a common practice for e-mail users to register some list of e-mail addresses to get notified in case someone from that list sends him a message. such systems help users to be informed about the received e-mails from important sources as soon as possible. usually this kind of 56 mailto:and.mkhitaryan@gmail.com mailto:arthur@sci.am mailto:ananas@sci.am a. mkhitaryan, a. petrosyan and a. nanassian 57 mechanisms are embedded in mail server, thus being implemented inside the server of mail providers because the server is the only resource, which has complete information about the sent/received e-mails. such mechanisms are expensive and not targeted especially for small providers as they face all the complexity of software and hardware for sending sms notifications. 2. solution in this article a mechanism is considered, which provides the possibility to use a separated and shared platform for e-mail providers (such as unimail), which solves the above-described problem. for such systems it is necessary to have information about the recent history of received mails of users (fig. 1). the purpose is to design a shared system so that it can have the possibility to integrate mail providers [2], [3]. to be able to realize such a system, the e-mail provider should provide the necessary information. that could be done in different ways:  design a filter software in mail server, which will have access to all components of all e-mails and by parsing the necessary content (i.e., the fields from and to) sent to the separated system. sending process also could be done in different ways. for such systems, mechanisms based on database server/client or other protocols are mostly being used. the advantage of this solution is that it has the message itself and can process all the fields, even the body and the subject. on the other hand, email messages are the property of the sender and receiver only, which could contain private information. a software, which has access to the private information of the e-mail and has an ability to send data to other machines is an additional risk for the admins of the servers of mail providers. this solution could be used for the systems where the contents of fields “from”, “to”, “body”, “subject”, etc., are necessary. however, any software, which is using this mechanism should be well tested and certified.  the other way, which is currently well known to admins is “log forwarding”. it sends the log files generated by some software to another machine. the log file contains information about the current processes of particular software. each line of the log has a standard format, which contains the date (in server’s time zone), the name of application, some details about the current action of the software, etc. [5]. specifically, for the mail server, the generated log contains the fields “from”, “to” and the time when the message was sent. those three components are necessary and sufficient information to send a notification to the receiver in case he/she asked about it before. the advantage of this method is that it is easy to configure, does not require access to private files. in many servers “log fig. 1. connection of mail server and unimail server. 58 notification mechanisms by initiative of e-mail receiver in systems based on e-mail/sms technologies forwarding” is already in use, thus in this case a new effort will not be required. however, it has limited information about the e-mail message and could not be used in systems, which require more information rather than “from” and “to” fields. log file of software, which is configured server side with active users (i.e., mail server) will contain lots of information written by different threads, which make the file size very big. forwarding such huge information periodically could lead to heavy traffic and slow down the workflow of software. the notifier applications require limited information from the log file. the only required lines are the log about e-mail, which indicates the mail sending/receiving and the log that confirms the mail has passed all the checking filters (i.e., spam filter). one of the most popular tools for log forwarding is called rsyslog. it has an easy to use interface and is embedded in many unix-like computer systems by default. rsyslog provides the ability to filter logs by specifying the format. for example: if ($programname ==′ programname′) and ($msg contains ′from =<′) and($msg contains ′to =< ′) … according to the example, only the logs containing the fragments ‘from=<’ and ‘to=<’ will be processed. in real example, in the log file with 20000 lines there are 1705 logs containing the fragments. the size of the log file has been reduced by 91.475 percent. a similar filtering mechanism could be realized in the unimail server in case the log forwarders have not done it on their side. this will improve the performance. another problem is the memory usage of forwarded logs in unimail server. the memory should be cleaned periodically. for such situations there is a mechanism called “log rotation”. that allows you to specify rotation rules, such as the size of log file, the period of rotation, the number of rotated files to keep, etc. in unix-like computer systems there is an already integrated mechanism for “log rotation”. for notification systems where receiver gets notified on his/her own initiative, the required information is the “from” and “to” fields of the e-mail, thus the limitations of the second considered solution cannot obstruct its usage. such a mechanism has been implemented in info-communicational system unimail [4]. as a complete resource, unimail provides services to email users to notify the receivers of messages on the initiative of either the sender or the receiver. the mandatory condition is that the mail provider of the user should be configured and integrated within the unimail server to share the important information for notifications. in unimail the second considered solution was implemented. to use it, the user just needs to combine “white list”. the user will be notified of each received email message from any member of the white list. to use the provided services, where the initiator is the receiver of email, the user should be authenticated in unimail server. unimail system provides complete command list, which allows users to register, combine black and white lists, send an sms notification parallel to the e-mail, get notified about the received emails from important sources, send sms messages by email, etc. some commands with descriptions:  *notice to…(phone numbers)* parallel to the letter send sms notification to the addressee/addressees  *sms to… (phone numbers)* send an sms to the addressee/addressees a. mkhitaryan, a. petrosyan and a. nanassian 59  *add white list* include the addressees in the "white list"  *del white list* exclude the addressees from the "white list" conclusion except for many indisputable advantages of e-mail and sms technologies, both have unique disadvantages. e-mail is a typical “on demand” info-communicational resource, while sms is not. that feature of e-mail could be overcome by using e-mail and sms technologies in a united system. to be able to create a separate system, some data is required from the mail server. for notification systems, a model of log forwarding is suggested based on the comparison of the other mechanisms. references [1] д. геворкян, а. нанасян и к. хачатрян «новые web ресурсы asnet.am», материалы конф. csit-2011, ереван, сс. 311-313, 2011. [2] d. gevorkyan, k.khachatryan, a.nanassian, a.petrosyan, g.petrosyan, v.sahakyan and e.vardanyan, “mail informerselective incoming instand phone notification system”, proceedings of international conference computer science and information technologies, yerevan, armenia, pp. 466-467, 2009. [3] а. нанасян и к. хачатрян, «mail2sms.asnet.am – система оповещения о входящих письмах», материалы конф. csit-2013, ереван, 2013. [4] e. mateev, a. mkhitaryan, a. nanasyan, v. sahakyan and a. petrosyan, “hybrid infocommunication email sms unimail system”, proceedings of international conference computer science and information technologies, yerevan, armenia, pp. 389-391, 2009. [5] r. gerhards “the syslog protocol rfc 5424”, ietf-related tools, standalone or hosted on tools.ietf.org, 2009. submitted 20.07.2018, accepted 28.11.2018. 60 notification mechanisms by initiative of e-mail receiver in systems based on e-mail/sms technologies էլ․ նամակ ստացողի նախաձեռնությամբ ծանուցման համակարգեր հիմնված էլ․փոստի և sms-ի տեխնոլոգիաների վրա ա. մխիթարյան, ա․ պետրոսյան և ա․նանասյան ամփոփում աշխատանքում դիտարկվել է բջջային և համակարգչային ցանցերում տեքստային հաղորդագրությունների փոխանակման մեխանիզմների կիրառմամբ ծանուցման համակարգերի ստեղծման նախադրյալները։ էլեկտրոնային հաղորդագրության ստացողին իր իսկ նախաձեռնությամբ ծանուցման համակարգերի առանձնահատկությունները՝ ip և gsm տեխնոլոգիաների օգտագործմամբ: համեմատվել են առանձնացված համակարգի նախագծման տեխնիկական մոդելների առանձնահատկությունները: механизмы уведомления по инициативе почтового получателя на базе технологий e-mail и sms а․ мхитарян, а. петросян и а․ нанасян аннотация рассмотрены механизмы обмена сообщениями в компьютерных и сотовых сетях с целью отправки уведомления получателю электронной почты по его собственной инициативе с использованием технологий ip и gsm. сравниваются особенности технических моделей для проектирования разделенной системы. mathematical problems of computer science 50, 61-66, 2018. community detection-based recommendation framework karen k. mkhitaryan institute for informatics and automation problems of nas ra e-mail: karenmkhitaryan@gmail.com abstract recommender system is a type of information filtering system predicting users preferences about items, aiming to generate personalized recommendations. various recommendation approaches exist in the literature that differ in terms of methodology and types of systems they can be utilized on. in recent years some attempts have been made to incorporate community detection methods into recommender systems to make the process of recommendation generation more accurate in terms of rating or preference prediction and efficient in terms of computational complexity. in this paper we propose a community detection-based approach for recommender system, which is more reasonable in certain applications. keywords: community detection, recommender systems. 1. introduction in this era of big data it is hard to imagine an area, which does not deal with the collection and analysis of data. recommender systems being part of information filtering system is such area, aiming to suggest relevant information to users based on available data. applying recommendation techniques help companies to increase revenues and customer satisfaction, make more personalized user profiles, etc. several real world examples of rs include movie recommendation (movielens), song recommendation (last.fm) and product recommendation (amazon). one of the most popular recommendation techniques is collaborative filtering (cf), which predicts users interests i.e., how the user will rate the item, by analyzing the data about the users and their preferences. two popular types of cf are memory-based and model-based approaches [1]. in memory-based or neighborhood-based approach, ratings are predicted by applying similarity measure on user-item rating data, while in model-based approaches machine learning and data mining techniques (e.g., clustering, singular value decomposition, bayesian networks, etc.) are used to predict users preferences. there are also hybrid cf algorithms that use both memory-based and model-based approaches, which can improve prediction performance. moreover, various approaches from other disciplines such as machine learning and network science were incorporated into recommender systems for the purpose to increase efficiency of recommendation generation. community detection is a research area from network science, providing tools to partition 61 62 community detection-based recommendation framework the relational data (network) composed of entities (nodes) and interactions among them (edges) into subgroups, called communities or clusters that have dense connections inside and sparse connections with other subgroups in the network. some efforts have been made to combine community detection and recommender system techniques. in [2] authors showed the principal stages in community-based recommender system and how recommender systems can benefit from detection of communities. in [3] deng et. al incorporated community detection into svd++ model which has improved accuracy compared with svd model. abdrabbah et. al [4] proposed dynamic communitybased collaborative filtering approach combining both collaborative filtering and dynamic community detection techniques. proposed approach outperforms both item-based collaborative filtering and collaborative filtering based on static community detection. in this work we propose community detection based recommendation framework which is more reasonable in certain applications compared with traditional recommendation techniques. the paper is organized as follows: in section 2 we overview the popular recommendation approaches and present our proposed framework in section 3. 2. recommender systems recommender systems play an important role in suggesting relevant information to users, which are utilized in many areas such as recommending books, research articles, songs and products. modern literature provides many recommendation techniques and algorithms designed to predict users preferences. currently recommender systems are mostly developed using the collaborative filtering, content-based and hybrid model approaches that we discuss in the following subsections. (fig. 1) fig. 2. different implementations of recommender systems 2.1 collaborative filtering collaborative filtering (cf) is used to make predictions about x user’s interest by analyzing the tastes or preferences of other users similar to user x. it has three main approaches k. mkhitaryan 63 which are memory-based or neighborhood-based cf, model-based cf and hybrid cf. memory-based approach memory-based cf considers the similarities of users and items and is divided into two subsections: user-user filtering and item-item filtering. in user-user filtering, the idea is to find similar users to the given user x and recommend items to x those similar users liked. in contrast, item-item filtering finds users that liked an item and recommends other items that those users liked. below we demonstrate a simple example how user-user cf is working. consider a user-item rating matrix r where each rij element shows the rating user i gave to book j (table 1). table 1: user-rating matrix. r book 1 book 2 book 3 book 4 user 1 5 3 1 2 user 2 4 1 ? 1 user 3 5 5 1 3 user 4 4 ? 3 1 user 5 1 ? ? 2 in the matrix we see unknown ratings denoted by ”?”. the goal of both user-user and item-item filtering is to predict the unknown ratings using the r matrix. in the first step, similarities of users are calculated using similarity measure (e.g., cosine similarity, pearson correlation, manhattan distance, euclidean distance, etc.). in our example we use cosine similarity defined as: cosine(x, y) = ∑ i rxiryi√∑ i r 2 xi √∑ i r 2 yi , (1) where rxi and ryi are the ratings users x and y gave to item i respectively. symmetric matrix u of similarities between users is shown in table 2. table 2: cosine similarity values between users. u user 1 user 2 user 3 user 4 user 5 user 1 1 0.943 0.971 0.785 0.644 user 2 0.943 1 0.852 0.785 0.632 user 3 0.971 0.852 1 0.658 0.635 user 4 0.785 0.785 0.658 1 0.526 user 5 0.644 0.632 0.635 0.526 1 next, suppose we want to predict how user 5 will rate the books 2 and 3, which are missing in table 1. to do this we calculate the weighted average of item ratings and user similarities. r(user 5, book2) = 3 ∗ 0.644 + 1 ∗ 0.632 + 5 ∗ 0.635 0.644 + 0.632 + 0.635 = 3.003. (2) the same way r(user 5, book 3) = 1.582. finally as r(user 5, book 2) > r(user 5, book 3), book 2 will be recommended to user 5. 64 community detection based recommendation framework item-item filtering approach works in similar way considering item-item similarities instead of user-user similarities. model-based and hybrid approaches in model-based approaches machine learning and data mining algorithms are used to develop models predicting the rating user will give to item. general approaches include clustering algorithms (k-nearest neighbors), matrix factorization (singular value decomposition) and deep learning (multi-layered neural nets). hybrid approaches combine both memory-based and model-based approaches and can improve prediction performance and data sparsity issues. however, hybrid approaches may have higher computational complexity. 2.2 content-based filtering unlike collaborative filtering approach, which is based on large amount of ”collaboration” data between users and items where recommendation engine does not rely on the ”understanding” of the product itself, content-based approaches are based on the item descriptions and the users taste. recommendation engines developed on the principle of content-based approach, learn about users preferences and recommend items that have similar features to items the user likes. bayesian classifiers, decision trees, neural networks can be used to develop content-based filtering algorithms. 3. proposed framework as we saw in previous section, different cf approaches take ”collaboration” data between users and items as input and predict how the user will rate the items, which are not previously rated by him/her. on the other hand, content-based approaches recommend similar items to what user liked based on the description of an item itself. both collaborative filtering and content-based filtering approaches rely on similarities of users and items, which is not necessarily typical to some recommendation tasks. in proposed framework, recommendation to target user is done by considering preferences of users lying in the same community in the network constructed using the similarities of users based on predefined features. recommendations are generated by the implementation of the following steps: (fig. 3.): fig. 2. steps to make recommendation. step 1 feature selection recommendation process starts with identifying features, which are a set of attributes that describe the user (e.g., age, favorite music genre). features help to discriminate users based on their preferences, however their selection highly depends on the nature of recommender system it will be implemented on. k. mkhitaryan 65 step 2 user similarity calculation and graph construction in this step similarity between users is calculated. some popular measures of similarity are cosine similarity, euclidean distance, manhattan distance, pearson correlation coefficient, etc., which can be used to compare users based on predefined features. after the measure is applied and the similarity scores between users are obtained, the next step is a construction of a weighted network or graph where nodes and edges represent the users and similarity scores between them, respectively. step 3 community detection after a weighted network is obtained, community detection algorithms designed for weighted networks such as louvain [5] or fast greedy modularity optimization [6] can be applied to partition the network into community structure. step 4 recommendation in the final step, recommendation is done using the preferences of users about the items in the community of a target user, i.e., items that majority of users preferred in the community are recommended to the user. this framework also enables to tackle some drawbacks that exist in collaborative filtering and content-based filtering techniques. running algorithms on systems composed of millions or billions of users and items cause scalability issues, however in our framework community detection algorithms do not have too large complexity (o(n log n) for louvain and o(n log2 n) for fast greedy). another issue typical to content-based filtering is over-specialization i.e., recommendation of items that are very close to what user likes and is aware of. this problem was also tackled in this framework as item similarities and description of an item are not taken into consideration. 4. conclusion in this paper we discussed several popular recommendation algorithms that are utilized in recommender systems. we propose community detection-based recommendation framework, the use of which is justified in certain applications. the framework also enables to eliminate several drawbacks that are typical to traditional recommendation techniques such as scalability and over specialization. acknowledgement i would like to thank prof. mariam haroutunian for her support in this work. references [1] f. ricci, l. rokach and b. shapira, “introduction to recommender systems handbook”, springer, pp. 1-35, 2011. [2] f. gasparetti, a. micarelli and g. sansonetti, “community detection and recommender systems”, encyclopedia of social network analysis and mining, pp. 1-14, 2017. 6 6 community detection based recommendation framework [3 ] w . d e n g , r . p a t il, l . n a jja r , y . s h i a n d z. ch e n , \ in c o r p o r a t in g co m m u n it y d e t e c t io n a n d clu s t e r in g te c h n iqu e s in t o co lla b o r a t ive filt e r in g mo d e l" , p rocedia computer science, vo l. 3 1 , p p . 6 6 -7 4 , 2 0 1 4 . [4 ] s . b . a b d r a b b a h , r . a ya c h i, n . b . a m o r , \ co lla b o r a t ive filt e r in g b a s e d o n d yn a m ic co m m u n it y d e t e c t io n " , d ynak , 2 0 1 4 . [5 ] v . d . b lo n d e l, j. gu illa u m e , r . l a m b io t t e a n d e . l e fe b vr e , \ fa s t u n fo ld in g o f c o m m u n it ie s in la r g e n e t wo r ks " , j ournal of statistical m echanics: theory and e xperiment, vo l. 2 0 0 8 , 2 0 0 8 . [6 ] a . cla u s e t , m. e . j. n e wm a n , a n d c. mo o r e , \ fin d in g c o m m u n it y s t r u c t u r e in ve r y la r g e n e t wo r ks " , p hys. r ev. e , vo l. 7 0 , n o . 0 6 6 1 1 1 , 2 0 0 4 . submitted 10.08.2018, accepted 18.11.2018. ð³ù³ûýùý»ñç ñ³ûïý³µ»ñù³ý íñ³ ñçùýí³í ëáñññ¹³ïí³ï³ý ùçç³í³ûñ î. øëçã³ñû³ý ²ù÷á÷áõù êáñññ¹³ïí³ï³ý ñ³ù³ï³ñ·»ñá çýýáñù³óç³ûç ùß³ïù³ý ñ³ù³ï³ñ·»ñç ï»ë³ï »ý, áñáýù ï³ýë³·áõß³ïáõù »ý û·ï³ï»ñ»ñç ý³ëáýïñáõãûáõýý»ñá ¨ ëï»õíáõù »ý ³é³ç³ñïáõãûáõýý»ñ: ¶ñ³ï³ýáõãû³ý ù»ç ·áûáõãûáõý áõý»ý µ³½ù³åçëç ³é·áñçãùý»ñ, áñáýù ï³ñµ»ñíáõù »ý çñ»ýó ùáï»óáõùý»ñáí: ²ûë ñá¹í³íáõù ¹çï³ñïí³í »ý áñáß ñ³ûïýç ëáñññ¹³ïíáõãûáõý çñ³ï³ýóýáõ ³é·áñçãùý»ñ ¨ ³é³ç³ñïí³í ¿ ñ³ù³ûýùý»ñç ñ³ûýïý³µ»ñù³ý ù»ãá¹ý»ñç íñ³ ñçùýí³í ëáñññ¹³ïí³ï³ý ñ³ù³ï³ñ·ç ùá¹»é, áñý ³í»éç ñçùý³íáñí³í ¿ áñáß³ïç ïçñ³éáõãûáõýý»ñáõù: øçç³í³ûñá ñý³ñ³íáñáõãûáõý ¿ ï³éçë ý³¨ í»ñ³óý»é ùç ß³ñù ã»ñáõãûáõýý»ñ, áñáýù µýáñáß »ý ³í³ý¹³ï³ý ëáñññ¹³ïí³ï³ý ù»ãá¹ý»ñçý, çýãåçëçù »ý, ûñçý³ï` ñ³ßíáõ³ï³ý µ³ñ¹áõãûáõýá ù»í³í³í³é ïíû³éý»ñç ¹»åùáõù ¨ ëáñññ¹³ïíáõãûáõýý»ñç ý»õ ù³ëý³·çï³óáõùá: ðåêîìåíäàòåëüíàÿ ñðåäà íà îñíîâå îáíàðóæåíèÿ ñîîáùåñòâ ê. ìõèòàðÿí àííîòàöèÿ ðåêîìåíäàòåëüíûå ñèñòåìû ýòî âèä ñèñòåì îáðàáîòêè èíôîðìàöèè, êîòîðûå ïðåäñêàçûâàþò ïðåäïî÷òåíèÿ ïîëüçîâàòåëåé è ñîçäàþò ïðåäëîæåíèÿ. â ëèòåðàòóðå ñóùåñòâóåò ìíîæåñòâî àëãîðèòìîâ, êîòîðûå îòëè÷àþòñÿ ïî ñâîèì ïîäõîäàì. â ýòîé ñòàòüå ðàññìàòðèâàþòñÿ íåêîòîðûå èç íàèáîëåå ïîïóëÿðíûõ ðåêîìåíäàòåëüíûõ àëãîðèòìîâ è ïðåäëàãàåòñÿ ìîäåëü ñèñòåìû íà îñíîâå ìåòîäîâ îáíàðóæåíèÿ ñîîáùåñòâ, êîòîðàÿ áîëåå îáîñíîâàíà â îïðåäåëåííûõ ïðèëîæåíèÿõ. ñðåäà òàêæå ïîçâîëÿåò óñòðàíèòü ðÿä íåäîñòàòêîâ, òèïè÷íûõ äëÿ òðàäèöèîííûõ ìåòîäîâ ðåêîìåíäàöèè, òàêèå êàê, íàïðèìåð, âû÷èñëèòåëüíàÿ ñëîæíîñòü â ñëó÷àå áîëüøèõ äàííûõ è óçêàÿ ñïåöèàëèçàöèÿ ðåêîìåíäàöèé. 06_karen_61-66 sbornik_karen k_abstract d:\user\sbornik_38_pdf\34.dvi mathematical problems of computer science 38, 80{81, 2012. a fixed p oint t heor em for q-lattices y u .m.mo vs is ya n , d .s .d a vid o va department of mathematics and mechanics yerevan state university, yerevan, armenia e-mail: yurimovsisyan@yahoo.com i ntr oduction and pr eliminar ies b . k n a s t e r 's a n d a . ta r s ki's s e t -t h e o r e t ic a l ¯ xe d p o in t t h e o r e m is we ll kn o wn [1 ]. a g e n e r a liz a t io n o f t h is r e s u lt is t h e la t t ic e -t h e o r e t ic a l ¯ xe d p o in t t h e o r e m ( n a m e d ta r s ki's ¯ xe d p o in t t h e o r e m ) [2 ]. in [3 ] ta r s ki's t h e o r e m is g e n e r a liz e d fo r s e m ila t t ic e s . in t h e p r e s e n t wo r k a ¯ xe d p o in t -like t h e o r e m is p r o ve d fo r q-la t t ic e s . th e c o n c e p t o f a q-la t t ic e wa s in t r o d u c e d in [4 ]. th e a lg e b r a ( l; \; [) is c a lle d q-s e m ila t t ic e , if it s a t is ¯ e s t h e fo llo win g id e n t it ie s : 1 .a\b = b \ a( c o m m u t a t ivit y) ; 2 . a \ ( b \ c) = ( a \ b ) \ c ( a s s o c ia t ivit y) ; 3 . a \ ( b \ b) = a \ b ( we a k id e m p o t e n c e ) . th e a lg e b r a ( l; \; [ ) wit h t wo b in a r y o p e r a t io n s is c a lle d q-la t t ic e , if t h e r e d u c t s ( l; \) a n d ( l; [) a r e q-s e m ila t t ic e s a n d t h e fo llo win g id e n t it ie s , a\ ( b[a) = a\a, a[ ( b\a ) = a[a ( we a k a b s o r p t io n ) , a \ a = a [ a ( e qu a liz a t io n ) a r e va lid . fo r e xa m p le , ( z n f0 g; \; [) , wh e r e x \ y = j ( x; y ) j a n d x [ y = j[x; y]j, fo r wh ic h ( x; y ) a n d [x; y] a r e t h e g r e a t e s t c o m m o n d ivis io n ( g c d ) a n d t h e le a s t c o m m o n m u lt ip le ( lc m ) o f x a n d y , is a q-la t t ic e , wh ic h is n o t a la t t ic e , s in c e x \ x 6= x a n d x [ x 6= x. th e r e la t io n q µ l £ l is c a lle d a qu a s io r d e r if it is r e ° e xive a n d t r a n s it ive . l e t q b e a qu a s io r d e r o n t h e s e t l 6= ;; t h e n eq = q \ q¡1 µ l £ l is a n e qu iva le n c e . th e r e la t io n q=eq wh ic h is in d u c e d fr o m q o n t h e s e t l=eq in t h e fo llo win g m a n n e r : ( a; b ) 2 q=eq $ aqb; 8a 2 a,8b 2 b , wh e r e a; b 2 l=eq, is a n o r d e r . fu r t h e r , t h e o r d e r q=eq is d e n o t e d b y ·q a n d t h e c la s s o f e qu iva le n c e wh ic h in c lu d e s t h e e le m e n t x is d e n o t e d b y [x] 2 l=eq. th e fu n c t io n k : l=eq ! l is c a lle d c h o ic e fu n c t io n , if k ( [a]) 2 [a] fo r e a c h [a] 2 l=eq. th e p a ir ( l; q) is c a lle d inf sup-qu a s io r d e r e d s e t , if fo r e a c h t wo c la s s e s o f e qu iva le n c e s [a]; [b] 2 l=eq t h e r e e xis t inf ( [a]; [b]) = [a] \ [b] a n d sup ( [a]; [b]) = [a] [ [b], i.e . if ( l=eq; ·q ) is a la t t ic e . a n inf sup-qu a s io r d e r e d s e t ( l; q ) is c a lle d c o m p le t e , if fo r e a c h ; 6= y 2 l=eq t h e r e e xis t inf ( y ) 2 l=eq a n d sup( y ) 2 l=eq , i.e . if ( l=eq; ·q ) is a c o m p le t e la t t ic e . l e t ( l; q ) b e a n infsup-qu a s io r d e r e d s e t , k : l=eq ! l is a n a r b it r a r y c h o ic e fu n c t io n a n d fo r a n y t wo e le m e n t s x; y 2 l we h a ve :x \ y = k ( sup( [x]; [y]) ) ; x [ y = k ( inf ( [x]; [y]) ) ; t h e n t h e a lg e b r a ( l; \; [) is a q-la t t ic e . l e t ( l; \; [) b e a q-la t t ic e , t h e n t h e r e la t io n aqb $ a \ b = a \ a is a qu a s io r d e r o n t h e s e t l, t h e fu n c t io n k : l=eq ! l , wh ic h is d e ¯ n e d in t h e fo llo win g m a n n e r k ( [a]) = a \ a is a c h o ic e fu n c t io n a n d t h e p a ir ( l; q) is a n infsup-qu a s io r d e r e d s e t , wh e r e inf ( [a]; [b]) a n d sup ( [a]; [b]) a r e d e ¯ n e d b y t h e fo llo win g r u le s : inf ( [a]; [b]) = [a \ b]; sup( [a]; [b]) = 8 0 yu. movsisyan, d. davidova 8 1 [a [ b]: mo r e o ve r , fo r t h e o p e r a t io n s \ a n d [ we h a ve : x \ y = k ( inf ( [x]; [y]) ) ; x [ y = k ( sup( [x]; [y]) ) : th e fu n c t io n ' : l ! l o f a c o m p le t e infsup-qu a s io r d e r e d s e t ( l; q) is c a lle d m o n o t o n e if it fo llo ws fr o m xqy t h a t '( x) q'( y ) . th e fu n c t io n ' : l ! l o f a c o m p le t e inf sup-qu a s io r d e r e d s e t ( l; q ) is c a lle d h o m o m o r p h is m , if ' is a h o m o m o r p h is m o f t h e c o r r e s p o n d in g q-la t t ic e ( l; \; [ ) in t o it s e lf. th e p o in t x o f a n inf sup-qu a s io r d e r e d s e t ( l; q) is c a lle d a ¯ xe d p o in t o f t h e fu n c t io n ' : l ! l, if '( x) = x. th e fu n c t io n ' : l ! l o f a c o m p le t e infsup-qu a s io r d e r e d s e t ( l; q) is c a lle d a n t im o n o t o n e , if it fo llo ws fr o m xqy t h a t '( y ) q'( x) . th e fu n c t io n ' : l ! l o f a c o m p le t e infsup-qu a s io r d e r e d s e t ( l; q) is c a lle d a n t ih o m o m o r p h is m , if t h e fu n c t io n ' is a h o m o m o r p h is m fo r t h e c o r r e s p o n d in g q-la t t ic e ( l; \; [) in t o it s e lf. n o t e , t h a t if ' : l ! l is a h o m o m o r p h is m ( a n a n t ih o m o m o r p h is m ) o f a n infsupqu a s io r d e r e d s e t ( l ;q) in t o it s e lf, t h e n t h e in d u c e d fu n c t io n ~' : l=eq ! l=eq , wh ic h is d e ¯ n e d in t h e fo llo win g m a n n e r ~'( [x]) = ['( x ) ] is a h o m o m o r p h is m ( a n a n t ih o m o m o r p h is m ) . th e p o in t s x; y o f a n infsup-qu a s io r d e r e d s e t ( l; q) wit h t h e p r o p e r t y xqy a r e c a lle d a lt e r n a t ive ¯ xe d p o in t s o f t h e fu n c t io n ' : l ! l, if '( x ) = y a n d '( y ) = x. th e a lt e r n a t ive ¯ xe d p o in t s x; y o f t h e fu n c t io n ' : l ! l o f a n infsup-qu a s io r d e r e d s e t ( l; q ) in t o it s e lf a r e c a lle d e xt r e m e , if fo r e a c h a lt e r n a t ive ¯ xe d p o in t s a; b o f t h e fu n c t io n ' we h a ve xqaqbqy. m ain r esult t heor em 1 ) e a c h h o m o m o r p h is m ' : l ! l o f t h e c o m p le t e inf sup-qu a s io r d e r e d s e t ( l; q ) h a s a lt e r n a t ive ¯ xe d p o in t s . mo r e o ve r , if [®] = supf[x] 2 l=eqj[x] ·q ~'( [x]) g 2 l=eq is t h e g r e a t e s t ¯ xe d p o in t o f t h e fu n c t io n ~', t h e n fo r e a c h ¯ xe d p o in t a o f t h e fu n c t io n ' we h a ve aq( ®\® ) . s im ila r ly, if [¯] = inff[x] 2 l=eqj[x] ·q ~'( [x]) g 2 l=eq is t h e lo we r ¯ xe d p o in t o f t h e fu n c t io n ~', t h e n fo r a n y ¯ xe d p o in t a o f t h e fu n c t io n ' we h a ve ( ¯ \ ¯ ) qa. mo r e o ve r , t h e infsup-qu a s io r d e r e d s e t f ix ( ') = fx 2 lj'( x) = xg is a c o m p le t e inf sup-qu a s io r d e r e d s e t . 2 ) e a c h a n t ih o m o m o r p h is m ' : l ! l o f t h e c o m p le t e infsup-qu a s io r d e r e d s e t ( l; q ) h a s e xt r e m e a lt e r n a t ive ¯ xe d p o in t s . a n a p p lic a t io n o f t h is t h e o r e m in s e m a n t ic o f lo g ic p r o g r a m m in g is c o n s id e r e d . r e fe r e n c e s [1 ] b . k n a s t e r , a . ta r s ki, u ¶n t h e o r e m e s u r le s fo n c t io n s d 'e n s e m b le s , a n n . s o c . p o lo n . ma t h ., 1 9 2 8 , v. 6 , p . 1 3 3 -1 3 4 . [2 ] a . ta r s ki, a la t t ic e -t h e o r e t ic a l ¯ xe d p o in t t h e o r e m a n d it s a p p lic a t io n s , p a c ī c jo u r n a l o f ma t h e m a t ic s , 1 9 5 5 , v. 5 , p . 2 8 5 -3 0 9 . [3 ] j. b e r m a n a n d w . j. b lo k, ge n e r a liz a t io n s o f ta r s ki's ¯ xe d p o in t t h e o r e m fo r o r d e r va r ie t ie s o f c o m p le t e m e e t s e m ila t t ic e s , or d e r , 1 9 8 9 , v. 5 , p . 3 8 1 -3 9 2 . [4 ] i. ch a jd a , l a t t ic e s in qu a s io r d e r e d s e t s , a c t a p o la c ky u n ive r s it y, olo m o u c , 1 9 9 2 , v. 3 , p . 6 -1 2 . 8 1 mathematical problems of computer science 40, 68---75, 2013. 68 concealment of targeted regions in digital images on mobile devices gevorg a. karapetyan and hakob g. sarukhanyan institute for informatics and automation problems of nas ra e-mail: gevorgka@gmail.com, hakop@ipia.sci.am abstract in this paper we present a framework for robust and effective inpainting of targeted regions in digital images. in framework the modification of frequency selective extrapolation algorithm is used, which provides good quality of concealment. the results of the concealment algorithm of the framework are being compared with the results of the state of art inpainting methods which are included in opencv library available for mobile devices. the framework is implemented for android os. for selection of targeted regions in digital images the advance of bézier curves is used. the effectiveness of the framework has been tested on various digital images. keywords: image inpainting, regions concealment, selective extrapolation, bezeir curve, mobile devices. 1. introduction mobile computing technologies are growing very rapidly. new mobile devices have multicore processors, which give ability to perform computer vision algorithms with high complexity on mobile devices. this paper describes the implementation of inpainting algorithm on a mobile phone with android os. one of the major uses of mobile devices is taking images via camera of device. before sharing the images via social networks users usually filter the images for enhancing the image quality as well as editing the image to remove or hide targeted regions. since the number of shared images of active user is large the application which he or she uses should be user friendly and have high performance. thus, we have designed an application with user friendly interface and optimized the accurate inpainting algorithm for working fast on a mobile device. for mobile applications several inpainting algorithms [5, 6] are included in opencv (open source computer vision library) library [7]. these algorithms can be used in mobile applications via opencv library. opencv was built to provide a common infrastructure for computer vision applications and to accelerate the use of machine perception in commercial products. opencv library has more than 2500 optimized algorithms, which includes a comprehensive set of both classic and state-of-the-art computer vision and machine learning algorithms. we have explored inpainting methods which are included in opencv lib. these methods are method by alexandru telea [5] and navier-stokes [6] based method. telea inpainting algorithm is based on propagating an image smoothness estimator along the g. karapetyan, h. sarukhanyan 69 image gradient. a similar approach is used in [8], where the image smoothness information is estimated by the image laplacian and is propagated along the isophotes directions, estimated by the image gradient rotated 90 degrees. in telea algorithm the image smoothness is estimated as a weighted average over the known adjacent areas of the pixel to be inpainted. into missing regions the information is being propagated via fast marching method (fmm) [9]. the other method in opencv is inpainting using the navier-stokes equations. in this approach the isophotes are being propagated with the usage of ideas from classical fluid dynamics [6]. algorithms provide good results for very small areas and high performance, but have issues with structured and large regions. in our framework the inpainting is done via a modified frequency selective extrapolation (fse) [1-4] algorithm, which provides good results in structured and homogeneous regions. the algorithm removes targeted areas from digital image with high accuracy. the areas are being selected with the help of bezeir curves [4]. the mobile screen sensor handles gesture of user’s finger and obtains the targeted area mask. based on targeted areas and mask is being generated and is passed to inpainting algorithm. the algorithm conceals targeted area with usage elements from adjacent areas. the inpainting algorithm is being implemented on c++ programing. the user interface is developed via java programming language which uses static lib developed on c++. the paper is organized as follows: in section 2 targeted areas selection is being described. then in section 3 the concealment method is being described. in section 4 the experiment results are being shown and compared with inpainting algorithm described in opencv 4.3 [7]. the work concludes in section 5 where a brief summary of proposed framework and future work is described. 2. selection of targeted regions the binary mask which highlights the unwanted regions of the image is obtained with user interaction. user with finger selects the unwanted region. the selection curves are drawn with the help of bezeir curves. on screen touch we can obtain the coordinates of current point ).,( 000 yxp then on finger move we measure the distance of the current point from the starting pixel. if it equals to 5, then we store that value and mark it as ).,( 111 yxp .)()( 210 2 10 yyxxdistance  then the user continues the movement of his finger we calculate the distance of ongoing point from 1p . when the distance is equal to 5px we store that point as 2p . the set of points 210 ,, ppp are used for building )(tb bezeir quadratic curve, which is calculated below: ]1,0[,)1(2)1()( 2 2 10 2  tptpttpttb . fig. 1. bezeir curve built with usage of points concealment of targeted regions in digital images on mobile devices70 fig. 2. the result of selection of targeted image areas via bezeir curves: a) input image; b) selected targeted region then 2p is set as 0p and the curve building procedure continues till the user takes the finger up. after selection regions of interest for being concealed we generate binary mask which will be used in concealment algorithm. 3. concealment method image inpainting is the process of concealing missing data in digital images, via interpolation of the missing pixels using information of the adjacent areas. as an inpainting method we have used a modification of powerful frequency selective extrapolation algorithm [1-3], where the size of support area is being chosen content aware. fig. 3. a targeted area which should be concealed with help of b supporting area we choose  area, which is composed of corrupted region a and support areas elements from b . the image in  we denote ,,]},,[{ ,0,0 nm nmnmf  where m and n are width and height of area  . for estimation of corrupted regions the samples in support area b will be approximated by weighted linear combination of two-dimensional dft (discrete fourier transform) basis functions, which are defined over area  . the approximation will be done with usage of ],[ nmg parametric model. ,],[],[ ),( ,,   vklk lklk nmcnmg  where vk is the set of basis functions, lkc , is the expansion coefficient, lk , 2d dft basis function which is defined in entire area  . g. karapetyan, h. sarukhanyan 71 we assume that in any iteration v the signal is being approximated by ],[)( nmg v parametric model, which is calculated as: ,],[],[ ),( , )( , )(    vklk lk v lk v nmcnmg  where )( , v lkc is the expansion coefficient calculated in iteration .v the quality of extrapolation is controlled by calculation of the residual error signal in area  , which is calculated by ],,[]),[],[(],[ )()( nmwnmgnmfnmr vv  where ],[ nmw is the window function, which is 0 in highlighted area and is 1 in support area, i.e.       .),(,1 ,),(,0 ],[ bnm anm nmw for each iteration )(vbe error criterion is being calculated   .],[],[ ),( 2)()(    nm vv b nmgnmfe the iteration stops, when e drops below the pre-defined threshold. after termination of the iterative process the corrupted pixels in area a of are being replaced with the corresponding pixels from parametric model. the algorithm is working in frequency domain and the algorithm steps are as follows: 1. initially the ],[ nmg parametric model is equal to 0: .),(,0],[)0(  nmnmg and the residual error is equal to the input signal multiplied with the window function, .),(],,[],[],[)0(  nmnmwnmfnmr since the algorithm works in frequency domain, one dft is applied on ],,[0 nmg ],[ nmf and :],[0 nmr  ,],[],[ nmfdftnmf   ,],[],[ )0()0( nmgdftnmg vv    .],[],[ )0()0( nmrdftnmr vv   2. calculation of )1(  vae energy decrease computation, the difference of error criterion between v and )1( v iterations. 3. selection of ),( vu basis function indexes value pair, for which )1(  vae .maxarg),( )1(  vaevu 4. update of )1( , v lkc expansion coefficient update. 5. repeat steps 2 – 4 until the )1(  vbe drops predefined threshold 15min e or the number of iteration becomes more than max number of iterations. we used max number of iterations 100. 6. after termination apply invers discrete fourier transform on parametric model:  .],[],[ )( nmgidftnmg v concealment of targeted regions in digital images on mobile devices72 then replace the element from ],[ nmf which contain specular reflections with corresponding elements from ],[ nmg : ],,[],[ nmgnmf  where .],[ anm  4. implementation on mobile device the proposed inpainting algorithm is implemented on samsung galaxy s4 mobile device which has open source android os. the mobile device has octa core processor with 1,6 ghz quad + 1,2 ghz quad cpu speed. the user interface is developed on java and the inpainting algorithm on c++. the algorithm is used as a static library in java project. we take the input image from the gallery of camera of a mobile device. we have designed an interface for easy selection of targeted regions of the image, which should be concealed with data form adjacent areas. the experiments have been done on different images on structured and textured areas. fig. 5. a) original image with selected area which should be inpainted; b) inpainted image with navier-stokes method; c) inpainted image with telea method; d) inpainted image with modification of fse method. we have also compared inpainting algorithms which are included in opencv library with the algorithm included in our framework. in navier-stokes and telea algorithms the radius of circular neighborhood of each point was set 24. the parameter was chosen by the authors for achieving better psnr. in modification of fse are used 16x16 blocks; adaptive supporting area size varies from 3-16 px; fft size 64; max iterations number 100. the results of the comparison are shown in figure 4 and table 1. c) b) d) a) g. karapetyan, h. sarukhanyan 73 table ι. psnr and performance time comparison of proposed method with navier-stokes and telea inpainting methods navier-stokes telea modification of fse psnr 34.2db 33.7 db 38.75 db time 12704ms 15160ms 26731ms we can see in table1 the excellence in quality of the proposed algorithm above telea and navierstrokes methods. but the modification fse performs slower because of more detailed analyses of support area and iterative propagation. (a) (b) fig. 4. (a) original image, (b) concealed image figure 4 shows an example of restoration of an old image via our framework; (a) is the input image which contains a corrupted area and (b) is an output image, where the target areas have been concealed with high accuracy. in figure 5 an example of removing object form the image is shown; (a) an input image and (b) is an output image where the airplane has been removed. a) b) fig. 5. a) original image; b) inpainted image with removed object concealment of targeted regions in digital images on mobile devices74 5. conclusion we have implemented a mobile framework for user-interactive selection and concealment of targeted regions in digital images. the framework has been developed for android os. the experiments showed the effectiveness and robustness of the proposed framework also provided positive results in comparison with some state of the art inpainting methods which are included in opencv. the modification of fse provides good results in inpainting not of very large structured and textured areas. for very large regions the modification of fse is not effective: we are going to create a hybrid framework which will use an exemplar-based inpainting method for very large regions and modification of fse for smaller regions. also we are going to use opencl lib for achieving better performance. moreover, we are going to design an interface for third-party mobile applications which can use our inpainting library. acknowledgement gevorg karapetyan would like to thank the state committee of science of armenia for supporting this work by grant 13a-1b30. references [1] a. kaup, k. meisinger and t. aach, “frequency selective signal extrapolation with applications to error concealment in image communications”, international journal of electronic communication (ae), vol. 59, pp. 147–156, 2005. [2] g. karapetyan, “modification of fse method based on coefficients of homogeneity”, mathematical problems of computer science, vol. 35, pp. 109–115, 2011. [3] g. karapetyan and h. sarukhanyan, “on a modification of the frequency selective extrapolation method“, information models and analyses, vol. 2, pp.139–145, 2012. [4] t. farouki, “the bernstein polynomial basis: a centennial retrospective”, computer aided geometric design, vol. 29, issue 6, pp. 379 – 419, 2012 [5] a. telea, “an image inpainting technique based on the fast marching method”, journal of graphics tools 9.1, pp. 23-34, 2004. [6] m. ebrahimi and m. holst, “the navier-stokes-voight model for image inpainting”, [online]. available: http://arxiv.org/pdf/0901.4548v3.pdf [7] open source computer vision library home page, [online]. available: http://www.opencv.org [8] m. bertalmio, g. sapiro, v. caselles, and c. ballester. “image inpainting”, proceedings siggraph 2000, computer graphics proceedings, pp. 417—424, 2000. [9] j. a. sethian. “a fast marching level set method for monotonically advancing fronts”, proc. nat. acad. sci., vol. 9, no.4, pp. 1591—1595, 1996. [10] a. criminisi, p. perez and k. toyama, “object removal by exemplar-based inpainting” proceedings of the 2003 ieee computer society conference on computer vision and pattern recognition, pp. 721--728, 2003. submitted 26.08.2013, accepted 15.10.2013. g. karapetyan, h. sarukhanyan 75 թվային պատկերների նպատակային տիրույթների քողարկում շարժական սարքերում գ. կարապետյան և հ. սարուխանյան ամփոփում աշխատանքում ներկայացված է շարժական սարքերում թվային պատկերների նպատակային տիրույթների քողարկման կայուն և արդյունավետ համակարգ: համակարգում օգտագործված է հաճախականային ընտրովի էքստրապոլյացիայի փոփոխված մեթոդը, որն ապահովում է քողարկման բարձր ճշգրտությունը: մեր կողմից առաջարկված քողարկման ալգորիթմի արդյունքները համեմատված են քողարկման հայտնի մի շարք ալգորիթմների հետ, որոնք ներառված են շարժական սարքերի համար հասանելի opencv գրադարանում: մշակված համակարգը աշխատում է android օպերացիոն համակարգի միջավայրում: թվային պատկերներում նպատակային տիրույթների ընտրման համար օգտագործվում են բեզեի (bézier) կորերը: համակարգի արդյունավետությունը ցուցադրված է բազմաթիվ թվային պատկերների միջոցով: сокрытие целевых регионов в цифровых изображениях мобильных устройств г. карапетян и а. саруханян аннотация в работе представлена устойчивая и эффективная система для сокрытия целевых регионов в цифровых изображениях мобильных устройств. в системе используется модификация метода частотной выборочной экстраполяции, которая обеспечивает высокую точность сокрытия. предлагаемый нами алгоритм сокрытия сравнен с рядом известных алгоритмов, которые включены в библиотеку opencv, которая доступна для мобильных устройств. разработанная система работает в операционной системе android. в цифровых фотографиях для выделения целевых регионов используются кривые безье. эффективность системы продемонстрирована на многочисленных цифровых изображениях. mathematical problems of computer science 40, 76--84, 2013. 76 on identification of anomalies in multidimensional hydrogeochemical data as earthquake precursors evgueni a. haroutunian1, irina a. safaryan1, hrachya m. petrosyan2 and armine r. gevorkyan2 institute for informatics and automation problems of nas ra armenian national survey of seismic protection of the ministry of emergency situations of ra e-mail: eghishe@sci.am abstract we consider two nonparametric methods are discussed for identification of anomalies in multidimensional hydrogeochemical data with the goal of forecasting major seismic events. the first method is based on using the threshold copula function aimed at the definition of change moments in the structure of dependencies between the components of a multidimensional vector of observations. the second method is designed to transform multidimensional data to one-dimensional which allows to use rank score tests. keywords: seismic-forecasting anomalies, hydrogeochemical precursors of strong earthquake, rank score test, threshold copula, pooled sample method. 1. introduction the processes occurring in the upper layers of the earth's crust prior to large earthquakes, cause abnormal changes in a number of different geophysical and geochemical indicators. an important component of the seismic risk reduction is the continuous monitoring and evaluation of the current seismic hazard. the current seismic hazard assessment (sha) being carried out at nssp of ra is based on the data from multiparameter observational network. the processing of monitoring data is presented in monographs of h. petrosyan [1, 2]. hydrogeochemical parameters take a special place among the monitored data since the moving parts of the earth’s crust–ground waters and gas are considered as informative parameters of monitoring. in particular, it is recognized that the percentage of dissolved chemicals and gases observed in special wells changes prior to strong seismic events. such changes are observed quite stable for several months before the earthquake, and therefore can serve as its medium–term precursor. the regional geochemical network of nssp consists of a e. haroutunian, i. safaryan, h. petrosyan , a. gevorkyan 77 number of monitoring stations; probes are taken once a day and analyzed for 14 microcomponents. on the territory of the russian federation the monitoring of hydrogeochemical indicators is conducted by kamchatka branch of the geophysical survey of russian academy of sciences. more than 20 parameters are measured and analyzed with the interval between the scheduled observations in the range of 3 to 6 days. the statistical analysis of results of multidimensional hydrochemical series is presented in the paper of g. rjabinin and j. khatkevich [3] and in the monograph by a. ljubooshin [4]. in 1997-2000 a computer program implementing a nonparametric algorithm (based on rank score tests) was developed in ipia which was designed to detect changes in statistical properties of one-dimensional random sequences. the program was applied to treat the hydrogeochemical data for 1985-1995 from the stations kajaran, ararat and surenavan provided by nssp. theoretical fundamentals of the algorithm are stated in [5-7], the results of computer processing and the program description are in [8-10]. in this paper several new results presented at the conferences [11-12] in the direction outlined, are devoted to a comparative analysis of time series of helium taken from the three points of observation, as well as to changes of their joint distribution on the eve of major seismic events. the behavior study of the joint distribution function of several indicators in the periods between the strong earthquakes has several objectives:  forecasting of earthquake parameters such as time coordinates of the epicenter, magnitude, energy class, etc.; one-dimensional time series allow predicting only time;  derivation of a reliable precursor; for multivariate hydrogeochemical data the structure of time correlation is such a sign indicated in [2];  working out of a one-dimensional integrated indicator for the seismic hazard. for detection of changes in multivariable dependence structure in this paper threshold copulas investigated in [13-15] and for obtaining an integrated one-dimensional indicator the method of pooled sample is applied. 2 the comparative analysis of data on helium from three observation stations in identification of precursors with statistical methods an anomaly is interpreted as a change of statistical characteristics of the observed series of indicator values which is in some sense conformed to the physical model of a seismic event preparation. the most successful, as it is defined in [1], is the model of consolidation by dobrovolsky, according to which the cycle of a single earthquake has 4 phases: i – regular state, ii – development of heterogeneity (phase of consolidation), i.e. appearing of long-term and medium-term precursors, iii – phase of destruction which has two stages (the stage of a short term precursors and for-shocks, namely at this stage the earthquake occurs, and the stage of aftershocks) and iv – restoration of regular state, that is returning to the phase i. thus, we present the following working hypothesis. we assume that the data of observation is a sequence of independent random variables which before the earthquake contain at least two change-points (disorder moments), and at least one change-point after the earthquake. the first change-point corresponds to the beginning of the phase ii, the second− to the moment of coming to some quasi-stable level, i.e. to the middle of the phase ii, and the third – to the start of the phase iv. an identification of precursor consists in obtaining of robust estimates of the moments and . on identification of anomalies in multidimensional hydrogeochemical data as earthquake precursors78 the change in statistical properties is interpreted as a change of the distribution function for the elements of series. the parametric form of distribution and any type of change is not postulated a priori which assumes the application of nonparametric (distribution-free) statistical procedures to identification of these changes. schematically a number of moments of disorder accompanying the earthquake can be depicted as it is shown on figure 1. thus, as a rule, changes are associated with an increase or decrease in the percentage of micro-component, so it is assumed that = − , > 0 where the distribution function ( ) reflects the variations of background of the observed indicator. however, not always return to the regular state means that = in general = − and either > 0 or < 0. in some cases, identification of moments and is possible visually. for their identification with statistical methods a sequence of rank score statistics is applied, which is defined as follows: fig. 1. disorder moment series before and after a single earthquake let = # : ≤ , = 1,…, = 1,…, be ranks of observations ,…, and = 1 + 1⁄ , = . the sequence of rank score statistics is defined as follows:= − − , = 1,…, − 1. then under the condition of an appropriate choice of the score function the number of observation defined as = argmin is consistent estimate either for the moment or for depending on the position of the window in which the sequence of statistics is calculated. based on theoretical results [5-7] it can be stated that if in the linear plot of the sequence of statistics ( ) both global and local minimums immediately follow each other and go beyond the critical value then the anomaly is considered as identified. e. haroutunian, i. safaryan, h. petrosyan , a. gevorkyan 79 the appropriate choice of the score function means that for distributions differing in shift( ) = , then is called a wilcoxon statistics, for distributions differing in scale= ( − ) and is called a mood statistics. in general case distributions ( ) and ( ) differ both in shift and scale, while statistics is a linear combination of statistics of wilcoxon and mood, the coefficients of which can be obtained as sample estimates in retrospective analysis. an algorithm for producing such estimates will be elaborated and included in the new version of the program. time series of helium on the three monitoring stations ararat, kajaran, surenavan over the period 1985-1989 and behavior of sequences of wilcoxon statistics, computed in a window including several seismic events, are presented on figure 2. dates of earthquakes occurring in the region in that period are marked with vertical lines. on the left vertical axis values of indicator of helium and on the right axis the values of wilcoxon statistics are scaled. fig. 2. behavior of the sequence of wilcoxon statistics in window 01.01.85-25.10.88 and time series of helium over the period 1985-1989 for the stations ararat, surenavan, kajaran. on identification of anomalies in multidimensional hydrogeochemical data as earthquake precursors80 the comparative analysis showed that each observation point (ararat, kajaran, surenavan) is characterized by its precursors anomaly which differs by its appearance term. all the three monitoring stations respond to the earthquakes of the period 1985-1989. the wilcoxon test allows detecting seismic-forecasting anomalies (i.e. change-points and marking the periods of growing increase of the average value of helium indicator) before each earthquake. the station ararat most significantly reacted to the earthquake of 1988 (spitak), the stations kajaran and surenavan to the earthquake in 1986, though a delay was detected on the station surenavan. on the base of exclusively visual analyses of the time series it is not possible to obtain these conclusions. 3. threshold copula models for identification of seismic-forecasting anomalies an important index of upcoming seismic event is a change in the structure of dependence between different indicators of one station or one indicator for different stations. in [2] a graph of correlation dependencies between various hydrogeochemical indicators of different stations of the nssp is presented. in particular it is confirmed that significant correlation between different stations in data on helium is absent. a contemporary approach to assessment of the structure of dependence is based on the use of the copula function. for theoretical and practical applications of the copula function we refer to the monograph of nelsen [16]. one threshold copula, introduced in [13-15], can be used in case when the dependence between the components of two-dimensional data arises in time as a precursor of upcoming seismic event. we call a random variable homogeneous with respect to if for all pairs, on the plane the following conditional probabilities are equal: pr ≤ ≤⁄ =pr ≤ >⁄ . the concept of homogeneity is equivalent to the notion of independence. if there exists a unique value of = such that for all ∈r , pr ≤ ≤⁄ = pr ≤ ≤⁄ for ≤ ,pr ≤ >⁄ = pr ≤ >⁄ for > , and pr ≤ ≤⁄ ≠ pr ≤ >⁄ , then the statistical dependence between and is called one-threshold and the value is called a threshold. hypotheses on the homogeneity indicator of helium on stations ararat and surenavan with respect to that on the station kajaran for the period 01.01.85 03.02.86 were tested. the indicators on stations surenavan and kajaran are independent. meanwhile when comparing the stations ararat and kajaran we found that the value of helium threshold, which was equal to 1571, registered at the station kajaran dated 15.08.85 proves to be separating for the indicator at the station ararat. the two-dimensional histograms built for copulas, depicted on figure 4 present evidence that the point 15.08.85 is actually the moment of change in the structure of dependence. e. haroutunian, i. safaryan, h. petrosyan , a. gevorkyan 81 • ararat-kajaran 01.01.85 – 03.02.86 • earthquake 13.06.86 • change-point moment15.08.85 fig. 4. copula density functions: left for period 01.01.85 03.02.86, right for periods 01.01.85-15.08.85 and 16.08.85 – 03.02.86; data of earthquake: 13.05.86, change-point moment data 15.08.85. 4. pooled sample method if the observed series refer to the same physical quantity the method of mixed samples can be used as well to detect the anomalies preceding the earthquake. there exist different ways of mixing. we joined the data on helium over 14 months preceding to spitak earthquake in the following way: the odd numbers correspond to observations at the station kajaran, while for the first sample the even observations registered at the station surenavan and for the second sample the even observations multiplied to six corresponded to the ones registered at the station ararat that day. the pooled samples are presented on figure 5. then, using the wilcoxon statistics to the combined samples, we obtained that for the first sample a global minimum statistic falls at 541, which corresponds to the date 26.06.88 and for the second at 244, which corresponds to the date 11.02.88. thus, by using a mixed sample you can get a reliable estimate of the beginning of the accumulation of changes than by onedimensional samples. on identification of anomalies in multidimensional hydrogeochemical data as earthquake precursors82 (a) 26.06.88 400 600 800 1000 1200 1400 1600 1800 2000 2200 2400 -6 -5 -4 -3 -2 -1 0 1 2 3 4 (b) 11.02.88 1300 1400 1500 1600 1700 1800 1900 2000 2100 2200 -16 -14 -12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12 fig. 5. surenavan-kajaran pooled sample – (a), ararat-kajaran pooled sample – (b). 5. conclusion the presented results show the prospects of application of threshold copula methods and mixed sampling to determination of anomalies in multidimensional hydrogeochemical data occurring prior to earthquakes. further theoretical elaboration of algorithms and implementation of programs for their realization is planned. acknowledgement this work was supported in part by scs of mes of ra under thematic program no scs 13-1a295. references [1] h. m. petrosyan, earthquake testing and prognosis, yerevan, 2004. [2] h. m. petrosyan, precursors and prognosis of earthquakes on the territory of the republic of armenia, yerevan, 2009. [3] g. d. rjabinin and yu. m. khatkevich, “on the issue of possibility of the earthquake prediction with hydrogeochemical methods”, materials of the iv allrussia symposium on volcanology and paleo-volcanology, pp. 660-663, 2009. e. haroutunian, i. safaryan, h. petrosyan , a. gevorkyan 83 [4] a. a. ljubooshin, analysis of data of geophysical and ecological monitoring, moscow, nauka, 2007. [5] d. g. asatryan and i. a. safaryan. “nonparametric methods for detecting changes in the properties of random sequences”, in: detecting changes in random processes. ed. by l. telksnys, new york, pp.1-13, 1986. [6] e. a. haroutunian and i.a. safaryan. “nonparametric consistent assessment of moments of property changes of random sequences”, mathematical problems of computer science, vol. 17, pp. 76-85, 1997. [7] i. a. safaryan, “nonparametric estimation under gradual change of the random sequence”, statistical problems of control, vol. 83, pp.121-126, 1988. [8] e. a. haroutunian, i. a. safaryan, p. a. petrossian and h. v. nersessian, “earthquake precursor identification on the base of statistical analysis of hydrogeochemical time series”, mathematical problems of computer science, vol. 18, pp. 33-39, 1997. [9] h .v. nersessian and e. a. haroutunian, “statistical nonparametric analysis of hydrogeochemical data of the ararat station hssp of armenia”, mathematical problems computer science, vol. 19, pp. 40-44, 1988. [10] p. a. petrossian, “software implementation of the algorithm of detection of moments in the state changes of time series”, mathematical problems of computer science, vol. 19, pp. 32-39, 1988. [11] e. a. haroutunian, i. a. safaryan and a.o. yesayan, “application of special statistical algorithms of the hydrogeochemical time series monitoring with the goal of the earthquake prognosis”, proceedings of the conference dedicated to the 23-d anniversary of the spitak earthquake, mes, yerevan, 2011. [12] i. a. safaryan and a. o. yesayan, “copula-model applications to seismic-forecasting anomalies detection”, multivariate statistical analysis and econometrics, proceedings of viii international school-seminar, armenia – tsakhkadzor. m., cemi ras, p. 189, 2012. [13] e. a. haroutunian and i. a. safaryan. “copulas of two-dimensional threshold models”, mathematical problems of computer science, vol .31, pp. 40-48, 2008. [14] e. a. haroutunian and i. a. safaryan. “on certain threshold copulas estimators”, abstracts of internatioal conference on computer science and information technologies, yerevan, pp. 145-147, 2009. [15] e. a. haroutunian and i. a. safaryan. “on estimation of threshold parameter in threedimensional copulas model”, abstracts of internatioal conference on computer science and information technologies, yerevan, pp. 129-131, 2011. [16] r.v. nelsen, an introduction to copulas, springer, new york, 2006. submitted 10.09.2013, accepted 16.10.2013. on identification of anomalies in multidimensional hydrogeochemical data as earthquake precursors84 երկրաշարժերին նախորդող բազմաչափ հիդրոերկրաքիմիական տվյալներում անկայունության նույնականացման մասին ե. հարությունյան, ի. սաֆարյան, ի. պետրոսյան և ա. գևորգյան ամփոփում քննարկվում են հզոր սեյսմիկ իրադարձությունների կանխատեսման նպատակով բազմաչափ հիդրոերկրաքիմիական տվյալներում անկայունության նույնականացման երկու ոչ պարամետրիկական մեթոդներ: առաջինը հիմնված է դիտարկումների բազմաչափ վեկտորի բաղադրիչների միջև կախվածությունների կառուցվածքի փոփոխության պահի որոշման նպատակով շեմային կապակցիչի օգտագործման վրա: երկրորդ եղանակը կառուցված է ըստ բազմաչափ տվյալների միաչափերի վերածման միտման, որը թույլ է տալիս կարգային նշումների տեստերի օգտագործումը: об идентификации аномалий в многомерных гидрогеохимических данных предшествующих землетрясениям е. арутюнян, и. сафарян, н. петросян и а. геворкян аннотация обсуждаются два непараметрических метода идентификации аномалий в многомерных гидрогеохимических данных, с целью предсказания сильных сейсмических событий. первый метод основан на применении функции пороговой копулы для определения моментов изменений структуры зависимостей между компонентами многомерного вектора наблюдений. второй метод использует превращение многомерных данных в одномерные для применения критериев ранговых меток. начиная с начала 2000 года осуществляется внедрение ghis в здравоохранении, в рамках принятого проекта о реформирование информ mathematical problems of computer science 49, 66--73, 2018. construction of linear codes over rings 𝑍𝑍𝑚𝑚 correcting double ±1 or ±2 errors gurgen h. khachatrian* and hamlet k. khachatrian ** * american university of armenia **institute for informatics and automation problems of nas ra e-mail: gurgenkh@aua.am, hamletkh@ipia.sci.am abstract in this paper a construction of double ±1 and ±2 errors correcting linear optimal and quasi-optimal codes over rings 𝑍𝑍5, 𝑍𝑍7 and 𝑍𝑍9 is presented with the limitation that both errors have the same amplitude in absolute value. keywords: error correcting codes, codes over the rings 𝑍𝑍𝐴𝐴 , errors with magnitude ±1 and ±2. 1. introduction from a practical point of view the codes over rings 𝑍𝑍2𝑚𝑚 or 𝑍𝑍2𝑚𝑚+1 are interesting, because they can be used in 22𝑚𝑚 – qam (quadrature amplitude modulation) schemes. codes over finite rings, particularly over integer residue rings and their applications in coding theory, have been studied for a long time. errors occuring in the channel usually have a limited magnitude, and this effect is particularly applicable to flash memories. there have been a couple of papers regarding the optimal ±1 single error correcting codes over alphabet 𝑍𝑍𝑚𝑚 [1,2]. in this paper we will consider on errors with the magnitude ±1 or ±2. there have been a couple of papers regarding the ±1 and ±2 single error correcting codes [1, 3]. in this paper we will construct codes correcting double ±1 or ±2 errors with the limitation that both errors have the same amplitude in absolute value over the rings 𝑍𝑍𝑚𝑚. they are based on optimal codes with 4 parity check symbols correcting double errors over rings 𝑍𝑍𝑚𝑚 of value ±1 , adding only one parity check symbol. in this case, we cannot correct double errors with different magnitudes, for example, one error with a magnitude of +1 and the other one with a magnitude of +2. the optimality criteria for linear codes over the fixed ring 𝑍𝑍𝑚𝑚 can be considered in two ways (see [4]). first of all, recall that the code of the length n is optimal-1 if it has a minimum possible number of parity check symbols. secondly, the optimality-2 criteria for the code is that for a given number of parity check symbols, it has a maximum possible length. our constructed 66 mailto:gurgenkh@aua.am g. khachatrian and h. khachatrian 67 code will satisfy the first optimality criteria. in this case, we do not know codes, which satisfy the optimality criteria – 2. from [4] we know the optimal code (12, 8) correcting double ±1 errors over the ring 𝑍𝑍5. the linear code (12, 8) correcting double errors over the ring 𝑍𝑍5 of value ±1, presented in [4] satisfies the optimality criteria -1: 𝐻𝐻 = � 1 1 1 1 1 0 1 2 3 4 1 1 0 1 2 3 4 2 2 2 2 2 1 1 3 2 4 4 2 3 2 4 4 2 1 1 1 1 1 1 1 3 2 4 4 2 0 4 �. this code was given by the parity check matrix 𝐻𝐻, which has 8 information and 4 parity check symbols. in this case the number of combinations for each code word that can be corrected is (1 + 12 ∗ 2 + (12 𝑐𝑐ℎ𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜 2) ∗ 4) = 289. thus, we have that 289 ∗ 58 ≤ 512 and the cardinality of the best possible code is 512/289 < 59 . in this paper a construction of the optimal code (13, 8) over the ring 𝑍𝑍5 correcting either double ±1 or ±2 errors is presented. 2. construction of optimal 𝑪𝑪 (𝟏𝟏𝟏𝟏, 𝟖𝟖) linear code over ring 𝑍𝑍5 our purpose is to construct an optimal linear code over the ring 𝑍𝑍5 correcting double errors of the type ±1 or ±2 . our constructed code will correct only double errors with the same magnitude (if both errors have magnitude ±1 , or ±2). it is well known, that a linear code given by the parity check matrix 𝐻𝐻, can correct up to two errors of the type ±1, only when 𝐻𝐻 has a property according to which all the syndromes resulting from adding and subtracting operations between any two columns of the matrix 𝐻𝐻 are different: ± ℎ𝑖𝑖𝑖𝑖 ± ℎ𝑖𝑖𝑚𝑚 ≠ ± ℎ𝑖𝑖𝑖𝑖 ± ℎ𝑖𝑖𝑖𝑖 (𝑗𝑗, 𝑚𝑚) ≠ (𝑙𝑙, 𝑘𝑘). in this case, for correcting up to two errors of the type ±1 or ±2, the syndromes of our parity check matrix 𝐻𝐻 must satisfy the following condition too: ±2 ∗ ℎ𝑖𝑖𝑖𝑖 ± 2 ∗ ℎ𝑖𝑖𝑚𝑚 ≠ ±ℎ𝑖𝑖𝑖𝑖 ± ℎ𝑖𝑖𝑖𝑖 (𝑗𝑗, 𝑚𝑚) ≠ (𝑙𝑙, 𝑘𝑘). to construct this kind of matrix 𝐻𝐻, at first we will use the first 10 columns of the optimal linear code (12, 8) correcting double ±1 errors: � 1 1 1 1 1 0 1 2 3 4 0 1 2 3 4 2 2 2 2 2 3 2 4 4 2 3 2 4 4 2 1 1 1 1 1 3 2 4 4 2 �. construction of linear codes over rings 𝑍𝑍𝑚𝑚 correcting double ±1 or ±2 errors 68 for this matrix we know, that all syndromes ± ℎ𝑖𝑖𝑖𝑖 ± ℎ𝑖𝑖𝑚𝑚 ≠ ±ℎ𝑖𝑖𝑖𝑖 ± ℎ𝑖𝑖𝑖𝑖 and ±2 ∗ ℎ𝑖𝑖𝑖𝑖 ± 2 ∗ ℎ𝑖𝑖𝑚𝑚 ≠ ±2(ℎ𝑖𝑖𝑖𝑖 ± ℎ𝑖𝑖𝑖𝑖) are different. however, in this case there can be matches between the syndromes: ±2 ∗ ℎ𝑖𝑖𝑖𝑖 ± 2 ∗ ℎ𝑖𝑖𝑚𝑚 𝑎𝑎𝑎𝑎𝑎𝑎 ± ℎ𝑖𝑖𝑖𝑖 ± ℎ𝑖𝑖𝑖𝑖 . to solve this problem, we need to add one parity check symbol to the code. consequently, we need to add one additional row to the matrix, which will be the following row: [1 2 3 0 4 1 2 3 0 4]. after adding this row to the matrix, all the corresponding ±2 ∗ ℎ𝑖𝑖𝑖𝑖 ± 2 ∗ ℎ𝑖𝑖𝑚𝑚 ≠ ±ℎ𝑖𝑖𝑖𝑖 ± ℎ𝑖𝑖𝑖𝑖 syndromes will be different. now the code given by the parity check matrix below, can correct up to two errors of the type ±1 or ±2 : ⎣ ⎢ ⎢ ⎢ ⎡ 1 1 1 1 1 0 1 2 3 4 0 1 2 3 4 2 2 2 2 2 3 2 4 4 2 3 2 4 4 2 1 1 1 1 1 3 2 4 4 2 1 2 3 0 4 1 2 3 0 4⎦ ⎥ ⎥ ⎥ ⎤ . our constructed code has a length of 10, from which 5 are parity check symbols, and the other 5 information symbols. this constructed code (10, 5) over the ring 𝑍𝑍5 can correct up to two errors of the type ±1 and ±2, but it is not optimal in the sense that it has a minimal possible number of parity check symbols. to make this code optimal, we at least need to have a length of 13, therefore, we need to add 3 additional columns to the parity check matrix 𝐻𝐻. we have found the corresponding 3 columns by computer search: ⎣ ⎢ ⎢ ⎢ ⎡ 1 4 4 1 0 4 1 2 3 4 3 0 2 3 2⎦ ⎥ ⎥ ⎥ ⎤ . so, we obtain the parity check matrix 𝐻𝐻5 with a length of 13. the code given by this parity check matrix 𝐻𝐻5 can correct up to two errors of the type ±1 or ±2: 𝐻𝐻5 = ⎣ ⎢ ⎢ ⎢ ⎡ 1 1 1 1 1 0 1 2 3 4 1 4 4 0 1 2 3 4 2 2 2 2 2 1 0 4 3 2 4 4 2 3 2 4 4 2 1 2 3 1 1 1 1 1 3 2 4 4 2 4 3 0 1 2 3 0 4 1 2 3 0 4 2 3 2⎦ ⎥ ⎥ ⎥ ⎤ . now we can prove the following lemma: lemma 2.1. a linear code (13, 8) over the ring 𝑍𝑍5, given by the parity check matrix 𝐻𝐻5 with 5 parity check symbols, correcting up to two errors of the type ±1 or ±2, is optimal in the sense that it has a minimal possible number of parity check symbols. g. khachatrian and h. khachatrian 69 proof. it can be checked that a linear code over the ring 𝑍𝑍5, given by the parity check matrix h5, can correct up to two errors of the type ±1 or ±2. this means that the syndromes of parity check matrix h5 satisfy the condition: ±2 ∗ ℎ𝑖𝑖𝑖𝑖 ± 2 ∗ ℎ𝑖𝑖𝑚𝑚 ≠ ±ℎ𝑖𝑖𝑖𝑖 ± ℎ𝑖𝑖𝑖𝑖 (𝑗𝑗, 𝑚𝑚) ≠ (𝑙𝑙, 𝑘𝑘). in this case, we have 8 possible types of double errors:  +1 𝑎𝑎𝑎𝑎𝑎𝑎 + 1  +1 𝑎𝑎𝑎𝑎𝑎𝑎 − 1  −1 𝑎𝑎𝑎𝑎𝑎𝑎 + 1  −1 𝑎𝑎𝑎𝑎𝑎𝑎 − 1  +2 𝑎𝑎𝑎𝑎𝑎𝑎 + 2  +2 𝑎𝑎𝑎𝑎𝑎𝑎 − 2  −2 𝑎𝑎𝑎𝑎𝑎𝑎 + 2  −2 𝑎𝑎𝑎𝑎𝑎𝑎 − 2 additionally, we have 4 more cases for single errors of the type ±1 and ±2. in this case, the number of combinations for each code word that can be corrected is (1 + 13 ∗ 4 + (13 𝑐𝑐ℎ𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜 2) ∗ 8) = 677. thus, we have that 677 ∗ 58 ≤ 513 and the cardinality of the best possible code is 513/677 < 59 . 3. construction of 𝑪𝑪 (𝟏𝟏𝟏𝟏, 𝟏𝟏𝟏𝟏) and 𝑪𝑪 (𝟏𝟏𝟏𝟏, 𝟏𝟏𝟏𝟏) codes over rings 𝑍𝑍7 and 𝑍𝑍9 in this paragraph we will construct asymmetric low magnitude error correcting codes over different rings 𝑍𝑍𝑚𝑚. as it was mentioned above, in this paper we intend to construct codes correcting double ±1 and ±2 errors with the same magnitude. in the previous paragraph we constructed the optimal code c (13, 8) over the ring 𝑍𝑍5 based on the optimal code c (12, 8) correcting only double ±1 errors by adding only one parity check symbol. codes constructed during this paragraph will not be optimal in the sense that they have a minimal possible number of parity check symbols. 3.1 construction of code 𝑪𝑪 (𝟏𝟏𝟏𝟏, 𝟏𝟏𝟏𝟏) over ring 𝑍𝑍7 it was shown in [5] that the optimal code c (16, 12) over the ring 𝑍𝑍7, correcting double ±1 errors, has the following parity check matrix: � 1 1 1 1 1 1 1 0 1 2 3 4 5 6 1 1 6 5 4 3 2 1 0 2 2 2 2 2 2 2 1 1 4 3 6 6 3 4 2 4 3 6 6 3 4 2 1 6 1 1 1 1 1 1 1 4 3 6 6 3 4 2 0 0 �. to construct the code c (17, 12), correcting double ±1 and ±2 errors with the same magnitude, we need to take the first 14 columns of the parity check matrix: construction of linear codes over rings 𝑍𝑍𝑚𝑚 correcting double ±1 or ±2 errors 70 � 1 1 1 1 1 1 1 0 1 2 3 4 5 6 6 5 4 3 2 1 0 2 2 2 2 2 2 2 4 3 6 6 3 4 2 4 3 6 6 3 4 2 1 1 1 1 1 1 1 4 3 6 6 3 4 2 � and add one parity check symbol to this code. consequently, we need to add one additional row to the parity check matrix, which will be the following row: [2 3 4 0 1 6 5 2 3 4 0 1 6 5]. now the code given by the following parity check matrix, can correct up to two errors of the type ±1 and ±2: ⎣ ⎢ ⎢ ⎢ ⎡ 1 1 1 1 1 1 1 0 1 2 3 4 5 6 6 5 4 3 2 1 0 2 2 2 2 2 2 2 4 3 6 6 3 4 2 4 3 6 6 3 4 2 1 1 1 1 1 1 1 4 3 6 6 3 4 2 2 3 4 0 1 6 5 2 3 4 0 1 6 5⎦ ⎥ ⎥ ⎥ ⎤ . our newly constructed code given by this parity check matrix has a length of 14, from which 5 are parity check symbols and 9 information symbols. we want to have the same number of information symbols as in code c (16, 12) with 4 parity check symbols and 12 information symbols. so, we need to add 3 additional columns to the matrix to have 12 information symbols. we found the corresponding 3 columns by computer search: ⎣ ⎢ ⎢ ⎢ ⎡ 6 6 6 1 2 3 5 4 1 3 1 4 5 5 4⎦ ⎥ ⎥ ⎥ ⎤ . so, we obtain the parity check matrix 𝐻𝐻7 , with a length of 17: 𝐻𝐻7 = ⎣ ⎢ ⎢ ⎢ ⎡ 1 1 1 1 1 1 1 0 1 2 3 4 5 6 6 6 6 6 5 4 3 2 1 0 2 2 2 2 2 2 2 1 2 3 4 3 6 6 3 4 2 4 3 6 6 3 4 2 5 4 1 1 1 1 1 1 1 1 4 3 6 6 3 4 2 3 1 4 2 3 4 0 1 6 5 2 3 4 0 1 6 5 5 5 4⎦ ⎥ ⎥ ⎥ ⎤ . the code given by this parity check matrix 𝐻𝐻7 with 5 parity check symbols and 12 information symbols, can correct up to two errors of the type ±1 and ±2 with the same magnitude. in this case the number of combinations for each code word that can be corrected is (1 + 17 ∗ 4 + (17 𝑐𝑐ℎ𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜 2) ∗ 8) = 1157. as such, using this code 𝐶𝐶(17, 12) over the ring 𝑍𝑍7, correcting double ±1 and ±2 errors with the same magnitude, we can correct all 1157 errors, which can occur in the code word. this code does not satisfy the optimality criteria-1, since an equation 717/1157 < 713 is not satisfied in g. khachatrian and h. khachatrian 71 this case which means that the minimum number of parity check symbols should be 4 not 5, i.e., we have one extra parity check symbol. we will refer to this kind of codes as quasi-optimal. 3.2 construction of code 𝑪𝑪 (𝟏𝟏𝟏𝟏, 𝟏𝟏𝟏𝟏) over ring 𝑍𝑍9 again, from [5] we know the optimal code c (20, 16) over the ring 𝑍𝑍9, correcting double ±1 errors with 4 parity check symbols and 16 information symbols, which was given below presented by the following parity check matrix : � 1 1 1 1 1 1 1 1 8 7 6 5 4 3 2 1 1 1 2 4 7 6 5 4 3 2 1 0 2 2 2 2 2 2 2 2 1 1 2 4 7 3 2 4 4 2 3 7 7 3 2 4 4 2 3 7 1 1 2 4 1 1 1 1 1 1 1 1 7 3 2 4 4 2 3 7 6 3 7 2 �. to construct the code c (21, 16), correcting double ±1 and ±2 errors with the same magnitude, we need to take the first 16 columns of the matrix above in the same way as we did in the previous construction for the code c (17, 12) over the ring 𝑍𝑍7: � 1 1 1 1 1 1 1 1 8 7 6 5 4 3 2 1 7 6 5 4 3 2 1 0 2 2 2 2 2 2 2 2 7 3 2 4 4 2 3 7 7 3 2 4 4 2 3 7 1 1 1 1 1 1 1 1 7 3 2 4 4 2 3 7 � and add one parity check symbol to this code. consequently, we need to add one additional row to the matrix, which will be the following row: [2 3 4 0 1 8 7 3 2 3 4 0 1 8 7 3]. now the code given by the following parity check matrix, can correct up to two errors of the type ±1 and ±2 with the same magnitude: ⎣ ⎢ ⎢ ⎢ ⎡ 1 1 1 1 1 1 1 1 8 7 6 5 4 3 2 1 7 6 5 4 3 2 1 0 2 2 2 2 2 2 2 2 7 3 2 4 4 2 3 7 7 3 2 4 4 2 3 7 1 1 1 1 1 1 1 1 7 3 2 4 4 2 3 7 2 3 4 0 1 8 7 3 2 3 4 0 1 8 7 3⎦ ⎥ ⎥ ⎥ ⎤ . our newly constructed code given by this parity check matrix has a length of 16, from which 5 are parity check symbols, and 11 information symbols. again, we want to have the same number of information symbols as in code c (20, 16) with 4 parity check symbols and 16 information symbols correcting double ±1 errors. so, we need to add 5 additional columns to the matrix to have 16 information symbols. we found the corresponding 5 columns by computer search: construction of linear codes over rings 𝑍𝑍𝑚𝑚 correcting double ±1 or ±2 errors 72 ⎣ ⎢ ⎢ ⎢ ⎡ 6 6 8 8 8 2 4 0 2 8 2 3 2 2 6 3 7 1 7 1 3 7 1 7 1⎦ ⎥ ⎥ ⎥ ⎤ . the code given by this parity check matrix 𝐻𝐻9 with 5 parity check symbols and 16 information symbols, can correct up to two errors of the type ±1 and ±2 with the same magnitude: 𝐻𝐻9 = ⎣ ⎢ ⎢ ⎢ ⎡ 1 1 1 1 1 1 1 1 8 7 6 5 4 3 2 1 6 6 8 8 8 7 6 5 4 3 2 1 0 2 2 2 2 2 2 2 2 2 4 0 2 8 7 3 2 4 4 2 3 7 7 3 2 4 4 2 3 7 2 3 2 2 6 1 1 1 1 1 1 1 1 7 3 2 4 4 2 3 7 3 7 1 7 1 2 3 4 0 1 8 7 3 2 3 4 0 1 8 7 3 3 7 1 7 1⎦ ⎥ ⎥ ⎥ ⎤ . in this case the number of error combinations for each code word that can be corrected is (1 + 21 ∗ 4 + (21𝑐𝑐ℎ𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜 2) ∗ 8) = 1765. as such, using this code 𝐶𝐶(21, 16) over the ring 𝑍𝑍9 correcting double ±1 and ±2 errors with the same magnitude, we can correct all 1765 errors which can occur in the code word. again, in this case, similar to the 𝐶𝐶(17, 12) code over the ring 𝑍𝑍7 we have a quasioptimal code. 4. conclusion in this paper a construction of double ±1 and ±2 error correcting linear optimal code over the ring 𝑍𝑍5 and quasi-optimal codes over the rings 𝑍𝑍7 and 𝑍𝑍9 are constructed. we plan to investigate if the approach presented in this paper can be extended to construct codes for larger alphabets as well as to construct quasi-optimal codes with higher code rates. references [1] s. martirossian, “single error correcting close packed and perfect codes”, proc.1st intas int. seminar coding theory and combinatorics, armenia, pp. 90-115, 1996. [2] h. kostadinov, n. manev and h. morita, “on ±1 error correctable codes”, ieice trans.fundamentals, vol. e93-a, pp. 2578-2761, 2010. [3] a. j. han vinck and h. morita, “codes over the ring of integers modulo m”, ieice trans.fundamentals , vol. e81-a, pp. 2013-2018, 1998. [4] g. khachatrian and h. morita, “construction of optimal ±1 double error correcting linear codes over ring z5”, 3th international workshop on advances in communications, boppard, germany, pp. 10-12, may 2014. g. khachatrian and h. khachatrian 73 [5] g. khachatrian and h. khachatrian ”construction of double ±1 error correcting linear optimal codes over rings 𝑍𝑍7 and 𝑍𝑍9”, mathematical problems of computer science, vol. 45, pp. 106--110, 2016. submitted 26.10.2017, accepted 07.02.2018. ±𝟏𝟏 և ±𝟏𝟏 մեծությամբ կրկնակի սխալներ ուղղող գծային կոդերի կառուցումներ տարբեր մեծության 𝑍𝑍𝑚𝑚 օղակներում գ. խաչատրյան և հ. խաչատրյան ամփոփում այս հոդվածի շրջանակներում ներկայացված են 𝑍𝑍5, 𝑍𝑍7 և 𝑍𝑍9 օղակներում ±1 կամ ±2 միևնույն մեծությամբ կրկնակի սխալներ ուղղող օպտիմալ և քվազի-օպտիմալ կոդերի կառուցումներ: построение кодов исправляющих двойные ошибки размера ±𝟏𝟏 и ±𝟏𝟏 в 𝑍𝑍𝑚𝑚 кольцах с разными величинами г. хачатрян и г. хачатрян аннотация в данной статье представлено построение оптимальных и квази-оптимальных линейных кодов в кольцах 𝑍𝑍5, 𝑍𝑍7 и 𝑍𝑍9, исправляющих двойные ошибки размера ±1 и ±2 с одинаковыми величинами. construction of linear codes over rings ,𝑍-𝑚. correcting double ±1 or ±2 errors gurgen h. khachatrian* and hamlet k. khachatrian ** in this paper a construction of double ±1 and ±2 errors correcting linear optimal and quasi-optimal codes over rings ,𝑍-5., ,𝑍-7. and ,𝑍-9. is presented with the limitation that both errors have the same amplitude in absolute value. ,,1-4-4-1-0-4-1-2-3-4-3-0-2-3-2... mathematical problems of computer science 40, 85--95, 2013. 85 comparative analysis of attack graphs levon h. aslanyan, daryoush alipour and minoosh heidari institute for informatics and automation problems of nas ra e-mail: lasl@sci.am abstract it is well-known that nowadays computers and networks that are unique in their computational and service provision power have also major weaknesses and vulnerabilities that can be exploited by outsiders in compromising the valuable data and knowledge. network administrators and network security analysts must be aware of different properties of current software solutions and diversity of problems regarding the possible protection of network assets. this means that they must know and use the latest and newest types of vulnerabilities, techniques and tools. “attack graphs” present formalized network maps and help with analysis of possible vulnerabilities that may exist in the network. hence, in this paper we will describe some basic concepts that can be used to understand and generate the attack graphs. keywords: network security, network vulnerability, attack graph. 1. introduction defending large scale networks is very difficult. the outside interest to information, conflicting relations and business objectives draw to special type of activities compiled round the term hacker. many of the applied systems are created just to provide the necessary work with information. protecting the system becomes an additional burden. a defender in such a situation must be able to locate all paths into the network and prevent attackers from using them at the moment when an attacker needs to find only one unprotected path. a network defender has the advantage of intimate knowledge of the network such as: the ways traffic may move through it, the services running on it, and the vulnerabilities in those services. a defender can use that knowledge to improve situational awareness. attack graphs are one way to leverage those data. there are many different papers on attack graphs and many representations, but the core idea remains the same: an attack graph shows the ways an attacker can compromise a network or host. defenders can then use the attack graph to identify critical bottlenecks and work to secure those bottleneck hosts and services first. attack graphs are a valuable tool to network defenders, illustrating paths an attacker can use to gain access to a targeted network. defenders can focus their efforts on patching the vulnerabilities and configuration errors that allow the attackers to have the greatest amount of access [1]. comparative analysis of attack graphs86 the remainder of this paper is organized as follows: section 2 provides an overview of vulnerability, section 3 describes the attack graph, and section 4 draws analysis and conclusions. 2. vulnerability in computer security, vulnerability is a flaw or weakness in a network that can be exploited by one or more threats to violate the system's security policy. network vulnerabilities have the potential of being exploited in a way that may lead to the use of the computer to achieve the intruder’s desired goal. hence, exploits give an attacker to take advantage of a flaw, action or vulnerability in a network. 2.1 terms and standards for information security vulnerability domain there are some best practices and standards to classify vulnerabilities that have been discussed in this section. 2.1.1 cve names common vulnerabilities and exposures (cve®) is a dictionary of common names (i.e. cve identifiers) for publicly known information security vulnerabilities. cve is an international information security community effort and it is now the industry standard for vulnerability and exposure names [2]. 2.1.2 osvdb references osvdb is an independent and open sourced web-based vulnerability database created for the security community. common vulnerabilities and exposures (cve) simply provides a standardized name for vulnerabilities, much like a dictionary. osvdb is a database that provides a wealth of information about each vulnerability. where appropriate, entries in the osvdb refer to their respective cve names. in addition, over the past 8 years, osvdb has imported over 23,000 vulnerabilities that cannot be found in cve [3]. 2.1.3 cvss scoring cvss stands for the common vulnerability scoring system and is a vendor agnostic, industry open standard designed to convey vulnerability severity and help determine urgency and priority of response. it solves the problem of multiple, incompatible scoring systems and is usable and understandable by anyone. cvss is a vulnerability scoring system designed to provide an open and standardized method for rating it vulnerabilities. cvss helps organizations prioritize and coordinate a joint response to security vulnerabilities by communicating the base, temporal and environmental properties of a vulnerability. first (the forum of incident response and security teams) hosts a special interest group to update and promote cvss and provides a central repository for cvss documentation [4]. 2.1.4 nvd nvd is the u.s. government repository of standards based on vulnerability management data represented with the use of the security content automation protocol (scap). these data enable automation of vulnerability management, security measurement, and compliance. nvd includes databases of security checklists, security related software flaws, misconfigurations, product names, and impact metrics [5]. 2.2 vulnerability and exposure a vulnerability is a state in a computing system (or set of systems) that allows an attacker to execute commands as another user, to conduct a denial of service, and so on. l. aslanyan, d. alipour, m. heidari 87 an information security "vulnerability" is a mistake in software that can be directly used by a hacker to gain access to a system or network. an information security "exposure" is a system configuration issue or a mistake in software that allows access to information or capabilities that can be used by a hacker as a stepping-stone into a system or network. an "exposure" describes a state in a computing system (or set of systems) that is not a vulnerability, but allows an attacker to conduct information gathering activities, to hide activities, and so on. 2.3 vulnerability and exposure examples examples of vulnerabilities include: •phf (remote command execution as user "nobody") •rpc.ttdbserverd (remote command execution as root) •world-writeable password file (modification of system-critical data) •default password (remote command execution or other access) •denial of service problems that allows an attacker to cause a blue screen of death •smurf (denial of service by flooding a network) examples of exposures include: •running services such as finger (useful for information gathering, though it works as advertised) •inappropriate settings for windows nt auditing policies (where "inappropriate" is enterprise-specific) •running services that are common attack points (e.g., http, ftp, or smtp) •use of applications or services that can be successfully attacked by brute force methods (e.g. use of trivially broken encryption, or a small key space). 2.4 vulnerability scanners a vulnerability scanner is a software designed to assess computers and networks for weaknesses and vulnerabilities. vulnerability scanners are divided into two groups: network-based scanners and host-based scanners. a network-based scanner is installed on a computer that scans a number of other hosts on the network, such as: port scanners(nmap, nessus), web application security scanner, network vulnerability scanner (boomscan). a host-based scanner is installed in the host, such as: database security scanner. 3. attack graph attack graph ([6--8], [11]) is an integral part of modeling the overview of network security. system administrators use attack graphs to determine how vulnerable their systems are and to determine what security measures to deploy to defend their systems [9]. each attack graph shows a set of scenarios of penetrating a computer network. a penetration scenario actually defines the order of steps that an intruder should take to achieve his goal, and each step is characterized to show which host must be abused [10]. major discussions in this area are generating an attack graph, and analyzing it for intrusion detection and hardening the system-critical. 3.1 a simple example consider the example network shown in figure 1. there are two target hosts, machine 1 and machine 2, and a firewall separating them from the rest of the internet. as shown, each host is comparative analysis of attack graphs88 running two of three possible services (ftp, sshd, a database). there are four possible atomic attacks, identified numerically as follows: (0) sshd buffer overflow, (1) ftp .rhosts, (2) remote login, and (3) local buffer overflow. the ftp .rhosts attack needs to find the target host with two vulnerabilities: a writable home directory and an executable command shell are assigned to the ftp user name. the local buffer overflow exploits a vulnerable version of the xterm executable. fig. 1. example network. service description common exploits sshd sshd (secure shell daemon) is the daemon program for ssh. sshd listens for connections from clients. 'ssh' client and 'sshd' server, provide secure encrypted communications between two untrusted hosts over an insecure network. sshd buffer overflow (sshd_bof) it gives a remote user a root shell on the target machine. ftp server file transfer protocol is a standard network protocol used to transfer files from one host to another one. an ftp server is a software application on networks that provides lots of software to download. ftp remote host (ftp-rhosts) using an ftp vulnerability, the intruder creates an .rhosts file in the ftp home directory and takes a remote login trust between his machine and the target one. rsh the remote shell (rsh) is a command line computer program. 'rsh' like 'ssh' can execute commands on remote systems. the remote shell (rsh) the intruders log in from one machine to another, using an existing remote login trust between two hosts, and gets a user shell without password. database server a database server is a software that provides database services on a computer or a network. buffer overflow (local_bof) l. aslanyan, d. alipour, m. heidari 89 3.1.1 different paths for the above example attack  sshd_bof(0,1) → ftp_rhosts(1,2) → rsh(1,2) → local_bof(2) the first assumed attack path starts with sshdbof(0,1). this indicates a buffer over exploit executed from machine 0 (the workstation) against machine 1 (the file server).sshd_bof(0,1)exploit is that the attacker can execute an arbitrary code on the file server. the ftp_rhosts(1,2) exploit is now possible, meaning that the attacker exploits a particular ftp vulnerability to anonymously upload a list of trusted hosts from machine 1 (the file server) to machine 2 (the database server). rsh(1,2)means the attacker can leverage this new trust to remotely execute shell commands on the database server, without providing a password. a local buffer over exploit is then possible on the database server, which runs in the context of a privileged process. the result is that the attacker can execute a code on the database server with full privileges. other possible attack paths can be viewed as either of the following:  ftp_rhosts(0,1) → rsh(0,1) → ftp_rhosts(1,2) → rsh(1,2) → local_bof(2)  ftp_rhosts(0,2) → rsh(0,2) → local_bof(2) 3.1.2. attack graph from machine 0 to db server in this section, we construct an attack graph of the example network so that each state transition corresponds to a single atomic attack by the intruder. a state in the model represents the state of the system between atomic attacks. the intruder launches his attack starting from a single computer, machine 1. his eventual goal is to disrupt the functioning of the database. for which, the intruder needs fig. 2. attack graph of above example. root access on the database machine 2. 3.2 types and views in this part, we describe various forms of attack graphs that are listed in some past papers. comparative analysis of attack graphs90 3.2.1 exploit-dependency attack graph in the exploit dependency graph, each exploit or dependency appears only once, and no edges appear between independent exploits. for example, in the exploit dependency graph, each of the three exploits ftp_rhosts(0,1), sshd_bof(0,1), and ftp_rhosts(0,2) appears only once, and since these exploits are independent, there are no edges between them. fig. 3. exploit-dependency attack graph. 3.2.2 state-enumeration attack graph attack graphs represented transitions of a state machine, where states are network security attributes and state transitions are attacker exploits, resulting in graphs that enumerate transition paths through the state space. these state-based graphs have a property that one can simply follow a path through the graph to generate an attack path (sequence of exploits leading from the initial state to the goal one). but such graphs have serious scalability problems, as they can grow exponentially with the number of state variables. fig. 4. state-enumeration attack graph. l. aslanyan, d. alipour, m. heidari 91 3.2.3 condition-oriented attack graph in a condition-oriented attack graph, a node represents a subset of the network state, and an edge represents an exploit (or group of exploits) that moves the network from one state to another one. graph vertices represent conditions, which are connected by edges that represent exploits. fig. 5. a condition-oriented attack graph. a state is a network attribute or a set of network attributes. network attributes include hosts, host connectivity, and available software at hosts, access rights at hosts, and any other network characteristic deemed relevant to the modeler. there are various types of “condition-oriented attack graph” that have small differences between them: finite state machine (fsm) attack graph, coordinated attack graph, full attack graph, host-compromised attack graph, predictive attack graph, node predictive attack graph. 3.2.4 exploit-oriented attack graph an exploit-oriented attack graph is the reverse of a condition-oriented graph with respect to nodes and edges. state is represented in the edges of the graph and the exploits are represented in nodes of the graph. exploit-oriented attack graphs may be referred to as exploit dependency graphs. a common representation of exploit-oriented attack graphs is to have unlabeled edges. the exploit-oriented attack graph's initial state(s) and the goal state(s) of the network are special nodes. initial states are exploit nodes with null preconditions and true post-conditions goal states are exploit nodes with true preconditions and null post-conditions there are various types of “exploit-oriented attack graph” that have small differences between them: condition-exploit-oriented attack graph, multiple prerequisites attack graph, logical attack graph, hybrid-oriented attack graph. comparative analysis of attack graphs92 fig. 6. a simple example network (a) , and its multiple prerequisites attack graph (b). 3.2.5 attack graph with probabilities numbers are estimated probabilities of occurrence for individual exploits, based on their relative difficulty. fig. 7. attack graph with probabilities. probabilities propagated through attack graph when one exploit should follow another in a path, this means both are needed to eventually reach the goal, so their probabilities are multiplied: p(a and b) = p(a)p(b). when a choice of paths is possible, either is sufficient for reaching the goal: p(a or b) = p(a) + p(b) – p(a)p(b). l. aslanyan, d. alipour, m. heidari 93 fig. 8. probabilities propagated through attack graph. 3.2.6 attack graph aggregation machines and the exploits among them can be aggregated to a machine-exploit set if they form a connected sub graph, which allows machines to be aggregated across protection domains. complexity for "state-transition graph" is exponential and for "exploit-dependency graph" is quadratic, but still too complex for easy understanding. 100 exploits could have up to 10000 edges. using hierarchical graph aggregation with abstraction is a solution for managing complexity. 4. conclusions and future works in this paper, we have proposed an overview and approach to definitions and survey of attack graphs. the main purpose of this approach is to emphasize the importance of attack graph as a high-performance network security solution. many related research should be done in the future: comparison of generating algorithms, review types of tool kits, and the methods to analyze attack graph will be further studied. comparative analysis of attack graphs94 fig. 9. attack graph aggregation. references [1] k. ingols, r. lippmann and k. piwowarski, practical attack graph generation for network defense, mit lincoln laboratory, 2006. [2] common vulnerabilities and exposures (cve®), the standard for information security vulnerability names, [online]. available: http://cve.mitre.org [3] open sourced vulnerability database, [online]. available: http://osvdb.org/ [4] common vulnerability scoring system (cvss-sig), [online]. available: http://www.first.org/cvss [5] national vulnerability database version 2.2, nist, usa, [online]. available: http://nvd.nist.gov/ [6] s. jha, o. sheyner and j.m. wing, minimization and reliability analyses of attack graphs. school of computer science carnegie mellon university, 2002. [7] s. noel, l. wang, a. singhal and s. jajodia, “measuring security risk of networks using attack graphs”, international journal of next-generation computing, vol. 1, no. 1, pp. 135-147, july 2010. [8] s. noel and s. jajodia, “managing attack graph complexity through visual hierarchical aggregation”, ccs workshop on visualization and data mining for computer security’04, october 29, fairfax, virginia, usa, 10p., 2004. [9] f. chen, et al., “an atomic-domains-based approach for attack graph generation”, world academy of science, engineering and technology, vol. 56, pp. 775-781, 2009. [10] m. jamali and v. ashraf, “attack graph analysis using parallel algorithm”, 5th symposium on advances in science & technology, 7p., 2011. [11] n. c. idika, characterizing and aggregating attack graph-based security metrics, purdue university, west lafayette, indiana, 2010. l. aslanyan, d. alipour, m. heidari 95 submitted 30.08.2013, accepted 11.10.2013. հարձակման գրաֆների համեմատական վերլուծություն լ. ասլանյան, դ. ալիփոուռ և մ. հեյդարի ամփոփում հայտնի է, որ ժամանակակից համակարգիչներն ու ցանցերը, որոնք առանձնակի հզոր են իրենց հաշվողական և ծառայությունների մատուցման հնարավորություններով, ունեն նաև խնդիրներ` կապված դրանց խոցելիության և դրա հետ կապված տվյալների արտաքին բացահայտման հնարարավոր լինելու պարագայի հետ: ցանցային կառավարիչները և ցանցերի վերլուծման մասնագետները պետք է տեղյակ լինեն ընթացիկ համակարգերի խնդիրների հետ կապված համակարգերի և ցանցերի պաշտպանման մասին: “հարձակման գրաֆները” ներկայացնում են ցանցերի հնարավոր հարձակումների ֆորմալացված պատկերը: ներկա աշխատանքում դիտարկվում և վերլուծվում են անհրաժեշտ գաղափարները, որոնք առաջանում են հարձակումների գրաֆների ձևավորման և կիրառման ընթացքում: сравнительный анализ графов атак л. асланян, д. алипоур и м. гейдари аннотация современные компъютеры являются мощными вычислительными системами, а также системами предоставления вспомогательных информационных услуг. вместе с тем, эти системы уязвимы с точки зрения внешних атак, что может являться причиной потери различных ценных информационных ресурсов. системные администраторы обязаны знать всю информацию об уязвимости прикладных систем связанных с защитой сетей. графы атак это удачный формализм, который предоставляет всю картину возможных атак для данной сети. настоящая работа посвящена анализу и изучению средств проектирования и внедрения технологии графов атак. d:\user\...\main.dvi mathematical problems of computer science 49, 110{114, 2018. on m ultiple h ypotheses lao t esting with rejection of decision for t wo dependent objects e vg u e n i h a r o u t u n ia n , a r a m y e s a ya n a n d n a r in e h a r u t yu n ya n institute for informatics and automation problems of nas ra e-mail: eghishe@sci.am, armfrance@yahoo.fr, narineharutyunyan57@gmail.com abstract multiple statistical hypotheses testing with possibility of rejecting of decision is considered for model consisting of two dependent objects characterized by joint discrete probability distribution. the matrix of error probabilities exponents (reliabilities) of asymptotically optimal tests is studied. keywords: multiple hypotheses testing, optimal tests, rejection option, two object. 1 . in t r o d u c t io n th is p a p e r is d e vo t e d t o t h e s t u d y o f c h a r a c t e r is t ic s o f lo g a r it h m ic a lly a s ym p t o t ic a lly o p t im a l ( l a o) h yp o t h e s e s t e s t in g wit h p o s s ib ilit y o f r e je c t io n o f d e c is io n fo r t h e m o d e l o f t wo d e p e n d e n t o b je c t s wit h jo in t p t o b a b ilit y d is t r ib u t io n s ( p d s ) . th e c o r r e s p o n d e n c e p r e s e n t s a c o m p le m e n t t o p r o b le m s s t u d ie d in [1 , 2 ], wh e r e t h e c o m p o n e n t s o f r a n d o m ve c t o r c h a r a c t e r iz in g t wo o b je c t s we r e in d e p e n d e n t , s o it wa s p o s s ib le t o c o n s id e r t h e t e s t p r o c e d u r e wit h s e p a r a t e t e s t s fo r t wo o b je c t s . in [2 ] t wo d i®e r e n t m o d e ls a r e s t u d ie d : ¯ r s t , wh e n r e je c t io n wa s a llo we d o n ly fo r o n e o f t h e o b je c t s a n d s e c o n d , wh e n r e je c t io n wa s a llo we d fo r b o t h o b je c t s . th e p r o b le m wit h a n a lo g o u s s t a t e m e n t fo r o n e a r b it r a r ily va r yin g o b je c t wit h s id e in fo r m a t io n wa s e xa m in e d in [3 ]. it is wo r t h t o r e c a ll t h e p r e vio u s r e s u lt s o n l a o t e s t in g o f m a n y h yp o t h e s e s p u b lis h e d in [4 -6 ]. me t h o d s a n d b a s ic r e s u lt s o f l a o t e s t in g a r e a ls o p r e s e n t e d in b o o ks [7 -9 ] 2 . p r o b le m fo r m u la t io n a n d r e s u lt l e t p ( x ) b e a s p a c e o f a ll p r o b a b ilit y d is t r ib u t io n s g( x ) o n ¯ n it e s e t x . l e t ( x1; x2 ) b e r a n d o m ve c t o r t a kin g va lu e s in t h e s e t x £ x wit h o n e o f m 2; m ¸ 2 jo in t p d s gm1;m2 2 p ( x £ x ) , m1; m2 = 1 ; m. l e t ( x1; x2 ) 4 = ( ( x11; x 2 1 ) ; :::; ( x 1 n; x 2 n ) ; :::; ( x 1 n ; x 2 n ) ) , x 1 n; x 2 n 2 x ; n = 1 ; n, b e a ve c t o r o f r e s u lt s o f n in d e p e n d e n t o b s e r va t io n s o f t h e ve c t o r ( x1; x2 ) , it is c a lle d a s a m p le . th e s t a t is t ic ia n h a s t o d e t e r m in e u n kn o wn p d s fr o m t h e s e t o f h yp o t h e s e s : hm1;m2 : g = gm1;m2, m1; m2 = 1 ; m o r wit h d r a w t o d o a n y ju d g e m e n t u s in g o b t a in e d s a m p le . 1 1 0 e. haroutunian, a. yesayan and n. harutyunyan 1 1 1 w e c a ll t h is p r o c e d u r e a c o m p o u n d t e s t a n d d e n o t e it b y ©n . th e t e s t ©n c a n b e d e ¯ n e d b y t h e d ivis io n o f t h e s p a c e x n £ x n in t o m 2 + 1 d is jo in t s u b s e t s , wh e r e am1;m2, m1; m2 = 1 ; m , c o n t a in s a ll ve c t o r s ( x1; x2 ) fo r wh ic h t h e h yp o t h e s is hm1;m2 is a d o p t e d , a n d am +1 c o n t a in s a ll ve c t o r s fo r wh ic h we r e fu s e t o t a ke a c e r t a in a n s we r . l e t ®l1;l2jm1;m2 ( ©n ) b e t h e p r o b a b ilit y o f t h e e r r o n e o u s a c c e p t a n c e o f t h e h yp o t h e s is hl1;l2 b y t h e t e s t ©n p r o vid e d t h a t t h e h yp o t h e s is hm1;m2 is t r u e , wh e r e ( m1; m2 ) 6= ( l1; l2 ) , m1; m2; l1; l2 = 1 ; m, ®l1;l2jm1;m2 ( ©n ) = g n m1;m2 ( al1;l2 ) : w h e n t h e h yp o t h e s is hm1;m2 is t r u e , b u t we d e c lin e t h e d e c is io n c o n c e r n in g t o t h e h yp o t h e s e s , t h e c o r r e s p o n d in g p r o b a b ilit y o f e r r o r is : ®m +1;m+1jm1;m2 ( ©n ) = g n m1;m2 ( am +1 ) : th e p r o b a b ilit y n o t t o a c c e p t a t r u e h yp o t h e s e s hm1;m2, m1; m2 = 1 ; m is t h e fo llo win g : ®m1;m2jm1;m2 ( ©n ) = x (l1;l2) 6=(m1;m2); l1;l2=1;m ; (l1;l2)=(m+1;m+1) ®l1;l2jm1;m2 ( ©n ) : ( 1 ) w e s t u d y t h e c o r r e s p o n d in g r e lia b ilit ie s el1;l2jm1;m2 ( ©) o f t h e s e qu e n c e o f t e s t s ©, el1;l2jm1;m2 ( ©) 4 = lim n!1 ¡ 1 n lo g ®l1;l2jm1;m2 ( ©n ) ; m1; m2; l1; l2 = 1 ; m; ( l1; l2 ) = ( m + 1 ; m + 1 ) : ( 2 ) d e ¯ n it io n s ( 1 ) a n d ( 2 ) im p ly t h a t em1;m2jm1;m2 ( ©) = m in (l1;l2) 6=(m1;m2) el1;l2jm1;m2 ( ©) ; m1; m2; l1; l2 = 1 ; m; ( l1; l2 ) = ( m + 1 ; m + 1 ) : ( 3 ) w e c a ll t h e t e s t s e qu e n c e ©¤ l a o fo r t h e m o d e l wit h t wo o b je c t s if fo r t h e g ive n p o s it ive va lu e s o f c e r t a in p a r t o f e le m e n t s o f t h e r e lia b ilit y m a t r ix e ( ©¤ ) t h e p r o c e d u r e ©¤ p r o vid e s m a xim a l va lu e s fo r a ll o t h e r e le m e n t s o f it . fo r m = 2 t h e m a t r ix will b e a s fo llo ws : e ( ©) = 0 bbb@ e1;1j1;1 e1;2j1;1e2;1j1;1 e2;2j1;1 e3;3j1;1 e1;1j1;2 e1;2j1;2e2;1j1;2 e2;2j1;2 e3;3j1;2 e1;1j2;1 e1;2j2;1e2;1j2;1 e2;2j2;1 e3;3j2;1 e1;1j2;2 e1;2j2;2e2;1j2;2 e2;2j2;2 e3;3j2;2 1 ccca : w it h t h e g ive n e le m e n t s e1;1j1;1; e1;2j1;2; e2;1j2;1; e2;2j2;2 we d e ¯ n e t h e r e g io n s o f a c c e p t a n c e o f t h e t e s t . in t h e g e n e r a l c a s e o f m h yp o t h e s e s fo r g ive n r e lia b ilit ie s e1;1j1;1; e1;2j1;2; e2;1j2;1; :::; em;mjm;m we d e ¯ n e t h e fo llo win g r e g io n s : rm1;m2 4 = fq : d ( qjjgm1;m2 ) · em1;m2jm1;m2g; m1; m2 = 1 ; m; ( 4 ) 1 1 2 on multiple hypotheses lao testing with rejection of decision for two dependent objects rm+1;m +1 4 = fq : d ( qjjgm1;m2 ) > em1;m2jm1;m2 ; m1; m2 = 1 ; mg; ( 5 ) e¤m1;m2jm1;m2 = e ¤ m1;m2jm1;m2 ( em1;m2jm1;m2 ) 4 = 4 = em1;m2jm1;m2; m1; m2 = 1 ; m; ( 6 ) e¤l1;l2jm1;m2 = e ¤ l1;l2jm1;m2 ( el1;l2jl1;l2 ) 4 = in f q2r l1;l2 d ( qjjgm1;m2 ) ; l1; l2; m1; m2 = 1 ; m; ( m1; m2 ) 6= ( l1; l2 ) ( 7 ) e¤m +1;m+1jm1;m2 = e ¤ m+1;m+1jm1;m2 ( e1;1j1;1; e1;2j1;2; :::; em;mjm;m ) 4 = in f q2r m +1;m+1 d ( qjjgm1;m2 ) ; m1; m2 = 1 ; m: ( 8 ) l e t u s d e n o t e b y ( m1; m2 ) ¡ t h e s e t o f a ll p a ir in d ic e s in r o w o f ( m1; m2 ) va r yin g fr o m ( 1 ; 1 ) t ill p r e vio u s o f ( m1; m2 ) a n d b y ( m1; m2 ) + t h e s e t o f a ll p a ir in d ic e s in r o w o f ( m1; m2 ) va r yin g fr o m n e xt o f ( m1; m2 ) t ill ( m; m ) . t heor em: if all distributions gm1;m2 = fgm1;m2 ( x1; x2 ) ; x1; x2 2 x g, m1; m2 = 1 ; m , are di®erent in the sense that d ( gl1;l2jjgm1;m2 ) > 0 , and the positive numbers e1;1j1;1; e2;2j2;2; :::; em;mjm;m are such that the following inequalities hold e1;1j1;1 < m in l1;l2=1;m ; (l1;l2) 6=(1;1) d ( gl1;l2jjg1;1 ) ; ( 9 ) em1;m2jm1;m2 < m in [ m in (l1;l2)2(m1;m2)¡ e¤l1;l2jm1;m2 ( el1;l2jl1;l2 ) ; m in (l1;l2)2(m1;m2)+ d ( gl1;l2jjgm1;m2 ) ]; m1; m2 = 1 ; m; ( m1; m2 ) 6= ( 1 ; 1 ) ; ( m1; m2 ) 6= ( m; m ) ; ( 1 0 ) em;mjm;m < m in l1;l2=1;m ; (l1;l2) 6=(m;m ) e¤l1;l2jm;m ( el1;l2jl1;l2 ) ; ( 1 1 ) then there exists a lao sequence of tests, all elements of the reliability matrix of which e ¤ = fe¤l1;l2jm1;m2g are positive and are de¯ned in ( 6 ) ¡ ( 8 ) . w hen one of the inequalities ( 9 ) ¡ ( 1 1 ) is violated, then at least one element of the matrix ( 6 ) ¡ ( 8 ) is equal to 0 . th e p r o o f o f t h e t h e o r e m c o n s is t s in p r e s e n t a t io n o f t h e p r o b le m fo r t wo o b je c t s a s a p r o b le m fo r o n e c a p a c io u s o b je c t . if we r e n u m e r a t e a s fo llo ws ( 1 ; 1 ) = 1 ; ( 1 ; 2 ) = 2 ; :::; ( 1 ; m ) = m; ( 2 ; 1 ) = m + 1 ; :::; ( 2 ; m ) = 2 m; :::; ( m; m ) = m 2 a n d d e n o t e ( x1; x2 ) = y , x £ x = y we will h a ve p r o b le m o f m 2 h yp o t h e s e s t e s t in g fo r o n e o b je c t wit h p o s s ib ilit y o f d e c is io n r e je c t io n . s o u s in g t h is n u m e r a t io n we will h a ve t h e c o r r e s p o n d in g e r r o r p r o b a b ilit ie s a n d r e lia b ilit ie s fo r l = 1 ; m 2 + 1 ; ; m = 1 ; m 2, wh e n we a p p ly th e o r e m 2 o f [3 ]. ge n e r a liz a t io n o f t h e r e s u lt is p o s s ib le in m a n y d ir e c t io n s . e. haroutunian, a. yesayan and n. harutyunyan 1 1 3 refer ences [1 ] e . a . h a r o u t u n ia n , p . m. h a ko b ya n a n d a . o. y e s s a ya n , \ on m u lt ip le h yp o t h e s e s l a o t e s t in g wit h r e je c t io n o f d e c is io n fo r m a n y in d e p e n d e n t o b je c t s " , p roceedings of international conference csit 2011, p p . 1 1 7 { 1 2 0 , y e r e va n 2 0 1 1 . [2 ] e . h a r o u t u n ia n , p . h a ko b ya n , a . y e s s a ya n a n d n . h a r u t yu n ya n \ on mu lt ip le h yp o t h e s e s l a o te s t in g w it h l ib e r t y o f r e je c t io n o f d e c is io n fo r two in d e p e n d e n t ob je c t s " ,... [3 ] e . h a r o u t u n ia n , p . h a ko b ya n a n d a . y e s s a ya n , \ ma n y h yp o t h e s e s l a o t e s t in g wit h r e je c t io n o f d e c is io n fo r a r b it r a r ily va r yin g o b je c t " , transactions of iiap of nas of r a and of ysu, m athematical p roblems of computer science, vo l. 3 5 , p p . 7 7 -8 5 , 2 0 1 1 . [4 ] e . a . h a r o u t u n ia n , \ ma n y s t a t is t ic a l h yp o t h e s e s : in t e r d e p e n d e n c e o f o p t im a l t e s t 's e r r o r p r o b a b ilit ie s e xp o n e n t s " , ( in r u s s ia n ) , a b s t r a c t o f t h e r e p o r t o n t h e 3 r d a llu n io n s c h o o l-s e m in a r , \p rogram-algorithmical software for applied multi-variate statistical analysis", ts a kh ka d z o r , p a r t 2 , p p . 1 7 7 { 1 7 8 , 1 9 8 8 . [5 ] e . a . h a r o u t u n ia n , \ l o g a r it h m ic a lly a s ym p t o t ic a lly o p t im a l t e s t in g o f m u lt ip le s t a t is t ic a l h yp o t h e s e s " , p roblems of control and information theory, vo l. 1 9 ( 5 -6 ) , p p . 4 1 3 { 4 2 1 , 1 9 9 0 . [6 ] e . h a r o u t u n ia n a n d p . h a ko b ya n , \ mu lt ip le h yp o t h e s e s l a o t e s t in g fo r m a n y in d e p e n d e n t o b je c t " , in t e r n a t io n a l jo u r n a l \ s c h o la r ly r e s e a r c h e xc h a n g e " , vo l. 2 0 0 9 , p p . 1 -6 , 2 0 0 9 . [7 ] i. cs is z ¶a r a n d j. k äo r n e r , information theory: coding theorems for discrete memoryless systems, s e c o n d e d it io n , ca m b r id g e u n ive r s it y p r e s s , 2 0 1 1 . [8 ] r . e . b la h u t , p rinciples and p ractice of information theory, a d d is o n -w e s le y, r e a d in g , ma , 1 9 8 7 . [9 ] t. m. co ve r a n d j. a . th o m a s , e lements of information theory. s e c o n d e d it io n , n e w y o r k, w ile y, 2 0 0 6 . submitted 10.10.2017, accepted 15.02.2018. ´³½ù³ïç í³ñï³íý»ñç áñáßáõùçó ññ³å³ñù³ùµ è²ú ëïáõ·ù³ý ù³ëçý »ñïáõ ï³ëû³é ûµû»ïïý»ñç ¹»åùáõù º. ð³ñáõãûáõýû³ý, ². ºë³û³ý ¨ ü. ð³ñáõãûáõýû³ý ²ù÷á÷áõù ð³ù³ï»õ áý¹ñ³ï ñ³í³ý³ï³ý³ûçý µ³ßëù³ùµ µýáõã³·ñíáõ »ñïáõ ï³ëû³é ûµû»ïïý»ñç ýï³ïù³ùµ ¹çï³ñïíáõù ¿ µ³½ù³ïç í³ñï³íý»ñç áñáßáõùçó ññ³å³ñù³ý ñý³ñ³íáñáõãû³ùµ ëïáõ·áõùá: àõëáõùý³ëçñí»é ¿ ëë³éç ñ³í³ý³ï³ýáõãûáõýý»ñç ³ëçùåïáïáñ»ý ûåïçù³é óáõóçãý»ñç (ñáõë³éçáõãûáõýý»ñç) ù³ïñçóá: 1 1 4 on multiple hypotheses lao testing with rejection of decision for two dependent objects î lao òåñòèðîâàíèè ìíîãèx ãèïîòåç ñ îòêàçîì îò ðåøåíèÿ äëÿ äâóõ çàâèñèìûõ îáúåêòîâ å. àðóòþíÿí, à. åñàÿí è í. àðóòþíÿí àííîòàöèÿ äëÿ ìîäåëè ñîñòîÿùeé èç äâóõ çàâèñèìûõ îáúåêòîâ, xàðàêòåðèçóåìûx ñîâìåñòíûì äèñêðåòíûì ðàñïðåäåëåíèåì âåðîÿòíîñòåé ðàññìàòðèâàåòñÿ òåñòèðîâàíèå ìíîãèx ñòàòèñòè÷åñêèõ ãèïîòåç ñ âîçìîæíîñòüþ îòêàçà îò ðåøåíèÿ. èçó÷åíà ìàòðèöà àñèìïòîòè÷åñêè îïòèìàëüíûõ ýêñïîíåíò âåðîÿòíîñòåé îøèáîê (íàäåæíîñòåé). mathematical problems of computer science 59, 27–34, 2023. doi: 10.51408/1963-0099 udc 510.64 the relationship between the proof complexities of linear proofs in quantified sequent calculus and substitution frege systems hakob a. tamazyan yerevan state university, yerevan, armenia e-mail: hakob.tamazyan@ysu.am abstract it has formerly been proved that there is an exponential speed-up in the number of lines of the quantified propositional sequent calculus over substitution frege systems when considering proofs as trees. this paper shows that a linear proof of any quantifierfree tautology in quantified propositional sequent calculus can be transformed into a linear proof of the same tautology in a substitution frege systems with no more than polynomially increasing proof lines and size. keywords: sequent systems, frege systems, proof size, number of proof lines, exponential speed-up. article info: received 23 march 2023; sent for review 2 april 2023; accepted 19 may 2023. 1. introduction the existence of a propositional proof system that has proofs of polynomial size for all tautologies is equivalent to the equation np = co-np [1]. this observation has gained attention in recent years, leading to the examination of new proof systems. through the discovery of new systems, the computational power of existing ones is gaining a greater understanding. a hierarchy of proof systems has been established based on two complexity measures (size and lines), and the relationships between these systems are being explored. alessandra carbone in [2] compared the number of derivation lines in the form of a tree in some propositional calculus systems and revealed a distinctive property of the quantified propositional sequent calculus (qpk system). namely, for some sequences of formulas, the qpk system has an exponential speed-up by lines with respect to the substitution sequent calculus (spk system) and substitution frege systems (sf systems) when proofs are considered as trees. it was shown in [3] that the lines of linear proofs of the same formulae families in all three systems are the same by order. later, in [4], the same result was achieved if one considers the sizes of linear proofs of the same formulae families for comparison. 27 28 the relationship between the proof complexities of linear proofs in qpk and substitution sf in this paper, the relationship between the proof complexities of linear proofs in qpk and sf has been investigated for all quantifier-free tautologies: it turns out that qpk system has no significant advantage over sf when only linear proofs are considered. specifically, after the transformation of linear qpk-proof of a quantifier-free tautology into a linear sfproof of the same tautology by some algorithm, both complexities (the number of lines and sizes) of linear proofs in sf can increase polynomially at most. 2. preliminaries first and foremost, lets define several proof systems according to [1, 5, 6]. the frege system f uses a denumerable set of propositional variables, a finite, complete set of propositional connectives. it has a finite set of inference rules defined by a figure of the form a1a2...am b (the rules of inference with zero hypotheses are the schemes of axioms). f must be sound and complete, i.e., for each rule of inference a1a2...am b every truth-value assignment, satisfying a1a2...am, also satisfies b, and f must prove every tautology. the substitution frege system sf is defined by adding to f the substitution rule a(p) a(b) where simultaneous substitution of the formula b is allowed for the variable p. the lk sequent calculus was introduced by gentzen [7] for first-order logic. each line in lk-proof is a sequent: a sequent is written in the form: a1, . . . , an → b1, . . . , bm where a1, . . . , an and b1, . . . , bm are formulas. we denote these sequences of formulas by capital greek letters γ, ∆, etc. as a quantifier symbol in lk, we will include only the universal quantification ∀. the existential quantification symbol ∃ will be added by the following definition: (∃x)a(x) ≡ ¬(∀x)¬a(x). the inference rules of the sequent calculus lk are as follows: • initial sequents are sequents of the following form: a → a where a is any formula. • structural rules: weakening : left γ → ∆ a, γ → ∆ weakening : right γ → ∆ γ → ∆, a exchange : left γ1, a, b, γ2 → ∆ γ1, b, a, γ2 → ∆ exchange : right γ → ∆1, a, b, ∆2 γ → ∆1, b, a, ∆2 contraction : left γ1, a, a, γ2 → ∆ γ1, a, γ2 → ∆ contraction : right γ → ∆1, a, a, ∆2 γ → ∆1, a, ∆2 h. tamazyan 29 • logical rules: ¬ : left γ → ∆, a ¬a, γ → ∆ ¬ : right a, γ → ∆ γ → ∆, ¬a ∧ : left a, b, γ → ∆ a ∧ b, γ → ∆ ∧ : right γ → ∆, a γ → ∆, b γ → ∆, a ∧ b ∨ : left a, γ → ∆ b, γ → ∆ a ∨ b, γ → ∆ ∨ : right γ → ∆, a, b γ → ∆, a ∨ b ⊃: left γ → ∆, a b, γ → ∆ a ⊃ b, γ → ∆ ⊃: right a, γ → ∆, b γ → ∆, a ⊃ b • the cut rule: γ → ∆, a a, γ → ∆ γ → ∆ let us denote by pk the sequent calculus lk, where the rules are restricted to propositional logic. the substitution system spk is defined as the propositional sequent calculus pk with an additional substitution rule: sbp γ → ∆, a(p) γ → ∆, a(b) , where simultaneous substitution of the formula b is allowed for the variable p, and p does not appear in γ, ∆. the quantifier system qpk is defined as the propositional sequent calculus pk, where new quantification rules are added: ∀ : left a(b), γ → ∆ (∀q)a(q), γ → ∆ ∀ : right γ → ∆, a(p) γ → ∆, (∀q)a(q) where b is any formula such that no free variable occurrence in b becomes bounded in a(b), and with the restriction that the atom p does not occur freely in the lower sequents of ∀ : right. notice that the the following two inferences can be derived in qpk system using the definition of the quantifier ∃: ∃ : left a(p), γ → ∆ (∃q)a(q), γ → ∆ a(p), γ → ∆ γ → ∆, ¬a(p) γ → ∆, (∀q)(q) ¬(∀q)¬a(q), γ → ∆ ∃ : right γ → ∆, a(b) γ → ∆, (∃q)a(q) γ → ∆, a(b) ¬a(b), γ → ∆ (∀q)(q), γ → ∆ γ → ∆, ¬(∀q)¬a(q) 30 the relationship between the proof complexities of linear proofs in qpk and substitution sf 3. main results for a given linear proof in qpk with n number of lines and proof size s, one can always find a linear proof in spk of the same tautology having o(n2) lines and o(s5) proof size. first of all, notice that for any linear proof in spk, there exists a linear proof in qpk of the same tautology with the same number of lines. the sequent (∀p)a(p), γ → ∆, a(b) is provable for all a, b, and the sequent γ → ∆, (∀p)a(p) is derivable from ∆ → ∆, a(p). hence, after combining them through a cut rule, one derives γ → ∆, a(b). here we examine the relationship between these systems in the opposite scenario. lemma. for n, m ≥ 0 and p not appeared in γ, ∆, the following inference γ, a1(p), . . . , an(p) → ∆, an+1(p), . . . , an+m(p) γ, a1(b), . . . , an(b) → ∆, an+1(b), . . . , an+m(b) can be achieved in spk system with o(n + m) lines using the substitution rule only once. proof. first, let’s prove these additional inferences: 1. γ → ∆, ¬a a, γ → ∆ γ → ∆, ¬a a → a γ → ∆, ¬a ¬a, a → a, γ → ∆ 2. γ → ∆, a ∨ b γ → ∆, a, b γ → ∆, a ∨ b a → a b → b γ → ∆, a ∨ b a → a, b b → b γ → ∆, a ∨ b a → a, b b → a, b γ → ∆, a ∨ b a ∨ b → a, b γ → ∆, a, b 3. γ, a ∧ b → ∆ γ, a, b → ∆ γ, a ∧ b → ∆ a → a b → b γ, a ∧ b → ∆ a, b → a b → b γ, a ∧ b → ∆ a, b → a a, b → b γ, a ∧ b → ∆ a, b → a ∧ b γ, a, b → ∆ h. tamazyan 31 the final proof will look like this: γ, a1(p), . . . , an(p) → ∆, an+1(p), . . . , an+m(p) γ, a1(p) ∧ a2(p), . . . , an(p) → ∆, an+1(p), . . . , an+m(p).... γ, a1(p) ∧ . . . ∧ an(p) → ∆, an+1(p), . . . , an+m(p).... γ, a1(p) ∧ . . . ∧ an(p) → ∆, an+1(p) ∨ . . . ∨ an+m(p) γ → ∆, an+1(p) ∨ . . . ∨ an+m(p), ¬(a1(p) ∧ . . . ∧ an(p)) γ → ∆, an+1(p) ∨ . . . ∨ an+m(p) ∨ ¬(a1(p) ∧ . . . ∧ an(p)) γ → ∆, an+1(b) ∨ . . . ∨ an+m(b) ∨ ¬(a1(b) ∧ . . . ∧ an(b)) γ → ∆, an+1(b) ∨ . . . ∨ an+m(b), ¬(a1(b) ∧ . . . ∧ anb) γ, a1(b) ∧ . . . ∧ an(b) → ∆, an+1(b) ∨ . . . ∨ an+m(b).... γ, a1(b), . . . , an(b) → ∆, an+1(b), . . . , an+m(b) note that in this proof the substitution rule is applied only once. theorem 1. for a given linear proof in qpk of some quantifier-free tautology with n number of lines, there exists a linear proof in spk of the same tautology having o(n2) number of lines. proof. suppose p is a given linear proof in qpk. since p is the proof of a quantifierfree tautology, if a formula with a quantifier appears in the proof, then it must disappear at some point in the next lines. these formulas can appear either by quantification rules or by weakening rules, and the cut rule is the only inference rule capable of removing a formula from the sequent. notice that if we apply the cut rule to two sequents and some formula a with a quantifier is removed, then it is impossible that both of these sequents got this quantifier by the ∀ : left rule. first of all, we will remove all applications of the ∀ : left rule in the proof of p . let (∀q)a(q) be some formula or subformula in the proof. suppose it appeared by ∀ : right rule that infers γ → ∆, (∀q)a(q) from γ → ∆, a(p). since p does not occur free in sequent γ → ∆, (∀q)a(q), instead of the ∀ : right rule, we can apply the substitution rule to γ → ∆, a(p) and substitute p with some new variable k that did not appear throughout the proof. if (∀q)a(q) appeared by weakening rules, we will replace it with the formula a(k), where k is again some new variable that did not appear throughout the proof. according to the previously mentioned claim, the formula (∀q)a(q) should have been removed at some point via the cut rule. therefore, just before the application of cut rule, we will substitute the variable k with the corresponding matching formula to be able to apply the cut rule successfully. this substitution is allowed since k does not appear in the remaining formulas of the sequent. this removal of formulas with quantifiers from the proof can have the following effects. firstly, since these formulas have been replaced with different ones, the contraction rule can not be applied to these replacements anymore, as they can differ from each other. therefore, instead of applying the contraction rule to them, in the next lines we will apply the same inference rules to both of them. as these formulas should disappear in one of the next lines by the cut rule, we will apply the cut-elimination rule twice so that both of them 32 the relationship between the proof complexities of linear proofs in qpk and substitution sf will be removed. there are o(n) applications of the contraction rule, then after this change, the number of lines will become o(n2). however, according to the lemma, the number of applications of the substitution rule will not change and will remain o(n). secondly, the ∀ : left rule that transformed some sequent a(b), γ → ∆ into (∀q)a(q), γ → ∆, will not be applied to the proof, and the formula b will appear in the next lines. hence, there might be an application of the substitution rule in these next lines that substitutes some variable x into some formula c so that x also appears in the formula b. this means that besides the formula c, there can also be other formulas with the variable x in the sequent. therefore, to fix this, we will apply the substitution to these formulas too. considering that the number of applications of the ∀ : left rule was o(n) and removing each application of the contraction rule adds just one formula to the sequent, the number of such formulas in the sequent will be o(n). therefore, according to the lemma, each such substitution will require o(n) additional lines. since there are o(n) applications of the substitution rule, this change will add o(n2) number of lines to our proof. this will conclude the transformation process, and the transformed spk proof will have o(n2) lines. theorem 2. for a given linear proof in qpk of some quantifier-free tautology with a proof size s, there exists a linear proof in spk of the same tautology having o(s5) proof size. proof. suppose p is a given linear proof in qpk with n number of lines and proof size s. let p ′ be the transformed spk proof according to the process described above. to calculate its size, let’s dive into the transformation process step by step. we replaced each application of the ∀ : right rule with a substitution rule to substitute one variable with another. the formulas with quantifiers that appeared by weakening rules have been replaced by formulas with the same size. afterwards, we added a substitution before the application of the cut rule to match the corresponding formula. all these steps change the number of proof lines and the proof size linearly. let’s denote them by n′, s′, respectively. moreover, we removed all applications of the ∀ : left rule. therefore, if some application of the ∀ : left rule transformed the sequent a(b), γ → ∆ into (∀q)a(q), γ → ∆, then after the removal, the formula b will appear in the next lines. this will increase the proof size by at most n′ · |a(b)|, where |a(b)| is the size of the formula a(b). removing the ith application of the ∀ : left rule increases the proof size by at most n′ ·|ai(bi)|, then removing all of them will add no more than ∑ i n′ · |ai(bi)| = n′ · ∑ i |ai(bi)| ≤ n′ · s′ ≤ s′2 to the proof size. as s′ is o(s), after this step, the proof size will be o(s2) and the number of lines will remain o(n). removing applications of the contraction rule has the following two effects on the proof size. first of all, it will keep the eliminated formula in a sequent, so it will appear in the next lines. the added proof size can be calculated completely like the previous method. since the number of applications of the contraction rule is o(n) and the proof size is o(s2), this change will make the proof size o(s3). the number of lines will remain o(n). the second effect of removing applications of the contraction rule will be applying the same inference rules to both formulas. since the proof size is o(s3), then applying the same h. tamazyan 33 inference rule to the previously eliminated formula can increase the proof size by o(s3). the number of applications of the contraction rule is o(n), and since n ≤ s, the overall proof size will become o(s4). finally, the removal of the ∀ : left rule causes some substitution steps to also substitute the same variable in several other formulas of the same sequent. notice that all these substitution steps were ∀ : right rule replacements that substitute one variable with another, as otherwise we won’t face such a problem. each such substitution that simultaneously substitutes the same variable in these sequent formulas required o(n) lines. if the ith such substitution is applied to the sequent si, then this change will overall add no more than∑ i c · n · |si| = c · n · ∑ i |si| ≤ c · s · ∑ i |si| to the proof size, where |si| is the size of the sequent si and c is some constant. ∑ i |si| is smaller than the current proof size, therefore the transformed spk proof will have o(s5) size. corollary. since the system spk is polynomially equivalent to the system sf, there is a transformation of a linear proof of any quantifier-free tautology in qpk into a linear proof in the system sf that increases the proof lines and size at most polynomially. 4. conclusion this work described an algorithm according to which any qpk linear proof can be transformed into a sf linear proof by increasing its lines and size to at most a polynomial extent. the obtained results show that the qpk system does not have a substantial advantage over the system sf in terms of linear proofs. references [1] s. a. cook and a. r. reckhow, “the relative efficiency of propositional proof systems”, symbolic logic, vol. 44, pp. 36–50, 1979. [2] a. carbone, “quantified propositional logic and the number of lines of tree-like proofs”, studia logica, vol. 64, pp. 315–321, 2000. [3] h. a. tamazyan and a. a. chubaryan, “on proof complexities relations in some systems of propositional calculus, mathematical problems of computer science, vol. 54, pp. 138–146, 2020. [4] l. a. apinyan and a. a chubaryan, “on sizes of linear and tree-like proofs for any formulae families in some systems of propositional calculus”, mathematical problems of computer science, vol. 57, pp. 47–55, 2022. [5] p. pudlák, the lengths of proofs, in s. buss (ed.), handbook of proof theory, elsevier, vol. 137, pp. 547-637, 1998. [6] j. kraj́ıc̆ek, proof complexity, encyclopedia of mathematics and its applications, cambridge university press, vol. 170, 2019. [7] g. gentzen, “die widerspruchsfreiheit der reinen zahlentheorie”, mathematische annalen, vol. 112, pp. 493–565, 1936. 3 4 the relationship between the proof complexities of linear proofs in qpk and substitution sf ¶í³ûçý ³ñï³íáõùý»ñç µ³ñ¹áõãûáõýý»ñç ï³åá í³í³éçãý»ñáí ë»ïí»ýóç³é ñ³ù³ï³ñ·áõù ¨ ï»õ³¹ñù³ý ï³ýáýáí üñ»·»ç ñ³ù³ï³ñ·»ñáõù ð³ïáµ ². â³ù³½û³ý ºñ¨³ýç å»ï³ï³ý ñ³ù³éë³ñ³ý, ºñ¨³ý, ð³û³ëï³ý e-mail: hakob.tamazyan@ysu.am ²ù÷á÷áõù ñâÿçü ìåæäó ñëîæíîñòÿìè äîêàçàòåëüñòâ ëèíåéíûõ âûâîäîâ â ñèñòåìå ñåêâåíöèàëüíîãî èñ÷èñëåíèÿ ñ êâàíòîðàìè è ñèñòåìàõ ôðåãå ñ ïðàâèëîì ïîäñòàíîâêè àêîá à. òàìàçÿí åðåâàíñêèé ãîñóäàðñòâåííûé óíèâåðñèòåò, åðåâàí, àðìåíèÿ e-mail: hakob.tamazyan@ysu.am àííîòàöèÿ ðàíåå áûëî äîêàçàíî, ÷òî ñóùåñòâóåò ýêñïîíåíöèàëüíîå óñêîðåíèå êîëè÷åñòâà øàãîâ â ñèñòåìå ñåêâåíöèàëüíîãî èñ÷èñëåíèÿ âûñêàçûâàíèé ñ êâàíòîðàìè ïî ñðàâíåíèþ ñ ñèñòåìàìè ôðåãå ñ ïðàâèëîì ïîäñòàíîâêè, êîãäà ìû ðàññìàòðèâàåì âûâîäû â âèäå äåðåâüåâ. ýòà ñòàòüÿ ïîêàçûâàåò, ÷òî ëèíåéíûé âûâîä ëþáîé áåñêâàíòîðíîé òàâòîëîãèè â ñèñòåìå ñåêâåíöèàëüíîãî èñ÷èñëåíèÿ âûñêàçûâàíèé ñ êâàíòîðàìè ìîæíî ïðåâðàòèòü â ëèíåéíûé âûâîä òîé æå òàâòîëîãèè â ñèñòåìàõ ôðåãå ñ ïðàâèëîì ïîäñòàíîâêè ñ íå áîëåå ÷åì ïîëèíîìèàëüíî âîçðàñòàþùèì êîëè÷åñòâîì øàãîâ è äëèíîé âûâîäà. êëþ÷åâûå ñëîâà: ñåêâåíöèàëüíûå ñèñòåìû, ñèñòåìû ôðåãå, äëèíà âûâîäà, êîëè÷åñòâî øàãîâ âûâîäà, ýêñïîíåíöèàëüíîå óñêîðåíèå. ü³ëïçýáõù ³å³óáõóí»é ¿, áñ í³í³éçãý»ñáí ë»ïí»ýóç³é ñ³ù³ï³ñ·áõù ³éï³ ¿ ù³ûé»ñç ù³ý³ïç ¿ùëåáý»ýóç³é ³ñ³·³óáõù ï»õ³¹ñù³ý ï³ýáýáí üñ»·»ç ñ³ù³ï³ñ·»ñç ýï³ïù³ùµ, »ñµ ¹çï³ñïáõù »ýù í³é³ûçý ³ñï³íáõùý»ñá: ²ûë ñá¹í³íá óáõûó ¿ ï³éçë, áñ ³é³ýó í³í³éçãý»ñç, ó³ýï³ó³í ýáõûý³µ³ýáõãû³ý ·í³ûçý ³ñï³íáõùá í³í³éçãý»ñáí ë»ïí»ýóç³é ñ³ù³ï³ñ·áõù ñý³ñ³íáñ ¿ í»ñ³í»é ýáõûý ýáõûý³µ³ýáõãû³ý ·í³ûçý ³ñï³íù³ý ï»õ³¹ñù³ý ï³ýáýáí üñ»·»ç ñ³ù³ï³ñ·»ñáõù` áõý»ý³éáí ³ñï³íù³ý ù³ûé»ñç ù³ý³ïç ¨ »ñï³ñáõãû³ý ³é³í»é³·áõûý µ³½ù³ý¹³ù³ûçý ³×: ´³ý³éç µ³é»ñ` ë»ïí»ýóç³é ñ³ù³ï³ñ·»ñ, üñ»·»ç ñ³ù³ï³ñ·»ñ, ³ñï³íù³ý »ñï³ñáõãûáõý, ³ñï³íù³ý ù³ûé»ñç ù³ý³ï, ¿ùëåáý»ýóç³é ³ñ³·³óáõù: 03_chub_59_27_34 03 mpcs_2004_e_davtyan_eng.dvi mathematical problems of computer science 23, 2004, 5{11. on the constr uction of cluster systolic ar r ays¤ e d m o n m. d a vt ya n institue for informatics and automation problems of nas of ra e-mail edmon@ipia.sci.am abstract the paper presents three approaches to the construction of cluster systolic arrays. a careful analysis of these approaches based on the comparison of the running times of corresponding systolic arrays was carried out. a method to minimize the running time is proposed. refer ences [1 ] ma r ia n n e d e lo r m e . a n in t r o d u c t io n t o ce llu la r a u t o m a t a . ce llu la r a u t o m a t a : a p a r a lle l mo d e l, ma t h e m a t ic s a n d it s a p p lic a t io n s , k lu we r , ju ly 1 9 9 8 . [2 ] k .v .s h a h b a z ya n , y u .h .s h o u ko u r ia n . l o g ic a lly d e ¯ n a b le l a n g u a g e s o f co m p u t a t io n s in o n e cla s s o f flo w e ve n t s t r u c t u r e s . in ta s g r a n t " w e a k a r it h m e t ic s " 2 0 0 0 -4 4 7 , ( 2 0 0 2 ) . [3 ] e .d a vt ya n . on t h e mo d e llin g o f on e cla s s o f s ys t o lic s t r u c t u r e s o n a p c clu s t e r . in p r o c e e d in g s o f cs it-2 0 0 3 , p p . 3 4 0 -3 4 4 . îé³ëï»ñ³ûçý ëçëïáéçï ½³ý·í³íý»ñç ï³éáõóù³ý ù³ëçý ¾. ø. ¸³íãû³ý ²ù÷á÷áõù ²ßë³ï³ýùáõù ¹çï³ñïí³í ¿ ïé³ëï»ñ³ûçý ëçëïáéçï ½³ý·í³íý»ñç ï³éáõóù³ý »ñ»ù ùáï»óáõù: î³ï³ñí³í ¿ ³û¹ ùáï»óáõùý»ñç ñ³ù»ù³ïáõãûáõý áëï ñ³ù³å³ï³ëë³ý ëçëïáéçï ½³ý·í³íý»ñç ³ßë³ï³ýùý»ñç ï³ï³ñù³ý å³ù³ý³ïý»ñç: ²é³ç³ñïí³í ¿ ³ßë³ï³ýùç ï³ï³ñù³ý å³ù³ý³ïá ùçýçùç½³óýáõ ù»ãá¹: ¤this research is supported by intas 0447, istc 823 grants and 04.10.31 target program of ra. 5 mathematical problems of computer science 53, 29–38, 2020. udc 519.6 estimating time of driver arrival with gradient boosting algorithms and deep neural networks henrik t. sergoyan american university of armenia e-mail: henriksergoyan@gmail.com abstract customer experience and resource management determine the degree to which transportation service providers can compete in today’s heavily saturated markets. the paper investigates and suggests a new methodology to optimize calculations for estimated time of arrival (from now on eta, meaning the time it will take for the driver to reach the designated location) based on the data provided by gg collected from rides made in 2018. gg is a transportation service providing company, and it currently uses the open source routing machine (osrm) which exhibits significant errors in the prediction phase. this paper shows that implementing algorithms such as xgboost, catboost, and neural networks for the said task will improve the accuracy of estimation. paper discusses the benefits and drawbacks of each model and then considers the performance of the stacking algorithm that combines several models into one. thus, using those techniques, final results showed that mean squared error (mse) was decreased by 54% compared to the current gg model. keywords: gg, estimated time of driver arrival, osrm, xgboost, catboost, neural networks. 1 introduction eta calculation is a relatively significant product differentiator for firms competing in the automated ride service hailing industry. eta computation is particularly difficult for cities with underdeveloped traffic and road management services such as yerevan, thus requiring novel methodologies and datasets to design models that can be deployed for use by relatively fickle customers. gg is one of the major transportation service providers in yerevan and this paper uses data provided by gg to predict eta. while gg currently estimates the time of arrival, it does so with a significant error. this paper discusses optimization techniques to calculate estimated arrival time with more accurate and flexible algorithms. 29 30 estimating time of driver arrival with gradient boosting algorithms and deep neural networks 1.1 problem setting and description gg has an application that enables drivers to receive orders to transport customers and goods. having chosen a pick-up location through the app, the customer gets an eta for the driver. the task is to improve eta estimation so that it is more accurate than the baseline osrm model, which gg currently employs. to do so, the paper uses several joint machine learning algorithms such as xgboost, catboost, and neural networks. the dataset provided by gg consists of more than 2.4m orders occured (received) in 2018. the number of unique drivers exceeds 6000. in addition to spatio-temporal dataset, the hourly data about weather conditions were used to better capture possible traffic jams. 1.2 structure of the paper the paper consists of five central sections. the first section provides the problem statement and description, the overall summary of the paper structure representing the current method gg uses for estimating the arrival time of the driver while describing its drawbacks. after giving full understanding about the reason for further investigations, the next 3 sections, give detailed information and clarification of alternative models and outline advantages and disadvantages of each of them showing the results run on the 1-year information containing data. based on the preceding sections the fifth part of the paper shows comparisons of all 3 suggested models and interprets the most optimal solution of the problem based on 3 methods. 1.3 description of osrm the open source routing machine is a routing engine that works to find the shortest paths in the road networks, and it supports linux, freebsd, windows and mac os x platforms. the primary function of the environment is to compute and output the shortest path between the origin and the destination points, on the basis of which the system can compute the estimated time within a couple of milliseconds for the transport to reach a particular destination. by implementing contraction hierarchies, pure route computation takes the minimal time of those calculations. the most significant part of the calculation is finding the route and transmitting the geometry over the network. the decision-making capacity of the system is based on routing algorithms with road network data from the osm (openstreetmap) project (see [1]). the osrm optimization algorithm prioritizes the speed with which calculations are made. data obtained from osm consist of 3 main components; namely nodes, ways and relations between them. nodes determine the geometric structure of the path. ways are 2d polylines consisting of line segments. several lines can share the same node if they intersect with each other. a relation relates nodes or ways to turn restrictions, routes, and other features [2]. in these graphs, nodes represent directions of an osm segment and graph edges connect graph nodes by describing the transition from one specific point to another. here is how it works: h. sergoyan 31 fig. 1. osrm visual implementation. even though this method seems to expedite the process of calculating eta, the major drawback is that it does not capture the impact of weather conditions, traffic conditions and other real-life situations that depend on human factors. thus, this paper concentrates on improving eta calculation using information concerning weather and concrete data generated by drivers on actual routes. 2 machine learning approach to the problem 2.1 related work managing traffic among the urban population is becoming an increasingly significant issue for major municipalities. thus, one of the fundamental problems to be solved is the efficient management of road traffic by minimizing congestion and assisting travelers with real-time information. some urban areas currently use datasets that include information concerning gps coordinates of origins and destinations, travel time, travel distance, pickup date, trip start-end times and the total fare to organize the road management. by analyzing data, route optimizing environments provide information for travelers related to the optimal routes, road conditions, and locations of incidents [3], [4]. ishan jindal, tony (zhiwei) qin, xuewen che, matthew nokleby, jieping ye studied this topic and used the unified neural network approach to estimate travel time and distance for a taxi trip. the data they used were structurally similar to those provided by gg, so this paper takes its results as an additional benchmark and aims to improve upon it [5]. their research considers the waiting time at intersections for travel time estimations. the method used is a particular case of the path-based approach, where they add predictions of waiting time at intersections of sub-paths including the neighbor-based method by averaging the travel time for all the samples in the training data that have the same origin, destination, and time of day. thus, the paper focuses on predicting travel time and distance from a source to a target as a function of the time of day based on nyc travel trip historical data. having a large amount of training data, the authors build a unified neural network learning model, which jointly learns the travel time and distance between the origin and destination [5]. to understand what the travel time is, one needs to think of the time taken by the driver to move from the initial location to the final location including velocity changes or stops 32 estimating time of driver arrival with gradient boosting algorithms and deep neural networks made by him/her depending on some conditions. travel distance is the path taken by the driver to reach from the primary point to the final location. thus, the travel time depends on the origin and destination at a particular time of the day [4]. 2.2 xgboost: extreme gradient boosting xgboost is a decision-tree-based ensemble machine learning algorithm that uses a gradient boosting framework. in prediction problems involving unstructured data (images, text, etc.), artificial neural networks tend to outperform all other algorithms or structures. however, when it comes to small-to-medium structured/tabular data, decision-tree-based algorithms are considered best-in-class (see [4]). xgboost algorithm was developed as a research project at the university of washington. tianqi chen and carlos guestrin presented their paper at sigkdd conference in 2016 and caught the machine learning world by fire. since its introduction, this algorithm has not only been credited with winning numerous kaggle competitions but also for being the driving force under the hood for several cutting-edge industry applications. xgboost is an ensemble learning method, and the main principle behind it is that a group of weak learners come together to form a keen learner. bagging and boosting are two widely used ensemble learners [4], [6]. 2.2.1 bagging while decision trees are one of the most easily interpretable models, they exhibit highly variable behavior. therefore, several decision trees are being generated in parallel and form the base learners of bagging technique. data sampled with replacement is fed to these learners for training. the final prediction is the averaged output from all the learners. the main principle behind the ensemble model is that a group of weak learners come together to form a strong learner [6]. 2.2.2 boosting in boosting, the trees are built sequentially so that each subsequent tree aims to reduce the errors of the previous tree. each tree learns from its predecessors and updates the residual errors. hence, the tree that grows next in the sequence will learn from an updated version of the residuals [6]. using xgboost, in this case, has several advantages. first of all, xgboost has a feature of out-of-core computing that helps to handle massive datasets that do not fit into memory. moreover, because of a block structure in its system design, xgboost can make use of multiple cores on the cpu and gpu. data are sorted and stored in in-memory units called blocks. that enables the data layout to be reused by subsequent iterations, instead of computing it again. finally, xgboost has an option to penalize complex models through both l1 and l2 regularizations. regularization helps in preventing overfitting [6]. 2.3 catboost: categorical boosting catboost (categorical boosting) is a state-of-the-art open-source gradient boosting on decision trees library developed by yandex. the algorithm is spawned from xgboost, but it has its advantages compared to all other gradient boosting algorithms. although xgboost h. sergoyan 33 became a game-changing algorithm in machine learning, it indeed, has its drawbacks. for example, xgboost has a problem of dealing with categorical features, and therefore, one-hot encoding is used, which adds a new binary feature for each category. however, in the case of high cardinality features, such as user id, partner id, which were present in the data, such a technique leads to an infeasibly large number of new features. catboost deals with this issue and is now considered the most innovative algorithm for processing categorical features. at the end of the paper, it is shown how well catboost deals with categorical features in the data by looking at the final accuracies of the models [7]. 2.4 nn: neural networks neural networks were used as the last method to predict the time of arrival of a driver. the main advantage of neural nets is that they can detect patterns that no other algorithms can discover. the logic behind it differs significantly from the aforementioned boosting algorithms [8]. • the network architecture includes an input layer, a hidden layer(s) and an output layer. it is also called mlp (multi-layer perceptron) because of the multiple layers. • the hidden layer can be seen as a distillation layer that concentrates some of the essential patterns from the inputs and passes them onto the next layer to view. it renders the network faster and more efficient by identifying only the critical information from the inputs and leaving out redundant information. • the activation function serves for two notable purposes [8]: ◦ captures non-linear relationships between the inputs; ◦ helps to convert the input into a more useful output. 2.5 feature engineering and final results to train the above-mentioned models, annual data generated by gg taxi orders were used. the data consists of more than 2m unique orders received in 2018. the features of the data were the driver id, the coordinates where he takes the order, the coordinates of the order denoted by the user, the time-stamp when the order was created, the actual duration of the trip, and the estimated duration of the trip by osrm. however, several extra features were added to the data, such as dummy variables of weekdays or weather information. it is important to mention that weather data contains hourly historic information about weather conditions, such as temperature, humidity, pressure, wind speed and facts about raining or snowing. the data contained outliers and missing values, and each of them happened because of failure of a mobile program. however, all this misleading information was identified and removed. 34 estimating time of driver arrival with gradient boosting algorithms and deep neural networks 2.5.1 training process and evaluation the evaluation of the proposed model and all the considered baselines are on the mean absolute error (mae). the task of each model was to minimize mae. mae = n∑ i=1 |yi −zi| n cross validation was used to prevent overfitting while training the xgboost and catboost models. moreover, as these algorithms require specifications of a lot of parameters, grid search was used to identify the best parameters for each model. below are bar graphs depicting the importance of features used to train xgboost and catboost algorithms. both models indicate that the most important feature is the distance. however, the remaining features are represented as having varying levels of importance for both respective models. for example, catboost illustrates that the second most important feature is partner id, while the second most important feature for xgboost was the hour of the day in which the trip occurred. it is worthwhile to mention that both models indicate that osrm estimates (”from start to take”) are important features. this makes considerable sense, since osrm estimates include information concerning the waiting time at traffic lights, the number of turns, the angle of turns, and the mean velocity between turns. fig. 2. importance of features for xgboost algorithm. h. sergoyan 35 fig. 3. importance of features for catboost algorithm. neural networks built with keras library require some intuitive approach to identify the number of neurons, hidden layers, and types of activation functions for each layer. based on experimentation, the optimal number of neurons and the most appropriate activation function to minimize mae were determined. as the last stroke of the brush, the predictions of those trained models and actual time were combined in one dataset and a linear regression model was trained on the said dataset. in machine learning, this process is usually called stacking, where it is expected that the ensemble of several models provides a better result than each of those models independently. however, this example clearly illustrates that this is not always the case. although our stacked model outperforms xgboost and neural networks, it still has a slightly higher mae than catboost regressor. this fact can be interpreted in the following way: our three models mostly pick up and model the same information quite similarly and therefore, we did not get much out of an ensemble. below the performance of each model for train and test sets is presented. table 1: mae results for each method. methods train test osrm 205.796 205.787 xgboost 98.77 98.96 catboost 92.34 94.97 nn 98.09 98.34 stack 93.03 95.01 3 conclusion in this paper, we proposed a new approach that formulates the eta calculation as a pure regression problem. as such, we train catboost, xgboost, and neural networks and successfully improve upon benchmarks set by the osrm architecture used by gg and the unified neural network approach. the catboost model outperformed xgboost and neural network. thus, while osrm exhibits errors of 205.796 seconds and 205.787 seconds for 36 estimating time of driver arrival with gradient boosting algorithms and deep neural networks test and train data respectively, catboost model exhibits errors of 92.34 and 94.97 seconds on train and test data respectively. we believe that by increasing the number and quality of features and using more powerful models we can register significant improvements in predictive performance. references [1] m. dodge and r. kitchin, “crowdsourced cartography: mapping experience and knowledge”, environment and planning a: economy and space, vol. 45, no. 1, pp. 19–36, 2013. [2] s. huber and ch. rust, “calculate travel time and distance with openstreetmap data using the opensource routing machine (osrm)”, the state journal, vol. 16, no. 2, pp. 416–423, 2016. [3] y. li and k. fu and zh. wang and c. shahabi and j. ye and y. liu, “multi-task representation learning for travel time estimation”, proceedings of the 24th acm sigkdd international conference on knowledge discovery and data mining, pp. 1695–1704, 2018. [4] zh. wang and k. fu and j. ye, “learning to estimate the travel time”, proceedings of the 24th acm sigkdd international conference on knowledge discovery and data mining, pp. 858–866, 2018. [5] i. jindal and t. qin and x. chen and m. nokleby and j. ye, “a unified neural network approach for estimating travel time and distance for a taxi trip”, arxiv:1710.04350v1, [stat.ml], 2017. [6] t. chen and c. guestrin, “xgboost: a scalable tree boosting system”, proceedings of the 22nd acm sigkdd international conference on knowledge discovery and data mining, pp. 785–794, 2016. [7] l. prokhorenkova and g. gusev and a. vorobev and a. v. dorogush and a. gulin, “catboost: unbiased boosting with categorical features”, proceedings of the 32nd international conference on neural information processing systems, pp. 6639–6649, 2018. [8] d. wang and j. zhang and w. cao and j. li and y. zheng, “when will you arrive? estimating travel time based on deep neural networks”, aaai, pp 2500–2507, 2018. submitted 10.12.2019, accepted 18.04.2020. h. sergoyan 3 7 ì³ñáñ¹ç å³ù³ýù³ý å³ù³ý³ïç ùáï³ñïáõùá áõå»õ³óí³í ·ñ³¹ç»ýïç ³é·áñçãùý»ñç ¨ ëáñá ý»ûñáý³ûçý ó³ýó»ñç ùççáóáí ð»ýñçï î. ê»ñ·áû³ý ºñ¨³ýç å»ï³ï³ý ñ³ù³éë³ñ³ý e-mail: henriksergoyan@gmail.com ²ù÷á÷áõù îöåíêà âðåìåíè ïðèáûòèÿ âîäèòåëÿ ñ ïîìîùüþ àëãîðèòìîâ ãðàäèåíòíîãî óñèëåíèÿ è ãëóáîêèõ íåéðîííûõ ñåòåé ãåíðèõ ò. ñåðãîÿí àìåðèêàíñêèé óíèâåðñèòåò àðìåíèè e-mail:henriksergoyan@gmail.com àííîòàöèÿ îáñëóæèâàíèå êëèåíòîâ è óïðàâëåíèå ðåñóðñàìè îïðåäåëÿþò ñòåïåíü, â êîòîðîé ïîñòàâùèêè òðàíñïîðòíûõ óñëóã ìîãóò êîíêóðèðîâàòü íà ñåãîäíÿøíèõ ñèëüíî íàñûùåííûõ ðûíêàõ. â ýòîé ðàáîòå èññëåäóåòñÿ è ïðåäëàãàåòñÿ íîâàÿ ìåòîäîëîãèÿ îïòèìèçàöèè äëÿ ðàñ÷åòîâ ïðåäïîëàãàåìîãî âðåìåíè ïðèáûòèÿ âîäèòåëÿ íà îñíîâå ïðîøëîãîäíèõ äàííûõ, ïðåäîñòàâëåííûõ gg, ñîáðàííûõ èç ïîåçäîê âîäèòåëåé êîìïàíèè. gg ÿâëÿåòñÿ êîìïàíèåé, ïðåäîñòàâëÿþùåé òðàíñïîðòíûå óñëóãè, è â íàñòîÿùåå âðåìÿ îíà èñïîëüçóåò the open source routing machine (osrm), êîòîðûé íà ýòàïå ïðîãíîçèðîâàíèÿ äåìîíñòðèðóåò ð³×³ëáñ¹ý»ñç ëå³ë³ñïáõùá ¨ é»ëáõñëý»ñç ï³é³í³ñáõùá áñáßáõù »ý ïñ³ýëåáñï³ûçý í³é³ûáõãûáõýý»ñ ù³ïáõóáõ áýï»ñáõãûáõýý»ñç ùñó³ïó»éáõ ï³ñáõáõãû³ý ³ëïç׳ýá ³ûëûñí³ ù»í ¨ ñ³·»ó³í ßáõï³ý»ñáõù: ²ûë ³ßë³ï³ýùý áõëáõùý³ëçñáõù ¨ ³é³ç³ñïáõù ¿ ýáñ ù»ãá¹³µ³ýáõãûáõý í³ñáñ¹ç å³ù³ýù³ý ï¨áõáõãûáõýá ñ³ßí³ñï»éáõ ñ³ù³ñ` ñçùýí»éáí gg-ç 2018 ãí³ï³ýç áõõ¨áñáõãûáõýý»ñçó ñ³í³ùí³í ïíû³éý»ñç íñ³: gg-ý ïñ³ýëåáñï³ûçý í³é³ûáõãûáõýý»ñ ù³ïáõóáõ áýï»ñáõãûáõý ¿ ¨ ý»ñï³ûáõùë û·ï³·áñíáõù ¿ the open source routing machine (osrm) ñ³ù³ï³ñ·á, áñá ï³ýë³ï»ëù³ý å³ù³ý³ï ³ßë³ïáõù ¿ ¿³ï³ý ëë³éý»ñáí: ²ûë ³ßë³ï³ýùáõù óáõûó ¿ ïñí³í, áñ ýßí³í ³é³ç³¹ñ³ýùç ñ³ù³ñ ³ûýåçëç ³é·áñçãùý»ñç çñ³ï³ý³óáõùá, çýãåçëçù »ý xgboost, catboost ¨ ëáñá ý»ûñáý³ûçý ó³ýó»ñá, ïµ³ñ»é³íç å³ù³ý³ïç ×ß·ñïáõãûáõýá: ²ßë³ï³ýùáõù ùýý³ñïíáõù »ý ûáõñ³ù³ýãûáõñ ùá¹»éç ³é³í»éáõãûáõýý»ñá ¨ ã»ñáõãûáõýý»ñá ¨ ³ûýáõñ»ï»õ ¹çï³ñïíáõù ¿ ïáõï³ï³ûçý ³é·áñçãùç ï³ñµ»ñ³ïá, áñá ùç³íáñáõù ¿ ùç ù³ýç ùá¹»éý»ñá ù»ïç ù»ç: ²ûëåçëáí, ³û¹ ï»ëýçï³ûç ïçñ³éù³ùµ ëï³óí³í í»ñçý³ï³ý ³ñ¹ûáõýùý»ñá óáõûó »ý ïí»é, áñ ùçççý ù³é³ïáõë³ûçý ß»õáõùá gg-áõù ïçñ³éíáõ ý³ëáñ¹ ùá¹»éç ýï³ïù³ùµ ýí³½»é ¿ 54%-áí: ´³ý³éç µ³é»ñ` gg, í³ñáñ¹ç å³ù³ýù³ý ï¨áõáõãûáõý, osrm, xgboost, catboost, ý»ûñáý³ûçý ó³ýó»ñ: 3 8 estimating time of driver arrival with gradient boosting algorithms and deep neural networks äîâîëüíî ñóùåñòâåííûå îøèáêè. ýòà ðàáîòà ïîêàçûâàåò, ÷òî ðåàëèçàöèÿ àëãîðèòìîâ, òàêèõ êàê xgboost,catboost è neural networks äëÿ óêàçàííîé çàäà÷è, ïîâûñèò òî÷íîñòü ðàñ÷åòà. â ñòàòüå îáñóæäàþòñÿ ïðåèìóùåñòâà è íåäîñòàòêè êàæäîé ìîäåëè, à çàòåì ðàññìàòðèâàåòñÿ ïðîèçâîäèòåëüíîñòü àëãîðèòìà ñóììèðîâàíèÿ, êîòîðûé îáúåäèíÿåò íåñêîëüêî ìîäåëåé â îäíó. òàêèì îáðàçîì, îêîí÷àòåëüíûå ðåçóëüòàòû, ïîëó÷åííûå ñ èñïîëüçîâàíèåì ýòèõ ìåòîäîâ, ïîêàçàëè, ÷òî ñðåäíÿÿ êâàäðàòè÷åñêàÿ îøèáêà óìåíüøèëàñü íà 54% ïî ñðàâíåíèþ ñ òåêóùåé ìîäåëüþ gg. êëþ÷åâûå ñëîâà: gg, ðàñ÷åòíîå âðåìÿ ïðèáûòèÿ âîäèòåëÿ, osrm, xgboost, catboost, neural networks 04_sergoyan_53 (1) 04 4_mossine-dominant_35-40.dvi mathematical problems of computer science 49, 35{40, 2018. degr ee sequences and dominating cycles in 2-connected gr aphs mo s s in e s . k o u la kz ya n institute for informatics and automation problems of nas ra e-mail: zhora@ipia.sci.am abstract let g be a graph on n vertices and minimum degree ± with degree sequence ± = d1 · d2 · ::: · dn. the minimum degree sum of two nonadjacent vertices in g is denoted by ¾2. let c be the circumference the order (the number of vertices) of a longest cycle, and p be the order of a longest path in g. in 1952, dirac proved: (1) if g is a 2-connected graph, then c ¸ minfn; 2d1g; (2) every graph with d1 ¸ n2 is hamiltonian. recently, these results were improved by nikoghosyan in terms of degree sequences: (3) if g is a 2-connected graph, then c ¸ minfn; d± + d±+1g; (4) every graph with d± + d±+1 ¸ n is hamiltonian. in this paper we present the dominating cycle versions of these theorems: (i) if g is a 2-connected graph, then either c ¸ d± + d¾2 or c ¸ p ¡ 1 (that is g has a dominating cycle); (ii) every 2-connected graph with d± + d±+1 ¸ p ¡ 1 has a dominating cycle. the results are sharp. keywords: hamilton cycle, dominating cycle, circumference, minimum degree, degree sums, degree sequence. 1 . in t r o d u c t io n w e c o n s id e r o n ly ¯ n it e u n d ir e c t e d g r a p h s wit h n e it h e r lo o p s n o r m u lt ip le e d g e s . a g o o d r e fe r e n c e fo r a n y u n d e ¯ n e d t e r m s is [1 ]. th e s e t o f ve r t ic e s o f a g r a p h g is d e n o t e d b y v ( g) , a n d t h e s e t o f e d g e s b y e ( g) . l e t n b e t h e o r d e r ( t h e n u m b e r o f ve r t ic e s ) o f g, c t h e o r d e r o f a lo n g e s t c yc le ( c a lle d c ir c u m fe r e n c e ) in g a n d p t h e o r d e r o f a lo n g e s t p a t h . th e m in im u m d e g r e e s u m o f t wo n o n a d ja c e n t ve r t ic e s in g is d e n o t e d b y ¾2. in p a r t ic u la r , t h e m in im u m d e g r e e in g is d e n o t e d b y ±. l e t d1; d2; :::; dn b e t h e d e g r e e s e qu e n c e in g wit h ± = d1 · d2 · ::: · dn. w e u s e n ( v ) t o d e n o t e t h e s e t o f a ll n e ig h b o r s o f a ve r t e x v a n d d ( v ) = jn ( v ) j t o d e n o t e t h e d e g r e e o f ve r t e x v. a g r a p h g is h a m ilt o n ia n if g c o n t a in s a h a m ilt o n c yc le , t h a t is a s im p le s p a n n in g c yc le . a c yc le c o f g is c a lle d a d o m in a t in g c yc le if e ve r y e d g e o f g h a s a t le a s t o n e o f it s e n d ve r t ic e s o n c, o r , e qu iva le n t ly, if g ¡ v ( c ) c o n t a in s n o e d g e s . w e wr it e a c yc le q wit h a g ive n o r ie n t a t io n b y ¡! q . fo r x; y 2 v ( q) , we d e n o t e b y x ¡! q y t h e s u b p a t h o f q in t h e c h o s e n d ir e c t io n fr o m x t o y. fo r x 2 v ( q ) , we d e n o t e t h e s u c c e s s o r a n d t h e p r e d e c e s s o r o f x o n ¡! q ( if s u c h ve r t ic e s e xis t ) b y x+ a n d x¡, r e s p e c t ive ly. fo r u µ v ( q) , we d e n o t e u + = fu+ju 2 ug a n d u¡ = fu¡ju 2 ug. w e s a y t h a t t h e ve r t e x 3 5 3 6 degree sequences and dominating cycles in 2-connected graphs z1 p r e c e d e s t h e ve r t e x z2 o n a p a t h ¡! q if z1, z2 o c c u r o n ¡! q in t h is o r d e r a n d in d ic a t e t h is r e la t io n s h ip b y z1 á z2. w e will wr it e z1 ¹ z2 wh e n e it h e r z1 = z2 o r z1 á z2. l e t ¡! p = v1v2:::vp b e a lo n g e s t p a t h in g. cle a r ly, n ( v1 ) [ n ( vp ) µ v ( p ) . a vin e o f le n g t h m o n p is a s e t fli = wi ¡! l izi : 1 · i · mg o f in t e r n a lly-d is jo in t p a t h s s u c h t h a t ( a ) v ( li ) \ v ( p ) = fwi; zig ( i = 1 ; :::; m) , ( b ) v1 = w1 á w2 á z1 ¹ w3 á z2 ¹ w4 á ::: ¹ wm á zm¡1 á zm = vp o n p . th e fo llo win g r e s u lt g u a r a n t e e s t h e e xis t e n c e o f a t le a s t o n e vin e o n ¡! p in a 2 -c o n n e c t e d g r a p h . t he vine lemma [2 ]. if g is a 2 -c o n n e c t e d g r a p h a n d p a p a t h in g, t h e n t h e r e is a t le a s t o n e vin e o n p . in 1 9 5 2 , d ir a c [2 ] o b t a in e d t h e ¯ r s t lo we r b o u n d fo r t h e c ir c u m fe r e n c e fo r 2 -c o n n e c t e d g r a p h s a n d t h e ¯ r s t s u ± c ie n t c o n d it io n fo r h a m ilt o n c yc le s in t e r m s o f m in im u m d e g r e e ±. t heor em a [2 ]. if g is a 2-connected graph, then c ¸ m in fn; 2 ±g = m in fn; 2 d1g. t heor em b [2 ]. e ve r y g r a p h wit h ± = d1 ¸ n2 is h a m ilt o n ia n . th e o r e m s a a n d b we r e im p r o ve d in [3 ] in t e r m s o f d e g r e e s e qu e n c e s . t heor em c [3 ]. if g is a 2-connected graph, then c ¸ m in fn; d± + d±+1g. t heor em d [3 ]. e very graph with d± + d±+1 ¸ n is hamiltonian. in t h is p a p e r we p r e s e n t t h e d o m in a t in g ve r s io n s o f th e o r e m s c a n d d . p r oposition 1 [4 ]. l et g be a connected graph with c ¸ p ¡ 1 . then every longest cycle in g is a dominating cycle. t heor em 1. if g is a 2-connected graph, then c ¸ m in fp ¡ 1 ; d± + d¾2g. th e n e xt r e s u lt fo llo ws fr o m th e o r e m 1 im m e d ia t e ly a s a s u ± c ie n t c o n d it io n fo r t h e e xis t e n c e o f a d o m in a t in g c yc le . t heor em 2. if g is a 2-connected graph with d± + d¾2 ¸ p ¡ 1 , then c ¸ p ¡ 1 . if g = k±+1 + k±, t h e n d± = ±, d2± = d¾2 = 2 ± = ¾2 a n d c = 2 ± = ¾2 = p ¡ 1 . th is g r a p h e xa m p le s h o ws t h a t t h e c o n c lu s io n " e it h e r c ¸ d± + d¾2 o r c ¸ p ¡ 1 " in th e o r e m 3 c a n n o t b e r e p la c e d b y " e it h e r c ¸ d± + d¾2 o r c ¸ p " . n e xt , le t g = ±k2 + k±¡1. th e n d± = d2± = d¾2 = ±, d2±+1 = d¾2+1 = 3 ± ¡ 2 a n d c = 3 ± ¡ 3 = p ¡ 2 . th is g r a p h e xa m p le s h o ws t h a t t h e c o n c lu s io n " e it h e r c ¸ d± + d¾2 o r c ¸ p¡ 1 " in th e o r e m 1 c a n n o t b e r e p la c e d m. koulakzyan 3 7 b y " e it h e r c ¸ d± + d¾2+1 o r c ¸ p ¡ 1 " . th u s , th e o r e m 1 is b e s t p o s s ib le . 2 . p r o o fs p r oof of t heor em 1. l e t ¡! p = v1v2:::vp b e a lo n g e s t p a t h in g. cle a r ly, n ( v1 ) [ n ( vp ) µ v ( p ) : a s s u m e t h a t ( a 1 ) p is c h o s e n s o t h a t d( v1 ) is m a xim u m . ( a 2 ) p is c h o s e n s o t h a t d( vp ) is m a xim u m s u b je c t t o ( a 1 ) . l e t x1; x2; :::; xt b e t h e e le m e n t s o f n ( v1 ) o c c u r r in g o n ¡! p in a c o n s e c u t ive o r d e r , wh e r e t = d ( v1 ) ¸ ±. n e xt , le t y1; y2; :::; yf b e t h e e le m e n t s o f n ( vp ) o c c u r r in g o n ã¡ p in a c o n s e c u t ive o r d e r . if e it h e r xt = vp o r yf = v1, t h e n we c a n fo r m a p a t h lo n g e r t h a n p , a c o n t r a d ic t io n . h e n c e , xt 6= vp a n d yf 6= v1. ob s e r ve t h a t fo r e a c h i 2 f1 ; 2 ; :::; tg, x¡i ã¡ p v1xi ¡! p vp is a lo n g e s t p a t h in g, im p lyin g t h a t n ( x¡i ) µ v ( p ) ( i = 1 ; 2 ; :::; t) : b y a s ym m e t r ic a r g u m e n t , n ( y+i ) µ v ( p ) ( i = 1 ; 2 ; :::; f ) : case 1. xt ¹ yf . l e t fli = wi ¡! l izi : 1 · i · mg b e a vin e o f m in im a l le n g t h m o n ¡! p . s in c e p is a lo n g e s t p a t h in g, we h a ve l1; lm 2 e ( g) . n e xt , s in c e m is m in im a l, we h a ve xt á z2, xt á w3 a n d wm¡1 á yf , zm¡2 á yf . ch o o s e z¤1 2 v ( p ) s u c h t h a t w2 á z¤1 a n d jv ( w2 ¡! p z¤1 ) j is m in im a l. a n a lo g o u s ly, c h o o s e w¤m 2 v ( p ) s u c h t h a t w¤m á zm¡1 a n d jv ( w¤m ¡! p zm¡1 ) j is m in im a l. p u t h = p [ m¡1[ i=2 li [ fv1z¤1 ; vpw¤mg: b y d e le t in g t h e fo llo win g p a t h s wi ¡! p zi¡1 ( i = 3 ; 4 ; :::; m ¡ 1 ) ; w2 ¡! p z¤1; w ¤ m ¡! p zm¡1 fr o m h ( e xc e p t fo r t h e ir e n d ve r t ic e s ) , we o b t a in a c yc le c wit h a t le a s t d( v1 ) + d ( vp ) + 1 ve r t ic e s . s in c e t h e ve r t ic e s x¡1 ; x ¡ 2 ; :::; x ¡ t ; y + 1 ; y + 2 ; :::; y + f a r e p a ir wis e d is t in c t , we h a ve d( v1 ) = m a xfd ( x¡1 ) ; d( x¡2 ) ; :::; d( x¡t ) ; d( y+1 ) ; d ( y+2 ) ; :::; d( y+f ) g 3 8 degree sequences and dominating cycles in 2-connected graphs ¸ m a xfd1; d2; :::; dt+fg = dt+f = dd(v1)+d(vp) ¸ d¾2 ; d( vp ) = m a xfd( y+1 ) ; d( y+2 ) ; :::; d( y+f ) g ¸ m a xfd1; d2; :::; dfg = df = dd(vp) ¸ d±; im p lyin g t h a t c ¸ d( v1 ) + d( vp ) + 1 > d± + d¾2 : case 2. yf á xt. case 2.1. n ( v1 ) \ n + ( vp ) 6= ;. l e t v 2 n ( v1 ) \ n + ( vp ) , t h a t is v1v; vpv¡ 2 e ( g) . s in c e v1v ¡! p vpv ¡ã¡p v1 is a c yc le o f o r d e r p, a n d g is c o n n e c t e d , e it h e r p < jv ( g ) j, a n d we c a n fo r m a p a t h lo n g e r t h a n p ( a c o n t r a d ic t io n ) o r p = jv ( g ) j, im p lyin g t h a t c = p. case 2.2. n ( v1 ) \ n + ( vp ) = ;. case 2.2.1. n¡ ( v1 ) \ n + ( vp ) 6= ;. l e t v 2 n¡ ( v1 ) \ n + ( vp ) , t h a t is z = x¡i = y+j fo r s o m e i 2 f 1 ; :::; tg a n d j 2 f 1 ; :::; fg. cle a r ly, v1z +¡!p vpz¡ ã¡ p v1 is a c yc le o f o r d e r p ¡ 1 , t h a t is c ¸ p ¡ 1 . case 2.2.2. n¡ ( v1 ) \ n + ( vp ) = ;. s in c e yf á xt, we c a n c h o o s e t wo in t e g e r s 1 · a · t a n d 1 · b · f s u c h t h a t yb á xa, a n d jv ( yb ¡! p xp ) j is m in im u m . p u t c = v1xa ¡! p vpyb ã¡ p v1: cle a r ly, ( n ( v1 ) [ n + ( vp ) ) ¡ y+b µ v ( c ) : h e n c e , c ¸ jv ( c ) j ¸ j ( n ( v1 ) [ n + ( vp ) ) ¡ y+b j + jfv1gj = jn ( v1 ) j + jn + ( vp ) j = jn ( v1 ) j + jn ( vp ) j = d( v1 ) + d ( vp ) : b y t h e h yp o t h e s is , t h e fo llo win g ve r t ic e s x¡1 ; x ¡ 2 ; :::; x ¡ t ; y + 1 ; y + 2 ; :::; y + f a r e p a ir wis e d is t in c t . b y ( a 1 ) a n d ( a 2 ) , d ( v1 ) = m a xfd ( x¡1 ) ; d( x¡2 ) ; :::; d( x¡t ) ; d( y+1 ) ; d( y+2 ) ; :::; d( y+f ) g ¸ m a xfd1; d2; :::; dt+fg = dt+f = dd(v1)+d(vp) ¸ d¾2 ; d ( vp ) = m a xfd ( y+1 ) ; d ( y+2 ) ; :::; d( y+f ) g ¸ m a xfd1; d2; :::; dfg = df = dd(vp) ¸ d±; im p lyin g t h a t c ¸ d ( v1 ) + d( vp ) ¸ d± + d¾2: m. koulakzyan 3 9 refer ences [1 ] j. a . b o n d y a n d u . s . r . mu r t y, graph theory with applications, ma c m illa n , l o n d o n a n d e ls e vie r , n e w y o r k 1 9 7 6 . [2 ] g. a . d ir a c , \ s o m e t h e o r e m s o n a b s t r a c t g r a p h s " , p roc. l ondon, m ath. soc., vo l. 2 , p p . 6 9 -8 1 , 1 9 5 2 . [3 ] zh . g. n iko g h o s ya n , \ d e g r e e s e qu e n c e s a n d lo n g c yc le s in gr a p h s " , a r x iv:1 7 1 1 .0 4 1 3 4 ( 2 0 1 7 ) 9 p a g e s . [4 ] k . oz e ki a n d t. y a m a s h it a , \ l e n g t h o f lo n g e s t c yc le s in a g r a p h wh o s e r e la t ive le n g t h is a t le a s t t wo " , graphs and combin., vo l. 2 8 , p p . 8 5 9 -8 6 8 , 2 0 1 2 . submitted 22.08.2017, accepted 14.12.2017. ²ëïç׳ý³ûçý ñ³çáñ¹³ï³ýáõãûáõýý»ñ ¨ ¹áùçý³ýï óçïé»ñ 2-ï³å³ïóí³í ·ñ³ýý»ñáõù ø. øáõé³ù½û³ý ²ù÷á÷áõù ¸çóáõù g-ý n ·³·³ã³ýç ·ñ³ý ¿ ± ýí³½³·áõûý ³ëïç׳ýáí ¨ ± = d1 · d2 · · dn ³ëïç׳ý³ûçý ñ³çáñ¹³ï³ýáõãû³ùµ: ²ù»ý³»ñï³ñ óçïéç »ñï³ñáõãûáõýá ýß³ý³ïíáõù ¿ c-áí, çëï ³ù»ý³»ñï³ñ ßõã³ûç »ñï³ñáõãûáõýá (·³·³ãý»ñç ù³ý³ïá) p-áí: 1952-çý ¸çñ³ïý ³å³óáõó»ó. (1) »ã» g-ý 2-ï³å³ïóí³í ·ñ³ý ¿, ³å³ c ¸ m in fn; 2 d1g, (2) ï³ù³û³ï³ý ·ñ³ý, áñá µ³í³ñ³ñáõù ¿ d1 ¸ n2 å³ûù³ýçý, ñ³ùçéïáýû³ý ¿: ì»ñç»ñë ³ûë ³ñ¹ûáõýùý»ñá é³í³óí»óçý ³ëïç׳ý³ûçý ñ³çáñ¹³ï³ýáõãûáõýý»ñç 黽íáí` (3) ï³ù³û³ï³ý 2-ï³å³ïóí³í ·ñ³ýáõù c ¸ m in fn; d± + d±+1g, (4) d± + d±+1 ¸ n å³ûù³ýçý µ³í³ñ³ñáõ ï³ù³û³ï³ý ·ñ³ý ñ³ùçéïáýû³ý ¿ (zh.g. nikoghosyan, degree sequences and long cycles in graphs, arxiv:1711.04134): ü»ñï³ ³ßë³ï³ýùáõù µ»ñíáõù »ý (3) ¨ (4) ã»áñ»ùý»ñç ï³ñµ»ñ³ïý»ñá ¹áùçý³ýï óçïé»ñç ñ³ù³ñ: êï³óí³í ³ñ¹ûáõýùý»ñá é³í³óý»éç ã»ý: 4 0 degree sequences and dominating cycles in 2-connected graphs ñòåïåííûå ïîñëåäîâàòåëüíîñòè è äîìèíàíòíûå öèêëû â 2-ñâÿçíûõ ãðàôàõ ì. êóëàêçÿí àííîòàöèÿ ïóñòü g ÿâëÿåòñÿ n âåðøèííûì ãðàôîì ñ ìèíèìàëüíîé ñòåïåíüþ ± è ñòåïåííîé ïîñëåäîâàòåëüíîñòüþ ± = d1 · d2 · · dn. äëèíà äëèííåéøåãî öèêëà îáîçíà÷àåòñÿ ÷åðåç c, à äëèíà äëèííåéøåé öåïè (÷èñëî å¸ âåðøèí) ÷åðåç p. â 1952 ãîäó äèðàê äîêàçàë: (1) åñëè g ÿâëÿåòñÿ 2-ñâÿçíûì ãðàôîì, òî c ¸ m in fn; 2 d1g; (2) åñëè ãðàô óäîâëåòâîðÿåò óñëîâèþ d1 ¸ n2 , òî îí ÿâëÿåòñÿ ãàìèëüòîíîâûì. íåäàâíî ýòè ðåçóëüòàòû áûëè óëó÷øåíû â òåðìèíàõ ñòåïåííûõ ïîñëåäîâàòåëüíîñòåé: (3) åñëè g ÿâëÿåòñÿ 2-ñâÿçíûì ãðàôîì, òî c ¸ m in fn; d± + d±+1g; (4) åñëè ãðàô óäîâëåòâîðÿåò óñëîâèþ d± + d±+1 ¸ n, òî îí ÿâëÿåòñÿ ãàìèëüòîíîâûì (zh.g. nikoghosyan, degree sequences and long cycles in graphs, arxiv:1711.04134). â íàñòîÿùåé ðàáîòå ïðåäñòàâëÿþòñÿ âåðñèè òåîðåì (3) è (4) äëÿ äîìèíàíòíûõ öèêëîâ. ïîëó÷åííûå ðåçóëüòàòû íåóëó÷øàåìû. microsoft word intelligent agent serve.doc mathematical problems of computer science 25, 2006, 64-70. 1 the research is supported partly by netint: 04-77-7173 project, http://www.intas.be and state principal program of armenia on scientific computations. 64 intelligent agent server (netint) system david a. karapetyan institute for informatics and automation problems of nas of ra e-mail david@dm-lab.sci.am abstract the paper describes intelligent agent server system, which employs software agents. a detailed description of software agents as well some examples of the system application, particularly intrusion detection in local networks is given. references [1] fipa (federation of intelligent physical agents home page http://www.cselt.stet.it/fipa/fipa_rationale.htm. [2] michael w. and n. jennings (1995), "agent theories, architectures, and languages: a survey," in wooldridge and jennings eds., intelligent agents, berlin: springerverlag, pp. 1-22. [3] hayes-roth, b. (1995). “architecture for adaptive intelligent systems” artificial intelligence: special issue on agents and interactivity, 72, pp. 329-365. [4] kulin a., salmonsen g. and aslanyan l., security policy adaptation reinforced through agents, international seminar “conversion potential of armenia and istc programs”, yerevan. [5] aslanyan l., markaryan k., pasic a., sahakyan h., intrusion detection intelligence, proceeding of the conference computer science & information technologies, yerevan, pp. 429-433. ´³ý³ï³ý³ûçý ³·»ýï ë»ñí»ñ ñ³ù³ï³ñ· ¸. î³ñ³å»ïû³ý ²ù÷á÷áõù ²ßë³ï³ýùá ýï³ñ³·ñáõù ¿ ´³ý³ï³ý³ûçý ³·»ýï ë»ñí»ñ ñ³ù³ï³ñ·á, áñá ñçùýí³í ¿ íñ³·ñ³ûçý ³·»ýïý»ñç íñ³: ´»ñí³í »ý íñ³·ñ³ûçý ³·»ýïý»ñç ñ³ïïáõãûáõýý»ñç ýï³ñ³·ñáõãûáõýý»ñá ¨ ëï»õíí³í ñ³ù³ï³ñ·ç ïçñ³éáõãû³ý ùç ù³ýç ûñçý³ïý»ñ, ù³ëý³íáñ³å»ë ý»ñëáõåù³ý ñ³ûïý³µ»ñáõùá ñ³ßíáõ³ï³ý éáï³é ó³ýóáõù. vahan_margaryan.dvi mathematical problems of computer science 26, 2006, 21{27. on restr iction optimal fixpoints v a h a n k . ma r g a r ya n department of informatics and applied mathematics of yerevan state university e-mail vahan80a@netsys.am abstract optimal ¯xpoints extract maximum consistent information from recursive programs. however, optimal ¯xpoints, although they always exist for a recursive operator, aren't necessarily computable. we have introduced a modi¯ed notion of the optimal ¯xpoint, where the recursive operators are restricted to computable inputs. existence results for and properties of this ¯xpoint are summarized in the article. refer ences [1 ] z. ma n n a , a . s h a m ir . th e o p t im a l a p p r o a c h t o r e c u r s ive p r o g r a m s . communications of the acm 1 1 :8 2 4 -8 3 1 [2 ] v . ma r g a r ya n . on c e r t a in p r o p e r t ie s o f c o m p u t a b le o p t im a l ¯ xp o in t s . p roceedings of csit-2003, [3 ] a . p a t t e r s o n . im p lic it p r o g r a m m in g a n d th e l o g ic o f co n s t r u c t ib le d u a lit y, p h.d . thesis, university of illinois at urbana-champaign, 1 9 9 8 ê³ñù³ý³÷³ïù³ý ûåïçù³é ³ýß³ñå ï»ï»ñç ù³ëçý ì. ø³ñ·³ñû³ý ²ù÷á÷áõù úåïçù³é ³ýß³ñå ï»ï»ñá ³é³í»é³·áõûý çýýáñù³óç³ý »ý å³ñáõý³ïáõù é»ïáõñëçí íñ³·ñç ù³ëçý: ê³ï³ûý ¹ñ³ýù ï³ñáõ »ý éçý»é áã ñ³ßí³ñï»éç: ø»ýù ý»ñùáõí»é »ýù ûåïçù³éáõãû³ý ÷á÷áëí³í ·³õ³÷³ñ, áñç ¹»åùáõù ¹çï³ñïíáõù ¿ ùç³ûý ñ³ßí³ñï»éç ýáõýïóç³ý»ñç íñ³ ûå»ñ³ïáñç ·áñíáõáõãûáõýá: ðá¹í³íáõù ý»ñï³û³óí³í »ý ³ûý ³ýß³ñå ï»ïç ·áûáõãû³ý »í ñ³ïïáõãûáõýý»ñç í»ñ³µ»ñû³é ³ñ¹ûáõýùý»ñç ³ù÷á÷áõù: 2 1 microsoft word mod petri2t.doc ìàòåìàòè÷åñêèå âîïðîñû êèáåðíåòèêè è âû÷èñëèòåëüíîé òåõíèêè 26, 2006, 48–53. 1 the research is supported partly by intas: 04-77-7173 project, http://www.intas.be and state principal program of armenia on scientific computations. 48 модифицированные сети петри, описание поведения с помощью формальных языков гоар р. петросян ереванский государственный университет аннотация в работе вводится понятие модифицированной сети петри. по заданной ксграмматике (контекстно-свободной грамматике) строится эквивалентная по порождаемому языку модифицированная сеть петри, являющася расширением стандартной сети петри с помощью сдерживаюших позиций. литература [1] питерсон д., “теория сетей петри и моделирование систем”. москва, мир, 1984г. [2] котов в. е. “ сети петри”. москва, мир, 1984г. [3] ахо а., ульман д., “теория синтаксического анализа, перевода и компиляции”. перевод под редакцией курочкина, т1-т3. [4] гордеев а. в., молчанов а. ю., “системное программное обеспечение”. учебник, санкт-петербург, 2002г. ä»ïñçç ó¨³÷áëí³í ó³ýó»ñ, í³ñùç ýï³ñ³·çñá ýáñù³é 黽áõý»ñç û·ýáõãû³ùµ ¶. ä»ïñáëû³ý ²ù÷á÷áõù ²ßë³ï³ýùáõù ùïóíáõù »ý ä»ïñçç ó¨³÷áëí³í ó³ýó»ñ ñ³ëï³óáõãûáõýý»ñá: îñí³í î² ù»ñ³ï³ýáõãû³ùµ (ïáýï»ïëïçó ³ýï³ë ù»ñ³ï³ýáõãûáõý) ï³éáõóíáõù ¿ ¹áõñë µ»ñí³í 黽íáí ñ³ù³ñå»ù ä»ïñçç ó¨³÷áëí³í ó³ýó, áñá ñ³ý¹çë³ýáõù ¿ ä»ïñçç ëï³ý¹³ñï ó³ýóç áý¹é³ûýáõùá ë³ñù³ý³÷³ïáõ ¹çñù»ñç û·ýáõãû³ùµ: microsoft word article.doc ìàòåìàòè÷åñêèå âîïðîñû êèáåðíåòèêè è âû÷èñëèòåëüíîé òåõíèêè 24, 2005, 147-157. 147 некоторые методы сжатия данных и их индексов в субд мигран с. григорян èíñòèòóò ïðîáëåì èíôîðìàòèêè è àâòîìàòèçàöèè íàí ðà e-mail mihran.grigoryan@buy.am àííîòàöèÿ в данной статье рассматриваются некоторые методы сжатия табличных данных и индексных структур в субд. приводятся сравнительные характеристики этих методов, и предлагаются их некоторые модификации. литература [1] ватолин д., ратушняк а., смирнов м., юкин в. методы сжатия данных. устройство архиваторов, сжатие изображений и видео. – м.: диалог-мифи, 2002. – 384 с. [2] alsberg p. a. space and time savings through large data base compression and dynamic restructuring. proc. ieee 63(8):1114-1122, august 1975. [3] buchsbaum a. l., caldwell d. f., church k. w., fowler g. s., and muthukrishnan s. engineering the compression of massive tables: an experimental approach. proc. 11th annual acm-siam symposium on discrete algorithms, pp. 175-184, 2000. [4] cannane a., williams h. e., and zobel j. a general-purpose compression scheme for databases. proc. ieee data compression conference, p. 519, 1999. [5] chan c.y. and ioannidis y.e. an efficient bitmap encoding scheme for selection queries. proc. acm sigmod intl' conference, philadelphia, pennsylvania, june 1999, pp. 215-226. [6] goldstein j. improved query processing and data representation techniques. a dissertation submitted in partial fulfillment of the requirements for the degree of doctor of philosophy (computer sciences) at the university of wisconsin – madison. 1999. [7] goldstein j., ramakrishnan r., and shaft u.. compressing relations and indexes. proc. ieee conf. on data engineering, orlando, fl, usa, pp. 370-379, 1998. [8] goyal k., ramamritham k., datta a., thomas h. indexing and compression in data warehouses. technical report, indian institute of technology, bombay, april 1999. [9] iyer b. r. and wilhite d. data compression support in databases. in proceedings of the 20th international conference on very large data bases, santiago, chile, pp. 695-704. 1994. [10] johnson t. performance measurements of compressed bitmap indices. proceedings of 25th international conference on very large data bases, september 7-10, 1999 (vldb'99), edinburgh, scotland, uk, pp. 278-289. [11] mysql ab (2004). mysql reference manual for version 4.0.18. [12] stockinger k. multi-dimensional bitmap indices for optimising data access within object oriented databases at cern. phd. nov. 2001. некоторые методы сжатия данных и их индексов в субд 148 îðôð – áõù ïíû³éý»ñç ¨ çý¹»ùëý»ñç ë»ëù³ý ùç ù³ýç ³é·áñçãù ø. ¶ñç·áñû³ý ²ù÷á÷áõù ²ûë ñá¹í³íáõù ¹çï³ñïíáõù »ý ùç ß³ñù ³õûáõë³ï³ûçý ïíû³éý»ñç ë»õù³ý ù»ãá¹ý»ñ: îñíáõù »ý ýñ³ýó ñ³ù»ù³ï³ï³ý µýáõã³·ñ»ñá ¨ µ»ñíáõù »ý ùç ß³ñù ï³ï³ñ»é³·áñíáõùý»ñ: d:\sbornik\...\article.dvi mathematical problems of computer science 24, 2005, 86{88. on i nter val e dge color ings of h ar ar y gr aphs h 2 n¡ 2 ; 2 n ¤ r a fa ye l r . k a m a lia n , p e t r o s a . p e t r o s ya n institute for informatics and automation problems of nas of ra e-mail rrkamalian@yahoo.com, pet petros@yahoo.com abstract the problems of existence and construction of interval edge colorings of harary graphs h2n¡2;2n are investigated. bounds are found for the possible number of colors in interval edge colorings of h2n¡2;2n. refer ences [1 ] f. h a r a r y, gr a p h th e o r y, a d d is o n -w e s le y, r e a d in g , ma , 1 9 6 9 . [2 ] m.n .s . s wa m y, k . th u la s ir a m a n , gr a p h s , n e t wo r ks a n d a lg o r it h m s , jo h n w ile y & s o n s , n e w y o r k, 1 9 8 1 . [3 ] d .b . w e s t , in t r o d u c t io n t o gr a p h th e o r y, p r e n t ic e -h a ll, n e w je r s e y, 2 0 0 1 . [4 ] r .r . k a m a lia n , in t e r va l e d g e co lo r in g s o f gr a p h s , d o c t o r a l d is s e r t a t io n , n o vo s ib ir s k, 1 9 9 0 . [5 ] r .r . k a m a lia n , p .a . p e t r o s ya n , on lo we r b o u n d fo r w ( k2n ) , ma t h e m a t ic a l p r o b le m s o f co m p u t e r s c ie n c e , v o l. 2 3 , y e r e va n , 2 0 0 4 , p p .1 2 7 -1 2 9 . ¤the work was partially supported by 04.10.31 target program of ra. 8 6 r.r. kamalian, p.a. petrosyan 8 7 ê³é³ñçç h2n-2,2n ·ñ³ýý»ñç ùçç³ï³ûù³ûçý ïáõ³ûçý ý»ñïáõùý»ñç ù³ëçý è.è. ø³ù³éû³ý, ä.². ä»ïñáëû³ý ²ù÷á÷áõù ¸çï³ñïí³í »ý ê³é³ñçç h2n¡2;2n ·ñ³ýý»ñç ùçç³ï³ûù³ûçý ïáõ³ûçý ý»ñïáõùý»ñç ·áûáõãû³ý ¨ ï³éáõóù³ý ñ³ñó»ñ, ¨ ëï³óí³í »ý ·ý³ñ³ï³ï³ýý»ñ ³û¹ ý»ñïáõùý»ñáõù û·ï³·áñííáõ ·áõûý»ñç ñý³ñ³íáñ ãíç ñ³ù³ñ: mathematical problems of computer science 58, 32–41, 2022. doi:10.51408/1963-0090 udc 519.218 comparison of model-free algorithms for clustering garch processes garik l. adamyan yerevan state university e-mail: garik.adamyan@ysu.am abstract in this paper, we evaluate several model-free algorithms for clustering time series datasets generated by garch processes. in extensive experiments, we generate synthetic datasets in different scenarios. then, we compare k-means (for euclidian and dynamic time warping distance), k-shape, and kernel k-means models with different clustering metrics. several experiments show that the k-means model with dynamic time warping distance archives comparably better results. however, the considered models have significant shortcomings in improving the clustering accuracy when the amount of information (the minimum length of the time series) increases, and in performing accurate clustering when data is unbalanced or clusters are overlapping. keywords: time series clustering, garch process, dynamic time warping, k-means, k-shape. article info: received 02 may 2022; received in revised form 20 july 2022; accepted 29 september 2022. 1. introduction time series clustering has been used in diverse scientific disciplines to discover patterns and extract valuable information from complex and massive datasets. these algorithms have a wide range of applications in many research areas, for instance, in finance, biology, and robotics [1]. time series clustering approaches can be classified as feature-based, shape-based, and model-based [1]. it is noteworthy that these methods are based on dissimilarity measures on time series data, according to which the time series data points are grouped by some clustering method (for instance, pam). in general, shape-based methods use linear and non-linear transformations to align time series samples and calculate dissimilarity measures on aligned samples. additionally, shape-based algorithms process the time series data directly without making any statistical assumptions about the underlying data generating processes. on the contrary, model-based methods make statistical assumptions on time series generating processes. in general, modelbased approaches assume that time series samples are generated from specific models (for 32 g. adamyan 33 instance, arima [2], mixtures of arimas [3]). time series samples are transformed into fitted models, and then a suitable distance and a clustering algorithm are applied to the estimated model parameters. although several benchmarking results on different real-world datasets for nonparametric clustering methods can be found in ([4], [5], [6]), the comparison of nonparametric clustering methods on time series data generated from garch processes is not well studied. in this paper, we are interested in non-parametric models evaluation of time series data generated from the well-known garch process, which is the actual choice for modeling the volatility of returns on financial assets. we simulate multiple garch models with different data generating scenarios and compare several non-parametric time series clustering models. motivated by [4], for comparison we choose well-known partition-based time series clustering models: k-means, k-means with dynamic time warping and dtw barycenter averaging, k-shape and kernel k-means models. furthermore, we can find open-source implementations of these algorithms [7]. although the main focus in the field of time series clustering comparison remains clustering accuracy metrics, in this work we also explore a number of other challenges of model-free methods. in particular, we study the ability of the above-mentioned modelfree methods to cluster garch processes with imbalanced, overlapping clusters and also examine the impact of increasing information on clustering accuracy. 2. related work in time series analysis research, benchmarking and numerical comparison have been recognized as integral steps to justify theoretical results. the importance of numerical comparison is emphasized in [8], where the authors reimplemented many time-series classification algorithms and compared them in 50 real-world datasets. the authors note that most reported methods have insignificant improvements regarding the variance of the evaluation metrics. this empirical evidence reclaimed the statement of the importance of the time series benchmark datasets and the empirical evaluation of the suggested methods. among the works that compare time series clustering models based on real-world datasets, we can mention ([4], [5], [6]) works. in [4], authors compare several partition, density, and hierarchical clustering methods to cluster all time series datasets available in the university of california riverside (ucr) archive [9]. they conclude that the overall performance of the eight compared algorithms is quite similar with high dependence on the evaluation dataset. the method of comparing time series clustering algorithms with synthetic, generated datasets also attracts a lot of attention among scholars. in addition to the actual clusters being known, this comparison method gives additional flexibility to examining the behavior of algorithms in different situations. in particular, scholars discussed the difference between stationary and non-stationary time series [10], the presence of noise in time series samples [11], the presence of noise clusters in time series dataset [11]. 34 comparison of model-free algorithms for clustering garch processes 3. clusters of garch the garch process is introduced in [12] for statistical modeling of the volatility of returns on financial assets. the garch model has many extensions such as asymmetric garch [13], threshold garch [14]. the garch(p,q) model is defined as follows: yt = µt + ϵt ϵt = σtet, where et i.i.d e(et) = 0, var(et) = 1, σ2t = ω + p∑ i=1 αiϵ 2 t−i + q∑ j=1 βjσ 2 t−j, where ω > 0, αi ≥ 0, i = 1, 2, ..., p, βj ≥ 0, i = 1, 2, ..., q. the garch(p, q) model admits a strictly stationary solution with a finite variance if and only if p∑ i=1 αi + q∑ j=1 βj < 1. (1) moreover, this strictly stationary solution is also unique. [15] for the evaluation of non-parametric models, we chose constant zero mean specification for the garch model because it is advised to standardize input data prior to clustering. in addition, we choose the innovations et as standard gaussian innovations. so µt = 0 and et ∼ n(0,1). in order to measure the clustering accuracy, we need to define the ground truth clusters of garch processes. let n, k, t ∈ n where k is the number of clusters, n is the number of samples and t is the time sample size of each series. in this paper, we consider samples with a fixed time size t, because some of the models (ex. km-e) support samples with fixed length. we denote by p i = (ω, α1, α2, ..., αpi, β1, β2, ..., βqi) the vector of all parameters for the given garch(pi,qi) model. let {p i}ki=1 be a family of garch process parameters, where k is a number of clusters. assume that each p i(i = 1, 2, ..., k) is unique and all the parameters satisfy (1) in order to provide a strict stationary solution of the corresponding model. we are given n samples of time series yi = {yit}ti=1, where each sample is generated from one of the k garch processes. definition 1. we say that yi and yj samples are from the same cluster if they are generated from the same garch process. in other words, a cluster of garch processes is a set of samples that are generated with the same parameters. the uniqueness of the parameters p i and definition 1 imply that the given sample belongs to exactly one cluster. g. adamyan 35 4. evaluation models for evaluation, we choose well-known non-parametric time series clustering models such as k-means with euclidean (km-e) and dynamic time warping metrics (km-dtw), k-shape, and kernel k-means with fast global alignment kernel (kkm-gak) models. km-e uses euclidean distance, for cluster assignment and means averaging for the barycenter (centroid) computation. it is known that the euclidean distance metric is not the most accurate metric for measuring time series similarities. firstly, to use euclidean distance, we need to take into account the order of elements in the time series; secondly, the euclidean distance does not consider a phase shift between two curves or a length difference between the series. in this paper, we consider this model for comparison with more complex approaches. km-dtw uses dynamic time warping [16] for cluster assignment and dtw barycenter averaging (dba)[17] algorithm for averaging time series within the same cluster. k-shape [18] is a partitional clustering algorithm that relies on an iterative refinement procedure similar to the one used in k-means. to measure the distance between time series, k-shape uses a normalized version of the cross-correlation measure to consider the shapes of time series while comparing them. during the iterative procedure, this model minimizes the sum of squared distances between the sequences of time series. kernel k-means[19] is an alternative clustering algorithm that uses kernel functions as a nonlinear mapping from the input space to a higher dimensional space. by using kernels, kernel k-means can separate clusters in higher dimensional space, even if the input data is not non-linearly separable in the input space. for treating time series data, practitioners usually used global alignment kernels [20]. we will refer to this algorithm kkm-gak. the problem is to generate synthetic datasets and evaluate non-parametric models for clustering time series processes generated by the garch model. 5. assessment metrics in practice, the use of clustering methods is due to working with unlabeled datasets. as a result, we can find evaluation metrics that can evaluate clustering models without having labeled data. these types of metrics are called internal. by the method of our data generating process, we can use external measures, which assume that ground truth labels are available. examples of this type of metrics are the rand index (ri) [21], the adjusted rand index (ari) [22], the adjusted mutual information (ami)[23]. following the evaluation made in [4] in our study, we choose the adjusted rand index, because the values of this metric are consistently low for random cluster assignments and do not depend on the number of clusters. 6. experiments to evaluate non-parametric models, we simulate random datasets with different setups. in the first experiment, we measure the ability of the models of clustering different numbers of clusters. for this purpose, we generate datasets for 2, 4, 8, and 10 clusters, respectively. for each number of clusters, we generate random parameter families, which satisfy (1) for guaranteeing a unique and stationary solution of processes. for the purpose of generating a family of parameters, we constrain the maximum length of p and q by 5. this constraint is inherited from the common choice of garch models with fewer parameters. for every 36 comparison of model-free algorithms for clustering garch processes parameter vector p i (cluster), we generate samples for the given cluster and separate them into training and testing parts (30% testing) and repeat this process for averaging purposes. in table 1, we present the results of the first experiment evaluated with the ami metric. we can see that the km-dtw model outperforms other models. in the second experiment, table 1: ami score for different n clusters n clusters km-e km-dtw k-shape kkm-gak 2 0.003+-0.001 0.325+-0.403 0.004+-0.009 0.003+-0.002 4 0.004+-0.001 0.463+-0.129 0.02+-0.007 0.002+-0.001 6 0.018+-0.016 0.578+-0.151 0.043+-0.021 0.001+-0.0005 8 0.006+-0.003 0.498+-0.077 0.005+-0.011 0.001+-0.0005 10 0.005+-0.01 0.624+-0.03 0.062+-0.022 0.0001+-0.00005 we measure the clustering quality in scenarios when the amount of information increases. we generate datasets with 5 clusters and 100 samples in each cluster. we set t = 1000 and consider 5 intervals on the time axis. we train and evaluate models in the first interval and consequently add information. from the second experiment, we can see that the km-dtw model outperforms other models, but we do not observe increased accuracy as a result of adding information. there is a significant increase in the accuracy of the km-dtw model when the number of samples increases from 200 to 400, but further increases in the number of samples do not improve the accuracy of the model. the k-shape model also shows a slight improvement in accuracy when the number of samples increases from 800 to 1000. given that model-based methods rely on ml/quasy ml estimates of the parameters of garch models and also the asymptotic properties of these estimates, this experiment may suggest that model-based methods have the potential to increase clustering accuracy as information increases. the results of the second experiment are displayed in fig. 1. fig. 1. ami for different time intervals. g. adamyan 37 fig. 2 shows the results of the third experiment. in this experiment, we measure the ability of the km-dtw model to cluster an imbalanced dataset. for the fairness of the experiment, we generate time series samples with the garch(1,1) process and ensure that parameters satisfy (1). in addition, we constrain the l2 norm of generated parameters to obtain non-overlapping clusters. we generate a dataset with different sample ratios and increase the ratio to 1. in the figure, we can observe that the best model for other experiments km-dtw is dependent on cluster imbalance. this experiment shows that the claim made in [24] that centroid-based methods should be adapted to unbalanced scenarios also holds in the domain of time series clustering. fig. 2. results for clustering imbalanced dataset. moreover, we measure the effect of the l2 norm of generated parameters in clustering accuracy. we generate parameters for garch(1,1) process so that the parameters satisfy the current restriction on the l2 norm. throughout the experiment, we increase the bounds of the l2 norm. during each step, we generate a balanced dataset with t = 500, c = 2, and 100 samples per cluster. we train models ten times for averaging purposes. we can observe that the km-dtw model depends on clusters overlapping and increasing the bounds of parameters l2 norm results in improvement of ami. this problem is directly related to the ability of the similarity measure used in the km-dtw algorithm to distinguish realizations of the garch process with parameters that are close to each other with the l2 norm. 7. conclusion and future work in this work, several non-parametric clustering algorithms for clustering time series datasets generated by garch processes are evaluated. we generate multiple datasets and conduct multiple experiments to evaluate the k-means (with euclidean and dynamic time warping distance), k-shape, and kernel k-means models. in the first experiment, we evaluate the ability of models to cluster different numbers of clusters. the results of the first experiment 38 comparison of model-free algorithms for clustering garch processes fig. 3. garch parameters vector l2 norm versus ami score. are displayed in table 1. in the second experiment, we measure the clustering quality in the scenarios when the amount of information increases. we generate a dataset with 1000 time length and increase the information set. the results of the second experiment are shown in fig. 1. during both experiments, the km-dtw model shows better results. in the third experiment, we measure the ability of the km-dtw model to cluster imbalanced datasets by generating multiple datasets with imbalanced samples in the cluster. the results are provided in fig. 2. in the fourth experiment, we measure the ability of the km-dtw model to cluster overlapping clusters. we constrain the norm of the parameters of the garch(1,1) model and evaluate the km-dtw model. the experiment shows that km-dtw is highly dependent on the norm of the generated parameters. the results of the fourth experiment are shown in fig. 3. we hope that our findings can motivate scholars to examine the discussed issues related to clustering accuracy, cluster overlapping, and available information effect. we think that already designed garch-based clustering methods have the potential to overcome these problems, so it is important to conduct similar experiments to show this. moreover, as a direct application of our findings, it is worth applying clustering algorithms to the real-world financial dataset. references [1] s. aghabozorgi, a. seyed shirkhorshidi, and t. ying wah, “time-series clustering a decade review,” information systems, vol. 53, p. 1638, 2015. 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[7] r. tavenard, j. faouzi, g. vandewiele, f. divo, g. androz, c. holtz, m. payne, r. yurchak, m. rußwurm, k. kolar, and e. woods, “tslearn, a machine learning toolkit for time series data,” journal of machine learning research, vol. 21, no. 118, pp. 1–6, 2020. [8] e. keogh and s. kasetty data mining and knowledge discovery, vol. 7, no. 4, pp. 349– 371, 2003. [9] y. chen, e. keogh, b. hu, n. begum, a. bagnall, a. mueen, and g. batista, “the ucr time series classification archive,” july 2015. www.cs.ucr.edu/ eamonn/timeseriesdata/. [10] s. p. daz and j. a. vilar, “comparing several parametric and nonparametric approaches to time series clustering: a simulation study,” journal of classification, vol. 27, no. 3, pp. 333–362, 2010. [11] p. durso, l. de giovanni, r. massari, and d. di lallo, “noise fuzzy clustering of time series by autoregressive metric,” metron, vol. 71, no. 3, p. 217243, 2013. [12] t. bollerslev, “generalized autoregressive conditional heteroskedasticity,” journal of econometrics, vol. 31, no. 3, p. 307327, 1986. [13] l. hentschel, “all in the family nesting symmetric and asymmetric garch models,” journal of financial economics, vol. 39, no. 1, p. 71104, 1995. [14] j. park, j. baek, and s. hwang, “persistent-threshold-garch processes: model and application,” statistics amp; probability letters, vol. 79, no. 7, p. 907914, 2009. [15] a. m. lindner, “stationarity, mixing, distributional properties and moments of garch(p, q)processes,” handbook of financial time series, p. 4369, 2009. [16] dynamic time warping, pp. 69–84. berlin, heidelberg: springer berlin heidelberg, 2007. [17] f. petitjean, a. ketterlin, and p. ganarski, “a global averaging method for dynamic time warping, with applications to clustering,” pattern recognition, vol. 44, no. 3, pp. 678–693, 2011. [18] j. paparrizos and l. gravano, “k-shape,” proceedings of the 2015 acm sigmod international conference on management of data, 2015. [19] i. s. dhillon, y. guan, and b. kulis, “kernel k-means,” proceedings of the 2004 acm sigkdd international conference on knowledge discovery and data mining kdd ’04, 2004. [20] m. cuturi, “fast global alignment kernels,” in icml, 2011. [21] l. hubert and p. arabie, “comparing partitions,” journal of classification, vol. 2, no. 1, p. 193218, 1985. 4 0 comparison of model-free algorithms for clustering garch processes [2 2 ] j. m. s a n t o s a n d m. e m b r e c h t s , \ on t h e u s e o f t h e a d ju s t e d r a n d in d e x a s a m e t r ic fo r e va lu a t in g s u p e r vis e d c la s s i¯ c a t io n ," arti¯cial neural networks icann 2009, p . 1 7 5 1 8 4 , 2 0 0 9 . [2 3 ] s . r o m a n o , n . t h e v in h , j. c. b a ile y, a n d k . m. v e r s p o o r , \ a d ju s t in g fo r c h a n c e c lu s t e r in g c o m p a r is o n m e a s u r e s ," j ournal of m achine l earning (j m l r ), p p . 4 6 3 5 { 4 6 6 6 , vo l. 1 7 , n o . 1 , 2 0 1 6 . [2 4 ] b . k r a wc z yk, \ l e a r n in g fr o m im b a la n c e d d a t a : op e n c h a lle n g e s a n d fu t u r e d ir e c t io n s ," p rogress in arti¯cial intelligence, vo l. 5 , n o . 4 , p . 2 2 1 { 2 3 2 , 2 0 1 6 . garch åñáó»ëý»ñç ïé³ëï»ñç½³óç³ûç ñ³ù³ñ ùá¹»éý»ñçó ³ýï³ë ³é·áñçãùý»ñç ñ³ù»ù³ïáõãûáõý ¶³ñçï è. ²¹³ùû³ý ºñ¨³ýç å»ï³ï³ý ñ³ù³éë³ñ³ý, ºñ¨³ý, ð³û³ëï³ý e-mail: garik.adamyan@ysu.am ²ù÷á÷áõù ðá¹í³íáõù ù»ýù ·ý³ñ³ïáõù »ýù ùç ù³ýç ùá¹»éý»ñçó ³ýï³ë ïé³ëï»ñç½³óç³ûç ³é·áñçãùý»ñç garch åñáó»ëý»ñáí ·»ý»ñ³óí³í å³ù³ý³ï³ûçý ß³ñù»ñç ïíû³éý»ñç ïé³ëï»ñ³íáñù³ý áõý³ïáõãûáõýá: è³ûý³í³í³é ÷áñó»ñç áýã³óùáõù ù»ýù ·»ý»ñ³óýáõù »ýù ëçýã»ïçï ïíû³éý»ñç ñ³í³ù³íáõý»ñ ï³ñµ»ñ ëó»ý³ñý»ñáí: ²ûýáõñ»ï¨, ù»ýù ñ³ù»ù³ïáõù »ýù k-means ùá¹»éý»ñá (¾íïéç¹»ëû³ý ¨ å³ù³ý³ïç ¹çý³ùçï ÷áë³ï»ñåù³ý ù»ïñçï³ý»ñáí), k-shape ¨ kernel k-means ùá¹»éý»ñç ï³ñµ»ñ ïé³ëï»ñ³ûçý ã³÷çãý»ñáí: øç ù³ýç ÷áñó»ñá óáõûó »ý ï³éçë, áñ k-means ùá¹»éá å³ù³ý³ïç ¹çý³ùçï ÷áë³ï»ñåù³ý ù»ïñçï³ûáí óáõûó ¿ ï³éçë ñ³ù»ù³ï³µ³ñ ³í»éç é³í ³ñ¹ûáõýùý»ñ: ²ûýáõ³ù»ý³ûýçí, ¹çï³ñïí³í ùá¹»éý»ñý áõý»ý ½·³éç ã»ñáõãûáõýý»ñ çýýáñù³óç³ûç (å³ù³ý³ï³ûçý ß³ñùç ýí³½³·áõûý »ñï³ñáõãûáõýá) ù³ý³ïç ³í»é³óù³ý ñ»ï ïé³ëï»ñ³íáñù³ý ×ß·ñïáõãû³ý µ³ñóñ³óù³ý ñ»ï ï³åí³í, çýãå»ë ý³¨ ïíû³éý»ñç ³ýñ³í³ë³ñ³ïßéáõãû³ý ï³ù ïé³ëï»ñç ñ³ùáýïýù³ý ¹»åùáõù ×ß·ñçï ïé³ëï»ñ³íáñáõù çñ³ï³ý³óý»éáõ ñ³ñóáõù: ´³ý³éç µ³é»ñ` å³ù³ý³ï³ûçý ß³ñù»ñç ïé³ëï»ñç½³óç³, garch åñáó»ëý»ñ, å³ù³ý³ïç ¹çý³ùçï ÷áë³ï»ñåáõù, k-means, k-shape. g. adamyan 4 1 ñðàâíåíèå áåçìîäåëüíûõ àëãîðèòìîâ êëàñòåðèçàöèè garch-ïðîöåññîâ ãàðèê ë. àäàìÿí åðåâàíñêèé ãîñóäàðñòâåííûé óíèâåðñèòåò, åðåâàí, àðìåíèÿ e-mail: garik.adamyan@ysu.am àííîòàöèÿ â ýòîé ñòàòüå ìû îöåíèâàåì íåêîòîðûå áåçìîäåëüíûå àëãîðèòìû êëàñòåðèçàöèè íàáîðîâ äàííûõ âðåìåííûõ ðÿäîâ, ñãåíåðèðîâàííûõ garch ïðîöåññàìè. â îáøèðíûõ ýêñïåðèìåíòàõ ìû ãåíåðèðóåì ñèíòåòè÷åñêèå íàáîðû äàííûõ äëÿ ðàçëè÷íûõ ñöåíàðèÿõ. çàòåì ìû ñðàâíèâàåì ìîäåëè k-means (ñ ìåòðèêàìè åâêëèäîâîé è äèíàìè÷åñêîé òðàíñôîðìàöèè âðåìåííîé øêàëû), ìîäåëè kshape è kernel k-means ñ ðàçëè÷íûìè ìåòðèêàìè êëàñòåðèçàöèè. íåñêîëüêî ýêñïåðèìåíòîâ ïîêàçûâàþò, ÷òî ìîäåëü k-means ñ ìåòðèêîé äèíàìè÷åñêîé òðàíñôîðìàöèè âðåìåííîé øêàëû äàåò ñðàâíèòåëüíî ëó÷øèå ðåçóëüòàòû. îäíàêî ðàññìîòðåííûå ìîäåëè èìåþò ñóùåñòâåííûå íåäîñòàòêè â ïîâûøåíèè òî÷íîñòè êëàñòåðèçàöèè ïðè óâåëè÷åíèè êîëè÷åñòâà èíôîðìàöèè (ìèíèìàëüíîé äëèíû âðåìåííîãî ðÿäà), à òàêæå ïðè íåñáàëàíñèðîâàííîñòè äàííûõ èëè ïåðåêðûòèè êëàñòåðîâ. êëþ÷åâûå ñëîâà: êëàñòåðèçàöèÿ âðåìåííûõ ðÿäîâ, ïðîöåññ garch, äèíàìè÷åñêàÿ äåôîðìàöèÿ âðåìåíè, k-means, k-shape. 03_adamyan_33 3333 d:\sbornik\...\ciss3.dvi mathematical problems of computer science 26, 2006, 82{90. on t esting of h ypotheses for m any i ndependent objects p a r a n d z e m m. h a ko b ya n institue for informatics and automation problems of nas of ra e-mail par h@ipia.sci.am abstract the problem of many hypotheses testing for a model consisting of three independent objects is considered. it is supposed that m probability distributions are known and each object independently of others follows to one of them. the matrix of asymptotic interdependencies (reliability{reliability functions) of all possible pairs of the error probability exponents (reliabilities) in optimal testing of this model is studied. this problem was introduced (and solved for the case with two given probability distributions) by ahlswede and haroutunian. the model with two independent objects with m hypotheses was examined by haroutunian and hakobyan. refer ences [1 ] e . a . h a r o u t u n ia n , " l o g a r it h m ic a lly a s ym p t o t ic a lly o p t im a l t e s t in g o f m u lt ip le s t a t is t ic a l h yp o t h e s e s " , p roblems of control and information theory, vo l. 1 9 ( 5 -6 ) , p p . 4 1 3 { 4 2 1 , 1 9 9 0 . [2 ] r . f. a h ls we d e a n d e . a . h a r o u t u n ia n , " te s t in g o f h yp o t h e s e s a n d id e n t i¯ c a t io n " , e lectronic notes on d iscrete m athematics, vol. 21, p p . 1 8 5 { 1 8 9 , 2 0 0 5 . [3 ] e . a . h a r o u t u n ia n a n d p . m. h a ko b ya n , " on lo g a r it h m ic a lly a s ym p t o t ic a lly o p t im a l h yp o t h e s is t e s t in g o f t h r e e d is t r ib u t io n s fo r o f in d e p e n d e n t o b je c t s " , m athematical p roblems of computer science vol. 24, p p . 7 6 { 8 1 , 2 0 0 5 . [4 ] e . a . h a r o u t u n ia n a n d p . m. h a ko b ya n , " on l a o t e s t in g o f m u lt ip le h yp o t h e s e s fo r p a ir o f o b je c t s " , m athematical p roblems of computer science vol. 25, p p . 9 3 { 1 0 1 , 2 0 0 6 . [5 ] l . b ir g ¶ e , " v it e s s e s m a xim a ls d e d ¶ e c r o is s a n c e d e s e r r e u r s e t t e s t s o p t im a u x a s s o c ie ¶ s " . z. w a h r s c h . ve r w. ge b ie t e , vo l. 5 5 , p p . 2 6 1 { 2 7 3 , 1 9 8 1 . [6 ] i. cs is z ¶a r a n d j. k äo r n e r , information theory: coding theorems for d iscrete m emoryless systems, a c a d e m ic p r e s s , n e w y o r k, 1 9 8 1 . 8 2 p. m. hakobyan 8 3 ´³½ù³ïç ³ýï³ë ûµû»ïïý»ñç ýï³ïù³ùµ í³ñï³íý»ñç ëïáõ·ù³ý ù³ëçý ö. ð³ïáµû³ý ²ù÷á÷áõù ¸çï³ñïí³í ¿ µ³½ù³ïç í³ñï³íý»ñç ëïáõ·ù³ý ëý¹çñá »ñ»ù ³ýï³ë ûµû»ïïý»ñçó ï³½ùí³í ùá¹»éç ñ³ù³ñ: m ( ¸ 2 ) ñ³í³ý³ï³ý³ûçý µ³ßëáõù»ý»ñá h³ûïýç »ý, ¨ ûµû»ïïý»ñçó ûáõñ³ù³ýãûáõñá ³ýï³ëáñ»ý µ³ßëí³í áëï ¿ ¹ñ³ýóçó ù»ïç: ²ûë ùá¹»éç ñ³ù³ñ áõëáõùý³ëçñí»é ¿ ûåïçù³é ï»ëï³íáñù³ý ¹»åùáõù µáéáñ ñý³ñ³íáñ ½áõû·»ñç ëë³éý»ñç ñ³í³ý³ï³ýáõãûáõýý»ñç óáõóçãý»ñç (ñáõë³éçáõãûáõýý»ñç) ÷áëï³ëí³íáõãûáõýá: ø»í ãíáí ³ýï³ë ûµû»ïïý»ñçó ï³½ùí³í ùá¹»éç µ³½ù³ïç í³ñï³íý»ñáí ¹»åùá ýáõûýå»ë ùýý³ñïíáõù ¿ : microsoft word va.doc mathematical problems of computer science 25, 2006, 27-38. 27 a rule system for translation of universal networking language (unl) expressions into armenian vahan avetisyan institute for informatics and automation problems of nas of ra e-mail va@ipia.sci.am abstract in this paper a system of rules is described for automatic translation (deconversion) of universal networking language (unl) expressions into armenian sentences. some classification of rules is proposed for provision of proper coverage of armenian grammar. also some approaches to the representation of formalized grammar and morphology of the armenian language are described. references [1] v. avetisyan, v. sahakyan, l. petrosyan, r. urutyan, a manukyan, l. hovsepyan. development of armenian language module of unl. convergences’03, alexandria, egypt, 2003 [2] v. avetisyan, a. manukyan. development environment for the armenian language server of unl. proceedings of the international conference on computer science and information technologies, abstracts. yerevan. 2003. [3] a. manukyan, v. avetisyan. armenian morphology model for unl language server. proceedings of the international conference on computer science and information technologies, abstracts. yerevan. 2003. [4] a. manukyan. formalized model for ancient armenian verb//proceedings of the international conference on computer science and information technologies, abstracts. yerevan. 2001. [5] g. jahukyan. the universal linguistic model. yerevan. 2000. [6] v. avetisyan, t. grigoryan. the unl toolbox cicling2005 mexico, mexico. 2005 [7] v. avetisyan, r. urutyan, l. hovsepyan, s. tioyan. development of deconversion rules for generation of armenian sentences from unl. proceedings of the international conference on computer science and information technologies, abstracts. yerevan. 2005 [8] h. uchida, m. zhu, tarcisio della senta, the unl, “a gift for a millenium”, unu/ias, tokyo, 1999. [9] u. hiroshi, m. zhu, “the universal networking language beyond machine translation” undl foundation, september 26, 2001. [10] t. dhanabalan, k.saravanan, t.v. geetha, tamil to unl enconverter. goa. india. [11] t. dhanabalan, t.v. geetha. unl deconverter for tamil. convergences’03, alexandria, egypt, 2003. [12] r. martins, r. hasegawa, v.nunnes, m.graças. hermeto a nl-unl enconverting environment. convergences’03, alexandria, egypt, 2003. a rule system for translation of universal networking language (unl) expressions into armenian 28 [13] kuntal dey, pushpak bhattacharyya. unl based analysis and generation for bengali case structure constructs. convergences’03, alexandria, egypt, 2003. [14] uw manual, unl center, undl foundation, june 2003. [15] h. uchida, m. zhu, the universal networking language (unl) specifications version 7 june 2005. [16] deconverter specification version 2.6, unl centre/undl foundation, 2002.unl center. enconverter specifications. version 3.3 tokyo, 2002. unl ³ñï³ñ³ûïáõãûáõýý»ñçó ñ³û»ñ»ý ý³ë³¹³ëáõãûáõýý»ñç ë»ñù³ý ï³ýáýý»ñç ñ³ù³ï³ñ· ì. ²í»ïçëû³ý ²ù÷á÷áõù ²ûë ñá¹í³íáõù ý»ñï³û³óí³í ¿ unl ³ñï³ñ³ûïáõãûáõýý»ñçó ñ³û»ñ»ý ý³ë³¹³ëáõãûáõýý»ñç ë»ñù³ý ï³ýáýý»ñç ñ³ù³ï³ñ·: ²é³ç³ñïí³í ¿ ï³ýáýý»ñç ¹³ë³ï³ñ·áõùª ñ³û»ñ»ýç ù»ñ³ï³ýáõãû³ý µ³í³ñ³ñ í³íïáõûã ï³éáõó»éáõ ñ³ù³ñ: üï³ñ³·ñí³í »ý ý³¨ ñ³û»ñ»ýç 󨳵³ýáõãû³ý ¨ ß³ñ³ñûáõëáõãû³ý ó¨³ûý³óí³í ýï³ñ³·ñù³ý áñáß³ïç ùáï»óáõùý»ñ: d:\user\sbornik_38_pdf\25.dvi mathematical problems of computer science 38, 61{65, 2012. m any h ypotheses p ar allel distr ibuted detection of the p air of families of p r obability distr ibutions fa r s h in h o r m o z i n e ja d 1 a n d e vg u e n i h a r o u t u n ia n 2 1islamic azad university, ahvaz branch, iran 2institute for informatics and automation problems, nas of ra e-mails: hormozi-nejad@iauahvaz.ac.ir, evhar@ipia.sci.am 1 in t r o d u c t io n th e r e is a c o n s id e r a b le lit e r a t u r e o n t h e p r o b le m s o f d is t r ib u t e d d e t e c t io n a n d d e c is io n in e n g in e e r in g c o n t e xt s [4 , 5 , 6 ]. th e d e c e n t r a liz e d o r d is t r ib u t e d d e t e c t io n p r o b le m wa s ¯ r s t fo r m u la t e d a n d s t u d ie d b y te n n e y a n d s a n d e ll [7 ]. it c o n s is t s o f n g e o g r a p h ic a lly d is p e r s e d s e n s o r s , o n e -wa y c o m m u n ic a t io n lin ks , a n d a fu s io n c e n t e r . e a c h s e n s o r m a ke s a n o b s e r va t io n ( d e n o t e d b y xi; i = 1 ; n ) o f a r a n d o m s o u r c e , qu a n t iz e s xi in t o a n m -a r y m e s s a g e ui = gi ( xi ) , a n d t h e n t r a n s m it s ui; i = 1 ; n t o t h e fu s io n c e n t e r . u p o n r e c e ip t o f u1; u2; :::; un t h e fu s io n c e n t e r m a ke s a g lo b a l d e c is io n u0 = d ( u1; u2; :::; un ) a b o u t t h e n a t u r e o f t h e r a n d o m s o u r c e . th e m e s s a g e s u1; u2; :::; un a r e a ll t r a n s m it t e d t o t h e fu s io n c e n t e r wh ic h d e c la r e s h yp o t h e s is hi; i = 1 ; n t o b e t r u e , b a s e d o n a d e c is io n r u le d. h a r o u t u n ia n [1 ] in ve s t ig a t e d t h e p r o b le m o f l a o t e s t in g o f m u lt ip le s t a t is t ic a l h yp o t h e s e s . th e m o d e l o f t h e t wo -s t a g e l a o t e s t in g in m u lt ip le h yp o t h e s e s fo r a p a ir o f fa m ilie s o f d is t r ib u t io n s is in ve s t ig a t e d in [2 , 3 ]. in t h is p a p e r t h e p r o b le m o f p a r a lle l d is t r ib u t e d d e t e c t io n o f t wo -s t a g e m u lt ip le h yp o t h e s e s l a o t e s t in g t o d e t e c t b e t we e n h yp o t h e s e s c o n s is t in g o f t h e p a ir fa m ilie s o f p r o b a b ilit y d is t r ib u t io n s ( p d s ) is s t u d ie d . e a c h s e n s o r o b s e r va t io n x t a ke s va lu e s in t h e s e t x . a d e t e r m in is t ic m -a r y qu a n t iz e r is a m e a s u r a b le m a p p in g g fr o m t h e o b s e r va t io n s p a c e x t o t h e m e s s a g e s p a c e u = f1 ; 2 ; :::; mg. r a n d o m va r ia b le ( r v ) x c h a r a c t e r iz in g t h e s t u d ie d o b je c t t a ke s va lu e s in t h e s e t x a n d p ( x ) is t h e s p a c e o f a ll d is t r ib u t io n s o n x . th e r a n d o m s o u r c e h a ve s h yp o t h e t ic a l p r o b a b ilit y d is t r ib u t io n s ( p d s ) o f x t h a t d ivid e d in t wo d is jo in t fa m ilie s o f d is t r ib u t io n s . th e ¯ r s t fa m ily in c lu d e s r h yp o t h e s e s p1; p2; :::; pr a n d t h e s e c o n d fa m ily c o n s is t s o f s ¡ r h yp o t h e s e s pr+1; pr+2; :::; ps. th e d is t r ib u t io n o f x u n d e r h yp o t h e s e s hi is d e n o t e d b y pi; i = 1 ; n. th e d is t r ib u t io n s o f t h e m e s s a g e p r o d u c e d b y g a r e d e n o t e d b y pi(g), a n d it is o b t a in a b le fr o m pi a n d g. 2 th e on e -s t a g e l a o mu lt ih yp o t h e s e s te s t in g o f d is t r ib u t e d d e t e c t io n w e c a ll t h e p r o c e d u r e o f m a kin g d e c is io n o n t h e b a s e o f n-s a m p le t h e t e s t án wh e n it is o n e -s t a g e . th e s t a t is t ic ia n m u s t d e t e c t o n e a m o n g s h yp o t h e s e s . w e s t u d y t h e p r o b a b ilit ie s o f t h e e r r o n e o u s a c c e p t a n c e o f h yp o t h e s is hl p r o vid e d t h a t hs is t r u e ®ljs ( á n ) 4 = 6 1 6 2 many hypotheses parallel distributed detection of the pair of families of pds p ns ( u0 = l ) ; l; s = 1 ; s; l 6= s a n d if t h e h yp o t h e s is hs is t r u e , b u t it is n o t a c c e p t e d , t h e n t h e p r o b a b ilit y o f e r r o r is ®sjs ( á n ) 4 = p ns ( u0 6= s) = p l 6=s ®ljs ( á n ) ; s = 1 ; s: co r r e s p o n d in g \ r e lia b ilit ie s " , a r e d e ¯ n e d fo r in ¯ n it e s e qu e n c e o f t e s t s á a s fo llo ws : eljs ( á) 4 = lim s u p n !1 f¡ 1 n lo g ®ljs ( á n ) g; l; s = 1 ; s: fo r c o n s t r u c t io n o f t h e n e c e s s a r y l a o t e s t fo r p r e lim in a r ily g ive n p o s it ive va lu e s e1j1; :::; es¡1js¡1, we d e ¯ n e : rs 4 = n q : d ( qjjps(g) ) · esjs o ; s = 1 ; s ¡ 1 ; rs 4 = n q : d ( qjjps(g) ) > esjs; s = 1 ; s ¡ 1 o ; e¤sjs 4 = esjs; s = 1 ; s ¡ 1 ; e¤ljs 4 = in f q:d(qjjpl( g ) )·e¤ljl d ( qjjps(g) ) ; l; s = 1 ; s; s 6= l; e¤sjs 4 = m in l 6=s e¤ljs: ( 1 ) if a ll d is t r ib u t io n s ps; s = 1 ; s, a r e d i®e r e n t s u c h t h a t t h e fo llo win g in e qu a lit ie s h o ld e1j1 < m in l=2;s d ( pl(g)jjp1(g) ) ; esjs < m in " m in l=1;s¡1 e¤ljs; m in l=s+1;s d ( pl(g)jjps(g) ) # ; s = 2 ; s ¡ 1 ; ( 2 ) t h e n t h e r e e xis t s a lao s e qu e n c e o f t e s t s , a ll e le m e n t s o f t h e r e lia b ilit ie s m a t r ix e ¤ = fe¤ljsg o f wh ic h a r e p o s it ive a n d a r e d e ¯ n e d in ( 1 ) . w h e n o n e o f t h e in e qu a lit ie s ( 2 ) is vio la t e d , t h e n a t le a s t o n e e le m e n t o f t h e m a t r ix e ¤ is e qu a l t o z e r o . 3 th e two -s t a g e l a o te s t in g o f d is t r ib u t e d d e t e c t io n s u p p o s e n = n1 + n2 b e s u c h t h a t n1 = dãne; n2 = [( 1 ¡ ã ) n]; 0 < ã < 1 ; a n d x = ( x1; x2 ) ; x 2 x n ; x n = x n1 £ x n2: a n d we h a ve ve c t o r s o f m e s s a g e s a s u = ( u 1; u 2 ) ; u 2 u n ; u n = u n1 £ u n2 : th e t wo -s t a g e t e s t o n t h e b a s e o f n-s a m p le we d e n o t e b y ©n = ( 'n11 ; ' n2 2 ) is t h e p a r a lle l d is t r ib u t e d d e t e c t io n s ys t e m d e p ic t e d in fig u r e 1 . th e ¯ r s t s t a g e is fo r c h o ic e o f a fa m ily o f p d s , it is e xe c u t e d b y a n o n -r a n d o m iz e d t e s t 'n11 ( u 1 ) u s in g m e s s a g e s s a m p le u 1. th e n e xt s t a g e is a n o n -r a n d o m iz e d t e s t ' n2 2 ( u 2; u 0 ) b a s e d o n m e s s a g e s s a m p le u 2 a n d t h e o u t c o m e o f t h e ¯ r s t fu s io n c e n t e r u 0. fir s t s t a g e o f t wo -s t a g e t e s t in g o f d is t r ib u t e d d e t e c t io n is a s fo llo ws : l e t u s in t r o d u c e t wo s e t s o f in d ic e s d1 = f1 ; rg a n d d2 = fr + 1 ; sg a n d a p a ir o f d is jo in t fa m ilie s o f p d s a r e p1 = fps; s 2 d1g a n d p2 = fps; s 2 d2g: l e t ®0ijj ( ' n1 1 ) ; i 6= j; i; j = 1 ; 2 ; b e t h e p r o b a b ilit y o f t h e e r r o n e o u s a c c e p t a n c e o f t h e i-t h fa m ily o f p d s p r o vid e d t h a t t h e j-t h fa m ily o f p d s is t r u e ( t h a t is t h e c o r r e c t p d is in t h e j-t h fa m ily) ®0ijj ( ' n1 1 ) 4 = m a x s2d j p n1s ( u 0 = i ) ; i 6= j; i; j = 1 ; 2 : w e c o n s id e r r e lia b ilit ie s o f t h e in ¯ n it e s e qu e n c e o f t e s t s e0ijj ( '1 ) 4 = lim s u p n1!1 n ¡ 1 n1 lo g ®0ijj ( ' n1 1 ) o ; i; j = 1 ; 2 : t heor em 1. if all distributions ps; s = 1 ; s, are di®erent and the positive values e 0¤ 1j1; is such that e 0¤ 1j1 < m in s2d 1; l2d 2 d ( pl(g)jjps(g) ) ; f. hormozi nejad and e. haroutunian 6 3 another element of the reliabilities matrix e 0¤ 2j2 of which de¯ned as follows: e 0¤ 2j2 < m in s2d 2 in f q: m in l2d 1 d(qjjpl( g ) )·e 0¤ 1j1 d ( qjjps(g) ) : fig u r e 1 : th e t wo -s t a g e m u lt ip le h yp o t h e s e s d is t r ib u t e d d e t e c t io n s ys t e m s e c o n d s t a g e o f t h e t wo -s t a g e t e s t in g o f d is t r ib u t e d d e t e c t io n is a s fo llo ws : th e t e s t 'n22 ( u 2; u 0 ) c a n b e d e ¯ n e d b y u s in g t h e ¯ r s t fu s io n c e n t e r u0 a n d t h e t h e s e c o n d m e s s a g e s s a m p le u 2. th e p r o b a b ilit y o f t h e fa lla c io u s a c c e p t a n c e a t t h e s e c o n d s t a g e o f t e s t o f p d pl, wh e n ps is c o r r e c t a n d i-t h fa m ily o f p d s is a c c e p t e d , is ®00ljs ( ' n2 2 ) 4 = p n2s ( u 00 = lju0 = i) ; l 6= s; i = 1 ; 2 ; l 2 di: th e p r o b a b ilit y t o r e je c t ps, wh e n it is t r u e a n d i-t h fa m ily o f p d s is a c c e p t e d , is ®00sjs ( ' n2 2 ) 4 = p n2s ( u 00 6= sju0 = i) = x l 6=s ®00ljs ( ' n2 2 ) ; s 2 di; i = 1 ; 2 : co r r e s p o n d in g r e lia b ilit ie s fo r t h e s e c o n d s t a g e o f t e s t , a r e e00ljs ( '2 ) 4 = lim s u p n2!1 n ¡ 1 n2 lo g ®00ljs ( ' n2 2 ) o ; l; s = 1 ; s: t heor em 2. if at the ¯rst stage of test the ¯rst family of p d s is accepted, then for given positive and ¯nite values e00sjs, s = 1 ; r ¡ 1 of the reliabilities matrix e 00 ( '2 ) , let us investigate the regions: r00s = n q : d ³ q k ps(g) ´ · e00sjs o ; s = 1 ; r ¡ 1 ; r00r = n q : d ³ q k ps(g) ´ > e00sjs; s = 1 ; r ¡ 1 o ; 6 4 many hypotheses parallel distributed detection of the pair of families of pds and the following values of elements of the future reliabilities matrix e 00 ( '¤2 ) of the l ao test sequence: e00¤sjs = e 00 sjs; s = 1 ; r ¡ 1 ; e00¤ljs = in f q2r 00 l d ³ q k ps(g) ´ ; l; s = 1 ; r; l 6= s; e¤rjr 4 = m in l 6=r e¤ljr; w hen the following compatibility conditions are valid e001j1 < m in s=2;r d ( ps(g) k p1(g) ) ; e00sjs < m in [ m in l=1;s¡1 e00¤ljs ; m in l=s+1;r d ( pl(g) k ps(g) ) ]; 2 · s · r¡ 1 ; then there exists a l ao sequence of test '¤2, elements of reliabilities matrix e 00 ( '¤2 ) of which are de¯ned above and are positive. e ven if one of the compatibility conditions is violated, then e 00 ( '¤2 ) has at least one element equal to zero. if in t h e ¯ r s t s t a g e o f t e s t , t h e s e c o n d fa m ily o f p d s is a c c e p t e d , t h e n fo r s ¡ r ¡ 1 g ive n p o s it ive va lu e s e00sjs, s = r + 1 ; s ¡ 1 o f r e lia b ilit ie s m a t r ix e 00 ( '¤2 ) , t h e p r o c e d u r e is a n a lo g o u s . r e lia b ilit ie s a n d e r r o r p r o b a b ilit ie s o n t h e t wo -s t a g e t e s t in g o f d is t r ib u t e d d e t e c t io n a r e in c o m in g : ®000ljs ( © n ) 4 = p ns ( u 00 = l ) ; l; s = 1 ; s; l 6= s; ®000sjs ( ©n ) 4 = p ns ( u 00 6= s) = x l 6=s ®000ljs ( © n ) ; s = 1 ; s; e000ljs ( ©) 4 = lim s u p n!1 f ¡ 1 n lo g ®000ljs ( © n ) g; l; s = 1 ; s: s o we c a n c o n s id e r e r r o r p r o b a b ilit ie s a s fo llo ws ®000ljs ( © ¤n ) = p n1s ( u 0 = i ) ¢ p n2s ( u 00 = lju0 = i) ; l; s 2 di; i = 1 ; 2 ( 3 ) ®000ljs ( © ¤n ) = p n1s ( u 0 = j ) ¢ p n2s ( u00 = lju0 = j ) ; s 2 di; l 2 dj; i; j = 1 ; 2 ; i 6= j ( 4 ) a c c o r d in g t o ( 3 ) { ( 4 ) a n d d e ¯ n it io n o f r e lia b ilit ie s we o b t a in e000ljs ( © ¤ ) = ( 1 ¡ ã ) e00¤ljs ; l; s 2 di; i = 1 ; 2 ; e000ljs ( © ¤ ) = ãeijjs + ( 1 ¡ ã ) e00¤ljs ; s 2 di; l 2 dj ; i; j = 1 ; 2 ; i 6= j; e000sjs ( © ¤ ) = m in l 6=s e000ljs ( © ¤ ) ; s 2 di; i = 1 ; 2 : w e c h a r a c t e r iz e t h e o p t im a l e r r o r e xp o n e n t s in a p a ir o f s t a g e s a n d we p r o vid e c o m p a t ib ilit y c o n d it io n s fo r t h is t o h a p p e n a n d it is in ve s t ig a t e d d e s c r ip t io n o f c h a r a c t e r is t ic s o f l a o h yp o t h e s e s t e s t in g o f d is t r ib u t e d d e t e c t io n a n d t h e g o a l is t o m a ke a d e c is io n o n t h e m a n y p o s s ib le h yp o t h e s e s , b a s e d o n t h e m e s s a g e s r e c e ive d a t t h e fu s io n c e n t e r s . f. hormozi nejad and e. haroutunian 6 5 r e fe r e n c e s [1 ] h a r o u t u n ia n e .a . \ l o g a r it h m ic a lly a s ym p t o t ic a lly o p t im a l t e s t in g o f m u lt ip le s t a t is t ic a l h yp o t h e s e s ." p roblems of control and information theory, vo l 1 9 , n o s 5 -6 , p p . 4 1 3 -4 2 1 , 1 9 9 0 . [2 ] h a r o u t u n ia n e .a ., h a ko b ya n p .m. a n d h o r m o z i n e ja d f. \ on t wo -s t a g e lo g a r it h m ic a lly a s ym p t o t ic a lly o p t im a l t e s t in g o f m u lt ip le h yp o t h e s e s c o n c e r n in g d is t r ib u t io n s fr o m t h e p a ir o f fa m ilie s ." transactions of iiap of nas of r a and of ysu, m athematical p roblems of computer science, vo l. 3 7 , p p . 3 4 -4 2 , 2 0 1 2 . [3 ] h o r m o z i n e ja d f., h a r o u t u n ia n e .a . a n d h a ko b ya n p .m. \ on l a o t e s t in g o f m u lt ip le h yp o t h e s e s fo r t h e p a ir o f fa m ilie s o f d is t r ib u t io n s ." p roceeding of the conference \computer science and information technologies", y e r e va n , a r m e n ia , p p . 1 3 5 -1 3 8 , 2 0 1 1 . 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[7 ] te n n e y r . r . a n d s a n d e ll n . r ., d e t e c t io n wit h d is t r ib u t e d s e n s o r s , ie e e trans. aerosp. e lectron. syst., vo l. 1 7 , p p . 5 0 1 -5 1 0 , 1 9 8 1 . d:\sbornik\...\transactions.dvi mathematical problems of computer science 31, 28{39, 2008. lower b ound for e capacity of discr ete m emor yless channel with t wo-sided state i nfor mation ma r ia m e . h a r o u t u n ia n a n d a r t h u r r . mu r a d ya n institute for informatics and automation problems of nas of ra. e-mail: armar@ipia.sci.am abstract we study the channel with two-sided state information, a discrete memoryless channel with ¯nite input and output alphabets and random state sequence. partial information about the state sequence is available to the encoder and decoder. applications of this study include watermarking, data hiding, communication in presence of partially known interferers. the capacity of this model was obtained by cover and chiang in [1]. in this paper the random coding bound of e-capacity is derived for considered model which can be called also generalized channel with state information, as it includes four possible situations of the channel with random parameter. refer ences [1 ] t. m. co ve r a n d m. ch ia n g , \ d u a lit y b e t we e n c h a n n e l c a p a c it y a n d r a t e d is t o r t io n wit h t wo -s id e d s t a t e in fo r m a t io n " , ie e e transactions on information theory, vo l. 4 8 , n o . 6 , p p . 1 6 2 9 -1 6 3 8 , 2 0 0 2 . 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[1 6 ] i. cs is z ¶a r , \ th e m e t h o d o f t yp e s " , ie e e transactions on information theory, vo l. 4 4 , n o . 6 , p p . 2 5 0 5 -2 5 2 3 , 1 9 9 8 . ºñïïáõù³ýç íç׳ïý»ñáí áý¹ñ³ï ³é³ýó ñçßáõáõãû³ý ï³åáõõáõ e-áõý³ïáõãû³ý ëïáñçý ·ý³ñ³ï³ï³ýá ø. ð³ñáõãûáõýû³ý ¨ ². øáõñ³¹û³ý ²ù÷á÷áõù ø»ýù áõëáõùý³ëçñáõù »ýù »ñïïáõù³ýç íç׳ïý»ñáí ï³åáõõç, ³ûý ¿ í»ñç³íáñ ùáõïùç ¨ »éùç ³ûµáõµ»ýý»ñáí áý¹ñ³ï ³é³ýó ñçßáõáõãû³ý ï³åáõõç, áñá ï³ëí³í ¿ íç׳ïý»ñç å³ï³ñ³ï³ý ñ³çáñ¹³ï³ýáõãûáõýçó: ìç׳ïý»ñç ñ³çáñ¹³ï³ýáõãû³ý ù³ëý³ïç çýýáñù³óç³ý ñ³ë³ý»éç ¿ ïá¹³íáñçãçý ¨ ³å³ïá¹³íáñçãçý: ²ûë áõëáõùý³ëçñáõãûáõýý»ñá ïçñ³éíáõù »ý çýýáñù³óç³ý ã³ùóýáõ, çñ³ýßáõ ñ³ù³ï³ñ·»ñáõù: ²ûë ùá¹»éç áõý³ïáõãûáõýá ëï³óí»é ¿ îáí»ñç ¨ âç³ý·ç ïáõùçó: ²ûë ñá¹í³íáõù ¹áõñë ¿ µ»ñíáõù e-áõý³ïáõãû³ý å³ï³ñ³ï³ý ﻹ³íáñù³ý ·ý³ñ³ï³ï³ýá ¹çï³ñïí³í ùá¹»éç ñ³ù³ñ, áñá ï³ñ»éç ¿ ³ýí³ý»é ý³¨ íç׳ïý»ñáí áý¹ñ³ýñ³óí³í ï³åáõõç, ù³ýç áñ ³ûý ý»ñ³éáõù ¿ å³ï³ñ³ï³ý å³ñ³ù»ïñ»ñáí ï³åáõõáõù ãáñë ñý³ñ³íáñ çñ³íç׳ïý»ñá: 69 mathematical problems of computer science 59, 69–81, 2023. doi: 10.51408/1963-0103 udc 004.725, 004.852 research of model increasing reliability intrusion detection systems timur v. jamgharyan national polytechnic university of armenia, yrevan, armenia e-mail: t.jamgharyan@yandex.ru abstract the paper presents the results of the using, a recurrent neural network to detect malicious software as part of the snort intrusion detection system.the research was conducted on datasets generated on the basis of athena, dyre, engrat, grum, mimikatz, surtr malware exploiting vulnerability cve-2022-20685 in the snort intrusion detection system. processing of input traffic data was carried out before the frag-3 and modbus preprocessors. the method of k nearest neighbors was used as a mathematical apparatus. the simulation of the developed software at different iterations. all research results are presented in https://github.com/t-jn keywords: machine learning, dataset, malware, preprocessor, metasploit, k nearest neighbors method, intrusion detection system. article info: received 8 january 2023; send to review 7 february 2023; accepted 7 march 2023. 1. introduction the intrusion detection systems (ids) include many different software components designed to detect various types of traffic with an embedded malicious component. detection is carried out according to a set of rules that are configured based on the threat model and security policies. the security architecture of the network infrastructure (ni) is built taking into account possible attacks according to various models։ triad cia (confindentiality, integrity, availability, cia), parker's hexad [1]. network ids, unlike host ids, detect attacks directed at the network segment and contain a set of complementary rules and security scripts that can neutralize an attack on the network. unlike host-based ids, network-based ids require more computing resources due to the fact that a larger set of rules and detectors is activated during their operation [2]. when using host ids in the infrastructure for a fleet of computing systems running linux os, can disable https://github.com/t-jn research of model increasing reliability intrusion detection systems 70 the rules for windows (or another os), but hardly possible for a network ids, since different operating systems are used in the infrastructure. modern ids are able to detect various types of attacks at different levels of the osi (open system interconnection, osi) model: bad traffic, system scanning, the use of known exploits to attack over various protocols, various backdoors, various known malware [3]. a significant limitation of systems for analyzing network traffic and the state of ni is the algorithmic and functional determinism inherent in them. an important issue of infrastructure security is the reliability of the processed data of the ids itself (data reliability – is, the property of the processed data not to have hidden errors [4]). the processing of data streams in the ids itself is determined by the functioning algorithms, data presentation formats, and the formalization of signature classifiers. protecting the ids signature database (both remote and local) is also one of the most important tasks. if the signatures database has been attacked for availability, then when a new vulnerability appears, the ids will not receive the necessary signature and the infrastructure perimeter will become vulnerable [5]. the development of m2m (machine-to-machine, m2m) and ml (machine learning, ml) technologies has increased the capabilities of both attack and defense tools. various researchers are conducting research on increasing (improving) various parameters of ids with ml [6, 7, 8]. one of the parameters that improves when using ml modules as part of a standard ids is its variability. unlike deterministic ids, ids with ml are capable of forming a multi-criteria sample on the basis of which the detector operation scheme is formed within the given constraints. but ids with ml have certain limitations when integrating them into the ni architecture. in particular, ml ids are very sensitive to various implementations of «noise attacks» («noise attack» is a variant of an availability attack in which a large number of random and meaningless fragmented packets are sent to the attacked system, some of which contain malware [9]). a dangerous consequence of a «noise attack» on a ml network ids is that attackers «attack» it for a long time with streams of datasets that cause false positives, «teach» the ml ids discriminator to be immune to this type of traffic (creating a cyclic chain of operations: false positive--true negative--false negative--true positive, which overload both the ids itself and the siem system (security information and event management, siem). various manufacturers combine ids modules into different classes, which allows you to quickly reconfigure the ids itself for specific tasks. in particular, for snort open source ids, there are many different types of preprocessors (frag-3, stream, performance monitor, smtp, pop, imap, ssh, dns, dce/rpc, sip preprocessors, reputation preprocessor, modbus preprocessor) each of which is functionally is responsible for handling the given protocol and/or data type.  ids preprocessor is a software module that receives data from the network traffic decoding module and outputs them to the input of intrusion detection modules. as stated in the article «attacks on machine learning systems» [10], the most vulnerable part of the ml ids is the traditional ids component (the deterministic part of the ids). ml systems, like any other, will be hacked using vulnerabilities in these traditional components. the use of ml at the preprocessor level is due to the fact that when developing an ids with ml, it is not enough to create a functioning model that can detect a threat not described in a set of rules (signatures) or generate new ones based on «known» signatures, but it is also necessary to protect the ids itself from probable infection with malware that can compromise the reliability of the results issued by ids․the choice of using a neural network at the preprocessor level is also due to the fact that the ids, which has a neural network in its component composition after the preprocessor, is able to protect the ni, since malware not detected by standard datasets (described in the signature/rule database) will be detected with varying probability neural network. but with a «noise attack», the target is the ids itself, which, when taken out of the reliable functioning mode, will no longer detect malware. undescribed at the preprocessor level, t. jamgharyan 71 malicious data embedded in ids can be detected using performance preprocessors that evaluate various kinds of statistics. but the problem is that, having determined the type of network ids, attackers can design an attack taking into account the work of preprocessors, and malware embedded in the ids itself will not go beyond the allowable statistical deviations. a lot of research has been devoted to the task of applying machine learning as part of ids, but only a small part of them explores the use of machine learning at the preprocessor level. this limitation, in particular, is due to the fact that the «response» of the neural network is probabilistic in nature and it is necessary to introduce clear boundaries for the neural network itself. otherwise, the neural network will be an event generator, which will be classified as an attack by the ids detection modules. thus, there is a recursion to the problem of stability and integrity of both the ids and the ni as a whole [11]. this research explores the potential of a recurrent neural network (rnn) to detect malware at the preprocessor level. the choice in the research of rnn from the entire set of neural networks is determined by the fact that rnn form a directed sequence between elements, which allows processing a series of events in time (this characteristic allows granular processing of fragmented datasets). the relevance of the work lies in the ever-increasing role of ids with ml in the ni security architecture and the increasing security requirements of the ids itself. the use of a neural network at the preprocessor level will increase the reliability of malware detection results without affecting the main ids signature database, which will reduce the attack surface for the ids itself. the novelty of the research lies in the application of the k nearest neighbors (k nearest neighbors, knn) method to detect malware in ids before preprocessors.  the k nearest neighbors method is a metric algorithm for classifying objects. malicious software athena, dyre, engrat, grum, mimikatz, surtr obtained from publicly available sources was used as calibration data [12--15]. the choice of the knn method is determined by the fact that it is necessary to minimize the value of the preprocessor error, and for this it is necessary to carry out a preliminary grouping and classification of unknown input datasets in normalized traffic.  traffic normalization modification of packets of protocols of the transport, and network levels for their subsequent processing by ids detection modules. 2. formulation and description the problem it is necessary to detect a malicious dataset in normalized traffic. the mathematical model construction was carried out on the basis of the formulas obtained in the sources [16,17]. there are network traffic 𝑋 inputs that contain malware fragments (1). 𝑋𝑚 = {(𝑥1, 𝑦1 ), … , (𝑥𝑚 , 𝑦𝑚 )}, (1) where, 𝑥𝑚network traffic datasets that do not contain malicious components, 𝑦𝑚network traffic datasets containing malicious components, 𝑚number of the analyzed packet of the input dataset. on the set of input traffic data sets, the distance function 𝑥𝜌(𝑦, 𝑦′) is given. the greater the value of the distance function, the less similar the entities are 𝑦, 𝑦′, where 𝑦′the minimum size of a malware dataset that can be uniquely identified and classified with respect to 𝑦. for any entity 𝜐 in the data package, arrange the objects 𝑥𝑖 in ascending order (2). 𝜌(𝜐, 𝑥1;𝜐 ) ≤ 𝜌(𝜐, 𝑥2;𝜐 ) ≤ ⋯ ≤ 𝜌(𝜐, 𝑥𝑚;𝜐 ), (2) research of model increasing reliability intrusion detection systems 72 where 𝑥𝑖;𝜐 the set of network traffic data that is the 𝑖-th neighbor of the entity 𝜐. similarly for the 𝑖 -th neighbor of the entity 𝜐 in the dataset 𝑦𝑖;𝜐. using the formula (3 from the source [17], we determine the malicious knn components for the traffic arriving in the ni. 𝛼(𝜐) = arg max 𝑦∈𝑌 ∑[𝑦(𝑥𝑖;𝜐 ) = 𝑦] 𝑚 𝑖=1 𝜔(𝑖, 𝜐), (3) where, 𝜔(𝑖, 𝜐)a given weight function that evaluates the degree of importance of the 𝑖-th neighbor for the classification of the entity 𝜐. by changing the 𝜔(𝑖, 𝜐) value, you can get different versions of the k nearest neighbors method (4). 𝜔(𝑖, 𝜐) = [𝑖 ≤ 𝑘]. (4) when 𝜔(𝑖, 𝜐) = [𝑖 = 1] malware is detected only in the given single value 𝜔. that is, the rnn is only able to detect the malware datasets it was trained on. a graphical representation of a rnn is shown in fig. 1. fig. 1. recurrent neural network. attackers can load malware into the ids itself not in a single package, but in fragments (using the built-in frag-3 preprocessor as an internal attack tool), then the research task of grouping and classifying malware fragments arises. standard ids do not cope with this task very effectively, but ml ids, in the presence of a training set, are able to solve this problem. the disadvantage of ml ids is that they can produce unreliable results if the preprocessor responsible for a particular type of traffic/protocol is «damaged» as a result of a «noise attack». a particular danger lies in the fact that any traffic entering the ids preprocessors (both ml and deterministic) is not checked for malicious components, since the task of the preprocessor is to «reformat» traffic for processing by detectors. 3. task statement it is necessary to develop and programmatically implement an algorithm and, based on it, software that integrates a rnn capable of solving the problem of grouping and classification with the ids preprocessor. t. jamgharyan 73 4. boundary conditions 1. the smallest fragment of the malware file (𝜉) that can be classified 𝜉 = 20𝑏𝑦𝑡𝑒 (detection was carried out using context-piecewise hashing (context triggered piecewise hashing, ctph), which is discussed in detail in [18]. 2. the delay in the processed module should not cause a «signal race». traffic from the output of the preprocessor module to the input of the detection modules must be sent synchronously. as part of this condition, an additional restriction has been introduced only udp (user datagram protocol, udp) traffic is processed. 3. the hardware must support the parallel computing mode. the developed software connects the rnn to frag-3 and modbus preprocessors (frag3 preprocessor for defragmenting an ip packet, modbus preprocessor for processing data from a variety of devices operating in scada networks (supervisory control and data acquisition, scada).since the frag-3 preprocessor is designed to build packages, using a trained rnn can neutralize the process of «assembling» malicious packages inside the ids, increasing the level of reliability of its functioning. on fig.2 shows a diagram of the snort ids with the proposed data processing software implemented on rnn. fig. 2. snort ids with developed data processing software. research of model increasing reliability intrusion detection systems 74 5. description of the module the network traffic coming from the decoders is directed to the preprocessor processing module (standard operation of the snort ids). the traffic that should processed by the frag 3 and modbus preprocessors is sent to the developed module based on the rnn. after processing according to the developed algorithm, this traffic is again sent to the standard detection modules. the task of the module is to carry out the primary «cut-off» of possible malware and protect the ids itself from being modified by malware. the developed algorithm is shown in fig. 3. fig. 3. developed algorithm. algorithm operation the software that searches for fragmented malware receives network traffic datasets from a decoder (snort ids a low-level interceptor) as input. only traffic that must be processed by the frag-3 and modbus preprocessors is subject to processing. step 1. converting received datasets to «data frame». this conversion is necessary to speed up the work of the rnn, since the traffic not processed by the developed module goes directly to t. jamgharyan 75 the preprocessor module and the processing delay should not exceed the boundary conditions (boundary condition 2). step 2 phase 1. calculation of the distance from the target object, which must be classified to each of the sample objects (traffic). computing a distance metric between likely malware datasets. all calculations are performed in parallel mode (boundary condition 3),  2.1 k=0 calculation of the distance metric and detection of malicious datasets is not performed, since the classification of malicious and non-malicious datasets is impossible,  2.2 k=1 the distance between malicious and non-malicious datasets is constant (k=const). only those malicious datasets that fall within the specified distance metric are detected,  2.3 k=m continuous detection mode.upper limit: the value of m that the hardware can handle,  2.4 k>m malicious datasets are not detected,  2.5 k 0 𝑗, 𝑖 = 0, 𝑗 > 0 min { 𝐷(𝑖, 𝑗 − 1) + 1, 𝑗 > 0, 𝑖 > 0 𝐷(𝑖 − 1,𝑗) + 1 + 𝑚(𝑀[𝑖],𝑁[𝑗]), } (1) levenshtein editorial distance calculation equation, where, 𝐷 the editorial distance, 𝑀, 𝑁the length of strings obtained as a result of ctph over some alphabet (in this case hex), 𝑖 remove step from the first line, 𝑗-insert into the first line. 𝜙 = 𝑇𝑃 × 𝑇𝑁 − 𝐹𝑃 × 𝐹𝑁 √(𝑇𝑃 + 𝐹𝑃)(𝑇𝑃 + 𝐹𝑁)(𝑇𝑁 + 𝐹𝑃)(𝑇𝑁 + 𝐹𝑁) , (2) where, 𝜙 matthews correlation 𝑇𝑃 true positive, 𝑇𝑁 -true negative, 𝐹𝑃 -false positive, 𝐹𝑁 false negative. a capsule neural network was chosen as a transfer learning model. the choice of the capsule network is due to the following reasons:  the capsule network does not require a large amount of training data, which is critical for this research,  the capsule network explores hierarchical relationships, which allows detecting possibly probable versions, in the presence of a primary code (a fragment of the main code) of malware,  the capsule network allows searching even in obfuscated source code with a minimum malware representativeness value, research of obfuscated malware with a capsule neural network 70  the capsule network is the most easily adaptable to changing the learning algorithm compared to other neural networks. 2. diagrams of neural networks fig.1. diagram of a capsule neural network. the nonlinearity function of the capsule network is determined by (equation 3) [59]. 𝝂𝒊 = ||𝑠𝑖|| 2 1 + ||𝑠𝑖|| 2 𝑠𝑖 ||𝑠𝑖|| , (3) where, 𝑠𝑗the result obtained in the previous step, 𝝂𝒊 the result obtained after applying the nonlinearity. the left side of the equation performs additional compression, and the right side of the equation performs unity scaling of the output vector. the trained convolutional neural network (fig. 2) was chosen as a test to check the reliability of the output data. as «weight coefficients» of the convolutional neural network, the value of ctph was calculated the used ssdeep software. fig. 2. diagram of a convolutional neural network. t. jamgharyan 71 verification of the results obtained from both neural networks was carried out using public malware detection services [60]-[61]. the developed software algorithm is shown in fig.3. fig. 3. algorithm of the developed software. research of obfuscated malware with a capsule neural network 72 algorithm operation: operations on the input data of the research.  the dataset generated from the malware source code was obfuscated using various tools [62][63] and prepared for training a capsule neural network (dataset 1).  the same non-obfuscated dataset (dataset 2) generated from the malware source code was prepared to train a convolutional neural network. a total of 1000 annotated datasets of various sizes (20.40, 80, 128, 256, 512, 1024 bytes) were prepared for mimikatz, athena, engrat, grum, surtr, dyre software. steps 1, 2: input of the initial malware dataset into the trained neural networks and the conversion module, step 3: converting the source dataset to javascript object notation (json) format and setting the ctph step size, step 4: calculation of the edit distance by the levenshtein method, step 5: computation ctph using ssdeep software, step 6: comparison of the values calculated by the levenshtein method and using the ssdeep software, step 7:filtering the training datasets of neural networks from «noise» (the full implementation of this part of the algorithm is presented in [33]), step 8: training capsular neural network, step 9 training convolutional neural network, step 10 compute the matthews correlation and resize the training datasets.  𝜙 = −1 the received output data of both neural networks go beyond the value tolerance  𝜙 = 1 the resulting outputs of both neural networks are correct (within the permissible deviation value)  𝜙 = 0 the resulting output of both neural networks is random steps 11, 12: reconfiguring the training datasets and resizinge the ctph. table 1 presents the results of calculating the value of ctph and the editorial distance between the hashes of the obfuscated source code of mimikatz software using capsular, convolutional neural networks, as well as ssdeep software. table 2 shows the results of calculating the value of the context-piecewise hash of the obfuscated compiled source code and the editorial distance between the hashes of the mimikatz software using capsular, convolutional neural networks, and also the ssdeep software. in the research, datasets used a comparison between files 20-40, 20-80, 20-128, 20-256, 20512, 20-1024 bytes, as well as combinations of 40-512, 40-1024, 128-512, 128 -1024 bytes for mimikatz, athena, engrat, grum, surtr, dyre malware. t. jamgharyan 73 3. results table 1.the results of computing the value of ctph and the editorial distance between the hashes of the obfuscated source code of mimikatz software f il e n u m b e r in t h e d a ta se t mimikatz file hash values (20 byte) mimikatz file hash values (512 byte) e d it o ri a l d is ta n c e p e rc e n ta g e o f m a lw a re s a m p le s c a lc u la te d u si n g s с d e e p percentage of malware samples computed using convolutional neural networks percentage of malware samples computed using capsule neural network training epoch training epoch i ii iii i ii iii 1. b9be58b87140f922969c 905236829d2436c34400 ef73afe0b3862206e112 400dc97a6920c1240ca2 36 10 4 2 9 8 19 39 2. e1077e747c9486dce1bf da820c078fe300a901fb 081cdfaf631a003a5a5d fa678b52af5c0eb2cbd3 36 13 6 8 7 16 27 42 3. d86c9ca3861e333dc337 6fc5565943551389edd6 72840526d3cecbba084e ef91aed9c52cd94855d5 35 25 18 9 9 24 52 78 4. bd72fda18edc004d5181 b57e48a757ac2ed94444 783e9520a25faca4f815 2dfc092d7d67e359c5f6 35 28 21 22 24 8 10 12 5. 8ab1d3267a46f953c73b 4154b1a261a8e02493d8 ad523321e582956d7b51 e9f4bc3763d9305231dc 30 11 3 7 3 34 54 82 6. dc990c540fc50debf0cd c178101ab107acaef9fe f2ba969ed8f8ecc7ce57 c54c39de5333cf0d6a8e 36 23 11 16 21 16 28 65 7. b137df3d2083c226f985 c0494a9cef753034ac6d f7fd9ed34bc6ead485bd 5e7c1b9f9f13f30fddba 34 13 10 9 12 16 27 46 8. 9efa06fa6567be9554db 5c351da39c9c084306e0 f7fd9ed34bc6ead485bd 5e7c1b9f9f13f30fddba 33 21 15 15 17 31 46 79 9. 4f5ec65628d2bde662a4 08854a41caea98c0f44f f5cd09b85a44df103b21 ea9c4d02c564fcb19191 35 64 32 30 42 38 37 48 10. 5329b04a348368967844 f421453563001ad4ab89 37a56e3a4acbef542099 4c0d7864125e53f5aaa3 36 22 8 11 16 27 48 61 11. 95a56dfdfd7c8550afb8 ab2474916bb63e58bb15 37a56e3a4acbef542099 4c0d7864125e53f5aaa3 33 16 12 13 15 27 41 68 12. aececb9dccd29fd5dd9 c0559ad62afb84af374b2 51168e0c2ab45361cf05 834a721cd4aba48098be 34 19 11 12 18 36 49 73 13. 14791ec8ec19ca534367 c54f008b8439eea89f09 497a16d6dd757f05fb88 4994c71bea880e87ad18 35 11 18 29 25 37 49 68 14. dbfb0b8c0a28ea8bade 6306f9e8589ee1c310a39 c6ca0e98e0a66c45838f b254aec474553850ab91 34 16 14 21 29 52 58 71 15. c91e176518b7e42450e2 c28d45bf31a1b3178240 7ad0cc0f4ba8c767fac7 f0a4f7ec192b3a60ec9e 36 18 16 19 28 29 43 68 16. 04b66940a08ac7adb0cd f19382a8169d0c256c09 5db88a72cdcfe90ff987 1eae5bf8d2b617d73b0a 37 26 11 19 36 39 56 73 17. 67b4a269a360b994d776 9e4b40220c8b59c219b0 fa926a049a1d9d72126b d07f1a1b87326b5e355b 34 41 27 11 29 26 58 61 18. c2cdacd22e871ecef12b 0cbc8caf4559eecfa084 817c64fed50532e58dd2 1a8812c65fe10a250bd0 36 16 15 16 26 31 46 74 19. 4202fc70b1301ec50b1f 64ca525de6d31825787d 38bc177d79492834356f 1cce4f9120599f41e952 36 18 17 19 21 28 37 49 20. 20b5c47533cb97d72f9 0895ea1ffe27695063e54 818b59add29456248836 864d46c146d9d930d8a2 37 19 8 16 34 24 37 58 research of obfuscated malware with a capsule neural network 74 in training epochs 1-3, the results of the capsular neural network are better than the results of the convolutional neural network and ssdeep software, except for file №4 in the dataset, which is included in the statistical error. тable 2. the results of calculating the value of ctph and editorial distance between hashes of the compiled mimikatz source code. f il e n u m b e r in t h e d a ta se t mimikatz file hash values (20 byte) mimikatz file hash values (512 byte) e d it o ri a l d is ta n c e p e rc e n ta g e o f m a lw a re sa m p le s c a lc u la te d u si n g sс d e e p percentage of malware samples computed using convolutional neural networks percentage of malware samples computed using capsular neural network training epoch training epoch i ii iii i ii iii 1. d7e4e9abedd0949b8bcf f30c7abbdad97b182be8 51f028f6b078f51583e0 a048d9bc577b6a4e17b9 37 25 23 31 42 17 19 23 2. 2c0e9d614fab60e18bd4 2e99659974a3d298a9ae 7f966e5a707dd69c13b5 de45c9765a9be437e642 35 16 18 14 22 8 11 9 3. f76606cb6fae082991eb 271af5ab7629d592cb04 fb96549631c835eb239c d614cc6b5cb7d295121a 32 28 27 36 45 16 17 14 4. 14da593832768f0a08e8 ecd46363936eef096dcc 72ac7a00a3c2a0a825cd 016d71b0d587c6cc3f46 36 23 16 22 34 18 20 16 5. 7f01a23afa1bcecdfdbb 25b953c4f15366eaba51 35139ef894b28b73bea0 22755166a23933c7d9cb 37 37 34 41 48 27 29 23 6. 1ca12a53c82cdd508054 bdcdbe5256ccdd44c13c 918b1c05e576f4b90fce 15a06bc3442d72852a3c 35 48 44 53 61 34 31 28 7. a7f0499bf3eb6180d4da 748426822404e46dea13 4759f2ba1ba20f493664 dbf5e36c1a1ec0d75658 36 15 11 13 12 8 2 3 8. aec2a4accb7ca456a57a c4426e8f51c2e6a8b143 902a2d132f213700b5de fbefe7567f68ca8e234a 35 19 18 26 29 16 10 13 9. 582d2ceff8f4f493f3a9 d45c71286255946a7d37 b2fd9a1405ba74fc360e 1784961176b2b88bf5c9 37 39 28 48 57 25 23 12 10. a25a87930b155282e138 35142ad63cea1994d02d c47419fdd4d6f146e430 64b9ddb859a250404500 36 53 47 40 57 34 29 47 11. 2f7b14912dddcf7c1c7a ebb49955cb5bf0ab3257 b521d7652866027a7e5b 43c6269d7c81ffb5a86e 36 28 30 37 44 14 19 23 12. fd5fd2f7953cf5630f74 c2933b378d4381367ddd 9de4bfa1fdb6c90637d3 5492ec14ee10a3967997 33 56 49 53 67 42 48 34 13. e88dac72cd8ac64360d9 5fb15e8ea9aaa8794f8c 1eb796fd1ff7dda036fc a37d0f31aab19dedab1a 37 24 29 48 52 17 23 15 14. efa91cc773ee2c32ba51 2ffce8db8a3760bda564 99828f68be57c53ff954 5f79e32bdb36050bf93b 32 19 27 29 37 13 18 28 15. f9980d6122acf1bf54a6 8e49d15507fbc3ce7c1f 2400b40333821b00b5d0 b67f20f5f0e30ebf02dd 36 56 44 58 63 37 34 39 16. c5d4d95ce32029e1150a 20d2f836b7b2c6e49546 dfb380d8b0709104c606 978092c7164160f32887 37 29 27 35 38 21 17 34 17. 5156507d0b07bd9eaafe 56815e1a04a0eaa1a8e9 bd951f174a8f0f211c62 bc1869d69f581788ee59 37 48 27 44 56 25 38 14 18. 14fd3fa5756432336c73 656c76f4751aa6f707f9 b9acd4446a9ee133799f a3d8f3e35e001c616776 37 16 24 36 38 8 10 11 19. f1d8238c9141f46246bf 2193908b1be6f87b09f8 f1513655d577bf56bcf86 2b1851e66bb683d373c 33 56 48 56 61 32 27 46 20. 50effcaad368f00bfc71 2105a708ff917f9f95d0 49a48ed249c7b82959aa 85b9470938bbcc9c45cc 36 36 27 38 46 16 28 31 t. jamgharyan 75 in epochs 1-3 of training for compiled software, the results of the capsule neural network are worse worse than the results of the convolutional neural network and ssdeep software. fig. 4. ctph results of the obfuscated mimikatz fig. 5. ctph results of obfuscated and compiled source code. mimikatz source code. the use of a convolutional neural network is not always justified, since the degree of detection is comparable to the degree of detection by ssdeep software. the use of a capsule neural network for malware detection is justified in the presence of the source code (even in an obfuscated state), since even after the first training epoch, the detection results are not worse (and in most cases better) than the detection results using ssdeep and a trained convolutional neural network. tables 3 and 4 present the results of the studies of the operation of capsule and convolutional neural networks, based on datasets obtained from the obfuscated mimikatz source code with three training epochs and a variable block size of ctph. table 3. number of detected threats. n u m b e r o f d a ta se ts (d a ta se t 1 ) n u m b e r o f d a ta se ts (d a ta se t 2 ) the number of samples detected and classified as threats on different sizes (20, 40, 128 bytes) and three epochs (i, ii, iii) of training by a capsule neural network the number of samples detected and classified as a threat at different sizes (20, 40, 128 bytes) and three epochs (i, ii, iii) of training by a convolutional neural network number of detected but mismatched malware samples * ctpn size (byte) 20 40 128 20 40 128 training epoch i ii iii i ii iii i ii iii i ii iii i ii iii i ii iii i ii iii 100 100 7 7 9 11 13 12 12 15 18 3 3 4 4 6 6 9 10 1 1 200 200 10 11 11 12 14 16 17 17 21 5 4 6 6 8 5 8 5 6 1 2 300 300 12 12 14 16 18 23 28 29 22 8 7 8 8 9 11 13 15 16 1 1 2 350 350 12 13 15 15 16 18 21 26 25 7 7 11 10 12 18 16 18 19 2 2 3 450 450 14 16 19 19 22 26 29 34 38 10 9 11 12 16 18 18 21 20 2 1 4 500 500 14 16 18 19 21 27 29 33 36 11 10 13 16 15 15 17 19 19 2 2 4 600 600 22 25 29 30 34 35 39 41 44 14 15 11 19 24 26 20 25 26 3 3 3 800 800 37 41 46 48 52 55 57 57 60 22 26 27 29 34 37 39 44 45 5 4 6 950 950 42 42 46 47 58 60 66 68 68 28 29 28 31 33 39 42 46 49 4 4 4 1000 1000 42 43 47 50 51 59 61 65 69 34 33 35 30 35 39 49 52 55 5 6 3 research of obfuscated malware with a capsule neural network 76 *the number of detected but mismatched malware samples separately detected by both neural networks. these samples were output to a special dataset and verified by publicly available malware detection resources. table 4. number of detected threats. fig. 6 shows a report from the virustotal service when examining one of the mimikatz malware samples detected by neural networks. in particular, the virustotal service did not detect either the file type or whether ctph (based on ssdeep) belongs to a particular type of malware. fig. 6.virustotal service report. n u m b e r o f d a ta se ts (d a ta se t 1 ) n u m b e r o f d a ta se ts (d a ta se t 2 ) the number of samples detected and classified as threats at different sizes (256, 512, 1024 bytes) and three epochs (i, ii, iii) of training by a capsular neural network the number of samples detected and classified as a threat at different sizes (20, 40, 128 bytes) and three epochs (i, ii, iii) of training by a convolutional neural network number of detected but mismatched malware samples * ctpn size (byte) 256 512 1024 256 512 1024 training epoch i ii iii i ii iii i ii iii i ii iii i ii iii i ii iii i ii iii 100 100 18 14 16 14 16 19 8 12 14 7 11 14 9 11 14 7 8 11 1 1 200 200 18 12 12 14 18 19 11 13 10 3 4 3 5 8 11 5 9 14 1 1 2 300 300 17 19 16 14 17 12 10 21 23 9 11 10 8 12 9 8 8 13 2 2 350 350 18 18 21 18 21 23 23 27 27 9 15 17 12 18 14 14 11 12 2 2 3 450 450 22 26 28 29 29 34 20 23 25 12 15 13 20 16 16 17 29 13 2 5 3 500 500 23 24 29 31 33 30 28 21 32 16 12 15 22 22 25 28 26 25 3 7 7 600 600 28 31 30 32 35 39 34 38 41 20 24 21 24 28 25 29 34 31 5 6 6 800 800 37 37 39 41 46 39 42 46 49 31 28 34 34 25 27 39 32 34 7 9 11 950 950 48 53 53 52 58 56 64 65 56 34 30 31 35 38 38 39 42 45 11 9 10 1000 1000 47 52 51 56 61 60 64 66 68 40 42 46 42 44 44 47 49 51 8 11 12 t. jamgharyan 77 tables 5 and 6 present the results of the studies of the operation of capsule and convolutional neural networks, based on data sets from the obfuscated compiled code of the mimikatz software. table 5. number of detected threats. table 6. number of detected threats n u m b e r o f d a ta se ts (d a ta se t 1 ) n u m b e r o f d a ta se ts (d a ta se t 2 ) the number of samples detected and classified as threats at different sizes (20, 40, 128 bytes) and three epochs (i, ii, iii) of training by a capsular neural network the number of samples detected and classified as a threat at different sizes (20, 40, 128 bytes) and three epochs (i, ii, iii) of training by a convolutional neural network number of detected but mismatched malware samples * ctpn size (byte) 20 40 128 20 40 128 training epoch i ii iii i ii iii i ii iii i ii iii i ii iii i ii iii i ii iii 100 100 2 1 2 3 2 3 3 4 4 2 2 3 3 3 4 5 2 3 200 200 3 2 3 3 4 2 2 3 3 1 1 2 2 3 2 4 3 4 300 300 3 4 4 4 4 5 3 5 5 2 3 3 4 3 4 4 4 4 1 350 350 3 3 4 4 5 5 5 6 6 3 3 3 3 4 5 5 4 4 1 1 450 450 4 5 5 5 6 6 6 8 9 3 4 4 4 5 6 5 7 7 1 500 500 3 5 5 5 6 8 8 9 11 4 4 5 5 7 9 9 10 10 2 2 600 600 5 6 6 6 8 9 11 11 12 5 4 7 7 9 11 10 11 10 1 1 1 800 800 7 6 7 7 8 11 13 14 14 6 8 9 8 8 9 8 11 13 2 1 2 950 950 9 9 10 11 9 11 12 15 15 8 10 10 11 13 15 14 15 17 2 2 3 1000 1000 11 13 14 14 14 15 17 19 18 10 11 11 11 13 16 18 21 23 2 4 4 n u m b e r o f d a ta se ts (d a ta se t 1 ) n u m b e r o f d a ta se ts (d a ta se t 2 ) the number of samples detected and classified as threats at different sizes (256, 512, 1024 bytes) and three epochs (i, ii, iii) of training by a capsular neural network the number of samples detected and classified as a threat at different sizes (256, 512, 1024 bytes) and three epochs (i, ii, iii) of training by a convolutional neural network number of detected but mismatched malware samples * ctpn size (byte) 256 512 1024 256 512 1024 training epoch i ii iii i ii iii i ii iii i ii iii i ii iii i ii iii i ii iii 100 100 9 11 12 12 14 14 15 16 16 8 8 10 11 13 12 11 11 10 1 200 200 10 12 13 14 13 13 15 15 12 11 10 11 12 11 13 12 13 14 1 1 300 300 11 12 12 15 17 18 19 18 18 10 12 13 12 14 14 15 14 14 350 350 11 11 12 12 12 16 15 11 14 14 12 13 15 15 15 18 19 21 1 2 450 450 13 12 13 13 15 15 16 17 18 11 12 13 14 16 16 15 17 19 2 3 3 500 500 12 14 14 14 15 14 15 11 12 11 10 11 13 14 12 15 15 16 1 2 600 600 10 11 12 10 12 12 12 14 13 9 10 11 12 10 10 14 15 14 1 2 2 800 800 12 14 15 15 16 17 17 18 18 16 14 15 15 16 17 18 21 19 2 3 3 950 950 12 13 12 14 15 15 16 18 19 12 12 13 14 15 16 12 15 16 2 3 4 1000 1000 12 12 13 13 15 16 16 17 18 11 10 12 15 16 17 18 19 20 2 2 3 research of obfuscated malware with a capsule neural network 78 given the malware source code (or fragment), the capsule neural network performs better than the convolutional neural network in detecting obfuscated malware. but when compiled, the detection performance of the capsular neural network decreases. also, both neural networks separately detected a small set of data and software fragments classified as malware. figures. [7]-[12] show a visualization of the output data of a capsule neural network with 3 training epochs and ctpn datasets, 20, 40, 80, 128, 256, 512 bytes. fig. 7. visualization of malware detection results fig. 8. visualization of malware detection by capsule neural network. by capsule neural network. (i training epoch, ctph size 20 bytes) (i training epoch, ctph size 40 bytes) fig. 9. visualization of malware detection results fig. 10. visualization of malware detection by capsule neural network. by capsule neural network. (ii training epoch, ctph size 80 bytes) (ii training epoch, ctph size 128 bytes) fig. 11. visualization of malware detection results fig. 12. visualization of malware detection results by capsule neural network. by capsule neural network. (iii training epoch, ctph size 256 bytes) (iii training epoch, ctph size 512 bytes) t. jamgharyan 79 with an increase in the size of the ctph files (interval 256, 512, 1024 bytes) for training the capsule network, the increase in the detection of the number of malware code fragments is insignificant (0.3-0.5%, fig. 7, fig. 8, table 6) in contrast to files 20 , 40, 128 bytes (12-14% increase). but increasing the size of the ctph file allows increasing the editorial distance (figure 9-12) to granularly group malware by type. 4. conclusion this paper proposes the use of transfer learning of a capsule neural network to detect obfuscated malware. convolutional and capsule neural networks were trained on the same datasets. the source codes of mimikatz, athena, engrat, grum, surtr, dyre malware were used as datasets. when building an intrusion detection system using neural networks, their complex application is necessary. annotated malware datasets are critical when training neural networks. the use of transfer learning of a capsule neural network to detect malware is justified if the source code of the malware or its fragments (preferably the first versions) is available. in this case, the neural network detects malware, even with its high degree of obfuscation. but in the absence of source code, the effectiveness drops, yielding to «standard» means of detecting malware. the use of the ctph method for generating «weight» coefficients of a neural network is most effective with a small file size of ctph. increasing the editorial distance increases the selectivity of detecting different types of malware. references [1] d. ashok kumar and s. r.venugopalan, “intrusion detection systems: a review” international journal of advanced research in computer science, vol. 8, no 8, pp.356-370, 2017. 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[online]. available https://www.guardsquare.com/proguard կապսուլային նեյրոնային ցանցով օբֆուսկացված վնասաբեր ծրագրային ապահովման հետազոտում թիմուր վ․ ջամղարյան հայաստանի ազգային պոլիտեխնիկական համալսարան e-mail: t.jamgharyan@yandex.ru ամփոփում ներխուժման հայտնաբերման և կանխարգելման համակարգերը ցանցային ենթակառուցվածքի անվտանգության ապահովման անբաժանելի բաղադրիչն են: «դասական» ներխուժման հայտնաբերման և կանխարգելման համակարգերը չեն կարողանում հայտնաբերել այնպիսի սպառնալիքներ, որոնք նկարագրված չեն համակարգի կանոններում։ բացի այդ, նաև բաց խնդիր է համարվում օբֆուսկացիայի ենթարկված վնասաբեր ծրագրային ապահովման հայտնաբերումը։ ծրագրային ապահովման և ցանցային ենթակառուցվածքի անվտանգությունով զբաղվող հետազոտողները, փորձում են նշված խնդիրը լուծել մեքենայական ուսուցման միջոցով։ հետազոտությունում ներկայացված են փոխանցման ուսուցման մեթոդով ուսուցանված կապսուլային նեյրոնային ցանցի ցուցաբերած արդյունքները վնասաբեր ծրագրային ապահովման հայտնաբերելու հարցում։ հետազոտությունը իրականացվել է վնասաբեր ծրագրային ապահովման ելակետային կոդի հիման վրա, կիրառելով համատեքստա-մասնատված հեշավորման մեթոդը։ վնասաբեր ծրագրային ապահովման ելակետային կոդերը ստացվել են հանրահասանելի աղբյուրներից։ կապսուլային նեյրոնային ցանցի ուսումնասիրության արդյունքները համեմատվել են նախապես ուսուցանված փաթույթային նեյրոնային ցանցի և վնասաբեր ծրագրային ապահովման հայտնաբերելու հանրահասանելի համացանցային ծառայությունների միջոցով։ մշակված ծրագրային ապահովման ելակետային կոդերը, նախապես ուսուցանված մոդելը, տվյալների հավաքածուների մի մասը, հոդվածում չներառված հետազոտության արդյունքները հասանելի են https://github.com/t-jn կայքում։ բանալի բառեր՝ կապսուլային նեյրոնային ցանց, անորոշ հեշավորում, ներխուժման հայտնաբերման համակարգ, խմբագրական հեռավորույուն, ցանցային ենթակառուցվածք: https://www.virustotal.com/ https://www.herdprotect.com/ https://docs.microsoft.com/ru-ru/visualstudio/ide/dotfuscator/capabilities?view=vs-2022 https://docs.microsoft.com/ru-ru/visualstudio/ide/dotfuscator/capabilities?view=vs-2022 https://www.guardsquare.com/proguard https://github.com/t-jn t. jamgharyan 83 исследование обфусцированного вредоносного программного обеспечения с помощью капсульной нейронной сети тимур в. джамгарян национальный политехнический университет армении e-mail: t.jamgharyan@yandex.ru аннотация системы обнаружения и предотвращения вторжений являются неотьемлимым компонентом безопасности сетевой инфраструктуры. классические системы обнаружения и предотвращения вторжений не в состоянии обнаружить угрозу не описанную в наборе правил. также нерешенной полностью задачей является: задача обнаружения вредоносного программного обеспечения подвергнутого обфускации. исследователи в сфере безопасности программного обеспечения и сетевой инфраструктуры пытаются решить данные задачи с помощью машинного обучения. в работе представлены результаты исследования использования трансферного обучения капсульной нейронной сети для обнаружения вредоносного программного обеспечения. исследование проводилось на основе исходного кода вредоносного программного обеспечения с использованием метода контекстно-кусочного хеширования. исходные коды вредоносного программного обеспечения были получены из общедоступных источников программного обеспечения. проверка результатов обучения капсульной нейронной сети проводилась с использованием обученной сверточной нейронной сети и общедоступных источников тестирования вредоносного программного обеспечения. исходные коды разработанного программного обеспечения, часть наборов данных для обучения нейросети, результаты исследования не внесенные в статью представлены по адресу https://github.com/t-jn ключевые слова: капсульная нейронная сеть, нечеткое хэширование, система обнаружения вторжений, редакционное расстояние, трансферное обучение. https://github.com/t-jn microsoft word operation of alternative.doc îçµ»éý»ïçï³ûç ¨ ñ³ßíáõ³ï³ý ï»ëýçï³ûç ù³ã»ù³ïçï³ï³ý ñ³ñó»ñ 26, 2006, 54–63. 54 äñáó»ëý»ñç ³ûéáýïñ³ýù³ûçý ñ³ù³ïóù³ý ·áñíáõáõãûáõýá ¨ ýñ³ çñ³ï³ý³óáõùá è¨áý ð. ð³ûñ³å»ïû³ý ºñ¨³ýç ä»ï³ï³ý ð³ù³éë³ñ³ý e-mail levon.hayrapetyan@gmail.com ²ù÷á÷áõù ²ßë³ï³ýùáõù ¹çï³ñïí³í ¿ åñáó»ëý»ñç ³ûéáýïñ³ýù³ûçý ñ³ù³ïóù³ý ·áñíáõáõãûáõýá áñáß³ïç ûµû»ïïá ïáõùýáñáßí³í ùá¹»éáõù: ²ûý ïñí³í åñáó»ëý»ñá ùç³íáñáõù ¿ ¨ ï³éáõóáõù ¿ ù»ï ýáñ åñáó»ë, áñç ï³ï³ñáõùá ñ³ù³ñå»ù ¿ ³õµûáõñ åñáó»ëý»ñç ëçýëñáý ï³ï³ñù³ýá: ²ßë³ï³ýùáõù ë³ñù³ýí³í ¿ ³é·áñçãù, áñá ïñí³í åñáó»ëý»ñç ñ³ù³ñ ï³éáõóáõù ¿ ýñ³ýó` å³ù³ý³ïç ï»ë³ï»ïçó ûåïçù³é ñ³ù³ïóáõùá: ¶ý³ñ³ïí³í ¿ ³é·áñçãùç µ³ñ¹áõãûáõýá ¨ ýï³ñ³·ñí³í »ý çñ³ï³ý³óù³ý ù³ýñ³ù³ëý»ñá: ¶ñ³ï³ýáõãûáõý [1]. grady booch et.al., unified modeling language user guide, pearson education, 1999. [2]. l. hayrapetyan. “alternative combination of linear processes”. in proceedings of csit’2005, armenia, september 2005, pp 65-69. [3]. donald knuth. “the art of computer programming, volume 3: sorting and searching”, third edition. addison-wesley, 1997. isbn 0-201-89685-0. [4]. p. raulefs. “the virtual factory”, ifip world computer congress’94, v.2, pp.18-30, 1994. [5]. p. raulefs; s. shoukourian; a. grigoryan. “transformation of hammock type processes”, in proceedings of hpc‘2002, scs international advanced simulation technologies conference astc’2002, usa, april 2002, pp. 288-293. [6]. s. shoukourian, a. avagyan, d. tavangarian, “combination of separate processes in a distributed environment. a case of study.” in proceedings of hpc’2000, scs international advanced simulation technologies conference astc’2000, usa, april 2000, pp. 280-285. [7]. b. l. van der waerden, algebra, springer-verlag, 1971 è. ð. ð³ûñ³å»ïû³ý 55 operation of alternative combination of processes and its implementation l. hayrapetyan abstract in this paper an object oriented model for processes is used to describe the operation of alternative combination of processes. for specified processes this operation constructs a new process, execution of which is equivalent to the synchronized execution of source processes. an algorithm is proposed that constructs the optimal (by time) combination of given source processes. also, complexity of the algorithm is evaluated and implementation details are described. d:\sbornik\...\article.dvi mathematical problems of computer science 26, 2006, 28{32. on i nter val color ings of complete k¡ par tite gr aphs kkn r a fa ye l r . k a m a lia n , p e t r o s a . p e t r o s ya n institue for informatics and automation problems (iiap) of nas of ra e-mail rrkamalian@yahoo.com, pet petros@yahoo.com abstract problems of existence, construction and estimation of parameters of interval colorings of complete k-partite graphs kkn are investigated. refer ences [1 ] f. h a r a r y, gr a p h th e o r y, a d d is o n -w e s le y, r e a d in g , ma , 1 9 6 9 . [2 ] v .g. v iz in g , th e c h r o m a t ic in d e x o f a m u lt ig r a p h , k ibernetika 3 ( 1 9 6 5 ) , p p . 2 9 -3 9 . [3 ] a .a . zyko v, th e o r y o f ¯ n it e g r a p h s , n o vo s ib ir s k, n a u ka , 1 9 6 9 . [4 ] a .s . a s r a t ia n , r .r . k a m a lia n , in t e r va l c o lo r in g s o f e d g e s o f a m u lt ig r a p h , appl. m ath. 5 ( 1 9 8 7 ) , p p . 2 5 -3 4 . [5 ] r .r . k a m a lia n , in t e r va l e d g e co lo r in g s o f gr a p h s , d o c t o r a l d is s e r t a t io n , th e in s t it u t e o f ma t h e m a t ic s o f t h e s ib e r ia n b r a n c h o f t h e a c a d e m y o f s c ie n c e s o f u s s r , n o vo s ib ir s k, 1 9 9 0 . [6 ] p .a . p e t r o s ya n , in t e r va l e d g e c o lo u r in g s o f c o m p le t e g r a p h s a n d n -c u b e s , abstracts of 5th international algebraic conference in ukraine, od e s s a , 2 0 0 5 , p . 1 5 4 . [7 ] r . d ie s t e l, gr a p h th e o r y, s p r in g e r -v e r la g , n e w y o r k, 2 0 0 0 . [8 ] d .g. h o ®m a n , c.a . r o d g e r , th e c h r o m a t ic in d e x o f c o m p le t e m u lt ip a r t it e g r a p h s , j . graph theory 1 6 , ( 1 9 9 2 ) , p p . 1 5 9 -1 6 3 . kkn éñçí k ïáõù³ýç ·ñ³ýý»ñç ùçç³ï³ûù³ûçý ý»ñïáõùý»ñç ù³ëçý è. ø³ù³éû³ý, ä. ä»ïñáëû³ý ²ù÷á÷áõù ²ßë³ï³ýùáõù ñ»ï³½áïíáõù »ý kkn éñçí k ïáõù³ýç ·ñ³ýý»ñç ùçç³ï³ûù³ûçý ý»ñïáõùý»ñç ·áûáõãû³ý, ï³éáõóù³ý ¨ ãí³ûçý å³ñ³ù»ïñ»ñç ·ý³ñ³ïù³ý ñ³ñó»ñ: 2 8 mathematical problems of computer science 58, 7–19, 2022. doi:10.51408/1963-0088 udc 519.6 analytical inversion of tridiagonal hermitian matrices yuri r. hakopian and avetik h. manukyan yerevan state university, yerevan, armenia e-mail: yuri.hakopian@ysu.am, avetiq.manukyan1@ysumail.am abstract in this paper we give an algorithm for inverting complex tridiagonal hermitian matrices with optimal computational efforts. for matrices of a special form and, in particular, for toeplitz matrices, the derived formulas lead to closed-form expressions for the elements of inverse matrices. keywords: inverse matrix, tridiagonal matrix, hermitian matrix, toeplitz matrix. article info: received 21 aprile 2022; received in revised form 15 july 2022; accepted 23 august 2022. 1. introduction tridiagonal matrices are encountered in many areas of applied mathematics. such matrices are of great importance in finite difference and finite element methods for differential equations. the construction of cubic splines is reduced to solving systems with tridiagonal matrices. symmetric matrices are reduced to tridiagonal matrices by the similarity householder transformation (see [1, 2, 3], for instance). other examples can be cited. there is a well-known fast numerical method for solving systems with tridiagonal matrices. at the same time, the analytical matrix inversion is also of certain interest (see [4, 5, 6], for instance). for tridiagonal matrices of special types, this leads to closed-form expressions for the elements of inverse matrices [7, 8, 9, 10]. this is undoubtedly useful in theoretical considerations. further, explicit formulas can be a part of more general computational procedures. there are other reasons as well. in this article, we focus our attention on complex hermitian tridiagonal matrices. we will construct a fairly simple computational procedure, consisting of a sequence of recurrence relations, leading to the calculation of the elements of the inverse matrix. in special cases, in particular for toeplitz tridiagonal hermitian matrices, the procedure can become the basis for deriving closed-form expressions for the elements of the inverse matrix. we note right away that throughout this article z stands for the complex conjugate of the complex number z. 7 8 analytical inversion of tridiagonal hermitian matrices let a nonsingular tridiagonal hermitian matrix a =   a1 b1 b1 a2 b2 0 ... ... ... 0 bn−2 an−1 bn−1 bn−1 an   (1) be given, where ai, i = 1, 2, . . . , n are real numbers and bi ̸= 0 for i = 1, 2, . . . , n − 1. in accordance with the accepted notation, a = a∗. we assume that n > 3. the requirement that the subdiagonal (superdiagonal) elements of the matrix be nonzero is not restrictive. indeed, if some of these elements are equal to zero, the problem of computing the inverse matrix is decomposed into several similar problems for tridiagonal matrices of lower order. 2. preliminary calculations let a−1 = [xij]n×n. this matrix is also hermitian. in our considerations we will use the notation x(j) ≡ [x1 j x2 j . . . xn j]t , j = 1, 2, . . . , n for the columns of the inverse matrix. the matrix a can be represented as a product a = db (2) of the matrices d = diag [b1, b1, b2, . . . , bn−2, bn−1] (3) and b =   p 1 1 f2 g2 0 1 f3 g3 ... ... ... 0 1 fn−1 gn−1 1 q   , (4) where fi = ai bi−1 , gi = bi bi−1 , i = 2, 3, . . . , n − 1; p = a1 b1 , q = an bn−1 . (5) having a nonsingular matrix b defined in (4), let us consider the following system of linear algebraic equations pµ1 + µ2 = α µi−1 + fiµi + giµi+1 = 0, 2 ≤ i ≤ n − 1 µn−1 + qµn = 0, (6) where we will set the right-hand side α of the first equation a little later. it is easy to verify that regardless of the choice of α, the recursively defined quantities µn = 1 , µn−1 = −q , µi−1 = −fiµi − giµi+1 , i = n − 1, n − 2, . . . , 2 (7) yu. hakopian and a. manukyan 9 satisfy all equations of the system (6), starting with the second one. then, we choose the quantity α as follows: α = pµ1 + µ2. (8) remark 1 since, by assumption, the matrix b is nonsingular (it follows from (2)), then α ̸= 0. indeed, otherwise we would have obtained that the homogeneous system (6) has a nontrivial solution. further, α = a1 b1 µ1 + µ2 = 1 b1 (a1µ1 + b1µ2). therefore a1µ1 + b1µ2 ̸= 0 as well. thus, α = b−11 t −1, (9) where t ≡ (a1µ1 + b1µ2)−1. (10) let us introduce the vector r(1) ≡ [µ1 µ2 . . . µn]t , the components of which are specified in (7). as follows from (4), (6) and (9), br(1) = [α 0 . . . 0]t = αe(1) = b−11 t −1e(1), where e(1) ≡ [1 0 . . . 0]t . further, on the basis of factorization (2) of the matrix a, we obtain the equality ar(1) = dbr(1) = b−11 t −1de(1) = t−1e(1); (11) here we have used the obvious equality de(1) = b1e (1) (see (3)). the equality (11) allows to compute the first column of the inverse matrix a−1. indeed, from this equality we find that a−1e(1) = tr(1). since a−1e(1) = x(1), then x(1) = tr(1), or xi1 = tµi, i = 1, 2, . . . , n. (12) thus, we have found the first column of the inverse matrix. similarly, we can calculate the last column of the matrix a−1. for this purpose, let us consider the linear system pν1 + ν2 = 0 νi−1 + fiνi + giνi+1 = 0, 2 ≤ i ≤ n − 1 νn−1 + qνn = β, (13) where we will set the right-hand side β of the last equation later. regardless of the choice of β, the recursively defined quantities ν1 = 1 , ν2 = −p , νi+1 = − 1 gi (νi−1 + fiνi) , i = 2, 3, . . . , n − 1 (14) 10 analytical inversion of tridiagonal hermitian matrices satisfy the first n−1 equations of the system (13). then we choose the quantity β as follows: β = νn−1 + qνn. (15) since the matrix b is nonsingular, then β ̸= 0 (see remark 1). substituting the expression of the quantity q given in (5) into (15) yields β = νn−1 + an bn−1 νn = 1 bn−1 (bn−1νn−1 + anνn). thus, β = bn−1 −1 θ−1, (16) where θ ≡ (bn−1νn−1 + anνn)−1. now let us introduce the vector r(n) ≡ [ν1 ν2 . . . νn]t , the components of which are specified in (14). from (4), (13) and (16) we find that br(n) = [0, . . . 0 β]t = βe(n) = bn−1 −1 θ−1e(n), where e(n) ≡ [0 . . . 0 1]t . having the factorization (2) of the matrix a, we obtain the equality ar(n) = dbr(n) = bn−1 −1 θ−1de(n) = θ−1e(n). from here, a−1e(n) = θr(n). since a−1e(n) = x(n), then x(n) = θr(n), or xin = θνi, i = 1, 2, . . . , n. (17) let us refine the last expression. from (12), xn1 = tµn = t. further, according to (17), x1n = θν1 = θ. since a −1 is a hermitian matrix, then x1n = xn1. consequently, θ = t, and we come to the conclusion that xin = tνi, i = 1, 2, . . . , n. (18) so, we have found the first and the last columns of the hermitian matrix a−1. these are expressions (12) and (18). taking into account that ν1 = 1 and µn = 1, we write these elements in the form of xi1 = tµiν1, xin = t µnνi, i = 1, 2, . . . , n. (19) moreover, the diagonal elements x11 = tµ1ν1 and xnn = t µnνn are real numbers. therefore, we can write xnn = tµnνn as well. looking ahead, we say that in the next section we will prove that the quantities tµiνi, i = 2, 3, . . . , n − 1 (20) are the remaining diagonal elements of the matrix a−1. to do this, here we first establish that the quantities (20) are real numbers (naturally, without assuming that they are somehow related to the matrix a−1). let us introduce into consideration the quantities ri ≡ bi−1(tµiνi−1) + bi−1(tµi−1νi), i = 2, 3, . . . , n − 2. (21) yu. hakopian and a. manukyan 11 lemma 1. the quantity r2 is a real number. proof. since ν1 = 1 and ν2 = −p (see (2.13)), then r2 = t(b1µ2ν1 + b1µ1ν2) = tb1(µ2 − pµ1). further, taking into account the equalities (8) and (9), we get r2 = tb1(α − 2pµ1) = tb1α − 2pb1(tµ1) = 1 − 2a1(tµ1). the quantities a1 and tµ1 are real numbers, so r2 is also a real number. 2 lemma 2. the quantities ri from (21) satisfy the relations ri = −ri−1 − 2ai−1(tµi−1νi−1), i = 3, 4, . . . , n − 2. (22) proof. from (6) we have the equality µi−2 + fi−1µi−1 + gi−1µi = 0. using formulas (5), let us write this equality in the form of bi−2µi−2 + ai−1µi−1 + bi−1µi = 0. multiplying both parts of the last equality by tνi−1, we get that bi−1(tµiνi−1) = −bi−2(tµi−2νi−1) − ai−1(tµi−1νi−1). (23) similarly, from (13) we have the equality νi−2 + fi−1νi−1 + gi−1νi = 0, which can be written as follows: bi−2νi−2 + ai−1νi−1 + bi−1νi = 0. multiplying both parts of this equality by tµi−1 yields bi−1(tµi−1νi) = −bi−2(tµi−1νi−2) − ai−1(tµi−1νi−1). (24) the relation (22) follows directly from the equalities (23) and (24). 2 lemma 3. the quantities tµiνi, i = 2, 3, . . . , n − 1 are real numbers. proof. consider first the quantity tµ2ν2. since pµ1 + µ2 = α and ν2 = −p (see (6) and (14)), then tµ2ν2 = t(pµ1 − α)p = (pp)(tµ1) − tαp. further, using the equality (9), we obtain that tµ2ν2 = (pp)(tµ1) − p b1 = (pp)(tµ1) − a1 b1b1 . thus, the quantity tµ2ν2 is a real number. 12 analytical inversion of tridiagonal hermitian matrices next, consider the quantity tµ3ν3. as follows from (6) and (13), µ3 = − a2 b2 µ2 − b1 b2 µ1, ν3 = − a2 b2 ν2 − b1 b2 ν1. proceeding from these equalities, we get that tµ3ν3 = 1 b2b2 [ a22(tµ2ν2) + b1b1(tµ1ν1) + a2r2 ] . the quantities tµ1ν1 and tµ2ν2 are real numbers. according to lemma 1, the quantity r2 is also a real number. therefore, tµ3ν3 is a real number as well. further reasoning will be carried out by the method of mathematical induction on i. suppose that for some value of i, where 3 ≤ i ≤ n − 2, it is already known that the quantities tµkνk, k ≤ i and rk, k ≤ i − 1 are real numbers. from (6) and (13) we have µi+1 = − ai bi µi − bi−1 bi µi−1, νi+1 = − ai bi νi − bi−1 bi νi−1. then tµi+1νi+1 = 1 bibi [ a2i (tµiνi) + bi−1bi−1(tµi−1νi−1) + airi ] . hence, by virtue of the assumptions made and taking into account the assertion of lemma 2, we arrive at a conclusion that the quantity tµi+1νi+1 is a real number. 2 remark 2 we have established that the quantities tµiνi, i = 1, 2, . . . , n are real numbers. therefore, tµiνi = t µiνi. 3. the elements of the inverse matrix above we obtained the expressions (19) for the elements of the first and the last columns of the inverse matrix, as well as some auxiliary statements. based on these results, here we derive formulas for the remaining elements of the inverse matrix. let 2 ≤ j ≤ n − 1. we introduce into consideration the vector r(j) ≡ [ t µjν1 , . . . , t µjνj−1, tµjνj, tµj+1νj , . . . , tµnνj]t , (25) where the quantities µi and νi are specified in (7) and (14), respectively. multiplying the matrix b defined in (4) and the vector r(j) yields br(j) = z(j), (26) where the components of the vector z(j) = [z (j) 1 z (j) 2 . . . z (j) j−1 δj z (j) j+1 . . . z (j) n−1 z (j) n ] t are calculated as follows: z (j) 1 = t µj(pν1 + ν2), z (j) i = t µj(νi−1 + fiνi + giνi+1), 2 ≤ i ≤ j − 1, δj = t µjνj−1 + fj(tµjνj) + gj(tµj+1νj), z (j) i = t(µi−1 + fiµi + giµi+1)νj, j + 1 ≤ i ≤ n − 1, z(j)n = t(µn−1 + qµn)νj. yu. hakopian and a. manukyan 13 having equations (6) and (13), we conclude that z (j) i = 0 for 1 ≤ i ≤ j −1 and j +1 ≤ i ≤ n. thus, z(j) = [0 . . . 0 δj 0 . . . 0] t = δje (j), (27) where e(j) = [0 . . . 0 1 0 . . . 0]t (the unit is located on jth place). it remains to clarify the quantity δj. taking into account remark 2, we have δj = t µjνj−1 + fj(t µjνj) + gj(tµj+1νj) = t µj(νj−1 + fjνj) + gj(tµj+1νj). (28) since νj−1 + fjνj = −gjνj+1 (see (13)), then δj = gj(tµj+1νj − t µjνj+1), 2 ≤ j ≤ n − 1. (29) let us get one more representation of the quantity δj. since gjµj+1 = −µj−1 −fjµj (see (6)), then from(28) it follows that δj = t µjνj−1 − tµj−1νj + fj(t µjνj − tµjνj). from here, according to remark 2, we obtain δj = t µjνj−1 − tµj−1νj, 2 ≤ j ≤ n − 1. (30) assuming that 3 ≤ j ≤ n − 1, we can write the expression (30) in the form of δj = 1 gj−1 gj−1(tµjνj−1 − t µj−1νj). comparing with the record (29), we arrive at the relation δj = 1 gj−1 δj−1, 3 ≤ j ≤ n − 1. (31) based on the relation (31), one can easily show that δj =   bj−1 −1 b1δ2 , if j is odd, bj−1 −1 b1δ2 , if j is even. (32) finally, let us calculate the quantity δ2. according to the representation (30), we have δ2 = t µ2ν1 − tµ1ν2 = t µ2 + tµ1p = t µ2 + t µ1 p = t (µ2 + p µ1) = t α = b1 −1 , (33) (see (6) and (9)). thus, from (32) and (33) we conclude that δj = bj−1 −1 , j = 2, 3, . . . , n − 1. (34) summing up the results, from (27) and (34) we come to the equality z(j) = bj−1 −1 e(j). (35) 14 analytical inversion of tridiagonal hermitian matrices proceeding from the factorization (2) of the matrix a and using the equalities (26) and (35), we have ar(j) = dbr(j) = dz(j) = bj−1 −1 de(j) = e(j) (note that de(j) = bj−1e (j), which follows from (3)). further, a−1e(j) = r(j). since a−1e(j) = x(j), then x(j) = r(j). the components of the vector r(j) are given in (25). thus, xij = t µjνi, i = 1, 2, . . . , j − 1 and xij = tµiνj, i = j, j + 1, . . . , n. (36) combining formulas (36) with those of (12) and (18) yields xij =   t µjνi, i = 1, 2, . . . , j − 1, tµiνj, i = j, j + 1, . . . , n for j = 1, 2, . . . , n. (37) note the following. since the matrix a−1 is also hermitian, then in reality we only need to calculate the lower triangular part of this matrix. summarizing the considerations of sections 2 and 3, let us write the following procedure to calculate the elements of the inverse matrix a−1 = [xij]n×n for nonsingular matrix a given in (1). procedure inv 3d hermitian 1. input elements a1, a2, . . . , an and b1, b2, . . . , bn−1 of the matrix a (see (1)). 2. calculate the quantities fi, gi, p and q (see (5)): fi = ai bi−1 , gi = bi bi−1 , i = 2, 3, . . . , n − 1; p = a1 b1 , q = an bn−1 . 3. calculate recursively the quantities µi (see (7)): µn = 1 , µn−1 = −q , µi = −fi+1µi+1 − gi+1µi+2 , i = n − 2, n − 3, . . . , 1. 4. calculate recursively the quantities νi (see (14)): ν1 = 1 , ν2 = −p , νi = − 1 gi−1 (νi−2 + fi−1νi−1) , i = 3, 4, . . . , n. 5. calculate the quantity t (see (10) and remark 1): t = (a1µ1 + b1µ2) −1. 6. calculate the lower triangular part of the matrix a−1 (see (37)): xij = tµiνj, i = j, j + 1, . . . , n ; j = 1, 2, . . . , n . yu. hakopian and a. manukyan 15 7. set the upper triangular part of the matrix a−1 (see (37)): xij = xji, i = 1, 2, . . . , j − 1 ; j = 2, 3, . . . , n . 8. output the matrix a−1 = [xij]n×n. end procedure the procedure inv 3d hermitian can be useful for the following purposes. firstly, it can be used as a basis of numerical algorithms for inverting nonsingular tridiagonal hermitian matrices. in this case, it is easy to make sure that computing the lower triangular part of the matrix a−1 requires 0.5n2 + o(n) arithmetical operations with complex numbers. secondly, for matrices of special types, the procedure can be used for deriving closed form expressions for the elements of inverse matrices. the next section is devoted to this issue. 4. toeplitz tridiagonal hermitian matrices let us consider a matrix a =   a b b a b 0 ... ... ... 0 b a b b a   (38) of order n, where a is a real number and b ̸= 0. additionally, we assume that |a| ≥ 2|b|. (39) condition (39) ensures the nonsingularity of the matrix (38) (see [11], for instance). for the matrix we are considering, the quantities calculated in item 2 of the procedure inv 3d hermitian are as follows: fi = a b , gi = b b , i = 2, 3, . . . , n − 1; p = a b , q = a b . further, in item 3 of the procedure, the quantities µi are calculated. in our case, we have second-order recurrent relations bµi + aµi+1 + bµi+2 = 0 , i = n − 2, n − 3, . . . , 1, where µn = 1, µn−1 = −a/b. the solution of this problem is well known (see [2, 6], for instance). as a result of calculations, we get that µi = (−1)n−i b r [( a + r 2b )n+1−i − ( a − r 2b )n+1−i] , i = 1, 2, . . . , n if |a| > 2|b| (40) and µi = (−1)n−i (n + 1 − i) ( a 2b )i−n , i = 1, 2, . . . , n if |a| = 2|b|, (41) where r ≡ √ a2 − 4|b|2. 16 analytical inversion of tridiagonal hermitian matrices in a similar way, we find expressions for the quantities νi determined in item 4 of the procedure. these quantities satisfy the following second-order recurrent relations: bνi−2 + aνi−1 + bνi = 0 , i = 3, 4, . . . , n, where ν1 = 1, ν2 = −a/b. making calculations, we find that νi = (−1)i−1 b r [( a + r 2b )i − ( a − r 2b )i] , i = 1, 2, . . . , n if |a| > 2|b| (42) and νi = (−1)i−1 i ( a 2b )i−1 , i = 1, 2, . . . , n if |a| = 2|b|. (43) in item 5 of the procedure, the quantity t is calculated. using the expressions (40) and (41), we get t = (−1)n−1 r b 2 [( a + r 2b )n+1 − ( a − r 2b )n+1]−1 if |a| > 2|b| (44) and t = (−1)n−1 n + 1 2 a ( a 2b )n−1 if |a| = 2|b|. (45) finally, in items 6 and 7 of the procedure, the elements xij of the inverse matrix a −1 are calculated. if |a| > 2|b|, then we use the formulas (40), (42) and (44). for the values j = 1, 2, . . . , n, we obtain that xij = (−1)j−i r [( a + r 2b )i − ( a − r 2b )i] [(a + r 2b )n+1−j − ( a − r 2b )n+1−j] [( a + r 2b )n+1 − ( a − r 2b )n+1] (46) if i = 1, 2, . . . , j − 1 and xij = (−1)i−j r [( a + r 2b )n+1−i − ( a − r 2b )n+1−i] [(a + r 2b )j − ( a − r 2b )j] [( a + r 2b )n+1 − ( a − r 2b )n+1] (47) if i = j, j + 1, . . . , n. as an example, consider the matrix a =   5 2i −2i 5 2i 0 ... ... ... 0 −2i 5 2i −2i 5   . according to the expressions (46) and (47) we find that xij =   (2i − 2−i)(2n+1−j − 2−n−1+j) 3(2n+1 − 2−n−1) ii−j, i = 1, 2, . . . , j − 1, (2n+1−i − 2−n−1+i)(2j − 2−j) 3(2n+1 − 2−n−1) ii−j, i = j, j + 1, . . . , n, yu. hakopian and a. manukyan 17 where the symbol i stands for the imaginary unit. now consider the case |a| = 2|b|. for the values j = 1, 2, . . . , n, using the formulas (41), (43) and (45), we find that xij =   (−1)j−i (n + 1 − j)i n + 1 2 a ( a 2b )i−1 ( a 2b )j−1 , i = 1, 2, . . . , j − 1, (−1)i−j (n + 1 − i)j n + 1 2 a ( a 2b )i−1 ( a 2b )j−1 , i = j, j + 1, . . . , n. (48) for the matrix a =   2 i −i 2 i 0 ... ... ... 0 −i 2 i −i 2   , the expressions (48) take the following form: xij =   (−1)j (n − j + 1)i n + 1 ii+j, i = 1, 2, . . . , j − 1, (−1)j (n − i + 1)j n + 1 ii+j, i = j, j + 1, . . . , n, j = 1, 2, . . . , n. 5. conclusion in this paper, we have constructed the computational procedure inv 3d hermitian for inversion of tridiagonal hermitian matrices. this procedure can be used as a numerical algorithm with an optimal number of arithmetic operations (see the comment on the procedure at the end of section 3). in certain cases, the procedure can also be used to derive closed-form expressions for the elements of inverse matrices. in this regard, toeplitz tridiagonal hermitian matrices in section 4 were considered. references [1] g. h. golub and ch. f. van loan, matrix computations, the john hopkins university press, 1996. [2] d. kincaid and w. cheney, numerical analysis, brooks/cole, pacific grove, ca, 1991. [3] d. s. watkins, fundamentals of matrix computations, a wiley intercience publ., 2010. [4] b. buchberger and g. a. yemel’yanenko, ”methods for inverting tridiagonal matrices”, j. comput. math. and math. physics, vol. 13, no.3, pp. 546-554, 1973 (in russian). [5] m. el-mikkawy and a. karawia, ”inversion of general tridiagonal matrices”, applied math. letters, vol. 19, pp. 712-720, 2006. [6] v.p. il’in and yu. i. kuznetsov, tridiagonal matrices and their applications, (in russian), nauka, 1985. 1 8 analytical inversion of tridiagonal hermitian matrices [7 ] g.y . h u a n d r .f. o'co n n e ll, " a n a lyt ic a l in ve r s io n o f s ym m e t r ic t r id ia g o n a l m a t r ic e s " , j . p hys. a: m ath. gen., vo l. 2 9 , p p . 1 5 1 1 -1 5 1 3 , 1 9 9 6 . [8 ] y . h u a n g a n d w .f. mc co ll, " a n a lyt ic in ve r s io n o f g e n e r a l t r id ia g o n a l m a t r ic e s " , j . p hys. a: m ath. gen., vo l. 3 0 , p p . 7 9 1 9 -7 9 3 3 , 1 9 9 7 . [9 ] j. w . l e wis , " in ve r s io n o f t r id ia g o n a l m a t r ic e s " , numer. m ath., vo l. 3 8 , p p . 3 3 3 -3 4 5 , 1 9 8 2 . [1 0 ] r . a . u s m a n i, " in ve r s io n o f ja c o b i's t r id ia g o n a l m a t r ix" , computers m ath. applic., vo l. 2 7 , n o . 8 , p p . 5 9 -6 6 , 1 9 9 4 . [1 1 ] r . h o r n a n d ch . jo h n s o n , m atrix analysis, ca m b r id g e u n ive r s it y p r e s s , 1 9 8 6 . ºñ»ù³ýïûáõý³·í³ûçý ñ»ñùçïû³ý ù³ïñçóý»ñç ³ý³éçïçï ñ³ï³¹³ñóáõù úáõñç è. ð³ïáµû³ý ¨ ²í»ïçù ð. ø³ýáõïû³ý ºñ¨³ýç å»ï³ï³ý ñ³ù³éë³ñ³ý, ºñ¨³ý, ð³û³ëï³ý e-mail: yuri.hakopian@ysu.am, avetiq.manukyan1@ysumail.am ²ù÷á÷áõù ðá¹í³íáõù ïñíáõù ¿ »ñ»ù³ýïûáõý³·í³ûçý ñ»ñùçïû³ý ù³ïñçóý»ñç ñ³ï³¹³ñóù³ý ³é·áñçãùá, áñç ãí³ûçý çñ³ï³ý³óáõùá å³ñ³ýçáõù ¿ ûåïçù³é ãíáí ãí³µ³ý³ï³ý ·áñíáõáõãûáõýý»ñ: ð³ßíáõ³ï³ý åñáó»¹áõñ³ý çñ»ýçó ý»ñï³û³óýáõù ¿ ñ³ï³¹³ñó ù³ïñçóç ï³ññ»ñç ñ³ßíù³ýá ñ³ý·»óýáõ ³ý¹ñ³¹³ñó ³éýãáõãûáõýý»ñç ñ³çáñ¹³ï³ýáõãû³ý: ð³ïáõï ïçåç ù³ïñçóý»ñç ñ³ù³ñ ¨, ù³ëý³íáñ³å»ë, ïûáåéçóû³ý »ñ»ù³ýïûáõý³·í³ûçý ñ»ñùçïû³ý ù³ïñçóý»ñç ñ³ù³ñ, ëï³óí³í ³éýãáõãûáõýý»ñá ñ³ý·»óýáõù »ý ñ³ï³¹³ñó ù³ïñçóç ï³ññ»ñç ñ³ù³ñ µ³ó³ñ³ûï µ³ý³ó¨»ñç: àíàëèòè÷åñêîå îáðàùåíèå òðåõäèàãîíàëüíûõ èçîáðàæåíè þðèé ð. àêîïÿí è àâåòèê à. ìàíóêÿí åðåâàíñêèé ãîñóäàðñòâåííûé óíèâåðñèòåò, åðåâàí, àðìåíèÿ e-mail: yuri.hakopian@ysu.am, avetiq.manukyan1@ysumail.am àííîòàöèÿ â ñòàòüå äàåòñÿ àëãîðèòì îáðàùåíèÿ òðåõäèàãîíàëüíûõ ýðìèòîâûõ ìàòðèö, ÷èñëåííàÿ ðåàëèçàöèÿ êîòîðîãî îñóùåñòâëÿåòñÿ çà îïòèìàëüíîå ÷èñëî àðèôìåòè÷åñêèõ îïåðàöèé. âû÷èñëèòåëüíàÿ ïðîöåäóðà ïðåäñòàâëÿåò ñîáîé ´³ý³éç µ³é»ñ` ñ³ï³¹³ñó ù³ïñçó, »ñ»ù³ýïûáõý³·í³ûçý ù³ïñçó, ñ»ñùçïû³ý ù³ïñçó, ïûáåéçóû³ý ù³ïñçó: yu. hakopian and a. manukyan 1 9 ïîñëåäîâàòåëüíîñòü ðåêóððåíòíûõ ñîîòíîøåíèé, ïðèâîäÿùèõ ê âû÷èñëåíèþ ýëåìåíòîâ îáðàòíîé ìàòðèöû. äëÿ ìàòðèö ñïåöèàëüíîãî òèïà è, â ÷àñòíîñòè, äëÿ ò¸ïëèöåâûõ òðåõäèàãîíàëüíûõ ýðìèòîâûõ ìàòðèö, ïîëó÷åííûå ñîîòíîøåíèÿ ïðèâîäÿò ê ÿâíûì ôîðìóëàì äëÿ ýëåìåíòîâ îáðàòíîé ìàòðèöû. êëþ÷åâûå ñëîâà: îáðàòíàÿ ìàòðèöà, òðåõäèàãîíàëüíàÿ ìàòðèöà, ýðìèòîâà ìàòðèöà, ò¸ïëèöåâà ìàòðèöà. 01_hakopian_18_19 01 d:\sbornik\...\stsas.dvi mathematical problems of computer science 23, 2004, 47{53. on a softwar e t ool for i mplementation of systolic algor ithms in the cluster e nvir onment ¤ e d m o n m. d a vt ya n institue for informatics and automation problems of nas of ra e-mail edmon@ipia.sci.am abstract in this paper a software tool sas (systolic algorithm simulator) is designed to model the work of a one-dimensional systolic array of n cells on a homogenous computational cluster of m ¿ n processors. as a result of the program execution, a cluster-based programming module is obtained, where computational resourses of the system are used in an e®ective way. refer ences [1 ] e .d a vt ya n . on t h e mo d e llin g o f on e cla s s o f s ys t o lic s t r u c t u r e s o n a p c clu s t e r . in p r o c e e d in g s o f cs it-2 0 0 3 , p p . 3 4 0 -3 4 4 . [2 ] e d m o n m. d a vt ya n . on t h e co n s t r u c t io n o f clu s t e r s ys t o lic a r r a ys . tr a n s a ction s o f iia p n a s r a ,y e r e va n , 2 0 0 4 ( in t h is is s u e ) . [3 ] p a r o s h a b d u la . d e c id a b le a n d u n d e c id a b le p r o b le m s in s ys t o lic cir c u it v e r ī c a t io n . a cm in t e r n a t io n a l w o r ks h o p o n fo r m a l v l s i d e s ig n , mia m i, flo r id a , ja n u a r y 1 9 9 1 . [4 ] âîåâîäèí â.â., âîåâîäèí âë.â. ïàðàëëåëüíûå âû÷èñëåíèÿ. ñïá.: áõâ ïåòåðáóðã, 2002. 608 ñ.: isbn 5-94157-160-7. [5 ] êîðíååâ â. ïàðàëëåëüíîå ïðîãðàììèðîâàíèå â mpi. ìîñêâà-èæåâñê: èíñòèòóò êîìïüþòåðíûõ èññëåäîâàíèé, 2003, 203 ñ. [6 ] l . b o a s s o n , p . ce g ie ls ki, i. gu e s s a r ia n , y u . ma t iya s e vic h . w in d o w-a c c u m u la t e d s u b s e qu e n c e ma t c h in g p r o b le m is l in e a r . cs it co n fe r e n c e 2 0 0 1 , y e r e va n , a r m e n ia , s e p t e m b e r 1 7 -2 0 , p p . 7 4 -8 7 . [7 ] m p ic h : h t t p :/ / www-u n ix.m c s .a n l.g o v/ m p i/ m p ic h [8 ] m p ic h -g m : h t t p :/ / www.m yr i.c o m / s c s ¤this research is supported by intas 0447, istc 823 grants and 04.10.31 target program of ra. 4 7 4 8 on a software tool for implementation of systolic algorithms in the cluster environment îé³ëï»ñç ùçç³í³ûñáõù ëçëïáéçï ³é·áñçãùý»ñç çñ³ï³ý³óù³ý íñ³·ñ³ûçý ·áñíçù³ûçý ùççáóç ù³ëçý ¾. ø. ¸³íãû³ý ²ù÷á÷áõù øß³ïí³í ¿ sas (systolic algorithm simulator) íñ³·ñ³ûçý ·áñíçù³ûçý ùççáóá, áñá ùá¹»é³íáñáõù ¿ n »ñï³ñáõãûáõý áõý»óáõ ùç³ã³÷ ëçëïáéçï ½³ý·í³íç ³ßë³ï³ýùá m ¿ n åñáó»ëáñý»ñçó µ³õï³ó³í ñ³ù³ë»é ñ³ßíáõ³ï³ý ïé³ëï»ñç íñ³: ìñ³·ñç ³ßë³ï³ýùç ³ñ¹ûáõýùáõù ëï³óíáõù ¿ ïé³ëï»ñç íñ³ ³ßë³ïáõ íñ³·ñ³ûçý ùá¹áõé, áñá ¿ý»ïïçí ï»ñåáí ¿ û·ï³·áñíáõù ñ³ù³ï³ñ·ç ñ³ßíáõ³ï³ý é»ëáõñëý»ñá: microsoft word mod petri2t.doc ìàòåìàòè÷åñêèå âîïðîñû êèáåðíåòèêè è âû÷èñëèòåëüíîé òåõíèêè 25, 2006, 39–44. 1 the research is supported partly by intas: 04-77-7173 project, http://www.intas.be and state principal program of armenia on scientific computations. 39 взаимосвязь языков модифицированых сетей петри с некоторыми классами формальных языков гоар р. петросян ереванский государственный университет аннотация в работе приводится об эквивалентности кс-грамматик (контекстно-свободный) и модифицированных сетей петри. модифицированная сеть петри-это расширение стандартной сети петри с помощью сдерживаюших позиций. для кс языка {r /   ,  = {a, b}}, который не является языком сетей петри, построена модифицированная сеть петри, для которой l(c) = {r / ,  = {a, b}}. с помощью графов характеризуется взаимосвязь языков модифицированных сетей петри с некоторыми классами формальных языков. литература [1] питерсон д., “теория сетей петри и моделирование систем”. москва, мир, 1984г. [2] котов в. е. “сети петри”. москва, мир, 1984г. [3] ахо а., ульман д., “теория синтаксического анализа, перевода и компиляции”. перевод под редакцией курочкина, т1-т3. [4] гордеев а. в., молчанов а. ю., “системное программное обеспечение”. учебник, санкт-петербург 2002г. [5] петросян г. р., “модифицированные сети петри: описание поведения с помощью формальных языков”. математические вопросы кибернетики и вычислительной теxники, ереван, 2006. ä»ïñçç ó¨³÷áëí³í ó³ýó»ñç 黽áõý»ñç ÷áëï³å³ïóí³íáõãûáõýá áñáß ¹³ë»ñç ýáñù³é 黽áõý»ñç ñ»ï ¶. ä»ïñáëû³ý ²ù÷á÷áõù ²ßë³ï³ýùáõù µ»ñí³í ¿ î² ù»ñ³ï³ýáõãûáõýý»ñç ¨ ä»ïñçç ó¨³÷áëí³í ó³ýó»ñç ñ³ù³ñå»ùáõãûáõýá: ä»ïñçç ó¨³÷áëí³í ó³ýóá ¹³ ä»ïñçç ëï³ý¹³ñï ó³ýóç áý¹é³ûýáõùý ¿ ë³ñù³ý³÷³ïáõ ¹çñù»ñç û·ýáõãû³ùµ: î² (ïáýï»ïëïçó ³ýï³ë) 黽íç ñ³ù³ñ {r /   ,  = {a, b}}, áñá ãç ñ³ý¹çë³ýáõù ä»ïñçç ó³ýó»ñç 黽áõ, ï³éáõóí³í ¿ ä»ïñçç ó¨³÷áëí³í ó³ýó, áñç ñ³ù³ñ l (c) = {r / ,  = {a, b}}. ¶ñ³ýý»ñç û·ýáõãû³ùµ µýáõã³·ñíáõù ¿ ä»ïñçç ó¨³÷áëí³í ó³ýó»ñç 黽áõý»ñç ¨ áñáß ¹³ë»ñç ýáñù³é 黽áõý»ñç ÷áëï³å³ïóí³íáõãûáõýá: d:\sbornik\...\article_eng.dvi mathematical problems of computer science 31, 40{48, 2008. copulas of t wo-dimensional t hr eshold m odels e vg u e n i a . h a r o u t u n ia n a n d ir in a a . s a fa r ya n institute for informatics and automation problems of nas of ra. e-mail: evhar@ipia.sci.am abstract a representation of two-dimensional random vector bivariate distribution by copula is proposed for the case when one of components is categorizing for the other. the regression function and the bounds for the spearman rank correlation coe±cient are derived. refer ences [1 ] h . jo e , \ mu lt iva r ia t e m o d e ls a n d d e p e n d e n c e c o n c e p t s " , chapman and hall, l ondon, 1 9 9 7 . 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[1 4 ] j. k . ou s t e r h o u t , " s c r ip t in g : h ig h e r -l e ve l p r o g r a m m in g fo r t h e 2 1 s t ce n t u r y" , ie e e computer magazine , ma r c h , 1 9 9 8 . [1 5 ] a . v . a h o a n d r . s e t h i a n d j. d . u llm a n , " co m p ile r s p r in c ip le s , t e c h n iqu e s , a n d t o o ls " , a d d is o n -w e s le y, 1 9 8 6 . [1 6 ] t. w . p r a t t a n d m. v . ze lko wit z , " p r o g r a m m in g l a n g u a g e s : d e s ig n a n d im p le m e n t a t io n " , p r e n t ic e -h a ll, 1 9 9 6 . [1 7 ] j. k . ou s t e r h o u t , " tc l a n d t h e tk to o lkit " , a d d is o n -w e s le y, 1 9 9 4 . [1 8 ] b . s t r o u s t r u p , " th e c++ p r o g r a m m in g l a n g u a g e ( 3 r d e d it io n ) " , p u b l. a d d is o n w e s le y, 1 9 9 7 . [1 9 ] d . fla n a g a n , " ja va in a n u t s h e ll ( 2 n d e d it io n ) " , p u b l. o'r e illy & a s s o c ia t e s , 1 9 9 7 . [2 0 ] j. r . l e vin e a n d t. ma s o n a n d d . b r o wn , " ja va in a n u t s h e ll ( 2 n d e d it io n ) " , a d d is o n w e s le y, 1 9 9 2 . 1 4 constraint management in database applications via checker framework îíû³éý»ñç ñ»ýù»ñç ïçñ³é³ï³ý íñ³·ñ»ñç ë³ñù³ý³÷³ïáõùý»ñç ùß³ïáõù ëïáõ·çãý»ñç ñ³ù³ï³ñ·ç ïçñ³éù³ùµ ². æ. ²ë³ïñû³ý ²ù÷á÷áõù êáõûý ñá¹í³íáõù ý»ñï³û³óíáõù ¿ ë³ñù³ý³÷³ïáõùý»ñç ùß³ïù³ý ùáï»óáõù »ñù»í æýï»·ñí³í êë»ù³ý»ñç ü³ë³·íù³ý (vlsi design) ñ»ýù»ñç ïçñ³é³ï³ý íñ³·ñ»ñáõù »í ñáëù»ñáõù: àëï ³é³ç³ñïíáõ ùáï»óù³ýª ë³ñù³ý³÷³ïáõùý»ñá ý»ñï³û³óíáõù »ý eva ïçåç½³óí³í ëïñçåï³ûçý 黽íç ùççáóáí »í å³ñíáõù »ý ëïáõ·çãý»ñç ù»ï³ñ»ýùáõù: ìñ³·ñ»ñç »í ëïáõ·çãý»ñç ù»ï³ñ»ýùç ÷áë³½¹»óáõãûáõýá çñ³ï³ý³óíáõù ¿ êïáõ·çãý»ñç ð³ù³ï³ñ·ç ùççáóáí, áñá ñ³ý¹çë³ýáõù ¿ ñ»ýùç õ»ï³í³ñù³ý ñ³ù³ï³ñ·ç »ýã³ñ³ù³ï³ñ·: êïáõ·çãý»ñç ñ³ù³ï³ñ·á å³ß-ï³ëßë³ß-ý³ß-ïáõ ¿ ë³ñù³ý³÷³ïáõùý»ñç ëï»õíù³ý, ÷á÷áëù³ý »í çýçù³ý ñ³ù³ñ, çýãå»ë ý³»í ³å³ñáíáõù ¿ ùççáóý»ñª íñ³·ñ»ñáõù ë³ñù³ý³÷³ïáõùý»ñç ùß³ïù³ý ³ñ¹ûáõý³í»ï ï³½ù³ï»ñåù³ý ñ³ù³ñ: d:\sbornik\...\article.dvi mathematical problems of computer science 25, 2006, 9{11. an i nequality related to the p air s of m atchings of a gr aph r a fa ye l r . k a m a lia n * , v a h a n v . mkr t c h ya n * * *institute for informatics and automation problems of nas ra *department of informatics and applied mathematics, yerevan state university e-mails rrkamalian@yahoo.com, vahanmkrtchyan2002@yahoo.com abstract for a given graph disjoint pairs of matchings the union of which contains as many edges as possible are considered. it is shown that the relation of the cardinality of a maximum matching to the cardinality of the largest matching in those pairs does not exceed 3=2. a conjecture is posed which states that this coe±cient can be replaced by 5=4 . finally, a family of graphs is presented which shows that the abovementioned coe±cient can not be replaced by a constant which is smaller than 5=4. r eferences [1 ] v . v . mkr t c h ya n , on t r e e s wit h a m a xim u m p r o p e r p a r t ia l 0 -1 c o lo u r in g c o n t a in in g a m a xim u m m a t c h in g , d iscrete m athematics 306, 2 0 0 6 , p p . 4 5 5 -4 5 8 . [2 ] f. h a r a r y, \ gr a p h th e o r y" , a d d is o n -w e s le y, r e a d in g , ma , 1 9 6 9 . [3 ] f. h a r a r y, m. d . p lu m m e r , on t h e c o r e o f a g r a p h , p roc. l ondon m ath. soc. 1 7 ( 1 9 6 7 ) , 3 0 5 -3 1 4 . [4 ] l . l o va s z , m. d . p lu m m e r , ma t c h in g th e o r y, annals of d iscrete m ath. 2 9 , n o r t h h o lla n d , 1 9 8 6 . [5 ] d . b . w e s t , in t r o d u c t io n t o gr a p h th e o r y, p r e n t ic e -h a ll, in c .,1 9 9 6 . ¶ñ³ýáõù ½áõ·³ïóáõùý»ñç ½áõû·»ñçý ³éýãíáõ ùç ³ýñ³í³ë³ñáõãû³ý ù³ëçý è. ø³ù³éû³ý, ì. øïñïãû³ý ²ù÷á÷áõù ¸çï³ñïí»é »ý ·ñ³ýç ãñ³ïíáõ ½áõ·³ïóáõùý»ñç ³ûý ½áõû·»ñá, áñáýó ùç³íáñáõùá å³ñáõý³ïáõù ¿ ñý³ñ³íáñçý ã³÷ ß³ï ïáõ: òáõûó ¿ ïñí»é, áñ ·ñ³ýç ù³ùëçù³é 9 1 0 an inequality related to the pairs of matchings of a graph ½áõ·³ïóù³ý ñ½áñáõãû³ý ñ³ñ³µ»ñáõãûáõýá ³û¹ ½áõû·»ñáõù ³ù»ý³ß³ï ãíáí ïáõ»ñ å³ñáõý³ïáõ ½áõ·³ïóù³ý ñ½áñáõãû³ýá ãç ·»ñ³½³ýóáõù 3/2-á: ²é³ç³ñïí»é ¿ í³ñï³í, ñ³ù³ó³ûý áñç ³ûë ·áñí³ïçóá ï³ñ»éç ¿ ÷áë³ñçý»é 5/4-áí: ¶ïýí»é ¿ ·ñ³ýý»ñç ùç áýï³ýçù, áñá óáõûó ¿ ï³éçë, áñ áñù³ý ¿é ù»í éçýç ·ñ³ýá, í»ñáñçßû³é ·áñí³ïçóá ñý³ñ³íáñ ã¿ ÷áë³ñçý»é 5/4-çó ÷áùñ ãíáí: 1_mikayel_evoyan.dvi mathematical problems of computer science 39, 5{12, 2013. on k-switching of m appings on finite fields mika ye l g. e vo ya n 1, go h a r m. k yu r e g ya n 2 a n d me ls ik k . k yu r e g ya n 1 institute for informatics and automation problems of nas ra1 institute of algebra and geometry, otto-von-guericke university magdeburg2 e-mail: michael.evoyan@gmail.com, gohar.kyureghyan@ovgu.de, melsik@ipia.sci.am abstract the switching construction was used in several recent papers to construct special mappings on ¯nite ¯elds. in this paper we generalize the concept of switching to a k-switching with 1 · k · n. we present some general properties of k-switching and describe permutations produced using k-switching. keywords: mappings of ¯nite ¯elds, switching, permutation. 1 . in t r o d u c t io n l e t f : fqn ! fqn a n d ( °1; : : : ; °n ) b e a n fq-b a s is o f fqn . th e u n iqu e ly d e t e r m in e d fu n c t io n s fi : fqn ! fq; 1 · i · n; s u c h t h a t f ( x ) = f1 ( x ) ¢ °1 + : : : + fn ( x ) ¢ °n; a r e c a lle d t h e coordinate functions o f f wit h r e s p e c t t o t h e b a s is ( °1; : : : ; °n ) . th e component functions o f f o ve r t h e s u b ¯ e ld fq a r e t h e fu n c t io n s t rqn=q ( ®f ( x ) ) wit h ® 2 f¤qn , wh e r e t rqn=q is a t r a c e m a p p in g o f fqn in t o fq g ive n b y t rqn=q = x + x q + : : : + xq n¡1 : th e s e t o f c o m p o n e n t fu n c t io n s o f a m a p p in g c o in c id e s wit h o n e o f it s c o o r d in a t e fu n c t io n s : p r oposition 1: any component function over fq of a mapping f : fqn ! fqn is a coordinate function with respect to some fq-basis, and vice versa. p r oof. r e c a ll t h a t a n y b a s is ( °1; : : : ; °n ) h a s a u n iqu e d u a l b a s is ( ¹°1; : : : ; ¹°n ) d e ¯ n e d b y t rqn=q ( °i ¹°j ) = ( 1 if i = j; 0 if i 6= j: fo r a ll 1 · i; j · n. in p a r t ic u la r fo r a n y a 2 fqn t h e c o e ± c ie n t s ai in t h e lin e a r c o m b in a t io n a = pn i=1 ai°i a r e g ive n b y ai = t rqn=q ( ¹°ia ) : 5 6 on k-switching of mappings on finite fields co n s e qu e n t ly, t h e c o o r d in a t e fu n c t io n fi ( x ) o f f ( x ) wit h r e s p e c t t o ( °1; : : : ; °n ) is t h e c o m p o n e n t fu n c t io n t rqn=q ( ¹°if ( x ) ) . on t h e o t h e r h a n d , a c o m p o n e n t fu n c t io n t rqn=q ( ®f ( x) ) is a c o o r d in a t e fu n c t io n o f f ( x ) wit h r e s p e c t t o t h e d u a l b a s is o f a n y b a s is c o n t a in in g ®. a m a p p in g f : fqn ! fqn is c a lle d a s wit c h in g o f g : fqn ! fqn if t h e r e is a n fq-b a s is ( °1; : : : ; °n ) s u c h t h a t a ll b u t t h e ¯ r s t c o o r d in a t e fu n c t io n s o f f a n d g a r e e qu a l, i.e . f ( x ) = f1 ( x ) ¢ °1 + : : : + fn ( x) ¢ °n; a n d g ( x ) = g1 ( x ) ¢ °1 + : : : + gn ( x ) ¢ °n; wit h f1 ( x ) 6= g1 ( x ) a n d fi ( x ) = gi ( x ) fo r a ll 2 · i · n. s wit c h in g wa s u s e d t o p r o d u c e in t e r e s t in g c la s s e s o f s p e c ia l m a p p in g s o f ¯ n it e ¯ e ld s in [2 ]{ [8 ]. co n s t r u c t io n o f b ije c t ive m a p p in g s b y c h a n g in g t wo c o o r d in a t e fu n c t io n s o f t h e id e n t it y m a p p in g we r e s t u d ie d in [5 , 7 ]. in t h is p a p e r we g e n e r a liz e t h e c o n c e p t o f s wit c h in g t o a k-s wit c h in g wit h 1 · k · n a n d d e s c r ib e p e r m u t a t io n s p r o d u c e d u s in g k-s wit c h in g . 2 . k-s wit c h in g de¯nition 1: l et 1 · k · n be an integer. a mapping f : fqn ! fqn is called a kswitching of g : fqn ! fqn over fq if k is the minimal integer such that there is an fq-basis ( °1; : : : ; °n ) with respect to which all but the ¯rst k coordinate functions of f and g are equal, i.e. f ( x ) = f1 ( x ) ¢ °1 + : : : + fn ( x) ¢ °n; and g ( x ) = g1 ( x ) ¢ °1 + : : : + gn ( x ) ¢ °n; with fj ( x ) 6= gj ( x ) for all 1 · j · k and fi ( x ) = gi ( x) for all k + 1 · i · n. n o t e t h a t 1 -s wit c h in g r e d u c e s t o a s wit c h in g d e ¯ n e d a b o ve . cle a r ly, fo r a n y t wo d i®e r e n t m a p p in g s f a n d g t h e r e is a n in t e g e r 1 · k · n s u c h t h a t f is a k-s wit c h in g o f g. mo r e o ve r , if f is a k-s wit c h in g o f g, t h e n a ls o g is a k-s wit c h in g o f f . remar k 1: l et 1 · k < k0 · n. if the mapping f : fqn ! fqn is a k-switching of g : fqn ! fqn, then there is an fq-basis of fqn with respect to which exactly k0 coordinate functions of f and g di®er. indeed, let f ( x ) = kx i=1 fi ( x) °i + nx i=k+1 ai ( x ) °i; and g( x ) = kx i=1 gi ( x ) °i + nx i=k+1 ai ( x ) °i; with respect to an fq-basis ( °1; : : : ; °n ) and fi ( x ) 6= gi ( x ) for all 1 · i · k. then with respect to the fq-basis ( °1; : : : ; °k¡1; k0x j=k °j; ; °k+1; : : : ; °n ) ; m. evoyan, g. kyureghyan and m. kyureghyan 7 the coordinate functions of f and g are as follows: f ( x) = k¡1x i=1 fi ( x) °i + fk ( x ) 0 @ k0x j=k °j 1 a + k0x i=k+1 ( ai ( x ) ¡ fk ( x ) ) °i + nx i=k0+1 ai ( x ) °i; and g ( x) = k¡1x i=1 gi ( x ) °i + gk ( x ) 0 @ k0x j=k °j 1 a + k0x i=k+1 ( ai ( x) ¡ gk ( x ) ) °i + nx i=k0+1 ai ( x ) °i: in t h e fo llo win g fo r a s u b s e t s µ fqn we u s e hsi t o d e n o t e t h e fq-s u b s p a c e s p a n n e d b y s. t heor em 1: l et 1 · k · n and f; g : fqn ! fqn. then the following statements are equivalent: (i) f is a k-switching of g. (ii) the image set of the mapping f ¡ g : x 7! f ( x ) ¡ g( x ) spans a k-dimensional vector space over fq. p r oof. l e t f : fqn ! fqn b e a k-s wit c h in g o f g : fqn ! fqn . th e n t h e r e is a n fq-b a s is ( °1; : : : ; °n ) o f fqn s u c h t h a t f ( x ) = kx i=1 fi ( x) °i + nx i=k+1 ai ( x) °i; a n d g( x ) = kx i=1 gi ( x ) °i + nx i=k+1 ai ( x ) °i; wh e r e fi ( x ) 6= gi ( x) fo r a ll 1 · i · k. h e n c e f ( x ) ¡ g( x ) = kx i=1 ( fi ( x) ¡ gi ( x ) ) °i; s h o win g t h a t t h e d im e n s io n l o f himage ( f ¡ g) i is le s s o r e qu a l t o k. n o w le t ( ±1; : : : ; ±` ) b e a b a s is fo r himage( f ¡ g ) i, a n d ( ±1; : : : ; ±`; : : : ±n ) a b a s is fo r fqn . l e t hi; uj : fqn ! fq b e s u c h t h a t f ( x ) ¡ g ( x ) = x̀ i=1 hi ( x) ±i; a n d g ( x) = nx i=1 ui ( x ) ±i: th e n f ( x ) = x̀ i=1 ( ui ( x ) + hi ( x ) ) ±i + nx i=`+1 ui ( x ) ±i; a n d t h u s k · ` b y t h e d e ¯ n it io n o f k-s wit c h in g , c o m p le t in g t h e p r o o f. p r oposition 2: l et 1 · k · n and f : fqn ! fqn be a k-switching of g : fqn ! fqn : then f and g have exactly qn¡k ¡ 1 equal component functions over fq . 8 on k-switching of mappings on finite fields p r oof. l e t ® 2 fqn ; ® 6= 0 . th e n t rqn=q ( ®f ( x ) ) = t rqn=q ( ®g ( x ) ) h o ld s if a n d o n ly if t rqn=q ( ®( f ( x) ¡ g ( x ) ) ) is t h e c o n s t a n t z e r o fu n c t io n , o r , t h e e qu iva le n t , im a g e s e t o f t h e m a p p in g f ¡ g is c o n t a in e d in t h e h yp e r p la n e h® = fy 2 fqn : t rqn=q ( ®y ) = 0 g. n o t e t h a t image( f ¡ g) µ h® if a n d o n ly if t h e lin e a r s p a n himage ( f ¡ g ) i is a s u b s p a c e o f h®. b y th e o r e m 1 t h e d im e n s io n o f himage ( f ¡ g ) i is k. h e n c e , t h e r e a r e q n¡k¡1 q¡1 d i®e r e n t h yp e r p la n e s c o n t a in in g himage ( f ¡ g ) i. to c o m p le t e t h e p r o o f it r e m a in s t o r e c a ll t h a t h® = h®0 if a n d o n ly if ®0 = ® ¢ u wit h a n o n -z e r o u 2 fq. 3 . k-s wit c h in g o f t h e id e n t it y ma p p in g in t h is s e c t io n we c o n s id e r b ije c t ive k-s wit c h in g o f t h e id e n t it y m a p p in g . a s t h e n e xt o b s e r va t io n s h o ws t h e s t u d y o f k-s wit c h in g o f a n a r b it r a r y p e r m u t a t io n o n fqn c a n b e r e d u c e d t o o n e o f t h e id e n t it y m a p p in g s . l e t f : fqn ! fqn b e a k-s wit c h in g o f g : fqn ! fqn , a n d e it h e r f o r g b e a p e r m u t a t io n o n fqn . w it h o u t lo s s o f g e n e r a lit y, s a y f is a p e r m u t a t io n a n d d e n o t e it s in ve r s e m a p p in g b y f ¡1. th e n g( x ) = f ( x ) + kx i=0 fi ( x ) ¢ °i; ( 1 ) wit h r e s p e c t t o s o m e b a s is ( °1; : : : ; °n ) . n o t e t h a t ( 1 ) h o ld s if a n d o n ly if g ± f ¡1 ( x ) = x + kx i=0 fi ± f ¡1 ( x) ¢ °i; t h a t is wh e n g ± f ¡1 is a k-s wit c h in g o f t h e id e n t it y m a p p in g . h e n c e , u n d e r s t a n d in g o f t h e b e h a vio u r o f k-s wit c h in g o f t h e id e n t it y m a p p in g is a n im p o r t a n t s t e p fo r t h e g e n e r a l p r o b le m . th e r e m a in in g p a r t o f t h is s e c t io n is d e vo t e d t o a c la s s o f p e r m u t a t io n s o b t a in e d b y k-s wit c h in g u s in g t h e s o -c a lle d fu n c t io n s wit h a lin e a r t r a n s la t o r . ou r r e s u lt s g e n e r a liz e s e ve r a l c o n s t r u c t io n s o f p e r m u t a t io n s g ive n in [1 , 3 , 5 , 7 ]. a n o n -z e r o e le m e n t ® 2 fqn is c a lle d a n a-lin e a r t r a n s la t o r ( o r a-lin e a r s t r u c t u r e , c f. [3 ]) fo r t h e m a p p in g f : fqn ! fq if f ( x + u®) ¡ f ( x ) = ua; ( 2 ) fo r a ll x 2 fqn ; u 2 fq a n d s o m e ¯ xe d a 2 fq. th e fo llo win g t h e o r e m fr o m [3 ] a llo ws t o c o n s t r u c t fu n c t io n s wit h lin e a r t r a n s la t o r s e xp lic it ly. t heor em 2: l et g : fqn ! fqn and f ( x) = t rqn=q ( g ( x ) ) . then f has a linear translator if and only if there is a non-bijective fq-linear mapping l : fqn ! fqn such that f ( x ) = t rqn=q ( g( x ) ) = t rqn=q ( h ± l( x ) + ¯x ) ; for some h : fqn ! fqn and ¯ 2 fqn. in this case, any element from the kernel of l is a linear translator for f . m. evoyan, g. kyureghyan and m. kyureghyan 9 t heor em 3: l et 1 · k · n, ¸1; ¸2; ¢ ¢ ¢ ; ¸k 2 fqn be linearly independent over fq and fj : fqn ! fq, j = 1 ; : : : ; k. f urther, suppose ¸i is a bj;i-linear translator for fj, where i; j 2 f 1 ; 2 ; ¢ ¢ ¢ ; kg. set b := 0 bbbb@ 1 + b1;1 b1;2 ¢ ¢ ¢ b1;k b2;1 1 + b2;2 ¢ ¢ ¢ b2;k . .. bk;1 bk;2 ¢ ¢ ¢ 1 + bk;k 1 cccca ; and let f : fqn ! fqn be de¯ned as f ( x ) = x + ¸1f1 ( x ) + ¸2f2 ( x ) + ¢ ¢ ¢ + ¸kfk ( x ) : then f ( x ) = f ( y ) for some x; y 2 fqn if and only if x = y + ¸1a1 + ¸2a2 + : : : + ¸kak; and 0 bb@ a1 ... ak 1 cca 2 fq n belongs to the kernel of b. in particular, the mapping f is a qn¡r-to-1 on fqn where r is the rank of the matrix b. p r oof. l e t x; y 2 fqn b e s u c h t h a t f ( x) = f ( y ) . th e n , b y t h e d e ¯ n it io n o f f x + ¸1f1 ( x) + ¸2f2 ( x ) + ¢ ¢ ¢ + ¸kfk ( x ) = y + ¸1f1 ( y ) + ¸2f2 ( y ) + ¢ ¢ ¢ + ¸kfk ( y ) ; a n d t h u s x = y + ¸1a1 + ¸2a2 + : : : + ¸kak; fo r s o m e e le m e n t s ai 2 fq. ob s e r ve , t h a t wh e n ai 2 fq, t h e n f ã y + kx i=1 ¸iai ! = y + kx i=1 ¸iai + kx j=1 ¸j fj ã y + kx i=1 ¸iai ! = y + kx i=1 ¸iai + kx j=1 ¸j ã fj ( y ) + kx i=1 bj;iai ! = y + kx j=1 ¸j fj ( y ) + kx i=1 ¸iai + kx j=1 ¸j ã kx i=1 bj;iai ! = f ( y ) + kx j=1 ¸j 0 @( bj;j + 1 ) aj + kx i=1;i 6=j bj;iai 1 a : to c o m p le t e t h e p r o o f it r e m a in s t o n o t e t h a t t h e lin e a r in d e p e n d e n c e o f ¸j im p lie s f ( y ) + kx j=1 ¸j 0 @( bj;j + 1 ) aj + kx i 6=j;i=1 bj;iai 1 a = f ( y ) ; if a n d o n ly if a ll c o e ± c ie n t s ( bj;j + 1 ) aj + kx i 6=j;i=1 bj;iai = 0 ; 1 0 on k-switching of mappings on finite fields o r e qu iva le n t ly if 0 bb@ a1 ... ak 1 cca is in t h e ke r n e l o f t h e m a t r ix b. th e o r e m 1 r e d u c e s t o th e o r e m 8 fr o m [5 ] wh e n k = 1 . fo r k = 2 ; 3 , it e xt e n d s th e o r e m 1 0 fr o m [5 ] a n d th e o r e m 1 , 2 fr o m [7 ] c o n s id e r a b ly. remar k 2: l et ( °1; : : : ; °k ) be linear independent over fq and f1 ( x ) ; : : : ; fk ( x ) non-zero functions from fqn into fq. suppose the k-switching of the identity mapping x + kx j=1 fj ( x) °j; is a permutation on fqn. note that in this case the ( k ¡ 1 ) -switching x + k¡1x j=1 fj ( x ) °j; must not be a permutation on fqn . this follows easily from theorem 3 . indeed, there are non-singular matrices over fq whose leading principal ( n ¡ 1 ) £ ( n ¡ 1 ) minor is 0. cor ollar y 1: w ith the notations of theorem 3 , the mapping f is bijective on fqn if and only if the matrix b has a full rank. l et b¡1 be the inverses matrix of b. d e¯ne the functions hj : fqn ! fq, j = 1 ; : : : ; k by 0 bb@ h1 ( x ) ... hk ( x) 1 cca := b ¡1 ¢ 0 bb@ f1 ( x ) ... fk ( x) 1 cca : then the inverse mapping of f ( x) is given by f ¡1 ( x ) = x ¡ kx j=1 ¸jhj ( x) : p r oof. l e t bj b e t h e j t h r o w in t h e m a t r ix b. th e c a lc u la t io n s in t h e p r o o f o f th e o r e m 3 s h o w t h a t f 0 @x ¡ kx j=1 ¸jhj ( x ) 1 a = x + kx j=1 ¸j fj ( x ) ¡ kx j=1 ¸jbj ¢ 0 bb@ h1 ( x ) ... hk ( x) 1 cca = x + kx j=1 ¸j fj ( x ) ¡ kx j=1 ¸jbj ¢ b¡1 ¢ 0 bb@ f1 ( x ) ... fk ( x ) 1 cca = x + kx j=1 ¸j fj ( x ) ¡ kx j=1 ¸jfj ( x ) = x: th e in ve r s e m a p p in g o f t h e p e r m u t a t io n f o b t a in e d in co r o lla r y 1 is a k-s wit c h in g o f t h e id e n t it y m a p p in g a s we ll. th e n e xt p r o p o s it io n s h o ws t h a t t h is p r o p e r t y h o ld s fo r a ll p e r m u t a t io n s o b t a in e d via s wit c h in g fr o m t h e id e n t it y m a p p in g . m. evoyan, g. kyureghyan and m. kyureghyan 1 1 p r oposition 3: if a permutation f : fqn ! fqn is a k-switching of the identity mapping of fqn, then its inverse is a k-switching of the identity mapping as well. p r oof. l e t ( °1; : : : ; °k ) b e lin e a r in d e p e n d e n t o ve r fq a n d f1 ( x ) ; : : : ; fk ( x ) : fqn ! fq b e n o n -z e r o fu n c t io n s . fu r t h e r s u p p o s e t h a t t h e m a p p in g f ( x) = x + kx j=1 fj ( x) °j; is a p e r m u t a t io n o f fqn . if f ¡1 is t h e in ve r s e m a p p in g o f f , t h e n x = f ± f ¡1 ( x ) = f ¡1 ( x) + kx j=1 ( fj ± f ¡1 ) ( x ) °j : th u s f ¡1 ( x ) = x ¡ kx j=1 ( fj ± f ¡1 ( x ) ) °j; im p lyin g t h e r e s u lt . refer ences [1 ] a . a kb a r y, d . gh io c a a n d q. w a n g , \ on c o n s t r u c t in g p e r m u t a t io n s o f ¯ n it e ¯ e ld s " , f inite f ields appl., vo l. 1 7 , p p . 5 1 -6 7 , 2 0 1 1 . [2 ] l . b u d a g h ya n , c. ca r le t a n d g. l e a n d e r , \ co n s t r u c t in g n e w a p n fu n c t io n s fr o m kn o wn o n e s " , f inite f ields appl., vo l. 1 5 , p p . 1 5 0 -1 5 9 , 2 0 0 9 . [3 ] p . ch a r p in a n d g. k yu r e g h ya n , \ w h e n d o e s g ( x) + °tr ( h ( x) ) p e r m u t e fpn ?" , f inite f ields appl., vo l. 1 5 , p p . 6 1 5 -6 3 2 , 2 0 0 9 . [4 ] y . e d e l a n d a . p o t t , \ a n e w a lm o s t p e r fe c t n o n lin e a r fu n c t io n wh ic h is n o t qu a d r a t ic " , adv. m ath. commun., vo l. 3 , p p . 5 9 -8 1 , 2 0 0 9 . [5 ] g. k yu r e g h ya n , \ co n s t r u c t in g p e r m u t a t io n s o f ¯ n it e ¯ e ld s via lin e a r t r a n s la t o r s " , j . combin. theory ser. a, vo l. 1 1 8 , p p . 1 0 5 2 -1 0 6 1 , 2 0 1 1 . [6 ] g. k yu r e g h ya n a n d y in ta n , \ on a fa m ily o f p la n a r m a p p in g s " , e nhancing cryptographic primitives with techniques from error correcting codes, nato sci. p eace secur. ser. d inf. commun. secur. 23, ios , a m s t e r d a m , p p . 1 7 5 -1 7 8 , 2 0 0 9 . [7 ] m. k yu r e g h ya n a n d s . a b r a h a m ya n , \ a m e t h o d o f c o n s t r u c t in g p e r m u t a t io n p o lyn o m ia ls o ve r ¯ n it e ¯ e ld s " , int. j . information theories and applications, vo l. 1 7 , p p . 3 2 8 -3 3 4 , 2 0 1 0 . [8 ] a . p o t t a n d y . zh o u , \ s wit c h in g c o n s t r u c t io n o f p la n a r fu n c t io n s o n ¯ n it e ¯ e ld s " , arithmetic of ¯nite ¯elds, l ecture notes in comput. sci. 6087, s p r in g e r , b e r lin , p p . 1 3 5 1 5 0 , 2 0 1 0 . submitted 14.12.2012, accepted 11.02.2013. 1 2 on k-switching of mappings on finite fields ì»ñç³íáñ ¹³ßï»ñç íñ³ ³ñï³å³ïï»ñáõùý»ñç k-÷áë³ñïáõùý»ñç ù³ëçý ø. ¾íáû³ý, ¶. îûáõñ»õû³ý ¨ ø. îûáõñ»õû³ý ²ù÷á÷áõù öáë³ñïáõùý»ñá ïçñ³éíáõù »ý í»ñç»ñë éáõûë ï»ë³í í»ñç³íáñ ¹³ßï»ñç íñ³ ûáõñ³ñ³ïáõï ³ñï³å³ïï»ñáõùý»ñç ï³éáõóù³ýá ýíçñí³í ³ßë³ï³ýùý»ñáõù: ²ûë ³ßë³ï³ýùáõù áý¹ñ³ýñ³óíáõù ¿ ÷áë³ñïù³ý ñ³ëï³óáõãûáõýá ùçý㨠k -÷áë³ñïáõù, áñï»õ · k · n , çýãå»ë ý³¨ ý»ñï³û³óíáõù »ý k-÷áë³ñïáõùý»ñç áý¹ñ³ýáõñ ñ³ïïáõãûáõýý»ñ ¨ ýï³ñ³·ñíáõù ¿ k-÷áë³ñïáõùý»ñç ùççáóáí ï»õ³÷áëáõãûáõýý»ñç ï³éáõóù³ý ù»ãá¹: î k-îáìåíàõ îòîáðàæåíèé íà êîíå÷íûõ ïîëåé ì. ýâîÿí, ã. êþðåãÿí è ì. êþðåãÿí àííîòàöèÿ êîíñòðóêöèÿ îáìåíà èñïîëüçóåòñÿ â íåñêîëüêèõ íåäàâíèõ ðàáîòàõ ïî ïîñòðîåíèþ ñïåöèôè÷åñêèõ îòîáðàæåíèé íà êîíå÷íûõ ïîëåé. â ýòîé ñòàòüå ïîíÿòèå îá îáìåíå îáîáùåíî äî k-îáìåíà ñ · k · n . òàêæå ïðåäñòàâëåíû íåêîòîðûå îáùèå ñâîéñòâà k-îáìåíà è îïèñàí ìåòîä ïîëó÷åíèÿ ïåðåñòàíîâîê ñ èñïîëüçîâàíèåì k-îáìåíà. d:\user\sbornik_38_pdf\32.dvi mathematical problems of computer science 38, 76, 2012. on t ur ing completeness of one m inimal set of b uilt-in functions for functional p r ogr amming languages g. a . ma r t ir o s ya n chair of programming and information technologies, ysu e-mail: gevorg.martirosyan@gmail.com ma n y fu n c t io n a l p r o g r a m m in g la n g u a g e s o p e r a t e o n s-expressions. th e s e t s o f b u ilt in fu n c t io n s o f t h o s e la n g u a g e s c o n t a in car; cdr; cons; atom; eq; if then else fu n c t io n s . it is s h o wn t h a t tu r in g c o m p u t a b le fu n c t io n s d e ¯ n e d o n s-expressions c a n b e p r e s e n t e d in s u c h fu n c t io n a l p r o g r a m m in g la n g u a g e s wh ic h h a ve car; cdr; cons; atom; eq; if then else b u ilt -in fu n c t io n s . in o t h e r wo r d s , if t h e s e t o f b u ilt -in c o n s t a n t s o f a fu n c t io n a l p r o g r a m m in g la n g u a g e c o n t a in s a ll t h e s e fu n c t io n s , t h e n t h a t la n g u a g e is tu r in g c o m p le t e . th e fo llo win g t wo r e s u lt s a r e o b t a in e d fo r t h e m in im a lit y o f t h e s e t o f b u ilt -in fu n c t io n s © =fcar; cdr; cons; atom; eq; if then elseg. 1 . © is m in im a l fo r fu n c t io n a l p r o g r a m m in g la n g u a g e s wh ic h u s e m o r e t h a n t wo a t o m s . 2 . th e fu n c t io n eq is r e p r e s e n t a b le in a fu n c t io n a l p r o g r a m m in g la n g u a g e wh ic h u s e s o n ly t wo a t o m s a n d t h e s e t ©n feqg o f b u ilt -in fu n c t io n s ; t h e s e t o f b u ilt -in fu n c t io n s ©n feqg is m in im a l fo r fu n c t io n a l p r o g r a m m in g la n g u a g e s wh ic h u s e o n ly t wo a t o m s a n d it is t h e o n ly p r o p e r s u b s e t o f t h e s e t ©, wh ic h is m in im a l fo r s u c h la n g u a g e s . r eferences 1 . s .a . n ig iya n , " fu n c t io n a l l a n g u a g e s " , p rogramming and computer software, v o l. 1 7 . p p . 2 9 0 -2 9 7 , 1 9 9 2 . 2 . s .a . n ig iya n , " on in t e r p r e t a t io n o f fu n c t io n a l p r o g r a m m in g la n g u a g e s " , p rogramming and computer software, v o l. 1 9 . p p . 7 1 -7 8 , 1 9 9 3 . 3 . l .e . b u d a g h ya n , \ n e c e s s a r y a n d s u ± c ie n t c o n d it io n o f c o m p le t e n e s s o f c o m p u t a t io n r u le fo r s t r o n g ly t yp e d fu n c t io n a l p r o g r a m s " , p r o c e e d in g s o f t h e co n fe r e n c e o n co m p u t e r s c ie n c e a n d in fo r m a t io n te c h n o lo g ie s ( cs it-2 0 0 5 ) , y e r e va n , 2 0 0 5 , p . 1 6 -1 9 7 6 d:\user\sbornik_38_pdf\26.dvi mathematical problems of computer science 38, 66{67, 2012. on a p r oper ty of the n-dimensional cube r a fa ye l k a m a lia n 1, a r p in e k h a c h a t r ya n 2 1 institute for informatics and automation problems, national academy of sciences of the republic of armenia, 0014, armenia, email: rrkamalian@yahoo.com 2 ijevan branch of yerevan state university, 4001, armenia, email: khachatryanarpine@gmail.com w e s h o w t h a t in a n y s u b s e t o f ve r t ic e s o f t h e n-d im e n s io n a l c u b e wh ic h c o n t a in s a t le a s t 2 n¡1 +1 ve r t ic e s ( n ¸ 4 ) , t h e r e a r e fo u r ve r t ic e s t h a t in d u c e a c la w, o r t h e r e a r e e ig h t ve r t ic e s t h a t in d u c e t h e c yc le o f le n g t h e ig h t . w e c o n s id e r ¯ n it e g r a p h s g = ( v; e ) wit h ve r t e x s e t v a n d e d g e s e t e. th e g r a p h s c o n t a in n o m u lt ip le e d g e s o r lo o p s . th e n-d im e n s io n a l c u b e is d e n o t e d b y qn, a n d a c la w is t h e c o m p le t e b ip a r t it e g r a p h k1;3. mo r e o ve r , t h e ve r t e x o f a d e g r e e t h r e e in a c la w is c a lle d a c la w-c e n t e r . n o n -d e ¯ n e d t e r m s a n d c o n c e p t s c a n b e fo u n d in [1 ]. th e m a in r e s u lt o f t h e p a p e r is t h e fo llo win g : t heor em 1. l et n ¸ 4 and let v 0 µ v ( qn ) . if jv 0j ¸ 2 n¡1 + 1 , then at least one of the following two conditions holds: (a) there are four vertices in v 0 that induce a claw; (b) there are eight vertices in v 0 that induce a simple cycle. p r oof. ou r p r o o f is b y in d u c t io n o n n. s u p p o s e t h a t n = 4 . cle a r ly, wit h o u t lo s s o f g e n e r a lit y, we c a n a s s u m e t h a t jv 0j = 9 . co n s id e r t h e fo llo win g p a r t it io n o f t h e ve r t ic e s o f q4: v1 = f( 0 ; ®2; ®3; ®4 ) : ®i 2 f0 ; 1 g; 2 · i · 4 g; v2 = f( 1 ; ®2; ®3; ®4 ) : ®i 2 f0 ; 1 g; 2 · i · 4 g: cle a r ly, t h e s u b g r a p h s o f q4 in d u c e d b y v1 a n d v2 a r e is o m o r p h ic t o q3. d e ¯ n e : v 01 = v1 \ v 0; v 02 = v2 \ v 0: w e s h a ll a s s u m e t h a t jv 01j ¸ jv 02j. w e s h a ll c o m p le t e t h e p r o o f o f t h e b a s e o f in d u c t io n b y c o n s id e r in g t h e fo llo win g c a s e s : ca s e 1 : jv 01j = 8 a n d jv 02j = 1 . cle a r ly, a n y ve r t e x fr o m v 01 is a c la w-c e n t e r . ca s e 2 : jv 01j = 7 a n d jv 02j = 2 . it is n o t h a r d t o s e e t h a t v 01 c o n t a in s a c la w-c e n t e r . ca s e 3 : jv 01j = 6 a n d jv 02j = 3 . a g a in , it is a m a t t e r o f d ir e c t ve r ī c a t io n t h a t v 0 c o n t a in s a c la w-c e n t e r . ca s e 4 : jv 01j = 5 a n d jv 02j = 4 . co n s id e r t h e s u b g r a p h g1 o f q4 in d u c e d b y v 01 . cle a r ly, if g1 c o n t a in s a ve r t e x o f a d e g r e e t h r e e , t h e n t h is ve r t e x is a c la w-c e n t e r . th e r e fo r e , wit h o u t 6 6 r. kamalian, a. khachatryan 6 7 lo s s o f g e n e r a lit y, we c a n a s s u m e t h a t a n y ve r t e x in g1 h a s a d e g r e e a t m o s t t wo . it is n o t h a r d t o s e e t h a t t h is im p lie s t h a t g1 c o n t a in s n o is o la t e d ve r t e x. mo r e o ve r , s in c e jv 01j = 5 , we c a n c o n c lu d e t h a t g1 is a c o n n e c t e d g r a p h , a n d , c o n s e qu e n t ly, it is t h e p a t h o f le n g t h fo u r . n o w, le t a1; a2; a3 b e t h e in t e r n a l ve r t ic e s o f g1, a n d le t b1; b2 b e t h e e n d -ve r t ic e s o f g1. cle a r ly, we c a n a s s u m e t h a t n e it h e r o f a1; a2; a3 h a s a n e ig h b o u r in v 0 2 . s in c e jv2j = 8 a n d jv 02j = 4 , we h a ve t h a t t h e r e a r e ¯ ve p o s s ib ilit ie s fo r v 02 . w e in vit e t h e r e a d e r t o c h e c k t h a t in fo u r o f t h e s e c a s e s o n e c a n ¯ n d a c la w-c e n t e r in v 02 , a n d in t h e ¯ n a l c a s e v 0 h a s a ve r t e x z s u c h t h a t v 0nfzg in d u c e s a s im p le c yc le . n o w, le t u s a s s u m e t h a t t h e s t a t e m e n t is t r u e fo r n ¡ 1 , a n d a s u b s e t v 0 o f t h e ve r t ic e s o f qn s a t is ¯ e s t h e in e qu a lit y jv 0j ¸ 2 n¡1 + 1 . co n s id e r t h e fo llo win g p a r t it io n o f t h e ve r t ic e s o f qn: v1 = f( 0 ; ®2; :::; ®n ) : ®i 2 f0 ; 1 g; 2 · i · ng; v2 = f( 1 ; ®2; :::; ®n ) : ®i 2 f0 ; 1 g; 2 · i · ng: cle a r ly, t h e s u b g r a p h s o f qn in d u c e d b y v1 a n d v2 a r e is o m o r p h ic t o qn¡1. mo r e o ve r , it is n o t h a r d t o s e e t h a t a t le a s t o n e o f t h e fo llo win g t wo in e qu a lit ie s is t r u e : jv1 \ v 0j ¸ 2 n¡2 + 1 a n d jv2 \ v 0j ¸ 2 n¡2 + 1 . th u s t h e p r o o f fo llo ws fr o m t h e in d u c t io n h yp o t h e s is . fo r t h e c a s e o f n = 3 we h a ve : p r oposition 1. l et v 0 µ v ( q3 ) and let jv 0j ¸ 6 . then at least one of the following two conditions holds: ² there are four vertices in v 0 that induce a claw; ² there are six vertices in v 0 that induce a simple cycle. acknowledgement. w e wo u ld like t o t h a n k zh o r a n iko g h o s ya n a n d v a h a n mkr t c h ya n fo r t h e ir a t t e n t io n t o t h is wo r k. r e fe r e n c e s [1 ] w e s t d .b . introduction to graph theory. p r e n t ic e -h a ll, n e w je r s e y, 1 9 9 6 . d:\sbornik\...\ipia-art.dvi mathematical problems of computer science 24, 2005, 34{41. on i nter pr eter s of logic p r ogr amming systems s e m yo n a . n ig iya n a n d a r a m m. h a m b a r d z u m ya n department of system programming,yerevan state university, e-mail nigiyan@ysu.am, me76@front.ru abstract we introduce the notions of totally resolving and totally complete interpreters for horn programming languages. we prove the existence of totally complete interpreter (an interpreter which gives all the answers for a query if the query is a logical consequence of the program) for any horn programming language and existence of totally resolving interpreter (an interpreter which gives all the answers for any program and query) for languages whose programs have ¯nite templates of their least models. we also consider problems of total completeness and total resolvability for prolog interpreter from viewpoint of some (natural) program transformations and prove that it is not possible to make the interpreter totally complete. refer ences [1 ] n ig iya n s . a ., k h a c h o ya n l . o. transformations of l ogic p rograms. p r o g r a m m in g a n d co m p u t e r s o ft wa r e , v o l. 2 3 , n o . 6 , p p . 3 0 2 -3 0 9 , 1 9 9 7 . [2 ] n ig iya n s . a , k h a c h o ya n l . o. on ¢ -equivalence problem of logic programs. r e p o r t s o f n a t io n a l a c a d e m y o f s c ie n c e s o f a r m e n ia , v o l. 9 9 , n o . 2 , p p . 9 9 -1 0 3 ( in r u s s ia n ) , 1 9 9 9 . [3 ] clo c ks in w . f., me llis h c. s . p rogramming in p rolog. b e r lin : s p r in g e r -v e r la g , 1 9 8 4 . [4 ] l lo yd j. w . f oundations of l ogic p rogramming. b e r lin : s p r in g e r -v e r la g , 1 9 8 4 . [5 ] n ig iya n s . a . the p rolog interpreter from the viewpoint of l ogical semantics. p r o g r a m m in g a n d co m p u t e r s o ft wa r e , v o l. 2 0 , n o . 2 , p p . 6 9 -7 5 , 1 9 9 4 . [6 ] h a m b a r d z u m ya n a . m. the completeness and solvability p roblems for simple m onadic p r ol og interpreter. p r o c e e d in g s o f t h e co n fe r e n c e o n co m p u t e r s c ie n c e a n d in fo r m a t io n te c h n o lo g ie s , y e r e va n , p p . 3 6 -3 8 , 1 9 9 9 . 3 4 s. a. nigiyan and a. m. hambardzumyan 3 5 îñ³ù³µ³ý³ï³ý íñ³·ñ³íáñù³ý ñ³ù³ï³ñ·»ñç çýï»ñåñ»ï³ïáñý»ñç ù³ëçý ê. ². üç·çû³ý, ². ø. ð³ùµ³ñóáõùû³ý ²ù÷á÷áõù ²ßë³ï³ýùáõù ý»ñï³û³óíáõù »ý ïáï³é éáõí»éç ¨ ïáï³é éñçí çýï»ñåñ»ï³ïáñý»ñç ñ³ëï³óáõãûáõýý»ñá ðáñýç íñ³·ñ³íáñù³ý 黽áõý»ñç ñ³ù³ñ: ²å³óáõóíáõù ¿ ïáï³é éñçí çýï»ñåñ»ï³ïáñç (çýï»ñåñ»ï³ïáñ, áñá ï³éçë ¿ ñ³ñóù³ý µáéáñ å³ï³ëë³ýý»ñá, »ã» ñ³ñóáõùá ñ³ý¹çë³ýáõù ¿ íñ³·ñç ïñ³ù³µ³ý³ï³ý ñ»ï¨³ýù) ·áûáõãûáõý᪠ï³ù³û³ï³ý ðáñýç íñ³·ñ³íáñù³ý 黽íç ñ³ù³ñ, ¨ ïáï³é éáõí»éç çýï»ñåñ»ï³ïáñç (çýï»ñåñ»ï³ïáñ, áñá ï³ù³û³ï³ý íñ³·ñç ¨ ñ³ñóù³ý ñ³ù³ñ ï³éçë ¿ µáéáñ å³ï³ëë³ýý»ñá) ·áûáõãûáõýá ³ûý 黽áõý»ñç ñ³ù³ñ, áñáýó íñ³·ñ»ñç ÷áùñ³·áõûý ùá¹»éý»ñç ß³µéáýý»ñá í»ñç³íáñ »ý: ü³¨ ¹çï³ñïíáõù »ý ïáï³é éñçíáõãû³ý ¨ ïáï³é éáõí»éçáõãû³ý ñ³ñó»ñá äðàèà-ç çýï»ñåñ»ï³ïáñç ñ³ù³ñª íñ³·ñ»ñç ¨ ñ³ñóáõùý»ñç áñáß (µý³ï³ý) ó¨³÷áëáõãûáõýý»ñç ï»ë³ï»ïçó, ¨ ³å³óáõóíáõù ¿, áñ çýï»ñåñ»ï³ïáñá ñý³ñ³íáñ ã¿ ¹³ñóý»é ïáï³é éáõí»éç: начиная с начала 2000 года осуществляется внедрение ghis в здравоохранении, в рамках принятого проекта о реформирование информ mathematical problems of computer science 49, 115--122, 2018. object-oriented modeling of matching to systemic classifiers sedrak v. grigoryan, nairi p. hakobyan and hovhannes s. vrtanesyan institute for informatics and automation problems of nas ra e-mail: addressforsd@gmail.com, hakobyannairi@gmail.com, hovhannesvrtanesyan@gmail.com abstract in this paper we present a version of object-oriented implementation of models of constructive regularized mental systems, mentals, and systemic classifiers introduced in [1] as well as algorithms for matching them to situations. we experiment the adequacy of the models and algorithms for the chess representing kernels of the class of combinatorial problems, where space of solutions can be represented by reproducible game trees (rgt). keywords: modelling, systemic classifiers, mental, matching, chess. 1. introduction 1.1. a number of researches ([1, 3]) search for systemic solutions of combinatorial problems (such as chess), as it is stated that those problems cannot be adequately solved by parametric methods [2]. we follow the line of approach given in [1], trying to provide a programmatic implementation of the model, suggest matching algorithms for these implementations, as well as provide evidence of their adequacy. 1.2. as it is described in [1], mental systems represent realities, in particular utilities, but have varying effectiveness with respect to the goals and are processed to support utilization and gaining benefits from utilities. it is stated that a mighty way of enhancement of effectiveness of mss, and thus, cognizers, is the regularization of classifiers induced by mental doers and mental systems, while classifiers cl of members x of communities c are regularized in c if accompanied by ontological in c methods, instructions allowing x regularly provide positive samples of inputs of cl, as well as let the members of c do the same by communicating with x. doers, we assume, are realities having inoutput parts and for available inrealities, i.e, realities at the input parts, either elaborating certain output realities or staying passive. classifiers of roots and utilities are identified as root and induced goals. 115 mailto:addressforsd@gmail.com mailto:hakobyannairi@gmail.com mailto:hovhannesvrtanesyan@gmail.com object-oriented modeling of matching to systemic classifiers 116 1.3. to model mentals and systemic classifiers, we consider a class of regularized competition problems, where the space of solutions can be represented by reproducible game trees (rgt). rgt is a class of problems, which includes the following requirements: a. there are (a) interacting actors (players, competitors, etc.) performing (b) identified types of actions in the (c) specified types of situations; b. there are identified utilities, goals for each actor; c. actions for each actor are defined. [4] 1.4. we consider the implementation of the given in [1] models of mentals and systemic classifiers for rgt problems, particularly for the kernel rgt problem, chess, intensively studied since shannon’s pioneer work in 1949 [9]. in what follows, we present an implementation for the models of systemic classifiers induced by mentals, provide algorithms of matching for the given models, as well as the ways to experiment their adequacy for the presentation of chess mental systems. 2. implementation of systemic classifiers 2.1. natural languages are systemic and comprehensive by their coverage of msystems, but classifiers are not constructive and model only fuzzy ones, because they determine not the positives of msystems, but only ids of positives and ids of rels between them [1]. oop languages are covering mdoers as well but are more systemic with respect to algorithms since they involve attribute/parent/doing relationships corresponding to have/be/do ones in natural languages [10]. we follow the line of object-oriented implementation of systemic classifiers based on have/be/do relationships that in contrast with the previous versions of implementations [5, 6] aim to be comprehensive with respect to specification of mentals [1]. fig. 1. oop (left) and have/be/do (right) presentations. fig. 2. nuclears for chess. s. grigoryan, n. hakobyan and h. vrtanesyan 117 2.1.1 we start the implementation of systemic classifiers from nuclears, which are presented as oop classes containing only one rule. it is similar to primitive types in oop. for instance, in chess, we define the following as nuclears: coordinates, color, and figure type. nuclears are similar to nuclear abstracts defined in [5]. 2.1.2 to make the matching algorithms easier, we introduce basic classifiers, which we define as the minimal units to appear in the situation. in chess, fields of the chess boards, which contain or do not contain a figure are basic classifiers. these classifiers can only have nuclears as attributes. we create -have relations between minimal classifiers and their attributes. in case of parent existence (for simplicity, the current implementation allows having only one parent for each type of classifier) we create a -be relation between minimal classifiers and their parent. 2.1.3 more complex systemic classifiers, which include as attributes any other types of classifiers, we name compositeclassifiers. similar to minimal classifiers, they can have only one parent with a -be relation and -have relations with their attributes. they are similar to oop classes. as in oop classes, this type of classifiers can also be virtual. in oop this is achieved by having one undefined method, we achieved it by having undefined attributes. an attribute is called undefined if in its description there is at least one rule that is not specified (not specified rules have the value ‘?’). only for compositeclassifiers we implement negation. there are 2 types of negations: negation of general concept and negation for a specific instance. in the first case, compositeclassifier is generally negated if there is no such concept at all. for example, if we want to check whether there is a ‘check’ concept on the chess board, this is a general negation. in the second case, negation refers only to the exact instance, say, to the concept ‘there is no figure on e4 field’. compositeclassifiers are similar to complex abstracts defined in [5] but provide more flexible presentations. 2.1.4 we develop sets similar to arrays in oop and similar to sets described in [6] with restriction that they can be only continuous and represent sets of classifiers. for simplicity, we assume that sets consist only of compositeclassifiers (since they can only represent any other type of classifier), that are connected to the sets with -have relations. sets have 3 rules on their instances: 1. impose restrictions on lower and upper bounds; 2. define continuity with respect to attributes instances; 3. indicate the direction. 2.1.5 actions are similar to oop functions that describe some algorithm. they are connected with -do relation with an actor that makes an action (precondition). in contrast to the actions described in [5], we represent actions as a composition of the main precondition classifier, which is the actor, other precondition classifiers and rules describing the algorithm, how to change the precondition and provide an instance of classifier as an output. fig. 3. presentation of "empty line" set, which is virtual, where orange colored nodes are sets. fig. 4. presentation of actions, which perform similar to methods in oop. object-oriented modeling of matching to systemic classifiers 118 2.1.6 dynamic classifiers describe the transition from one state to another similar to [7]. for example, in chess it is impossible to describe the concept ‘king cannot escape’ with classifiers that describe static chess concepts. these types have 2 attributes – precondition and postcondition, both of them are compositeclassifiers and are connected with -have relations. dynamicclassifiers, the same as previously described classifiers, can have only one parent with be relation. 2.1.7 goals also have 2 attributes – precondition and postcondition with -have relations. they can also have one parent with -be relation. in addition, there is also an evaluating function, which is defined by rules and maximal depth of tree to evaluate. 2.1.8 plans include goals as attributes connected by -have relations, and the goals are ordered by priorities. 2.1.9 corresponders have a compositeclassifier as an attribute with a -have relation and actions with a -do relation. some real life concepts, like factories, cars implementation require both functionality and attributes existence. for instance, the concept ‘car’ should have 4 wheels (attribute) and must be able to be driven. 2.2 experimenting models for chess we will demonstrate the algorithm on the example of the chess concept ‘check’ (shown in figure 3). ‘check’ is a compositeclassifier. attributes of ‘check’ are ‘king’ and ‘field is under attack’. ‘king’ is a minclassifier and asfollows from its definition, there are nuclears as attributes – 2 ‘coordinates’, ‘figure type’ and ‘figure color’. ‘king’ has a parent ‘figure’. ‘field is under attack’ is a virtual compositeclassifier, where attributes are ‘field’, ‘attacker’, and it has children ‘field is under attack of pawn’, ‘field is under attack of bishop’ etc. ‘field’ is a minclassifier with all the described nuclears with attributes. ‘field’ is a parent for ‘figure’. ‘attacker’ is a ‘figure’. we’ll define one of ‘field is under attack’’s children, the others are defined equally. fig. 5 a goal from plan of "mate by rook". fig. 6. presentation of a machine tool, which has both actions to perform and attributes. fig. 7. systemic presentation of "check" chess concept in solver18. blue nodes present composites (including virtual "field is under attack" concept), red nodes present mindoers, green nodes present nuclears and blue nodes present actions. s. grigoryan, n. hakobyan and h. vrtanesyan 119 ‘field is under attack of pawn’ is a compositeclassifier with attributes ‘field’ and ‘attacker’, where ‘attacker’ is a ‘pawn’. 3. matching mental system to situations fig. 8. presentation of situations. we represent situations as instances of nuclears (for example, for chess it has a figure color: “white, black, no color”, figure type: “pawn, bishop, knight, rook, queen, king, no figure”, coordinates nuclear has two children – x and y both having 1-8 coordinates). nuclears instances are specified in groups with groupids. a set of nuclear instances with group ids describes the situation as described in [8], but, on the contrary, here we describe the side that will act in the given situation. 3.1 general matching 3.1.1 matching nuclears matching nuclears is performed by checking its rule, if the rule is satisfied it’s being matched, and an instance of active nuclear with the given value is sent forward through the system. 3.1.2 matching minimal classifiers matching of minclassifiers is done by matching its attributes. 3.1.3 matching composite classifiers matching composites is done by matching its attributes and local rules checking. if all the attributes are matched and rules are satisfied, then the composite is matched. virtual composites are created and matched as described in [8], usages, which are the uses of virtually specified composite classifiers, are matched by their virtual specification matching and additional rules checking, while specifications matched as regular composites trigger matching of their parent pure virtual composite classifiers. 3.1.4 matching sets is done by the checking of each instance of its composite element. if the given numbers of instances that satisfy all the rules defined in set are received, the set instance is considered as matched. basically the set activation is described in [8]. 3.1.5 matching actors and actions actions are activated with the following steps: 1. actor is activated and other precondition attributes are activated, 2: action is activated and can be applied to the situation and modify it. 3.1.6 matching dynamic classifiers matching dynamic classifiers basically described in [6], dynamics are matched when precondition composite is matched, and postcondition for all of the final situations is satisfied. 3.1.7 matching goals goals, as described above, have a precondition, which is composite, this shall be matched in order to consider the goal classifier. when the precondition is matched, the given depth of the tree is generated and final situations are evaluated with the given evaluation function. all the instances of goals that match the postcondition are considered as matched, and their evaluation value is assigned. 3.1.8 matching corresponders matching corresponders is performed by both checking its attributes matching and by checking its action, i.e., comparing the expected postcondition with the output for the given precondition. object-oriented modeling of matching to systemic classifiers 120 in addition, matching of any type of classifiers to situations can be done by matching of their children, i.e., any child matching triggers matching its parent (e.g. “king”, “queen” or any other type of figure matching means matching “figure” as well). general matching process is done through the system for the given situation. at the first step, it finds matching some classifiers, at the next step it continues doing the same until there are no more matched classifiers. after each step matching negated classifiers, dynamics and goals are performed, as described below: 3.2 matching one systemic classifier in some cases, it is just needed to find a certain concept on the situation, say, to see if there’s a check on the chess board or not. check classifier matching is searching exactly for it. in this case, all other classifiers, which are not relevant to the check classifier, are ignored in the system when processing the situation. 3.2.1 negated classifiers matching negated classifiers are composite ones, there are two types of negations in the system, the regular negation shows only a flag in the composite classifier, and when this flag is active and there is no such concept in the situation, then the appropriate classifier instance with all empty values is fired as it is the other case of negation is having a certain negated concept in the situation, then in contrast to the general negation, this needs the main attribute of negated composite to be activated first. then if after finishing the processing of the situation positive presentation of the concept is not matched, the negation instance with the given main attribute is considered as matched. e.g., if “e4 field is not attacked” concept is being searched, then the field with the value e4, which can be defined as a main attribute there is being checked if under attack. if no “field under attack”, where the field is “e4” found, then e4 field is not under attack, which is what we needed. 2.3 matching algorithm by the example of chess let’s discuss an example of matching on an example of ‘check’ described in the previous chapter. on field c3 white king is matched, on field c8 black rook is matched and on field h8 black king is matched. on fields c1-c7, a8, b8, d8-g8 matches ‘field is under attack of rook’, which triggers matching its parent ‘field is under attack’. rook on c8, the free line between rook and white king activates ‘field under attack of rook’, in its turn its triggers matching ‘field under attack’ the ‘field’ attribute of which is the same as ‘king’, consequently, the rule in chess ‘check’ compositeclassifier is also satisfied and ‘check’ is matched. fig. 9. situation where "check" concept is matched. s. grigoryan, n. hakobyan and h. vrtanesyan 121 4. conclusions in the following work models for constructive regularized mental systems were described and their implementations were discussed. 1. object-oriented models for systemic classifiers discussed in [1] were developed, particularly nuclears’, goals’, corresponders’ and other types of classifiers’ models were given, and their adequacy evidence is demonstrated for the example of chess concepts. 2. algorithms for matching situations to each type of systemic classifiers are provided, where there are two types of matching algorithms, a. matching situations to the whole system of classifiers, where all the matched classifiers are activated and b. matching situations to certain classifiers, where only instances of expected classifiers are searched. adequacy of matching algorithms is experimented for the example of chess concepts. acknowledgements authors express their deep gratitude to professor edward pogossian for supervising the work. references [1] e. pogossian, “towards adequate constructive models of mental systems”, international conference in computer sciences and information technologies, yerevan, armenia, pp. 6, 2017. [2] m. botvinnik, “computers in chess: solving in exact search problems”, springer series, in symbolic computation, with appendixes, springer-verlag, new york, 1984. [3] m. botvinnik, about solving approximate problems, (in russian), s. radio, moscow, 1979. [4] e. pogossian, v. vahradyan and a. grigoryan, “on competing agents consistent with expert knowledge”, lecture notes in computer science, ais-adm-07: the international workshop on autonomous intelligent systems agents and data mining, st. petersburg, russia, june 6-7, pp. 229-241, 2007. [5] k. khachatryan and s. grigoryan, “java programs for presentation and acquisition of meanings in ssrgt games”, proceedings of seua annual conference, yerevan, armenia, pp. 127-135, 2013. [6] s. grigoryan, research and development of algorithms and programs of knowledge acquisition and their effective application to resistance problems, pp. 111, yerevan, armenia, 2016. [7] s. grigoryan, “dynamic knowledge integration into hbd knowledge”, international conference in computer sciences and information technologies, yerevan, armenia, pp. 3, 2017. [8] k. khachatryan and s. grigoryan, “java programs for matching situations to the meanings of ssrgt games”, proceedings of seua annual conference, yerevan, armenia, pp. 135-141, 2013. [9] c. shannon, “programming a computer for playing chess”, philosophical magazine, vol. 41, no. 314, 1950. [10] e. pogossian, “on modeling cognition”, international conference in computer sciences and information technologies, yerevan, armenia, pp 194-198, 2011. submitted 11.09.2017, accepted 14.02.2018. object-oriented modeling of matching to systemic classifiers 122 սիստեմիկ դասակարգիչների համապատասխանեցման օբյեկտ-կողմնորոշված մոդելավորում ս. գրիգորյան, ն. հակոբյան և հ. վրթանեսյան ամփոփում աշխատանքում ներկայացնում ենք կառուցողական կանոնակարգված մտավոր համակարգերի, մտավորների (mentals) և սիստեմիկ դասակարգիչների օբյեկտկողմնորոշված մոդելների մի իրականացում, որոնք ներկայացված են [1]-֊ում, մշակված են ալգորիթմներ դրանք իրավիճակներին համապատասխանեցնելու համար։ մոդելների և ալգորիթմների համապատասխանությունը մենք փորձարկում ենք շախմատի համար, որը ներկայացնում է միջուկ կոմբինատոր խնդիրների մի դասի համար, որոնց լուծումների բազմությունը վերարտադրելի ծառ է (rgt): объектно-ориентированное моделирование соответствия системным классификаторам с. григорян, н. акобян и о. вртанесян аннотация в даной работе представлена версия оъектно-ориентированной реализации моделей конструктивно регуляризованных ментальных систем, mentals и системных классификаторов, представленных в [1], а также алгоритмов их сопоставления с ситуациями. адекватность моделей и алгоритмов мы экспериментурием для шахмат, представляющее собой ядро класса комбинаторных задач, где пространство решений представляет собой воспроизводимое дерево. mathematical problems of computer science 49, 41–48, 2018. dynamic task scheduling based on abelian sandpile and rotor-router models hayk e. nahapetyan and suren s. poghosyan institute for informatics and automation problems of nas ra e-mail: hayknahapetyan@yahoo.com, psuren55@yandex.ru abstract this study is dedicated to the possible usage of self-organized criticality models in large-scale computing systems for load balancing and energy-awareness. methods and software tools aimed at modeling and visualization of dynamic tasks scheduling in virtual distributed systems constructed over sandpile and rotor-router models, are also presented. keywords: asm, rotor-router decentralized systems, dynamic task scheduling. 1. introduction the concept of self-organized criticality was first introduced by bak, tang and wiesenfeld in 1987 [3], and gave rise to growing interest in the study of self-organizing systems. bak et al. argued that in many natural phenomena, the dissipative dynamics of the system is such that it drives the system to a critical state, thereby leading to ubiquitous power law behaviors. the sandpile models, being a class of cellular automata, are among the simplest theoretical models, which exhibit self-organized criticality. a special subclass of interest consists of so called abelian sandpile models (asm). the abelian sandpile and rotor-router models were discovered several times by researchers in different communities operating independently. the abelian sandpile model was invented by dhar [1], where the rotor-router model, a deterministic analogue of random walk, was first defined by priezzhev et al. under the name of eulerian walkers [2]. the abelian property means that the final stable state of the ca is independent of the order in which the updates of cells are carried out. this property plays a key role during the numerical, as well as analytical studies of the asm [4] – [7]. there are a number of solutions for tasks scheduling and load balancing based on the sandpile model [8]–[10]. anyway, for large-scale real-time computing systems, critical performance constraints are imposed by the environment, and the correctness depends not only on the logical result of the computation, but also on the time at which the results are produced with keeping energy-awareness. newer solutions based on rotor-router model may provide a better solution in the area. besides, the paper presents appropriate software packages for simulating and verifying large-scale cluster systems relied on sandpile and rotor-router-based tasks scheduling. the scheduler and load balancer possessing the above characteristics are constructed on an agent system in which the agents render the cells of a cellular automaton. thus, agents 41 42 dynamic task scheduling based on abelian sandpile and rotor-router models are essential components of the architecture, where the topology for interconnecting and where collaborating the agents is another issue for consideration. as a model for simulation, a 2-dimensional lattice is investigated, where every agent denotes itself as a computer node with a private computing resource. depended on the assigned workload, the node itself makes a decision whether to migrate tasks to adjacent/neighboring nodes or not. detailed description of the proposed model is given in the third section. 2. sandpile model consider an undirected graph g = (v, e) described with the set of vertices v = {v1, v2, . . . , vn} and the set of edges e. each vertex vi ∈ v is assigned a variable hi which takes integer values and represents the height of the sand at that vertex. hmaxi denotes the maximal allowed height for the vertex vi in the graph g. for a d-dimensional lattice, we take hmaxi = 2d + 1. ct denotes the set of heights hi, which determines the configuration of the system at a given discrete time t . a configuration is called stable, if all heights satisfy hi < h max i . the vertex vi is called closed, if h max i = deg(vi), where deg(vi) indicates the degree of vi. the dynamics of the system is defined by the following rules. consider a stable configuration ct at a given time t . we add a grain of sand to a random vertex vi ∈ v by setting hi to hi + 1 (we assume that the vertex is chosen randomly with a uniform distribution on the set v ). this new configuration, if stable, defines ct+1. if hi ≥ h max i , then the vi becomes unstable and topples losing h max i grains of sand, while all neighbors of vi receive one grain. note that if the vertex is open, then the system loses grains. during the toppling of the closed vertices, the number of grains is conserved. note also that toppling of a vertex may cause some of its neighboring vertices to become unstable. in this case, those vertices also topple according to the same toppling rule. once all unstable vertices are toppled, a new stable configuration ct+1 is obtained. if the finite connected graph g has at least one open vertex, then all vertices become stable after a finite number of topplings. moreover, the new stable configuration is independent of the toppling order. let âi be an operator, which acts on sandpile configurations and adds a grain to vertex i. it can be easily shown that âiâj = âjâi. this is the reason why the sandpile model is called abelian. 2.1 rotor-router model to define the rotor-router model on a directed graph g, for each vertex of g, fix a cyclic ordering of the outgoing edges. to each vertex v we associate a rotor (v ) chosen from among the outgoing edges from v . a chip performs a walk on g according to the rotor-router rule: if the chip is at v , we first increment the rotor (v ) to its successor e = (v, w) in the cyclic ordering of outgoing edges from v, and then route the chip along e to w. if the chip ever reaches a sink, i.e., a vertex of g with no outgoing edges, the chip will stop there; otherwise, the chip continues walking forever. h. nahapetyan and s. poghosyan 43 fig 1. rotor-router example. 3. sand-scheduler in this section, we are going to describe a software tool that has a purpose of modeling and visualizing dynamic tasks scheduling in distributed systems. as already mentioned, the scheduling algorithm used by this tool is based on two well-known models: abelian sandpile model (asm) and rotor-router model. in original formulation of asm, each site on a finite grid has an associated value that corresponds to the slope of the pile. this slope builds up as ”grains of sand” (or ”chips”) are randomly placed onto the pile, until the slope exceeds a specific threshold value at which time that site collapses and transfers its sand grains to its adjacent sites, increasing their slope. bak, tang, and wiesenfeld considered the process of successive random placement of sand grains on the grid; each such placement of sand at a particular site may have no effect,or it may cause avalanches, which may have a cascading effect on many sites. the original interest behind the model stemmed from the fact that in simulations on lattices, it is attracted to its critical state, at which point the correlation length of the system and the correlation time of the system go to infinity, without any fine tuning of a system parameter. in the sandpile model dropping another grain of sand onto the pile may cause nothing to happen, or it may cause the entire pile to collapse in a massive slide. we use the above mentioned property of sandpile in order to dynamically schedule tasks, based on the background process of avalanches that is visible in debug enabled state. in our workload, tasks may arrive in a group of up to 7 tasks and be assigned to some node in the system. these tasks in a group are typically a set of multiple instances of the same sequential program. that is why the tasks in a group are independent of each other and can be executed in parallel. all the tasks in a group have only one important property, assigned by ti, which shows the number of rounds needed for the task to be executed in a system. all the tasks of a given group have the same ti. preliminaries for the sandpile-based dynamic scheduling problem are the following: 44 dynamic task scheduling based on abelian sandpile and rotor-router models fig 2. sand-scheduler “debug” enabled. • each task has its own required execution time ti. • there is a set p of homogeneous processors, where |p | = nxm (10x10 in our example). • each processor can execute at most k task simultaneously at any given time. (k = 3 in our example) • the nodes are connected between them and only interacting with the small subset of the neighbours (at most 4). • each node has a working queue q, |q| ≤ 4 that gets filled up when all the resources are taken. • the total execution time for any task is said to be ti = ti + si, where si is the time required for the task to be scheduled(find empty slot in nodes) and start its execution. 3.1 sandpile-based mode in this mode, we study the system in a critical state, and the state of the system is being reconfigured periodically. it is a cellular automaton, which models the process of dropping on grains of sand on a surface and the collapsing of grains due to the increase of the height of the slope. this process is going on regardless of the number of assigned tasks to the system. the tasks are pretty similar to the grains of sand but they do not participate in avalanches themselves. when the grains of the current node reach the maximum value, they start to topple. in case there are tasks assigned to the node at that moment and these tasks are not yet ready to be executed (all the computing resources of the node are reserved), they will be toppled with the sands and move to another node along with the corresponding grain of sand. the transition rule in this model is triggered when the height of the current grain is bigger than the configured value for the system (4 in our study). this process of avalanches is going on indefinitely, meanwhile, this system is dynamically balancing the distribution of tasks among all buckets. h. nahapetyan and s. poghosyan 45 fig 3. sand-scheduler debug mode. green executing tasks, red waiting tasks, blue fictive sands for simulating avalanches, black fictive sands that are in critical state. another option that this scheduler supports, is the fault-tolerance. during the execution, we can disable an arbitrary node or nodes, meanwhile, the scheduler will keep working without this kind of failures that are common in large-scale computational systems. the “energy aware mode is a modification of this scheduler in order to reduce the number of nodes that are powered on. depending on the number of non-scheduled tasks in the system at any given moment of time, only the required (min count of nodes needed for executing tasks at the same time) count of nodes are powered on. fig 4. sand-scheduler. 46 dynamic task scheduling based on abelian sandpile and rotor-router models 3.2 rotor-router mode the software package developed implements one more scheduling mode based on the rotorrouter model. relying on priezzev’s [12, 13] dhar’s studies [14], we can make sure that grains in rotor-router configuration are equally distributed. so, if we change grains with tasks we can be sure that load balancing will be provided in cluster systems. moreover, we are pushing forward a hypothesis that even for tasks with execution timing(the task will be removed after execution and free up space) in rotor-router configuration tasks will be equally distributed in the system, which is visible via software package described in this paper. fig 5.sand-scheduler. non of the tasks is executed yet. fig 6. sand-scheduler. first tasks have been executed. h. nahapetyan and s. poghosyan 47 4. conclusion in this paper, possible usage of asm and rotor-router model in cluster systems has been discussed. also, appropriate software tools have been developed for cluster simulation and for visualization of tasks dissemination. perspectives of this work are to deploy asm and rotorrouter-based algorithms for task distribution on real systems and obtain a comparative analysis between real-world solutions. 5. acknowledgement the authors are grateful to prof. yu. h. shoukourian and dr. y. alaverdyan for important discussions and critical remarks at all stages of the work. this work was supported by the state committee of science mes ra, in the frames of the research project no. 16yr-1b008. references [1] d. dhar, “self-organized critical state of sandpile automaton models”, phys. rev. lett., vol. 64, no. 14, pp. 1613–1616, 1990. [2] b. priezzhev, d. dhar, a. dhar and s. krishnamurthy, “eulerian walkers as a model of self-organized criticality”, phys. rev. lett., vol. 77, pp. 50795082, 1996. [3] p. bak, c. tang and k. wiesenfeld,“self-organized criticality: an explanation of the 1/f noise”,phys. rev. lett., vol.59, no. 4, pp. 381384, 1987. [4] v. s. poghosyan, s. y. grigorev, v. b. priezzhev and p. ruelle, “pair correlations in the sandpile model: a check of logarithmic conformal field theory”, phys. lett. b, vol. 659, pp. 768772, 2008. [5] su. s. poghosyan, v. s. poghosyan, v. b. priezzhev and p. ruelle, “numerical study of correspondence between the dissipative and fixed-energy abelian sandpile models”, phys.rev. e, 84, 066119, 2011. [6] v. s. poghosyan, s. s. poghosyan and h. e. nahapetyan, “the investigation of models of self-organized systems by parallel programming methods based on the example of an abelian sandpile model”, proc. csit conference 2013, yerevan armenia, sept. 23-27, pp. 260-262, 2013. [7] h. nahapetyan, j.-pierre jessel, s. poghosyan and y. shoukourian,“a multi user and multi purpose ca simulator”,phys. rev. lett., vol.59, no. 4, pp. 381384, 1987. proc. csit conference 2017, yerevan armenia, sept. 23-27, pp. 260-262. [8] y. rabani, a. sinclair, and r. wanka, “local divergence of markov chains and the analysis of iterative load-balancing schemes”, in ieee symp. on foundations of computer science, pp. 694705, 1998. [9] j. l. j. laredo, p. bouvry, f. guinand, b. dorronsoro and c. fernandes, “the sandpile scheduler”, cluster computing vol.17, pp 191204, 2014. [10] j. gsior and f. seredyski, “a sandpile cellular automata-based scheduler and load balancer”, journal of computational science, vol.21, pp. 460-468, 2017. [11] l. levine and y. peres, “asymptotics for rotor-router aggregation and the divisible sandpile”, potential analysis, 30: 1. https://doi.org/10.1007/s11118-008-9104-6 4 8 dynamic task scheduling based on abelian sandpile and rotor-router models [1 2 ] a . m. p o vo lo t s ky, v . b . p r ie z z h e v a n d r . r . s h c h e r b a ko v, \ d yn a m ic s o f e u le r ia n wa lke r s " , p hysical review e , vo l.5 8 , d oi:h t t p s :/ / d o i.o r g / 1 0 .1 1 0 3 / p h ys r e ve .5 8 .5 4 4 9 [1 3 ] v . b . p r ie z z h e v, \ s e lf-o r g a n iz e d c r it ic a lit y in s e lf-d ir e c t in g wa lks " , a r x iv:c o n d m a t / 9 6 0 5 0 9 4 [1 4 ] d . d h a r , \ th e o r e t ic a l s t u d ie s o f s e lf-o r g a n iz e d c r it ic a lit y" , p hysica a: statistical m echanics and its applications, vo l. 3 6 9 , n o . 1 , p p . 2 9 -7 0 , 2 0 0 6 submitted 04.09.2017, accepted 15.01.2018. ¸çý³ùçï ³é³ç³¹ñ³ýùý»ñç åé³ý³íáñáõù` ñçùýí³í ²µ»éû³ý ³í³½³ïáõûïç ¨ rotor-router ùá¹»éý»ñç íñ³ ð. ü³ñ³å»ïû³ý ¨ ê. äáõáëû³ý ²ù÷á÷áõù ²ûë áõëáõùý³ëçñáõãûáõýá ýíçñí³í ¿ ëáßáñ³í³í³é ñ³ßíáõ³ï³ý ñ³ù³ï³ñ·»ñáõù çýùý³ï³½ù³ï»ñå ïñçïçï³ï³ý ùá¹»éý»ñç ñý³ñ³íáñ û·ï³·áñíù³ýá í³ýñ³µ»éýí³íáõãû³ý µ³ßëù³ý ¨ ¿ý»ñ·³ëý³ûáõáõãû³ý ýå³ï³ïý»ñáí: ü»ñï³û³óí³í »ý ù»ãá¹ý»ñ ¨ íñ³·ñ³ûçý ·áñíçùý»ñ, áñáýù ùçïí³í »ý ³í³½³ïáõûïç ¨ éáïáñ-éááõï»ñ ùá¹»éý»ñç ñçù³ý íñ³ ï³éáõóí³í íçñïáõ³é µ³ßëí³í ñ³ù³ï³ñ·»ñáõù ¹çý³ùçï ëý¹çñý»ñç åé³ý³íáñù³ýá ¨ ï»ë³µ»ñù³ýá: äèíàìè÷åñêîå ïëàíèðîâàíèå çàäà÷, îñíîâàííûõ íà ìîäåëÿõ àáåëåâîé ïåñ÷àíîé êó÷è è ðîòîð-ðîóòåðà ã. íàãàïåòÿí è ñ. ïîãîñÿàí àííîòàöèÿ èññëåäîâàíèå ïîñâÿùåíî âîçìîæíîìó èñïîëüçîâàíèþ ñàìîîðãàíèçóþùèõñÿ êðèòè÷íûõ ìîäåëåé â øèðîêîìàñøòàáíûõ ñèñòåìàõ äëÿ áàëàíñèðîâêè íàãðóçêè è ýíåðãîñáåðåæåíèÿ. òàêæå ïðåäñòàâëåíû ìåòîäû è ïðîãðàììíûå ñðåäñòâà, ïðåäíàçíà÷åííûå äëÿ ìîäåëèðîâàíèÿ è âèçóàëèçàöèè ïëàíèðîâàíèÿ äèíàìè÷åñêèõ çàäà÷ â âèðòóàëüíûõ ðàñïðåäåëåííûõ ñèñòåìàõ, ïîñòðîåííûõ íà ìîäåëÿõ ïåñ÷àíîé êó÷è è ðîòîð-ðîóòåðà. 051 hayk'sabstract_1 d:\sbornik\...\aritcle.dvi mathematical problems of computer science 24, 2005, 104{106. a n ote on m atching cover ed gr aphs v a h a n v . mkr t c h ya n department of informatics and applied mathematics, yerevan state university e-mail vahanmkrtchyan2002yahoo.com abstract a graph is called matching covered if for its every edge there is a maximum matching containing it. it is shown that line-extremal matching covered graphs contain a perfect matching. refer ences [1 ] h a r a r y f., gr a p h th e o r y, a d d is o n -w e s le y, r e a d in g , ma , 1 9 6 9 . [2 ] l o va s z l ., p lu m m e r m.d ., ma t c h in g th e o r y, a n n a ls o f d is c r e t e ma t h . 2 9 , n o r t h h o lla n d , 1 9 8 6 . [3 ] w e s t d . b ., in t r o d u c t io n t o gr a p h th e o r y, p r e n t ic e -h a ll, in c .,1 9 9 6 . ¶ñ³éáõù ½áõ·³ïóáõùý»ñáí í³íïí³í ·ñ³ýý»ñç ù³ëçý ì.ì. øïñïãû³ý ²ù÷á÷áõù ¶ñ³ýá ïáãíáõù ¿ ½áõ·³ïóáõùý»ñáí í³íïí³í, »ã» ýñ³ ó³ýï³ó³í ïáõç ñ³ù³ñ ·áûáõãûáõý áõýç ³ûý å³ñáõý³ïáõ ù³ùëçù³é ½áõ·³ïóáõù: òáõûó ¿ ïñí»é, áñ ½áõ·³ïóáõùý»ñáí í³íïí³í ·ñ³ýý»ñá, áñáýù ¿ùëïñ»ù³é »ý ïáõç ñ»é³óáõù ·áñíáõáõãû³ý ýï³ïù³ùµ, å³ñáõý³ïáõù »ý ï³ï³ñû³é ½áõ·³ïóáõù: 1 0 4 mathematical problems of computer science 54, 7–17, 2020. udc 519.1 a note on hamiltonian bypasses in digraphs with large degrees samvel kh. darbinyan institute for informatics and automation problems of nas ra e-mail: samdarbin@iiap.am abstract let d be a 2-strongly connected directed graph of order p ≥ 3. suppose that d(x) ≥ p for every vertex x ∈ v (d)\{x0}, where x0 is a vertex of d. in this paper, we show that if d is hamiltonian or d(x0) > 2(p − 1)/5, then d contains a hamiltonian path, in which the initial vertex dominates the terminal vertex. keywords: digraph, cycle, hamiltonian cycle, hamiltonian bypass. 1. introduction in this paper, we consider finite digraphs (directed graphs) without loops and multiple arcs. every cycle and path are assumed to be simple and directed. we shall assume that the reader is familiar with the standard terminology on digraphs and refer the reader to [1]. a cycle (path) in a digraph d passing through all the vertices of d is called hamiltonian. a digraph containing a hamiltonian cycle is called a hamiltonian digraph. a hamiltonian path in a digraph d in which the initial vertex dominates the terminal vertex is called a hamiltonian bypass. there are numerous sufficient conditions for the existence of a hamiltonian cycle in digraphs (see, e.g., [1, 2, 3]). it is natural to consider an analogous problem for the existence of a hamiltonian bypass. it was proved in [4] [9] that a number of sufficient conditions for a digraph to be hamiltonian is also sufficient for a digraph to contain a hamiltonian bypass (with some exceptions which are characterized). in particular, theorems 1.4 and 1.5 were proved in [5] and [6], respectively. to formulate these theorems, we need the following definitions. definition 1: let d0 denote any digraph of order p ≥ 3, p is odd, such that v (d0) = a∪b, where a∩b = ∅, a is an independent set with (p + 1)/2 vertices, b is a set of (p−1)/2 vertices inducing an arbitrary subdigraph, and d0 contains all the possible arcs between a and b. definition 2: for any k ∈ [1, p − 2] let dp−k,k denote a digraph of order p ≥ 3, obtained from k∗p−k and k ∗ k+1 by identifying a vertex of the first with a vertex of the second. 7 8 a note on hamiltonian bypasses in digraphs with large degrees definition 3: by t5 we denote a tournament of order 5 with vertex set {x1, x2, x3, x4, y} and arc set {xixi+1 |i ∈ [1, 3]}∪{x4x1, x1y, x3y, yx2, yx4, x1x3, x2x4}. theorem 1: (benhocine [5]). let d be a 2-strong digraph of order p with minimum degree at least p− 1. then d contains a hamiltonian bypass, unless d is isomorphic to a digraph of type d0. theorem 2: (darbinyan [6]). let d be a strong digraph of order p ≥ 3. suppose that d(x) + d(y) ≥ 2p−2 for every pair of non-adjacent vertices x, y of v (d). then d contains a hamiltonian bypass, unless d is isomorphic to a digraph of the set d0 ∪{dp−k,k, t5, c3}. the author [10] proved the following results. theorem 3: (darbinyan [10]). for every integer p ≥ 8 there is a 2-strong non-hamiltonian digraph of order p, which has p− 1 vertices of degrees at least p. theorem 4: (darbinyan [9]). let d be a 2-strong digraph of order p ≥ 3 with the minimum degree at least p−4. if p−1 vertices of d have degrees at least p, then d is hamiltonian. theorem 5: (darbinyan [10]). let d be a strong digraph of order p ≥ 3. suppose that d(x) + d(y) ≥ 2p− 1 for every pair of non-adjacent vertices x, y ∈ v (d) \{z0}, where z0 is some vertex in v (d). then d contains a cycle of length at least p− 1. the following corollary follows from theorem 5. corollary 1: let d be a strong digraph of order p ≥ 3. if p − 1 vertices of v (d) have degrees at least p, then d is hamiltonian or contains a cycle of length p− 1 (in fact, d has a cycle that contains all the vertices with degrees at least p). remark 1: for the proof of theorem 3, it suffices to consider a digraph h(n) of order n ≥ 8, which is defined as follows: v (h(n)) := {x0, x1, x2, . . . , xn−4, y1, y2, y3} and a(h(n)) := {yiyj |i 6= j}∪{xixi+1 |0 ≤ i ≤ n− 4}∪{yixj |1 ≤ i ≤ 3, 1 ≤ j ≤ n− 6} ∪{xixj |1 ≤ j < i ≤ n− 4}∪{xn−4yi, xn−6 |1 ≤ i ≤ 3}∪{xixn−5 |1 ≤ i ≤ n− 7} ∪{x0xn−5, xn−5x0, xn−4x0, xn−6xn−4}. note that theorem 3 disproves a conjecture of thomassen ([2]. every 3-strong digraph of order p with minimum degree at least p + 1 is strongly hamitonian-connected). in this paper, we prove the following theorem. theorem 6: let d be a 2-strong digraph of order p ≥ 3. suppose that d(x) ≥ p for every vertex x ∈ v (d)\{x0}, where x0 is a vertex of d. if d is hamiltonian or d(x0) > 2(p−1)/5, then d contains a hamiltonian bypass. s. darbinyan 9 2. terminology and notation in this paper, we consider finite digraphs without loops and multiple arcs. for a digraph d, we denote by v (d) the vertex set of d and by a(d) the set of arcs in d. the order of d is the number of its vertices. let x, y be distinct vertices in d. the arc of a digraph d directed from x to y is denoted by xy (we say that x dominates y). for disjoint subsets a and b of v (d) we define a(a → b) as the set {xy ∈ e(d) |x ∈ a, y ∈ b}, a(a, b) = a(a → b)∪a(b → a). the notation a → b denotes that every vertex of a dominates every vertex of b. a 7→ b means that a → b and there is no arc from a vertex of b to a vertex of a. if x ∈ v (d) and a = {x}, we write x instead of {x}. the out-neighborhood of a vertex x is the set n+(x) = {y ∈ v (d) |xy ∈ a(d)} and n−(x) = {y ∈ v (d) |yx ∈ a(d)} is the in-neighborhood of x. similarly, if a ⊆ v (d), then n+(x, a) = {y ∈ a |xy ∈ a(d)} and n−(x, a) = {y ∈ a |yx ∈ a(d)}. the out-degree of x is d+(x) = |n+(x)| and d−(x) = |n−(x)| is the in-degree of x. similarly, d+(x, a) = |n+(x, a)| and d−(x, a) = |n−(x, a)|. the degree of the vertex x in d is defined as d(x) = d+(x) + d−(x) (similarly, d(x, a) = d+(x, a) + d−(x, a)). the subdigraph of d induced by a subset a of v (d) is denoted by d[a]. the path (respectively, the cycle) consisting of the distinct vertices x1, x2, . . . , xm (m ≥ 2) and the arcs xixi+1, 1 ≤ i ≤ m − 1 (respectively, xixi+1, 1 ≤ i ≤ m − 1, and xmx1), is denoted by x1x2 · · ·xm (respectively, x1x2 · · ·xmx1). we say that x1x2 · · ·xm is a path from x1 to xm or is an (x1, xm)-path. the length of a cycle or a path is the number of its arcs. a cycle of length k, k ≥ 2, is denoted by ck. for a cycle ck := x1x2 · · ·xkx1, the subscripts considered modulo k, i.e., xi = xs for every s and i such that i ≡ s (mod k). if p is a path containing a subpath from x to y, we let p [x, y] denote that subpath. similarly, if c is a cycle containing vertices x and y, c[x, y] denotes the subpath of c from x to y. for a digraph d of order n, by d(n, 2) = [x1xn; x1x2x3 . . . xn] we denote a hamiltonian path in which the initial vertex x1 dominates the terminal vertex xn. a digraph d is strongly connected (or, just, strong) if there exists a path from x to y and a path from y to x for every pair of distinct vertices x, y. a digraph d is k-strongly connected (or, k-strong), if |v (d)| ≥ k + 1 and d[v (d) \ a] is strong for any set a of at most k−1 vertices. two distinct vertices x and y of a digraph d are adjacent if xy ∈ a(d) or yx ∈ a(d) (or both). by k∗n is denoted the complete digraph of order n. 3. preliminaries the following well-known simple lemmas 1-3 are the basis of our results and other theorems on directed cycles and paths in digraphs. they will be used extensively in the proof of our result. lemma 1: (häggkvist and thomassen [12]). let d be a digraph of order p ≥ 3 containing a cycle cm, 2 ≤ m ≤ p − 1. let x be a vertex not contained in this cycle. if d(x, v (cm)) ≥ m + 1, then for every k, 2 ≤ k ≤ m + 1, d contains a cycle of length k including x. the following lemma is a modification of a lemma by bondy and thomassen [13]. lemma 2: let d be a digraph of order p ≥ 3 containing a path p := x1x2 . . . xm, 2 ≤ m ≤ p− 1 and x be a vertex not contained in this path. if one of the following conditions holds: 10 a note on hamiltonian bypasses in digraphs with large degrees (i) d(x, v (p )) ≥ m + 2; (ii) d(x, v (p )) ≥ m + 1 and xx1 /∈ a(d) or xmx /∈ a(d); (iii) d(x, v (p )) ≥ m, xx1 /∈ a(d) and xmx /∈ a(d); then there is an i, 1 ≤ i ≤ m − 1, such that xix, xxi+1 ∈ a(d) i.e., x1x2 . . . xixxi+1 . . . xm is a path of length m d (we say that x can be inserted into p ). the following lemma is a simple extension of a lemma by bang-jensen, gutin and li [14]. lemma 3: let p = u1u2 . . . us be a path in a digraph d (possibly, s = 1) and let q = v1v2 . . . vt be a path (or q = v1v2 . . . vtv1 be a cycle) in d[v (d)\v (q)], t ≥ 2. suppose that for each ui, 1 ≤ i ≤ s, there is an arc vjvj+1 on q such that vjui, uivj+1 ∈ a(d). then there is a (v1, vt)-path (or a cycle) of length t + k − 1 (respectively, t + k), 1 ≤ k ≤ s, with vertex set {v1, v2, . . . , vt}∪{u1, u2, . . . , uk}. 4. proofs of the main results theorem 6: let d be a 2-strong digraph of order p ≥ 3. suppose that d(x) ≥ p for every vertex x ∈ v (d)\{x0}, where x0 is a vertex of d. if d is hamiltonian or d(x0) > 2(p−1)/5, then d contains a hamiltonian bypass. proof: suppose, on the contrary, that is d contains no hamiltonian bypass. we first will prove the following claim (note that in the proofs of claim 1 and case 1, we do not use the fact that d is 2-strong). claim 1: d has no cycle of length l through x0, where l = p− 1 or l = p− 2. proof: suppose that the claim is not true. assume that cp−1 = x1x2 . . . xp−1x1, x0 ∈ v (cp−1) and y /∈ v (cp−1). since d contains no hamiltonian bypass, for every i, 1 ≤ i ≤ p− 1, we have d+(y,{xi, xi+1}) ≤ 1 and d−(y,{xi, xi+1}) ≤ 1. therefore, 2d(y) = p−1∑ i=1 (d+(y,{xi, xi+1}) + d−(y,{xi, xi+1})) ≤ 2(p− 1), which contradicts that d(y) ≥ p. thus, d contains no cycle of length p− 1 through x0. now assume that d contains a cycle of length p− 2 through x0. let cp−2 = x1x2 . . . xp−2x1, x0 ∈ v (cp−2) and x, y /∈ v (cp−2). since d contains no cycle of length p − 1 through x0, from lemma 1 it follows that xy, yx ∈ a(d), d(x, v (cp−2)) = d(y, v (cp−2)) = p − 2 and there is a vertex xi such that the vertices x, xi are not adjacent and the arcs xi−1x, xxi+1 are in d. if yxi ∈ a(d), then d(p, 2) = [yxi; yxc[xi+1, xi]], if xiy ∈ a(d), then d(p, 2) = [xiy; c[xi, xi−1]xy], a contradiction. we may therefore assume that xi and y also are not adjacent. using this, lemmas 1, 2 and the fact that d contains no cycle of length p − 1 througt x0, we obtain that xi = x0, xi−1y, yxi+1 ∈ a(d) and the vertex x (y) is adjacent to every vertex in v (d) \{x0}. hence, we have that if xxi+2 ∈ a(d), then d(p, 2) = [yxi+1; yxc[xi+2, xi+1]], a contradiction. if xxi+2 /∈ a(d), then xi−2x ∈ a(d) and d(p, 2) = [xi−1y; c[xi−1, xi−2]xy], a contradiction. claim 1 is proved. now, we divide the proof into two cases to consider. case 1. d is hamiltonian. s. darbinyan 11 let cp = x1x2 . . . xpx1 be a hamiltonian cycle in d. since d contains no hamiltonian bypass, we have that xi+1xi /∈ a(d) for every i, 1 ≤ i ≤ p. using this, it is not difficult to check that if p ≤ 6, then d contains a hamiltonian bypass. we may therefore assume that p ≥ 7. claim 2: if x0 6= xi+1, then the vertices xi and xi+2 are not adjasent, where 1 ≤ i ≤ p. proof: suppose, on the contrary, that is for some i, 1 ≤ i ≤ p, x0 6= xi+1 and the vertices xi, xi+2 are adjasent. without loss of generality we may assume that i = 1. since x0 6= x2, we have d(x2) ≥ p and, by claim 1, x1x3 /∈ a(d). hence, x3x1 ∈ a(d). it is clear that x0 6= x1 or x0 6= x3. this together with claim 1 implies that xpx2 /∈ a(d) or x2x4 /∈ a(d). since there is no (x3, x1)-hamiltonian path, using lemma 2(ii) , we obtain that d(x2, v (d) \{x2}) = d(x2,{x1, x3}) + d(x2, v (d) \{x1, x2, x3}) ≤ 2 + p− 3 = p− 1, which contradicts that d(x2) ≥ p. it is not difficult to show that there are two distinct vertices xi and xi+k such that xi+kxi ∈ a(d) and x0 /∈ {xi+1, xi+2, . . . , xi+k−1}. we may assume that k is chosen so that k is the smallest possible. without loss of generality we may assume that i = 1. then d−(x1,{x2, x3, . . . , xk}) = 0. from claim 2 it follows that 3 ≤ k ≤ p− 2. assume first k = 3, i.e., x4x1 ∈ a(d). by claim 2, the vertices x2 and x4 (x1 and x3) are not adjacent since x0 /∈ {x2, x3}. now from xi+1xi /∈ a(d), d(x2) ≥ p and d(x3) ≥ p it follows that d(x2,{x5, x6, . . . , xp}) ≥ p− 2 and d(x3,{x5, x6, . . . , xp}) ≥ p− 2. hence, by lemma 2, the vertex x2 (x3) can be inserted into x5x6 . . . xp. then, by lemma 3, there is an (x4, x1)-hamiltonian path, which is a contradiction as x4x1 ∈ a(d). assume next that k ≥ 4. by claim 2, the vertices xi and xi+2, where 1 ≤ i ≤ k − 1 are not adjacent. from the minimality of k it follows that if 1 ≤ i < j ≤ k + 1, then xjxi ∈ a(d) if and only if j = k + 1 and i = 1. from the minimality of k ≥ 4 and claim 1 it follows that for each xi ∈{x1, x2, . . . , xk−2}, d(xi,{xi+2, xi+3}) = d(xk−1,{xk+1}) = 0. (1) also we need to show the following claim. claim 3: suppose that 1 ≤ i < j − 1 ≤ k. then xixj ∈ a(d) if and only if i = 1 and j = k + 1. proof: for a proof by contradiction, suppose that xmxn ∈ a(d), where 1 ≤ m < n−1 ≤ k and m 6= 1 or n 6= k+1. without loss of generality, we may assume that n−m is the minimum possible. from (1) it follows that n−m ≥ 4, i.e., |{xm+1, . . . , xn−1}| = n−m−1 ≥ 3. note that r := xmxnxn+1 . . . xpx1x2 . . . xm is a cycle of length p−n + m + 1 ≤ p− 3 through x0. by the minimality of k and n−m, for every y ∈{xm+1, . . . , xn−1} we have d(y,{xm+1, . . . , xn−1}) ≤ 2 and d(y, v (r)) ≥ p− 2. therefore, by lemma 1, every vertex y ∈ {xm+1, . . . , xn−1} can be inserted into r. now using lemma 3, we obtain a cycle of length p− 1 through x0, which contradicts claim 1. from claim 3 and the minimality of k ≥ 4 it follows that d(x2,{x2, x3, . . . , xk}) = d(xk,{x2, x3, . . . , xk}) = 1 12 a note on hamiltonian bypasses in digraphs with large degrees and for every i, 3 ≤ i ≤ k − 1, d(xi,{x2, x3, . . . , xk}) = 2. therefore, d(xi, v (q)) ≥ p − 2, where 2 ≤ i ≤ k and q := xk+1xk+2 . . . xpx1. note that |v (q)| = p − k + 1. if k ≥ 5, then |v (q)| ≤ p − 4, and, by lemma 2(i), every vertex xi, 2 ≤ i ≤ k, can be inserted into q. if k = 4, then |v (q)| = p−3, d(x2, v (q)) ≥ p−1, d(x4, v (q)) ≥ p−1 and d(x3, v (q)) ≥ p−2. since d(x3,{x1, x5}) = 0, again using lemma 2, we obtain that each vertex xi ∈{x2, x3, x4} can be inserted into q. therefore, by lemma 3, there is an (xk+1, x1)-hamiltonian path, which contradicts our initial supposition since xkx1 ∈ a(d). the discussion of case 1 is completed. case 2. d is not hamiltonian. (*) observe that by claime 1, in this case every cycle through x0 in d has length at most p− 3. then, by corollary 1, d contains a cycle of length p − 1. let cp−1 = x1x2 . . . xp−1x1 be a cycle of length p − 1 in d. by claim 1, x0 ∈ v (cp−1). for this case, we first give the following claim and lemma. claim 4: let p := x1x2 . . . xp−1 be an (x1, xp−1)-path of length p−2 through x0 in d. then x1xp−1 /∈ a(d). proof: for a proof by contradiction, suppose that x1xp−1 ∈ a(d). let x /∈ v (p ). then d(x) ≥ p since x 6= x0. since d contains no hamiltonian bypass, it follows that x cannot be inserted into p . now using lemma 2(i) and d(x) ≥ p, we obtain that xp−1x and xx1 ∈ a(d). therefore, x1x2 . . . xp−1xx1 is a hamiltonian cycle in d, which contradicts the hypothesis of this case. lemma 4: d contains no cycle of length p− 3 through x0. proof: suppose that the lemma is not true. let c := x1x2 . . . xp−4x0x1 be a cycle of length p − 3 through x0 in d and let b := v (d) \ v (c). by claim 1, d contains no cycle of length p − 1 and p − 2 through x0. this together with lemma 1 implies that for every y ∈ b, p ≤ d(y) = d(y, v (c)) + d(y, b) ≤ p− 3 + d(y, b). therefore, d(y, b) ≥ 3. this implies that d[b] is hamiltonian since |b| = 3, in particular, d[b] is strong. we now consider the folowing two cases. case (a). there exists a vertex y ∈ b, which is adjacent to every vertex xi for all i, 1 ≤ i ≤ p− 4. let yuzy be a hamiltonian cycle in d[b]. if y and x0 are adjacent then using the observation (*), it is not difficult to show that either d−(y, v (c)) = 0 or d+(y, v (c)) = 0. without loss of generality, we assume that d+(y, v (c)) = 0. then v (c) 7→ y. this together with claim 1 implies that a(b → v (c)) = ∅, which contradicts that d is 2-strong. we may therefore assume that y and x0 are not adjacent. if x1y ∈ a(d), then {x1, x2, . . . , xp−4}→ y. therefore, a(b → v (c) \{x1}) = ∅. this means that d[v (d) \ {x1}] is not strong, i.e., d is not 2-strong, a contradiction. now assume that x1y /∈ a(d). then yx1 ∈ a(d) since y and x1 are adjacent. similarly, xp−4y ∈ a(d). then by the above observation (*), d(x0, b) = 0. let xky ∈ a(d) with 2 ≤ k ≤ p− 4 and k be the minimum possible. it is not dificult to show that {xk, xk+1, . . . , xp−4}→ y →{x1, x2, . . . , xk−1}. assume first that k ≤ p − 5. then by claim 1, d−(xp−4, b) = 0. if xp−4z ∈ a(d), then d(p, 2) = [xp−4z; xp−4x0x1 . . . xp−5yuz], a contradiction. therefore, xp−4z /∈ a(d). thus, s. darbinyan 13 d(z,{x0, xp−4}) = 0 and d(z,{x1, x2, . . . , xp−5}) ≥ p−4. again using lemma 2(i) and (*), we obtain that xp−5z ∈ a(d). therefore, xp−4x0x1 . . . xp−5zy is a path of length p − 2 through x0 and xp−4y ∈ a(d), which contradicts claim 4. assume next that k = p− 4. then y →{x1, x2, . . . , xp−5}. hence, if p− 5 ≥ 2, then for the converse digraph of d we have the considered former case. for 4 ≤ p ≤ 6, this completes the discussion of case (a). case (b). for every y ∈ b there exists a vertex xk with 1 ≤ k ≤ p− 4 such that y and xk are not adjacent. to complete the proof of lemma 4 in this case, we first prove the following claims. claim 5: xk−1y /∈ a(d) or yxk+1 /∈ a(d). proof: suppose, on the contrary, that xk−1y ∈ a(d) and yxk+1 ∈ a(d). note that d(xk) ≥ p since xk 6= x0. using observation (*), we obtain that d(xk, b) = 0. now consider the cycle r := x0x1 . . . xk−1yxk+1 . . . xp−4x0 of length p − 3 through x0. by claim 1, xk cannot be inserted into r (for otherwise we obtain a cycle of length p − 2 througth x0). therefore by lemma 1, p ≤ d(xk) = d(xk, b) + d(xk, v (r)) ≤ p− 3, a contradiction. claim 6: (i) if xk−1y ∈ a(d), then xk+1y /∈ a(d). (ii) if yxk−1 ∈ a(d), then yxk+1 /∈ a(d). proof: (i) suppose that the claim is not true. then {xk−1, xk+1} → y. by claim 5, yxk+1 /∈ a(d). since y cannot be inserted into the path c[xk+1, xk−1] and yxk+1 /∈ a(d), from lemma 2(ii) it follows that d(y, v (c)) = p− 4. assume first that there is a vertex xs 6= xk such that y and xs also are not adjacent. let s be chosen so that |v (c[xk, xs])| is the minimum possible. note that xs /∈{xk−1, xk+1}. write p1 := c[xk+1, xs−1] and p2 := c[xs+1, xk−1]. then p− 4 = d(y, v (c)) = d(y, v (p1) + d(y, v (p2) ≤ |v (p1)| + |v (p2)| + 1 = p− 4. this implies that d(y, v (p1)) = |v (p1)| and d(y, v (p2)) = |v (p2)| + 1. now using lemma 2, we obtain xs−1y ∈ a(d) and yxs+1 ∈ a(d). by claim 5, xs = x0 and d−(xk+1, b) = d(xs, b) = 0. rcall that zyuz is a hamiltonian cycle in d[b]. if xkz ∈ a(d), then d(p, 2) = [xkz; c[xk, xk−1]yuz], a contradiction. therefore, z and xk are not adjacent. now using lemma 2, d(z) ≥ p, d(z,{xk, xs}) = 0 and the fact that d+(z,{xk+1, xk+2}) = 0, we obtain xk+1z ∈ a(d). therefore, c[xk+1, xk−1]yuz is a path of length p− 2 thruogh x0 and xk+1z ∈ a(d), which contradicts claim 4. assume next that y is adjacent to every vertex of v (c) \ {xk}. then by claim 1, v (c) \{xk} 7→ y since xk+1y ∈ a(d) and yxk+1 /∈ a(d). again using observation (*), we obtain that a(b → v (c)) = ∅, which contradicts that d is 2-strong. this completes the proof of claim 6(i). for the proof of claim 6(ii), it suffices to consider the converse digraph of d. claim 6 is proved. claim 7: if yxk−1 ∈ a(d), then xk+1y /∈ a(d). proof: for a proof by contradiction, suppose that yxk−1 ∈ a(d) and xk+1y ∈ a(d). claim 6 implies that xk−1y /∈ a(d) and yxk+1 /∈ a(d). since y cannot be inserted into c[xk+1, xk−1], using lemma 2(iii), we obtain d(y, v (c[xk+1, xk−1]) ≤ p − 5, which contradicts that d(y, v (c)) ≥ p− 4. claim 7 is proved. claim 8: (i) the vertices y and xk+1 are adjacent; (ii) the vertices y and xk−1 are adjacent. proof: (i) suppose that the claim is not true, i.e., d(y,{xk, xk+1}) = 0. write q := 14 a note on hamiltonian bypasses in digraphs with large degrees c[xk+2, xk−1]. then d(y, v (q)) = p − 4. therefore by lemma 2, yxk+2 ∈ a(d) and xk−1y ∈ a(d) since y cannot be inserted into q. assume first that xk+1 6= x0. we know that d(xk) ≥ p and d(xk+1) ≥ p. using obveration (*), it is not difficult to show that d(xk, b) = d(xk+1, b) = 0. therefore, d(xk, v (q)) ≥ p−2 and d(xk+1, v (q)) ≥ p−2. these together with lemmas 2 and 3 imply that the vertices xk and xk+1 both can be inserted into q. as a consequence, we obtain a cycle of length p − 2 through x0, which contradicts claim 1. assume next that xk+1 = x0. then d(x0, b) = d −(xk, b) = 0. if xkz ∈ a(d), then d(p, 2) = [xkz; c[xk, xk−1]yuz], a contradiction. if xku ∈ a(d), then c[xk, xk−1]yz is a path of length p − 2 through x0 and xku ∈ a(d), which contradicts claim 4. we may therefore assume that d(z,{xk, xk+1}) = d(u,{xk, xk+1}) = 0. therefore, d(z, v (q)) = p − 4, zxk+2 and xk−1z ∈ a(d). now using claims 1 and 5, we obtain that there is a vertex xs such that {y, z} → v (c[xk+2, xs]) and v (c[xs, xk−1]) → {y, z}. whitout loss of generality, assume that |v (c[xk+2, xs])| ≥ 2 (for otherwise we consider the converse digraph of d). then d(p, 2) = [yxk+2; yuzc[xk+3, xk+2]], a contradiction. this contradiction completes the proof of claime 8(i). by the same arguments one can prove claim 8(ii). claim 8 is proved. now we return to the proof of the lemma. from claim 8 it follows that y is adjacent to xk−1 and xk+1. therefore, only the following cases are possible: (i) xk−1y and yxk+1y ∈ a(d), (ii) {xk−1, xk+1} → y, (iii) y → {xk−1, xk+1}, (iv) xk+1y and yxk−1 ∈ a(d). on the other hand, claims 5, 6 and 7 imply that none of these cases holds. this contradiction completes the discussion of case (b). lemma 4 is proved. now we are ready to complete the proof of the theorem in case 2. since d is not hamiltonian, by corollary 1, d contains a cycle of length p−1. let r := x1x2 . . . xp−1x1 be a cycle of length p − 1 in d. then by claim 1 and lemma 4, we know x0 /∈ v (r) and for every i, j, 1 ≤ i, j ≤ p− 1 the following hold: d−(x0,{xi}) + d+(x0,{xi+1, xi+2, xi+3, xi+4}) ≤ 1, d+(x0,{xj}) + d−(x0,{xj−1, xj−2, xj−3, xj−4}) ≤ 1. therefore, d−(x0) + 4d +(x0) ≤ p − 1 and 4d−(x0) + d+(x0) ≤ p − 1. these mean that 5d(x0) ≤ 2p − 2, i.e., d(x0) ≤ 2(p − 1)/5, which contradicts that d(x0) > 2(p − 1)/5. the theorem is proved. corollary 2: (benhocine [5]). every strong digraph d of order p ≥ 3 and with minimum degree at least p contains d(p, 2). proof: by the famous theorem of ghoula-houri, d is hamiltonian. therefore, from the proof of theorem 6 in case (a), it follows that d contains a hamiltonian bypass. perhaps the following proposition will be useful for conjecture 1 (see, in section conclusion). proposition 1: let d be a non-hamiltonian 2-strong digraph of order p ≥ 3. suppose that d(x) ≥ p for every vertex x ∈ v (d) \{x0}, where x0 is a vertex of d. if p = x1x2 . . . xp−2 is an (x1, xp−2)-path of length p− 3 through x0 in d, then x1xp−2 /∈ a(d). s. darbinyan 15 proof: for a proof by contradiction, suppose that x1xp−2 ∈ a(d). write v (d) \ v (p ) = {y1, y2}. we know that d(y1) ≥ p and d(y2) ≥ p since x0 ∈ v (p ). from claim 4 it follows that yi cannot be inserted into p . on the other hand, since d contains no cycle of length p − 1 through x0, we have that yix1 /∈ a(d) or xp−2yi /∈ a(d). now using lemma 2(ii), we obtain d(yi, v (p )) = p − 2 and y1y2, y2y1 ∈ a(d). without loss of generality, assume that y1x1 /∈ a(d). then by lemma 2(ii), xp−2y1 ∈ a(d). since d is not hamiltonian and contains no cycle of length p−1 through x0, it follows that d−(x1,{y1, y2}) = 0. then xp−2y2 ∈ a(d). if x1y1 ∈ a(d) (or x1y2 ∈ a(d)), then it is not difficult to show that d(p, 2) = [x1y1; x1x2 . . . xp−2y2y1] (or d(p, 2) = [x1y2; x1x2 . . . xp−2y1y2]) is in d, a contradiction. therefore, d+(x1,{y1, y2}) = 0. thus, d(x1,{y1, y2}) = 0. this together with lemma 2 and d(yi, v (p )) = p − 2 implies that {y1, y2} → x2. then by claim 1, x1 = x0 since x2x3 . . . xp−2y1y2x2 is a cycle of length p− 1. write q := x2x3 . . . xp−2. then |v (q)| = p − 3, d(y1, v (q)) ≥ p − 2 and d(y2, v (q)) ≥ p−2. since x0 →{x2, xp−2}, by claim 4 we have that neither y1 nor y2 can be inserted into q. then by lemma 2(iii), we obtain that d(y1, v (q)) = d(y2, v (q)) = p − 2 and the arcs y2y1, xp−2y2, y1x2 are in a(d). we claim that the vertex y1 (y2) is adjacent to each vertex of v (q). assume that this is not the case. let d(y1,{xi}) = 0, where 3 ≤ i ≤ p−3. from lemma 2(iii), d(y1, v (q)) = p−2 and the fact that the vertex y1 cannot be inserted into q it follows that xi−1y1, y1xi+1 ∈ a(d). since y2y1, y1y2 ∈ a(d), it is easy to see that d(y2,{xi}) = 0 and xi−1y2, y2xi+1 ∈ a(d). they imply that the vertex xi can be inserted neither into s := x2 . . . xi−1 nor into t := xi+1 . . . xp−2. then it is easy to see that d(xi,{x0}) = 2 and p− 2 ≤ d(xi, v (s)) + d(xi, v (t )) ≤ |v (s)| + |v (t )| + 2 = p− 2. therefore, d(xi, v (s)) = |v (s)| + 1 and d(xi, v (t )) = |v (t )| + 1. again using lemma 2, we obtain xp−2xi and xix2 ∈ a(d). hence, x0xix2 . . . xi−1y1y2xi+1 . . . xp−2 is an (x0, xp−2)hamiltonian path, a contradiction as x0xp−2 ∈ a(d). this proves that y1 (y2) is adjacent to every vertex in v (q). therefore, there is an integer l, 2 ≤ l ≤ p− 2, such that {xl, xl+1, . . . , xp−2}→{y1, y2}→{x2, x3, . . . , xl}. (2) if 3 ≤ l ≤ p− 3, then by (2), x0xp−2y1x3 . . . xp−3y2x2 is an (x0, x2)-hamiltonian path, which is a contradiction as x0x2 ∈ a(d). if l = 2 or l = p−2, then a({y1, y2}→ v (d)\{x2}) = ∅ or a(v (d) \{xp−2}→{y1, y2}) = ∅ when l = 2 or l = p− 2, respectively. this means that d is not 2-strong, a contradiction. proposition 1 is proved. 5. conclusion in the current article, we examined the existence of a hamiltonian bypass in 2-strong digraphs of order p, in which p−1 vertices have degrees at least p. we proved that if such digraphs are hamiltohian or have the minimal degree more than 2(p − 1)/5, then such digraphs contain a hamiltonian bypass. if we consider the digraph h(n) (by remark 1, h(n) is 2-strong, d(x0) = 4 and is not hamiltonian), then we see that d(n, 2) = [y1x1; y1y2y3x2x3 . . . xn−4x0x1] is a hamiltonian bypass. by the above arguments, we believe that the following conjecture is true. conjecture 1: let d be a 2-strong digraph of order p. if p − 1 vertices in v (d) have degrees at least p, then d contains a hamiltonian bypass. 16 a note on hamiltonian bypasses in digraphs with large degrees acknowledgements the author would like to thank the referees of the paper for careful reading and many helpful remarks. references [1] j. bang-jensen, g. gutin, digraphs: theory, algorithms and applications, springer, 2001. [2] j.-c. bermond and c. thomassen, “cycles in digraphs-a survey”, journal of graph theory, vol. 5, pp. 1-43, 1981. [3] d. kühn and d. osthus, “a survey on hamilton cycles in directed graphs”, european journal of combinatorics, vol. 33, pp. 750-766, 2012. [4] a. benhocine and a.p. wojda, “bypasses in digraphs”, ars combinatoria, vol. 16, pp. 85-94, 1983. [5] a. benhocine, “on the existence of a specified cycle in digraphs with constraints on degrees”, journal of graph theory, vol. 8, pp. 101-107, 1984. [6] s.kh. darbinyan, “on hamiltonian bypasses in digraphs satisfying meyniel-like condition”, transactions of iiap of nas ra, mathematical problems of computer science, vol. 20, pp. 7-19, 1998. [7] s.kh. darbinyan, “on the specified cycles in oriented graphs”, akademy nauk armyan. ssr dokllady, vol. 84, no. 2, pp. 51-55, 1987 (in russian). [8] s.kh. darbinyan and i.a. karapetyan, “on hamiltonian bypasses in one class of hamiltonian digraphs”, mathematical problems of computer science, vol. 41, pp. 23-37, 2014. [9] s.kh. darbinyan, “on hamiltonian bypasses in digraphs with the condition of y. manoussakis”, ”2015 computer science and information technologies (csit), yerevan, doi:10.1109/csitechnol.2015.7358250, pp. 53-63, 2015. [10] s.kh. darbinyan, “on hmiltonian and strongly hamiltonconnected digraphs”, akademy nauk armyan. ssr doklady, vol. 91, no. 1, pp. 3-6, 1990.(arxiv.1801.05166v1). [11] s.kh. darbinyan, “a sufficient condition for a digraph to be hamiltonian”, akademy nauk armyan. ssr doklady,(in russian), vol. 91, no. 2, pp. 57-59, 1990. [12] r. häggkvist and c. thomassen, “on pancyclic digraphs”, journal combinatorial theory ser. b, vol. 20, pp. 20-40, 1976. [13] j.a. bondy and c. thomassen, “a short proof of meyniel’s theorem”, discrete mathematics, vol. 19, pp. 195-197, 1977. [14] j. bang-jensen, g. gutin and h. li, “sufficient conditions for a digraph to be hamiltonian”, journal of graph theory, vol. 22, no. 2, pp. 181-187, 1996. submitted 03.08.2020, accepted 01.12.2020. s. darbinyan 1 7 ø»ï ýï³ï³éáõù ù»í ³ëïç׳ýý»ñáí ïáõùýáñáßí³í ·ñ³ýý»ñáõù ñ³ùçéïáýû³ý ßñç³ýóáõùý»ñç ù³ëçý ê³ùí»é ê. ¸³ñµçýû³ý ðð ¶²² æýýáñù³ïçï³ûç ¨ ³íïáù³ï³óù³ý åñáµé»ùý»ñç çýëïçïáõï e-mail: samdarbin@iiap.sci.am ²ù÷á÷áõù ü»ñï³ ³ßë³ï³ýùáõù ³å³óáõóí»é ¿ ñ»ï¨û³é ã»áñ»ùá: â»áñ»ù: ¸çóáõù d -ý 2-áõå»õ ï³å³ïóí³í p-·³·³ã³ýç ïáõùýáñáßí³í ·ñ³ý ¿, áñç p ¡ 1 ·³·³ãý»ñç ³ëïç׳ýý»ñá ÷áùñ ã»ý p ãíçó: ºã» d -ý ñ³ùçéïáýû³ý ¿ ï³ù d-ç ÷áùñ³·áõûý ³ëïç׳ýá ù»í ¿ 2 ( p ¡ 1 ) = 5 ãíçó, ³å³ ³û¹ ·ñ³ýá å³ñáõý³ïáõù ¿ ßñç³ýóáõù: îäíà çàìåòêà î ãàìèëüòîíîâûõ îáõîäàõ â îðãðàôàõ ñ áîëüøèìè ñòåïåíüÿìè ñàìâåë õ. äàðáèíÿí èíñòèòóò ïðîáëåì èíôîðìàòèêè è àâòîìàòèçàöèè íàí ðà e-mail: samdarbin@iiap.sci.am àííîòàöèÿ â íàñòîÿùåé ðàáîòå äîêàçàíà ñëåäóþùàÿ òåîðåìà: òåîðåìà: ïóñòü d åñòü 2-ñèëüíî ñâÿçíûé p-âåðøèííûé îðãðàô, â êîòîðîì p ¡ 1 âåðøèíû èìåþò ñòåïåíü íå ìåíüøå ÷åì p. åñëè d ãàìèëüòîíîâ èëè èìååò ìèíèìàëüíóþ ñòåïåíü áîëüøå ÷åì 2 ( p ¡ 1 ) =5 , òî d ñîäåðæèò ãàìèëüòîíîâ îáõîä. êëþ÷åâûå ñëîâà: îðãðàô, öèêë, ãàìèëüòîíîâ öèêë, ãàìèëüòîíîâ îáõîä. ñ³ùçéïáýû³ý ßñç³ýóáõù: ´³ý³éç µ³é»ñ` îáõùýáñáßí³í ·ñ³ý, óçïé, ñ³ùçéïáýû³ý óçïé, ñ³ùçéïáýû³ý 01_darbinyan_54_7_17 samveldarbinyan d:\sbornik\...\paper.dvi mathematical problems of computer science 26, 2006, 64{71. ¶øæê åçï³ýçáõãû³ý ·ý³ñ³ïù³ý ù³ã»ù³ïçï³ï³ý ùá¹»éý»ñá ì³½·»ý ê. î³ñ³å»ïû³ý èáõë-ð³ûï³ï³ý å»ï³ï³ý ñ³ù³éë³ñ³ý e-mail vazgen.karapetyan@pontesolutions.com ²ù÷á÷áõù ²ûë ³ßë³ï³ýùá µ³õï³ó³í ¿ »ñïáõ ñçùý³ï³ý ù³ëçó. ý³ë ý»ñï³û³óí³í ¿ åçï³ýç »éùç (yield) ëý¹çñá å³ù³ý³ï³ïçó ·»ñ-ù»í çýï»·ñ³é³ûçý ëë»ù³ý»ñç (¶øæê) ³ñï³¹ñáõãûáõýáõù: ²ûýáõñ»ï¨ ý»ñï³û³óíáõù »ý ³ñï³¹ñ³ï³ý ëáï³ýç ï»ë³ï³ñ³ñ ïßéç ·ý³ñ³ïù³ý ù³ã»ù³ïçï³ï³ý íç׳ﳷñ³ï³ý ùá¹»éý»ñá: ²ßë³ï³ýùáõù ñçùý³ï³ý áõß³¹ñáõãûáõýá ¹³ñóíáõù ¿ ³ýï³ýáý ¹»ý»ïïý»ñç ³éï³ûáõãû³ùµ å³ûù³ý³íáñí³í ëáï³ýçý: ¶ñ³ï³ýáõãûáõý [1 ] a lb e r t v . fe r r is -p r a b h u . 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s s io n o f s e c r e t kn o wle d g e ( p a s s wo r d , s e c r e t ke y, e t c .) . mo r e o ve r , p a s s wo r d c a n o ft e n b e g u e s s e d , s in c e t e n d t o u s e p a s s wo r d s wh ic h a r e e a s y t o r e m e m b e r . b io m e t r ic s is t h e t e c h n o lo g y t h a t is u s e d t o u n iqu e ly id e n t ify a s p e c ī c h u m a n b e in g . it is p r im a r ily u s e d t o p r o vid e s e c u r it y fo r p e r s o n a l o r b u s in e s s a s s e t s . a b io m e t r ic s s ys t e m m u s t ¯ r s t s t o r e a p e r s o n 's b io m e t r ic d a t a . th e n , wh e n s o m e o n e t r ie s t o a c c e s s a p e r s o n a l o r b u s in e s s s ys t e m , t h e s t o r e d b io m e t r ic d a t a is c o m p a r e d t o t h e d a t a o f t h e p e r s o n c u r r e n t ly a c c e s s in g t h e s ys t e m . if t h e d a t a m a t c h e s u p , t h e p e r s o n c a n h a ve a c c e s s t o t h e p r o t e c t e d in fo r m a t io n . b io m e t r ic s ys t e m s o ®e r a s o lu t io n t o m o s t o f t h e p r o b le m s m e n t io n e d a b o ve . th e y c o u ld e it h e r s u b s t it u t e d fo r t r a d it io n a l s ys t e m s o r u s e d t o r e in fo r c e t h e m . b io m e t r ic s ys t e m s a r e b a s e d o n p h ys ic a l o r b e h a vio r a l c h a r a c t e r is t ic s o f h u m a n b e in g s , like fa c e s , ¯ n g e r p r in t s , vo ic e , ir is e s . th e r e s u lt s o f t h e m e a s u r e m e n t o f t h e s e c h a r a c t e r is t ic s a r e c a lle d b io m e t r ic d a t a . b io m e t r ic d a t a h a ve t h e a d va n t a g e t h a t p o t e n t ia lly t h e y a r e u n iqu e id e n t ī e r s o f h u m a n b e in g . a b io m e t r ic s s ys t e m m u s t ¯ r s t s t o r e a p e r s o n 's b io m e t r ic d a t a . th e n , wh e n s o m e o n e t r ie s t o a c c e s s a p e r s o n a l o r b u s in e s s s ys t e m , t h e s t o r e d b io m e t r ic d a t a is c o m p a r e d t o t h e d a t a o f t h e p e r s o n c u r r e n t ly a c c e s s in g t h e s ys t e m . if t h e d a t a m a t c h e s u p , t h e p e r s o n c a n h a ve a c c e s s t o t h e p r o t e c t e d in fo r m a t io n . th e a t t r a c t ive p r o p e r t y o f u n iqu e n e s s , t h a t h o ld s fo r b io m e t r ic s , a ls o r e s u lt s in it s m a jo r we a kn e s s . u n like p a s s wo r d s , b io m e t r ic in fo r m a t io n , if c o m p r o m is e d o n c e , c a n n o t b e c a n c e le d a n d e a s ily r e p la c e d b y o t h e r b io m e t r ic in fo r m a t io n , s in c e p e o p le o n ly h a ve lim it e d r e s o u r c e s o f b io m e t r ic d a t a . r e qu ir e m e n t s fo r b io m e t r ic s ys t e m s s h o u ld in c lu d e s e c u r e s t o r a g e a n d s e c u r e c o m m u n ic a t io n o f b io m e t r ic d a t a in t h e a p p lic a t io n s wh e r e t h e y a r e u s e d . th e p r o b le m o f b io m e t r ic id e n t i¯ c a t io n is t r a n s fe r r e d t o in fo r m a t io n -t h e o r e t ic a l m o d e l b y w ille m s e t a l [1 ]. th e m o d e l o f a b io m e t r ic s ys t e m is s h o wn in fig u r e 1 . 5 6 m. haroutunian, l. ter-vardanyan 5 7 e nr olling phase i denti¯cation phase figur e. 1. m odel of biometr ic identi¯cation system b io m e t r ic a l id e n t i¯ c a t io n in g e n e r a l in vo lve s t wo p h a s e s . in a n e n r o llm e n t p h a s e a ll in d ivid u a ls a r e o b s e r ve d a n d fo r e a c h in d ivid u a l a r e c o r d is a d d e d t o a d a t a b a s e . th is r e c o r d c o n t a in s e n r o llm e n t -d a t a , i.e . a n o is y ve r s io n o f t h e b io m e t r ic a l d a t a c o r r e s p o n d in g t o t h e in d ivid u a l. in t h e id e n t i¯ c a t io n p h a s e a n u n kn o wn in d ivid u a l is o b s e r ve d a g a in . th e r e s u lt in g id e n t ī c a t io n -d a t a , a n o t h e r n o is y ve r s io n o f t h e b io m e t r ic a l d a t a o f t h e u n kn o wn in d ivid u a l, is c o m p a r e d t o ( a ll) t h e e n r o llm e n t -d a t a in t h e d a t a b a s e a n d t h e s ys t e m h a s t o c o m e u p wit h a n e s t im a t e o f t h e in d ivid u a l. e s s e n t ia l in t h is p r o c e d u r e is t h a t b o t h in t h e e n r o llm e n t -p h a s e a n d in t h e id e n t ī c a t io n -p h a s e n o is y ve r s io n s o f t h e b io m e t r ic a l d a t a a r e o b t a in e d . th e a c t u a l b io m e t r ic a l d a t a o f e a c h in d ivid u a l r e m a in u n kn o wn . w ille m s e t a l [1 ] in ve s t ig a t e d t h e fu n d a m e n t a l p r o p e r t ie s o f b io m e t r ic id e n t ī c a t io n s ys t e m . it h a s b e e n s h o wn t h a t it is n o t p o s s ib le t o id e n t ify r e lia b ly m o r e p e r s o n s t h a n c a p a c it y wh ic h is a n in h e r e n t c h a r a c t e r is t ic o f a n y id e n t i¯ c a t io n s ys t e m . th e y d e r ive d t h e c a p a c it y o f s u c h s ys t e m . w e in ve s t ig a t e t h e e xp o n e n t ia lly h ig h r e lia b ilit y c r it e r io n in b io m e t r ic id e n t i¯ c a t io n s ys t e m s . in o t h e r wo r d s we in t r o d u c e a n e w p e r fo r m a n c e c o n c e p t o f b io m e t r ic id e n t i¯ c a t io n e-c a p a c it y, wh ic h t a ke s in t o a c c o u n t a s t r o n g e r r e qu ir e m e n t o n id e n t i¯ c a t io n fa u lt e ve n t s wit h e xt r e m e ly s m a ll p r o b a b ilit y ( 2 ¡ne in s t e a d o f " ) . in t e r m s o f p r a c t ic a l a p p lic a t io n s a n e xp o n e n t ia l d e c r e a s e in e r r o r p r o b a b ilit y ( n a m e ly, in u n wa n t e d id e n t i¯ c a t io n fa u lt s ) is m o r e d e s ir a b le . w e in ve s t ig a t e t h e e-c a p a c it y fu n c t io n , wh ic h is t h e g e n e r a liz a t io n o f t h e c a p a c it y, a s it t e n d s t o c a p a c it y, wh e n e t e n d s t o 0 . u p p e r a n d t h e lo we r b o u n d s fo r id e n t i¯ c a t io n e-c a p a c it y fo r m a xim a l a n d a ve r a g e e r r o r p r o b a b ilit ie s a r e c o n s t r u c t e d in [3 ]. w h e n e ! 0 we d e r ive u p p e r a n d lo we r b o u n d s o f t h e c h a n n e l c a p a c it y, wh ic h c o in c id e wit h t h e c a p a c it y o b t a in e d in [1 ]. w h e n e ! 0 we d e r ive t h e lo we r a n d u p p e r b o u n d s o f t h e c h a n n e l c a p a c it y, wh ic h c o in c id e wit h t h e c a p a c it y o b t a in e d in [1 ]. a s im ila r r e s u lt is o b t a in e d fo r t h e b io m e t r ic id e n t ī c a t io n s ys t e m wit h r a n d o m p a r a m e t e r , wh ic h is m o r e r e a lis t ic fo r a p p lic a t io n s . 5 8 information-theoretic approach to biometric identi¯cation problem r e fe r e n c e s [1 ] f. w ille m s , t. k a lke r , j. go s e lin g , a n d j.-p . l in n a r t z , \ on t h e c a p a c it y o f a b io m e t r ic a l id e n t i¯ c a t io n s ys t e m " , international symposium on information theory, y o ko h a m a , ja p a n , p . 8 2 , 2 0 0 3 . [2 ] s . p a n ka n t i, r . m. b o lle a n d a . ja in , \ b io m e t r ic s -th e fu t u r e o f id e n t i¯ c a t io n " , ie e e computer, vo l. 3 3 , n o . 2 , p p . 4 6 4 9 , fe b r u a r y, 2 0 0 2 . [3 ] m. h a r o u t u n ia n , a . mu r a d ya n a n d l . te r v a r d a n ya n , " u p p e r a n d lo we r b o u n d s o f b io m e t r ic id e n t i¯ c a t io n e c a p a c it y" , transactions of iiap of nas of r a, m athematical p roblems of computer science,v3 6 , p p .1 -1 0 , 2 0 1 2 . [4 ] e . a . h a r o u t u n ia n , \ on b o u n d s fo r e-c a p a c it y o f d mc" , ie e e transactions on information theory, vo l. 5 3 , n o . 1 1 , p p . 4 2 1 0 -4 2 2 0 , 2 0 0 7 . [5 ] e . a . h a r o u t u n ia n , m. e . h a r o u t u n ia n a n d a . n . h a r u t yu n ya n , " r e lia b ilit y c r it e r ia in in fo r m a t io n t h e o r y a n d in s t a t is t ic a l h yp o t h e s is t e s t in g " , f oundations and trends in communications and information theory, vo l. 4 , n o 2 -3 , p p . 9 7 -2 6 3 , 2 0 0 8 . [6 ] m. e . h a r o u t u n ia n , \ e s t im a t e s o f e-c a p a c it y a n d c a p a c it y r e g io n s fo r m u lt ip le -a c c e s s c h a n n e l wit h r a n d o m p a r a m e t e r " , l ecture notes in computer science, vo l. 4 1 2 3 , s p r in g e r v e r la g , p p . 1 9 6 -2 1 7 , 2 0 0 6 . [7 ] m. e . h a r o u t u n ia n , s . a . to n o ya n , \ r a n d o m c o d in g b o u n d o f in fo r m a t io n h id in g ec a p a c it y" , p roc. of ie e e international symposium on information theory, p . 5 3 6 , u s a , ch ic a g o , 2 0 0 4 . [8 ] t. m. co ve r a n d j. a . th o m a s , e lements of information theory, w ile y, n e w y o r k, 1 9 9 1 . [9 ] i. cs is z ¶a r a n d j. k äo r n e r , information theory: coding theorems for d iscrete m emoryless systems, a c a d e m ic p r e s s , n e w y o r k, 1 9 8 1 . mathematical problems of computer science 51, 82–89, 2019. udc 519.872 the queue distribution in multiprocessor systems with the waiting time restriction artur p. vardanyan and vladimir g. sahakyan institute for informatics and automation problems of nas ra e-mail: artvardanyan@asnet.am, vladimir.sahakyan@sci.am abstract a queueing system model is considered, consisting of m (m ≥ 1) servicing devices and a maximum number of tasks with n (n ≥ 1) in the waiting queue. each task is characterized by three random parameters (ν,β,ω), where ν is the number of servicing devices required to perform the task, β is the maximum time required to complete the task and ω is the possible time that the task can wait before assigning to run, after which it leaves the system without service. tasks are accepted for service in the order of their entry into the system, i.e., fifo (first-in-first-out) discipline is used. in paper the equations are obtained for the state probabilities of the system in the stationary mode, which can serve as an assessment for real multiprocessor systems using mpi and openmp technologies. keywords: multiprocessor cluster-type system, cluster computing, queueing theory, waiting time restriction. the optimal use of processor time in multiprocessor cluster-type systems depends on many factors: the method of receiving and queuing the task, determining the order of execution, the possibility of dynamically distributing computing resources, the ability to move the task during different phases of execution to the minimum necessary environment or stop the execution with the possibility of continuing, etc.. the reception of a task in the system for execution plays an important role in the organization of this process. the ability to interact distributed processes in certain periods of time requires synchronization and simultaneous execution both in one and different computer systems. therefore, accepting a task from the queue for execution imposes the responsibility on the scheduler for ensuring its timely execution. at the same time, tasks arriving for execution may be ”impatient”, that is, they leave the queue after a certain waiting time. in this paper, the probabilities of the queue state are obtained for the exponential distributions of the task of receipt, execute, and failure of service. such models play an important role on multiprocessor systems using mpi and openmp technologies [1]. suppose that a task stream enters a computing system consisting of m processors (m ≥ 1). each task is characterized by three random parameters (ν,β,ω), where ν is the number of 82 1. introduction a. vardanyan and v. sahakyan 83 computational resources(processors, cores, cluster nodes, etc.,) required to perform the task, β is the maximum time required to complete the task and ω is the possible time that the task can wait before assigning to run, after which it leaves the system without service [2]. by using david kendall’s notation(which is widely used to describe elementary queueing systems)[3], the system under consideration can be represented as m|m|m|n. so, the system parameters are described: m the maximum number of computational resources; n the maximum permissible number of tasks in the queue; α a random value of the time interval between neighboring entrances, which has the probability distribution: p(α < t) = 1 −e−at, where a is the intensity of the incoming stream; β a random value of the task execution time, which has the probability distribution: p(β < t) = 1 −e−bt, where b is the intensity of service; ω a random value of the permissible waiting time for a task in the queue, which has the probability distribution: p(ω < t) = 1 −e−wt, where w is the intensity of the failure of service for a task from the queue; ν a random value of the number of required computational resources for performing a task, which has the probability distribution: p(ν ≤ k) = k m ,k = 1, 2, ...,m. tasks will be accepted for service in the order of their entry into the system, i.e., fifo discipline is used (first-in-first-out). those tasks that arrive at the time of full occupation of the queue (there are already n tasks in the queue) receive a denial of service. to obtain a system of equations, we need the values of some probabilistic characteristics. by p(i,k) is denoted the probability that k processors will be occupied by i tasks: p(i,k) = p ( i∑ j=1 νj = k ) . p(i,k) = 1 mi ( k − 1 i− 1 ) , 1 ≤ i ≤ k ≤ m. 2. basic notations and lemmas lemma 2.1:the probability that k processors will be occupied by i tasks, can be obtained in the following way: 84 the queue distribution in multiprocessor systems with the waiting time restriction proof. to prove the lemma we use the mathematical induction technique. the method of induction requires two cases to be proved. the first case, called the base case, proves that the property holds for i = 1: p(1,k) = 1 m ( k − 1 0 ) = 1 m . the statement is true because if i = 1, then p(1,k) = p(ν = k) = 1 m . the second case, called the induction step, proves that if the property holds for number i−1, then it holds for the next natural number i: p(i,k) = k−1∑ j=i−1 p(i− 1,j)p(1,k − j) = 1 m k−1∑ j=i−1 1 mi−1 ( j − 1 i− 2 ) = 1 mi k−1∑ j=i−1 ( j − 1 i− 2 ) (1) from combinatorics we know this equality [4]:( i i ) + ( i + 1 i ) + ... + ( i + k − 1 i ) = ( i + k i + 1 ) (2) considering (2) equality to count (1), we get the formula, which was mentioned in lemma 2.1.: p(i,k) = 1 mi ( k − 1 i− 1 ) . p ( i∑ j=1 νj ≤ k ) = 1 mi ( k i ) , 1 ≤ i ≤ k ≤ m. proof. to prove the lemma we use the formula, which we got in lemma 2.1. p ( i∑ j=1 νj ≤ k ) = k∑ j=i p(i,j) = k∑ j=i 1 mi ( i− 1 j − 1 ) = 1 mi k∑ j=i ( i− 1 j − 1 ) to calculate the last sum, we again use the (2) equality and as a result we get that p ( i∑ j=1 νj ≤ k ) = 1 mi ( k i ) . p ( k∑ i=1 νi ≤ s < k+1∑ i=1 νi ) = 1 mk+1 ( m− s−k k + 1 )( s k ) , 1 ≤ k ≤ s ≤ m. lemma 2.2: the probability that i tasks will occupy no more than k processors, can be obtained in the following way: lemma 2.3: a. vardanyan and v. sahakyan 85 proof. it’s obvious that: p ( k∑ i=1 νi ≤ s < k+1∑ i=1 νi ) = s∑ j=k p ( k∑ i=1 νi = j ) p (νk+1 > s− j). primarily, we use the obvious fact that p (νk+1 > s− j) = m−s + j m , and then we use the formula, which we got in lemma 2.1. for the first probability in sum, as a result we get: p ( k∑ i=1 νi ≤ s < k+1∑ i=1 νi ) = 1 mk+1 ( s∑ j=k (m−s) ( j − 1 k − 1 ) + s∑ j=k j ( j − 1 k − 1 )) = = 1 mk+1 ( (m−s) ( s k ) + k ( s + 1 k + 1 )) = 1 mk+1 ( m− s−k k + 1 )( s k ) . to analyze our system we need to identify the following basic notation: pi,j(t) the probability that at the moment of time t there are i tasks in service, and in the queue j tasks wait for service; due to the finite number of possible states of the system (m∗n+1) with long-term operation, the system goes into a steady mode of operation, i.e., in a stationary state [5]. in this case, the limiting values pi,j(t) are considered as t tends to infinity, which will be denoted by pi,j. by li,j we denote the state of the system when i tasks are serviced and j tasks are waiting in the queue. cases when the system can pass li,j state from the other state are presented in the following scheme: li−k+1,j+k li,j li,j−1 li,j+1 or li−k,j+k+1 q(1)(i,j) q(2)(i,j) q(3)(i,j,k) where q(1)(i,j), q(2)(i,j), q(3)(i,j) are probabilities for appropriate passing and when the passing is from li−k+1,j+k, then k = 1, 2, ...,min(i,n − j), but when the passing is from li−k,j+k+1, then k = 0, ..., i − 1. note that if j = 0, then there won’t be the passing from li,j−1 and if j = n, then there won’t be the passing from li,j+1 or li−k,j+k+1. we also assume that at the passing from li−k,j+k+1 the first task from the queue leaves the queue and at the passing from li,j+1 not the first task, but another task from the queue leaves the queue. obviously, q(1)(i,j) = min(i,n−j)∑ k=0 ( (i−k + 1)b a + (i−k + 1)b + (j + k)w pi−k+1,j+kp(i,k) ) , 3. the equations for system state 86 the queue distribution in multiprocessor systems with the waiting time restriction where p(i,k) = 0 if i = 0, but if 0 < i ≤ m, then p(i,k) is the following conditional probability: p(i,k) = p ( i−k∑ s=1 νs + i+1∑ s=i−k+2 νs ≤ m, i−k∑ s=1 νs + i+2∑ s=i−k+2 νs > m / i−k+1∑ s=1 νs ≤ m, i−k+2∑ s=1 νs > m ) , where we consider that νi−k+1 is the number of required computational resources required to service the task that was being left the system(it was serviced) and due to which the system has changed its state, q(2)(i,j) = a a + ib + (j − 1)w pi,j−1, q(3)(i,j) = jw a + ib + (j + 1)w pi,j+1 + i−1∑ k=0 ( w a + (i−k)b + (j + k + 1)w pi−k,j+k+1p(i,k) ) , where p(i,k) = 0 if i = 0, but if 0 < i ≤ m, then p(i,k) is the following conditional probability: p(i,k) = p ( i−k∑ s=1 νs + i+1∑ s=i−k+2 νs ≤ m, i−k∑ s=1 νs + i+2∑ s=i−k+2 νs > m / i−k∑ s=1 νs ≤ m, i−k+1∑ s=1 νs > m ) , where we consider that νi−k+1 is the number of required computational resources required to service the task that was being left the system(it left the queue) and due to which the system has changed its state. in this case, the equations for system state are given in the following way: pi,j = ηjq (1)(i,j) + θjq (2)(i,j) + ηjq (3)(i,j), (3) where 0 ≤ i ≤ m, 0 ≤ j ≤ n and ηj = { 0, for j = n 1, for 0 ≤ j < n , θj = { 0, for j = 0 1, for 0 < j ≤ n . note that if i = 0, then p0,j = 0 for 0 ≤ j ≤ n. a. vardanyan and v. sahakyan 87 to calculate p(i,k) probability, we first perform a simple transformation, then use the conditional probability formula: p(i,k) = p ( i+1∑ s=1 νs ≤ m + νi−k+1 < i+2∑ s=1 νs / i−k+1∑ s=1 νs ≤ m < i−k+2∑ s=1 νs ) = = p (∑i+1 s=1 νs ≤ m + νi−k+1 < ∑i+2 s=1 νs, ∑i−k+1 s=1 νs ≤ m < ∑i−k+2 s=1 νs ) p (∑i−k+1 s=1 νs ≤ m < ∑i−k+2 s=1 νs ) . by using lemma 2.3. we can calculate the probability, which is in the denominator of the last fraction: p ( i−k+1∑ s=1 νs ≤ m < i−k+2∑ s=1 νs ) = i−k + 1 mi−k+2 ( m + 1 i−k + 2 ) . before the calculation of the probability, which is in the numerator of the fraction, it is denoted by δk, then it is calculated in the following way: δk = m−k+1∑ u=i−k p ( i−k∑ s=1 νs = u ) p̃u, where k = 1, 2, ...,min(i,n− j) and p̃u = p ( i+1∑ s=i−k+2 νs ≤ m−u < i+2∑ s=i−k+2 νs,νi−k+1 ≤ m−u < νi−k+1 + νi−k+2 ) . obviously, in the last probability we deal with independent probabilities and with the help of lemma 2.3. for p̃u we get the following formula: p̃u = (m−u)(m + u + 1) ( (m + 1)k + u ) 2(k + 1)mk+3 ( m−u k ) . (4) by using lemma 2.1. as a result we get the following formula for δk: δk = 1 mi−k m−k+1∑ u=i−k ( u− 1 i−k − 1 ) p̃u, (5) where p̃u is calculated by formula (4). so, we get formula for p(i,k) probability: p(i,k) = mi−k+2 (i−k + 1) ( m+1 i−k+2 )δi. (6) note that we can calculate the probability p(i,k) in the same way as p(i,k). thus, according to equation (3), probabilistic equation of the state of the system is given. as we know, if i = 0 for all 0 ≤ j ≤ n p0,j = 0, and please note that m∑ i=0 n∑ j=0 pi,j = 1. the probability that the system will refuse a new arrival task is denoted by r. 8 8 the queue distribution in multiprocessor systems with the waiting time restriction corollary 3.1. r = mx i=1 a a + ib + nw pi;n: 4 . co n c lu s io n in c la s s ic a l qu e u e in g s ys t e m s , o n e t a s k d o e s n o t r e qu ir e m o r e t h a n o n e s e r vic in g d e vic e , b u t in t h is p a p e r we s u g g e s t a qu e u e in g s ys t e m m o d e l t h a t d i®e r s fr o m o t h e r qu e u e in g s ys t e m s . in t h e s u g g e s t e d n e w m o d e l it m a y t a ke m o r e t h a n o n e s e r vic in g d e vic e s t o p e r fo r m o n e t a s k. s u c h a qu e u e in g s ys t e m m o d e l c a n p la y a n im p o r t a n t r o le o n m u lt ip r o c e s s o r s ys t e m s u s in g mp i a n d op e n mp t e c h n o lo g ie s . in p a p e r fo r t h e e xp o n e n t ia l d is t r ib u t io n s o f t h e t a s k o f r e c e ip t , e xe c u t e , a n d fa ilu r e o f s e r vic e , t h e p r o b a b ilis t ic e qu a t io n o f s t a t e o f t h e s ys t e m is o b t a in e d in t h e s t a t io n a r y m o d e . [1 ] n . d a h n o u n , m ulticore d sp : f rom algorithms to r eal-time implementation on the tm s320c66x soc, w il e y , b r is t o l, 2 0 1 8 . [2 ] v . s a h a kya n , y . s h o u ko u r ia n a n d h . a s t s a t r ya n ,\ a b o u t s o m e qu e u e in g m o d e ls fo r c o m p u t a t io n a l g r id s ys t e m s " , transactions of iiap nas r a, m athematical p roblems of computer science, vo l. 4 6 , p p .5 5 -5 8 , 2 0 1 6 . [3 ] l . k le in r o c k, queueing systems: volume 1 theory, w ile y in t e r s c ie n c e , n e w y o r k, 1 9 7 5 . [4 ] í. ß. âèëåíêèí, êîìáèíàòîðèêà, íàóêà, ìîñêâà, 1 9 6 9 . [5 ] á. â. ãíåäåíêî è è. í. êîâàëåíêî, ââåäåíèå â òåîðèþ ìàññîâîãî îáñëóæèâàíèÿ, íàóêà, ìîñêâà, 1 9 8 7 . submitted 04.02.2019, accepted 26.04.2019. ð»ñãç µ³ßëáõùá µ³½ù³åñáó»ëáñ³ûçý ñ³ù³ï³ñ·áõù ëå³ëù³ý å³ù³ý³ïç ë³ñù³ý³÷³ïù³ý ¹»åùáõù ìé³¹çùçñ ¶. ê³ñ³ïû³ý ¨ ²ñãáõñ ä. ì³ñ¹³ýû³ý ðð ¶²² æýýáñù³ïçï³ûç ¨ ³íïáù³ï³óù³ý åñáµé»ùý»ñç çýëïçïáõï e-mail: vladimir.sahakyan@sci.am, artvardanyan@asnet.am ²ù÷á÷áõù ¸çï³ñïí³í ¿ ½³ý·í³í³ûçý ëå³ë³ñïù³ý ñ³ù³ï³ñ·ç ùá¹»é, áñáõù áñå»ë ëå³ë³ñïáõ ë³ñù»ñ í»ñóíáõù ¿ m ( m ¸ 1 ) åñáó»ëáñçó µ³õï³ó³í µ³½ù³åñáó»ëáñ³ûçý references v. sahakyan and a. vardanyan 8 9 ñ³ù³ï³ñ·á, ñ»ñãáõù å³ñ³ýçý»ñç ³é³í»é³·áõûý ãáõûé³ïñ»éç ù³ý³ïá n ( n ¸ 1 ) ¿: ð³ù³ï³ñ·áõù ûáõñ³ù³ýãûáõñ å³ñ³ýç µýáõã³·ñíáõù ¿ ( º; ¯; ! ) å³ñ³ù»ïñ»ñ, áñï»õ ºý å³ñ³ýçç ëå³ë³ñïù³ý ñ³ù³ñ ³ýññ³å»ßï é»ëáõñëý»ñç(åñáó»ëáñý»ñç) ù³ý³ïý ¿, ¯ ý å³ñ³ýçç ëå³ë³ñïù³ý ³é³í»é³·áõûý å³ù³ý³ïý ¿, çëï ! ý ³ûý ³é³í»é³·áõûý å³ù³ý³ïý ¿, áñ ³û¹ å³ñ³ýçá ï³ñáõ ¿ ñ»ñãáõù ëå³ë»é ùçý㨠ëå³ë³ñïù³ý áõõ³ñïí»éá: ²ñ¹ûáõýùáõù ûáõñ³ù³ýãûáõñ å³ñ³ýç áõý»ýáõù ¿ ëå³ëù³ý å³ù³ý³ïç ë³ñù³ý³÷³ïáõù: ²ßë³ï³ýùáõù ëï³óí³í »ý ñ³ù³ï³ñ·ç íç׳ïá µýáõã³·ñáõ ñ³í³ý³ï³ý³ûçý ñ³í³ë³ñáõùý»ñ, »ñµ ñ³ù³ï³ñ·á ï³ûáõý íç׳ïáõù ¿: ðàñïðåäåëåíèå î÷åðåäè â ìíîãîïðîöåññîðíîé ñèñòåìå ïðè îãðàíè÷åíèè íà âðåìÿ îæèäàíèÿ âëàäèìèð ã. ñààêÿí è àðòóð ï. âàðäàíÿí èíñòèòóò ïðîáëåì èíôîðìàòèêè è àâòîìàòèçàöèè íàí ðà e-mail: vladimir.sahakyan@sci.am, artvardanyan@asnet.am àííîòàöèÿ ðàññìîòðåíà ìîäåëü ñèñòåìû ìàññîâîãî îáñëóæèâàíèÿ, ñîñòîÿùàÿ èç m ( m ¸ 1 ) îáñëóæèâàþùèõ ïðèáîðîâ è ñ ìàêñèìàëüíûì êîëè÷åñòâîì çàäàíèé â î÷åðåäè îæèäàíèÿ n ( n ¸ 1 ) . êàæäîå çàäàíèå õàðàêòåðèçóåòñÿ òðåìÿ ñëó÷àéíûìè ïàðàìåòðàìè ( º; ¯; ! ) , ãäå º ÷èñëî òðåáóåìûõ îáñëóæèâàþùèõ ïðèáîðîâ, íåîáõîäèìûõ äëÿ âûïîëíåíèÿ çàäàíèÿ, ¯ -âðåìÿ, òðåáóåìîå äëÿ âûïîëíåíèÿ çàäàíèÿ, ! äîïóñòèìîå âðåìÿ ïðåáûâàíèÿ çàäàíèÿ â î÷åðåäè äî íà÷àëà åãî âûïîëíåíèÿ, ïîñëå êîòîðîãî îíî ïîêèäàåò ñèñòåìó áåç îáñëóæèâàíèÿ. çàäàíèÿ ïðèíèìàòüñÿ íà îáñëóæèâàíèå â ïîðÿäêå ïîñòóïëåíèÿ èõ â ñèñòåìó, ò.å. èñïîëüçóåòñÿ äèñöèïëèíà fifo (first-in, first-out). â ðàáîòå ïîëó÷åíû óðàâíåíèÿ äëÿ âåðîÿòíîñòåé ñîñòîÿíèÿ ñèñòåìû â ñòàöèîíàðíîì ðåæèìå, êîòîðûå ìîãóò ñëóæèòü îöåíêîé äëÿ ðåàëüíûõ ìíîãîïðîöåññîðíûõ ñèñòåì, èñïîëüçóþùèõ òåõíîëîãèè mpi è openmp. êëþ÷åâûå ñëîâà: ìíîãîïðîöåññîðíàÿ ñèñòåìà êëàñòåðíîãî òèïà, êëàñòåðíûå âû÷èñëåíèÿ, òåîðèÿ ìàññîâîãî îáñëóæèâàíèÿ, îãðàíè÷åíèå âðåìåíè îæèäàíèÿ. ´³ý³éç µ³é»ñ՝ µ³½ù³åñáó»ëáñ³ûçý ïé³ëï»ñ³ûçý ïçåç ñ³ù³ï³ñ·, ïé³ëï»ñ³ûçý ñ³ßí³ñï, ù³ëë³û³ï³ý ëå³ë³ñïù³ý ï»ëáõãûáõý, ëå³ëù³ý å³ù³ý³ïç ë³ñù³ý³÷³ïáõù: sbornik_karen abstract_artur 07_artur_petrosyan's article 61 mathematical problems of computer science 52, 61--65, 2019. udc 004 identity infrastructure boost concept for eduroam service arthur s. petrosyan, gurgen s. petrosyan, robert n. tadevosyan and kevork kh. arsalanian institute for informatics and automation problems of nasra e-mail: arthur@sci.am, gurgen@sci.am, robert@sci.am, kevork.arsalanian@sci.am abstract this paper presents the concept of authenticating eduroam users via their organization's imap server. the concept described is based on the fact that most organizations, even lacking their own personnel identity database, have at least an email service in place and, thus, have a working imap server for their organization domain name. thus, the existing email username/password within a particular organization can be used as an identity source for eduroam authentication. the paper describes the components required to implement the concept, as well as some expected limitations for this solution. the concept is planned to be implemented in asnet-am network in order to stimulate more rapid use of eduroam service in armenia. keywords: eduroam, wifi, wireless, authentication, authorization, identity, imap, pam 1. introduction eduroam (education roaming) [1] is a secure, global wireless network roaming access service developed for the international research and education community. it allows users (researchers, teachers, students, staff) from different institutions to securely gain wifi internet access, while being within the wifi coverage area of any eduroam-enabled institution around the globe. the eduroam principle is based on the fact that the user’s authentication is done by the user’s home institution, whereas the authorization decision allowing access to the network resources is done by the visited network. eduroam is based on the ieee 802.1x standard technology and a hierarchy of authorizing radius servers [2]. the role of the radius hierarchy is to forward user credentials to the user’s home institution, where they can be verified and validated. when a user requests authentication, the user’s realm determines where the request is routed to. the realm is the suffix of the user-name, delimited with ‘@’, and is derived from the organization’s domain name. so, identity infrastructure boost concept for eduroam service 62 each institution participating in eduroam has its institutional radius server (irs) connected to the federation level radius server (flrs) of the country where the institution is located. the flrs is normally operated by the national research and education network (nren) of that territory. these federation-level servers have a complete list of the participating eduroam institutions in that country. this is sufficient to guarantee roaming operations. in case of armenia, asnet-am is the armenian nren acting as a national roaming operator (nro). international roaming in eduroam is operated by means of two top-level radius servers deployed in europe, which forward the users request to the right territory. 2. concept architecture instead of using the simple “one-password-for-all” principle to provide wifi internet access, eduroam service instead requires connecting users to provide their personal credentials (username/password) from home institution in order to gain wifi access. for territories, where research and education organizations would like to participate in eduroam, but lack their own personnel identity database or have it for internal purpose use only, it might take long time to launch eduroam. for such cases we propose to use organization’s corporate email service, since it generally means they would have working imap server for their organization domain name. thus, one possible way of quickly starting to use the eduroam service with minimal administrative overhead is to use the existing email username/password within a particular organization as an identity source for eduroam authentication. institutions that would like their staff members or students to use their institution’s email username/password as an identity source for eduroam authentication, should install and configure their irs in a specific way. first irs should be properly registered at flrs within the nro. specific configuration should include the radius server module to do the authentication via imap server (fig. 1.). fig.1. a. petrosyan, g. petrosyan, r. tadevosyan and k. arsalanian 63 current version of freeradius [3] does not include authentication against imap servers. but there are several solutions that can be used to overcome that limitation. one of them is described in [4] and requires pam-imap pluggable authentication module to be used. in this case freeradius authentication will go through linux pluggable authentication modules (pam) [5] solution to reach the imap server. pam provide dynamic authentication support for applications and services in a linux. pam-imap module should be configured to connect to a particular remote imap server for authentication. in this concept freeradius server, using pam-imap module will treat the remote users as local to the linux system running the radius server. another approach is described in [6] and is based on several python 3 modules (imaplib,sys,ssl). it requires creating an external imap connector an application or script that can be invoked by the freeradius server process on demand for verifying the user name and password pairs against external imap server. the connector gets the user name/password and returns back an exit status value (return value) on completion. if that value is zero, the freeradius sever process considers the user credentials verified. both solutions support secure imap connection method – imaps, which is mostly used nowadays. but there are some limitations for the use of the concept described above. 3. limitations since freeradius has no access to a cleartext password when authenticating via the methods mentioned above, only pap can be used as an inner (phase two) authentication method. this typically means clients (wifi devices) need to be configured to use ttls/pap. ttls was not originally supported out-the-box by windows operating systems, but windows 10 now includes a ttls supplicant. other modern operating systems now support ttls too. another limitation is scalability. imap servers are slow authenticators, compared to those based on ldap or active directory. so setting up an imap connection is rather slow (about 2 seconds) and more resource intensive than other authentication methods. this means that the imap approach likely does not scale too much. of course pam solution can provide some caching of credentials, which may improve this. 4. advantages the concept described in this paper has several advantages. first advantage is that in case of authenticating eduroam users via their organization's imap server, users will likely refrain from sharing their credentials with anyone, since it would potentially give others access to their mailbox too. another advantage of this concept is that it allows to easily manage eduroam access simultaneously with mailbox access. so, if the user leaves organization and his/her mailbox is being disabled or removed, then that user automatically is being restricted from using eduroam too. identity infrastructure boost concept for eduroam service 64 5. conclusion this solution may be interesting for cases, where organization decides to restrict access to the authentication database or does not yet have such a database at all. it enables to launch the eduroam service by authenticating the users of the radius server against publicly available service like imap, and the existing email username/password within a particular organization’s domain name can be used as an identity source for eduroam authentication. references [1] [online]. available: https://www.eduroam.org/ [2] [online]. available: ieee 802.1x remote authentication dial in user service (radius) usage guidelines [3] [online]. available: https://freeradius.org/ [4] [online]. available: https://github.com/asnet-am/eduroam-imap-playbook [5] [online]. available: http://www.linux-pam.org/ [6] [online]. available: https://vessokolev.blogspot.com/2018/09/imap-connector-for-freeradiusto.html submitted 02.09.2019, accepted 28.11.2019. eduroam ծառայության ենթակառուցվածքների զարգացումը խթանող գաղափար արթուր ս. պետրոսյան, գուրգեն ս. պետրոսյան, ռոբերտ ն․ թադևոսյան և գէորգ խ. արսալանյան հհ գաա ինֆորմատիկայի և ավտոմատացման պրոբլեմների ինստիտուտ e-mail: arthur@sci.am, gurgen@sci.am, robert@sci.am, kevork.arsalanian@sci.am ամփոփում հոդվածում նկարագրված է eduroam ենթակառուցվածքի զարգացումը խթանող գաղափար, որը իրականացվելու է asnet-am ցանցում՝ հայաստանում eduroam ծառայության առավել արագ օգտագործմանը նպաստելու համար: նկարագրված գաղափարը հիմնված է այն փաստի վրա, որ որոշ կազմակերպություններ չունեն անձնակազմի ինքնության տվյալների բազա, սակայն ունեն էլ. փոստի ծառայություն և այդպիսով ունեն imap սերվեր իրենց կազմակերպության դոմենային տիրույթի անվանման համար: այսպիսով, որոշակի կազմակերպության ներսում առկա էլ. a. petrosyan, g. petrosyan, r. tadevosyan and k. arsalanian 65 փոստի անունը/գաղտնաբառը կարող է օգտագործվել որպես eduroam ծառայության վավերացման ինքնության աղբյուր: հոդվածը նկարագրում է գաղափարի իրականացման համար անհրաժեշտ բաղադրիչները, ինչպես նաև՝ առաջարկվող լուծման որոշ սահմանափակումներ: բանալի բառեր` eduroam, wifi, անլար, նույնականացում, թույլտվություն, ինքնություն, imap, pam концепция ускоренного развития идентификационной инфраструктуры для сервиса eduroam артур с. петросян, гурген с. петросян, роберт н․ тадевосян и кеворк х. арсаланян институт проблем информатики и автоматизации нан ра e-mail: arthur@sci.am, gurgen@sci.am, robert@sci.am, kevork.arsalanian@sci.am аннотация в статье представлена концепция ускоренного внедрения инфраструктуры eduroam в организациях. описанная концепция основана на том факте, что даже если организации не имеют собственной базы данных идентификации персонала, они имеют по крайней мере службу электронной почты и, таким образом, имеют рабочий сервер imap для доменного имени своей организации. таким образом, существующее имя пользователя/пароль электронной почты в конкретной организации может использоваться как источник идентификации для аутентификации eduroam. в статье описываются компоненты, необходимые для реализации концепции, а также некоторые ожидаемые ограничения и преимущества описанного решения. концепцию планируется реализовать в сети asnet-am, для стимуляции ускоренного расширения использования услуги eduroam в армении. ключевые слова: eduroam, wifi, беспроводная связь, аутентификация, авторизация, идентификация, imap, pam d:\user\sbornik_38_pdf\35.dvi mathematical problems of computer science 38, 82{83, 2012. remar ks on e volutionar y h amiltonian gr aph t heor y zh .g. n iko g h o s ya n ¤ institute for informatics and automation problems national academy of sciences e-mail: zhora@ipia.sci.am w e p r e s e n t a n a p p r o p r ia t e c o m p le m e n t t o t h e la r g e p a le t t e o f e vo lu t io n a r y t h e o r ie s ( s u c h a s e vo lu t io n a r y p s yc h o lo g y, e c o n o m ic s , c o m p u t a t io n , a lg o r it h m , p r o g r a m m in g , g a m e t h e o r y, t h o u g h t a n d s o o n ) b y a n e w d is c ip lin e c o n c e r n in g m a t h e m a t ic s . th e c o n c e p t o f np -c o m p le t e n e s s wa s in t r o d u c e d in 1 9 7 1 b y s t e p h e n co o k [1 ], wh o c o n je c t u r e d t h a t np -c o m p le t e p r o b le m s a r e n o t s o lva b le in p o lyn o m ia l t im e . to d a y, t h is c o n je c t u r e s e e m s m u c h m o r e r e a s o n a b le m o t iva t e d b y t h e fa c t t h a t t h e d e ve lo p m e n t s a r is in g a r o u n d va r io u s np -c o m p le t e p r o b le m s h a ve u n d e r g o n e a n a t u r a l g r a d u a l g r o wt h a n d e vo lu t io n , g e n e r a t in g a g r e a t d ive r s it y. th e s e d e ve lo p m e n t s p r o vid e a n e xc lu s ive va lu a b le d o m a in b e yo n d b io lo g y wit h c o n t in u o u s ly g r o win g d ive r s it y a n d we ll d e s c r ib e d e n vir o n m e n t -o r ig in s g e n e s t r u c t u r e s r e la t io n s . w e fo c u s o n o n e o f t h e m o s t h e a vily s t u d ie d a r e a s in g r a p h t h e o r y, t h a t jo in s t o g e t h e r a n u m b e r o f n p -c o m p le t e c yc le p r o b le m s , c a lle d la r g e c yc le s t h e o r y a s im p li¯ e d ve r s io n o f we ll-kn o wn h a m ilt o n ia n g r a p h t h e o r y, t o s h o w t h a t t h e in d ivid u a ls ( c la im s , p r o p o s it io n s , le m m a s , c o n je c t u r e s , t h e o r e m s ) o n t h is s u b je c t e vo lve a n d a d a p t t o t h e ir e n vir o n m e n t g e n e r a t in g a g r e a t d ive r s it y b y a n it e r a t ive p r o c e s s fr o m s im p lic it y t o c o m p le xit y, fr o m p r im it ive b e g in n in g s ( s u c h a s " e ve r y c o m p le t e g r a p h is h a m ilt o n ia n " ) t o b e s t p o s s ib le t h e o r e m s b y c e r t a in h e r e d it a r y m e c h a n is m s . l a r g e c yc le s t h e o r y p la ys t h e r o le o f a g e n e r a l e n vir o n m e n t a n d va r io u s s t a t e m e n t s , in c lu d in g c la im s , p r o p o s it io n s , le m m a s c o n je c t u r e s a n d t h e o r e m s , p la y t h e r o le o f in d ivid u a ls in a p o p u la t io n . th is s im p li¯ e d a n d va lu a b le m o d e l h a s a n u m b e r o f a d va n t a g e s wit h r e s p e c t t o b io lo g y a n d c a n b e u s e fu l t o wa r d s b e t t e r u n d e r s t a n d in g t h e u n ive r s a l m e c h a n is m s t o e xp la in e vo lu t io n in a wid e va r ie t y o f d o m a in s o u t s id e o f b io lo g y. ( a 1 ) l a r g e c yc le s t h e o r y, o r ig in a t e d a b o u t 6 0 ye a r s a g o , e vo lve s m u c h m o r e r a p id ly t h a n livin g fo r m s o n e a r t h , o r ig in a t e d a b o u t 3 .7 b illio n ye a r s a g o . ( a 2 ) th e o r ig in s o f t h e o r e m s in la r g e c yc le s t h e o r y c a n b e s t r o n g ly d e t e r m in e d b y e xa c t b r a n c h in g s o f t h e t r e e o f d e ve lo p m e n t s . ( a 3 ) ge n e t ic u n it s a n d h e r e d it a r y m e c h a n is m s in la r g e c yc le s t h e o r y a r e m u c h m o r e s im p le r t h a n g e n e s t r u c t u r e s o f livin g fo r m s . w e d is t in g u is h t h e fo llo win g e vo lu t io n m e c h a n is m s in la r g e c yc le s t h e o r y: ( b 1 ) im p r o ve m e n t s ( ve r t ic a l e vo lu t io n ) , ( b 2 ) m o d ī c a t io n s ( h o r iz o n t a l e vo lu t io n ) , ( b 3 ) ve r t ic a l g e n e r a liz a t io n s ( ve r t ic a l e vo lu t io n le a p b a s e d o n in d u c t ive r e a s o n in g ) , 8 2 zh. nikoghosyan 8 3 ( b 4 ) h o r iz o n t a l g e n e r a liz a t io n s ( h o r iz o n t a l e vo lu t io n le a p b a s e d o n in d u c t ive r e a s o n in g ) , ( b 5 ) in vo lvin g n e w g e n e t ic u n it s ( g e n o m e e xt e n s io n ) . de¯nition 1. im p r o ve m e n t is o n e o f t h e fo llo win g p r o c e d u r e s : ( c1 ) r e la xin g o n e o f t h e c o n d it io n s in t h e o r e m s a n d p r e s e r vin g t h e c o n c lu s io n , ( c2 ) s t r e n g t h e n in g t h e c o n c lu s io n a n d p r e s e r vin g t h e c o n d it io n s . de¯nition 2. mo d i¯ c a t io n is o n e o f t h e fo llo win g p r o c e d u r e s : ( d 1 ) r e la xin g o f s o m e c o n d it io n s , a t t h e s a m e t im e s t r e n g t h e n in g s o m e o t h e r s , u n d e r t h e s a m e c o n c lu s io n , ( d 2 ) r e la xin g o f s o m e c o n d it io n s , a t t h e s a m e t im e r e la xin g t h e c o n c lu s io n , ( d 3 ) s t r e n g t h e n in g o f s o m e c o n d it io n s , a t t h e s a m e t im e s t r e n g t h e n in g t h e c o n c lu s io n . de¯nition 3. v e r t ic a l g e n e r a liz a t io n is a le a p in im p r o ve m e n t p r o c e s s b a s e d o n in d u c t ive r e a s o n in g t o wa r d ¯ n d in g b e s t p o s s ib le r e s u lt s . de¯nition 4. h o r iz o n t a l g e n e r a liz a t io n is a le a p in m o d i¯ c a t io n p r o c e s s b a s e d o n in d u c t ive r e a s o n in g t o wa r d ¯ n d in g b e s t p o s s ib le r e s u lt s . w e d e a l a s p e c ia l a t t e n t io n t o s o c a lle d " fu n d a m e n t a l t h e o r e m s " , b y o b s e r vin g t h a t a ll t h e o r e m s in la r g e c yc le s t h e o r y h a ve d e s c e n d e d fr o m a n u m b e r o f c o m m o n a n c e s t o r s via g e n e r a liz a t io n s , c a lle d fu n d a m e n t a l t h e o r e m s . r e m e m b e r , t h a t t h e t e r m " fu n d a m e n t a l r e s u lt " is u s e d in va r io u s ¯ e ld s o f s c ie n c e t o c h a r a c t e r iz e m a in ly t h e c e n t r a l a n d m o s t im p o r t a n t r e s u lt s in t h e a r e a . r e fe r e n c e s [1 ] s .a . co o k, th e co m p le xit y o f th e o r e m -p r o vin g p r o c e d u r e s , p r o c e e d in g s , th ir d a n n u a l a cm s ym p o s iu m o n t h e t h e o r y o f c o m p u t in g , a cm, n e w y o r k ( 1 9 7 1 ) 1 5 1 -1 5 8 . 8 3 d:\sbornik\...\tpel.dvi mathematical problems of computer science 32, 56{64, 2009. on reliability appr oach to i denti¯cation of p r obabilty distr ibutions of t wo statistically dependent objects a r a m o. y e s s a ya n institute for informatics and automation problems of nas of ra evhar@ipia.sci.am abstract the identi¯cation of the distributions of two objects is an answer to the question whether r1-th and r2-th distributions occured, or not on the ¯rst and the second objects, correspondigly . haroutunian and hakobyan solved the problem reliable identi¯cation of probability distributions for two independent objects. in this paper we present the solution of the problem of logarithmically asymptotically optimal identi¯cation of probability distributions for two statistically dependent objects. refer ences [1 ] e . a . h a r o u t u n ia n , \ l o g a r it h m ic a lly a s ym p t o t ic a lly o p t im a l t e s t in g o f m u lt ip le s t a t is t ic a l h yp o t h e s e s " , p roblems of control and information theory, vo l. 1 9 ( 5 -6 ) , p p . 4 1 3 { 4 2 1 , 1 9 9 0 . [2 ] r . f. a h ls we d e a n d e . a . h a r o u t u n ia n , \ on lo g a r it h m ic a lly a s ym p t o t ic a lly o p t im a l t e s t in g o f h yp o t h e s e s a n d id e n t ī c a t io n " . l e c t u r e n o t e s in co m p u t e r s c ie n c e , vo l. 4 1 2 3 , \ ge n e r a l th e o r y o f in fo r m a t io n tr a n s fe r a n d co m b in a t o r ic s " , s p r in g e r , p p . 4 6 2 { 4 7 8 , 2 0 0 6 . [3 ] e . a . h a r o u t u n ia n , \ r e lia b ilit y in m u lt ip le h yp o t h e s e s t e s t in g a n d id e n t ī c a t io n " . p r o c e e d in g s o f t h e n a to a s i, y e r e va n 2 0 0 3 , n a to s c ie n c e s e r ie s , iii: co m p u t e r a n d s ys t e m s s c ie n c e s , vo l. 1 9 8 , ios p r e s s , p p . 1 8 9 { 2 0 1 , 2 0 0 5 . [4 ] e . a . h a r o u t u n ia n a n d p . m. h a ko b ya n , \ on l a o t e s t in g o f m u lt ip le h yp o t h e s e s fo r p a ir o f o b je c t s " , m athematical p roblems of computer science, vo l. x x v , p p . 9 2 { 1 0 0 , 2 0 0 6 . [5 ] e . a . h a r o u t u n ia n a n d p . m. h a ko b ya n , \ on id e n t i¯ c a t io n o f d is t r ib u t io n s o f t wo in d e p e n d e n t o b je c t s " , m athematical p roblems of computer science, vo l. x x v iii, p p . 1 1 4 { 1 1 9 , 2 0 0 7 . 5 6 a. yessayan 5 7 [6 ] e . a . h a r o u t u n ia n a n d a . o. y e s s a ya n , \ on h yp o t h e s e s t e s t in g fo r t wo d i®e r e n t ly d is t r ib u t e d o b je c t s " . m athematical p roblems of computer science, vo l. x x v i, p p . 9 1 { 9 6 , 2 0 0 6 . [7 ] e . a . h a r o u t u n ia n a n d a . o. y e s s a ya n , \ on lo g a r it h m ic a lly a s ym p t o t ic a lly o p t im a l h yp o t h e s is t e s t in g fo r p a ir o f s t a t is t ic a lly d e p e n d e n t o b je c t s " , m athematical p roblems of computer science, vo l. x x ix , p p . 9 7 { 1 0 3 , 2 0 0 7 . [8 ] e . a . h a r o u t u n ia n a n d a . o. y e s s a ya n , \ on o p t im a l h yp o t h e s is t e s t in g fo r p a ir o f s t o c h a s t ic a lly d e p e n d e n t o b je c t s " , m athematical p roblems of computer science, vo l. x x x i, p p . 4 9 { 5 9 , 2 0 0 8 . [9 ] e . a . h a r o u t u n ia n , m. e . h a r o u t u n ia n , a n d a . n . h a r u t yu n ya n , \ r e lia b ilit y c r it e r ia in in fo r m a t io n t h e o r y a n d in s t a t is t ic a l h yp o t h e s e s t e s t in g " , f oundations and trends in communications and information theory, vo l. 4 , n o . 2 -3 , 2 0 0 8 . ìç׳ﳷñáñ»ý ï³ëû³é ûµû»ïïý»ñç ñ³í³ý³ï³ý³ûçý µ³ßëáõùý»ñç ýáõûý³ï³ý³óù³ý ñáõë³éçáõãû³ý ùáï»óù³ý ù³ëçý ². ºë³û³ý ²ù÷á÷áõù ðá¹í³íáõù ëï³óí³í ¿ »ñïáõ íç׳ﳷñáñ»ý ï³ëû³é ûµû»ïïý»ñç ñ³í³ý³ï³ý³ûçý µ³ßëáõùý»ñç ³ëçùåïáïáñ»ý ûåïçù³é ýáõûý³ï³ý³óù³ý ëý¹ñç éáõíáõùá: ºñïáõ ³ýï³ë ûµû»ïïý»ñç ¹»åùáõù ëý¹çñá éáõíí»é ¿ñ [5]-áõù: начиная с начала 2000 года осуществляется внедрение ghis в здравоохранении, в рамках принятого проекта о реформирование информ mathematical problems of computer science 51, 108--118, 2019. udc 001.53 computer science in transcaucasia and baltic states: a comparative bibliometric analysis shushanik a. sargsyan1,2 and edita g. gzoyan1 1center for scientific information monitoring and analysis (csiam) institute for informatics and automation problems of nas ra 2yerevan state medical university after mkhitar heratsi, medical physics department e-mail: shushanik@ipia.sci.am abstract based on the web of science, incites database, this article will analyze the publication output of the transcaucasia and baltic states in computer science research field, their citations, as well as international collaboration in the field of computer science. the obtained results demonstrate that publications on computer science from the baltic states are nearly 4 times higher than publications from transcaucasia. among the baltic states, lithuania holds a primary position followed by estonia and latvia; while in the transcaucasia, the leading position is held by azerbaijan, and followed by armenia and georgia. the same picture can be seen in the case of citations on the works on computer science of the studied states. in the international collaboration framework, the european states are the most frequent collaboration countries of the baltic states. the same tendency can be seen in the case of georgia and armenia, while azerbaijan shows a dramatically different vector of scientific internationalization. keywords: computer science, transcaucasia, baltic states, web of science; incites, citations, bibliometric analysis, regional analysis. 1. introduction the collapse of the soviet union resulted in the major transformations of the political map of the eurasian continent. despite sharing historical, economic, social and other similarities during the soviet period, the 15 republics established as a result of the soviet breakup, chose separate ways of political, social, economic, as well as scientific development. already in 2004, all three baltic states (latvia, lithuania, and estonia) entered the european union, while for a considerable time the transcaucasia region (armenia, azerbaijan and georgia) remained pro-russian. recently, however, there has also been some shift in the geopolitical discourse of transcaucasia: georgia also took a course to europe, while armenia chose to become a member-state of the eurasian 108 sh. sargsyan and e. gzoyan 109 economic union. azerbaijan holds a neutral position, maintaining close political ties with turkey. having all these developments in the background, the aim of this article is to analyze the development of computer science in the studied states, to understand the vectors of their international collaboration. this article will present a bibliometric analysis of publications and citations in the field of computer science in the transcaucasia and the baltic states after their independence, their international collaboration, comparatively analyzing their scientific output in the field under study. although already well-established computer science is a relatively new and dynamically developed research field, nowadays, it is a highly interdisciplinary scientific field that has significant links with mathematics, physics, biology and even humanities. the field is also declared a priority in the baltic and caucasus states. during the soviet time, the studied countries had research institutions that were engaged in computer science. in 1958, nikita khrushchev announced the importance of cybernetics for defense and space industry. in furtherance of this, a lot of researchers from the soviet union were reeducated. lithuanian researchers were the most active in the baltic states. they established a computer plant in vilnius – sigma where the major computing equipment was developed for non-military use. the ruta 110 computer was designed and produced in “sigma” and was widely used in the soviet union [1]. in the caucasus region, major developments in this field occurred in armenia, which was the “silicon valley” in the field of it for the soviet union. in 1956, the yerevan computer research and development institute was established as a pioneer in the it and software industry in soviet armenia. in 1963, the institute developed the nairi computer for engineering purposes. another it research hub was established in 1957 initially as the computer centre, which later became the institute for informatics and automation problems of nas ra. the institute also played a great role in the fields of computer science and its applications. 2. data and method the study is based on the clarivate analytics’ web of science incites dataset. publications from armenia, republic of georgia, azerbaijan, lithuania, latvia and estonia were used for this analysis. the study period is 1991-2018, that is, after the establishment of their independence from the soviet union. the last update of the data was made on 04.02.2019. we took the computer science research area with the following sub-areas from the wos: artificial intelligence, cybernetics, hardware & architecture, information systems, interdisciplinary applications, software engineering and theory & methods. the following types of documents were used in the article: articles, meeting abstracts, note, proceedings papers and reviews. for the purposes of our article, the full counting method was used, so that every participatory country receives a full count for each article 1 score is assigned to each co-author country. thus, some scientific products are counted more than once. 3. discussion in the period 1991-2018, the 6 baltic and caucasus countries considered here produced a total of 7721 computer science publications in the wos core collection as detailed in table 1. among the countries studied, the baltic states are in a leading position by the gross number of publications. the share of publications within the countries of the studied blocks is interestingly computer science in transcaucasia and baltic states: a comparative bibliometric analysis 110 nearly the same: the three baltic states have somewhat nearly two thousand publications, while the number of publications from the caucasus states doesn’t exceed 500. table 1: gross number of publications from the individual baltic and transcaucasia states. wos cc (1991-2018). country publications lithuania 2702 estonia 2075 latvia 1813 azerbaijan 516 armenia 344 republic of georgia 312 as can be seen from table 1, lithuania is by far the most productive country, whose publications are approaching 3 thousands, followed by estonia and latvia. among the caucasus states azerbaijan has a leading position, while armenia and georgia have somewhat similar number of publications. fig. 1 shows the distribution of publications of the concerned countries by years. the figure shows the overall increase in the number of publication in all 6 countries, while in the case of the baltic states this growth is quite obvious after 2004, which is the year of accession of the baltic states to the eu, which presumably was the reason for the raise of publications from the baltic states. fig. 1: publications of the baltic and transcaucasia states from the wos in 5 years trends (1991-2018). 0 100 200 300 400 500 600 700 800 900 1000 1100 1200 lithuania estonia latvia azerbaijan armenia georgia sh. sargsyan and e. gzoyan 111 in the next part of our research, we looked into the number of citations received by the 6 countries studied. we have the same picture as with the publications. in the baltic states the leading role is ascribed to lithuania, followed by estonia and latvia. among the caucasus states, azerbaijan is on the forefront, followed by armenia and georgia. this implies that together with the quantitative element, the publications of baltic states are cited more. the quantity-citation link is seen also at the country level in fig. 2. fig. 2: citations received by the publications of the baltic and transcaucasia states from the wos (1991-2018). 3.1. scientific collaboration of the baltic and caucasus states in the field of computer science scientific collaboration became one of the identifying features of modern science. it has become an important tool to promote science and technology, increase scientific output, as well as its visibility and impact of scientific collaboration. of course, not all research collaboration ends up with a joint paper, especially in the field of computer science, where the result of collaboration can be a new computer program that is still considered to be the best indicator of scientific cooperation. there are different measurements of research collaboration [2], but for the purposes of our study we have chosen the co-authorship measurement method. co-authorship is a complex phenomenon and it is built on several decisions at both state and individual levels. at the state level, scientific cooperation is influenced by the policy priorities of the states, which are reflected in scientific cooperation agreements with individual countries or a block of countries. at the individual level, the decision of individual scientists matters. the level of international coauthorship also depends on the size of the particular country, the “proximity” between the countries [3]. meanwhile, the proximity is also a multi-layered phenomenon and includes the 44% 35% 13% 4% 2% 2% lithuania estonia latvia azerbaijan armenia georgia computer science in transcaucasia and baltic states: a comparative bibliometric analysis 112 geographical position of the states, historical, cultural ties, linguistic affinity, and political, economic factors. thematic proximity is another base for a collaboration decision [4]. all the mentioned factors can affect the collaboration decisions of individual scientists. taking this in the background, we studied the international scientific collaboration of the 6 countries concerned to find the preferences (fig. 3). fig. 3: collaboration of armenia in the field of computer science. fig. 3 presents the international scientific collaboration of armenia in the researched field. it is evident that the great portion of papers is an inter-country collaboration, which are the joint publications of armenian institutions. as we can see from the figure, usa is the main partner of armenia in the field of computer science, followed by russia and france. fig. 3 also shows that the remaining collaboration states are from europe. so, the western direction is quite obvious in case of armenia. collectively, the european countries are the main partners of armenia in computer science (61 publications). 107 31 17 14 12 10 7 7 6 5 0 20 40 60 80 100 120 sh. sargsyan and e. gzoyan 113 fig. 4: international collaboration of azerbaijan in the field of computer science. international scientific collaboration of azerbaijan showed a quite different pattern (see fig. 4). although here again the usa is among the first three main collaboration countries of azerbaijan, the first partner of azerbaijan is turkey, with which azerbaijani scientists have 54 joint publications in the field of computer science. interestingly, azerbaijan has quite passive links with russia, and the republic hasn’t accepted the european direction. among the eu countries there is collaboration only with poland and cyprus. meanwhile, azerbaijan is actively cooperating with canada, which appeared the fifth in the list of collaborative countries, and has some joint publications with south korea and malaysia. in case of armenia and georgia, these three countries are absent from the list of collaborating countries. among the three caucasus republics, azerbaijan has the most publications in the wos without international collaboration (136), which implies that their leading position among the other caucasus states is not the result of collaboration. 136 54 23 16 16 12 11 9 8 7 0 20 40 60 80 100 120 140 160 computer science in transcaucasia and baltic states: a comparative bibliometric analysis 114 fig. 5: international collaboration of the republic of georgia in the field of computer science. international scientific collaboration of georgia performed some similarities to armenia (see fig. 5). here again the usa is the main scientific partner in the researched field, and here again the european direction is quite obvious. although georgia has joint publications with russia, but their number is quite small (only 5). another distinctive feature among the caucasus states is the cooperation with iran in case of azerbaijan and georgia, while the cooperation of armenia with iran is quite limited to only 4 papers (it is not seen in the table, as we took the first 10 collaboration states). as for the international scientific collaboration of the baltic states, here the scientific preferences are much similar to each other. the three countries have accepted the european direction, which means that they are more actively collaborating with the other eu states. for all three countries, the usa is the second partner in the field. for latvia and estonia, germany is the main collaboration country, while estonian researchers prefer sweden. fig. 6 details the collaboration pattern of latvia. interestingly, russia is the forth in the list of partners of latvia, while it is absent from the list of the first 10 collaborating countries of the other two baltic states. 98 17 17 14 13 11 6 5 5 5 0 20 40 60 80 100 120 georgia usa italy spain germany finland austria iran united kingdom russia sh. sargsyan and e. gzoyan 115 fig. 6: international collaboration of latvia in the field of computer science. fig. 7 presents the lithuanian collaboration in computer science. apart from general features identified before, china appeared to be among the collaboration countries with 31 joint publications. fig. 7: international collaboration of lithuania in the field of computer science. 366 59 58 33 31 25 22 22 21 17 0 50 100 150 200 250 300 350 400 609 106 87 78 62 45 42 38 34 31 0 100 200 300 400 500 600 700 computer science in transcaucasia and baltic states: a comparative bibliometric analysis 116 fig. 8 presents the scientific collaboration of estonia. the main difference from the other two baltic states is the presence of australia among the partner states. fig. 8: international collaboration of estonia in the field of computer science. 4. conclusion we studied the publications of the baltic and caucasus states in the field of computer sciences from the wos database. we retrieved 7721 publications and also citations received by the 6 concerned states. based on our findings, the baltic states have nearly 3 times more publications on computer science than the caucasus states. among the baltic states, the leading position is held by lithuania, followed by estonia and latvia. in the caucasus region, the leader is azerbaijan, followed by armenia and georgia. despite its leading position in the field of computer science during the soviet times, armenia was unable to maintain its position, while lithuania, which was the forerunner among the baltic states during the soviet times, was able to keep its position. we noticed a sharp increase in publications from the baltic states after their accession to the eu. so, the eu accession of the baltic states significantly affected the number of their publications and citations in the field of computer science. as for the scientific collaboration, the countries studied showed some similar and also diverse behavior. the usa is an important partner for all 6 countries. the baltic states, armenia and azerbaijan are actively collaborating with the eu member-states, while the european direction is not popular with azerbaijan. the latter is quite actively working with turkey, with which azerbaijan has close political ties. as for the post-soviet countries, only armenia and latvia have a significant number of joint publications with russia, while the joint georgian-russian papers in the field are only 5. 737 119 111 86 80 70 66 64 62 51 0 100 200 300 400 500 600 700 800 sh. sargsyan and e. gzoyan 117 references [1] e. tyugu, “computing and computer science in the soviet baltic region”, in history of nordic computing 2, timo järvi petri paju (eds.), springer, pp. 2937, 2009. [2] u. finardi, “scientific collaboration between brics countries”, scientometrics, vol. 102, но. 2, pp. 1139–1166, doi 10.1007/s11192-014-1490-5, 2015. [3] m. zitt, e. bassecoulard and y. okubo, “shadows of the past in international cooperation: collaboration profiles of the top five producers of science”, scientometrics, vol. 47, no. 3, pp. 627–657, 628-629, 2000. [4] p. s. nagpaul, “exploring a pseudo-regression model of transnational cooperation in science”, scientometrics, vol. 56, no. 3, pp. 403–416, 2003. submitted 28.02.2019, accepted 09.04.2019. համակարգչային գիտությունները անդրկովկասում և բալթյան երկրներում. համեմատական գիտաչափական վերլուծություն շուշանիկ ա. սարգսյան1,2 և էդիտա գ. գզոյան1 1գիտական տեղեկատվության վերլուծության և մոնիթորինգի կենտրոն, հհ գաա ինֆորմատիկայի և ավտոմատացման պրոբլեմների ինստիտուտ 2երևանի մխիթար հերացու անվան պետական բժշկական համալսարան, բժշկական ֆիզիկայի ամբիոն e-mail: shushanik@ipia.sci.am ամփոփում հիմնվելով web of science, incites մատենագիտական շտեմարանի վրա` հոդվածը վերլուծում է համակարգչային գիտությունների ոլորտում անդրկովկասի և բալթյան պետությունների հրապարակումների, այդ հրապարակումների ստացած հղումների թիվը, ինչպես նաև համակարգչային գիտությունների բնագավառում ուսումնասիրվող երկրների միջազգային համագործակցությունը: ստացված արդյունքները ցույց են տալիս, որ համակարգչային գիտությունների բնագավառում բալթյան պետությունների հրապարակումները շուրջ 4 անգամ գերազանցում են անդրկովկասյան երկրների հրապարակումները: բալթյան պետություններից լիտվան առաջատար դիրք է զբաղեցնում հրապարակումների թվով, որին հետևում են էստոնիան և լատվիան: անդրկովկասյան երկրների շարքում հրապարակումների թվով առաջատարը ադրբեջանն է, ապա` հայաստանը և վրաստանը: համանման պատկեր է ստացվում նաև հղումների վերլուծության արդյունքում: computer science in transcaucasia and baltic states: a comparative bibliometric analysis 118 միջազգային համագործակցության ուսումնասիրությունը ցույց է տալիս, որ բալթյան պետությունները հիմնականում համագործակցում են եվրամիության անդամ-երկրների հետ: եվրոպական ուղղվածությունը նկատվում է նաև հայաստանի և վրաստանի միջազգային համագործակցություններում, մինչդեռ ադրբեջանի վեկտորը միանգամայն տարբեր է: բանալի բառեր` համակարգչային գիտություններ, անդրկովկաս, բալթյան պետություններ, web of science, incites, գիտաչափական վերլուծություն, տարածաշրջանային վերլուծություն: компьютерные науки в закавказье и странах балтии: сравнительный наукометрический анализ шушаник а. саргсян1,2 и эдита г. гзоян1 1центр анализа и мониторинга научной информации, институт проблем информатики и автоматизации нан ра 2ереванский государственный медицинский университет им. мхитара гераци, кафедра медицинской физики e-mail: shushanik@ipia.sci.am аннотация основываясь на библиометрические базы данных web of science, incites, статья анализирует число публикаций и цитирований на эти публикации в сфере компьютерных наук в странах закавказья и балтии, а также международное сотрудничество вышеуказанных стран в сфере компьютерных наук. полученные данные показывают, что в сфере компьютерных наук число публикаций стран балтии превышает аналогичный показатель стран закавказья около 4 раз. лидером по числу публикаций из стран балтии является литва, за которой следуют эстония и латвия. первое место в закавказье занимает азербайджан, второе место армения, а третье грузия. аналогичная картина складывается и при анализе числа цитирований. исследование международного сотрудничества показывает, что страны балтии сотрудничают в основном со странами ес. европейское направление международного сотрудничества наблюдается и в случае армении и грузии, в то время, как вектор международного сотрудничества азербайджана совершенно другой. ключевые слова: компьютерные науки, закавказье, страны балтии, web of science, incites, наукометрический анализ, региональный анализ. microsoft word dsa.doc îçµ»éý»ïçï³ûç ¨ ñ³ßíáõ³ï³ý ï»ëýçï³ûç ù³ã»ù³ïçï³ï³ý ñ³ñó»ñ 25, 2006, 45–52. 45 ´³½ù³ã³÷ ïíû³éý»ñç ï³éáõóí³íùç í»ñéáõíáõãû³ý ýáñ ù»ãá¹á ¨ ¹ñ³ çñ³ï³ý³óáõùá statistica íñ³·ñ³ß³ñç ùçç³í³ûñáõù ²ñë»ý ø³ý³ëû³ý ê. ²µáíû³ýç ³ýí³ý ºñ¨³ýç å»ï³ï³ý ù³ýï³í³ñå³ï³ý ñ³ù³éë³ñ³ý ðð¶²² e-mail arsen.manasyan@gmail.com ²ù÷á÷áõù ðá¹í³íáõù ý»ñï³û³óí³í ¿ statistica ñ³ù³ï³ñ·áõù ³ßë³ïáõ µ³½ù³ã³÷ ïíû³éý»ñç ï³éáõóí³íùý»ñç µ³ó³ñ³ûïù³ý ñ³ù³ñ ùß³ïí³í ù»ãá¹ç dsa íñ³·ñ³ûçý çñ³ï³ý³óáõùᣠø»ãá¹á ß³ñ³¹ñí³í ¿ [1]-áõù£ üáñ íñ³·çñá ï³ñáõ ¿ ïçñ³éí»é çýãå»ë áõëáõóù³ý ýå³ï³ïý»ñáí, ³ûýå»ë ¿é çñ³ï³ý ïíû³éý»ñç í»ñéáõíáõãû³ý ñ³ù³ñ£ ¶ñ³ï³ýáõãûáõý [1] ø³ý³ëû³ý ². ì., ìç׳ﳷñ³ï³ý çýï»ñ³ïïçí ·í³å³ïï»ñý»ñç ·áñíáýã³óç û·ï³·áñíù³ùµ ïíû³éý»ñç ñ³ù³ë»éáõãû³ý í»ñéáõíáõãû³ý ù³ëçý, ø³ã»ù³ïçï³ý µ³ñóñ³·áõûý ¹åñáóáõù, ñ³ýóýí³í ¿£ [2] lausen b., schumacher m., maximally selected rank statistics, biometrics 48, pp 7385, 1992. [3] hawkins d. m., fitting multiple change-point models to data. computational statistics and data analysis 37, pp 323-341, 2001 [4] schwartz p., threshold models for combination data from reproductive and developmental experiments. j. of amer. statist. assoc. 90, pp 862-870, 1995 [5] safaryan i. a., haroutunian e. a., manasyan a. v., two-dimensional sequence homogeneity testing against mixture alternative. mathematical problems of computer science 23, pp 67-79, 2004 [6] dias a., embrechs p., change-point analysis for dependence structures in finance and insurance, www.math.ethr.ch/~embrechs [7] shih y.-sh., tsai h.-w., variable selection bias in regression trees with constant fit. computational statistics and data analysis v. 45, pp 595-608, 2004 the new method of multidimensional sequence structure snalysis and its implementation in statistica system. a. manasyan abstract the article presents dsa software, which works in statistica environment and implements the method of multidimensional sequence structure analysis. the method was described in [1]. the new tool can be used in teaching purposes as well for real data analysis. d:\sbornik\...\exihs.dvi mathematical problems of computer science 26, 2006, 33{37. computation of i nfor mation h iding capacity and e-capacity lower b ounds¤ s m b a t a . to n o ya n institue for informatics and automation problems of nas of ra e-mail smbatt@ipia.sci.am abstract the binary{hamming case of the e-capacity and capacity results for information hiding system [1, 2] is evaluated for practical interests. a special parallel algorithm is elaborated and the computational software utilities are developed. the graphs, describing dependences of information hiding rate from the reliability and allowed distortion levels for the information hider and the attacker are obtained and presented. also the graphical view of the capacity, depending from the allowed distortion levels is plotted. refer ences [1 ] m. e . h a r o u t u n ia n a n d s . a . to n o ya n , " r a n d o m c o d in g b o u n d o f in fo r m a t io n h id in g e-c a p a c it y" , p roc. of ie e e intern. symp. infrom. theory, p . 5 3 6 , u s a , ch ic a g o , 2 0 0 4 . [2 ] m. e . h a r o u t u n ia n a n d s . a . to n o ya n , " on e s t im a t e s o f r a t e -r e lia b ilit y-d is t o r t io n fu n c t io n fo r in fo r m a t io n h id in g s ys t e m " , transactions of the institute for informatics and automation p roblems of the nas of r a. m athematical p roblems of computer scence 23, p p . 2 0 -3 1 , 2 0 0 4 . [3 ] p . mo u lin a n d j. a . o's u lliva n , " in fo r m a t io n -t h e o r e t ic a n a lys is o f in fo r m a t io n h id in g " , ie e e trans. inform. theory, vo l. 4 9 , n o . 3 , p p . 5 6 3 -5 9 3 , ma r . 2 0 0 3 . [4 ] n . me r h a v, " on r a n d o m c o d in g e r r o r e xp o n e n t s o f wa t e r m a r kin g s ys t e m s " , ie e e trans. inform. theory, vo l. 4 6 , n o . 2 , p p . 4 2 0 -4 3 0 , ma r . 2 0 0 0 . [5 ] n . me r h a v a n d a . s o m e kh -b a r u c h , " on t h e e r r o r e xp o n e n t a n d c a p a c it y g a m e s o f p r iva t e wa t e r m a r kin g s ys t e m s " , ie e e trans. inform. theory, vo l. 4 9 , n o . 3 , p p . 5 3 7 5 6 2 , ma r . 2 0 0 3 . ¤the work was supported by armenian target programm 04.10.31. 3 3 3 4 computation of information hiding capacity and e-capacity lower bounds îíû³éý»ñ ã³ùóýáõ ñ³ù³ï³ñ·ç áõý³ïáõãû³ý ¨ e-áõý³ïáõãû³ý ëïáñçý ·ý³ñ³ï³ï³ýý»ñç ñ³ßíáõùá ê. ². îáýáû³ý ²ù÷á÷áõù ºñïáõ³ï³ý ð»ùùçý·ç ¹»åùç ñ³ù³ñ ¹çï³ñïí»é »ý [1,2]-áõù ñ»ï³½áïí³í ïíû³éý»ñ ã³ùóýáõ ñ³ù³ï³ñ·ç áõý³ïáõãû³ý ¨ e-áõý³ïáõãû³ý ëïáñçý ·ý³ñ³ï³ï³ýý»ñá, »éý»éáí ïçñ³é³ï³ý ýß³ý³ïáõãûáõýçó: î³éáõóí»é ¨ íñ³·ñ³íáñí»é ¿ ñ³ßí³ñïý»ñç ï³ï³ñù³ý ñ³ù³ñ ½áõ·³ñ»é ³é·áñçãù: êï³óí³í ¨ ý»ñï³û³óí³í »ý ïíû³éý»ñ ã³ùóý»éáõ ³ñ³·áõãû³ý ·ñ³ýçïý»ñ᪠ï³ëí³í ñáõë³éçáõãûáõýçó ¨ ïíû³éý»ñ ã³ùóýáõç áõ ñ³ñó³ïíáõç ñ³ù³ñ ãáõûé³ïñ»éç ß»õù³ý ù³ï³ñ¹³ïý»ñçó: ü³¨ ï³éáõóí³í ¿ áõý³ïáõãû³ý ·ñ³ýçï³ï³ý å³ïï»ñ᪠ï³ëí³í ãáõûé³ïñ»éç ß»õù³ý ù³ï³ñ¹³ïý»ñçó: mathematical problems of computer science 58, 42–51, 2022. doi:10.51408/1963-0091 udc 519.6, 004.9 a brief comparison between white box, targeted adversarial attacks in deep neural networks grigor v. bezirganyan and henrik t. sergoyan department of mathematics, technical university of munich, muinch, german e-mail: grigor.bezirganyan@tum.de, henrik.sergoyan@tum.de abstract today, neural networks are used in various domains, in most of which it is critical to have reliable and correct output. this is why adversarial attacks make deep neural networks less reliable to be used in safety-critical areas. hence, it is important to study the potential attack methods to be able to develop much more robust networks. in this paper, we review four white box, targeted adversarial attacks, and compare them in terms of their misclassification rate, targeted misclassification rate, attack duration, and imperceptibility. our goal is to find the attack(s), which would be efficient, generate adversarial samples with small perturbations, and be undetectable to the human eye. keywords: adversarial attacks, robustness, machine learning, deep learning. article info: received 26 aprile 2022; received in revised form 4 july 2022; accepted 29 july 2022. 1. introduction nowadays, deep neural networks are becoming more and more popular to solve problems in various domains, including safety-critical areas such as medicine, self-driving cars, etc. unfortunately, techniques to fool deep learning models have recently come out to provide incorrect outputs [1]. particularly, in the image classification domain, an attacker can create an altered image, which will be misclassified by a model but will be classified correctly by a human. this altered image is often referred to as an adversarial example, and this process as an adversarial attack. to be protected against such attacks, researchers try to create methods to make the models more robust against such perturbations. studying adversarial attacks and their potential helps us develop better countermeasures against them. in this paper, we will discuss some of the adversarial algorithms and test them against an image classification model. we then compare the results of the experiments in terms of their misclassification rate, targeted misclassification rate, attack duration, and imperceptibility. 42 g. bezirganyan and h. sergoyan 43 in poisoning attacks, the attacker tries to insert fake samples (i.e., data samples with wrong labels) into the training dataset, which will make the model learn on those fake samples and output wrong results. this kind of attack is possible when the attacker has the means to import those fake samples into the training set. in contrast, in evasion attacks, the attacker does not need access to the dataset. in this case, the attacker creates adversarial samples, which are similar and hard to distinguish by a human from the original samples but are misclassified by the trained model. based on how much information the attacker has about the model, attacks can be classified into white-box, black-box, and gray-box attacks. in the white box scenario, the attacker has full knowledge about the model architecture and uses this knowledge to generate adversarial examples. in contrast, in the black-box setup, the attacker does not know the architecture. instead, the attacker observes the output of the model from the given input. some of the attacks assume access to the soft labels (i.e., probability or likelihood score of belonging to a class), while others try to generate examples based on only hard labels (i.e., class labels without the score). in the gray-box setting, the attacker has an access to the original model and trains a generative model on it. when the generative model is ready, the attacker uses that model to generate adversarial samples. hence, the original model is no more needed. recently, in [2] another category was introduced, called no-box attacks. in contrast to black-box attacks, the attacker cannot query the model, instead, he has a small number of samples from the same domain as the victim. the authors train an auto-encoder on those samples and then generate the adversarial examples using the features learned from the auto-encoder. in the targeted attack, the attacker tries to misclassify the given sample into a specific target label. in contrast, in non-targeted attacks, the attacker tries to classify the sample into any other class. in this paper, we try to overview some of the adversarial attack techniques and, running experiments in the same setting, compare them based on: • misclassification: what percentage of the adversarial samples were misclassified • targeted misclassification: what percentage of the adversarial samples were successfully misclassified to the target class • imperceptibility: how much the adversarial example looks like the original image • duration of the attack: how long it takes to generate an adversarial example 1.1 definitions and notations 1.1.1 poisoning attacks vs evasion attacks 1.1.2 attacker’s knowledge of the model 1.1.3 targeted vs non-targeted attacks 1.2 our goal and contribution 44 a brief comparison between white box, targeted adversarial attacks in deep neural networks in this paper, we will concentrate only on white-box and target attacks. in particular, we will discuss and experiment with the fast gradient sign method [1], projected gradient descent [3], autopgd [4], and fw + dual lmo [5]. we chose these attack methods as fgsm is one of the first and simplest methods, which is still popular today. pgd is the most popular, as even many new state-of-the-art methods are modified versions of the pgd attack. autopgd, being one of those variations, achieves state-of-the-art results according to the authors. and while these attacks use ℓp norms, we also chose fw + dual lmo as an example of an attack that uses another norm (wasserstain norm in this case). 2. attack mechanisms in this section, we will briefly overview the attacks, which will be used for experimentation further in the paper. in our attacks, we are given a set of input images x ∈ rn×n, and our goal is to craft an adversarial example x′ ∈ rn×n that will be misclassified by the deep learning model f : rn×n → n. since we are discussing targeted attacks, we want to misclassify the adversarial sample into our desired target class t ∈ n instead of the original class y ∈ n. furthermore, the perturbation we add to the image should be as small as possible, not to be detected by a human. so, we can formulate the problem in the following way: given a neural network f : rn×n → n, input image x ∈ rn×n with a label y ∈ n, a distance function || · || and a perturbation budget ϵ ∈ r find an x′ ∈ r such that f(x′) = t ̸= y s.t. ||x′ − x|| ≤ ϵ. (1) in our case, the distance functions will be l1, l2, l∞ distances or the wasserstein distance. since we can access the gradients of the network in the white-box setting, what most of the gradient-based attacks do, is to fix the network weight and maximize the loss by updating the image. for that, they add a small perturbation η ∈ rn×n to the original image: x′ = x + η the most efficient way to maximize the loss would be to add noise in the same direction as the gradients. [1] introduced an attack method, where they do exactly that: add a perturbation in a direction that will increase the loss function l between the adversarial example and the original label x′ = x + ϵ · sign(∇xl(θ, x, y)). (2) we can see that in this way the maximum allowed perturbation is added, while still being in the ϵ ball. for a targeted setting, the update step will become: x′ = x − ϵ · sign(∇xl(θ, x, t)) in other words, a perturbation is added to minimize the loss between the adversarial sample and the target class t. 2.1 fast gradient sign method (fgsm) g. bezirganyan and h. sergoyan 45 the projected gradient descent attack (pgd) or basic iterative method (bim) was introduced in [3], where they transformed the fgsm [1] one-step attack into an iterative one by performing the update step (2) multiple times with a small step size α ∈ rn×n. this will work better, as the fgsm adds the maximum allowed perturbation, but does not guarantee to maximize the loss within the allowed ϵ−ball. in contrast, in an iterative approach, the algorithm is more likely to find the maxima. to ensure that the adversarial sample remains in the ϵ neighborhood, pgd projects the sample back to the ϵ ball after each update step. in other words, it performs projected gradient descent (or ascent) on the input sample. the update steps for targeted and untargeted attacks will be as follows: x(i+1) = πϵ(x (i) + α · sign(∇x(i)l(θ, x (i), y))) (3) x(i+1) = πϵ(x (i) − α · sign(∇x(i)l(θ, x (i), t))) (4) so, the attacker tries to find a perturbation that either finds the maximum loss between x′ and y (3) (untargeted attack), or the minimum loss between x′ and t (4) (targeted attack). it has recently been suggested [4] that the cross-entropy loss and the fixed step size of the pgd attack [3] may be two reasons for its potential failure. they propose an alternative loss function and a new gradient-based method, auto-pgd, which does not require a fixed step size. they divide their method into two phases: an exploration phase and an exploitation phase. during the exploration phase, they search for good initial points, while in the exploitation phase, they try to maximize the accumulated knowledge. the step size value depends on the trend of optimization. if the objective function decreases rapidly, then the step size does not need to be changed, otherwise, if it decreases slowly, the step size is reduced. the wasserstein adversarial attack was introduced in [6]. here they proposed to use the wasserstein distance instead of the commonly used ℓp distances. for images, the wasserstein distance can be seen as the cost of redistributing pixel mass. for example, while rotations change ℓp norms dramatically, they only slightly change the wasserstein distance. so, what their algorithm does, is to do a pgd attack [3], but instead of projecting on an ℓp norm, they project on the wasserstein ball. however, since the projection onto the wasserstein ball is computationally expensive, they make an approximation by performing modified sinkhorn iterations [7]. [5] improved the algorithm by introducing an exact but still efficient projection operator. they also introduce an adversary generating method based on the frank-wolfe [8] method equipped with a suitable linear minimization oracle and show that it works very fast for wasserstein constraints. in this paper, we will use that frank-wolfe method (fw + dual lmo) for the experiments. 2.2 projected gradient descent 2.3 auto-projected gradient descent 2.4 wasserstein attack 46 a brief comparison between white box, targeted adversarial attacks in deep neural networks in this experiment, our goal is to run fgsm [1], pdg [3], autopgd [4], and fw + dual lmo [5] attacks on the same environment and compare them in terms of misclassification, targeted misclassification, attack duration, and imperceptibility. we are performing our experiments on a pre-trained resnet-18 [9] classifier on the cifar10 dataset [10], with initial 92.4% accuracy on the test set. we generate the adversarial examples on a server with an nvidia geforce gtx 1080-ti gpu. we use the adversarial robustness toolkit (art) [11] for fgsm [1] and pgd [3] and autopgd [4] attacks, and the original implementation by the authors for fw + dual lmo [5]. we run each of the adversarial attacks with a set of epsilon values in ϵ ∈ (0, 0.5] and for all target classes. we use ℓp norms for fgsm, pgd, and autopgd, and we use the wasserstein distance for the fw + dual lmo. all the other hyper-parameters are left to their default values. for the fw + dual lmo, in the original implementation, there was no option for targeted attacks. hence, we modified their implementation and added the option for target attacks. for that we converted the problem: maximize l(f(x′), y) subject to ||x′ − x|| ≤ ϵ to minimize l(f(x′), t) subject to ||x′ − x|| ≤ ϵ we log the duration of the attack, the misclassification rate, and the targeted misclassification rate for later comparison. the source code for the experiment can be found https : //github.com/bezirganyan/adversarialarenahere. we first look at the average misclassification and targeted misclassification scores that each of our models was able to achieve for some ϵ ∈ (0, 0.5]. in table 1, we can see average misclassification and targeted misclassification rates for the best epsilon of each attack. as we can see from the ℓp attacks, the ℓ∞ norm yields the highest scores in our setup. hence, from now on we will use the ℓ∞ norm for further comparisons. note that this does not mean that the ℓ∞ norm is better since we could get similar scores and similar perturbations for higher ϵ values under other norms, as the ℓ∞ attack will add a higher amount of perturbation under the same epsilon. furthermore, we can see that from the ℓp attacks in terms of targeted misclassification rate, the pgd, and autopgd attacks yield very high scores leaving the fgsm attack behind with a huge margin. in general, pgd and autopgd attacks behave almost identically in 3. experiments 3.1 goal 3.2 setup 4. results 4.1 targeted misclassification and misclassification rate g. bezirganyan and h. sergoyan 47 our experiments. we hypothesize that this is because we are testing on an undefended model, on which they both reach their maximum potential limit. the developers of the art framework confirmed that on their tests on defended models in an untargeted setting, autopgd behaved slightly better. we, hence, plan to test and compare the models on a defended model in our future work. in fig. 1, we can see the misclassification and targeted misclassification rates of the attacks for different epsilons and under the ℓ∞ norm. we can see that in terms of misclassification and targeted misclassification rates the pgd and autopgd attack perform best within the ℓp attacks by having around 90% misclassification rate even for very small epsilon. fig. 1. average misclassification and targeted misclassification rates for different ϵ values under ℓ∞ and wasserstein (fw) norms. furthermore, we can see that for the fgsm attack, the targeted misclassification does not increase monotonically. the reason for this can be that since the fgsm is not an iterative algorithm and performs just one step, it overshoots when the epsilon is too big and misses the target class. the fw+dual lmo attack performs best in terms of both misclassification and targeted misclassification rates. nevertheless, we cannot compare the amount of perturbation under ℓ∞ and wasserstein norms, since they imply different amounts of changes to the image. hence, we will need to combine these results with the visual ones to be able to make a fair comparison. in table 2, we can see the time duration needed to generate an adversarial example. being a simple one-step attack, fgsm leads the competition followed by the pgd and autopgd 4.2 duration 48 a brief comparison between white box, targeted adversarial attacks in deep neural networks table 1: average misclassification and average targeted misclassification rates for different norms attack norm miscl targ. miscl. autopgd ℓ1 0.0832 0.1100 pgd ℓ1 0.0810 0.1086 fgsm ℓ1 0.0879 0.1116 autopgd ℓ2 0.8968 0.9977 fgsm ℓ2 0.6157 0.3914 pgd ℓ2 0.8927 0.9925 autopgd ℓ∞ 0.9000 1.0000 fgsm ℓ∞ 0.9149 0.5515 pgd ℓ∞ 0.9022 1.0000 fw was 0.9000 1.0000 attacks. pgd, which performs much better than fgsm in terms of targeted misclassification rate, is around 71 times slower. the slowest is the fw + dual lmo attack, which performs around 400 times slower than the fgsm attack. in table 2, we can see the time duration needed to generate an adversarial example. being a simple one-step attack, fgsm leads the competition followed by the pgd and autopgd attacks. pgd, which performs much better than fgsm in terms of targeted misclassification rate, is around 71 times slower. the slowest is the fw + dual lmo attack, which performs around 400 times slower than the fgsm attack. table 2: duration of generating an adversarial example in seconds. fgsm pgd autopgd fw+dual lmo 0.7 50 87 338 one of the most important aspects of adversarial attacks is that they should be undetected by the human eye. hence, in this section, we study how detectable are the adversarial samples generated by the attacks. to visualize the results, we chose the smallest ϵ for each of our attacks, under which our model showed at least 80% misclassification. you can see the visualizations in the figures 2 and 3. we can see that in the examples generated by the fgsm attack, although the original image is still well visible, the perturbation is easily detectable to us. for pgd, autopgd, and fw + dual lmo attacks, however, the perturbations are 4.3 duration 4.4 imperceptibility g. bezirganyan and h. sergoyan 49 hardly visible. in fact, from fig. 3 it is noticeable that pgd and autopgd attacks apply small perturbations uniformly over the image. while the fw + dual lmo attack perturbs only small portions of the image, the perturbations are much more visible. fig. 2. adversarial samples on an image with original label 4 (deer). fig. 3. perturbations added to the image with original label 4 (deer). 5. conclusion and future work we compared different attack methods with different metrics. the champion of the comparison is the pgd attack. although being a very simple attack, it performs very well in terms of misclassification and targeted misclassification rates, is fast, and is almost nondetectable by the human eye in our experiments. autopgd, while yielding similar results, is much slower, and hence, comes in second place in our comparison. fw + dual lmo attack performed very well in terms of duration, misclassification, and targeted misclassification 50 a brief comparison between white box, targeted adversarial attacks in deep neural networks rates, but the perturbations were much more noticeable. the fgsm attack was the fastest with a high misclassification rate but came last in terms of imperceptibility. since we’ve covered only a small portion of attacks, we plan to extend the attack list by adding more well-known or state-of-the-art methods and extend the experiment domain to black-box attacks as well. furthermore, we plan to test these attacks on a defended model and compare their performances. particularly, we are interested to see the difference between autopgd and pgd attacks on a defended model. references [1] i. j. goodfellow, j. shlens, and c. szegedy, “explaining and harnessing adversarial examples,” in 3rd international conference on learning representations, iclr 2015 conference track proceedings, 2015. [2] q. li, y. guo, and h. chen, “practical no-box adversarial attacks against dnns,” preproceedings advances in neural information processing systems, vol. 33, 2020. [3] a. kurakin, i. j. goodfellow, and s. bengio, “adversarial machine learning at scale,” in 5th international conference on learning representations, iclr 2017 conference track proceedings, 2017. [4] f. croce and m. hein, “reliable evaluation of adversarial robustness with an ensemble of diverse parameter-free attacks,” arxiv preprint arxiv:2003.01690, 2020. [5] k. wu, a. wang, and y. yu, “stronger and faster wasserstein adversarial attacks,” in international conference on machine learning, pp. 10377–10387, pmlr, 2020. [6] e. wong, f. r. schmidt, and j. zico kolter, “wasserstein adversarial examples via projected sinkhorn iterations,” in 36th international conference on machine learning, icml 2019, 2019. [7] m. cuturi, “sinkhorn distances: lightspeed computation of optimal transport,” advances in neural information processing systems, vol. 26, pp. 2292–2300, 2013. [8] m. frank, p. wolfe, et al., “an algorithm for quadratic programming,” naval research logistics quarterly, vol. 3, no. 1-2, pp. 95–110, 1956. [9] k. he, x. zhang, s. ren, and j. sun, “deep residual learning for image recognition,” in proceedings of the ieee computer society conference on computer vision and pattern recognition, 2016. [10] a. krizhevsky, g. hinton, et al., “learning multiple layers of features from tiny images,” 2009. [11] m.-i. nicolae, m. sinn, m. n. tran, b. buesser, a. rawat, m. wistuba, v. zantedeschi, n. baracaldo, b. chen, h. ludwig, i. molloy, and b. edwards, “adversarial robustness toolbox v1.2.0,” corr, vol. 1807.01069, 2018. g. bezirganyan and h. sergoyan 5 1 êåçï³ï ïáõ÷áí, ãçñ³ë³íáñí³í ùñó³ïó³ûçý ñ³ñó³ïáõùý»ñç ñ³ù³éáï ñ³ù»ù³ïáõãûáõýá ëáñá ý»ûñáý³ûçý ó³ýó»ñáõù ¶ñç·áñ ´»½çñ·³ýû³ý ¨ ð»ýñçï ê»ñ·áû³ý øûáõýë»ýç î»ëýçï³ï³ý ð³ù³éë³ñ³ý e-mail: grigor.bezirganyan@tum.de, henrik.sergoyan@tum.de ²ù÷á÷áõù êðàòêîå ñðàâíåíèå ìåæäó ”áåëûì ÿùèêîì”, öåëåâûìè ñîñòÿçàòåëüíûìè àòàêàìè ïðîòèâíèêà â ãëóáîêèõ íåéðîííûõ ñåòÿõ ãðèãîð áåçèðãàíÿí è ãåíðèê ñåðãîÿí òåõíè÷åñêèé óíèâåðñèòåò ìþíõåíà e-mail: grigor.bezirganyan@tum.de, henrik.sergoyan@tum.de àííîòàöèÿ ñåãîäíÿ íåéðîííûå ñåòè èñïîëüçóþòñÿ â ðàçëè÷íûõ îáëàñòÿõ, â áîëüøèíñòâå èç êîòîðûõ âàæíî èìåòü íàäåæíûé è ïðàâèëüíûé âûâîä. âîò ïîýòîìó ñîñòÿçàòåëüíûå àòàêè äåëàþò ãëóáîêèå íåéðîííûå ñåòè ìåíåå íàäåæíûìè äëÿ èñïîëüçîâàíèÿ â îáëàñòÿõ, ãäå áåçîïàñíîñòü èìååò ðåøàþùåå çíà÷åíèå. ñëåäîâàòåëüíî, âàæíî èçó÷èòü ïîòåíöèàëüíûå ìåòîäû àòàêè, ÷òîáû èìåòü âîçìîæíîñòü ðàçðàáàòûâàòü ãîðàçäî áîëåå íàäåæíûå ñåòè. â ýòîé ñòàòüå ìû ðàññìàòðèâàåì ÷åòûðå ”áåëûõ ÿùèêà” öåëåíàïðàâëåííûå ñîñòÿçàòåëüíûå àòàêè è ñðàâíèâàåì èõ ñ òî÷êè çðåíèÿ ÷àñòîòû îøèáî÷íûõ êëàññèôèêàöèé, ÷àñòîòû öåëåâûõ îøèáî÷íûõ êëàññèôèêàöèé, äëèòåëüíîñòè àòàêè è íåçàìåòíîñòè. íàøà öåëü íàéòè àòàêè, êîòîðûå áûëè áû ýôôåêòèâíû è ãåíåðèðîâàëè áû ñîñòÿçàòåëüíûå âûáîðêè ñ íåáîëüøèìè âîçìóùåíèÿìè è íå îáíàðóæèâàëèñü áû ÷åëîâå÷åñêèì ãëàçîì. êëþ÷åâûå ñëîâà: ñîñòÿçàòåëüíûå àòàêè, íàäåæíîñòü, ìàøèííîå îáó÷åíèå, ãëóáîêîå îáó÷åíèå. ²ûëûñ ý»ûñáý³ûçý ó³ýó»ñý û·ï³·áñííáõù »ý ï³ñµ»ñ ³ëå³ñ»½ý»ñáõù, áñáýóçó ß³ï»ñáõù ï³ñ¨áñ ¿ áõý»ý³é ñáõë³éç ¨ ×ß·ñçï ³ñ¹ûáõýù: ²ñ³ ã» çýãáõ ùñó³ïó³ûçý ñ³ñó³ïáõùý»ñá ý»ûñáý³ûçý ó³ýó»ñá ¹³ñóýáõù »ý ³í»éç ùçã ñáõë³éç‘ µ³ñóñ ³ýíï³ý·áõãû³ý ù³ï³ñ¹³ï å³ñ³ýçáõ ïçñáõûãý»ñáõù: ð»ï¨³µ³ñ, ï³ñ¨áñ ¿ áõëáõùý³ëçñ»é ñ³ñó³ïù³ý ñý³ñ³íáñ ù»ãá¹ý»ñá` ³í»éç ï³ûáõý ¨ ³ýíï³ý· ó³ýó»ñ ùß³ï»éáõ ñ³ù³ñ: ²ûë ñá¹í³íáõù ù»ýù ùýý³ñïáõù »ýù ãáñë ëåçï³ï?ïáõ÷áí, ãçñ³ë³íáñí³í ùñó³ïó³ûçý ñ³ñó³ïáõùý»ñ ¨ ñ³ù»ù³ïáõù ¹ñ³ýù çñ»ýó ëë³é ¹³ë³ï³ñ·ù³ý ³ëïç׳ýç, ýå³ï³ï³ûçý ëë³é ¹³ë³ï³ñ·ù³ý ³ñ³·áõãû³ý ³ëïç׳ýç, ñ³ñó³ïù³ý ï¨áõáõãû³ý ¨ ³ýýï³ï»éçáõãû³ý ³éáõùáí: ø»ñ ýå³ï³ïý ¿ ·ïý»é ñ³ñó³ïáõù(ý»ñ), áñáýù ³ñ¹ûáõý³í»ï ïéçý»ý ¨ ïëï»õí»ý ùñó³ïó³ûçý ûñçý³ïý»ñ‘ ÷áùñ ß»õáõùý»ñáí ¨ ³ýýï³ï»éç ù³ñ¹áõ ³ãùç ñ³ù³ñ: ´³ý³éç µ³é»ñ` ùñó³ïó³ûçý ñ³ñó³ïáõùý»ñ, ï³ûáõýáõãûáõý, ù»ù»ý³û³ï³ý áõëáõóáõù, ëáñá áõëáõóáõù: 04_sergoyan_33__52__copy_ 04 mathematical problems of computer science 59, 35–44, 2023. doi: 10.51408/1963-0100 udc 004.75 data compression-aware performance analysis of dask and spark for earth observation data processing arthur g. lalayan institute for informatics and automation problems of nas ra, yerevan, armenia national polytechnic university of armenia, yerevan, armenia e-mail: arthurlalayan97@gmail.com abstract high-performance computing is a good choice for handling big earth observation data, allowing the processing of the data in a distributed and performance-efficient way using in-memory computing frameworks. the data compression technique reduces the amount of storage and network transfer time and improves processing performance. the article aims to investigate the effectiveness of widely used distributed data processing frameworks in conjunction with lossless data compression techniques, to find the optimal compression method and processing framework for specific earth observation workflows. normalized difference vegetation index has been evaluated for the territory of armenia, obtaining data from the sentinel satellite and considering the supported compression methods to compare the performance of in-memory dask and spark frameworks. experiments show that the zstandard compression method and the dask framework are the best choices for such workflows. keywords: earth observation, hpc, spark, dask, distributed computing, data compression. article info: received 29 january 2022; sent for review 7 february 2023; received in revised form 15 march 2023; accepted 17 april 2023. acknowledgement: the research was supported by the science committee of the republic of armenia and the university of geneva leading house by the projects entitled self-organized swarm of uavs smart cloud platform equipped with multi-agent algorithms and systems (nr. 21ag-1b052), remote sensing data processing methods using neural networks and deep learning to predict changes in weather phenomena (nr. 21scbrffr-1b009), and adc4sd: armenian data cube for sustainable development. 1. background and motivation earth observation (eo) satellite data are necessary for environmental monitoring and gathering vital information about various earth layers [1]. specifically, eo data are widely used 35 36 data compression-aware performance analysis of dask and spark for earth obser. data processing to monitor the atmosphere including air pollution [2] and temperature [3], the oceans considering sea pollution and ocean acidity [4], and ground, such as deforestation [5] and forest fire [6], as well as to detect climatic changes [7]. to facilitate work with eo data, australian researchers [8] have provided an open-source open data cube (odc) [9], which is deployed and widely used by several communities from different countries, including armenia [10]. nevertheless, the odc communities still encounter the big eo data processing challenge requiring high-performance computational (hpc) resources. for instance, the sentinel-2 satellite [11] provides approximately 200-300 gb, 3 tb, and 36 tb of daily, monthly, and annual data for the territory of armenia. handling this amount of data is a complex task. therefore, hpc is the right choice to improve data processing performance using distributed computing techniques. thus, the big eo data processing obstacle is coping with using open-source apache spark [12] and dask [13] frameworks, which can process data in parallel by dividing them into chunks, processing them in a distributed way using computational clusters, and aggregating the result. both frameworks have master-slave architecture, where slave nodes are worker nodes executing functions in parallel, and the master node is the driver or scheduler to manage them. spark ecosystem supports many projects in data streaming, sql analytics, and machine learning. spark is a multi-language engine that processes and analyzes data, while dask is a python library. therefore, spark has its ecosystem apis and memory models, while dask uses them from the python ecosystem. however, these frameworks have some differences and limitations in finding an optimal solution for eo data processing workflows. besides using hpc, the format of eo satellite images also has a crucial influence on performance. the data compression techniques can reduce storage usage and the number of i/o operations, improving processing performance. recent studies [14, 15] show that compression methods combined with hpc can significantly enhance the performance of big data workflows. one of the optimal satellite image formats is cloud optimized geotiff (cog) [16], which provides essential advantages compared to traditional formats, such as netcdf [17]. cog format provides an http range request to extract a part of the data. hence, when extracting eo data using cog, there is no need to download the entire image and then extract the area of interest as in the netcdf format. besides the mentioned benefit, both cog and netcdf formats support data compression methods. several studies [18, 19, 20] evaluate and compare the performance of the frameworks for particular cases, such as data-intensive neuroimaging pipelines [18], different applications of molecular dynamics [20], and scientific image analytics [19]. nevertheless, they did not consider performance-tuning techniques, such as data compression. the main objective of the article is to investigate the efficacy of widely used distributed data processing frameworks, such as dask and spark, in combination with lossless data compression methods, to enhance the performance of eo data processing. the methodology involved evaluating the approach on the armenian hybrid research computing platform, and the results obtained from the evaluation could be used by eo communities to make informed decisions about improving their data processing performance. 2. methodology a test-bed platform for eo data processing has been deployed to execute eo data processing functions and compare the performances in spark and dask. the platform is a container-based solution within the kubernetes system, enabling evaluating and comparing a. lalayan 37 the environments’ performance. it relies on the computational resources of the armenian hybrid research computing platform [21]. fig. 1 shows the architecture of the experimental platform. fig. 1. test-bed platform based on spark and dask. as the figure shows, each node scheduler/driver or worker/executor corresponds to a pod in kubernetes with some fixed computational resources. it is possible to configure the computational resource characteristics of nodes with kubernetes api. the jupyter notebook [22] corresponds to the frontend of the spark and dask cluster backend. it connects to dask and spark of master nodes, configures environments by providing the number of worker nodes and computational resources for each node, requests to process eo data using dask and spark clusters, and visualizes the output. dask and spark clusters fetch data from repositories of either local armenian datacube [23] or global eo data providers. armenian datacube [10] provides data from landsat 5, 7, 8 [24], and sentinel-2 satellites, and one of the global eo data providers is sentinel-2 cloud-optimized geotiffs [25]. the functionality evaluation of the dask and spark frameworks is quite interesting. dask is a flexible python library, which makes it easy to migrate and execute the oldwritten python code in a distributed manner. moreover, python is widely used in eo data workflows, and various useful libraries provide vital tools to make the work with eo data easier. however, working with eo data in spark is a little tricky because the execution of the old-written codes in the spark environment is impossible, as it supports apis of its ecosystem, therefore, the code adjustment is inevitable. the geopyspark library [26] makes working with eo data somewhat easier in spark. so the data processing function can be easily parallelized only in dask, considering the limitations and complexity of using spark. as eo data processing applications, the normalized difference vegetation index (ndvi) [27] was evaluated during the experiments, which provides information for monitoring the health of the vegetation. the formula of the index is presented in (1). ndv i = nir − red nir + red , (1) where red is the red band, and nir is the near-infrared band. all bands and the calculation result are matrices or images and the ndvi index is calculated from sentinel-2 satellite images. 38 data compression-aware performance analysis of dask and spark for earth obser. data processing several experiments were conducted with different parameters to evaluate the performances of dask and spark using the developed experimental platform. table 1 presents all parameters and their values. table 1: experimental parameters and their values. parameter name possible values environment dask and spark input data sizes 16, 32, 64 gbs number of workers 4, 8, 16, 32 applications ndvi compression methods none, deflate, lzw, packbits, and zstandard 3. experimental results data compression techniques reduce the actual size of data, resulting in savings in storage space, providing faster network transmission times, and improving the performance of processing. eo data repositories, which provide satellite images in cog format, such as sentinel-2 cogs, by default, use deflate compression method to reduce the downloading time of satellite images and save some storage space. besides the deflate method, several compression methods, either lossy or lossless, could be applied with cogs. the accuracy of the satellite image is essential, as the spatial resolution of the sentinel-2 image is 10m [10], which corresponds to the surface area measured on the ground represented by each pixel. therefore, the compression methods used for optimization should be lossless to ensure accurate results. the cog format supports several lossless compression methods, such as deflate [28], lzw [29], packbits [30], and zstandard [31]. eo band tiles come in three different sizes (light, medium, and heavy) by which the compression factor is estimated to understand the average compression ratio of the method. the light band tiles (coastal, water vapor, etc.) usually have up to 5-10 mb size, medium 50-70 mb (short-wave infrared (swir), vegetation red edge, etc.), and heavy 200-250 mb (red, nir, etc.). they consider all types of possible lossless compression methods. the compression ratio is calculated for each method by dividing the compressed data size by the original uncompressed data size. the compression ratios for various compression methods are presented in fig. 2. the figure shows that the best compression factor is provided by the zstandard method, whereas the worst one is provided by the packbits method. zstandard codec compresses the band image more than the deflate does, which is by default used by the sentinel-2 cogs repository. therefore, using zstandard instead of deflate will lead to more storage savings, and less network transfer time and i/o operations. the storage reduction, in this case, is 34 % compared with the uncompressed data and 16 % compared with deflate. the compression ratio of the packbits method for the heavy tiles is close to 1, which means that the method is useless for data size reduction since the actual size and compressed data size will be the same. besides the storage saving, further data processing is also essential, as a. lalayan 39 fig. 2. compression ratio of deflate, lzw, packbits, and zstandard methods for light, medium, and heavy tiles. high compression needs more cpu time to decompress into memory before processing. the majority of the time spent in computing ndvi is devoted to transferring satellite images over the network and loading them into memory, rather than performing calculations using the cpu. the comparison of the performances of dask and spark, considering different sizes of input data, compression methods, and 32 worker nodes is shown in fig. 3. the execution time of the cog tile compressed with the packbits method and without compression is almost the same, as packbits provides weak compression; thus, it uses little cpu time for decompression. the worst performance for both environments from the possible compression methods is deflate, whereas the best one is zstandard. hence, the best compression method for satellite images in cog format is zstandard, as it provides the highest compression ratio and optimal memory loading time. the performance improvement when using zstandard compared to uncompressed mode is achieved by reducing network transfer time. zstandard provides on average 2.15 and 1.82 times faster execution time compared with the uncompressed mode, approximately 4.72 and 3.99 times faster than the default selected deflate method provided by global satellite image repositories correspondingly for dask and spark environments. performance evaluation using dask and spark is quite interesting. for the default used deflate compression method provided by eo repositories, spark and dask show similar execution times; however, spark is a bit faster. the lzw compression method for the dask environment is better than deflate but worse than without compressing or compressing with zstandard. also, spark does not support the compression method. with uncompressed data, dask is faster than spark for 16 gb input, whereas, in cases of 32 gb and 64 gb, spark is faster. performance in dask using the zstandard compression method is an optimal choice. 40 data compression-aware performance analysis of dask and spark for earth obser. data processing fig. 3. comparison of dask and spark considering 16, 32, 64 gbs of input data and compression methods. 4. discussion the study showed that various data compression methods could reduce storage requirements and network transfer time at different scales. moreover, compressed data processing using multiple techniques in distributed environments such as spark and dask exhibited other execution times, with some compression methods outperforming uncompressed data processing time. the study aims to determine the optimal data compression method that balances performance and storage savings in the chosen distributed processing environments. the evaluation shows that the dask and zstandard combination is the best choice for the environment and compression method for eo satellite images. it provides the highest compression factor and performance compared to other supported compression methods. the armenian datacube was initially set up with a 2-terabyte storage capacity, which is limited. to manage this, only the essential bands for specific eo applications that researchers are interested in during a particular period are downloaded and stored. if the storage capacity is exceeded, the options are to scale vertically or add external storage. the zstandard compression technique was used in experiments to conserve 34 % of storage. this allows more data to be stored in the allocated datacube space. the zstandard compression method combined with the dask environment offers benefits such as improved data storage efficiency and eo data processing time. however, additional steps are required to achieve these benefits, such as converting analysis-ready data from the datacube to cloud optimized geotiff format and compressing them using the zstandard method. although this may increase the total execution time of downloading and preprocessing, it provides such benefits as enhanced processing time and storage savings. moreover, this efficient method of storing compressed data can be applied to other types of eo data repositories and datacubes. a. lalayan 41 in conclusion, data compression methods can effectively reduce the amount of eo data stored and improve processing performance. zstandard exhibits the best performance and storage efficiency for eo data among the available compression methods. additionally, the implementation of the dask environment speeds up distributed processing. 5. conclusion the study evaluates the performance of eo data processing in dask and spark, considering compression methods. experimental results show that dask and spark provide similar data processing performances. the mixture of the dask and zstandard compression methods is optimal, as the compression method provides the best compression factor of all possible lossless compression methods. it reduces the amount of used storage by 16 % and speeds up execution times by 4.72x and 3.99x in dask and spark, correspondingly compared with the deflate method, which is used by default from the eo data repositories. in further work, it is planned to store the data in armenian datacube compressed with the zstandard method and use the dask environment for data processing. references [1] o. r. young, m. onoda. “satellite earth observations in environmental problemsolving”, in book: satellite earth observations and their impact on society and policy, pp. 3-27, 2017. 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[3 1 ] y . co lle t , m. k u c h e r a wy, \ zs t a n d a r d co m p r e s s io n a n d t h e 'a p p lic a t io n / z s t d ' me d ia typ e " , r f c e ditor, usa, fe b r u a r y 2 0 2 1 . dask-ç ¨ spark-ç ï³ï³ñáõ³ï³ýç í»ñéáõíáõãûáõý` ñ³ßíç ³éý»éáí ïíû³éý»ñç ë»õùáõùá ºñïñç ¹çï³ñïù³ý ïíû³éý»ñç ùß³ïù³ý ñ³ù³ñ ²ñãáõñ ¶. è³é³û³ý ðð ¶²² æýýáñù³ïçï³ûç ¨ ³íïáù³ï³óù³ý åñáµé»ùý»ñç çýëïçïáõï, ºñ¨³ý, ð³û³ëï³ý e-mail: arthurlalayan97@gmail.com ²ù÷á÷áõù ´³ñóñ ï³ï³ñáõ³ï³ý ñ³ßí³ñïá é³í áýïñáõãûáõý ¿ »ñïñç ¹çï³ñïù³ý ù»í ïíû³éý»ñç ùß³ïù³ý ñ³ù³ñ, çýãá ãáõûé ¿ ï³éçë ïíû³éý»ñç ùß³ïáõùá µ³ßëí³í ¨ µ³ñóñ ³ñ¹ûáõý³í»ïáõãû³ùµ? û·ï³·áñí»éáí ñçßáõáõãû³ý ù»ç ñ³ßíáõ³ï³ý ñ³ñã³ïý»ñ: îíû³éý»ñç ë»õùù³ý ï»ëýáéá·ç³ý ýí³½»óýáõù ¿ å³ñ³ýçíáõ å³ñ»ëï³íáñù³ý í³í³éá ¨ ó³ýóç ÷áë³ýóù³ý å³ù³ý³ïá, çýãå»ë ý³¨ µ³ñ»é³íáõù ¿ ïíû³éý»ñç ùß³ïù³ý å³ù³ý³ïá: ðá¹í³íç ýå³ï³ïý ¿ áõëáõùý³ëçñ»é é³ûýáñ»ý û·ï³·áñííáõ ïíû³éý»ñç ùß³ïù³ý ßñç³ý³ïý»ñç ³ñ¹ûáõý³í»ïáõãûáõýá‘ ïíû³éý»ñç ³ýïáñáõëï ë»õùù³ý ï»ëýçï³ûç ñ»ï ñ³ù³ï»õ, ºñïñç ¹çï³ñïù³ý ñ³ïáõï ³ßë³ï³ýù³ûçý ñáëù»ñç ñ³ù³ñ ë»õùù³ý ûåïçù³é ù»ãá¹ ¨ ùß³ïù³ý ßñç³ý³ï ·ïý»éáõ ñ³ù³ñ: ´áõë³ï³ýáõãû³ý ýáñù³é³óí³í ï³ñµ»ñáõãû³ý çý¹»ùëá ·ý³ñ³ïí»é ¿ ð³û³ëï³ýç ï³ñ³íùç ñ³ù³ñ` û·ï³·áñí»éáí sentinel ³ñµ³ýû³ïç ïíû³éý»ñá ¨ ñ³ßíç ³éý»éáí ë»õùù³ý ³ç³ïóíáõ ù»ãá¹ý»ñá ñçßáõáõãû³ý ù»ç dask ¨ spark ßñç³ý³ïý»ñç ³ßë³ï³ýùç ñ³ù»ù³ïù³ý ñ³ù³ñ: öáñó»ñá óáõûó »ý ï³éçë, áñ zstandard ë»õùù³ý ù»ãá¹á ¨ dask ùçç³í³ûñá é³í³·áõûý áýïñáõãûáõýý »ý ýù³ý ³ßë³ï³ýù³ûçý ñáëù»ñç ñ³ù³ñ: ´³ý³éç µ³é»ñ` ºñïñç ¹çï³ñïáõù, hpc, spark, dask, µ³ßëí³í ñ³ßí³ñï, ïíû³éý»ñç ë»õùáõù: 4 4 data compression-aware performance analysis of dask and spark for earth observation data processing àíàëèç ïðîèçâîäèòåëüíîñòè dask è spark äëÿ îáðàáîòêè äàííûõ íàáëþäåíèÿ çåìëè ñ ó÷åòîì ñæàòèÿ äàííûõ àðòóð ã. ëàëàÿí èíñòèòóò ïðîáëåì èíôîðìàòèêè è àâòîìàòèçàöèè íàí ðà, åðåâàí, àðìåíèÿ e-mail: arthurlalayan97@gmail.com àííîòàöèÿ âûñîêîïðîèçâîäèòåëüíûå âû÷èñëåíèÿ ÿâëÿþòñÿ õîðîøèì âûáîðîì äëÿ îáðàáîòêè áîëüøèõ äàííûõ íàáëþäåíèÿ çåìëè, ïîçâîëÿÿ îáðàáàòûâàòü äàííûå ðàñïðåäåëåííûì è âûñîêîïðîèçâîäèòåëüíûì ñïîñîáîì ñ èñïîëüçîâàíèåì âû÷èñëèòåëüíûõ ïëàòôîðì â ïàìÿòè. òåõíîëîãèÿ ñæàòèÿ äàííûõ ñîêðàùàåò îáúåì õðàíèëèùà è âðåìÿ ïåðåäà÷è ïî ñåòè è ïîâûøàåò ïðîèçâîäèòåëüíîñòü îáðàáîòêè. öåëüþ ñòàòüè ÿâëÿåòñÿ èññëåäîâàíèå ýôôåêòèâíîñòè øèðîêî èñïîëüçóåìûõ ñèñòåì ðàñïðåäåëåííîé îáðàáîòêè äàííûõ â ñî÷åòàíèè ñ ìåòîäàìè ñæàòèÿ äàííûõ áåç ïîòåðü, ÷òîáû íàéòè îïòèìàëüíûé ìåòîä ñæàòèÿ è ñòðóêòóðó îáðàáîòêè äëÿ êîíêðåòíûõ ðàáî÷èõ ïðîöåññîâ íàáëþäåíèÿ çåìëè. íîðìàëèçîâàííûé ðàçíîñòíûé èíäåêñ ðàñòèòåëüíîñòè áûë îöåíåí äëÿ òåððèòîðèè àðìåíèè ñ èñïîëüçîâàíèåì äàííûõ ñî ñïóòíèêà sentinel è ñ ó÷åòîì ïîääåðæèâàåìûõ ìåòîäîâ ñæàòèÿ äëÿ ñðàâíåíèÿ ïðîèçâîäèòåëüíîñòè ôðåéìâîðêîâ dask è spark â ïàìÿòè. ýêñïåðèìåíòû ïîêàçûâàþò, ÷òî ìåòîä ñæàòèÿ zstandard è ôðåéìâîðê dask ÿâëÿþòñÿ íàèëó÷øèì âûáîðîì äëÿ òàêèõ ðàáî÷èõ ïðîöåññîâ. êëþ÷åâûå ñëîâà: íàáëþäåíèå çåìëè, hpc, spark, dask, ðàñïðåäåëåííûå âû÷èñëåíèÿ, ñæàòèå äàííûõ. 04_artur_59 04 d:\user\sbornik_38_pdf\39.dvi mathematical problems of computer science 38, 91{92, 2012. on the for malization of scienti¯c t heor ies ig o r d . za s la vs ky institute for informatics and automation problems of nas of ra e-mail: zaslav@ipia.sci.am th e g e n e r a l c o n c e p t o f t h e fo r m a liz a t io n is c o n s id e r e d . five le ve ls o f t h e fo r m a liz a t io n o f s c ie n t i¯ c t h e o r ie s a r e n o t e d a n d d e s c r ib e d . th e g e n e r a l c o n c e p t o f t h e fo r m a liz a t io n is c o n n e c t e d wit h t h e s ys t e m a t iz a t io n a n d t h e s p e c i¯ c a t io n o f h u m a n kn o wle d g e . th e fo r m a liz a t io n o f s o m e b r a n c h o f kn o wle d g e is d e ¯ n e d a s a r e p r e s e n t a t io n o f kn o wle d g e in t h is b r a n c h b y m e a n s o f a n o t h e r b r a n c h wh e n t h is r e p r e s e n t a t io n is c o n n e c t e d wit h s ys t e m a t iz a t io n a n d s p e c i¯ c a t io n o f kn o wle d g e ; a s a r u le s u c h a r e p r e s e n t a t io n is a p p r o xim a t ive a n d e s t r a n g e d fr o m t h e in it ia l b r a n c h . th e fo r m a liz a t io n o f d i®e r e n t b r a n c h e s o f s c ie n c e is t yp ic a l fo r s u c h s c ie n c e s a s , fo r e xa m p le , m a t h e m a t ic s a n d p h ys ic s , b u t it t a ke s p la c e a ls o in t h e h u m a n it ie s . fo r e xa m p le , t h e c r im in a l a n d t h e c ivil la w m a y b e c o n s id e r e d a s a fo r m a liz a t io n o f t h e h u m a n m o r a ls . th e wr it t e n n a t u r a l la n g u a g e m a y b e c o n s id e r e d a s a fo r m a liz a t io n o f t h e o r a l o n e . w e s h a ll s t u d y s o m e le ve ls in t h e fo r m a liz a t io n o f s c ie n t i¯ c t h e o r ie s . b e lo w ¯ ve le ve ls o f s u c h kin d will b e d e s c r ib e d . th e ¯ r s t o f t h e m is t h e \descriptional" le ve l. th e c r e a t io n o f a n y s c ie n t ī c t h e o r y b e g in s fr o m t h e o b s e r va t io n o f fa c t s . a c o lle c t io n o f t h e o b s e r ve d fa c t s is t h e c o n t e n t o f t h e d e s c r ip t io n a l le ve l. it g ive s a b a s is fo r t h e d e ve lo p m e n t o f t h e t h e o r y in fu t u r e . on t h is le ve l o n ly t h e c o lle c t io n o f fa c t s is p r e s e n t ; t h e t h e o r y ( in t h e o wn s e n s e o f t h is t e r m ) ye t d o e s n o t e xis t . th e s e c o n d le ve l m a y b e c h a r a c t e r iz e d a s a \linguistic" o n e . it is p e r fo r m e d wh e n t h e o b s e r ve d fa c t s a r e n u m e r o u s a n d m u lt ifo r m , a n d it is n e c e s s a r y t o s ys t e m a t iz e t h e m a n d t o in t r o d u c e s o m e s ys t e m o f n o t io n s g ivin g a c la s s ī c a t io n o f t h e o b s e r ve d fa c t s a n d o f t h e r e la t io n s b e t we e n t h e m . on t h is le ve l t h e language o f t h e t h e o r y is c r e a t e d . th e t h ir d le ve l m a y b e c h a r a c t e r iz e d a s a n \intuitive logical" o n e . it is p e r fo r m e d wh e n it is n e c e s s a r y t o s p e c ify a n d s ys t e m a t iz e t h e s t a t e m e n t s o f t h e t h e o r y a n d t o e s t a b lis h r e la t io n s b e t we e n t h e m . on t h e h ig h e s t s t a g e o f t h is le ve l s o m e c e n t r a l s t a t e m e n t s o f t h e t h e o r y s o m e t im e s a r e fo r m u la t e d s u c h t h a t a ll t h e s t a t e m e n t s in t h e c o n s id e r e d t h e o r y c a n b e d e d u c e d fr o m t h e m . s u c h c e n t r a l s t a t e m e n t s b e a r d i®e r e n t n a m e s in d i®e r e n t s c ie n c e s . th e y a r e , fo r e xa m p le , axioms a n d postulates in t h e g e o m e t r y, la ws ( fo r e xa m p le , n e wt o n 's la ws ) in t h e p h ys ic s . s o m e t im e s s u c h c e n t r a l s t a t e m e n t s a r e equations ( fo r e xa m p le , ma xwe ll's e qu a t io n s in t h e e le c t r o d yn a m ic s , s c h r yd in g e r 's e qu a t io n in t h e qu a n t u m m e c h a n ic s ) . b u t t h e m e t h o d s o f lo g ic a l d e d u c t io n o n t h is le ve l a r e t yp ic a l o n ly fo r a c o n s id e r e d t h e o r y; we h a ve , fo r e xa m p le , t h e \ g e o m e t r ic a l t h in kin g " in t h e g e o m e t r y, t h e \ m e c h a n ic a l t h in kin g " in t h e m e c h a n ic s . th e fo u r t h le ve l o f t h e fo r m a liz a t io n m a y b e c h a r a c t e r iz e d a s a \formal logical" o n e . it is p e r fo r m e d wh e n t h e m e t h o d s o f t h e in t u it ive lo g ic a l t h in kin g u s e d o n t h e p r e vio u s le ve l 9 1 9 2 on the formalization of scienti¯c theories b e c o m e n o t s a t is fa c t o r y fr o m t h e p o in t o f vie w o f t h e e xa c t n e s s a n d t h e r e lia b ilit y o f t h e s t a t e m e n t s o b t a in e d b y t h e d e d u c t io n . th e in t u it ive lo g ic a l d e d u c t io n is r e p la c e d o n t h is le ve l b y t h e fo r m a l lo g ic a l o n e . fo r e xa m p le , t h e p a s s in g o f t h e g e o m e t r y t o t h e fo r t h le ve l wa s p e r fo r m e d in 1 9 -t h c e n t u r y [1 ]. th e ¯ ft h le ve l o f t h e fo r m a liz a t io n m a y b e c h a r a c t e r iz e d a s a \syntactical" o n e . it is p e r fo r m e d wh e n t h e fo r m a l lo g ic a l d e d u c t io n is r e d u c e d t o t r a n s fo r m a t io n s o f c o m b in a t io n s o f fo r m a l s ym b o ls ( d e ¯ n e d o n ly b y t h e ir fo r m a l s t r u c t u r e ) . s u c h a le ve l is e s s e n t ia l fr o m t h e p o in t o f vie w o f s o m e d ir e c t io n s in t h e m a t h e m a t ic s ( fo r e xa m p le , fo r m a lis m [2 ]) ; it is e s s e n t ia l a ls o fo r a p p lic a t io n s o f t h e c o m p u t e r s c ie n c e t o t h e in ve s t ig a t io n s o f p e c u lia r it ie s o f t h e h u m a n t h in kin g . th e p r e d ic a t e c a lc u lu s [3 ] g ive s a g e n e r a l m e t h o d fo r p a s s in g t o t h e ¯ ft h le ve l fr o m t h e fo u r t h o n e . s o m e h ig h e r le ve ls o f t h e fo r m a liz a t io n m a y a ls o b e c o n s id e r e d , b u t t h e y a r e ye t n o t im p le m e n t e d , a n d we d o n o t c o n s id e r t h e m h e r e . r eferences 1 . d a vid h ilb e r t . gr u n d la g e n d e r ge o m e t r ie , s ie b e n t e a u ° a g e , l e ip z ig u n d b e r lin , 1 9 3 0 . 2 . d . h ilb e r t u n d p . b e r n a ys . gr u n d la g e n d e r ma t h e m a t ik i, zwe it e a u ° a g e , s p r in g e r v e r la g , b e r lin -h e id e lb e r g -n e w y o r k, 1 9 6 8 . 3 . h . e n d e r t o n . a ma t h e m a t ic a l in t r o d u c t io n t o l o g ic , 2 n d e d ., s a n d ie g o , h a r c o u r t , a c a d e m ic p r e s s , 2 0 0 1 . d:\sbornik\...\onmultiple.dvi mathematical problems of computer science 23, 2004, 36{46. on m ultiple h ypotheses t esting by i nfor med statistician for ar bitr ar ily var ying object and application to sour ce coding¤ e vg u e n i a . h a r o u t u n ia n a n d p a r a n d z e m m. h a ko b ya n institue for informatics and automation problems of nas of ra e-mails evhar@ipia.sci.am, par h@ipia.sci.am abstract the matrix of asymptotic interdependencies (reliability{reliability functions) of all possible pairs of the error probability exponents (reliabilities) in testing of multiple statistical hypotheses is studied for arbitrarily varying object with the current states sequence known to the statistician. the case of two hypotheses when state sequences are not known to the decision maker was studied by fu and shen, and when decision is founded on the known states sequence was considered by ahlswede, haroutunian and aloyan. in the same way as fu and shen we obtain from the main result rate-reliability and reliability-rate functions for arbitrarily varying source coding with side information. an illustrative example is presented. refer ences [1 ] r . f. a h ls we d e , " co lo r in g h yp e r g r a p h s : a n e w a p p r o a c h t o m u lt i-u s e r s o u r c e c o d in g " i, ii, j . combin. inform. and syst. sci., vo l. 4 , n o . 1 , p p . 7 6 -1 1 5 , 1 9 7 9 , vo l. 5 , n o . 3 , p p . 2 2 0 { 2 6 8 , 1 9 8 0 . [2 ] e . a . h a r o u t u n ia n , " ma n y s t a t is t ic a l h yp o t h e s e s : in t e r d e p e n d e n c e o f o p t im a l t e s t 's e r r o r p r o b a b ilit ie s e xp o n e n t s " , ( in r u s s ia n ) , a b s t r a c t o f t h e r e p o r t o n t h e 3 r d a llu n io n s c h o o l-s e m in a r , "p rogram-algorithmical software for applied multi-variate statistical analysis", ts a kh ka d z o r , p a r t 2 , p p . 1 7 7 { 1 7 8 , 1 9 8 8 . [3 ] e . a . h a r o u t u n ia n , " l o g a r it h m ic a lly a s ym p t o t ic a lly o p t im a l t e s t in g o f m u lt ip le s t a t is t ic a l h yp o t h e s e s " , p roblems of control and information theory, vo l. 1 9 ( 5 -6 ) , p p . 4 1 3 { 4 2 1 , 1 9 9 0 . [4 ] r . l . d o b r u s h in , p e r s o n a l c o m m u n ic a t io n , 1 9 8 7 . [5 ] r . f. a h ls we d e , e . a . h a r o u t u n ia n a n d e . v . a lo ya n , " on lo g a r it h m ic a lly a s ym p t o t ic a lly o p t im a l h yp o t h e s is t e s t in g fo r a r b it r a r ily va r yin g s o u r c e wit h s id e in fo r m a t io n " , p r e s e n t e d fo r p u b lic a t io n 2 0 0 4 . ¤the work was partially supported by intas, project 00{738. 3 6 e. a. haroutunian and p. m. hakobyan 3 7 [6 ] l . b ir g ¶ e , " v it e s s e s m a xim a ls d e d ¶ c r o is s e n c e d e s e r r e u r s e t t e s t s o p t im a u x a s s o c ie ¶ s " . z. w a h r s c h . ve r w. ge b ie t e , vo l. 5 5 , p p . 2 6 1 { 2 7 3 , 1 9 8 1 . [7 ] i. cs is z ¶a r a n d j. k äo r n e r , " in fo r m a t io n th e o r y: co d in g th e o r e m s fo r d is c r e t e me m o r yle s s s ys t e m s " , academic press., n e w y o r k, 1 9 8 1 . [8 ] i. cs is z ¶a r , " th e m e t h o d o f t yp e s " , ie e e trans. inform. theory, vo l. 4 4 , n o . 6 , p p . 2 5 0 5 { 2 5 2 3 , 1 9 9 8 . [9 ] i. cs is z ¶a r a n d g. l o n g o , " on t h e e r r o r e xp o n e n t fo r s o u r c e c o d in g a n d fo r t e s t in g s im p le s t a t is t ic a l h yp o t h e s e s " , studia sc. m ath. hungarica, vo l. 6 , p p . 1 8 1 { 1 9 1 , 1 9 7 1 . [1 0 ] v . a n a n t a r a m , " a la r g e d e r ia t io n s a p p r o a c h t o e r r o r e xp o n e n t in s o u r c e c o d in g a n d h yp o t h e s e s t e s t in g " , ie e e trans. inform. theory, vo l. 3 6 , n o . 4 , p p . 9 3 8 { 9 4 3 , 1 9 9 0 . [1 1 ] s . n a t a r a ja n , " l a r g e d e r iva t io n s h yp o t h e s is t e s t in g a n d s o u r c e c o d in g fo r ¯ n it e ma r ko v c h a in s " , ie e e trans. inform. theory, vo l. 3 1 , n o . 3 , p p . 3 6 0 { 3 6 5 , 1 9 8 5 . [1 2 ] f.-w . fu a n d s .-y . s h e n , " h yp o t h e s is t e s t in g fo r a r b it r a r ily va r yin g s o u r c e wit h e xp o n e n t ia l-t yp e c o n s t r a in t " , ie e e trans. inform. theory, vo l. 4 4 , n o . 2 , p p . 8 9 2 { 8 9 5 , 1 9 9 8 . î»õ»ï³ï³óí³í íç׳ﳷñç ïáõùçó ï³ù³û³ï³ýáñ»ý ÷á÷áëíáõ ûµû»ïïç ýï³ïù³ùµ µ³½ù³ïç í³ñï³íý»ñç ëïáõ·áõù³ý ¨ ³õµûáõñç ïá¹³íáñù³ý ñ³ù³ñ ïçñ³éáõãû³ý ù³ëçý º. ². ð³ñáõãûáõýû³ý ¨ ö. ø. ð³ïáµû³ý ²ù÷á÷áõù àõëáõùý³ëçñí»é ¿ µ³½ù³ïç í³ñï³íý»ñç ï»ëï³íáñù³ý áýã³óùáõù µáéáñ ñý³ñ³íáñ ½áõû·»ñç ëë³éý»ñç ñ³í³ý³ï³ýáõãáõýý»ñç ñáõë³éçáõãû³ý óáõóçãý»ñç ÷áëï³ëí³íáõãûáõýý»ñç ÷á÷áëíáõ ûµû»ïïç ñ³ù³ñ, áñç íç׳ïý»ñá ñ³ûïýç »ý íç׳ﳷñçý: ºñïáõ í³ñï³íý»ñç ¹»åùá, »ñµ áñáßáõù áý¹áõýáõçý ³ýñ³ûï ¿ íç׳ïý»ñç ñ³çáñ¹³ï³ýáõãûáõýá, ùýý³ñïí»é ¿ üáõç ¨ þ»ýç ïáõùçó, çëï ñ³ûïýç íç׳ïý»ñáí ï³ñµ»ñ³ïá ¹çï³ñïí»é ¿ ²éëí»¹»ç, ð³ñáõãûáõýû³ýç ¨ ²éáû³ýç ïáõùçó: æýãå»ë üáõý ¨ þ»ýá, ù»ýù ýáõûýå»ë ëï³ó»é »ýù ïáõùý³ïç çýýáñù³óç³ûáí ï³ù³û³ï³ýáñ»ý ÷á÷áëíáõ ³õµûáõñç ñ³ù³ñ ³ñ³·áõãûáõý-ñáõë³éçáõãûáõý ¨ ñáõë³éçáõãûáõý-³ñ³·áõãûáõý ýáõýïóç³ý»ñá: ü»ñï³û³óí³í ¿ å³ñ½³µ³ýáõ ûñçý³ï: d:\sbornik\...\conmanag.dvi mathematical problems of computer science 23, 2004, 80{99. consistency m anagement i n database systems: review a r m e n j. a s a t r ya n institute for informatics and automation problems of nas of ra unicad cjsc e-mail armen.asatryan@unicad.am abstract by its de¯nition, a database must serve as a faithful and incorruptible repository of data. applications that consult the database expect a "warranty" that the database is supplying the correct values. this survey brie°y presents the approaches of integrity constraint management in database systems. it re°ects the various research activities in this ¯eld. we focus on central approaches, concepts, methods, and systems in this area. refer ences [1 ] ma lc o lm a t kin s o n , fr a n »c o is b a n c ilh o n , d a vid d e w it t , k la u s d it t r ic h , d a vid ma ie r a n d s t a n le y zd o n ik, " th e ob je c t -or ie n t e d d a t a b a s e s ys t e m ma n ife s t o " , p roceedings of the f irst international conference on d eductive and object-oriented d atabases, k yo t o , ja p a n , p p . 2 2 3 -2 4 0 , 1 9 8 9 , citeseer.nj.nec.com/atkinson89objectoriented.html. [2 ] r a ke s h a g r a wa l, s h a u l d a r a n d n a r a in h . ge h a n i, " th e o++ d a t a b a s e p r o g r a m m in g l a n g u a g e : im p le m e n t a t io n a n d e xp e r ie n c e " , icd e , p p . 6 1 -7 0 , 1 9 9 3 , citeseer.nj.nec.com/dar93database.html. [3 ] r . a g r a wa l a n d n . h . ge h a n i, " od e ( ob je c t d a t a b a s e a n d e n vir o n m e n t ) : t h e la n g u a g e a n d t h e d a t a m o d e l" , p p . 3 6 -4 5 , 1 9 8 9 , citeseer.nj.nec.com/agrawal89ode.html. [4 ] v ¶ e r o n iqu e b e n z a ke n a n d x a vie r s c h a e fe r , " s t a t ic in t e g r it y co n s t r a in t ma n a g e m e n t in ob je c t -or ie n t e d d a t a b a s e p r o g r a m m in g l a n g u a g e s via p r e d ic a t e tr a n s fo r m e r s " , l ecture notes in computer science, vo l. 1 2 4 1 , p p . 6 0 -??, 1 9 7 7 , citeseer.nj.nec.com/article/benzaken97static.html. [5 ] s t e fa n o ce r i, p ie r o fr a t e r n a li a n d s t e fa n o p a r a b o s c h i, " co n s t r a in t ma n a g e m e n t in ch im e r a " , ie e e d ata e ng. b ull., vo l. 1 7 , n o . 2 , p p . 4 -8 , 1 9 9 4 . [6 ] s t e fa n o ce r i, p ie r o fr a t e r n a li, s t e fa n o p a r a b o s c h i a n d l e t iz ia ta n c a , " a u t o m a t ic g e n e r a t io n o f p r o d u c t io n r u le s fo r in t e g r it y m a in t e n a n c e " , acm trans. d atabase syst., a cm p r e s s , vo l 1 9 , n o . 3 , p p . 3 6 7 -4 2 2 , 1 9 9 4 . 8 0 a. j. asatryan 8 1 [7 ] s t e fa n o ce r i a n d r a in e r ma n t h e y, " ch im e r a : a mo d e l a n d l a n g u a g e fo r a c t ive d ood s ys t e m s " , e ast/w est d atabase w orkshop ,p p . 3 -1 6 , 1 9 9 4 , citeseer.nj.nec.com/ceri94chimera.html. [8 ] s . ce r i a n d j. w id o m , " d e r ivin g p r o d u c t io n r u le s fo r co n s t r a in t ma in t e n a n c e " , p roceedings of the 16th vl d b conference, b r is b a n e , a u s t r a lia , d . mc l e o d a n d r . s a c ks d a vis a n d h . s c h e k, p p . 5 6 6 -5 7 7 , 1 9 9 0 , citeseer.nj.nec.com/ceri90deriving.html. [9 ] s . ce r i a n d j. w id o m , " d e r ivin g p r o d u c t io n r u le s fo r in c r e m e n t a l v ie w ma in t e n a n c e " ,p roceedings of the 17th conference on very l arge d atabases, ( l o s a lt o s ca ) , b a r c e lo n a , mo r g a n k a u fm a n , 1 9 9 1 , citeseer.nj.nec.com/ceri91deriving.html. [1 0 ] c. j. d a t e , " d a t a b a s e s ys t e m s " , a d d is o n -w e s le y, 2 0 0 0 . [1 1 ] u m e s h wa r d a ya l, e r ic n . h a n s o n a n d je n n ife r w id o m , " a c t ive d a t a b a s e s ys t e m s " , m odern d atabase systems, p p . 4 3 4 -4 5 6 , 1 9 9 5 , citeseer.nj.nec.com/dayal94active.html. [1 2 ] k . r . d it t r ic h a n d s . ga t z iu a n d a . ge p p e r t , " th e a c t ive d a t a b a s e ma n a g e m e n t s ys t e m ma n ife s t o : a r u le b a s e o f a a d b ms fe a t u r e s " , p roceedings of the 2nd international w orkshop on r ules in d atabase systems, s p r in g e r , vo l. 9 8 5 , p p . 3 -2 0 , 1 9 9 5 , citeseer.nj.nec.com/dittrich95active.html. [1 3 ] m. fit t in g , " fir s t -or d e r l o g ic a n d a u t o m a t e d th e o r e m p r o vin g " , s p r in g e r v e r la g , 1 9 9 0 . [1 4 ] p ie r o fr a t e r n a li a n d l e t iz ia ta n c a , " a s t r u c t u r e d a p p r o a c h fo r t h e d e ¯ n it io n o f t h e s e m a n t ic s o f a c t ive d a t a b a s e s " , acm trans. d atabase syst., a cm p r e s s , vo l. 2 0 , n o . 4 , p p . 4 1 4 -4 7 1 , 1 9 9 5 , http://doi.acm.org/10.1145/219035.219042. [1 5 ] a n d r e a s ge p p e r t a n d k la u s r . d it t r ic h ," s p e c i¯ c a t io n a n d im p le m e n t a t io n o f co n s is t e n c y co n s t r a in t s in ob je c t -or ie n t e d d a t a b a s e s ys t e m s : a p p lyin g p r o g r a m m in g -b yco n t r a c t " , d atenbanksysteme in b uro, technik und w issenschaft, p p . 3 2 2 -3 3 7 , 1 9 9 5 , citeseer.nj.nec.com/geppert95speci¯cation.html. [1 6 ] s t e lla ga t z iu a n d a n d r e a s ge p p e r t a n d k la u s r . d it t r ic h , " th e s a mos a c t ive d b ms p r o t o t yp e " , p p . 4 8 0 -4 8 0 , 1 9 9 5 , citeseer.nj.nec.com/gatziu94samo.html. [1 7 ] n . h . ge h a n i a n d h . v . ja g a d is h , " od e a s a n a c t ive d a t a b a s e : co n s t r a in t s a n d tr ig g e r s " , p roceedings of the 17th conference on very l arge d atabases, ( l o s a lt o s ca ) , b a r c e lo n a , mo r g a n k a u fm a n , 1 9 9 1 , citeseer.nj.nec.com/gehani91ode.html. 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[3 5 ] j. w id o m a n d s . j. fin ke ls t e in , " s e t -o r ie n t e d p r o d u c t io n r u le s in r e la t io n a l d a t a b a s e s ys t e m s " , p p . 2 5 9 -2 7 0 , 1 9 9 0 , citeseer.nj.nec.com/widom90setoriented.html. ²ùµáõç³ï³ýáõãû³ý ³å³ñáíáõùá ïíû³éý»ñç ñ»ýù»ñáõù. ³ïý³ñï ². æ. ²ë³ïñû³ý ²ù÷á÷áõù àëï çñ ë³ñù³ýù³ý ïíû³éý»ñç ñ»ýùá å»ïù ¿ í³é³ûç áñå»ë ïíû³éý»ñç §íëï³ñ»éç¦ »í §³ýëáó»éç¦ ßï»ù³ñ³ý »í å»ïù ¿ ³å³ñáíç ïíû³éý»ñç ïáé»ïïáõãûáõý᪠ñ»ýùç ñ»ï ³ßë³ïáõ ïçñ³é³ï³ý íñ³·ñ»ñç ñ³ù³ñ: êáõûý ³ïý³ñïáõù ý»ñï³û³óíáõù »ý ïíû³éý»ñç ñ»ýù»ñáõù ³ùµáõç³ï³ýáõãû³ý ë³ñù³ý³÷³ïáõý»ñç ùß³ïù³ý ùç ß³ñù ùáï»óáõùý»ñ: ²ûý ³ñï³óáéáõù ¿ ³ûë µý³·³í³éáõù ý»ñï³ûáõùë ï³ï³ñíáõ ñ»ï³½áï³ï³ý áõõõáõãûáõýý»ñá: article_with_style.dvi mathematical problems of computer science 31, 49{59, 2008. on optimal h ypothesis t esting for p air of stochastically coupled objects. e vg u e n i a . h a r o u t u n ia n a n d a r a m o. y e s s a ya n institute for informatics and automation problems of nas of ra e-mail: eghishe@sci.am abstract the paper is devoted to hypotheses testing for a model consisting of two stochastically coupled objects. it is supposed that l1 possible probability distributions are known for the ¯rst object and the second object is distributed according to one of l1 £ l2 given conditional distributions depending on the distribution index and the current observed state of the ¯rst object. the matrix of interdependencies of all possible pairs of the error probability exponents in asymptotically optimal tests of distributions of both objects is studied. the case of two objects which cannot have the same probability distribution from two possible variants was considered by ahlswede and haroutunian. this case for three hypotheses and the model of two statistically dependent objects for two hypotheses were examined by haroutunian and yessayan. refer ences [1 ] e . a . h a r o u t u n ia n , \ l o g a r it h m ic a lly a s ym p t o t ic a lly o p t im a l t e s t in g o f m u lt ip le s t a t is t ic a l h yp o t h e s e s " , p roblems of control and information theory, vo l. 1 9 ( 5 -6 ) , p p . 4 1 3 { 4 2 1 , 1 9 9 0 . 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[7 ] i. cs is z ¶a r a n d j. k äo r n e r , information theory: coding theorems for d iscrete m emoryless systems, a c a d e m ic p r e s s , n e w y o r k, 1 9 8 1 . 4 9 5 0 on optimal hypothesis testing for pair of stochastically coupled objects. [8 ] t. m. co ve r a n d j. a . th o m a s , e lements of information theory. w ile y, n e w y o r k, 1 9 9 1 . [9 ] i. cs is z ¶a r a n d p . c. s h ie ld s , \ in fo r m a t io n t h e o r y a n d s t a t is t ic s : a t u t o r ia l" . f oundations and trends in communications and information theory, vo l.1 , n o .4 , 2 0 0 4 . [1 0 ] e . a . h a r o u t u n ia n , m. e . h a r o u t u n ia n , a n d a . n . h a r u t yu n ya n , \ r e lia b ilit y c r it e r ia in in fo r m a t io n t h e o r y a n d in s t a t is t ic a l h yp o t h e s e s t e s t in g " , f oundations and trends in communications and information theory, vo l. 4 , n o . 2 -3 , 2 0 0 8 . [1 1 ] e . a . h a r o u t u n ia n a n d a . o. y e s s a ya n , \ on lo g a r it h m ic a lly a s ym p t o t ic a lly o p t im a l h yp o t h e s is t e s t in g fo r p a ir o f s t a t is t ic a lly d e p e n d e n t o b je c t s " , m athematical p roblems of computer science, vo l. x x ix , p p . 9 7 { 1 0 3 , 2 0 0 7 . êïáë³ëïçïáñ»ý ï³ëí³í ûµû»ïïý»ñç ½áõû·ç ýï³ïù³ùµ í³ñï³íý»ñç ûåïçù³é ëïáõ·ù³ý ù³ëçý º. ². ð³ñáõãûáõýû³ý ¨ ². ú. ºë³û³ý ²ù÷á÷áõù èáõíí³í ¿ ëïáë³ëïçïáñ»ý ï³ëû³é »ñïáõ ûµû»ïïý»ñç ýï³ïù³ùµ í³ñï³íý»ñç éá·³ñçãùáñ»ý ³ëçùåïáïáñ»ý ûåïçù³é ëïáõ·ù³ý ëý¹çñá: ²é³ççý ûµû»ïïá ï³ñáõ ¿ µ³ßëí³í éçý»é ïñí³í l1 ñ³í³ý³ï³ý³ûçý µ³ßëáõùý»ñçó ù»ïáí, çëï »ñïñáñ¹áª ï³ëí³í ³é³ççýç µ³ßëáõùçó ¨ ¹çï³ñïíáõ å³ñçý ýñ³ íç׳ïçó, ïñí³í l1 £ l2å³ûù³ý³ï³ý ñ³í³ý³ï³ý³ûçý µ³ßëáõùý»ñçó ù»ïáí: àõëáõùý³ëçñí»é ¿ ûµû»ïïý»ñç ýï³ïù³ùµ í³ñï³íý»ñç ï»ëï³íáñù³ý ëë³éý»ñç ñ³í³ý³ï³ýáõãûáõýý»ñç óáõóçãý»ñç (ñáõë³éçáõãûáõýý»ñç) ÷áëï³ëí³íáõãûáõýý»ñç ù³ïñçóá: 84 mathematical problems of computer science 58, 84–90, 2022. doi: 10.51408/1963-0095 udc 004.62:004.946 data processing and persistence in virtual reality systems arman a. hovhannisyan national polytechnic university of armenia e-mail: aahovhannisyan1@gmail.com abstract data processing and persistence are key aspects of developing a virtual reality system. in this paper, an improvement is offered to the distance calculation algorithm of the unity engine. additionally, data persistence mechanisms provided by the unity engine are reviewed, and file system is selected as an appropriate option. storage of object coordinates to the file system is implemented. the results provide a baseline for developing a system for creating virtual stands for professional research. keywords: virtual reality, data management, file system, serialization. article info: received 29 march 2022; received in revised form 9 october 2022; accepted 17 november 2022. 1. introduction in virtual reality, data is represented both in primitive types (int, float, string) and complex types provided by the engine. object positions in space are determined in the cartesian coordinate system [1] (see fig. 1). a common task is to compare the distance of 2 points from a given point 𝐴(𝑥1, 𝑦1, 𝑧1). the unity engine scripting api [2] provides a complex data type called vector3 to store object coordinates, along with its vector3.distance() method to calculate distance between 2 points. given the points (𝑥2, 𝑦2 , 𝑧2), 𝐶(𝑥3, 𝑦3 , 𝑧3), this method may be used to accomplish the task, comparing the following values: vector3.distance(b, a), vector3.distance(c, a). a more efficient solution may be applied using the formula of the distance between 2 points [1]: 𝑑𝐴𝐵 = √(𝑥2 − 𝑥1) 2 + (𝑦2 − 𝑦1) 2 + (𝑧2 − 𝑧1) 2 , (1) a. hovhannisyan 85 𝑑𝐴𝐶 = √(𝑥3 − 𝑥1) 2 + (𝑦3 − 𝑦1) 2 + (𝑧3 − 𝑧1) 2 (2) instead of comparing values for 𝑑𝐴𝐵 and 𝑑𝐴𝐶 , the radicands may be compared, saving cpu time on unnecessary calculations. fig. 1. object positions in space. to have persistent data between sessions, the user progress has to be stored on the disk. there are several methods of managing data storage, including sql database, playerprefs and os file system. on specific events during the runtime, which are to be defined, data containing all the current values have to be stored. these events may include user interaction, object state mutation, or events may be set to trigger on specific timestamps, e.g., every 10 seconds. then, on the next program run, these stored values have to be fetched and transmitted to the engine to render the objects in the same state and position, as they were when the last event was triggered. 2. persistent data to have persistent data between sessions, the user progress has to be stored on the disk. below are listed several methods of managing data storage. 1. sql database sql is useful when there is relational data. it supports queries to fetch related data sets. in our case, we have just objects that need to be memorized and then retrieved on the next run. such simple operations are easier to implement and faster in work on file system. sql is a dedicated software and isn’t an integrated part of operating systems, as file system is. also, a connection to sql service should be kept active during the runtime. 2. playerprefs playerprefs is a class provided by unity engine that stores player preferences between game sessions. it stores values in the os registry. though it is possible to store data using this data processing and persistence in virtual reality systems 86 method, it is not recommended to do so. this method should be used for data, that can be afforded to lose, such as user settings and preferences. sensitive and relatively big data should not be kept in registries. 3. file system to store data in files, it needs to be formatted in some way. it may be serialized [3] to binary format and written to a file. that data will then be successfully deserialized and used in the application. but since binary is not human-readable, it makes this format insufficient. moreover, it is not possible to edit the saved data manually. using json data type allows bypassing these problems. taking into account the points mentioned above, it was decided to handle data storage using file system and serialization, so every time data needs to be stored, it is serialized to json format and written to a file (see fig. 2). then, to restore the state in the application, the file is read and data is deserialized to object (see fig. 3). fig. 2. storing data. fig. 3. using stored data. 3. saving object position a specific and common example is persisting the object position. in this example, we have a cube placed on a table (see fig. 4). the origin (0, 0, 0) can be located on the ground. a. hovhannisyan 87 fig. 4. cube. to save the cube position after it is replaced, a class called savemanager is created, which contains 2 methods: save and load. these methods use a file called "position.dat" to write/read data. savemanager.cs public class savemanager { public static void save(vector3 pos) { string path = path.combine(application.persistentdatapath, "position.dat"); file.writealltext(path, jsonutility.tojson(pos)); } public static vector3 load() { string path = path.combine(application.persistentdatapath, "position.dat"); string result = file.readalltext(path); vector3 pos = jsonutility.fromjson(result); return pos; } } the save and load methods would then be invoked from a script, which is bound to the object. the save method would be bound to the xr grab interactable component [4] “select exited” event to save data every time the object is released. the load method would be invoked from the start method to set object positions from the saved data on a fresh program run. cubescript.cs public class cubescript : monobehaviour { // start is called before the first frame update void start() { vector3 position = savemanager.load(); transform.position = position; } public void save() { savemanager.save(transform.position); } } data processing and persistence in virtual reality systems 88 fig. 5. cube position changed via left controller. the saved file position.dat: {"x":-0.0030583017505705358,"y":0.838373601436615,"z":-2.0934112071990969} after restarting the program, we still have the cube in its new place (see fig. 5). now we can modify this file content, and set the coordinates to (0, 0, 0). {"x":0,"y":0,"z":0} after modifying and saving the file, and running the program again, we can see that the cube appears on the origin as expected (see fig. 6). fig. 6. cube position manually set to origin. 4. conclusion in this article, an improvement to the unity engine distance calculation algorithm was suggested. additionally, data types provided by the unity engine were reviewed. data storage options were compared and decided to use the os file system and data serialization. as an example, a cube position storage and loading were implemented. this method will be used also for custom complex data types to store, marking the class representing the data type as serializable. a. hovhannisyan 89 references [1] wikipedia, (2012) cartesian coordinate system. [online]. available: https://en.wikipedia.org/wiki/cartesian_coordinate_system [2] unity engine scripting api reference. [online]. available: https://docs.unity3d.com/scriptreference/ [3] microsoft docs, (2021) serialization (c#). [online]. available: https://docs.microsoft.com/en-us/dotnet/csharp/programming-guide/concepts/serialization/ [4] the khronos group inc., “the openxr specification”. [5] unity learning, (2020) create with vr. [online]. available: https://learn.unity.com/course/create-with-vr?uv=2020.3 տվյալների մշակումը և պահպանումը վիրտուալ իրականության համակարգերում արման ա․ հովհաննիսյան հայաստանի ազգային պոլիտեխնիկական համալսարան e-mail: aahovhannisyan1@gmail.com ամփոփում տվյալների մշակումը և պահպանումը վիրտուալ իրականության համակարգի զարգացման հիմնական ասպեկտներ են։ այս հոդվածում առաջարկվում է unity շարժիչի հեռավորության հաշվարկման ալգորիթմի լավարկում։ բացի այդ, դիտարկվում են unity շարժիչի կողմից տրամադրվող տվյալների պահպանման մեխանիզմները, և որպես նպատակահարմար տարբերակ, ընտրվում է ֆայլային համակարգը։ իրականացվում է օբյեկտի կոորդինատների պահպանումը ֆայլային համակարգում։ ստացված արդյունքները հիմք են հանդիսանում մասնագիտական հետազոտությունների վիրտուալ ստենդների ստեղծման համակարգի մշակման համար։ բանալի բառեր՝ վիրտուալ իրականություն, տվյալների կառավարում, ֆայլային համակարգ, սերիալիզացիա https://docs.unity3d.com/scriptreference/ https://docs.microsoft.com/en-us/dotnet/csharp/programming-guide/concepts/serialization/ data processing and persistence in virtual reality systems 90 обработка и сохранение данных в системах виртуальной реальности арман а. оганесян национальный политехнический университет армении e-mail: aahovhannisyan1@gmail.com аннотация обработка и сохранение данных являются ключевыми аспектами разработки системы виртуальной реальности. в данной статье предлагается улучшение алгоритма расчета расстояний unity engine. кроме того, рассматриваются механизмы сохранения данных, предоставляемые unity engine, и в качестве подходящего варианта выбирается файловая система. реализуется хранение координат объекта в файловой системе. результаты обеспечивают основу для разработки системы создания виртуальных стендов для профессиональных исследований. ключевые слова: виртуальная реальность, управление данными, файловая система, сериализация mailto:aahovhannisyan1@gmail.com d:\sbornik\...\infhidingsys.dvi mathematical problems of computer science 23, 2004, 20{31. on e stimates of rate-r eliability-distor tion function for i nfor mation h iding system¤ ma r ia m e . h a r o u t u n ia n a n d s m b a t a . to n o ya n institute for informatics and automation problems of nas of ra e-mails: armar@ipia.sci.am, smbatt@ipia.sci.am abstract the model of information hiding system, introduced and studied by p. moulin and j. a. o'sullivan [1] is explored. the rate-reliability-distortion function for this system is investigated. upper and lower estimates of rate-reliability-distortion function, called the random coding and the sphere packing bounds are constructed. the limit of random coding bound, when e ! 0, coincides with the information hiding capacity stated by p. moulin and j. a. o'sullivan. refer ences [1 ] p . mo u lin a n d j. a . o's u lliva n , " in fo r m a t io n -t h e o r e t ic a n a lys is o f in fo r m a t io n h id in g " , ie e e trans. inform. theory, vo l. 4 9 , n o . 3 , p p . 5 6 3 -5 9 3 , ma r . 2 0 0 3 . [2 ] f. a . p . p e t it c o la s , r . j. a n d e r s o n , a n d m. g. k u h n , " in fo r m a t io n h id in g { a s u r ve y," p roc. ie e e (special issue on identi¯cation and p rotection of m ultimedia information), vo l. 8 7 , p p . 1 0 6 2 -1 0 7 8 , ju ly 1 9 9 9 . [3 ] p . mo u lin , " th e r o le o f in fo r m a t io n t h e o r y in wa t e r m a r kin g a n d it s a p p lic a t io n t o im a g e wa t e r m a r kin g ," signal p rocessing, vo l. 8 1 , p p . 1 1 2 1 -1 1 3 9 , 2 0 0 1 . [4 ] e . a . h a r o u t u n ia n , " u p p e r e s t im a t e o f t r a n s m is s io n r a t e fo r m e m o r yle s s c h a n n e l wit h c o u n t a b le n u m b e r o f o u t p u t s ig n a ls u n d e r g ive n e r r o r p r o b a b ilit y e xp o n e n t " , ( in r u s s ia n ) , 3rd all-union conf. on theory of information transmission and coding, uzhgorod, p ublication house of uzbek academy of sciences, tashkent, p p . 8 3 { 8 6 , 1 9 6 7 . [5 ] m. e . h a r o u t u n ia n a n d s . a . to n o ya n , " r a n d o m c o d in g b o u n d o f in fo r m a t io n h id in g e-c a p a c it y" , p roc. of ie e e intern. symp. infrom. theory, p . 5 3 6 , u s a , ch ic a g o , 2 0 0 4 . [6 ] s . i. ge l'fa n d a n d m. s . p in s ke r , " co d in g fo r c h a n n e l wit h r a n d o m p a r a m e t e r s ," p roblems of control and information theory, vo l. 9 , n o . 1 , p p . 1 9 -3 1 , 1 9 8 0 . ¤the work was partially supported by intas grant 00-738 and by 04.10.31 target program of ra. 2 0 m. e. haroutunian and s. a. tonoyan 2 1 [7 ] m. e . h a r o u t u n ia n , " n e w b o u n d s fo r e-c a p a c it ie s o f a r b it r a r ily va r yin g c h a n n e l a n d c h a n n e l wit h r a n d o m p a r a m e t e r " trans. iiap nas r a and ysu, m athematical p roblems of computer sciences, vo l. 2 2 , p . 4 4 -5 9 , 2 0 0 1 . [8 ] m. e . h a r o u t u n ia n , " b o u n d s o f e-c a p a c it y fo r m u lt ip le -a c c e s s c h a n n e l wit h r a n d o m p a r a m e t e r " , s p e c ia l b o o k is s u e d in t h e fr a m e wo r k o f r e s e a r c h p r o je c t " ge n e r a l th e o r y o f in fo r m a t io n tr a n s fe r a n d co m b in a t o r ic s " a t zif, b ile fe ld u n ive r s it y, ge r m a n y, 2 0 0 4 . [9 ] n . me r h a v, " on r a n d o m c o d in g e r r o r e xp o n e n t s o f wa t e r m a r kin g s ys t e m s " , ie e e trans. inform. theory, vo l. 4 6 , n o . 2 , p p . 4 2 0 -4 3 0 , ma r . 2 0 0 0 . [1 0 ] n . me r h a v a n d a . s o m e kh -b a r u c h , " on t h e e r r o r e xp o n e n t a n d c a p a c it y g a m e s o f p r iva t e wa t e r m a r kin g s ys t e m s " , ie e e trans. inform. theory, vo l. 4 9 , n o . 3 , p p . 5 3 7 5 6 2 , ma r . 2 0 0 3 . [1 1 ] i. cs is z ¶a r a n d j. k äo r n e r , information theory: coding theorems for discrete memoryless systems, a c a d e m ic p r e s s , n e w y o r k, 1 9 8 1 . [1 2 ] i. cs is z ¶a r , " th e m e t h o d o f t yp e s " , ie e e trans. inform. theory, vo l. 4 4 , n o . 6 , p p . 2 5 0 5 { 2 5 2 3 , 1 9 9 8 . îíû³éý»ñ ã³ùóýáõ ñ³ù³ï³ñ·»ñç ñ³ù³ñ ³ñ³·áõãûáõý-ñáõë³éçáõãûáõý-ß»õáõù ýáõýïóç³ûç ·ý³ñ³ï³ï³ýý»ñç ù³ëçý ø. º. ð³ñáõãûáõýû³ý ¨ ê. ². îáýáû³ý ²ù÷á÷áõù ²ßë³ï³ýùáõù áõëáõùý³ëçñí³í ¿ ä. øáõéçýç ¨ æ. úªêáõéçí³ýç ïáõùçó ¹çï³ñïí³í ïíû³éý»ñ ã³ùóýáõ ñ³ù³ï³ñ·ç áý¹ñ³ýáõñ ùá¹»éá: ²û¹ ñ³ù³ï³ñ·ç ñ³ù³ñ ý»ñùáõíí³í ¿ ³ñ³·áõãûáõý-ñáõë³éçáõãûáõý-ß»õáõù ýáõýïóç³ûç ·³õ³÷³ñá, áñç ñ³ù³ñ ï³éáõóí³í »ý ëïáñçý ¨ í»ñçý ·ý³ñ³ï³ï³ýý»ñ, áñáýù ñ³ù³å³ï³ëë³ý³µ³ñ ïáãíáõù »ý å³ï³ñ³ï³ý ïá¹³íáñù³ý ¨ ëý»ñ³ý»ñç ÷³ã»ã³íáñù³ý ë³ñù³ýý»ñ: êïáñçý ·ý³ñ³ï³ï³ýá ë³ñù³ý³ûçý ¹»åùáõù, »ñµ e ! 0 , ñ³ùáýïýáõù ¿ ñ³ù³ï³ñ·ç áõý³ïáõãû³ý ñ»ï: 08_anahit_chubaryan's article l=2�ì=2,÷�“*,� "%c!%“/ *,k�!…�2,*, , "/÷,“ë,2�ëü…%l 2�.…,*, 52, 66-73, 2019. 66 удк 510.6 о количестве минимальных тавтологий и свойствах их выводов в ряде систем классической и неклассических логик aрсен a. aмбарцумян, гайк а. гаспарян, саркис а. ованнисян и анаит а. чубарян ереванский государственный университет e-mails: hambardzumyanarsen99@gmail.com, haykgasparyan012@gmail.com, saqohovhannisyan0199@gmail.com, achubaryan@ysu.am аннотация в настоящей работе для тавтологий заданной логики длины п доказано, что максимально возможное количество различных их минимальных тавтологий той же логики имеет экспоненциальную оценку от п, и доказано, что для каждой заданной в данной логике тавтологии существует такая минимальная тавтология, количество шагов вывода секвенциальной формы которой совпадает с наименьшим количеством шагов вывода секвенциальной формы заданной формулы в секвенциальных системах без правила сечения классической, интуиционистской, монотонной логик и логики иогансона. ключевые слова: минимальная тавтология; секвенциальные системы без правила сечения пропозициональных логик; количество шагов выводов; монотонность системы, строгая монотонность системы. 1. введение теория сложностей пропозициональных выводов является одной из активно исследуемых областей общей теории сложностей, так как доказано, что возможное существование полиномиально ограниченной теории выводов решит вопрос равенства известных классов сложностей np и conp [1]. в теории сложностей выводов пропозициональных систем важную роль играют минимальные тавтологии, т.е. тавтологии, которые не являются результатом подстановки в более короткие тавтологии. традиционно считалось, что минимальные тавтологии не могут выводиться сложнее результатов подстановок в них, т.е. должна быть некоторая «естественная монотонность» выводов. однако, оказалось, что многие «строгие» пропозициональные системы выводов двузначных и многозначных логик не монотонны ни по шагам ни по длине выводов [2,3], т.е. существуют минимальные тавтологии, которые выводятся сложнее, чем некоторые результаты подстановок в них. тем не менее, оказалось, что у некоторых таких результатов подстановки сушествуют иные минимальные тавтологии, которые выводятся не сложнее. a. aмбарцумян, г. гаспарян, с. ованнисян и а.чубарян 67 таким образом возник вопрос о монотонности и строгой монотонности пропозициональных систем подобные разновидности монотонностей исследованы для некоторых «слабых» систем выводов классической и неклассических логик в [4-6]. в связи с разнообразием множества минимальных тавтологий одной и той же тавтологии в настоящей работе 1) для тавтологий заданной логики длины п доказано, что максимально возможное количество различных минимальных тавтологий той же логики имеет экспоненциальную оценку от п, и 2) доказана монотонность пропозициональных секвенциальных систем без правила сечения для классической, интуиционистской, монотонной логик и логики иогансона. отметим, что строгая немонотонность этих систем, как и соответствующих систем с правилом сечения доказана в [7]. 2. предварительные понятия для представления основных результатов напомним некоторые понятия и обозначения. мы пользуемся общепринятыми oпределениями пропозициональной формулы, тавтологии в данной логике, секвенции, сукцедента, антецедента, главной формулы секвенции, секвенциальных систем без сечения классических и неклассических логик, сложностей выводов [8-10]. конкретный выбор языка для представления пропозициональной формулы, а значит, и системы доказательств, не имеет значения для наших рассмотрений, однако из технических соображений мы предполагаем, что он содержит пропозициональные переменные, логические связки ¬, &, ∨, ⊃ и пару скобок ( , ). в некоторых системах будут использованы также знаки τ«истина» и ⏊«ложь». длина формулы �, определяемая как количество всех вхождений в нее логических связок, обозначается через |�|. очевидно, что линейной функцией от |�| оценивается и полная длина формулы, понимаемая как количество всех символов. 2.1. описания рассматриваемых систем напомним ряд определений. секвенцией называется выражение γ → δ , где γ (антецедент) и δ (сукцедент) являются конечной (может быть пустой) последовательностью пропозициональных формул. следуя [8], определим следующие системы. схемой аксиом для классической системы (pc) является секвенция , γ → , где произвольная формула, а γ произвольная конечная (быть может пустая) последовательность формул. для произвольных формул� , � и последовательностей формулγиδ, логическими правилами вывода являются: ⊃→ � ⊃ �, γ → δ, a � ⊃ �, b, γ → δ � ⊃ �, γ → δ →⊃ a, γ → b, � ⊃ �, δ γ → � ⊃ �, δ ∨→ �∨�, �, �→�и�∨�, �, �→� �∨�, �→� →∨ �→�, �∨�, �или�→�, �∨�, � �→�∨� о количестве минимальных тавтологий и свойствах их выводов в ряде систем логик 68 & → �&�, a, γ → δили�&�, b, γ → δ �&�, γ → δ → & γ → a, �&�, δиγ → b, �&�, δ γ → �&�, δ ¬→ ¬�,�→ �, � ¬�, �→� → ¬ �, �→¬�, � �→¬�, � , где формулы � ⊃ �, � ∨ �, �&� и ¬� являются главными формулами секвенции. структурным правилом является γ→ δ ——————, где γ⊆γ’ ևδ⊆δ’ γ’ → δ’ схема аксиом пропозициональной интуиционистской системы (pi) и системы иогансона (pm) та же. в верхней секвенции (секвенциях) правилах введения логических функций в сукцеденте для систем pi и pm отсутствует главная формула, δ во всех правилах pi пуста или состоит из одной формулы, а для pm пуста [9]. пропозициональную систему (pmon) для монотонной логики, где используются только монотонные формулы (формулы, строящиеся с использованием только монотонных логических функций), определим, следуя [10]. схемами аксиом системы pmon являются p → p, ⏊ → γ, γ → τ, где p – пропозициональная переменная, а γконечная (быть может пустая) последовательность формул. для любых монотонных формул� , � и любых последовательностей монотонных формул γ, γ�, δ и δ� правилами вывода являются: ��� γ, α, α, δ → γ� γ, α, δ → γ� ��" γ, α, β, δ → γ� γ, β, α, δ → γ� ��$ γ → γ� γ, α → γ� ��% α, β, γ → δ α&β, γ → δ ��& α, γ → δиβ, γ� → δ� α ∨ β, γ, γ� → δ, δ� �'� γ� → γ, α, α, δ γ� → γ, α, δ �'" γ� → γ, α, β, δ γ� → γ, β, α, δ �'$ γ� → γ γ� → γ, α �'% γ → δ, α, β γ → δ, α ∨ β �'& γ → δ, a иγ� → δ�, β γ, γ� → δ, δ�, α&β обратим внимание, что порядок вхождения формул ни в сукцеденте, ни в антецеденте не существенен ни в одной из вышеописанных систем. мы пользуемся общепринятым понятием вывода в указанных системах. секвенция γ → δ называется c-доказуемой (i-доказуемой, mдоказуемой, mon доказуемой), если она выводима в системе pc (pi, pm, pmon). формула a называется cтавтологией (i-тавтологией, m-тавтологией), если секвенциальная форма→ a выводима в соответствующей системе pc (pi, pm). для монотонных формул a и b формула a⊃b называется monтавтологией, если ее секвенциальная форма � →b выводима в системе pmon. a. aмбарцумян, г. гаспарян, с. ованнисян и а.чубарян 69 2.2. некоторые характеристики минимальных тавтологий и логических систем выводов здесь мы исследуем некоторые свойства тавтологий различных логик и вышеприведенных систем выводов на основе одной из сложностных характеристик выводов tсложности, определяемой как количество различных секвенций в выводе. пусть ϕ является некоторой секвенциальной системой выводов фиксированной логики, а φ – некоторая тавтология данной логики. через ()�φ обозначим минимально возможное значение t-сложности всевозможных выводов соответствующей секвенциальной формы тавтологии φ в системе ϕ. определение 2.2.1: тавтология данной логики называется минимальной, если она не может быть получена подстановкой вместо ее переменных из более короткой тавтологии этой же логики. для произвольной минимальной тавтологии � фиксированной логики обозначим через s( � ) множество всех тавтологий этой же логики, являющихся результатом подстановки в �. определение 2.2.2: система выводов ϕ называется t-монотонной, если для каждой не минимальной тавтологии + этой системы существует такая минимальная тавтология φ этой же системы, что + принадлежит s(φ) и (, �+ = (, �� . определение 2.2.3: система выводов ϕ называется t-строго монотонной, если для каждой минимальной тавтологии φ и для каждой формулы + из s(φ) (, �� ≤(,�+ . 3. основные результаты здесь будут даны оценки максимального количества различных минимальных тавтологий для тавтологий длины n каждой из приведенных логик, а также будет доказана монотонность всех вышеперечисленных систем выводов. для формулировки результатов введем несколько обозначений. обозначим через тϕ(n) множество тавтологий длины n системы ϕ и ∀g∊тϕ(n) через kϕ(g) обозначим количество различных минимальных тавтологий формулы g в логике, определяемой системой ϕ. пусть mϕ(n) =max { kϕ(g) | ∀g∊тϕ(n) }: теорема 1: если ϕ одна из систем pc, pi, pm и pmon, то log3mϕ(n)=θ(n). доказательство основано на получении верхних и нижних оценок mϕ(n). для получения нижних оценок в системах pc, pi и pm для любого n ≥ 3 рассмотрим формулы gn = ((p11⊃p11)∨ ((p12⊃p12) ∨ (…∨ (p1k⊃p1k)...))&(...& ((pt1⊃pt1) ∨ ((pt2⊃pt2) ∨ (…∨ (ptk⊃ptk)...)), где n=2kt-1. нетрудно убедиться, что ∀i∊ {1...t} подформула gi=((pi1⊃pi1) ∨ ((pi2⊃pi2) ∨ (…∨ (pik⊃pik)...)) имеет k штук различных минимальных, откуда следует, что количество различных минимальных формулы gn равно k(n+1) / 2k . если через аn обозначим формулу p⊃gn, где p – пропозициональная переменная, отличная от переменных формулы gn, то количество различных минимальных формулы аn будет о количестве минимальных тавтологий и свойствах их выводов в ряде систем логик 70 kn / 2k. не трудно убедиться, что эта функция получает максимальное значение при k=3, а следовательно, mϕ(n)≥ 3n / 6 для систем pc, pi и pm, в каждой из которых выводимы и сама формула аn и все ее минимальные. обозначим через вn результат подстановки константы т в формулу аn вместо всех переменных подформулы gn, тогда количество различных минимальных тавтологий формулы вn, а значит и секвенции, получаемой из вn заменой знака ⊃ на → n, и выводимой в pmon, будет равно kn / 2k. таким образом, и для системы pmon получаем mϕ(n)≥ 3n /6, а, следовательно, для всех перечисленных систем ϕlog3mϕ(n)=0(n). для получения нетривиальной верхней оценки обратимся к общепринятому представлению пропозициональной формулы в виде дерева, вершинам которого специальным образом приписываются подформулы заданной формулы. заметим, что построение минимальных тавтологий данной тавтологии заключается в поиске максимального количества подформул, замена которых на переменные, не входящие в данную формулу, сохраняет тавтологичность в данной логике. отметим, что если в формуле уже выбрана некая подформула, которая должна быть заменена переменной, то ни одна из входящих в нее подформул более не может быть выбрана, что на дереве отражается следующим образом: если выбрана некая вершина, соответствующая формула которой выбрана для замены переменной, то уже ни одна вершина поддерева с корнем в выбранной вершине не может быть выбрана. если через f(n) обозначить максимально возможное количество подформул (верщин дерева), которые могут быть заменены переменными в формуле длины n (в дереве с n вершинами) и отвлечься от требования тавтологичности, то нетрудно убедиться, что для f(n) получим следующее рекуррентное соотношение: f(1) = 1 f(n) = max(f(n-1) + 1; max ((1 + f(i))(1 + f(n-i-1))). 1≤i≤n-2 учитывая, что 2n -1 является тривиальной верхней оценкой для функции f(n), а также полученную выше нижнюю оценку, будем искать решение указанного рекуррентного соотношения в виде cαn для наименьшего α в промежутке [31/6, 2] и некоторой константы с, для которых выполнено неравенство: (1 + f(i))(1 + f(n-i-1)))≤(cαi+ 1) ・(cαn-i-1 + 1) = c2αn-1 + c(αi + αn-i-1 ) + 1≤ cαn . методом приближения получаем f(n) =1 ((3/2)n). так как mϕ(n)≤f(n) в любой из выбранных логик, тоlog3mϕ(n)=1 (n), откуда и следует утверждение теоремы 1. теорема 2: секвенциальные системы pc, pi, pm и pmon монотонны. доказательство теоремы основано на следующем утверждении. лемма 1: пусть ϕ одна из систем pc, pi, pm и pmon, φ некоторая тавтология этой системы и + любая из ее минимальных тавтологий в этой системе, тогда вывод секвенциальной формы формулы + в системе ϕ является подвыводом одного из всевозможных выводов секвенциальной формы тавтологии φ в системе ϕ. доказательство леммы основано на разрешающей процедуре, описанной для pc и pi в пункте (d) теоремы 56 из [7] и выполнимой для всех вышеописанных систем. действительно, для любой данной секвенцииγ→ δ, составленных из пропозициональных формул, и для каждого выбора в качестве главной некоторой формулы из γ или из δ, содержащей логический символ, имеется только один или два несходных выбора посылок для вывода γ→ δ. процедура завершается в силу свойства подформульности, которым обладают все правила выводов перечисленных систем и в силу конечного числа a. aмбарцумян, г. гаспарян, с. ованнисян и а.чубарян 71 различных подформул выводимых секвенций. метод доказательства позволяет также строить всевозможные минимальные тавтологии заданной тавтологии. 4. заключение все обладающие свойством подформульности системы, исследованные в работах [1-6] и в настоящей работе оказались монотонными, ряд систем без свойства подформульности оказались немонотонными. заметим также, что все они не строго монотонны. остается открытым вопрос о монотонности систем фреге и секвенциальных систем с правилом сечения и о существовании строго монотонных систем, быть может и не полных. исследование выполнено при финансовой поддержке государственного комитета по науке мон ра в рамках научного проекта № 18t-1b034. литература [1] s. cook, r. reckhow, “the relative efficiency of propositional proofs systems”, journal of symbolic logic, vol. 44, pp. 36-50, 1979. [2] a. chubaryan, g. petrosyan, “frege systems are no monotonous”, evolutio, естественные науки, вып. 3, 12-14, 2016. [3] a. chubaryan, a. khamisyan, g. petrosyan, on some systems for two versions of many-valued logics and its properties, lambert academic publishing (lap), 2017. [4] с. м. саядян,.а. a. чубарян, o свойстве немонотонности некоторых систем выводов классического исчисления высказываний, днан ра, т. 118. no 1, 2025, 2018. [5] г. м. зограбян, с. м. саядян и а. а. чубарян, исследование свойства монотонности некоторых пропозициональных систем выводов классической и неклассических логик, днан ра, т.119, №1, сс. 33-39, 2019. [6] а. а. чубарян, с. м. саядян и г. м. зограбян, о свойствах монотонности и строгой монотонности пропозициональных систем резолюций классической и неклассических логик, sciences of europe, physics and mathematics, vol 2, no. 35, pp. 74-79, 2019. [7] a. chubaryan, a. karabakhtsyan and g. petrosyan, “on some properties of several proof systems for non classical propositional logics”, вестник рау, no. 1, pp. 5-17, 2018. [8] s.c.kleene, introduction to metamathemics, d.vannostrand company, inc,1952. [9] а. чубарян и о.болибекян, “о секвенциальных системах слабых арифметик”, днан армении, прикладная математика, 102, т. 3, 214-218, 2002. [10] a. atserias, n. galesi and r. gavalda, “monotone proofs of the pigeon hole principle”, mathematical logic quarterly, vol. 47, no. 4, pp. 461-474, 2001. о количестве минимальных тавтологий и свойствах их выводов в ряде систем логик 72 դասական և ոչ դասական տրամաբանությունների մինիմալ նույնաբանությունների քանակի և դրանց արտածումների հատկությունների մասին արսեն. ա. համբարձումյան, հայկ ա. գասպարյան, սարգիս ա. հովհաննիսյան և անահիտ ա. չուբարյան երևանի պետական համալսարան e-mail: hambardzumyanarsen99@gmail.com, haykgasparyan012@gmail.com, saqohovhannisyan0199@gmail.com, achubaryan@ysu.am ամփոփում սույն աշխատանքում ապացուցված է, որ տվյալ տրամաբանության п երկարությամբ նույնաբանությունների մինիմալ նույնաբանությունների մաքսիմալ հնարավոր քանակը կարող է լինել ցուցչային ֆունկցիա п-ից, ինչպես նաև ապացուցված է, որ դասական, ինտուիցիոնիստական, մինիմալ և մոնոտոն տրամաբանությունների յուրաքանչյուր նույնաբանության համար գոյություն ունի այնպիսի մինիմալ նույնաբանություն, որի սեքվենցիալ ձևի արտածման բարդությունը համընկնում է տրված բանաձևի սեքվենցիալ ձևի արտածման նվազագույն քայլերի հետ թվարկված տրամաբանությունների առանց հատույթի կանոնի սեքվենցիալ համակարգերում: բանալի բառեր` մինիմալ նույնաբանություններ, ասույթային տրամաբանության արտածումների սեքվենցիալ համակարգեր առանց հատույթի կանոնի, արտածման քայլեր,մոնոտոնիկ համակարգեր, խիստ մոնոտոնիկ համակարգեր: a. aмбарцумян, г. гаспарян, с. ованнисян и а.чубарян 73 on the numbers of minimal tautologies and properties of their proofs in classical and nonclassical logic arsen a.hambardzumyan, hayk a. gasparyan, sarkis a. hovhannisyan, anahit a. chubaryan yerevan state university e-mail: hambardzumyanarsen99@gmail.com, haykgasparyan012@gmail.com, saqohovhannisyan0199@gmail.com, achubaryan@ysu.am abstract it is proved in this paper that the number of minimal tautologies for any given logic tautology of size п can be an exponential function in п, and it is also proved that for every tautology of the given logic there is some minimal tautology such that the number of its sequential form proof steps is equal to minimal steps in the proof of sequential form for the given tautology in cut-free sequent systems for classical, intuitionistic, joganssons and monotone logics. keywords: minimal tautology, cut-free sequent proof systems of propositional logic, steps of proof, monotonous system, strong monotonous system. submitted 15.06.2019, accepted 22.10.2019. mathematical problems of computer science 39, 21--30, 2013. 21 new fingerprint image thinning algorithm davit a. kocharyan institute for informatics and automation problems of nas of ra e-mail: david.kocharyan@gmail.com abstract minutiae-based fingerprint recognition systems rely heavily on efficient and fast image enhancement algorithms. an image thinning is a very important stage of the image enhancement. a good thinning algorithm preserves the structure of the original fingerprint image, reduces the amount of data needed to process and helps to improve the feature extraction accuracy and efficiency. in this paper we describe and compare some of the most used fingerprint thinning algorithms. the results show that the faster algorithms have difficulty in preserving connectivity. zhang and suen’s algorithm gives the least processing time, while guo and hall’s algorithm produces the best skeleton quality. in this paper we propose a modified zhang and suen’s algorithm that is efficient and fast. some test results show that the proposed modification better preserves the structure and connectivity of the original fingerprint image. keywords: fingerprint recognition, image enhancement, image thinning; minutiae. 1. introduction fingerprint image thinning is a very important step in fingerprint recognition algorithms. in this step the ridge lines of the fingerprint image are transformed to a one-pixel thickness. this process is fundamental for fingerprint recognition algorithms [2], because the thinned images are easier to process and reduce the processing time. thinning does not change the structure of the fingerprint image: it preserves the locations of the fingerprint ridge and valley features, which makes easier to identify the global and local features of the fingerprint image (such as core, delta, minutiae points) that are used for fingerprint classification, recognition and matching [1]. an example of thinned fingerprint image is shown in figure 1 below: new fingerprint image thinning algorithm22 fig. 1. from left to right: original fingerprint image; binarized image and the corresponding thinned image. an effective and accurate thinning algorithm directly affects the fingerprint feature extraction, the matching accuracy and the results. the best known thinning algorithms fall into the following two categories: iterative and non-iterative [3]. iterative algorithms delete pixels on the boundary of a pattern repeatedly until only the unit pixelwidth thinned image remains. non-iterative distance transformation algorithms are not appropriate for general applications since they are not robust, especially for patterns with highly variable stroke directions and thicknesses. thinning, based on iterative boundary removal, can be divided into sequential and parallel algorithms. thinning is mostly done on the binarized image of the fingerprint. the mostly discussed and described thinning algorithms are based on parallel thinning, as they are fast and efficient. in this paper we intend to describe and compare the most used iterative fingerprint thinning algorithms (zhang-suen [4], guo-hall [2], abdulla et al [5], r. w. hall [6]), to understand their strengths and weaknesses and propose a modified and more efficient algorithm. 2. concepts the binary image i is described as a matrix ,nm  where ),( jix represents the binary value of the pixel ),,( ji equal to 1, if the pixel is black, or 0, if the pixel is white. any pixel which is at distance of 1 from the pixel ),( ji is considered a neighbor for that pixel. connectivity is defined as the number of neighbors to which the pixel is connected:  4-connectivity: the pixel is connected to every horizontal and vertical neighbor (see figure 2a).  8-connectivity: the pixels are connected to every horizontal, vertical and diagonal neighbor (see figure 2b). (a) (b) fig. 2. pixel connectivity: (a) 4-connectivity; (b) 8-connectivity. 3. known thinning algorithms in this chapter some known fingerprint thinning algorithms are described. d. kocharyan 23 1. zhang-suen’s algorithm the zhang-suen’s algorithm works using a 33 sized block. it is an iterative algorithm and it removes all the contour points of the image except those that belong to the skeleton. the algorithm is divided into two sub-iterations [4]. the algorithm is described below: 1. while points are deleted, do 2. for all ),( jip pixels, do 3. if (a) 6)(2 1  pb (b) 1)( 1 pa (c) one of the following is true: 1. 0642  ppp in odd iteration, 2. 0842  ppp in even iteration, (d) one of the following is true: 1. 0864  ppp in odd iteration, 2. 0862  ppp in even iteration, then 4. delete pixel ),( jip , where )( 1pa is the number of 0 to 1 transitions in the clockwise direction from 9p , )( 1pb is the number of non-zero neighbors of :1p .)( 9321 ppppb   1p is not deleted, if any of the above conditions is not met. the algorithm is fast, but fails to preserve such patterns that have been reduced to 22 squares: they are completely removed. it also has problems preserving connectivity with diagonal lines and identifying line endings. 2. guo-hall’s algorithm the algorithm works using a 22 sized block. )( 1pc is defined as the number of distinct 8connected components of 1p [2]. )( 1pb is defined as the number of non-zero neighbors of 1p : ,  and  symbols are defined as logical completing, and, and or, respectively. )( 1pn is defined as: )},(),(min{)( 11111 pnpnpn  where ).()()()()( ),()()()()( 9876543212 8765432911 pppppppppn pppppppppn   )( 11 pn and )( 12 pn divide the neighbors of 1p into four pairs and calculate the number of pairs that contain one or two non-zero elements. the algorithm is described below: 1. while points are deleted, do 2. for all ),( jip pixels, do 3. if (a) 1)( 1 pc (b) 3)(2 1  pn (c) one of the following is true: 1. 0)( 4532  pppp in odd iteration, 2. 0)( 8976  pppp in even iteration, new fingerprint image thinning algorithm24 then 4. delete pixel ),( jip . when ,1)( 1 pb 1p is an ending point and .1)( 1 pn but when ,2)( 1 pb 1p could also be a nonending point. the definition of )( 1pn preserves the ending points and removes the redundant pixel in the middle of the curve. guo-hall [5] algorithm is more precise than zhang-suen’s [4] algorithm, but it needs more computational time to execute. 3. abdulla et al’s algorithm the algorithm uses a 33 sized block and consists of two sub-iterations [5]. the first sub-iteration scans the image horizontally using a 43 sized block (fig. 3a). any two points, which are horizontally adjacent to each other and horizontally isolated from other pixels, are deleted. the second sub-iteration scans the image vertically using a 34 sized block (fig. 3b). any two points, which are vertically adjacent to each other and vertically isolated from other points, are deleted. (a) (b) fig. 3. (a) 43 sized block; (b) 34 sized block. the algorithm is described below: 1. while points are deleted, do 2. for all pixels ),( jip do 3. first iteration: 4. if (a) 161,1  psp or (b) 122,1  psp or (c) 1]()[(])()[( 7656592332  pppppppppp then 5. delete pixel 1p 6. where ,9231,1 pppsp  ,7562,1 pppsp  ,  and  are defined as logical completing, and, or, respectively. 7. if 1p is not deleted then 8. if 1)()( 125103  pppp then 9. delete pixel .4p 10. second iteration: 11. if (a) 141,2  psp or (b) 182,2  psp or d. kocharyan 25 (c) 1)]()[(])()[( 3455498778  pppppppppp then 12. delete pixel 1p . 13. where ,7891,2 pppsp  ,5432,2 pppsp  ,  and  are defined as logical completing, and, or, respectively: 14. if 1p is not deleted then 15. if 1)()( 105127  pppp then 16. delete pixel .6p 4. r. w. hall’s algorithm the algorithm [6] consists of two parallel sub-iterations, functions first identifying in parallel all deletable pixels and then deleting in parallel all those deletable pixels except certain pixels which should be maintained to preserve the connectivity in an image. the algorithm is described below: 1. while pixels are deleted, do 2. for all pixels ),( jip do 3. determine whether ),( jip should be deleted 4. if (a) 7)(1 1  pb (b) sp '1 8-neighborhood contains exactly one 4-connected component of 1’s. then 5. ),( jip should be deleted 6. for all ),( jip pixels, do 7. if (a) 162  pp and 4p is deletable, (b) 184  pp and 6p is deletable, (c) 654 ,, ppp are deletable, then 8. do not delete pixel ),( jip . the above-mentioned conditions preserve local connectivity, end-points and 22 sized patterns. 4. comparison of the algorithms during the comparison, the evaluation was based on the following criteria: connectivity, spurious branches, convergence to unit width and data reduction efficiency/computational cost. connectivity preservation of a fingerprint pattern is a crucial fingerprint recognition, as the disconnected patterns may produce false minutiae points. spurious branches also produce false minutiae points. some post processing operations may be applied to remove spurious branches, but it will cost extra processing operations and execution time. a perfect skeleton must be unitary, meaning that it does not contain any of the patterns given in figure 4: new fingerprint image thinning algorithm26 fig. 4. patterns of non-unitary skeletons. jang and chin [7] introduced a measure tm to compute the width of the thinned ms skeleton:   ,1 41 m k km t sarea qsarea m          where ][area is the operation that counts the number of pixels with the value of 1. if ,1tm then ms is a perfect unitary skeleton [7]. an effective thinning algorithm should also be fast. a measure to evaluate both the data reduction efficiency and the computational cost was defined by jang and chin [7] as: , ][ ][][ ,1min          sarean sareasarea m md where n is the number of parallel operations required to converge, and s is the original input image. this measure has a value between 0 and 1. the larger value means the higher efficiency [7]. to compare the above described algorithms, they have been applied to thin five different images, shown in figure 5. the results of the values tm and dm are given in table 1. fig. 5. five different fingerprint images used for comparing the thinning algorithms: (1) 276x408 pixels; (2) 408x480 pixels; (3) 264x264 pixels; (4) 336x336 pixels; (5) 420x600 pixels. d. kocharyan 27 table 1: results of the tests. image algorithm results 1  abdulla et.al  guo-hall  hall  zhang-suen 0.996 0.998 0.991 0.698 0.117 0.062 0.083 0.129 2  abdulla et.al  guo-hall  hall  zhang-suen 0.974 0.997 0.988 0.790 0.120 0.065 0.085 0.137 3  abdulla et.al  guo-hall  hall  zhang-suen 0.997 0.998 0.999 0.864 0.122 0.061 0.084 0.130 4  abdulla et.al  guo-hall  hall  zhang-suen 0.978 0.993 0.993 0.747 0.105 0.056 0.079 0.115 5  abdulla et.al  guo-hall  hall  zhang-suen 0.985 0.997 0.993 0.695 0.118 0.064 0.085 0.134 the results show that guo-hall’s algorithm best preserves the structure of the image, but the efficiency and speed are low, giving the result of ,062.0dm a comparatively low value. zhangsuen’s algorithm is most frequently used in literature and shows an average .129.0dm but in some cases it does not preserve the structure of the image and even removes some ridges and endpoints [8]. 5. proposed modification we propose a slight modification to the zhang-suen’s algorithm, as it is the most efficient one. the proposed modification improves and preserves the structure of the image and stops an unwanted removal of lines and end-points. in the original algorithm the end-points are detected by ,1)( 1 pa but it does not apply to diagonal ridges that have 2 pixel thickness, as in that case .2)( 1 pa the following conditions can be added to zhang-suen’s algorithm to eliminate those problems: in odd iterations: when ,2)( 1 pa the following conditions are checked: in odd iterations: when ,2)( 1 pa the following conditions are checked: 1. p4 x p6 = 1 and p9 = 0 or 2. p4 x p2 = 1 and = 1 in even iterations: when a(p1) = 2, the following conditions are checked: 1. p2 x p8 = 1 and p5 = 0 or 2. p6 x p8 = 1 and = 1 these conditions are added to avoid deleting diagonal lines and preserve connectivity. new fingerprint image thinning algorithm28 the modified algorithm is described below: 1. while points are deleted, do 2. for all p(i, j) pixels, do 3. if 2 ≤ b( ) ≤ 6 4. if a(p1) = 1 and (a) one of the following is true: 1. p2 x p4 x p6 = 0 in odd iteration, 2. p2 x p4 x p8 = 0 in even iteration, (b) one of the following is true: 1. p4 x p6 x p8 = 0 in odd iteration, 2. p2 x p6 x p8 = 0 in even iteration, then 5. delete pixel p(i, j). 6. else if a(p1) = 2 and (a) one of the following is true: 1. p4 x p6 = 1 and p9 = 0, in odd iteration, 2. p2 x p8 = 1 and p5 = 0, in even iteration, (b) one of the following is true: 1. p4 x p2 = 1 and = 1 in odd iteration, 2. p6 x p8 = 1 and = 1, in even iteration, then 5. delete pixel p(i, j). where a(p1) is the number of 0 to 1 transitions in the clockwise direction from p9, b(p1) is the number of non-zero neighbors of p1: .)( 9321 ppppb   p1 is not deleted, if any of the above conditions is not met. after adding the above-mentioned conditions, zhang suen’s algorithm preserves the structure and fairly maintains connectivity. a comparison of skeletons produces the original algorithm and the modified version shown in figure 6, where the corresponding minutiae points are also shown: fig. 6. left to right: modified and original versions of zhang suen’s algorithm d. kocharyan 29 a noticeable improvement in maintaining structure and connectivity can be seen. the modified algorithm has been applied to thin five different images, shown in figure 5. the results of the values and are given in the table below: table 2: results of the tests image results 1 0.897 0.130 2 0.943 0.132 3 0.976 0.143 4 0.935 0.113 5 0.896 0.136 the modified algorithm shows a noticeable improvement with average mt= 0.929 and an average= 0.130. 6. conclusion and future work in this paper we discussed the most frequently used fingerprint thinning algorithms and showed their comparisons. zhang suen’s [4] algorithm proves to be the most efficient one, and with the proposed modification it shows the best result among all with respect to the comparison criteria. as to the next step, a creation of a fingerprint recognition software solution based on minutiae matching and the proposed thinning algorithm are planned. references [1] d. maltoni and d.maio, handbook of fingerprint recognition, springer, 2009. 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[8] g. raju and y. xu, “study of parallel thinning algorithms,” ieee international conference on systems, man, and cybernetics, vol. 01, pp. 661–666, 1991. [9] s. prabhakar, a. k. jain and s. pankanti, “learning fingerprint minutiae location and type”, pattern recognition, vol. 36, no. 8, pp. 1847–1857, 2003. [10] k. mali and s. bhattacharya, “fingerprint recognition using global and local structures”, international journal on computer science and engineering, vol. 3, no. 1, pp. 161--172, 2011. submitted 15.11.2012, accepted 06.02.2013. new fingerprint image thinning algorithm30 նոր մատնահետքերի պատկերների բարակեցման ալգորիթմ դ. քոչարյան ամփոփում հատուկ կետերի հիման վրա մատնահետքերի ճանաչման համակարգերում մեծ դեր ունեն արագ և արդյունավետ պատկերների բարելավման ալգորիթմները: պատկերների բարելավման կարևոր փուլ է հանդիսանում պատկերի բարակեցումը: բարակեցման արդյունքում պետք է պահպանվեն օրիգինալ պատկերի կառուցվածքային առանձնահատկությունները: ալգորիթմի աշխատանքի ընթացքում նվազեցվում է մշակման համար պահանջվող տվյալների քանակը` դրանով բարձրացնելով մատնահետքի հատկանիշների դուրս բերման ճշգրտությունը և արդյունավետությունը: այս աշխատանքում համեմատվում են հայտնի և հաճախ օգտագործվող մատնահետքերի բարակեցման ալգորիթմները: արդյունքները ցույց են տալիս, որ արագագործ ալգորիթմները հիմնականում չեն պահպանում պատկերի կառուցվածքը և կապակցվածությունը: zhang և suen-ի ալգորիթմը ամենաարագագործն է, իսկ guo և hallի ալգորիթմը լավագույնն է պահպանում պատկերի կառուցվածքը: առաջարկվում է ալգորիթմ, որը հանդիսանում է zhang և suen-ի ալգորիթմի ձևափոխությունը, արագագործ է և արդյունավետորեն է պահպանում մատնահետքի պատկերի կառուցվածքը: փորձերի արդյունքները ցույց են տալիս, որ առաջարկվող ալգորիթմը զգալի առաջադիմություն է ցուցաբերում պատկերի կառուցվածքի և կապակցվածության պահպանման մեջ` միաժամանակ լինելով արագագործ և արդյունավետ: новый алгоритм утончения изображений отпечатков пальцев д. кочарян аннотация системы распознавания отпечатков пальцев, в основном опираются на алгоритмы, которые эффективно и быстро улучшают изображение. утончение изображения является очень важным этапом в улучшении изображения. хороший утончающий алгоритм сохраняет структуру оригинального изображения отпечатков пальцев, уменьшает количество данных необходимых для обработки и улучшает точность и эффективность извлечения характерной черты отпечаткa. в данной работе мы описываем и сравниваем некоторые из наиболее часто используемых алгоритмов утончения изображения отпечатков пальцев. результаты показывают, что быстрые алгоритмы плохо сохраняют соединительность. алгоритм zhang и suen обрабатывает в наикратчайший промежуток времени, а алгоритм guo и hall производит наилучшее качество скелета. в данной работе мы предлагаем модифицированный алгоритм zhang и suen, который работает эффективно и быстро. результаты тестов показывают, что предлагаемая модификация лучше сохраняет структуру и соединительность оригинального изображения отпечатков пальцев. microsoft word tpel1.doc mathematical problems of computer science 32,101--106, 2009. 101 detection and classification of objects by applying genetic programming nerses a. safaryan institute for informatics and automation problems of nas ra nerses.s@gmail.com abstract this article is devoted to discussion of the problem of detection and classification of objects in digital images by using genetic programming (gp). references [1]. c. harris and b. buxton, “evolving edge detectors with genetic programming,” proceedings of the genetic programming, 1st annual conference, cambridge, ma, usa, mit press, pp. 309-314, 1996. [2]. r. poli, “genetic programming for feature detection and image segmentation”, in evolutionary computation, t. c. forgarty (ed.), pp. 110-125, 1996. [3]. j. r. koza, genetic programming ii: automatic discovery of reusable programs, mit press, 1994. [4]. s. a. stanhope and j. m. daida, “genetic programming for automatic target classification and recognition in synthetic aperture radar imagery”, proceeding conference on evolutionary programming vii., pp. 735-744, 1998. [5]. b. bhanu and y. lin, “learning composite operators for object detection”, proceedings of the genetic and evolutionary computation conference, pp.1003-1010, 2002. úµû»ïïý»ñç ñ³ûïý³µ»ñáõùá ¨ ¹³ë³ï³ñ·áõùá ·»ý»ïçï íñ³·ñ³íáñù³ý ïçñ³éù³ùµ ü. ê³ý³ñû³ý ²ù÷á÷áõù ²ßë³ï³ýùá ýíçñí³í ¿ ·»ý»ïçï íñ³·ñ³íáñ³ù³ý ùççáóáí ãí³ûçý å³ïï»ñý»ñáõù ûµû»ïïý»ñç ñ³ûïý³µ»ñù³ý ¨ ¹³ë³ï³ñ·ù³ý ëý¹ñç ùýý³ñïù³ýá: microsoft word art.doc îçµ»éý»ïçï³ûç ¨ ñ³ßíáõ³ï³ý ï»ëýçï³ûç ù³ã»ù³ïçï³ï³ý ñ³ñó»ñ 26, 2006, 72–75. 72 ð³ñ³í³ûçý îáíï³ëç ·»ï»ñç ùáýçãáñçý·ç ïíû³éý»ñç µ³½³ûç õ»ï³í³ñáõùá »ñïñ³ï»õ»ï³ïí³ï³ý ï»ëýáéá·ç³ûç ïçñ³éù³ùµ ø»éçý» ì. ²ùçñë³ýû³ý ðð ¶²² ¾ïáéá·³ýááëý»ñ³ûçý ñ»ï³½áïáõãûáõýý»ñç ï»ýïñáý ²ù÷á÷áõù ²ßë³ï³ýùá ýíçñí³í ¿ ·»ï»ñç ùáýçãáñçý·ç ³ñ¹ûáõýùý»ñç ñ³í³ù³·ñù³ýá, ùß³ïù³ýý áõ ¿é»ïïñáý³ûçý ù³ñ﻽ý»ñç ï»ëùáí ³ñï³å³ïï»ñù³ýá: ²û¹ ýå³ï³ïáí ïçñ³éí»é ¿ arcview »ñïñ³ï»õ»ï³ïí³ï³ý íñ³·ñ³ûçý ÷³ã»ãá, áñá çýýáñù³óç³ûç ³ùµáõç³óù³ý ýáñ ï»ëýáéá·ç³ ¿: òáõûó ¿ ïñí»é, çýãå»ë ï³ñ»éç ¿ ³é³ç³¹ñí³í ëý¹ñç éáõíù³ý ýå³ï³ïáí áý¹é³ûý»é íñ³·ñ³ûçý ÷³ã»ãç ñý³ñ³íáñáõãûáõýý»ñá: ¶ñ³ï³ýáõãûáõý [1]. лебедева н. arcview 3.0 gisгигантский шаг вперед. arcreview, 1997, n 1, с. 4. [2]. тикунов в. математизация тематической картографии. владивосток, 1986, 24 с. [3]. цветков в. геоинфомационные системы и технологии. –м., 1998, 286 с. south caucasian river monitoring database management applying gis technology m. amirkhanyan abstract the article is devoted to river monitoring data collection and processing and their presentation as electronic maps. for such a purpose, arcview gis program was applied which is an innovative technology of data integration. it is shown how to expand program possibilities for the solution of stated problems. начиная с начала 2000 года осуществляется внедрение ghis в здравоохранении, в рамках принятого проекта о реформирование информ mathematical problems of computer science 49, 79--91, 2018. on the way to dominating cognition edward m. pogossian institute for informatics and automation problems of nasra e-mail:epogossi@aua.am abstract humans become powered enough to question the further types of their being in the universe but have not answers yet whether the solutions are in the corrections of their genomes , discovering of new types of organizations of humans or in transition to a new type of descends, humanoid machines or others. assuming that scenarios with dominating cognition are the most promising to resolve the challenges of evolvement of humans and are the most expected for the next stage of their being, while the kernel of effective cognition is universal for all types of further being of humans, we present constructive models of the kernel, question the completeness of representations and ways of its further study. keywords: mental systems, adequate, constructive modeling, cognition,negentropics. 1. introduction 1.1. 1. following the cosmological views [33], the vast majority of realities we classify as entropics while interpreting schrodinger [6] we classify as negentropics or negs, an island of not entropic realities r that can gain energy from others to preserve in space and time certain roots, i.e., realities, say, constituents or doings of r, determining the identity of r. roots of negs, as it succeeds from definitions, have to include, at least, means to gain energy and ones to preserve those means that, in turn, require means to represent and affect realities, i.e., sensors and effectors, then, means to classify utilities, damagers and uncertains, i.e., classifiers of favorable, damaging or uncertain yet with respect to (wrt) the roots realities, as well as controllers ofthe doings. 1.2.1. we classify cellular realities, cellulars, as a type of negentropics that include alive cells in the roots able to reproduce and compose themselves into the organism while classifying humans as a type of cellulars. 1.2.1. mental doings of humans are backing the root and induced goals. they are either genomic or cognized in the life time while cognized mainly by acquisition from the cultures of communities. 79 on the way to dominating cognition 80 mental doings are represented by mental doers, mdoers, while mental classifying by mdoers or systems of mdoers, msystems (mss). mdoers and mss, have, we assume, at least, ids and can be processed consciously, particularly, for explaining and understanding doings/doers in communications. 1.2.2. mdoers include algorithms and are based, we assume, on and do over ids of sensors, elaborate instructions for the effectors and are governed by the controllers. mss induce systemic classifiers, say, factories, computers, chess positions. mss, particularly, represent the goals, classify coexistence of nominated utilities/goals, i.e., represent relationships between them, can be processed to learn new utilities, to enhance effectiveness of mss or communicate them. 1.2.3. acquisition and accumulation of mss, as well as revelation of mss, particularly, by learning new utilities or by enhancement of effectiveness of ad hoc mss, comprise the balk of doings of a type of mss, cognizers. processing of mss is governed, we assume, by controllers. for example, controllers explain mss “humans” of the author by resolving it into this, ongoing text, namely, by corresponding english words to ids of constituents of the mss. that resolution, in general, could be started from any constituents of target mss, selected constituents, thus their ids could be chained in a variety of modes and details depending, particularly, on the intensions of the author. 1.2.4. a mighty way of enhancement of effectiveness of mss, and thus, cognizers, is the regularization of classifiers induced by mdoers and mss [47]. namely, classifiers cl of members x of communities c are regularized in c if accompanied by ontological in c methods, instructions allowing x regularly provide positive samples of inputs of cl, as well as let the members of c do the same by communicating with x. in constructive regularization, those samples can be provided deterministically and without any involvement of cellulars while, otherwise, they can be grown up from a priory given prototypes like cells or crystals, be the products of services to humans or machines. 1.3. accelerated mental power approaches humans to “homo deus” [34] with a chance of reliable preservation of cellular roots while, in parallel, to entire crashing of cellulars on the earth. the kernel of that strength is in the effectiveness of organizations of humans while the weakness is caused by the imperfectness of those organizations and their genomic heritage. humans become powered enough to question the further types of their negentropic being in the universe but have not answers yet whether the solutions are in the corrections of their genomes, discovering of new types of organizations of humans or in transition to a new type of descends, humanoid machines or others. it is worth reminding that nowadays autonomous agents, a type of constructed negentropics, are challenging the cellular being of humans at all, as well as recall that following the buddhist and indian sources, humans being lemuroids or atlantis in the far past could have not only cellular types of negentropicity [35]. 1.4. 1. in what follows, first, we present the modified specification of the models of mss, mentals, introduced in [47], refine systemic classifiers and constructive regularization of classifiers. we continue questioning the ways of proving that mentals can be adequate constructive models of mss. in addition to the approach in [47], based on examining performances of particular mss with the corresponded them mentals, we analyze the ways to achieve consistency of performances of structural models of connectivity neuron nets, for example, artificial neuron nets, with purely functional models of mss, mentals. assuming that the kernel of effective cognition is universal for all types of negs, thescenarios with dominating cognition are the most promising to resolve the challenges of evolvement of e. pogossian 81 humans and are the most expected for the next stage of their being, finally, we question the completeness of representation of the kernel and ways of its further study. 1.5. our models are based on and try to fuse findings of many outstanding researchers. we refer to some of their publications [1-35] to study them in depth, as well as refer to some works [36]-[42], which can add to understanding of our ideas and their approbations [43]-[48]. 2.constructed mental systems 2.1.1. doers, in general, are, we assume, realities having inoutput parts and for available inrealities, i.e., realities at the input parts, either elaborate certain output realities or stay passive. inoutrealities comprise their inoutdomains, or inout-doms. indoms wrt outputs are split into classes of equality, particularly, class of uncertainty (?) if inrealities don’t cause outputs. 2.1.2. doers are do-classifierscl if indoms are split into two classes +cl and ?cl; otherwise they are corresponders, cors. apparently, identifiers of do-classifiers cl by themselves are sufficient to indicate their classes of equality, i.e., the positives +cl, while classes of cors can be indicated by pairing those identifiers with corresponding outputs. 2.1.3. doers of type of classifiers are sensors if inrealities are not necessarily pre-classified, of type of cors are effectors if inrealities are necessarily classified while are controllers if both in outrealities are necessarily classified. 2.2.1.realities i{i} are identifiers, ids, of realities r{r} and z{z} wrt z if to any r,z unique ids i(r), i(z) are corresponded to any r,z certain classifiers are linked allowing by ids i to recall the corresponding r or z any r can address any z to recall any r, z. identified realities of given r, z paired with their ids are named nominals wrt z. z and r can coincide for, say, r presenting members of communities or their mdoers. controllers cns, are assumed, can assign ids to given mdoers aimed to control their processing and inoutinteractions with realities. realities of z can be interpreted as sets of cns controlling in certain ways realities of r analogically to servers of “star” types controlling networks of computers or, seemingly, analogically to unicellular controllers. 2.2.2. nominals wrt cns where realities of r are outputs of doers, particularly, sensors, controllers or effectors, are named otids while sets of otids of doers d are the alphabets of d. and sets of otids comprised from only some representatives of alphabets a1,a2, …,an of doers d1,d2,…, dn are words in a1,a2, …,an. 2.2.3. classifiers of n-tuples of nominals are n-place relationships named rels for n=2. rels (a,b) can be depended or not on the orders of their arguments. 2.3.1. systems h over nominals nls containing rels rls, i.e., rls a s t h e s t r u c t u r e s o f t r u t h va lu e s . de¯nition 1. l e t ¼m b e a fu z z y e qu a lit y o n m: a n ( n + 1 ) ¡ a r y fu z z y r e la t io n ½ o n a s e t m is c a lle d a n n¡ a r y fu z z y o p e r a t io n w.r .t . ¼m a n d ¼m n if we h a ve t h e fo llo win g c o n d it io n s e xt e n s io n a lit y: ( p ¼m n p0 ) ­ ( y ¼m y0 ) ­ ½( p; y ) · ½( p0; y0 ) 8p; p0 2 m n; 8y; y0 2 m; fu n c t io n a lit y: ½( p; y ) ­ ½( p; y0 ) · y ¼m y0 8p 2 m n; 8y; y0 2 m; fu lly-d e fe n d : _ y2m ½( p; y ) = 1 8p 2 m n; wh e r e ( a1; :::; an ) ¼m n ( b1; :::; bn ) = vn i=1 ( ai ¼m bi ) : w e s a y t h a t ½ is a fu z z y o p e r a t io n o n m wit h a r it y n: de¯nition 2 [1 ]. a n a lg e b r a wit h l -e qu a lit y o r l ¡a lg e b r a o f t yp e <¼; f > is a t r ip le t m = < m; ¼m ; f m > s u c h t h a t < m; f m > is a n a lg e b r a o f t yp e < f > a n d ¼m is a n l¡ e qu a lit y o n m s u c h t h a t e a c h f m 2 f m is c o m p a t ib le wit h ¼m ; i.e . ( a1 ¼m b1 ) ­ ::: ­ ( an ¼m bn ) · f m ( a1; ::; an ) ¼m f m ( b1; :::; bn ) fo r e a c h n¡ a r y f 2 f a n d e ve r y a1; b1; :::; an; bn 2 m: de¯nition 3. a n a lg e b r a wit h fu z z y o p e r a t io n s o f t yp e <¼; f > is a t r ip le t m =< m; ¼m ; fm > s u c h t h a t ( i) ¼m is a fu z z y e qu a lit y o n t h e s e t m; ( ii) fm is t h e s e t o f fu z z y o p e r a t io n s o n t h e s e t m: to s im p ly, we c a ll f-a lg e b r a s in s t e a d o f t h e a lg e b r a wit h fu z z y o p e r a t io n s . t heor em 1. l e t l b e a h e yt in g a lg e b r a ( ^ = ­ ) a n d m =< m; ¼m ; f m > b e a n l ¡ a lg e b r a o f t yp e <¼; f > : l e t f m : m n £m ! l wit h f m ( p; y ) = f m ( p) ¼m y fo r e ve r y n¡ a r y f m 2 f m a n d fo r a ll p 2 m n; y 2 m: th e n m =< m; ¼m ; f m > is a n f ¡a lg e b r a o f t yp e <¼; f > : de¯nition 4. l e t m =< m; ¼m ; fm > b e a n f ¡a lg e b r a o f t yp e <¼; f > : a n l ¡ r e la t io n ( b in a r y fu z z y r e la t io n ) µ o n m is c a lle d a c o n g r u e n c e o n m if ( i) µ is a fu z z y e qu iva le n c e o n m ; ( ii) µ is c o m p a t ib le wit h ¼m ; i.e . ( a ¼m b ) ­ ( a0 ¼m b0 ) ­ µ ( a; a0 ) · µ ( b; b0 ) fo r e ve r y a; a0; b; b0 2 m ; ( iii) ^ni=1µ ( ai; bi ) ­ µ ( y; y0 ) ­ f m ( a1; :::; an; y ) · f m ( b1; :::; bn; y0 ) fo r e ve r y n¡a r y fu z z y o p r a t io n f m 2 fm a n d a1; :::; an; y; y0 2 m: th e o r d in a r y s e t o f a ll c o n g r u e n c e s o n f ¡a lg e b r a m is d e n o t e d b y con( m) : de¯nition 5. l e t µ b e a c o n g r u e n c e o n a n f ¡ a lg e b r a m =< m; ¼m ; fm > o f t yp e <¼; f > : a fa c t o r a lg e b r a o f m b y µ is a n f ¡a lg e b r a m=µ =< m=µ; ¼m=µ; fm=µ > o f t yp e <¼; f > s u c h t h a t s. davidov, j. hatami 5 1 ( i) [a]µ ¼m=µ [b]µ = µ ( a; b) fo r e a c h [a]µ; [b]µ 2 m=µ; ( ii) f m=µ ( [a1]µ; :::; [an]µ; [y]µ ) = f m ( a1; :::; an; y ) fo r e ve r y n¡ a r y o p e r a t io n f m=µ 2 fm=µ a n d fo r a r b it r a r y a1; :::; an; y 2 m, wh e r e m=µ = f[a]µj a 2 mg a n d [a]µ = fa0j µ ( a; a0 ) = 1 g: de¯nition 6 [1 ]. l e t m a n d n b e l ¡ a lg e b r a s o f t yp e <¼; f > : a m a p p in g h : m ! n is c a lle d a m o r p h is m o f m t o n if ( i) h( f m ( a1; :::; an ) ) = f n ( h( a1 ) ; :::; h( an ) ) fo r e ve r y n¡a r y f 2 f a n d a r b it r a r y a1; :::; an 2 m; ( ii) a ¼m b · h( a ) ¼n h( b ) fo r e ve r y a; b 2 m: if a ¼m b = h( a ) ¼m h ( b ) fo r e ve r y a; b 2 m; t h e n h is c a lle d a n e m b e d d in g o f m t o n : de¯nition 7. l e t m =< m; ¼m ; fm > a n d n =< n; ¼n ; fn > b e t wo f ¡ a lg e b r a s o f t yp e <¼; f > : a m a p p in g h : m ! n is c a lle d a m o r p h is m o f m t o n if ( i) a ¼m b · h( a) ¼n h( b ) fo r a ll a; b 2 m; ( ii) f m ( a1; :::; an; y ) = f n ( h ( a1 ) ; :::; h( an ) ; h( y ) ) fo r e ve r y n¡a r y o p e r a t io n f m 2 fm a n d a r b it r a r y a1; :::; an; y 2 m: a m o n o m o r p h is m is a n in je c t ive m o r p h is m , a n e m b e d d in g is a m o r p h is m s u c h t h a t , fo r e ve r y a; b 2 m; a ¼m b = h( a ) ¼n h( b ) ; fo r a ll f m 2 fm wit h a r it y n a n d fo r a ll a1; :::; an 2 m: a n e p im o r p h is m wh ic h is a n e m b e d d in g is c a lle d a n is o m o r p h is m . t heor em 2. l e t l b e a h e yt in g a lg e b r a . l e t m a n d n b e t wo l ¡ a lg e b r a s o f t yp e <¼; f > a n d a m a p p in g h : m ! n b e a n e m b e d d in g o f m t o n : th e n h is a n e m b e d d in g o f m t o n de¯nition 8. fo r e ve r y f ¡a lg e b r a s m a n d µ 2 con( m) a m a p p in g hµ : m ! m=µ; wh e r e hµ ( a) = [a]µ fo r a ll a 2 m; is c a lle d a n a t u r a l m a p p in g . lemma 1. a n a t u r a l m a p p in g hµ fr o m f ¡ a lg e b r a s m t o a fa c t o r f ¡a lg e b r a m=µ is a n e p im o r p h is m . t heor em 3. ( ¯ r s t is o m o r p h is m t h e o r e m ) . l e t h : m ! n b e a n e p im o r p h is m o f f ¡ a lg e b r a s . th e n t h e r e is a n is o m o r p h is m g : m=µh ! n s u c h t h a t hµh ± g = h, wh e r e µh = f( a; b ) jh( a) ¼n h( b ) g: de¯nition 9. l e t m b e a n f ¡ a lg e b r a a n d á; µ 2 con( m) ; µ µ á: th e n we le t á=µ d e n o t e a n l ¡ r e la t io n o n m=µ d e ¯ n e d b y ( á=µ ) ( [a]µ; [b]µ ) = á( a; b ) fo r a ll a; b 2 m: t heor em 4. l e t m b e a n f ¡a lg e b r a a n d á; µ 2 con( m) ; µ µ á: th e n á=µ 2 con( m=µ ) : t heor em 5. ( s e c o n d is o m o r p h is m t h e o r e m ) . s u p p o s e m is a n f ¡ a lg e b r a a n d á; µ 2 con( m) ; µ µ á: th e n t h e m a p p in g h : ( m=µ ) = ( á=µ ) ! m=á d e ¯ n e d b y h( [[a]µ]á=µ ) = [a]á is a n is o m o r p h is m . r e fe r e n c e s [1 ] b ·elo h la ve k r ., v yc h o d il v ., fu z z y e qu a t io n a l lo g ic , s p r in g e r , 2 0 0 5 . [2 ] r o s e n fe ld a ., fu z z y g r o u p s , j. ma t h . a n a l. a p p l. 3 5 ( 3 ) ( 1 9 7 1 ) , 5 1 2 -5 1 7 . 5 2 algebras with fuzzy operations [3 ] b ·elo h la ve k r ., v yc h o d il v ., a lg e b r a s wit h fu z z y e qu a lit ie s , fu z z y s e t s a n d s ys t e m s , 1 5 7 ( 2 0 0 6 ) , 1 6 1 -2 0 1 . [4 ] k la wo n n f., fu z z y p o in t s , in : v . n o va k, i. p e r ¯ lie va ( e d s ) , d is c o ve r in g w o r ld wit h fu z z y l o g ic , p h ys ic a , h e id e lb e r g , ( 2 0 0 0 ) , 4 3 1 -4 5 3 . [5 ] d e m ir c i m., fu z z y fu n c t io n s a n d t h e ir fu n d a m e n t a l p r o p e r t ie s , fu z z y s e t s a n d s ys t e m s , 1 0 6 ( 1 9 9 9 ) , 2 3 6 -2 4 6 . [6 ] b o ·sn ja k i., ma d a r a s z r ., v o jo d ic g., a lg e b r a s o f fu z z y s e t s , fu z z y s e t s a n d s ys t e m s , 1 6 0 ( 2 0 0 9 ) , 2 9 7 9 -2 9 8 8 . [7 ] ig n ja t o vic j., cir ic m., b o g d a n o vic s ., fu z z y h o m o m o r p h is m s o f a lg e b r a s , fu z z y s e t s a n d s ys t e m s , 1 6 0 ( 2 0 0 9 ) , 2 3 4 5 -2 3 6 5 . [8 ] ·s o s t a k a .p ., fu z z y fu n c t io n s a n d e xt e n s io n o f t h e c a t e g o r y l -to p o f ch a n g-gogu e n l -t o p o lo g ic a l s p a c e s , 2 7 1 -2 9 4 . d:\sbornik\...\mac-random.dvi mathematical problems of computer science 24, 2005, 42{61. b ounds of e-capacity region for m ultiple-access channel with random p ar ameter ¤ ma r ia m e . h a r o u t u n ia n institue for informatics and automation problems of nas of ra e-mail armar@ipia.sci.am abstract the discrete memoryless multiple-access channel with random parameter is investigated. various situations, when the state of the channel is known or unknown on the encoders and decoder, are considered. some bounds of e-capacity and capacity regions for average error probability are obtained. refer ences [1 ] r . f. a h ls we d e ," mu lt y-wa y c o m m u n ic a t io n c h a n n e ls " , 2nd intern. sympos. inform. theory. tsahkadsor, armenia, 1971, b u d a p e s t : a ka d . k ia d o , p p . 2 3 { 5 2 , 1 9 7 3 . [2 ] r . f. a h ls we d e ," th e c a p a c it y r e g io n o f a c h a n n e l wit h t wo s e n d e r s a n d t wo r e c e ive r s " , annalsp robability, vo l. 2 . n o . 2 . p p . 8 0 5 { 8 1 4 , 1 9 7 4 . [3 ] e . a . h a r o u t u n ia n , m. e . h a r o u t u n ia n a n d a . e . a ve t is s ia n ," mu lt ip le -a c c e s s c h a n n e l a c h ie va b le r a t e s r e g io n a n d r e lia b ilit y" , izvestiya akademii nauk armenii, m atematika, vo l. 2 7 , n o . 5 , p p . 5 1 { 6 8 , 1 9 9 2 . [4 ] m. e . h a r o u t u n ia n , " on e-c a p a c it y r e g io n o f m u lt ip le -a c c e s s c h a n n e l" , ( in r u s s ia n ) izvestiya akademii nauk armenii, m atematika, vo l. 3 8 , n o . 1 , p p . 3 { 2 2 , 2 0 0 3 . [5 ] r . g. ga lla g e r , " a p e r s p e c t ive o n m u lt ia c c e s s c h a n n e ls " , ie e e trans. inform. theory, vo l. 3 1 , n o . 1 , p p . 1 2 4 { 1 4 2 , 1 9 8 5 . [6 ] a . g. d ya c h ko v, " r a n d o m c o n s t a n t c o m p o s it io n c o d e s fo r m u lt ip le a c c e s s c h a n n e ls " , p roblems of control and inform. theory, vo l. 1 3 , n o . 6 , p p . 3 5 7 { 3 6 9 , 1 9 8 4 . [7 ] j. p o ko r n y a n d h .m. w a llm e ie r , " r a n d o m c o d in g b o u n d a n d c o d e s p r o d u c e d b y p e r m u t a t io n s fo r t h e m u lt ip le -a c c e s s c h a n n e l" , ie e e trans. inform. theory, vo l. it-3 1 , p p . 7 4 1 { 7 5 0 , 1 9 8 5 . [8 ] y . s . l iu a n d b . l . h u g h e s , " a n e w u n ive r s a l c o d in g b o u n d fo r t h e m u lt ip le -a c c e s s c h a n n e l" , ie e e trans. inform. theory, vo l. it-4 2 , p p . 3 7 6 { 3 8 6 , 1 9 9 6 . ¤work was partially supported by intas grant 00-738. 4 2 m. e. haroutunian 4 3 [9 ] s . i. ge lfa n d , m. s . p in s ke r , " co d in g fo r c h a n n e l wit h r a n d o m p a r a m e t e r s " , p roblems of control and inform. theory, vo l. 8 , n o . 1 , p p . 1 9 { 3 1 , 1 9 8 0 . [1 0 ] e . a . h a r o u t u n ia n a n d m. e . h a r o u t u n ia n , " e-c a p a c it y u p p e r b o u n d fo r c h a n n e l wit h r a n d o m p a r a m e t e r " , p roblems of control and information theory, vo l. 1 7 , n o .2 , p p . 9 9 { 1 0 5 , 1 9 8 8 . [1 1 ] m. e . h a r o u t u n ia n , " b o u n d s o f e-c a p a c it y fo r t h e c h a n n e l wit h r a n d o m p a r a m e t e r " , p roblemi p eredachi informatsii, ( in r u s s ia n ) , vo l. 2 7 , n o . 1 , p p . 1 4 { 2 3 , 1 9 9 1 . [1 2 ] m. e . h a r o u t u n ia n , " n e w b o u n d s fo r e-c a p a c it ie s o f a r b it r a r ily va r yin g c h a n n e l a n d c h a n n e l wit h r a n d o m p a r a m e t e r " , trans. iiap nas r a and ysu, m athematical p roblems of computer sciences, v. 2 2 , p . 4 4 { 5 9 , 2 0 0 1 . [1 3 ] j. ja h n , " co d in g o f a r b it r a r ily va r yin g m u lt iu s e r c h a n n e ls " , ie e e trans. inform. theory, vo l. it-2 7 , n o . 2 , p p . 2 1 2 { 2 2 6 , 1 9 8 1 . [1 4 ] r . f. a h ls we d e , a r b it r a r ily va r yin g c h a n n e ls wit h s t a t e s s e qu e n c e kn o wn t o t h e s e n d e r , ie e e trans. inform. theory, vo l. it-3 2 , n o . 5 , p p . 6 2 1 { 6 2 9 , 1 9 8 6 . [1 5 ] r . f. a h ls we d e a n d n . ca i, " a r b it r a r ily va r yin g m u lt ip le a c c e s s c h a n n e ls " , p a r t 1 , ie e e trans. inform. theory, vo l. it-4 5 , n o . 2 , p p . 7 4 2 { 7 4 9 , 1 9 9 9 . [1 6 ] r . f. a h ls we d e a n d n . ca i, " a r b it r a r ily va r yin g m u lt ip le a c c e s s c h a n n e ls " , p a r t 2 , ie e e trans. inform. theory, vo l. it-4 5 , n o . 2 , p p . 7 4 9 { 7 5 6 , 1 9 9 9 . [1 7 ] a . d a s a n d p . n a r a ya n , " ca p a c it ie s o f t im e -va r yin g m u lt ip le -a c c e s s c h a n n e ls wit h s id e in fo r m a t io n " , ie e e transactions on information theory, vo l. 4 8 , n o . 1 , p p . 4 { 2 5 , 2 0 0 2 . [1 8 ] m. s . p in s ke r , " mu lt i-u s e r c h a n n e ls " , ii j oint swedish-soviet intern. workshop on inform. theory, gr ¶a n n a , s we d e n , p p . 1 6 0 -1 6 5 , 1 9 8 5 . [1 9 ] i. cs is z ¶a r a n d j. k ¶o r n e r , information theory. coding theorems for d iscrete m emoryless systems, b u d a p e s t : a ka d . k ia d o , 1 9 8 1 . [2 0 ] i. cs is z ¶a r , " th e m e t h o d o f t yp e s " , ie e e trans. inform. theory, vo l. it-4 4 , p p . 2 5 0 5 2 5 2 3 , 1 9 9 8 . öá÷áëíáõ å³ñ³ù»ïñáí µ³½ù³ùáõïù ï³åáõõáõ e-áõý³ïáõãû³ý ïçñáõûãç ·ý³ñ³ï³ï³ýý»ñá ø. º. ð³ñáõãûáõýû³ý ²ù÷á÷áõù ²ßë³ï³ýùáõù ñ»ï³½áïí³í ¿ ÷á÷áëíáõ å³ñ³ù»ïñáí áý¹ñ³ï ³é³ýó ñçßáõáõãû³ý µ³½ù³ùáõïù ï³åáõõçý: àõëáõùý³ëçñí³í »ý ï³ñµ»ñ ¹»åù»ñ, »ñµ ï³åáõõáõ íç׳ïá ñ³ûïýç ¿ ï³ù ³ýñ³ûï ïá¹³íáñçãçý ¨ ³å³ïá¹³íáñçãçý: êï³óí³í »ý áñáß³ïç ·ý³ñ³ï³ï³ýý»ñ e-áõý³ïáõãû³ý ¨ áõý³ïáõãû³ý ñ³ù³ñ, ùçççý ëë³éç ñ³í³ý³ï³ýáõãû³ý ¹»åùáõù: d:\sbornik\...\markov.dvi mathematical problems of computer science 32, 65{69, 2009. on optimal i denti¯cation of m ar kov chain distr ibution subject to the reliability cr iter ion evgueni a. haroutuniany and leader navaeiz yinstitute for informatics and automation problem of nas of ra ddag payame noor university, iran evhar@ipia.sci.am, ashkan l1380@yahoo.com abstract in this paper the identi¯cation of distribution of simple homogeneous stationary markov chain with a ¯nite number of states is studied. the problem has been formulated by ahlswede and haroutunian on identi¯cation of hypotheses and solved for the case of the sequence of independent observations. refer ences [1 ] r . f. a h ls we d e a n d e . a . h a r o u t u n ia n , \ on lo g a r it h m ic a lly a s ym p t o t ic a lly o p t im a l t e s t in g o f h yp o t h e s e s a n d id e n t i¯ c a t io n " , l ecture notes in computer science, vo l. 4 1 2 3 . \general theory of information transfer and combinatorics" springer, p p . 4 6 2 -4 7 8 , 2 0 0 6 . [2 ] e . a . h a r o u t u n ia n , \ r e lia b ilit y in m u lt ip le t e s t in g a n d id e n t i¯ c a t io n p r o b le m s " , nato science series iii: computer and systems sciences, vo l. 1 9 8 , p p . 1 8 9 { 2 0 1 , ios or e s s , 2 0 0 5 . [3 ] e . a . h a r o u t u n ia n , \ on a s ym p t o t ic a lly o p t im a l t e s t in g o f h yp o t h e s e s c o n c e r n in g ma r ko v c h a in " , (in r ussian). izvestia acad. nauk armenian ssr . seria m athem. vo l. 2 2 , n o . 1 . p p . 7 6 -8 0 , 1 9 8 8 . [4 ] l n a va e i, \ a p p lic a t io n o f l d t t o m a n y h yp o t h e s e s o p t im a l t e s t in g fo r ma r ko v c h a in " , m athematical p roblems of computer science, vo l. 3 1 , p p .7 3 -7 8 , 2 0 0 8 . [5 ] e . a . h a r o u t u n ia n , m. e . h a r o u t u n ia n a n d a . n . h a r u t yu n ya n ,\ r e lia b ilit y c r it e r ia in in fo r m a t io n t h e o r y a n d in s t a t is t ic a l h yp o t h e s is t e s t in g " , f oundations and trends in communications and information theory, vo l. 4 , n o . 2 -3 , 2 0 0 8 . 6 5 6 6 on optimal identi¯cation of markov chain distribution subject to the reliability criterion ø³ñïáíç ßõã³ûç µ³ßëù³ý ñáõë³éçáõãû³ý å³ûù³ýáí ûåïçù³é ýáõûý³ï³ý³óù³ý ù³ëçý º. ð³ñáõãûáõýû³ý, è. ü³í³ûç ²ù÷á÷áõù àôëáõùý³ëçñí³í ¿ í»ñç³íáñ ãíáí íç׳ïýáñáí ñ³ù³ë»é ëï³óçáý³ñ ø³ñïáíû³ý ßõã³ûç íç׳ﳷñ³ï³ý ýáõûý³ï³ý³óáõùá: ì³ñï³íý»ñç ýáõûý³ï³ý³óù³ý ñçùý³ëý¹çñá ó¨³ï»ñåí»é ¿ ²éëí»¹»ç ¨ ð³ñáõãûáõýû³ýç ïáõùçó ¨ éáõíí»é ³ýï³ë ¹çï³ñïáõùý»ñç ¹»åùç ñ³ù³ñ: d:\sbornik\...\untitled1.dvi mathematical problems of computer science 26, 2006, 91{96. on h ypothesis optimal t esting for t wo di®er ently distr ibuted objects e vg u e n i a . h a r o u t u n ia n a n d a r a m o. y e s s a ya n institue for informatics and automation problems of nas of ra e-mail evhar@ipia.sci.am abstract hypotheses identi¯cation for two objects having di®erent distributions from two given probability distrubutions was examined by r. ahlswede and e. haroutunian. we investigate a model with two objects having di®erent distributions from three possible distributions. the matrix of all possible pairs of asymptotical interdependencies of the realabilities (error probability exponents) for logarithmically asymptotically optimal testing is studied. refer ences [1 ] e . h a r o u t u n ia n " l o g a r it h m ic a lly a s ym p t o t ic a lly o p t im a l t e s t in g o f m u lt ip le s t a t is t ic a l h yp o t h e s e s " , p roblems of control and information theory, vo l. 1 9 ( 5 -6 ) , p p . 4 1 3 { 4 2 1 , 1 9 9 0 . [2 ] r . a h ls we d e a n d i. w e g e n e r " s e a r c h p r o b le m s " . w ile y, n e w yo r k, 1 9 8 7 . [3 ] e . h a r o u t u n ia n " r e lia b ilit y in m u lt ip le h yp o t h e s e s t e s t in g a n d id e n t i¯ c a t io n " . n a to s c ie n c e s e r ie s , iii: co m p u t e r a n d s ys t e m s s c ie n c e s -v o l 1 9 8 , p p . 1 8 9 -2 0 1 . p r o c e e d in g s o f t h e n a to a s i, y e r e va n 2 0 0 3 , 2 0 0 5 ios p r e s s . [4 ] e . h a r o u t u n ia n a n d p . h a ko b ya n " on lo g a r it h m ic a lly o p t im a l h yp o t h e s is t e s t in g o f t h r e e d is t r ib u t io n s fo r p a ir o f in d e p e n d e n t o b je c t s " , m athematical p roblems of computer sciences, vo l. x x iv , y e r e va n 2 0 0 5 , p p . 7 6 -8 1 . [5 ] i. cs is z ¶a r a n d p . c. s h ie ld s " in fo r m a t io n t h e o r y a n d s t a t is t ic s : a t u t o r ia l" . f oundations and trends in communications and information theory. v o lu m e 1 , is s u e 4 , 2 0 0 4 . [6 ] t. m. co ve r a n d j. a . th o m a s " e le m e n t s o f in fo r m a t io n t h e o r y" . w ile y, n e w y o r k, 1 9 9 1 . [7 ] r . e . b e c h h o fe r , j. k ie fe r a n d m. s o b e l " s e qu e n t ia l id e n t i¯ c a t io n a n d r a n kin g p r o c e d u r e s " , th e u n ive r s it y o f ch ic a g o , 1 9 6 8 . 9 1 9 2 on optimal testing of three hypotheses for two dependent objects î³ñµ»ñ µ³ßëáõùý»ñáí »ñïáõ ûµû»ïïý»ñç ýï³ïù³ùµ »ñ»ù í³ñï³íý»ñç ûåïçù³é ëïáõ·ù³ý ù³ëçý º. ð³ñáõãûáõýû³ý ¨ ². ºë³û³ý ²ù÷á÷áõù ¸çï³ñïí³í ¿ »ñïáõ ï³ëû³é ûµû»ïïý»ñçó ï³½ùí³í ùá¹»éç ñ³ù³ñ »ñ»ù í³ñï³íý»ñç ëïáõ·ù³ý ëý¹çñá: ºñ»ù ñ³í³ý³ï³ý³ûçý µ³ßëáõùý»ñá ñ³ûïýç »ý, ¨ ûµû»ïïý»ñá áý¹áõýáõù »ý ùçùû³ýó ãïñïýáõ µ³ßëáõùý»ñ ïñí³íý»ñçó: ²ûë ùá¹»éç ñ³ù³ñ áõëáõùý³ëçñí»é ¿ ûåïçù³é ï»ëï³íáñù³ý ¹»åùáõù µáéáñ ñý³ñ³íáñ ½áõû·»ñç ëë³éý»ñç ñ³í³ý³ï³ýáõãûáõýý»ñç óáõóçãý»ñç (ñáõë³éçáõãûáõý»ñç) ÷áëï³ëí³íáõãûáõýá: mathematical problems of computer science 39, 54--65, 2013. 54 a combinative approach to generalization of meanings karen s. khachatryan state engineering university of armenia e-mail: khkarens@gmail.com abstract we gain new meanings through the acquisition of communities, the revelation from experience and the creation of new meanings by combining the existing ones. we refine meanings by abstract classes combined by be-, have-, docategories of relationships [1, 3, 7]. in the paper we present a novel combinative algorithm constructing new meanings by generalization of the given sets of meanings. keywords: meanings, meaning processing, combinative generalization. 1. introduction one way of new meaning extraction is the combination of available ones. particularly, the computation of a (or the) least generalization of two or more meanings helps to build a meaning expressing their general properties. it is a fundamental problem in inductive inference occurring particularly in machine learning [2, 6]. it may help to start from a set of descriptions assumed to be examples of the same meaning and consider their least generalization as a working basis. the operation can also be used to organize a large set of descriptions in a hierarchical structure. generalization and specialization problems have been studied for different knowledge representation models. [6] summarizes and distinguishes the methods of generalization into two categories: a generalization by features and a structural logical (or conceptual) generalization. the first methods usually solve the problems of classification and formation of meanings. [6, 13, 15] review and analyze the methods of generalization by features. these include a method of potential function as a function for the class k which is built to have the maximal value on the set of objects ∈ [16]; and a generalization by features using either the voting method proposed in [13, 14] or the covering technique [17]: first it prepares the input training data for building of base classifiers by perturbing the original training data, and builds base classifiers on the perturbed data, then the proximity measure is determined for the object s to each of the classes by comparing the values of features of the object s from the specified subsets with the appropriate values of features of etalon objects. a wide set of generalization/specialization techniques have also been developed for conceptual graphs which are well studied language of knowledge representation and reasoning [2, 10]. these include techniques like the evaluation of the least upper bound (the least generalization) h as an irredundant form of the categorical product of two basic conceptual graphs (bg) b and g: = × ; the maximum join operation between two bgs g and h as a composition of the following steps: first it merges a concept node in g and a concept node in h, k. khachatryan 55 and then continues merging as far as possible neighbors of previously merged nodes [12]; and the extended join operation which generalizes maximal join operations by using the properties of compatible partitions of the concept node set of a bg [11]. in the solver of ssrgt class problems (problems, where space of solutions can be represented by reproducible game trees) [1, 3, 7], the task of finding the least generalization in its basic form takes two meanings, say a and b, and asks for a least generalization of a and b, i.e., a meaning k such that ≽ (k is a generalization of a) and ≽ and for all meanings k', if ′ ≽ and ′ ≽ then ′ ≽ . in other words, given two acquired meanings a and b, extract a new meaning k which will represent the general characteristics of both a and b, while taking as a criterion of the generalization the most specific meaning which can be extracted. further in the paper we give the definition of the skeleton of a meaning as a sub-graph which is necessary to compose and correctly activate the meaning from sub-meanings [5] and define the elementary generalization and specialization operations for it. next we define the generalization (specialization) as a sequence of elementary generalization (specialization) operations and finally propose an algorithm of evaluating a least generalization of two meanings. we represent the strategy of selecting the generalizable sub-meanings from two meanings and detail the generalization procedure. we show that the algorithm is able to find common parts of two meanings, i.e., if there are meanings "two pawns" and "two knights", then it extracts a new meaning "two figures" etc., dynamically generate and integrate a new meaning between the be connection chain, for example, it extracts and integrates "fieldundercheckofpawn" new meaning between "fieldundercheck", "fieldundercheckofpawnatpos1" and "fieldundercheckofpawnatpos2" meanings, extract a common interface (pattern), the algorithm is able to extract "fieldundercheck" meaning by analyzing the list of "fieldundercheck of specific figure" meanings. in the conclusion we summarize the main findings of the research. 2. the skeleton of a meaning [3] discusses the means of meanings acquisition and the algorithm of their integration into the internal graph of abstracts. within the graph, each meaning is represented as a sub-graph centered in the meaning node and having other nodes as sub-meaning while edges as one of be-, haveor do connections [1, 3, 5]. note that all edges are bidirectional, in other words, if there is a be connection between the nodes a and b indicating that a is a base type of (subsumes) b then also a reverse connection is built between b and a indicating that b is a sub type of (subsumed by) a. similarly, the reversed connections are built also for have and do edges. therefore, when referring to these bidirectional edges, we distinguish the roles for nodes as sources and destinations. we call a node source for be connection if it is the sub type, consequently, the base type becomes a destination. similarly, a source for have connection is the node that has the other node as an attribute. and, finally, for do connection the precondition node serves as a source while the action itself becomes a destination. what follows is that the above definition of meanings is quite wide and assuming that all meanings are using the same set of nucleus meanings and taking the allowed distance of a sub-meaning from the central node as big as we want, we can end up having almost the whole connected component (even a graph of abstracts itself) as a representation of a single meaning (regardless what meaning we pick). therefore, we define the skeleton of a meaning as a sub-graph which is necessary to compose and correctly activate [5] the meaning from sub-meanings. it is constructed from the meaning's graph by recursively traversing only the following set of edges and connected nodes: 1. if the node is a nucleus then stop further processing. a combinative approach to generalization of meanings56 2. if the node is neither a virtual nor a usage then as a next layer of the skeleton select only have connections where the role of the node is a source. this means, that none of be, do connections or have connections, where the node is a destination, will be considered. 3. if the node is a virtual or a usage then only be edges, where the node is a destination, must be selected for further processing. for virtual nodes, this includes the set of all specifications of the node. for the usage nodes, this is basically the base virtual node (it is being selected because of the reverse be connection) and, finally, applying steps 1 to 3 recursively on each new node will result in a complete skeleton of the meaning. 3. elementary generalization/specialization operations we consider the generalization of meanings as a generalization of their skeletons. hereafter saying a meaning we refer to the skeleton of a meaning: we will use an entire meaning notation to refer to the meaning with the original definition. let us describe the generalization of a single meaning before considering the computation of a least generalization of two or more meanings. a partial order is interpreted as a generalization or a relation: ≽ means that the meaning is a generalization of the meaning (or is a specialization of , subsumes or every entry having a meaning has also a meaning ). a generalization is a "unary" operation, i.e. it has a meaning as an input and a meaning as an output. we define the following elementary generalization operations: increase. increase the type of a sub-meaning. more precisely, given a meaning a, a submeaning x of a, and a type ≽ increase (a, x, t) is the meaning obtained from the a by increasing the type of x up to t. within our meaning representation model the increase operation means replacing the sub-meaning with one of the meanings in its be connection chain fig. 1 a). similarly, the increase operation is defined for the relations of sub-meanings. given a meaning a and a relation r for a sub-meaning x of a, and a relation ∈ , then increase(a, x, r, r) is the meaning obtained from the a by changing the relation r of x up to the relation r. the increase condition of a relation indicated that the new relation r should enclose the value range defined by the relation r. substract. given a meaning a, and a set of sub-meanings , ,…, of a, then( , , ,…, ) is the meaning obtained from a by deleting , ,…, sub meanings and any relation which has a reference to them (the result can be the empty meaning). similarly, the substract operation for relations is defined as follows: given a meaning a, and a set of relations , ,…, for the sub-meanings , ,…, , then( , , ,…, , , ,…, ) is the meaning obtained from a by dropping the relations , ,…, fig. 1 b). the elementary specialization operations are defined as inverse operations of the elementary generalization operations. they are as follows: restrict. given a meaning a, a sub-meaning x of a, and a type ≼ restrict(a, x, t) is the meaning obtained from a by decreasing the type of x to t. within our meaning representation model the restrict operation means replacing the sub-meaning with one of the meanings in its reverse be connection chain fig. 1 c). similarly, the restrict operation is defined for relations of sub-meanings. given a meaning a and a relation r for a sub-meaning x of a, and a relation∈ , then restrict(a, x, r, r) is the meaning obtained from the a by changing the relation r of x down to the relation r. the restrict operation indicated that the relation r should embrace the relation r. k. khachatryan 57 disjoint sum. given two disjoint meanings a and b, a+b is the union of a and b that is the meaning which has a and b as sub meanings. similarly, for relations, a new relation r can be defined for sub meanings fig. 1 c). we define a generalization/specialization relation (or subsumption) by a sequence of elementary operations definition. a meaning a is a generalization of a meaning b if there is a sequence of meanings = , ,…, = ( ), and, for all = 1,…, , is obtained from by a generalization operation. su bm ea ni ng s an d re la ti on s restrict sub-m eanings and relations generalize increase su bm ea ni ng s knight y.coord. in [1,8] x coord. in [1,8] color in [1,2] figure t. = 3 (knight) sub-m eanings figure y.coord. in [1,8] x coord. in [1,8] color in [1,2] figure t. in [1,6] (fig) a) generalize substract neighbor pawns p2 is pawn p1 is pawn p1.y = p2.y p1.x = p2.x+1 pawns on same vertical p2 is pawn p1 is pawn p1.y = p2.y b) sub-m eanings and relations specialize disjoint sum su bm ea ni ng s figure and knight k1 is knight f1 is figure knights on same vertical k1 is knight f1 is knight f1.y = k1.y c) fig. 1. elementary generalization and specialization operations. a) generalization of figure meaning from knight meaning by an elementary increase operation applied on figure type submeaning. b) generalization of pawns on same vertical meaning from neighbor pawns by applying substract elementary operation on 1. = 2. + 1 relations. c) specialization of figure and knight meaning to knight on same vertical by applying restrict elementary specialization operation on f1:figure sub-meaning and disjoint sum operation to add new relation between f1 and k1 sub-meanings: 1. = 1. . a combinative approach to generalization of meanings58 from the definitions it follows that b can have extra sub-meanings and relations which don't exist in a. 4. the generalization of two meanings finding a (or the) least generalization of two meanings a and b is to find a meaning k such that ≽ and ≽ and for all meanings k', if ′ ≽ and ′ ≽ , then ′ ≽ . in other words, the algorithm has to traverse both meanings (abstracts in the meaning graph) and compose a new abstract from the least generalizations of their sub-meaning pairs and relations. the latter one can be either memorized or dropped based on some post processing algorithm: for example, if we deal with interactive expert systems, then the system would rather ask the expert for further analysis. alternately, it can maintain a weighted graph of abstracts and evaluate durables during the time [1], however, the consideration of the post-processing algorithms is out of the scope of this paper, therefore, we suppose that an expert can be asked to approve or reject the new evaluated meaning. the paper aims to answer the following major questions which arise during the processing of the algorithm: how to select the pairs of sub-meanings (abstract's attributes) to be generalized into the new meanings? how to generalize them? how to verify/extract the relations (dependencies) between sub-meanings? 4.1. select the attribute pairs to generalize in this section we will describe the algorithm which extracts the sets of attribute pairs to compose least generalizations form user/expert defined meanings. note, that there can be more than one least generalization when considering both sub-meanings and relations. for example, let’s suppose there are a meaning a and a meaning b where a is composed of sub-meanings: , : and : (t:t notation means that t is a type of t) and there is a relation . = . + 1 defined for and attributes. on the other hand, b has two submeanings : and : with a relation . = . + 1. let’s also suppose that ≻ ( strictly subsumes ). what follows is that the algorithm can either specialize the relation and extract a least generalization composed of two pawns, or specialize the sub-meaning and extract a new meaning composed of figure and pawn by keeping the relation defined between them. in order to find the compatible pairs of sub-meanings, let us recall the structure of graph of abstracts and the semantic connections existing between the abstracts (the semantics of be, have and do connections). as defined in [3], there are the following types of ga nodes: nucleus smallest representation units of meanings, ar1 abstracts having only nucleus attributes (there can be at most one attribute of a given nucleus type), sets representing a group of abstracts with similar characteristics, composite abstracts a complex form of abstract representation, they can have any kind of attributes, virtual abstracts composite abstracts with attributes having undefined relations, and actions representing the action meanings. considering these categories of ga nodes, as a first step, the algorithm segregates attributes into different compatible groups and evaluates the pair extraction between the groups containing nodes having the same type. 4.2. generalization of nucleus abstracts property. two nucleus abstracts are generalizable if and only if they have the same type. the proof simply follows from the definition of nucleus abstracts: they represent different nucleus characteristics, hence, cannot be generalized further. k. khachatryan 59 we shall recall the structure of a nucleus abstract: it has a single attribute and value range defined for that attribute. for the generalization we shall assume one of the following operations: union of regulations finding the closest base type which covers both sets defined by regulations. both of these have a practical sense, however, we first try to find a common base nucleus type by traversing their be connection chain in order to avoid the expansion of new nucleus abstracts by unifying different nucleus values (this can lead to a creation of new types for all possible combinations of nucleus values). 4.1.generalization of ar1 abstracts according to the definition of ar1, it contains at most one element of a given nucleus type (fig. 2). therefore: property. the attributes of two ar1s can be generalized only if they have the same nucleus types, moreover, there is at most one possible pair for a given attribute. table 1 represents the mapping between a and b ar1s' attributes. as we see a.r1, b.s2 and b.p2 attributes do not have pairs, consequently, they are ignored in the generalized abstract and a new ar1 is constructed by generalizing the paired nucleus attributes ( 1: 2→, 1: 2→ and 1: 2→ in this example). the asset of ar1s is that there is no dependency defined between the attributes. the only dependency (belonging to the same id group) is implied in [3]. 4.2. generalization of set abstracts and actions the generalization of sets is done by generalizing the composite element of the set and uniting the min-max ranges of source sets. here actions' generalization is discussed only by their preconditions, which in their turn, are composites. 4.3. generalization of composite abstracts the least generalization of two meanings is, basically, the least generalization of two composite abstracts. intuitively, the effect of the least generalization is to find the biggest subgraphs of two meanings which can be merged into one. in this section we will present some general ideas behind the operation and will give the algorithm adopted for solver's meaning representation. definition. let a and b be two disjoint meanings and c and d two sub-meanings in a and b, respectively. a generalization of a and b is a least generalization of c and d sub-meanings. a way of extending a generalization of c in a and d in b consists of searching the neighbors (sub-meanings connected with them through relations) of c and d, then to check if these nodes can be generalized and so on. in other words, starting from a pair of generalizable sub-meanings, the idea is to search, in a greedy way, generalizable neighbors of previously identified generalizable nodes. the resulted least generalization is, thus, locally "maximal". in order to specify a least generalization operation, one has to define not only the conditions for generalization of sub-meanings and relations but also a strategy for exploring the meaning table 1. the mapping table of ar1s attributes. a nucleus types b x1 x x2 y1 y y2 r1 r z1 z z2 s s2 p p2 fig. 2. a and b ar1s. a x1 y1 r1 z1 b x2 s2 y2 k. khachatryan 59 we shall recall the structure of a nucleus abstract: it has a single attribute and value range defined for that attribute. for the generalization we shall assume one of the following operations: union of regulations finding the closest base type which covers both sets defined by regulations. both of these have a practical sense, however, we first try to find a common base nucleus type by traversing their be connection chain in order to avoid the expansion of new nucleus abstracts by unifying different nucleus values (this can lead to a creation of new types for all possible combinations of nucleus values). 4.1.generalization of ar1 abstracts according to the definition of ar1, it contains at most one element of a given nucleus type (fig. 2). therefore: property. the attributes of two ar1s can be generalized only if they have the same nucleus types, moreover, there is at most one possible pair for a given attribute. table 1 represents the mapping between a and b ar1s' attributes. as we see a.r1, b.s2 and b.p2 attributes do not have pairs, consequently, they are ignored in the generalized abstract and a new ar1 is constructed by generalizing the paired nucleus attributes ( 1: 2→, 1: 2→ and 1: 2→ in this example). the asset of ar1s is that there is no dependency defined between the attributes. the only dependency (belonging to the same id group) is implied in [3]. 4.2. generalization of set abstracts and actions the generalization of sets is done by generalizing the composite element of the set and uniting the min-max ranges of source sets. here actions' generalization is discussed only by their preconditions, which in their turn, are composites. 4.3. generalization of composite abstracts the least generalization of two meanings is, basically, the least generalization of two composite abstracts. intuitively, the effect of the least generalization is to find the biggest subgraphs of two meanings which can be merged into one. in this section we will present some general ideas behind the operation and will give the algorithm adopted for solver's meaning representation. definition. let a and b be two disjoint meanings and c and d two sub-meanings in a and b, respectively. a generalization of a and b is a least generalization of c and d sub-meanings. a way of extending a generalization of c in a and d in b consists of searching the neighbors (sub-meanings connected with them through relations) of c and d, then to check if these nodes can be generalized and so on. in other words, starting from a pair of generalizable sub-meanings, the idea is to search, in a greedy way, generalizable neighbors of previously identified generalizable nodes. the resulted least generalization is, thus, locally "maximal". in order to specify a least generalization operation, one has to define not only the conditions for generalization of sub-meanings and relations but also a strategy for exploring the meaning table 1. the mapping table of ar1s attributes. a nucleus types b x1 x x2 y1 y y2 r1 r z1 z z2 s s2 p p2 fig. 2. a and b ar1s. y2 z2 p2 k. khachatryan 59 we shall recall the structure of a nucleus abstract: it has a single attribute and value range defined for that attribute. for the generalization we shall assume one of the following operations: union of regulations finding the closest base type which covers both sets defined by regulations. both of these have a practical sense, however, we first try to find a common base nucleus type by traversing their be connection chain in order to avoid the expansion of new nucleus abstracts by unifying different nucleus values (this can lead to a creation of new types for all possible combinations of nucleus values). 4.1.generalization of ar1 abstracts according to the definition of ar1, it contains at most one element of a given nucleus type (fig. 2). therefore: property. the attributes of two ar1s can be generalized only if they have the same nucleus types, moreover, there is at most one possible pair for a given attribute. table 1 represents the mapping between a and b ar1s' attributes. as we see a.r1, b.s2 and b.p2 attributes do not have pairs, consequently, they are ignored in the generalized abstract and a new ar1 is constructed by generalizing the paired nucleus attributes ( 1: 2→, 1: 2→ and 1: 2→ in this example). the asset of ar1s is that there is no dependency defined between the attributes. the only dependency (belonging to the same id group) is implied in [3]. 4.2. generalization of set abstracts and actions the generalization of sets is done by generalizing the composite element of the set and uniting the min-max ranges of source sets. here actions' generalization is discussed only by their preconditions, which in their turn, are composites. 4.3. generalization of composite abstracts the least generalization of two meanings is, basically, the least generalization of two composite abstracts. intuitively, the effect of the least generalization is to find the biggest subgraphs of two meanings which can be merged into one. in this section we will present some general ideas behind the operation and will give the algorithm adopted for solver's meaning representation. definition. let a and b be two disjoint meanings and c and d two sub-meanings in a and b, respectively. a generalization of a and b is a least generalization of c and d sub-meanings. a way of extending a generalization of c in a and d in b consists of searching the neighbors (sub-meanings connected with them through relations) of c and d, then to check if these nodes can be generalized and so on. in other words, starting from a pair of generalizable sub-meanings, the idea is to search, in a greedy way, generalizable neighbors of previously identified generalizable nodes. the resulted least generalization is, thus, locally "maximal". in order to specify a least generalization operation, one has to define not only the conditions for generalization of sub-meanings and relations but also a strategy for exploring the meaning table 1. the mapping table of ar1s attributes. a nucleus types b x1 x x2 y1 y y2 r1 r z1 z z2 s s2 p p2 fig. 2. a and b ar1s. a combinative approach to generalization of meanings60 graphs. given two generalizable meaning nodes as a starting point, there may be several least generalizations, but computing one of them can be done in polynomial time, whereas computing the least generalization with a maximum number of nodes is np-hard (indeed it admits the homomorphism or injective homomorphism as a special case) [2]. in order to improve a least generalization obtained by a greedy approach, we propose a strategy for picking the starting nodes and exploring meanings' graphs. in the strategy we are excessively using the structure of the meanings' graph and particularly, the semantics of be connections (as we did in ar1s). definition. two abstracts are strongly compatible if they have a common node in the chain of their be connections. the be connection is one of the major relations defined between the abstracts (during the acquisition procedure). therefore the existence of a common base type indicates the importance of the connection between two abstracts. on the other hand, the lack of a common base type means that they had not strong connections during the acquisition procedure. it could also be possible that the post processing algorithm or an expert discarded evaluated generalizations for these abstracts, thus taking into account that the sub types have to satisfy also the restrictions defined in the base types, it can indicate that there is no acceptable generalization for these abstracts. property. the closest is the common node the more strongly compatible are abstracts. proof of the property follows from the semantics of be connection. in this connection there are two components, namely, base type and sub type. the sub type is constructed from the base type by inheriting from it and possibly adding more restrictions. however, any instance of the sub type will also satisfy the regularities defined in the base type, hence, the closest base type contains the most common characteristics. using this property we propose a substructure for the meanings' graph exploration strategy to pick attribute pairs standing closest in the be connection chain. let's suppose we have the hierarchy fig. 3. the type hierarchy. table 2. the initial mapping list. left container list right container a1121, a1121 a1121 a112 a1121 a11 a111 a1121, a12 a1 a111, a13, a1 a1121, a12, a21 a a111,a211,a13, a1 a12 a12 b1 b1 b1 b c2 c2 c2 c c1 a21 a21 a211 a21 a2 a211 c1 c1 a111 a111 a211 a211 a13 a13 fig. 4. a and b composite abstracts. a a1121 a12 b1 c2 a21 b c1 a111 a1 a13 a combinative approach to generalization of meanings60 graphs. given two generalizable meaning nodes as a starting point, there may be several least generalizations, but computing one of them can be done in polynomial time, whereas computing the least generalization with a maximum number of nodes is np-hard (indeed it admits the homomorphism or injective homomorphism as a special case) [2]. in order to improve a least generalization obtained by a greedy approach, we propose a strategy for picking the starting nodes and exploring meanings' graphs. in the strategy we are excessively using the structure of the meanings' graph and particularly, the semantics of be connections (as we did in ar1s). definition. two abstracts are strongly compatible if they have a common node in the chain of their be connections. the be connection is one of the major relations defined between the abstracts (during the acquisition procedure). therefore the existence of a common base type indicates the importance of the connection between two abstracts. on the other hand, the lack of a common base type means that they had not strong connections during the acquisition procedure. it could also be possible that the post processing algorithm or an expert discarded evaluated generalizations for these abstracts, thus taking into account that the sub types have to satisfy also the restrictions defined in the base types, it can indicate that there is no acceptable generalization for these abstracts. property. the closest is the common node the more strongly compatible are abstracts. proof of the property follows from the semantics of be connection. in this connection there are two components, namely, base type and sub type. the sub type is constructed from the base type by inheriting from it and possibly adding more restrictions. however, any instance of the sub type will also satisfy the regularities defined in the base type, hence, the closest base type contains the most common characteristics. using this property we propose a substructure for the meanings' graph exploration strategy to pick attribute pairs standing closest in the be connection chain. let's suppose we have the hierarchy fig. 3. the type hierarchy. a a1 a11 a111 a112 a1121 a12 a121 a13 a2 a21 a211 a212 a213 b b1 c1 table 2. the initial mapping list. left container list right container a1121, a1121 a1121 a112 a1121 a11 a111 a1121, a12 a1 a111, a13, a1 a1121, a12, a21 a a111,a211,a13, a1 a12 a12 b1 b1 b1 b c2 c2 c2 c c1 a21 a21 a211 a21 a2 a211 c1 c1 a111 a111 a211 a211 a13 a13 fig. 4. a and b composite abstracts. a13 a211 a combinative approach to generalization of meanings60 graphs. given two generalizable meaning nodes as a starting point, there may be several least generalizations, but computing one of them can be done in polynomial time, whereas computing the least generalization with a maximum number of nodes is np-hard (indeed it admits the homomorphism or injective homomorphism as a special case) [2]. in order to improve a least generalization obtained by a greedy approach, we propose a strategy for picking the starting nodes and exploring meanings' graphs. in the strategy we are excessively using the structure of the meanings' graph and particularly, the semantics of be connections (as we did in ar1s). definition. two abstracts are strongly compatible if they have a common node in the chain of their be connections. the be connection is one of the major relations defined between the abstracts (during the acquisition procedure). therefore the existence of a common base type indicates the importance of the connection between two abstracts. on the other hand, the lack of a common base type means that they had not strong connections during the acquisition procedure. it could also be possible that the post processing algorithm or an expert discarded evaluated generalizations for these abstracts, thus taking into account that the sub types have to satisfy also the restrictions defined in the base types, it can indicate that there is no acceptable generalization for these abstracts. property. the closest is the common node the more strongly compatible are abstracts. proof of the property follows from the semantics of be connection. in this connection there are two components, namely, base type and sub type. the sub type is constructed from the base type by inheriting from it and possibly adding more restrictions. however, any instance of the sub type will also satisfy the regularities defined in the base type, hence, the closest base type contains the most common characteristics. using this property we propose a substructure for the meanings' graph exploration strategy to pick attribute pairs standing closest in the be connection chain. let's suppose we have the hierarchy fig. 3. the type hierarchy. c c1 c2 table 2. the initial mapping list. left container list right container a1121, a1121 a1121 a112 a1121 a11 a111 a1121, a12 a1 a111, a13, a1 a1121, a12, a21 a a111,a211,a13, a1 a12 a12 b1 b1 b1 b c2 c2 c2 c c1 a21 a21 a211 a21 a2 a211 c1 c1 a111 a111 a211 a211 a13 a13 fig. 4. a and b composite abstracts. k. khachatryan 61 described in fig. 3. the be chain of the abstract a1121 is: 1121→ 112→ 11→ 1→ . this means that the top type is a, while the closest type is a112. let's suppose we want to generalize abstracts a and b from fig. 4. we compose the substructure by the following rules. for the left abstract we initialize a list of all nodes which appear in the be connection chain of each attribute (including itself) and keep reflexive mapping from the attribute to the base types in a list. note, that during the initialization, the most specific types for a given attribute are inserted first. if a base type with the similar id (name in our case) already exists in the list, then we simply increase the number of references to that element and add the pointing attribute to the element's left container. similarly, we iterate over all attributes of the right abstract and integrate all base types into the list with a difference that each attribute is added to the right container of the base type (table 2). as a next step we count the number of elements in the left and right containers of base types and sort the list in the increasing order of the cumulative element count in right and left containers (table 3). meanwhile, if the number of elements is the same for two entries, we keep the initial ordering (see 21→ 21 and 21→ 2 mapping in table 3). from the definition of the structure it follows that the base types appearing in the upper levels of the hierarchy will have a bigger number of connected elements. this is because each attribute from the bottom levels will increase also the number of connected elements of base types. therefore, we can argue that mappings’ entries appearing first for the given attribute are the closest base types and the later the mapping entry appears the further is the base type. this leads us to the selection algorithm of the best matches of attributes in two abstract. we start iterating over the elements of the sorted list and remove the mapping entries which contain only right or left container (table 4). from the remaining set, if there is a one to one mapping (like 2: 1→ or 21: 211→21), then these attributes are paired and are removed from the lists of bottom entries (table 5). if there is more than one attribute in one of table 3. the mapping list after sorting. left container list right container a1121, a1121 a1121 a112 c2 c2 a12 a12 a111 a111 a211 a211 a13 a13 b1 b1 b1 b c1 c1 c2 c c1 a21 a21 a211 a21 a2 a211 a1121 a11 a111 a1121, a12 a1 a111, a13, a1 a1121,a12, a21 a a111,a211,a13, a1 table 4. the list without pairless elements. left container list right container c2 c c1 a21 a21 a211 a21 a2 a211 a1121 a11 a111 a1121, a12 a1 a111, a13, a1 a1121,a12, a21 a a111,a211,a13, a1 table 5. the list after removing one to one mapping entries. left container list right container a12 a1 a13, a1 a12 a a13, a1 a combinative approach to generalization of meanings62 containers then we have an uncertainty, therefore new sets of pairs are created for all the possible combination chains. for the given example, two set of pairs will be generalized:2: 1→ , 21: 211→ 21, 1121: 111→ 11, : → }2: 1→ , 21: 211→ 21, 1121: 111→ 11, : → } as a result of this procedure we get a list of arrays of attribute pairs. each element of the array represents one possible least generalization of two input abstracts. we shall note here that some of the attributes might be ignored because of not having proper pairs (b1 in a, for example). the important achievement of this strategy is that we significantly reduced the number of possible least generalizations. it is only multiplied if there are attributes having the same types, but mainly, in the definition of a composite abstract the same attribute is not used multiple times (there are other types, like sets to be used for such kind of definitions). 4.4. generalize paired attributes at this point for each array of paired attributes we have to extract their least generalizations and consider the extraction of relations. finally, we have to compose a new abstract by putting together all these "building blocks". first of all, let us discuss the approaches of extraction least generalization of paired attributes. the key point here is that they have an evaluated common subsumer (a common base type). based on the type of the subsumer, i.e. whether it is a virtual abstract, thereby indicating the attributes being usage nodes [3], or not, the algorithm adopts different strategies. if the attributes are usages then the common subsumer is taken as a least generalization and only new relations are further analyzed for them. this is because the usage nodes do not add any additional attribute to the base type, rather, they only specify more restrictions. on the other hand, if the base type is not a virtual then the set of attributes, which exist in the base type, are extracted as a part of a generalization and the remaining ones are generalized further. more precisely, let and be paired attributes which have a common base type which is not virtual. in this case the least generalization contains the set of attributes defined in b merged with the generalization of / and / . an interesting property of the new generalized abstract from the pairs having the same base type is that it is either the base type itself or an abstract holding a place between the base type and sub types. in other words, from the property of be connection, it follows that there is a homomorphism from the common base type to the new generalized abstract. this action leads to the organization of a hierarchical structure between definitions. for example, by generalizing "fieldisundercheckofpawnpos1" and "fieldisundercheckofpawnpos2" meanings the algorithm can extract a new meaning "fieldisundercheckofpawn" and integrate it into the be connection chain between these meanings and their base type "fieldinundercheck" meaning. once the paired attributes of each abstract have their mappings in the generalized one, the algorithm starts evaluating and extracting relations defined between the attributes and integrate them into the generalized abstract. a relation/regulation between an abstract's attributes is called a dependency and has form, where attr is the name of the attribute, rop is a relational operator ( , , , , , ) and expr is an arithmetic expression. naturally, the generalization algorithm has to consider the existence and extract the analogical dependencies from two source abstracts. thence, the algorithm has to analyze each dependency and check: is it still valid on the generalized abstract? fig. 5. generalization of c1 and c2 paired abstracts. a combinative approach to generalization of meanings62 containers then we have an uncertainty, therefore new sets of pairs are created for all the possible combination chains. for the given example, two set of pairs will be generalized:2: 1→ , 21: 211→ 21, 1121: 111→ 11, : → }2: 1→ , 21: 211→ 21, 1121: 111→ 11, : → } as a result of this procedure we get a list of arrays of attribute pairs. each element of the array represents one possible least generalization of two input abstracts. we shall note here that some of the attributes might be ignored because of not having proper pairs (b1 in a, for example). the important achievement of this strategy is that we significantly reduced the number of possible least generalizations. it is only multiplied if there are attributes having the same types, but mainly, in the definition of a composite abstract the same attribute is not used multiple times (there are other types, like sets to be used for such kind of definitions). 4.4. generalize paired attributes at this point for each array of paired attributes we have to extract their least generalizations and consider the extraction of relations. finally, we have to compose a new abstract by putting together all these "building blocks". first of all, let us discuss the approaches of extraction least generalization of paired attributes. the key point here is that they have an evaluated common subsumer (a common base type). based on the type of the subsumer, i.e. whether it is a virtual abstract, thereby indicating the attributes being usage nodes [3], or not, the algorithm adopts different strategies. if the attributes are usages then the common subsumer is taken as a least generalization and only new relations are further analyzed for them. this is because the usage nodes do not add any additional attribute to the base type, rather, they only specify more restrictions. on the other hand, if the base type is not a virtual then the set of attributes, which exist in the base type, are extracted as a part of a generalization and the remaining ones are generalized further. more precisely, let and be paired attributes which have a common base type which is not virtual. in this case the least generalization contains the set of attributes defined in b merged with the generalization of / and / . an interesting property of the new generalized abstract from the pairs having the same base type is that it is either the base type itself or an abstract holding a place between the base type and sub types. in other words, from the property of be connection, it follows that there is a homomorphism from the common base type to the new generalized abstract. this action leads to the organization of a hierarchical structure between definitions. for example, by generalizing "fieldisundercheckofpawnpos1" and "fieldisundercheckofpawnpos2" meanings the algorithm can extract a new meaning "fieldisundercheckofpawn" and integrate it into the be connection chain between these meanings and their base type "fieldinundercheck" meaning. once the paired attributes of each abstract have their mappings in the generalized one, the algorithm starts evaluating and extracting relations defined between the attributes and integrate them into the generalized abstract. a relation/regulation between an abstract's attributes is called a dependency and has form, where attr is the name of the attribute, rop is a relational operator ( , , , , , ) and expr is an arithmetic expression. naturally, the generalization algorithm has to consider the existence and extract the analogical dependencies from two source abstracts. thence, the algorithm has to analyze each dependency and check: is it still valid on the generalized abstract? fig. 5. generalization of c1 and c2 paired abstracts. c p q c1 p1 q1 r c2 p2 a combinative approach to generalization of meanings62 containers then we have an uncertainty, therefore new sets of pairs are created for all the possible combination chains. for the given example, two set of pairs will be generalized:2: 1→ , 21: 211→ 21, 1121: 111→ 11, : → }2: 1→ , 21: 211→ 21, 1121: 111→ 11, : → } as a result of this procedure we get a list of arrays of attribute pairs. each element of the array represents one possible least generalization of two input abstracts. we shall note here that some of the attributes might be ignored because of not having proper pairs (b1 in a, for example). the important achievement of this strategy is that we significantly reduced the number of possible least generalizations. it is only multiplied if there are attributes having the same types, but mainly, in the definition of a composite abstract the same attribute is not used multiple times (there are other types, like sets to be used for such kind of definitions). 4.4. generalize paired attributes at this point for each array of paired attributes we have to extract their least generalizations and consider the extraction of relations. finally, we have to compose a new abstract by putting together all these "building blocks". first of all, let us discuss the approaches of extraction least generalization of paired attributes. the key point here is that they have an evaluated common subsumer (a common base type). based on the type of the subsumer, i.e. whether it is a virtual abstract, thereby indicating the attributes being usage nodes [3], or not, the algorithm adopts different strategies. if the attributes are usages then the common subsumer is taken as a least generalization and only new relations are further analyzed for them. this is because the usage nodes do not add any additional attribute to the base type, rather, they only specify more restrictions. on the other hand, if the base type is not a virtual then the set of attributes, which exist in the base type, are extracted as a part of a generalization and the remaining ones are generalized further. more precisely, let and be paired attributes which have a common base type which is not virtual. in this case the least generalization contains the set of attributes defined in b merged with the generalization of / and / . an interesting property of the new generalized abstract from the pairs having the same base type is that it is either the base type itself or an abstract holding a place between the base type and sub types. in other words, from the property of be connection, it follows that there is a homomorphism from the common base type to the new generalized abstract. this action leads to the organization of a hierarchical structure between definitions. for example, by generalizing "fieldisundercheckofpawnpos1" and "fieldisundercheckofpawnpos2" meanings the algorithm can extract a new meaning "fieldisundercheckofpawn" and integrate it into the be connection chain between these meanings and their base type "fieldinundercheck" meaning. once the paired attributes of each abstract have their mappings in the generalized one, the algorithm starts evaluating and extracting relations defined between the attributes and integrate them into the generalized abstract. a relation/regulation between an abstract's attributes is called a dependency and has form, where attr is the name of the attribute, rop is a relational operator ( , , , , , ) and expr is an arithmetic expression. naturally, the generalization algorithm has to consider the existence and extract the analogical dependencies from two source abstracts. thence, the algorithm has to analyze each dependency and check: is it still valid on the generalized abstract? fig. 5. generalization of c1 and c2 paired abstracts. c2 p2 q2 s k. khachatryan 63 is there an equivalent dependency in the pair abstract? first it is necessary to ensure that none of the dependent attributes is excluded to continue the further verification. this could happen if abstracts contain extra attributes which are missing in the generalized one. for example, let us suppose c, c1 and c2 have the attributes given in fig. 5. moreover, 1. 1 = 1. + 2, 1. 1 = 1. 1− 2 and 2. 2 = 2. 2− 2, 2. = 2. 2−3 are the dependencies defined for them. let us also suppose that the algorithm picked 1: 2→ pair for the processing and c is picked as the generalized abstract. thus, the dependencies1. 1 = 1. + 2 and 2. = 2. 2− 3 cannot be verified, because s and r attributes do not exist in the abstract c. therefore, these dependencies are dropped. the other two are analized further. the second step is to perform the referred attribute name replacement in dependency expressions. to do that the experssion is parsed and an expression tree is built. afterwards, each reference node is changed to point to the exact node in the generalized abstract: 1. 1→ . ,1. 1→ . , 2. 2→ . , 2. 2→ . , correspondingly. the final step is to extract the equivalent expressions from the set of dependent expressions. note, that we use the term equivalence instead of isomorphism because the free literals are the same within both expressions. to decide whether two arithmetic expressions are equivalent is an important problem in computational theory [9]. however, the general problem of equivalence checking, in digital computers, belongs to the np hard class of problems [8]. even though, there are different algorithms which are fast enough to be used in practice. from that point of view we have adopted the algorithm proposed in [4]. its technique is specifically designed to solve the problem of equivalence checking of arithmetic expressions obtained from high-level language descriptions, which consists of regular arithmetic operators (+, -, ×) and logical operators (and, or, not). the method uses interval analysis [10] to substantially prune the domain space of arithmetic expressions (and conditional expressions) and limit the evaluation effort to a sufficiently small number of minimally sized spaces within the domain of the expression. then, it is extended to the technique to incorporate the arbitrary use of logic operators and, or, and not within the arithmetic expressions. thus, applying the above technique equivalent dependencies are extracted and integrated into the generalized abstract. 4.5. extraction of a virtual abstract the difference between virtual and non virtual composite abstracts is that the first one has attributes with undefined relations. here we will discuss the extraction of a new virtual abstract by generalizing both from virtual and non virtual abstracts. the major difference of the procedure, compared to the procedure of extraction of composites, is the handling of the regulations when generalizing paired attributes. thus, the initial steps of finding the attribute pairs and the algorithm of verifying/extracting the relations between the attributes, are replicating the ones defined for composite abstracts, however, the handling of regulations are defined for an attribute in the topmost, nucleus, level drags in the peculiarities. here, if regulations differ then rather than applying increase elementary operation (up to a "*" symbol representing all applicable values) we replace them with a symbol representing an undefined relation: "?". this leads to a core difference between two generalizations. the first one is like a complete class and can be used to instantiate an object. however, the property of undefined relation drives the virtual generalization closer to the interfaces or abstract classes in oop languages (java, etc.). in other words, as a result of a generalization the algorithm is capable of extracting categorically new abstracts. for example it is possible to extract "fieldundercheck" virtual abstracts by generalizing "fieldundercheckofknightpositionpattern1" and "fieldundercheckofqueen positionpattern1" abstracts. a combinative approach to generalization of meanings64 5. conclusion in the paper we discussed the generalization and specialization operations for meanings within the be-, have-, dolinguistic representation model of ssrgt solver. particularly we defined increase and substract (restrict and disjoint sum) elementary operations and generalization (specialization) operation by means of their sequence. next we considered a (or the) least generalization of two acquired meanings and proposed a strategy for selecting the attribute (sub-meaning) pairs to be generalized into the new meaning. moreover, we claimed that the attributes pairs selected by our algorithm are strongly compatible. in other words, they are the closest pairs in the be connection chain. in order to deal with the extraction of relations/dependencies between attributes we adopted the algorithm of equivalence checking of arithmetic expressions from [4]. further we showed how the generalized meanings were integrated into the meaning hierarchy and represented the algorithm of extraction of virtual generalizations. our experiments showed that the proposed algorithms were able to: find common parts of two meanings, dynamically generate and integrate a new meaning between the be connection chain, extract a common interface (pattern). acknowledgements the author expresses his gratitude to professor edward pogossian for supervising the work as well as sedrak grigoryan, sipan babertsyan and vahan margaryan for very valuable discussions. references [1] e. pogossian, “on modeling cognition”. computer science and information technologies (csit11), yerevan, pp 194-198, 2011. 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[14] ю. и. журавлев, “корректные алгебры над множествами некорректных (эвристических) алгоритмов”, кибернетика, n. 2, с. 35-43, 1978. [15] д. а. поспелов, ситуационное управление: теория и практика, наука, 1986. [16] м. а. айзерман, э. м. браверман, л. и. розоноер, метод потенциальных функций в теории обучения машин, наука, 1970. [17] р. харалик, “структурное распознавание образов: гомоморфизмы и размещения”, кибернетический сборник, пер. с англ. n 19, с. 170-199, 1982. submitted 25.12.2012, accepted 20.02.2013. իմաստների ընդհանրացման համակցական մոտեցում կ. խաչատրյան ամփոփում նոր իմաստները ձեռք են բերվում հանրությունից հարցման, փորձի միջոցով բացահայտման և առկա իմաստների համակցման միջոցով: մենք կատարելագործում ենք իմաստները` ներկայացնելով նրանց աբստրակտ դասերի միջոցով, որոնք կապված են միմյանց հետ հարաբերությունների լինել-, ունենալ-, անել(be-, have-, do) կատեգորիաներով [1, 3, 7]: տրված իմաստների ընդհանրացման միջոցով հոդվածում ներկայացնում ենք իմաստների կառուցման նոր համակցված ալգորիթմ: комбинационный подход к обобщению смыслов к. хачатрян аннотация мы получаем новые смыслы за счет приобретения из общин, откровения от опыта и создания новых смыслов путем объединения имеющихся смыслов. мы уточняем смыслы с помощью абстрактных классов объединенных быть-, иметь-, делатькатегориями отношений [1, 3, 7]. в статье представляется новый комбинационный алгоритм для построения нового смысла обобщения данного набора смыслов. d:\user\sbornik_38_pdf\27.dvi mathematical problems of computer science 38, 68{69, 2012. distr ibutivity in symmetr ic constr uctive full lambek calculus mic h a ãl k o z a k pozna¶n supercomputing and networking center polish academy of sciences, pozna¶n, poland mkozak@man.poznan.pl in [4 ] we h a ve d e ve lo p e d a n a lg e b r a ic s e m a n t ic s fo r s ym m e t r ic c o n s t r u c t ive lo g ic o f p r o fe s s o r ig o r d . za s la vs ky [9 ] d e vo id o f s t r u c t u r a l r u le s a n d h a ve s h o wn h o w it is r e la t e d t o c yc lic in vo lu t ive fl { a lg e b r a s a n d n e ls o n fl ew { a lg e b r a s . b e c a u s e o f t h is a n a lo g y we c a lle d t h e o b t a in e d c a lc u lu s symmetric constructive full l ambek calculus ( symconfl ) a n d it s a lg e b r a ic m o d e ls symmetric constructive f l {algebras. w e p r o ve d t h a t t h e c la s s o f c yc lic in vo lu t ive fl { a lg e b r a s ( cyin fl) is m u t u a lly in t e r p r e t a b le wit h t h e c la s s o f s ym m e t r ic c o n s t r u c t ive fl { a lg e b r a s ( s ym co n fl) . mo r e o ve r , c o n s id e r in g symconfl wit h t h e b a s ic s t r u c t u r a l r u le s exchange ( e) , weakening ( w ) a n d contraction ( c) , we h a ve d e ve lo p e d a n a lo g o u s s e m a n t ic s fo r a ll va r ia n t s o f symconfls, wh e r e s is a n y s u b s e t o f fe; w; cg. in p a r t ic u la r , s in c e symconfl e wc is e xa c t ly s ym m e t r ic c o n s t r u c t ive lo g ic , t h e c la s s s ym co n flew c is it s a lg e b r a ic s e m a n t ic s . 1 l ike wis e , we ve r ī e d t h a t t h e m u t u a l in t e r p r e t a b ilit y h o ld s b e t we e n t h e c o m m u t a t ive s u b c la s s e s cyin fle a n d s ym co n fle, a n d t h e in t e g r a l s u b c la s s e s cyin flw a n d s ym co n flw . fo r t h e c o n t r a c t ive s u b c la s s e s cyin flc a n d s ym co n flc a s im ila r c o r r e s p o n d e n c e d o e s n o t h o ld . n e ve r t h e le s s , we p r o ve d t h e t e r m e qu iva le n c e b e t we e n t h e c la s s s ym co n flew c ( wh ic h we a ls o c a lle d t h e c la s s o f za s la vs ky fl ew c { a lg e b r a s ) a n d t h e c la s s o f n e ls o n fl ew { a lg e b r a s . a c c o r d in g t o t h e r e s u lt o f m. s p in ks a n d r . v e r o ® [7 , 8 ], wh o h a ve in t r o d u c e d t h e va r ie t y o f n e ls o n fl ew { a lg e b r a s a s t h e t e r m wis e e qu iva le n t d e ¯ n it io n o f n e ls o n a lg e b r a s [6 ], za s la vs ky fl ew c { a lg e b r a s a r e t e r m e qu iva le n t t o n e ls o n a lg e b r a s a s we ll. in t h is t a lk we a d d it io n a lly c o n s id e r va r ia n t s o f symconfl t h a t a llo ws o n e t o p r o ve t h e la w o f d is t r ib u t ivit y o f c o n ju n c t io n o ve r d is ju n c t io n . in s ym m e t r ic c o n s t r u c t ive lo g ic ( symconfl e wc ) t h is la w is p r o va b le , b u t it is b e yo n d t h e r a n g e o f t h a t s ys t e m wit h o u t we a ke n in g o r c o n t r a c t io n . w e u s e t h e m e t h o d in d e p e n d e n t ly e la b o r a t e d b y j.m. d u n n [1 ] a n d g. min t s [5 ], t h a t c o n s is t s in a llo win g a n a n t e c e d e n t o f a s e qu e n t t o b e a s t r u c t u r e b u ilt fr o m t wo kin d s o f s t r u c t u r e s in d u c t ive ly. w e h a ve d e ve lo p e d s u c h s ys t e m s fo r c yc lic in vo lu t ive d is t r ib u t ive fl { a lg e b r a s ( cyin dfl ) a n d t h e ir c o m m u t a t ive a n d in t e g r a l va r ia n t s [3 ]. u s in g t h e d e ¯ n it io n o f s ym m e t r ic c o n s t r u c t ive fl { a lg e b r a s a n d e xt e n d in g it wit h t h e la w o f d is t r ib u t ivit y we c a n a ls o e xp a n d t h e c o m p le t e n e s s t h e o r e m t o s ys t e m s wit h c o n t r a c t io n . 1we use the naming convention adopted for variants of full lambek calculus and their algebraic models, where subscripts stand for structural rules determining properties of fusion [2]. 6 8 m. kozak 6 9 r e fe r e n c e s [1 ] d u n n , j.m., a ge n t z e n s ys t e m fo r p o s it ive r e le va n t im p lic a t io n . jo u r n a l o f s ym b o lic l o g ic 3 8 , 3 5 6 -3 5 7 ( 1 9 7 3 ) . a b s t r a c t . [2 ] ga la t o s , n ., k o wa ls ki, t., jip s e n , p ., on o , h .: r e s id u a t e d l a t t ic e s : a n a lg e b r a ic glim p s e a t s u b s t r u c t u r a l l o g ic s . e ls e vie r ( 2 0 0 7 ) [3 ] k o z a k, m.: cyc lic in vo lu t ive d is t r ib u t ive fu ll l a m b e k ca lc u lu s is d e c id a b le . jo u r n a l o f l o g ic a n d co m p u t a t io n 2 1 , 2 3 1 { 2 5 2 ( 2 0 1 1 ) [4 ] k o z a k, m.: s t r o n g n e g a t io n in in t u it io n is t ic s t yle s e qu e n t s ys t e m s fo r r e s id u a t e d l a t t ic e s . ma n u s c r ip t ( 2 0 1 2 ) [5 ] min t s , g.: cu t e lim in a t io n th e o r e m fo r r e le va n t l o g ic s . jo u r n a l o f ma t h e m a t ic a l s c ie n c e s 6 , 4 2 2 { 4 2 8 ( 1 9 7 6 ) . tr a n s la t e d fr o m issledovanija po konstructivnoj mathematike i matematiceskoj logike v, izdatelstvo nauka, 1972. [6 ] r a s io wa , h .: n { l a t t ic e s a n d co n s t r u c t ive l o g ic wit h s t r o n g n e g a t io n . fu n d a m e n t a ma t h e m a t ic a e 4 6 , 6 1 { 8 0 ( 1 9 5 8 ) [7 ] s p in ks , m., v e r o ®, r .: co n s t r u c t ive l o g ic wit h s t r o n g n e g a t io n is a s u b s t r u c t u r a l l o g ic . i. s t u d ia l o g ic a 8 8 , 3 2 5 { 3 4 8 ( 2 0 0 8 ) [8 ] s p in ks , m., v e r o ®, r .: co n s t r u c t ive l o g ic wit h s t r o n g n e g a t io n is a s u b s t r u c t u r a l l o g ic . ii. s t u d ia l o g ic a 8 9 , 4 0 1 { 4 2 5 ( 2 0 0 8 ) [9 ] za s la vs ky, i.d .: s ym m e t r ic co n s t r u c t ive l o g ic ( in r u s s ia n ) . p u b lis h in g h o u s e o f a c a d e m y o f s c ie n c e s o f a r m e n ia s s r ( 1 9 7 8 ) article1_in_english.dvi mathematical problems of computer science 25, 2006, 53{56. i nter val colour ings of some regular gr aphs r a fa ye l r . k a m a lia n , p e t r o s a . p e t r o s ya n institue for informatics and automation problems of nas of ra e-mails rrkamalian@yahoo.com, pet petros@yahoo.com abstract a lower bound is obtained for the greatest possible number of colors in an interval colourings of some regular graphs. r eferences [1 ] f. h a r a r y, " gr a p h th e o r y" , a d d is o n -w e s le y, r e a d in g , ma ,1 9 6 9 . [2 ] v .g. v iz in g , th e c h r o m a t ic in d e x o f a m u lt ig r a p h , k ibernetika 3 ( 1 9 6 5 ) , p p . 2 9 -3 9 . [3 ] a .a . zyko v, th e o r y o f ¯ n it e g r a p h s , n o vo s ib ir s k, n a u ka , 1 9 6 9 . [4 ] a .s . a s r a t ia n , r .r . k a m a lia n , in t e r va l c o lo u r in g s o f e d g e s o f a m u lt ig r a p h , appl. m ath. 5 ( 1 9 8 7 ) , y e r e va n s t a t e u n ive r s it y, p p . 2 5 -3 4 . [5 ] r .r . k a m a lia n , \ in t e r va l e d g e c o lo u r in g s o f gr a p h s " , d o c t o r a l d is s e r t a t io n , th e in s t it u t e o f ma t h e m a t ic s o f t h e s ib e r ia n b r a n c h o f t h e a c a d e m y o f s c ie n c e s o f u s s r , n o vo s ib ir s k, 1 9 9 0 . [6 ] s .v . s e va s t ia n o v, on in t e r va l c o lo r a b ilit y o f e d g e s o f a b ip a r t it e g r a p h , m ethods of d iscr. anal. in s o lu t io n o f e xt r e m a l p r o b le m s . th e in s t it u t e o f ma t h e m a t ic s o f t h e s ib e r ia n b r a n c h o f t h e a c a d e m y o f s c ie n c e s o f u s s r , n o vo s ib ir s k, n 5 0 ( 1 9 9 0 ) , p p . 6 1 -7 2 . [7 ] i. h o lye r , th e np -c o m p le t e n e s s o f e d g e c o lo u r in g , siam j . comput. 1 0 , n 4 ( 1 9 8 1 ) , p p . 7 1 8 -7 2 0 . [8 ] s . co o k, th e c o m p le xit y o f t h e o r e m -p r o vin g p r o c e d u r e s . in p roc.3rd acm symp. o n th e o r y o f co m p u t in g , 1 9 7 1 , p p . 1 5 1 -1 5 8 . [9 ] r .m. k a r p , r e d u c ib ilit y a m o n g co m b in a t o r ia l p r o b le m s , in \ co m p le xit y o f co m p u t e r co m p u t a t io n s " ( r .e . mille r a n d j.w . th a t c h e r , e d s .) , n e w y o r k, 1 9 7 2 , p p . 8 5 -1 0 3 . [1 0 ] e . t. p a r ke r , e d g e -c o lo u r in g n u m b e r s o f s o m e r e g u la r g r a p h s , p roc. amer. m ath. soc. 3 7 , ( 1 9 7 3 ) , p p . 4 2 3 -4 2 4 . àñáß ñ³ù³ë»é ·ñ³ýý»ñç ùçç³ï³ûù³ûçý ý»ñïáõùý»ñ è. ø³ù³éû³ý, ä. ä»ïñáëû³ý ²ù÷á÷áõù ²ßë³ï³ýùáõù ëï³óí³í ¿ ëïáñçý ·ý³ñ³ï³ï³ý áñáß ñ³ù³ë»é ·ñ³ýý»ñç ùçç³ï³ûù³ûçý ý»ñïù³ý ù»ç û·ï³·áñííáõ ·áõûý»ñç ³é³í»é³·áõûý ñý³ñ³íáñ ãíç ñ³ù³ñ: 5 3 microsoft word 40.doc mathematical problems of computer science 38, 93--94, 2012. 93 about order 3 hypergroups over group which arisе from groups of order 18 pervaneh zolfaghari yerevan state university each (unitary) hypergroup m over a group h is isomorphic to a hypergroup over group associated with a complementary set m to a subgroup h of a group g. all unitary hypergroups of order 3 over group, associated with the complementary sets to a subgroup of symmetric groups s3 and s4 was described in [1] and [2]. according to [3] any hypergroup over group is reduced to an irreducible hypergroup over group. therefore to describe all hypergroups over group first of all it is necessary to find all irreducible hypergroups over group. let m be a hypergroup over a group h. then there exists a structural homomorphism from h to the sm. the hypergroup over group is irreducible if and only if the kernel of this homomorphism is trivial. suppose |m| = 3. then we have the following sorts of monomorphisms from h to sm. 1) trivial |h| = 1. in this case the hypergroup over group is a group. 2) |h| = 2. in this case the hypergroup m is isomorphic to a hypergroup associated with a subgroup of index 3 in s3. there exist three such hypergroups over group (up to isomorphism). one of this hypergroups is reducible and is reduced to a group. two others are irreducible. 3) |h| = 3. then the hypergroup over group is reduced to a group, because there does not exist a non-abelian group of order 9. 4) |h| = 6. every such hypergroup over group is associated with a complementary subset m to a subgroup h of index 3 in a group g of order 18. up to isomorphism, there exist three non-abelian groups g of order 18. (a) g =  a, b, c, a3 = b2 = c3 = e, ba = a2b, ca = ac, cb = bc  . this group is the direct product of its symmetric subgroup n =  a, b  s3, and cyclic subgroup  c  c3. it has four subgroups of index 3: one normal subgroup n and three conjugate cyclic subgroups, which are generated by c and one of elements of order 2 in n. (b) g =  a, b, a9 = b2 = baba = e  . then g = d9 is the dihedral group and is the semidirect product of the normal subgroup n =  a   c9, and cyclic subgroup  b   c2. this group has three subgroups of order 6. they are conjugate and isomorphic to s3. (c) g =  a, b, c, a3 = b3 = c2 = e, ba = ab, ca = ac, cb = bc  . 94 about order 3 hypergroups over group which arisе from groups of order 18 this group is a semidirect product of the normal subgroup n =  a, b  c3 c3 and cyclic subgroup  c  c2. it has 12 subgroups of index 3. they are generated by one element of order 3 and one element of order 2. two subgroups of order 6 are conjugate if and only if they have the same generating element of order 3. thus, there are four classes consisting of three conjugate subgroups of order 6 in case (a) for an arbitrary subgroup h of order 6 of the group g the structural homomorphism from h to sm has a nontrivial kernel, i. e. in this case we have not irreducible hypergroups over group. the description of order 3 hypergroups over group, arising in the cases (b) and (c), is more complicate. reference 1. zolfaghary p., the hypergroups of order 3, arising from symmetric group s3. fourth group theory conference of iran, payam noor university of isfahan, isfahan, iran, march 7-9, 2012 . 2. zolfaghary p., the order three right hypergroups over group, arising from symmetric group s4. thes. of conf. of amu, yerevan, 25 may -2 junе 2012. p. 94-96. 3. dalalyan s. h., the reducibility theory for hypergroups over group. thes. of conf. of amu, yerevan, 25 may -2 juin 2012. p. 22 24. d:\sbornik\...\c1.dvi mathematical problems of computer science 23, 2004, 59{66. constr uction of sequences of n -polynomials over finite fields of odd char acter istics ge vo r g m. h a m b a r d z u m ya n institute for informatics and automation problems of nas of ra e-mail gev@hylink.am abstract in this paper the method of construction of n-polynomials over fq with q ´ 1 (mod 4) is presented. for a suitably chosen initial n-polynomial f1 (x) 2 fq [x] of degree 2 n -polynomials fk (x) 2 fq [x] of degree 2k are constructed by the iterated application of following transformation:f (x) ! (2x) d eg(f) f ³ x+´2x¡1 2 ´ ; ´ 2 fq; ´ 6= 0. refer ences [1 ] co h e n s . d ., th e e xp lic it c o n s t r u c t io n o f ir r e d u c ib le p o lyn o m ia ls o ve r ¯ n it e ¯ e ld s , d esigns, codes and cryptography 2 p p . 1 6 9 -1 7 4 , 1 9 9 2 . [2 ] k yu r e g ya n m. k ., r e c u r r e n t m e t h o d s fo r c o n s t r u c t in g ir r e d u c ib le p o lyn o m ia ls o ve r fq o f o d d c h a r a c t e r is t ic s , f inite f ields and their applications 9 p p . 3 9 -5 8 , 2 0 0 3 . [3 ] me yn h ., e xp lic it n-p o lyn o m ia ls o f 2 -p o we r d e g r e e o ve r fin it e fie ld s , i., d esigns, codes and cryptography 6 p p . 1 0 7 -1 1 6 , 1 9 9 5 . n -µ³½ù³ý¹³ùý»ñç ñ³çáñ¹³ï³ýáõãûáõýý»ñç ï³éáõóáõùá ï»ýï µýáõã³·ñçãý»ñáí í»ñç³íáñ ¹³ßï»ñç íñ³ ¶. ø. ð³ùµ³ñóáõùû³ý ²ù÷á÷áõù ²ûë ñá¹í³íáõù ý»ñï³û³óí³í ¿ n -µ³½ù³ý¹³ù»ñç ï³éáõóù³ý »õ³ý³ï fq; q ´ 1 mod ( 4 ) í»ñç³íáñ ¹³ßï»ñáõù: ð³ù³å³ï³ëë³ý ï»ñåáí áýïñí³í 2 ³ëïç׳ýç f1 ( x ) 2 fq[x] n -µ³½ù³ý¹³ùç ñ³ù³ñ 2 k ³ëïç׳ýç f1 ( x) 2 fq[x] n -µ³½ù³ý¹³ù»ñá ï³éõóíáõù »ý f ( x ) ! ( 2 x ) deg(f)f ( x+´2x¡1 2 ) ; ´ 2 fq; ´ 6= 0 ó¨³÷áëáõãû³ý ïçñ³éáõùáí: 5 9 d:\sbornik\...\mirumyan.dvi mathematical problems of computer science 23, 2004, 32{35. some simpli¯cation of complete system of t r ansfor mations for p r oper e dge colour ings of b ipar tite gr aphs a lis a k . mir u m ya n institute for informatics and automation problems of nas of ra e-mail mirumyan@rambler.ru abstract in the investigation of a set of combinatorial objects, an important problem is to ¯nd a complete system of transformations. in this article a set of (k + 1)-colourings of an arbitrary bipartite graph are considered. it is prooved that 2-transformation is a complete system of transformations for this type of set. refer ences [1 ] a . s . a s r a t ia n a n d a . n . mir u m ya n , tr a n s fo r m a t io n o f e d g e co lo u r in g s o f a b ip a r t it e m u lt ig r a p h , a n d t h e ir a p p lic a t io n s , s o vie t ma t h . d o kl. vo l. 4 3 , 1 9 9 1 . [2 ] f. h a r a r y, gr a p h t h e o r y, a d d is o n -w e s le y, 1 9 6 9 . [3 ] a . s . a s r a t ia n , tr is t a n m. j. d e n le y & r . h a g g kvis t , b ip a r t it e g r a p h s a n d t h e ir a p p lic a t io n s , ca m b r id g e u n ive r s it y, 1 9 9 8 . [4 ] a . s . a s r a t ia n a n d a . n . mir u m ya n , on t r a n s fo r m a t io n s o f e d g e c o lo u r in g s o f t h e c o m p le t e b ip a r t it e g r a p h knn:, a ka d . n a u k r e s p u b . a r m e n ia d o kl. 9 5 , 1 9 9 5 . ºñïù³ë ·ñ³ý»ñç ë»÷³ï³ý ïáõç ·áõý³íáñù³ý ó¨³÷áëáõãûáõýý»ñç éñçí ñ³ù³ï³ñ·ç å³ñ½»óáõù ². ü. øçñáõùû³ý ²ù÷á÷áõù ð³ù³ïó³ï³ý ûµû»ïïý»ñç µ³½ùáõãû³ý ñ»ï³½áïù³ý áýã³óùáõù ï³ñ¨áñ ñçùý³ëý¹çñ ¿ ó¨³÷áëáõãûáõýý»ñç éñçí ñ³ù³ï³ñ·ç ·ïý»éá: 3 2 articlearsen0728f.dvi mathematical problems of computer science 23, 2004, 67{79. t wo-dimensional sequence h omogeneity t esting against m ixtur e alter native¤ ir in a a . s a fa r ya n , e vg u e n i a . h a r o u t u n ia n a n d a r s e n v . ma n a s ya n institue for informatics and automation problems of nas of ra e-mail evhar@ipia.sci.am abstract the behavior of linear rank statistics is investigated on models in which various subsequences of observations follow di®erent statistical distributions. such data can be interpreted both as models of a ¯nite number distribution mixtures and as dependence models. we apply data set simulation to obtain estimates of average and variance of used rank statistics. the modeled and asymptotic results are enough close. refer ences [1 ] l a u s e n b ., s h u m a c h e r h ., " ma xim a lly s e le c t e d r a n k s t a t is t ic s " , b io m e t r ic s , 4 8 , p p . 7 3 -8 5 , 1 9 9 2 . [2 ] h a r o u t u n ia n e ., s a fa r ya n i. " d is t r ib u t io n s m ixt u r e d ivis io n wit h a s t r a t ifyin g p a r a m e t e r " , s u b m it t e d fo r p u b lic a t io n . [3 ] s a fa r ya n i., h a r o u t u n ia n e . " a co m m o n a p p r o a c h t o t h e d is t r ib u t io n s m ixt u r e id e n t i¯ c a t io n a n d d e p e n d e n c e m o d e ls a n a lys is " , p r o c e e d in g s o f csit 2003, p p . 1 8 4 -1 8 6 . [4 ] h a r o u t u n ia n e . a n d s a fa r ya n i. " n o n p a r a m e t r ic c o n s is t e n t e s t im a t io n o f t h e c h a n g e m o m e n t o f r a n d o m s e qu e n c e p r o p e r t ie s " , transactions of institute for informatics and automation p roblems of nas of r a and of ysu, m athematical p roblems of computer science, vo l. 1 7 , p p . 7 6 -8 5 , 1 9 9 7 . [5 ] h o t h o r n t., l a u s e n b ., " on t h e e xa c t d is t r ib u t io n o f m a xim a lly s e le c t e d r a n k s t a t is t ic s " , comp. statist. and d ata anal., vo l. 4 3 , p p . 1 2 1 1 3 7 , 2 0 0 3 . ¤the work was partially supported by intas, project 00-738. 6 7 6 8 two-dimensional sequence homogeneity testing against mixture alternative ê³éýáõñ¹ç »ñïáýïñ³ýùç ñ³ý¹»å »ñïã³÷³ýç ñ³çáñ¹³ï³ýáõãû³ý ñ³ù³ë»éáõãû³ý ëïáõ·áõùá º. ð³ñáõãûáõýû³ý, æ. ê³ý³ñû³ý ¨ ². ø³ý³ëû³ý ²ù÷á÷áõù ð»ï³½áïí³í ¿ ·í³ûçý ï³ñ·³ûçý íç׳ï³ýçý»ñç í³ñùá ùá¹»éý»ñáõù, áñï»õ ¹çï³ñïáõùý»ñç ñ³çáñ¹³ï³ýáõãûáõýý»ñá »ýã³ñïíáõù »ý ï³ñµ»ñ íç׳ﳷñ³ï³ý µ³ßëáõùý»ñç: ²û¹åçëç ïíû³éý»ñá ï³ñ»éç ¿ ù»ïý³µ³ý»é ¨ áñå»ë í»ñç³íáñ ãíáí µ³ßëáõùý»ñç ë³éýáõñ¹ ùá¹»éý»ñ, ¨ áñå»ë ï³ëí³íáõãû³ý ùá¹»éý»ñ: ø»ýù ïçñ³éáõù »ýù ïíû³éý»ñç µ³½ùáõãû³ý ùá¹»é³íáñáõùª û·ï³·áñíí³í ï³ñ·³ûçý íç׳ï³ýçý»ñç ùçççýý»ñç ¨ óñí³íùý»ñç ·ý³ñ³ï³ï³ýý»ñç ëï³óù³ý ñ³ù³ñ: øá¹»é³íáñù³ý ¨ ³ëçùåïáï³ï³ý ³ñ¹ûáõýùý»ñá µ³í³ï³ý³ã³÷ ùáï »ý: d:\sbornik\...\tpel3.dvi mathematical problems of computer science 32, 78{85, 2009. i nter val t otal color ings of gr aphs with a spanning star p e t r o s a p e t r o s ya n yz a n d n e r s e s a . k h a c h a t r ya n y yinstitute for informatics and automation problems of nas of ra, zdepartment of informatics and applied mathematics, ysu pet petros@ipia.sci.am, xachnerses@gmail.com abstract an interval total t¡coloring of a graph g is a total coloring of g with colors 1; 2; : : : ; t such that at least one vertex or edge of g is colored by i; i = 1; 2; : : : ; t, and the edges incident to each vertex v together with v are colored by dg(v) + 1 consecutive colors, where dg(v) is the degree of a vertex v in g. in this paper we prove that if g = (v; e) is a graph containing the vertex u with dg(u) = jv j ¡ 1, k(g) = maxv2v (v 6=u)dg(v) < jv j ¡ 1 and g admits an interval total t¡coloring then t · jv j + 2k(g). we also show that this upper bound is sharp. further we determine all possible values of t for which the wheels have an interval total t¡coloring. refer ences [1 ] p . a . p e t r o s ya n , \ in t e r va l t o t a l c o lo r in g s o f c o m p le t e b ip a r t it e g r a p h s " , p roceedings of the csit conference, p p . 8 4 -8 5 , 2 0 0 7 . [2 ] p . a . p e t r o s ya n , \ in t e r va l t o t a l c o lo r in g s o f c e r t a in g r a p h s " , m athematical p roblems of computer science, vol. 31, p p . 1 2 2 -1 2 9 , 2 0 0 8 . [3 ] d . b . w e s t , in t r o d u c t io n t o gr a p h th e o r y, p r e n t ic e -h a ll, n e w je r s e y, 1 9 9 6 . [4 ] h . p . y a p , to t a l co lo r in g s o f gr a p h s , l e c t u r e n o t e s in ma t h e m a t ic s 1 6 2 3 , s p r in g e r v e r la g , 1 9 9 6 . îù³ëù³ûçý ³ëïõáí ·ñ³ýý»ñç ùçç³ï³ûù³ûçý éç³ï³ï³ñ ý»ñïáõùý»ñ ä. ä»ïñáëû³ý, ü. ê³ã³ïñû³ý ²ù÷á÷áõù g ·ñ³ýç éç³ï³ï³ñ ý»ñïáõùá 1 ; 2 ; : : : ; t ·áõûý»ñáí ï³ýí³ý»ýù ùçç³ï³ûù³ûçý éç³ï³ï³ñ t –ý»ñïáõù, »ã» ³ù»ý ùç i ·áõûýáí, i = 1 ; 2 ; : : : ; t, ý»ñïí³í ¿ ³éýí³½ý ù»ï ·³·³ã, ï³ù ïáõ, ¨ ûáõñ³ù³ýãûáõñ v ·³·³ãçý ïçó ïáõ»ñá, ¨ ³û¹ ·³·³ãá ý»ñïí³í ¿ dg ( v ) + 1 ñ³çáñ¹³ï³ý ·áõûý»ñáí, áñï»õ dg ( v ) -áí ýß³ý³ïí³í v ·³·³ãç ³ëïç׳ýá g ·ñ³ýáõù: ²ûë ³ßë³ï³ýùáõù ³å³óáõóí³í ¿, áñ »ã» g = ( v; e ) -ý, áñá å³ñáõý³ïáõù ¿ ³ûýåçëç u ·³·³ã, áñ dg ( u ) = jv j¡ 1 , k ( g) = m a xv2v (v 6=u)dg ( v ) < jv j¡ 1 ¨ g ·ñ³ýý áõýç ùçç³ï³ûù³ûçý éç³ï³ï³ñ tý»ñïáõù, ³å³ then t · jv j + 2 k ( g ) : ü³¨ óáõûó ¿ ïñí³í, 7 8 p. petrosyan, n. khachatryan 7 9 áñ ³ûë í»ñçý ·ý³ñ³ï³ï³ýá ñ³ë³ý»éç ¿: ²ûýáõñ»ï¨, ·ïýí»é »ý tç µáéáñ ñý³ñ³íáñ ³ñå»ùý»ñá, áñáýó ñ³ù³ñ ³ýçíý»ñá áõý»ý ùçç³ï³ûù³ûçý éç³ï³ï³ñ tý»ñïáõù: d:\sbornik\...\untitled12.dvi ìàòåìàòè÷åñêèå âîïðîñû êèáåðíåòèêè è âû÷èñëèòåëüíîé òåõíèêè 32, 107–111, 2009. î íàèìåíüøåì è íàèáîëüøåì âîçìîæíûõ ÷èñëàõ âåðøèí ñ èíòåðâàëüíûì ñïåêòðîì íà ìíîæåñòâå ïðàâèëüíûõ ðåáåðíûõ ðàñêðàñîê äåðåâà íàðèíå í. äàâòÿí èäæåâàíñêèé ôèëèàë åãó n a r in e d a vt ya n @m a il.r u àííîòàöèÿ íàéäåíû íàèìåíüøèå è íàèáîëüøèå âîçìîæíûå ÷èñëà âåðøèí ñ èíòåðâàëüíûì ñïåêòðîì íà ìíîæåñòâå ïðàâèëüíûõ ðåáåðíûõ ðàñêðàñîê ïðîèçâîëüíîãî äåðåâà. ëèòåðàòóðà [1 ] ô. õàðàðè , òåîðèÿ ãðàôîâ, ì., ìèð, 304 ñ., 1973. [2 ] à. ñ. àñðàòÿí, ð.ðêàìàëÿí., ”èíòåðâàëüíûå ðàñêðàñêè ðåáåð ìóëüòèãðàôà”, ïðèêëàäíàÿ ìàòåìàòèêà, åãó, âûï. 5, ñ. 25-34, 1987. [3 ] ð. ð. êàìàëÿí, èíòåðâàëüíûå ðåáåðíûå ðàñêðàñêè ãðàôîâ, äèññåðòàöèÿ íà ñîèñêàíèå ó÷åíîé ñòåïåíè êàíäèäàòà ôèç.-ìàò. íàóê, èì ñî àí ñññð, íîâîñèáèðñê, 103 ñ., 1990. [4 ] ð. ð. êàìàëÿí, èíòåðâàëüíûå ðàñêðàñêè ïîëíûõ äâóäîëüíûõ ãðàôîâ è äåðåâüåâ, ïðåïðèíò âö àí àðìÿíñêîé ññð, åðåâàí, 11 ñ, 1989. [5 ] ï. à. ïåòðîñÿí, èíòåðâàëüíî ðåàëèçóåìûå íàáîðû â íåêîòîðûõ êëàññàõ ãðàôîâ, äèññåðòàöèÿ íà ñîèñêàíèå ó÷åíîé ñòåïåíè êàíäèäàòà ôèç.-ìàò. íàóê, èïèà íàí ðà, åðåâàí, 130 ñ., 2006. ì³éç ïáõ³ûçý ×çßï ý»ñïáõùý»ñç µ³½ùáõãû³ý íñ³ ùçç³ï³ûù³ûçý ëå»ïïñáí ·³·³ãý»ñç ñý³ñ³íáñ ýí³½³·áõûý ¨ ³é³í»é³·áõûý ãí»ñç ù³ëçý ü. ¸³íãû³ý ²ù÷á÷áõù ¶ïýí³í »ý ï³ùû³ï³ý í³éç ïáõ³ûçý ×çßï ý»ñïáõùý»ñç µ³½ùáõãû³ý íñ³ ùçç³ï³ûù³ûçý ëå»ïïñáí ·³·³ãý»ñç ñý³ñ³íáñ ýí³½³·áõûý ¨ ³é³í»é³·áõûý ãí»ñá: 1 0 7 начиная с начала 2000 года осуществляется внедрение ghis в здравоохранении, в рамках принятого проекта о реформирование информ mathematical problems of computer science 51, 90--97, 2019. udc 004.93 object color estimation with dominant color in automated image semantic description generation tasks aghasi s. poghosyan institute for informatics and automation problems of nas ra e-mail: agasy18@gmail.com abstract the automated image tagging is an important part of modern search engines. the generated image tags can be constructed from object names and their attributes, for example, colors. this work presents an object color name detection real-time algorithm. it is applicable to any automatic object detection and localization systems. the presented algorithm is fast enough to run after the existing real-time object detection system, without adding visible overhead. the algorithm uses k-means to detect the dominant color and selects the correct name for the color via delta e (cie 2000). keywords: dominant color, image color palette, color name selection, object color detection, image semantic analysis. 1. introduction automated image semantic description generation is one of the fundamental problems in computer vision. one of the ways to generate a semantic description of the given image is the object detection and localization. there are many solutions, which can find objects by localizing and naming them. in recent years, neural networks have become a leading method for highquality object detection tasks. they have enough accuracy to be used in different areas, such as self-driving cars, robotics, automatic surveillance and also in web search engines. in web search engines, the object detectors are used for automated image tagging. these tags allow the user to find images without any annotations using a simple keyword search. the simplest way to automatically generate image tags is to detect objects in the image and use the names of objects as tags. this kind of tags do not allow the user to find images, which contain an object with a specific attribute. to solve this issue, we can add additional attributes for each object. one of the main attributes for each object is color. this kind of addition will allow users to search, for example, а “red car” or а “white cat” and find the corresponding images. 90 ag. poghosyan 91 this work presents an object color name detestation algorithm, which is applicable to automatic object detection systems. the presented algorithm is fast enough (28 ms on i7 cpu) to run after the existing object detection real-time system without adding visible overhead on each iteration. 2. related works and methodology the modern object detectors are based on convolutional neural networks (cnn), such as faster region-based convolutional neural network (faster r-cnn) [1], region-based fully convolutional network (r-fcn) [2], multibox [3], single shot detector (ssd) [4] and yolo: real-time object detection [5]: for example, networks can be used to generate image tags from the detected object names. there are networks which, can generate a single caption [6], [7] for the given image. there are also networks, which can generate the image caption and detect the objects [8], [9]. those works are not computing color for each object. all of these works can be used as object detectors for this work. in [10], the authors present a method, which assigns a color attribute to a recognized object. they are using an image segmentation method based on graph cuts. after segmentation, they are filtering the results by “attribute recommender” [10]. this approach is slightly slow for realtime systems. this work introduces a color name extraction method, which has a low computational cost. to exact color name for each detected object, we will detect the dominant color and then select the correct name for it. 3. dominant color detection at first sight, an image dominant color detection is a simple task, and we can compute the mean color for an image. but we will be disappointed, because we will get something close to gray (see fig. 1). fig. 1: image and its mean color. the second approach that we can apply, we can split the image as a grid, and then compute the mean value for each grid’s cell. this approach will be approximated to a downscaled image with an interpolation algorithm, which uses neighbor pixels, such as liner, bicubic or spline methods. object color estimation with dominant color in automated image semantic description generation tasks 92 fig. 2: scaled image with bicubic interpolation and central color. as we see in fig. 2, we can’t select the central pixel color as the image dominant color of the downscaled image. fig. 3: image color palette and dominant color. to solve this problem, we will use a clustering algorithm to extract a color palette from the image. at first step, we will downscale our image to 100x100 dimensionality via bicubic interpolation method (see fig. 3). this step will remove rarely encountered pixels by including them in the final pixel of the downscaled image. in addition, this technique will reduce the number of pixels, which will have a huge impact on the clustering algorithm. after that, we will cluster the image pixels in srgb (standard red green blue) color space via the k-means [11] clustering method. experimentally we found that our color palette is more meaningful when the cluster count is equal to 9. the k-means cauterization will give as cluster indices (labels, per pixel mappings to their cluster) and the cluster center values (means). to get the image color palette, we will sort by clusters size in descending order. each palette’s color will correspond to cluster centers (see fig. 3). the first color of the pallet will be selected as the dominant color for the image. ag. poghosyan 93 4. color name selection as we have a dominant color value in srgb color space, now we need to select a name for this color. selection will be made from a predefined color table (see table 1). we will compare each table’s color with our computed value and find the nearest one. but we can’t do this using the euclidean distance (1), because visually close colors are not located near in srgb space. one of the best algorithms to compute the color visual difference (distance) 𝛥𝛥𝛥𝛥 (2) between a sample color (𝐿𝐿2, 𝑎𝑎2, 𝑏𝑏2) and a reference color (𝐿𝐿1, 𝑎𝑎1, 𝑏𝑏1) is delta e (cie 2000) [12]: in equation (2) δ𝐿𝐿′, δ𝐶𝐶′, δ𝐻𝐻′ and 𝑅𝑅𝑇𝑇 values are computed by equations (3)-(4). we will take𝐾𝐾𝐿𝐿 = 1, 𝐾𝐾𝐶𝐶 = 1, 𝐾𝐾𝐻𝐻 = 1 by default. 𝑅𝑅𝐶𝐶 and 𝛥𝛥𝛥𝛥 values are computed by equations (5)-(10). 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑎𝑎𝑑𝑑𝑑𝑑𝑑𝑑 = �(𝑅𝑅1 − 𝑅𝑅2) 2 + (𝐺𝐺1 − 𝐺𝐺2)2(𝐵𝐵1 − 𝐵𝐵2)2 . (1) δ𝛥𝛥 = �� δ𝐿𝐿′ 𝐾𝐾𝐿𝐿𝑆𝑆𝐿𝐿 � 2 + � δ𝐶𝐶′ 𝐾𝐾𝐶𝐶𝑆𝑆𝐶𝐶 � 2 + � δ𝐻𝐻′ 𝐾𝐾𝐻𝐻𝑆𝑆𝐻𝐻 � 2 + 𝑅𝑅𝑇𝑇 � δ𝐶𝐶′ 𝐾𝐾𝐶𝐶𝑆𝑆𝐶𝐶 �� δ𝐻𝐻′ 𝐾𝐾𝐻𝐻𝑆𝑆𝐻𝐻 �. (2) δ𝐿𝐿′ = 𝐿𝐿2 − 𝐿𝐿1, δ𝐶𝐶′ = 𝐶𝐶2′ − 𝐶𝐶1′, (3) δ𝐻𝐻′ = 2�𝐶𝐶1 ′𝐶𝐶2 ′ sin � 𝛥𝛥ℎ′ 2 � , 𝑅𝑅𝑇𝑇 = −𝑅𝑅𝐶𝐶 sin(2𝛥𝛥𝛥𝛥) (4) 𝐻𝐻 ′ = � ℎ1′ + ℎ2′ + 360° 2 , if |ℎ1′ − ℎ2′ | > 180° ℎ1′ + ℎ2′ 2 , otherwise. (5) 𝑇𝑇 = 1 − 0.17 cos �𝐻𝐻 ′ − 30°� + 0.24 cos �2𝐻𝐻 ′ � + 0.32 cos �3𝐻𝐻 ′ + 6°� −0.20cos (4𝐻𝐻 ′ − 63°) (6) δℎ′ = � ℎ2′ − ℎ1′ if |ℎ2′ − ℎ1′ | ≤ 180° ℎ2′ − ℎ1′ + 360° else if |ℎ2′ − ℎ1′ | > 180°and ℎ2′ ≤ ℎ1′ ℎ2′ − ℎ1′ − 360° otherwise (7) δ𝛥𝛥 = 30 exp �−� 𝐻𝐻 ′ − 275° 25 � 2 �, 𝐿𝐿 ′ = 𝐿𝐿1 + 𝐿𝐿2 2 , 𝐶𝐶 ′ = 𝐶𝐶1′ + 𝐶𝐶2′ 2 (8) 𝑆𝑆𝐿𝐿 = 1 + 0.015(𝐿𝐿 ′ − 50)2 �20 + (𝐿𝐿 ′ − 50)2 , 𝑆𝑆𝐶𝐶 = 1 + 0.045𝐶𝐶 ′, 𝑆𝑆𝐻𝐻 = 1 + 0.015𝐶𝐶 ′ 𝑇𝑇 (9) 𝑅𝑅𝐶𝐶 = 2� 𝐶𝐶 ′7 𝐶𝐶 ′7 + 257 (10) object color estimation with dominant color in automated image semantic description generation tasks 94 𝐿𝐿1, 𝐶𝐶1′, ℎ1′ are computed by equations (11)-(12). also, the 𝐿𝐿2, 𝐶𝐶2′, ℎ2′ values are computed in the same way. we will use metric (2) to find the nearest color and get the color name from table 1. table 1: css color table. 4. conclusion we have developed a fast object color name extraction method. to test the algorithm on end to end system, we used a system, which generates the image caption, detects and localizes the objects [9]. for each detected object, we computed the object dominate color and found the correct name for that color. each color name was used as an attribute for that object (see fig. 4). for detections, which were predicated as humans, we did not use the color attribute. 𝐶𝐶1 = �𝑎𝑎1 2 + 𝑏𝑏1 2, 𝐺𝐺 = 1 2 �1 − � 𝐶𝐶 7 𝐶𝐶 7 + 257 �, 𝑎𝑎1′ = 𝑎𝑎1(1 + 𝐺𝐺) (11) 𝐶𝐶1′ = �𝑎𝑎1 ′ 2 + 𝑏𝑏1 2, ℎ1′ = ⎩ ⎪ ⎨ ⎪ ⎧arctan � 𝑏𝑏1 𝑎𝑎1 ′� , if arctan � 𝑏𝑏1 𝑎𝑎1 ′� ≥ 0, arctan � 𝑏𝑏1 𝑎𝑎1 ′� + 360°, otherwise. (12) ag. poghosyan 95 fig. 4: image caption generation, object detection, localization and color estimation, second row presents dominant color detection process. references [1] s. ren, k. he, r. girshick and j. sun, “faster r-cnn: towards real-time object detection with region proposal networks,” in advances in neural information processing systems 28, 2015. [2] j. dai, y. li, k. he and j. sun, “r-fcn: object detection via region-based fully convolutional networks,” in advances in neural information processing systems 29, 2016. [3] c. szegedy, s. reed, d. erhan, d. anguelov and s. ioffe, “scalable, high-quality object detection,” arxiv preprint arxiv:1412.1441, 2014. [4] w. liu, d. anguelov, d. erhan, c. szegedy, s. reed, c.-y. fu and a. c. berg, “ssd: single shot multibox detector,” in european conference on computer vision, pp. 21-37, 2016. [5] j. redmon, s. divvala, r. girshick and a. farhadi, “you only look once: unified, realtime object detection,” in proceedings of the ieee conference on computer vision and pattern recognition, pp. 779-788, 2016. [6] o. vinyals, a. toshev, s. bengio and d. erhan, "show and tell: a neural image caption generator," in proceedings of the ieee conference on computer vision and pattern recognition, pp. 3156-3164, 2015. [7] a. poghosyan and h. sarukhanyan, “short-term memory with read-only unit in neural image caption generator,” 11-th international conference computer science and object color estimation with dominant color in automated image semantic description generation tasks 96 information technologies, revised selected papers, ieee xplore, doi: 10.1109/csitechnol.2017.8312163, pp. 162-167, 2017. [8] a. karpathy and l. fei-fei, “deep visual-semantic alignments for generating image descriptions,” in proceedings of the ieee conference on computer vision and pattern recognition, pp. 3128—3137, 2015. [9] a. poghosyan and h. sarukhanyan, “image caption generation model based on object detector”, transactions of iiap nas ra, mathematical problems of computer science, vol. 50, pp. 5-14, 2018. [10] k. duan, d. parikh, d. crandall and k. grauman, “discovering localized attributes for fine-grained recognition,” ieee conference on computer vision and pattern recognition (cvpr), pp. 1346—1353, 2012. [11] j. a. hartigan and m. a. wong, “algorithm as 136: a k-means clustering algorithm” journal of the royal statistical society. series c (applied statistics), vol. 28, pp. 100108, 1979. [12] g. sharma, w. wu and e. n. dalal, “the ciede2000 color-difference formula: implementation notes, supplementary test data, and mathematical observations,” color research & application: endorsed by inter-society color council, the colour group (great britain), canadian society for color, color science association of japan, dutch society for the study of color, the swedish colour centre foundation, colour society of australia, centre français de la couleur, vol. 30, pp. 21-30, 2005. submitted 16.01.2019, accepted 22.04.2019. օբյեկտների գույնի որոշումը դոմինանտ գույնի կիրառմամբ պատկերների իմաստաբանական նկարագրերի ավտոմատ գեներացման խնդիրներում աղասի ս. պողոսյան հհ գաա ինֆորմատիկայի և ավտոմատացման պրոբլեմների ինստիտուտ e-mail: agasy18@gmail.com ամփոփում ժամանակակից փնտրող համակարգերում մեծ դեր ունի պատկերների ավտոմատացված պիտակավորումը։ պատկերի պիտակները ավտոմատ կարող են գեներացվել պատկերում գտնվող օբյեկտների անուններից և վերջիններիս հատկություններից, ինչպիսին, օրինակ, գույնն է։ այս աշխատանքում ներկայացված է օբյեկտների գույնի անվանման հայտնաբերման արագ ալգորիթմ, որը կիրառելի է ավտոմատացված օբյեկտները հայտնաբերող համակարգի հետ։ ներկայացված mailto:agasy18@gmail.com ag. poghosyan 97 ալգորիթմը բավական արագ է (28 միլիվայրկյան), որ կարելի է համակցել իրական ժամանակում օբյեկտներ հայտնաբերող համակարգին` առանց տեսանելի ծախսեր ավելացնելու։ ալգորիթմը կիրառում է k միջինների ալգորիթմը դոմինանտ գույնի հայտնաբերման և delta e (cie 2000) -ը` գույնի ճիշտ անվան ընտրության համար։ բանալի բառեր՝ դոմինանտ գույն, պատկերի գունապնակ, գույնի անվան ընտրություն, օբյեկտի գույնի հայտնաբերում, պատկերի իմաստաբանական վերլուծություն։ определение цвета объекта с использованием доминантного цветного в задачах автоматической генерации семантического описания изображений агаси с. погосян институт проблем информатики и автоматизации нан ра e-mail: agasy18@gmail.com аннотация автоматическая маркировка изображений является важной частью современных поисковых систем. сгенерированные маркировки изображений могут быть построены из имен объектов и их атрибутов, например, цвета. в данной работе представлен быстрый алгоритм определения имени цвета объекта. это применимо к любым системам автоматического обнаружения и локализации объектов. представленный алгоритм достаточно быстр (28 мс) для запуска с существующими системами обнаружения объектов в реальном времени, без видимых дополнительных расходов. алгоритм использует метод k-средних для обнаружения доминирующего цвета и delta e (cie 2000) для выбора правильного имени цвета. ключевые слова: доминирующий цвет, цветовая палитра изображения, выбор названия цвета, определение цвета объекта, семантический анализ изображения. mailto:agasy18@gmail.com udc 004.93 object color estimation with dominant color in automated image semantic description generation tasks 2. related works and methodology 3. dominant color detection 4. color name selection d:\user\...\main.dvi mathematical problems of computer science 49, 26{34, 2018. on a p r oblem of wang concer ning the h amiltonicity of b ipar tite digr aphs s a m ve l k h . d a r b in ya n a n d is ka n d a r a . k a r a p e t ya n institute for informatics and automation problems of nas ra e-mail: amdarbin@ipia.sci.am, isko@ipia.sci.am abstract r. wang (discrete mathematics and theoretical computer science, vol. 19(3), 2017) proposed the following problem. problem. let d be a strongly connected balanced bipartite directed graph of order 2a ¸ 8. suppose that d(x) ¸ 2a ¡ k, d(y) ¸ a + k or d(y) ¸ 2a ¡ k, d(x) ¸ a + k for every pair of vertices fx; yg with a common out-neighbour, where 2 · k · a=2. is d hamiltonian? in this paper, we prove that if a digraph d satis¯es the conditions of this problem, then (i) d contains a cycle factor, (ii) for every vertex x 2 v (d) there exists a vertex y 2 v (d) such that x and y have a common out-neighbour. keywords: digraph, cycle, hamiltonian cycle, bipartite balanced digraph, perfect matching. 1 . in t r o d u c t io n in t h is p a p e r , we c o n s id e r ¯ n it e d ir e c t e d g r a p h s ( d ig r a p h s ) wit h o u t lo o p s a n d m u lt ip le a r c s . a d ig r a p h d is c a lle d h a m ilt o n ia n if it c o n t a in s a h a m ilt o n ia n c yc le , i.e ., a c yc le t h a t in c lu d e s e ve r y ve r t e x o f d. th e ve r t e x s e t a n d t h e a r c s e t o f a d ig r a p h d a r e d e n o t e d b y v ( d ) a n d a( d ) , r e s p e c t ive ly. th e o r d e r o f a d ig r a p h d is t h e n u m b e r o f it s ve r t ic e s . a c yc le fa c t o r in d is a c o lle c t io n o f ve r t e x-d is jo in t c yc le s c1; c2; : : : ; cl s u c h t h a t v ( c1 ) [v ( c2 ) [: : :[v ( cl ) = v ( d ) . a d ig r a p h d is b ip a r t it e if t h e r e e xis t s a p a r t it io n x, y o f v ( d ) in t o t wo p a r t it e s e t s s u c h t h a t e ve r y a r c o f d h a s it s e n d -ve r t ic e s in d i®e r e n t p a r t it e s e t s . it is c a lle d b a la n c e d if jxj = jy j. th e r e a r e a n u m b e r o f c o n d it io n s t h a t g u a r a n t e e t h a t a b ip a r t it e d ig r a p h is h a m ilt o n ia n ( s e e , e .g ., [1 ]-[1 1 ]) . l e t u s r e c a ll t h e fo llo win g d e g r e e c o n d it io n s t h a t g u a r a n t e e t h a t a b a la n c e d b ip a r t it e d ig r a p h is h a m ilt o n ia n . t heor em 1.1. ( a d a m u s , a d a m u s a n d y e o [8 ]) l et d be a balanced bipartite digraph of order 2 a, where a ¸ 2 . then d is hamiltonian provided one of the following holds: (a) d ( u) + d ( v ) ¸ 3 a + 1 for every pair of non-adjacent distinct vertices u and v of d; (b) d is strongly connected and d( u ) + d( v ) ¸ 3 a for every pair of non-adjacent distinct vertices u and v of d; 2 6 s. darbinyan and i. karapetyan 2 7 (c) the minimal degree of d is at least ( 3 a + 1 ) = 2 ; (d) d is strongly connected, and the minimal degree of d is at least 3 a=2 . ob s e r ve t h a t th e o r e m 1 .1 im p o s e s a d e g r e e c o n d it io n o n a ll p a ir s o f n o n -a d ja c e n t ve r t ic e s . in t h e fo llo win g t h e o r e m s a d e g r e e c o n d it io n r e qu ir e s o n ly fo r s o m e p a ir s o f n o n -a d ja c e n t ve r t ic e s . t heor em 1.2. ( j. a d a m u s [9 ]) l et d be a strongly connected balanced bipartite digraph of order 2 a ¸ 6 . if d( x ) +d( y ) ¸ 3 a for every pair of vertices x, y with a common out-neighbour or a common in-neighbour, then d is hamiltonian. n o t ic e t h a t th e o r e m 1 .2 im p r o ve s th e o r e m 1 .1 . s o m e s u ± c ie n t c o n d it io n s fo r t h e e xis t e n c e o f h a m ilt o n ia n c yc le s in a b ip a r t it e t o u r n a m e n t a r e d e s c r ib e d in t h e s u r ve y p a p e r [3 ] b y gu t in . a c h a r a c t e r iz a t io n fo r h a m ilt o n ic it y fo r s e m ic o m p le t e b ip a r t it e d ig r a p h s wa s o b t a in e d in d e p e n d e n t ly b y gu t in [2 ] a n d h äa g g kvis t a n d ma n o u s s a kis [4 ]. t heor em 1.3. ( w a n g [1 0 ]) l et d be a strongly connected balanced bipartite digraph of order 2 a, where a ¸ 1 . suppose that, for every pair of vertices fx; yg with a common outneighbour, either d ( x) ¸ 2 a ¡ 1 and d( y ) ¸ a + 1 or d( y ) ¸ 2 a ¡ 1 and d( x ) ¸ a + 1 . then d is hamiltonian. b e fo r e s t a t in g t h e n e xt t h e o r e m we n e e d t o d e ¯ n e a b a la n c e d b ip a r t it e d ig r a p h o f o r d e r e ig h t . e xample 1. l e t d ( 8 ) b e a b ip a r t it e d ig r a p h wit h p a r t it e s e t s x = fx0; x1; x2; x3g a n d y = fy0; y1; y2; y3g, a n d t h e a r c s e t a ( d ( 8 ) ) c o n t a in s e xa c t ly t h e fo llo win g a r c s : y0x1, y1x0, x2y3, x3y2 a n d a ll t h e a r c s o f t h e fo llo win g 2 -c yc le s : xi $ yi, i 2 [0 ; 3 ], y0 $ x2, y0 $ x3, y1 $ x2 a n d y1 $ x3. it is e a s y t o s e e t h a t d( x2 ) = d( x3 ) = d ( y0 ) = d( y1 ) = 7 a n d d( x0 ) = d( x1 ) = d ( y2 ) = d( y3 ) = 3 ; a n d t h e d o m in a t in g p a ir s in d ( 8 ) a r e : fy0; y1g, fy0; y2g,fy0; y3g,fy1; y2g, fy1; y3g, fx0; x2g, fx0; x3g, fx1; x2g, fx1; x3g a n d fx2; x3g. n o t e t h a t e ve r y d o m in a t in g p a ir s a t is ¯ e s t h e c o n d it io n b1. s in c e x0y0x3y2x2 y1x0 is a c yc le in d ( 8 ) , it is n o t d i± c u lt t o c h e c k t h a t d ( 8 ) is s t r o n g . ob s e r ve t h a t d ( 8 ) is n o t h a m ilt o n ia n . in d e e d , if c is a h a m ilt o n ia n c yc le in d ( 8 ) , t h e n c wo u ld c o n t a in t h e a r c s x1y1 a n d x0y0. th e r e fo r e , c wo u ld c o n t a in t h e p a t h x1y1x0y0 o r t h e p a t h x0y0x1y1, wh ic h is im p o s s ib le s in c e n ¡ ( x0 ) = n ¡ ( x1 ) = fy0; y1g. n o t ic e t h a t t h e d ig r a p h d ( 8 ) d o e s n o t s a t is fy t h e c o n d it io n s o f w a n g 's t h e o r e m . t heor em 1.4. ( d a r b in ya n [1 1 ]) l et d be a strongly connected balanced bipartite digraph of order 2 a ¸ 8 . suppose that maxfd( x ) ; d( y ) g ¸ 2 a ¡ 1 for every pair of vertices x, y with a common out-neighbour. then d is hamiltonian unless d is isomorphic to the digraph d ( 8 ) (for de¯nition of d ( 8 ) , see e xample 1). 2 8 on a problem of wang concerning the hamiltonicity of bipartite digraphs fo r a ¸ 4 th e o r e m 1 .4 im p r o ve s w a n g 's t h e o r e m . a d ig r a p h d is c a lle d p a n c yc lic if it c o n t a in s c yc le s o f e ve r y le n g t h k, 3 · k · jv ( d ) j. a b a la n c e d b ip a r t it e d ig r a p h o f o r d e r 2 a is e ve n p a n c yc lic if it c o n t a in s c yc le s o f e ve r y le n g t h 2 k, 2 · k · a. th e r e a r e va r io u s s u ± c ie n t c o n d it io n s fo r a d ig r a p h ( u n d ir e c t e d g r a p h ) t o b e h a m ilt o n ia n , t h e y a r e a ls o s u ± c ie n t fo r t h e d ig r a p h ( u n d ir e c t e d g r a p h ) t o b e p a n c yc lic . r e c e n t ly, t h e fo llo win g r e s u lt s we r e p r o ve d . t heor em 1.5. ( d a r b in ya n [1 2 ]) l et d be a strongly connected balanced bipartite digraph of order 2 a ¸ 8 other than a directed cycle of length 2 a. if maxfd ( x ) ; d( y ) g ¸ 2 a ¡ 1 for every dominating pair of vertices fx; yg, then either d contains cycles of all even lengths less than or equal to 2 a or d is isomorphic to the digraph d ( 8 ) . t heor em 1.6. ( me s z ka [1 3 ]) l et d be a balanced bipartite digraph of order 2 a ¸ 4 with partite sets x and y . if d ( x) + d ( y ) ¸ 3 a + 1 for every pair of distinct vertices fx; yg either both in x or both in y , then d contains cycles of all even lengths less than or equal to 2 a. t heor em 1.7. ( d a r b in ya n [1 4 ]) l et d be a strongly connected balanced bipartite digraph of order 2 a ¸ 6 with partite sets x and y . if d( x ) + d( y ) ¸ 3 a for every pair of distinct vertices fx; yg either both in x or both in y , then d contains cycles of all even lengths less than or equal to 2 a. t heor em 1.8. ( a d a m u s [1 5 ]) l et d be a strongly connected balanced bipartite digraph of order 2 a ¸ 6 . if d ( x) + d( y ) ¸ 3 a for every pair of distinct vertices fx; yg with a common in-neighbour or a common out-neighbour, then d contains cycles of all even lengths less than or equal to 2 a or a directed cycle of length 2 a. de¯nition 1. l et d be a balanced bipartite digraph of order 2 a, where a ¸ 2 . f or any integer k ¸ 0 , we will say that d satis¯es the condition bk when d( x ) ¸ 2 a ¡ k; d( y ) ¸ a + k or d( x ) ¸ a + k; d( y ) ¸ 2 a ¡ k for any dominating pair of vertices fx; yg in d. in [1 0 ], w a n g p r o p o s e d t h e fo llo win g p r o b le m . p r oblem ( w a n g [1 0 ]) . l et d be a strongly connected balanced bipartite digraph of order 2 a ¸ 8 satisfying the condition bk with 2 · k · a= 2 . is d hamiltonian? b e fo r e s t a t in g t h e n e xt t h e o r e m s we n e e d t o d e ¯ n e a d ig r a p h o f o r d e r t e n . e xample 2. l e t d ( 1 0 ) b e a b ip a r t it e d ig r a p h wit h p a r t it e s e t s x = fx0; x1; x2; x3; x4g a n d y = fy0; y1; y2; y3; y4g s a t is fyin g t h e fo llo win g c o n d it io n s : th e in d u c e d s u b d ig r a p h hfx1; x2; x3; y0; y4gi is a c o m p le t e b ip a r t it e d ig r a p h wit h p a r t it e s e t s fx1; x2; x3g a n d fy0; y4g; fx1; x2; x3g ! fy1; y2; y3g; x4 $ y4; x0 $ y0, x3 $ y1 a n d xi $ yi+1 fo r a ll i 2 [1 ; 3 ]. d ( 1 0 ) c o n t a in s n o o t h e r a r c s . s. darbinyan and i. karapetyan 2 9 it is e a s y t o c h e c k t h a t t h e d ig r a p h d ( 1 0 ) is s t r o n g ly c o n n e c t e d a n d s a t is ¯ e s t h e c o n d it io n b0, b u t t h e u n d e r lyin g u n d ir e c t e d g r a p h o f d ( 1 0 ) is n o t 2 -c o n n e c t e d , a n d d ( 1 0 ) h a s n o c yc le o f le n g t h 8 . ( it fo llo ws fr o m t h e fa c t s t h a t d( x0 ) = d ( x4 ) = 2 , a n d x0 ( x4 ) is o n 2 -c yc le ) . s in c e x1y1x3y3x2y2x1 is a c yc le o f le n g t h 6 , x0 $ y0 a n d x4 $ y4, it is n o t d i± c u lt t o c h e c k t h a t a n y d ig r a p h o b t a in e d fr o m d ( 1 0 ) b y a d d in g a n e w a r c t h e o n e e n d -ve r t e x o f wh ic h is x0 o r x4 c o n t a in s a c yc le o f le n g t h e ig h t . mo r e o ve r , if t o a( d ) we a d d s o m e n e w a r c s o f t h e t yp e yixj , wh e r e i 2 [1 ; 3 ] a n d j 2 [1 ; 3 ], t h e n we a lwa ys o b t a in a d ig r a p h , wh ic h d o e s n o t s a t is fy t h e c o n d it io n b0. t heor em 1.9. ( [1 6 ], [1 7 ]) . l et d be a balanced bipartite digraph of order 2 a ¸ 1 0 other than a directed cycle of length 2 a. suppose that d satis¯es the condition b0, i.e., maxfd( x ) ; d( y ) g ¸ 2 a ¡ 2 for every dominating pair of vertices fx; yg. then d contains cycles of all lengths 2 ; 4 ; : : : ; 2 a ¡ 2 unless d is isomorphic to the digraph d ( 1 0 ) . cle a r ly, t h e e xis t e n c e o f a c yc le fa c t o r is a n e c e s s a r y c o n d it io n fo r a d ig r a p h t o b e h a m ilt o n ia n . in t h is n o t e we p r o ve t h e fo llo win g t h e o r e m . t heor em 1.10. l et d be a strongly connected balanced bipartite digraph of order 2 a ¸ 8 satisfying the condition bk with 2 · k · a= 2 . then d contains a cycle factor. 2 . te r m in o lo g y a n d n o t a t io n te r m in o lo g y a n d n o t a t io n n o t d e s c r ib e d b e lo w fo llo w [1 ]. if xy 2 a ( d ) , t h e n we s a y t h a t x d o m in a t e s y o r y is a n o u t -n e ig h b o u r o f x a n d x is a n in -n e ig h b o u r o f y. l e t x; y b e d is t in c t ve r t ic e s in a d ig r a p h d. th e p a ir fx; yg is c a lle d d o m in a t in g if t h e r e is a ve r t e x z in d s u c h t h a t xz 2 a( d ) a n d yz 2 a ( d ) . in t h is c a s e we s a y t h a t x is a p a r t n e r o f y a n d y is a p a r t n e r o f x. if x 2 v ( d ) a n d a = fxg we s o m e t im e s will wr it e x in s t e a d o f fxg. a ! b m e a n s t h a t e ve r y ve r t e x o f a d o m in a t e s e ve r y ve r t e x o f b. th e n o t a t io n x $ y d e n o t e s t h a t xy 2 a ( d ) a n d yx 2 a ( d ) . l e t n + ( x ) , n¡ ( x) d e n o t e t h e s e t o f o u t -n e ig h b o u r s , r e s p e c t ive ly t h e s e t o f in -n e ig h b o u r s o f a ve r t e x x in a d ig r a p h d. if a µ v ( d ) , t h e n n + ( x; a ) = a \ n + ( x ) , n¡ ( x; a ) = a\n ¡ ( x ) a n d n + ( a) = [x2an + ( x) , n¡ ( a) = [x2an¡ ( x) . th e o u t -d e g r e e o f x is d+ ( x ) = jn + ( x) j a n d d¡ ( x ) = jn¡ ( x) j is t h e in -d e g r e e o f x. s im ila r ly, d+ ( x; a) = jn + ( x; a) j a n d d¡ ( x; a ) = jn¡ ( x; a ) j. th e d e g r e e o f t h e ve r t e x x in d is d e ¯ n e d a s d( x ) = d+ ( x ) + d¡ ( x ) ( s im ila r ly, d( x; a) = d+ ( x; a ) + d¡ ( x; a ) ) . th e p a t h ( r e s p e c t ive ly, t h e c yc le ) c o n s is t in g o f t h e d is t in c t ve r t ic e s x1; x2; : : : ; xm ( m ¸ 2 ) a n d t h e a r c s xixi+1, i 2 [1 ; m ¡ 1 ] ( r e s p e c t ive ly, xixi+1, i 2 [1 ; m ¡ 1 ], a n d xmx1 ) , is d e n o t e d b y x1x2 ¢ ¢ ¢ xm ( r e s p e c t ive ly, x1x2 ¢ ¢ ¢ xmx1 ) . w e s a y t h a t x1x2 ¢ ¢ ¢ xm is a p a t h fr o m x1 t o xm o r is a n ( x1; xm ) -p a t h . give n a ve r t e x x o f a d ir e c t e d p a t h p o r a d ir e c t e d c yc le c, we d e n o t e b y x+ ( r e s p e c t ive ly, b y x¡ ) t h e s u c c e s s o r ( r e s p e c t ive ly, t h e p r e d e c e s s o r ) o f x ( o n p o r c ) , a n d in c a s e o f a m b ig u it y, we p r e c is e p o r c a s a s u b s c r ip t ( t h a t is x+p . . . ) . a d ig r a p h d is s t r o n g ly c o n n e c t e d ( o r , ju s t , s t r o n g ) if t h e r e e xis t s a n ( x; y ) -p a t h in d fo r e ve r y o r d e r e d p a ir o f d is t in c t ve r t ic e s x; y o f d. two d is t in c t ve r t ic e s x a n d y a r e a d ja c e n t if xy 2 a( d ) o r yx 2 a ( d ) ( o r b o t h ) . l e t h b e a n o n -t r ivia l p r o p e r s u b s e t o f ve r t ic e s o f a d ig r a p h d. a n ( x; y ) -p a t h p is a n h-b yp a s s if jv ( p ) j ¸ 3 , x 6= y a n d v ( p ) \ h = fx; yg. 3 0 on a problem of wang concerning the hamiltonicity of bipartite digraphs l e t d b e a b a la n c e d b ip a r t it e d ig r a p h wit h p a r t it e s e t s x a n d y . a m a t c h in g fr o m x t o y is a n in d e p e n d e n t s e t o f a r c s wit h o r ig in in x a n d t e r m in u s in y . ( a s e t o f a r c s wit h n o c o m m o n e n d -ve r t ic e s is c a lle d in d e p e n d e n t ) . if d is b a la n c e d , o n e s a ys t h a t s u c h a m a t c h in g is p e r fe c t if it c o n s is t s o f p r e c is e ly jxj a r c s . th e u n d e r lyin g u n d ir e c t e d g r a p h o f a d ig r a p h d is d e n o t e d b y ug( d ) , it c o n t a in s a n e d g e xy if xy 2 a ( d ) o r yx 2 a( d ) ( o r b o t h ) . 3 . ma in r e s u lt th e o r e m 1 .1 0 is t h e m a in r e s u lt o f t h is p a p e r . p r oof of theor em 1.10. l e t d b e a d ig r a p h s a t is fyin g t h e c o n d it io n s o f t h e t h e o r e m . or e in [1 8 ] ( s e c t io n 8 .6 ) h a s s h o wn t h a t a b a la n c e d b ip a r t it e d ig r a p h d wit h p a r t it e s e t s x a n d y h a s a c yc le fa c t o r if a n d o n ly if d c o n t a in s a p e r fe c t m a t c h in g fr o m x t o y a n d a p e r fe c t m a t c h in g fr o m y t o x. th e r e fo r e , b y t h e we ll-kn o wn k äo n in g -h a ll t h e o r e m ( s e e , e .g ., [1 9 ]) t o s h o w t h a t d c o n t a in s a p e r fe c t m a t c h in g fr o m x t o y , it s u ± c e s t o s h o w t h a t jn + ( s ) j ¸ jsj fo r e ve r y s e t s µ x. l e t s µ x. if jsj = 1 o r jsj = a, t h e n jn + ( s ) j ¸ jsj s in c e d is s t r o n g ly c o n n e c t e d . a s s u m e t h a t 2 · jsj · a ¡ 1 . w e c la im t h a t jn + ( s ) j ¸ jsj. s u p p o s e t h a t t h is is n o t t h e c a s e , i.e ., jn + ( s ) j · jsj ¡ 1 · a ¡ 2 . fr o m t h is a n d s t r o n g ly c o n n e c t e d n e s s o f d it fo llo ws t h a t t h e r e a r e t wo ve r t ic e s x; y 2 s a n d a ve r t e x z 2 n + ( s ) s u c h t h a t fx; yg ! z, i.e ., fx; yg is a d o m in a t in g p a ir . th e r e fo r e , b y c o n d it io n bk, d ( x) ¸ 2 a ¡ k a n d d ( y ) ¸ a + k o r d( x ) ¸ a + k a n d d( y ) ¸ 2 a ¡ k. w it h o u t lo s s o f g e n e r a lit y, we a s s u m e t h a t d( x ) ¸ 2 a ¡ k a n d d( y ) ¸ a + k. th e n 2 a ¡ k · d ( x) · 2 jn + ( s ) j + a ¡ jn + ( s ) j = a + jn + ( s ) j: th e r e fo r e , jn + ( s ) j ¸ a ¡ k a n d jsj ¸ a ¡ k + 1 . p r oposition 1. l e t fu; vg b e a d o m in a t in g p a ir o f ve r t ic e s o f d. th e n fr o m c o n d it io n bk a n d 2 · k · a=2 it fo llo ws t h a t d( u ) ¸ a + k a n d d( vu ) ¸ a + k, i.e ., if a ve r t e x z h a s a p a r t n e r in d , t h e n d ( z ) ¸ a + k. w e c la im t h a t e a c h ve r t e x in y nn + ( s ) h a s n o p a r t n e r in d. in d e e d , le t u b e a n a r b it r a r y ve r t e x in y n n + ( s ) . s in c e jsj ¸ a ¡ k + 1 , we h a ve d( u ) · jsj + 2 ( a ¡ jsj ) = 2 a ¡ jsj · a + k ¡ 1 ; wh ic h c o n t r a d ic t s p r o p o s it io n 1 . th is m e a n s t h a t u h a s n o p a r t n e r in d. w it h o u t lo s s o f g e n e r a lit y, a s s u m e t h a t s = fx1; x2; : : : ; xsg a n d n + ( s ) = fy1; y2; : : : ; ytg: r e c a ll t h a t e ve r y ve r t e x yi wit h t+1 · i · a h a s n o p a r t n e r in d. n o t e t h a t s ¸ t+1 , a¡s · a¡t¡ 1 a n d t h e r e is n o a r c fr o m a ve r t e x o f fx1; x2; : : : ; xsg t o a ve r t e x o f fyt+1; yt+2; : : : ; yag. fr o m t h is a n d s t r o n g ly c o n n e c t e d n e s s o f d it fo llo ws t h a t t h e r e is a ve r t e x xi1 s u c h t h a t yt+1xi1 2 a( d ) . s in c e yt+1 h a s n o p a r t n e r , it fo llo ws t h a t d¡ ( xi1; y nfyt+1g ) = 0 . th e r e fo r e , d( xi1 ) · a + 1 · a + k ¡ 1 s in c e k ¸ 2 . b y p r o p o s it io n 1 , t h is m e a n s t h a t t h e ve r t e x s. darbinyan and i. karapetyan 3 1 xi1 a ls o h a s n o p a r t n e r . s in c e d is s t r o n g ly c o n n e c t e d , t h e r e is a ve r t e x yi2 2 y s u c h t h a t xi1yi2 2 a ( d ) . th e n d¡ ( yi2; x n fxi1g ) = 0 , b e c a u s e o f t h e fa c t t h a t xi1 h a s n o p a r t n e r . th e r e fo r e , d( yi2 ) · a + 1 a n d h e n c e , yi2 a ls o h a s n o p a r t n e r . co n t in u in g t h is p r o c e s s , a s lo n g a s p o s s ib le , a s a r e s u lt we o b t a in a p a t h p = yt+1xi1yi2 xi2 : : : xil yil o r a c yc le c = yt+1xi1yi2xi2 : : : xil yt+1. it is n o t d i± c u lt t o s e e t h a t a ll t h e ve r t ic e s o f t h is p a t h ( c yc le ) h a ve n o p a r t n e r s . if t h e fo r m e r c a s e h o ld s , t h e n x1 is in p , wh ic h is a c o n t r a d ic t io n s in c e x1 h a s a p a r t n e r ( n a m e ly x2 ) . if t h e s e c o n d c a s e h o ld s , t h e n , s in c e e ve r y ve r t e x o f c h a s n o p a r t n e r in d, it fo llo ws t h a t t h e r e is n o a r c fr o m a ve r t e x o f v ( d ) n v ( c ) t o a ve r t e x o f v ( c ) , wh ic h c o n t r a d ic t s t h a t d is s t r o n g ly c o n n e c t e d . th is c o m p le t e s t h e p r o o f o f t h e e xis t e n c e o f a p e r fe c t m a t c h in g fr o m x t o y . th e p r o o f fo r a p e r fe c t m a t c h in g in t h e o p p o s it e d ir e c t io n is a n a lo g o u s . th is c o m p le t e s t h e p r o o f o f t h e t h e o r e m . 4 . r e m a r ks n o w u s in g th e o r e m 1 .1 0 , we p r o ve t h e fo llo win g r e s u lt s ( l e m m a s 3 .1 -3 .3 ) . lemma 3.1. l et d be a strongly connected balanced bipartite digraph of order 2 a ¸ 8 with partite sets x and y satisfying the condition bk, 2 · k · a= 2 . if d is not hamiltonian, then every vertex u 2 v ( d ) has a partner in d. p r oof of lemma 3.1: l e t d b e a d ig r a p h s a t is fyin g t h e c o n d it io n s o f t h e le m m a . fo r a p r o o f b y c o n t r a d ic t io n , s u p p o s e t h a t t h e r e is a ve r t e x x in d, wh ic h h a s n o p a r t n e r . b y th e o r e m 1 .1 0 , d h a s a c yc le fa c t o r , s a y c1, c2, ... , cl. th e n l ¸ 2 s in c e d is n o t h a m ilt o n ia n . w it h o u t lo s s o f g e n e r a lit y, we a s s u m e t h a t x 2 v ( c1 ) . it fo llo ws t h a t d¡ ( x+c1 ) = 1 . th e r e fo r e , d( x+c1 ) · a + 1 . b y p r o p o s it io n 1 , t h is m e a n s t h a t t h e ve r t e x x + c1 a ls o h a s n o p a r t n e r . s im ila r ly, we o b t a in t h a t d( x++c1 ) · a + 1 ( wh e r e x ++ c1 d e n o t e s t h e s u c c e s s o r o f x+c1 o n c1 ) a n d h e n c e , x ++ c1 a ls o h a s n o p a r t n e r in d. co n t in u in g t h is p r o c e s s , we c o n c lu d e t h a t e ve r y ve r t e x o f c1 h a s n o p a r t n e r in d. th is im p lie s t h a t t h e r e is n o a r c fr o m a ve r t e x o f a ( v ( d ) n v ( c1 ) t o a ve r t e x o f v ( c1 ) ) , wh ic h c o n t r a d ic t s t h a t d is s t r o n g ly c o n n e c t e d . th e le m m a is p r o ve d . lemma 3.2. l et d be a strongly connected balanced bipartite digraph of order 2 a ¸ 8 with partite sets x and y satisfying the condition bk, 2 · k · a=2 . if d is not a cycle, then d contains a non-hamiltonian cycle of length at least four. p r oof of lemma 3.2: l e t d b e a d ig r a p h s a t is fyin g t h e c o n d it io n s o f t h e le m m a . fo r a p r o o f b y c o n t r a d ic t io n , s u p p o s e t h a t d c o n t a in s a n o n -h a m ilt o n ia n c yc le o f le n g t h a t le a s t fo u r . if d is h a m ilt o n ia n , t h e n it is n o t d i± c u lt t o s h o w t h a t d c o n t a in s a n o n -h a m ilt o n ia n c yc le o f le n g t h a t le a s t 4 . s o we s u p p o s e , fr o m n o w o n , t h a t d is n o t h a m ilt o n ia n a n d c o n t a in s n o c yc le o f le n g t h a t le a s t 4 . b y th e o r e m 1 .1 0 , d c o n t a in s a c yc le fa c t o r . l e t c1; c2; : : : ; ct b e a m in im a l c yc le fa c t o r o f d ( i.e ., t is a s s m a ll a s p o s s ib le ) . th e n t h e le n g t h o f e ve r y ci is e qu a l t o t wo a n d t = a. l e t ci = xiyixi, wh e r e xi 2 x a n d yi 2 y . b y l e m m a 3 .1 , e ve r y ve r t e x o f d h a s a p a r t n e r . th is m e a n s t h a t fo r e ve r y ve r t e x x 2 v ( d ) , d( x ) ¸ a + k a n d d¡ ( x) ¸ k ¸ 2 , d+ ( x ) ¸ k ¸ 2 . w it h o u t lo s s o f g e n e r a lit y, we a s s u m e t h a t fx1; xjg wit h j 6= 1 is a d o m in a t in g p a ir a n d d( x1 ) ¸ 2 a ¡ k. 3 2 on a problem of wang concerning the hamiltonicity of bipartite digraphs l e t z b e t h e s u b s e t o f y wit h t h e m a xim u m c a r d in a lit y, s u c h t h a t e ve r y ve r t e x o f z t o g e t h e r wit h x1 fo r m s a c yc le o f le n g t h t wo . w it h o u t lo s s o f g e n e r a lit y, we a s s u m e t h a t z = fy1; y2; : : : ; ylg. th e n 2 a ¡ k · d ( x1 ) · 2 l + a ¡ l = a + l. h e n c e , l ¸ a ¡ k. s in c e d c o n t a in s n o c yc le o f le n g t h fo u r , it fo llo ws t h a t t h e ve r t ic e s y1 a n d xi, 2 · i · l, a r e n o t a d ja c e n t . th e r e fo r e , a + k · d( y1 ) · 2 a ¡ 2 l + 2 · 2 k + 2 ; i.e ., k ¸ a ¡ 2 . s in c e a= 2 ¸ k ¸ a ¡ 2 , we h a ve a ¸ 2 k ¸ 2 a ¡ 4 , a · 4 . if a = 4 , t h e n k = a=2 = 2 a n d l = a ¡ k = 2 . it is e a s y t o s e e t h a t d ( x1 ) = d ( y1 ) = 6 , t h e ve r t ic e s y1 a n d xi, 3 · i · 4 , fo r m a c yc le o f le n g t h t wo a n d x1y3 2 a ( d ) o r y3x1 2 a ( d ) . n o w it e a s y t o s e e t h a t d c o n t a in s a c yc le o f le n g t h fo u r . l e m m a 3 .2 is p r o ve d . fo r t h e n e xt le m m a we n e e d t h e fo llo win g le m m a d u e t o b o n d y. b ypass lemma ( l e m m a 3 .1 7 , b o n d y [2 0 ]) . l et d be a strong non-separable (i.e., ug ( d ) is 2-connected) digraph, and let h be a non-trivial proper subdigraph of d. then d contains an h-bypass. remar k: on e c a n p r o ve b yp a s s l e m m a u s in g t h e p r o o f o f th e o r e m 5 .4 .2 [1 ]. n o w we will p r o ve t h e fo llo win g le m m a . lemma 3.3. l et d be a strongly connected balanced bipartite digraph of order 2 a ¸ 8 with partite sets x and y satisfying the condition bk, where 2 · k · a=2 . then the following statements hold: (i) the underlying undirected graph ug ( d ) is 2-connected; (ii) if c is a cycle of length m, 2 · m · 2 a ¡ 2 , then d contains a c-bypass. p r oof of lemma 3.3. ( i) s u p p o s e , o n t h e c o n t r a r y, t h a t d is a s t r o n g ly c o n n e c t e d b a la n c e d b ip a r t it e d ig r a p h o f o r d e r 2 a ¸ 8 wit h p a r t it e s e t s x a n d y s a t is ¯ e s t h e c o n d it io n bk b u t ug ( d ) is n o t 2 -c o n n e c t e d . th e n v ( d ) = e [ f [ fug, wh e r e e a n d f a r e n o n -e m p t y s u b s e t s , e \ f = ;, u =2 e [ f a n d t h e r e is n o a r c b e t we e n e a n d f . s in c e d is s t r o n g , it fo llo ws t h a t t h e r e a r e ve r t ic e s x 2 e a n d y 2 f s u c h t h a t fx; yg ! u, i.e ., fx; yg is a d o m in a t in g p a ir . w it h o u t lo s s o f g e n e r a lit y, we a s s u m e t h a t x; y 2 x. th e n u 2 y . b y c o n d it io n bk, it is e a s y t o s e e t h a t 3 a · d( x ) + d( y ) · 4 + 2 je \ y j + 2 jf \ y j · 2 a + 2 ; wh ic h is a c o n t r a d ic t io n . th is p r o ve s t h a t ug( d ) is 2 -c o n n e c t e d . ( ii) th e s e c o n d c la im o f t h e le m m a is a n im m e d ia t e c o n s e qu e n c e o f t h e ¯ r s t c la im a n d b yp a s s l e m m a . l e m m a 3 .3 is p r o ve d . refer ences [1 ] j. b a n g -je n s e n a n d g. gu t in , d igraphs: theory, algorithms and applications, s p r in g e r , 2 0 0 1 . [2 ] g. gu t in , \ cr it e r io n fo r c o m p le t e b ip a r t it e d ig r a p h s t o b e h a m ilt o n ia n " , vestsi akad, navuk b ssr ser. f iz.-m at. navuk vo l. 1 , p p . 1 0 9 -1 1 0 , 1 9 8 4 . s. darbinyan and i. karapetyan 3 3 [3 ] g. gu t in , \ cyc le s a n d p a t h s in s e m ic o m p le t e m u lt ip a r t it e d ig r a p h s , t h e o r e m s a n d a lg o r it h m s : a s u r ve y" . j . graph theory vo l. 1 9 , n o . 4 , p p . 4 8 1 -5 0 5 , 1 9 9 5 . [4 ] r . h äa g g kvis t a n d y . ma n o u s s a kis , \ cyc le s a n d p a t h s in b ip a r t it e t o u r n a m e n t s wit h s p a n n in g c o n ¯ g u r a t io n s " . combinatorica, vo l. 9 , n o . 1 , p p . 3 3 -3 8 , 1 9 8 9 . [5 ] y . ma n o u s s a kis a n d i. millis , \ a s u ± c ie n t c o n d it io n fo r m a xim u m c yc le s in b ip a r t it e d ig r a p h s " , d iscrete m ath., vo l. 2 0 7 , p p . 1 6 1 -1 7 1 , 1 9 9 9 . [6 ] j. a d a m u s a n d l . a d a m u s , \ a d e g r e e c o n d it io n fo r c yc le s o f m a xim u m le n g t h in b ip a r t it e d ig r a p h s " , d iscrete m ath., vo l. 3 1 2 , p p . 1 1 1 7 -1 1 2 2 , 2 0 1 2 . [7 ] d . a m a r a n d y . ma n o u s s a kis , \ cyc le s a n d p a t h s o f m a n y le n g t h s in b ip a r t it e d ig r a p h s " , j . combin. theory ser. b 50, p p . 2 5 4 -2 6 4 , 1 9 9 0 . [8 ] j. a d a m u s , l . a d a m u s a n d a . y e o , \ on t h e me yn ie l c o n d it io n fo r h a m ilt o n ic it y in b ip a r t it e d ig r a p h s " , d iscrete m ath. and theoretical computer science,vo l. 1 6 , n o . 1 , p p . 2 9 3 -3 0 2 , 2 0 1 4 . [9 ] j. a d a m u s , \ a d e g r e e s u m c o n d it io n fo r h a m ilt o n ic it y in b a la n c e d b ip a r t it e d ig r a p h s " , graphs and combinatorics, vo l. 3 3 , n o . 1 , p p . 4 3 -5 1 , 2 0 1 7 . [1 0 ] r . w a n g , \ a s u ± c ie n t c o n d it io n fo r a b a la n c e d b ip a r t it e d ig r a p h t o b e h a m ilt o n ia n " . d iscrete m athematics and theoretical computer science, vo l. 1 9 , n o . 3 , 2 0 1 7 . [1 1 ] s . k h . d a r b in ya n , \ s u ± c ie n t c o n d it io n s fo r h a m ilt o n ia n c yc le s in b ip a r t it e d ig r a p h s " , a r x iv:1 6 0 4 .0 8 7 3 3 v1 [m a t m .co] 2 9 , 1 5 p a g e s , a p r 2 0 1 6 . [1 2 ] s . k h . d a r b in ya n , \ s u ± c ie n t c o n d it io n s fo r a b a la n c e d b ip a r t it e d ig r a p h t o b e e ve n p a n c yc lic " , d iscrete applied m athematics, vo l. 2 3 8 , p p . 7 0 -7 6 , 2 0 1 8 . [1 3 ] m. me s z ka , \ n e w s u ± c ie n t c o n d it io n s fo r b ip a n c yc lic it y o f b a la n c e d b ip a r t it e d ig r a p h s " , d iscrete m ath., s u b m it t e d fo r p u b lic a t io n . [1 4 ] s . k h . d a r b in ya n , \ a t h e o r e m o n e ve n p a n c yc lic b ip a r t it e d ig r a p h s " , a r x iv:1 8 0 1 .0 5 1 7 7 v1 [m a t m .co] 1 6 , 1 2 p a g e s , ja n 2 0 1 8 . [1 5 ] j. a d a m u s , \ a me yn ie l-t yp e c o n d it io n fo r b ip a n c yc lic it y in b a la n c e d b ip a r t it e d ig r a p h s " , a r x iv:1 7 0 8 .0 4 6 7 4 v2 [m a t m .co], 7 p a g e s , 2 2 a u g 2 0 1 7 . [1 6 ] s . k h . d a r b in ya n , \ on p r e -h a m ilt o n ia n c yc le s in b a la n c e d b ip a r t it e d ig r a p h s " , m athematical problems of computer science, vo l. 4 6 , p p . 7 -1 7 , 2 0 1 4 . [1 7 ] s . k h . d a r b in ya n a n d i. a . k a r a p e t ya n , \ a s u ± c ie n t c o n d it io n fo r p r e -h a m ilt o n ia n cyc le s in b ip a r t it e d ig r a p h s " , csit 2017 r evised selected p apers,ie e e conference proceeding,, p p . 1 0 1 -1 0 9 , d oi:1 0 .1 1 0 9 / cs itte c h n o l.2 0 1 7 .8 3 1 2 1 5 0 . [1 8 ] o. or e , \ th e o r y o f g r a p h s " , a m e r ic a n ma t h e m a t ic a l s o c ie t y, p r o vid e n c e , r .i., a m e r ic a n ma t h e m a t ic a l s o c ie t y co llo qu iu m p u b lic a t io n s , vo l. 3 8 , 1 9 6 2 . [1 9 ] c. b e r g e , graphs and hypergraphs, n o r t h -h o lla n d , a m s t e r d a m , 1 9 7 3 . [2 0 ] j. a . b o n d y, b asic graph theory: p aths and circuits, in h a n d b o o k o f c o m b in a t o r ic s 1-2, e ls e vie r , a m s t e r d a m , 1 9 9 5 . submitted 20.09.2017, accepted 16.01.2018. 3 4 on a problem of wang concerning the hamiltonicity of bipartite digraphs îáùýáñáßí³í »ñïù³ëýû³ ·ñ³ýç ñ³ùçéïáýû³ýáõãû³ý í»ñ³µ»ñû³é ì³ý·ç ëý¹ñç ù³ëçý ê. ¸³ñµçýû³ý ¨ æ. î³ñ³å»ïû³ý ²ù÷á÷áõù ì³ý·á (discrete mathematics and theoretical computer science, vol. 19(3) 2017) ³é³ç³ñï»é ¿ ñ»ï¨û³é ëý¹çññ: êý¹çñ: ¸çóáõù d-ý áõå»õ ï³å³ïóí³í 2 a-·³·³ã³ýç 2 a ¸ 8 ïáõýáñáßí³í »ñïù³ëýû³ ñ³í³ë³ñ³ïßéí³í ·ñ³ý ¿, áñáõù ·³·³ãý»ñç ó³ýï³ó³í fx; yg ñ³õãáõ ½áõû·ç ñ³ù³ñ ï»õç áõý»ý ñ»ï¨û³é ³ýñ³í³ë³ñáõãûáõýý»ñá. d( x ) ¸ 2 a¡k, ¨ d( y ) ¸ a+k ï³ù d( x ) ¸ a + k ¨ d( y ) ¸ 2 a ¡ k, áñï»õ 2 · k · a= 2 : ²ñ¹ûáù d-ý ñ³ùçéïáýû³ý ¿: ü»ñï³ ³ßë³ï³ýùáõù ³å³óáõóí³í ¿, áñ »ã» d ·ñ³ýá µ³í³ñ³ñáõù ¿ ì³ý·ç ëý¹ñç å³ûù³ýý»ñçý, ³å³ (i) d ·ñ³ýá å³ñáõý³ïáõù ¿ óçïé-ý³ïïáñ ¨ ³éýí³½ý ãáñë »ñï³ñáõãû³ùµ áã-ñ³ùçéïáýû³ý óçïé, (ii) d ·ñ³ýç ó³ýï³ó³í x ·³·³ãç ñ³ù³ñ ·áûáõãûáõý áõýç ³ûýåçëç ùç y ·³·³ã, áñ fx; yg á ñ³õãáõ ½áõû· ¿: î çàäà÷å âàíãà î ãàìèëüòîíîâîñòè äâóäîëüíûõ îðãðàôîâ ñ. äàðáèíÿí è è. êàðàïåòÿí àííîòàöèÿ âàíã (discrete mathematics and theoretical computer science, vol. 19(3) 2017) ïðåäëîæèëà ñëåäóþùóþ çàäà÷ó. çàäà÷à: ïóñòü d 2 a -âåðøèííûé 2 a ¸ 8 ñèëüíîñâÿçíûé ñáàëàíñèðîâàííûé äâóäîëüíûé îðãðàô, â êîòîðîì äëÿ ëþáîé ïàðû äîìèíèðóþùèõ âåðøèí fx; yg, d( x ) ¸ 2 a ¡ k, d( y ) ¸ a + k èëè d( y ) ¸ a + k, d ( y ) ¸ 2 a ¡ k, ãäå 2 · k · a=2 . ßâëÿåòñÿ ëè d ãàìèëüòîíîâûì? â íàñòîÿùåé ðàáîòå äîêàçàíî, ÷òî åñëè îðãðàô d óäîâëåòâîðÿåò óñëîâèÿì çàäà÷è âàíãà, òî (i) d ñîäåðæèò öèêë-ôàêòîð è íå-ãàìèëüòîíîâûé öèêë äëèíû ïî êðàéíåé ìåðå 4, (ii) äëÿ êàæäîé âåðøèíû x ñóùåñòâóåò òàêàÿ âåðøèíà y, ÷òî fx; yg ÿâëÿåòñÿ äîìèíèðóþùèì ïàðîì. article_with_style.dvi mathematical problems of computer science 31, 60{72, 2008. on m ultiple h ypotheses lao t esting by i nfor med statistician for ar bitr ar ily var ying m ar kov sour ce and on such sour ce coding reliability function n a ir a m. gr ig o r ya n state engineering university of armenia nar.gri@gmail.com abstract in this paper the problem of multiple hypotheses testing for arbitrarily varying markov source (avms) with state sequence known to the statistician is solved from the point of view of logarithmically asymptotically optimal (lao) testing. the matrix of asymptotic interdependencies of all possible pairs of the error probability exponents (reliabilities) in optimal testing for this model is studied. the lao test, assuming that exponents of some number of the error probabilities are given, ensure the best asymptotic exponents for the rest of them. we ¯nd lao test and the corresponding matrix of all error probability exponents. as an application to information theory, the e-optimal rate r(e) (the minimum rate r of the source sequences compression when the decoding error probability is less than expf¡n eg; e > 0) and the reliability function e(r) of avms coding are determined. refer ences [1 ] w . h o e ®d in g , \ a s ym p t o t ic a lly o p t im a l t e s t s fo r m u lt in o m ia l d is t r ib u t io n s ," a n n . ma t h . s t a t is t ., vo l. 3 6 , p p . 3 6 9 ¡ 4 0 1 , 1 9 6 5 . [2 ] i. cs is z ¶a r a n d g. l o n g o , \ on t h e e r r o r e xp o n e n t fo r s o u r c e c o d in g a n d fo r t e s t in g s im p le s t a t is t ic a l h yp o t h e s e s ," s t u d ia s c ie n t ia r u m ma t h e m . h u n g ., vo l. 6 , p p . 1 8 1 ¡1 9 1 , 1 9 7 1 . [3 ] l . b ir g ¶ e , \ v it e s s m a xim a ls d e d ¶ c r o is m e n c e d e s e r r e u r s e t t e s t s o p t im a u x a s s o c ie s " . z. w a h r s c h . ve r w. ge b ie t e , vo l. 5 5 , p p . 2 6 1 ¡ 1 7 3 , 1 9 8 1 . [4 ] e . a . h a r o u t u n ia n , \ l o g a r it h m ic a lly a s ym p t o t ic a lly o p t im a l t e s t in g o f m u lt ip le s t a t is t ic a l h yp o t h e s e s " , p roblems of control and information theory, vo l. 1 9 , n o . 5 -6 , p p . 4 1 3 ¡4 2 1 , 1 9 9 0 . [5 ] s . n a t a r a ja n , \ l a r g e d e via t io n s , h yp o t h e s e s t e s t in g , a n d s o u r c e c o d in g fo r ¯ n it e ma r ko v c h a in s " , ie e e trans. inform. theory, vo l 3 1 , n o . 3 , p p . 3 6 0 ¡3 6 5 , 1 9 8 5 . [6 ] f.-w . fu a n d s .-y . s h e n , \ h yp o t h e s is t e s t in g fo r a r b it r a r ily va r yin g s o u r c e wit h e xp o n e n t ia l-t yp e c o n s t r a in t " , ie e e trans. inform. theory, vo l. 4 4 , n o . 2 , p p . 8 9 2 ¡ 8 9 5 , 1 9 9 8 . [7 ] e . a . h a r o u t u n ia n , \ on a s ym p t o t ic a lly o p t im a l t e s t in g o f h yp o t h e s e s c o n c e r n in g ma r ko v c h a in " , ( in r u s s ia n ) , izvestiya akademii nauk armenii, m athematika, vo l. 2 3 , n o . 1 , p p . 7 6 ¡ 8 0 , 1 9 8 8 . 6 0 n. grigoryan 6 1 [8 ] e . a . h a r o u t u n ia n , \ on a s ym p t o t ic a lly o p t im a l c r it e r ia fo r ma r ko v c h a in s " , ( in r u s s ia n ) , the ¯rst w orld congress of b ernoulli society, s e c t io n 2 , vo l. 2 , n o . 3 , p p . 1 5 3 ¡ 1 5 6 , 1 9 8 9 . [9 ] e . a . h a r o u t u n ia n , \ a s ym p t o t ic a lly o p t im a l t e s t in g o f m a n y s t a t is t ic a l h yp o t h e s e s c o n c e r n in g ma r ko v c h a in " , ( in r u s s ia n ) , 5-th intern. vilnius conferance on p robability theory and m athem. statistics, vo l. 1 ( a -l ) , p p . 2 0 2 ¡2 0 3 , 1 9 8 9 . [1 0 ] r . f. a h ls we d e , e . v . a lo ya n a n d e . a . h a r o u t u n ia n , \ on lo g a r it h m ic a lly a s ym p t o t ic a lly o p t im a l h yp o t h e s is t e s t in g fo r a r b it r a r y va r yin g s o u r c e wit h s id e in fo r m a t io n " , l ecture notes in computer science, v o lu m e 4 1 2 3 , " ge n e r a l th e o r y o f in fo r m a t io n tr a n s fe r a n d co m b in a t o r ic s " , s p r in g e r , p p . 4 5 7 ¡ 4 6 1 , 2 0 0 4 . [1 1 ] e . a . h a r o u t u n ia n a n d p . m. h a ko b ya n , \ on mu lt ip le h yp o t h e s is t e s t in g b y in fo r m e d s t a t is t ic ia n fo r a r b it r a r ily va r yin g o b je c t a n d a p p lic a t io n t o s o u r c e c o d in g " , m athematical p roblems of computer science, vo l. 2 3 , p p . 3 6 ¡ 4 6 , 2 0 0 4 . [1 2 ] e . a . h a r o u t u n ia n , m. e . h a r o u t u n ia n a n d a . n . h a r u t yu n ya n , \ r e a la b ilit y c r it e r ia in in fo r m a t io n t h e o r y a n d in s t a t is t ic a l h yp o t h e s e s t e s t in g " , f oundation and trends in comunications and information theory, vo l. 4 , n o . 2 ¡3 , 2 0 0 8 . [1 3 ] e . a . h a r o u t u n ia n a n d n . m. gr ig o r ya n , \ on r e lia b ilt y a p p r o a c h fo r t e s t in g o f m a n y d is t r ib u t io n s fo r p a ir o f ma r ko v c h a in s " , m athematical p roblems of computer science, vo l. 2 9 , p p . 8 9 ¡ 9 6 , 2 0 0 7 . [1 4 ] i. cs is z ¶a r a n d j. k äo r n e r , \information theory, coding theorems for discrete memoryless systems", a c a d e m ic p r e s s , n e w y o r k, 1 9 8 1 . [1 5 ] m. gu t m a n , \ a s ym p t o t ic a lly o p t im a l c la s s i¯ c a t io n fo r m u lt ip le t e s t wit h e m p ir ic a lly o b s e r ve d s t a t is t ic s ," ie e e trans. inform. theory, vo l. 3 5 , n o . 2 , p p . 4 0 1 ¡ 4 0 8 , 1 9 8 9 . [1 6 ] p . ja c ke t a n d w . s z p a n ko vks i, \ ma r ko v t yp e s a n d m in im a x r e d u n d a n c y fo r ma r ko v s o u r c e s ," ie e e trans. inform. theory, vo l. 5 0 , n o . 7 , p p . 1 3 9 3 ¡ 1 4 0 2 , 2 0 0 4 . [1 7 ] k . ma r t o n , \ e r r o r e xp o n e n t fo r s o u r c e c o d in g wit h a ¯ e d e lit y c r it e r io n ," ie e e trans. inform. theory, vo l. 2 0 , n o . 2 , p p . 1 9 7 ¡ 1 9 9 , 1 9 7 4 . î»õ»ï³óí³í íç׳ﳷñç ïáõùçó ï³ù³û³ï³ýáñ»ý ÷á÷áëíáõ ù³ñïáíû³ý ³õµûáõñç í»ñ³µ»ñû³é µ³½ù³ïç í³ñï³íý»ñç ëïáõ·áõùá ¨ ³õµûáõñç ïá¹³íáñù³ý ñáõë³éçáõãû³ý ýáõýïóç³ûç ·ý³ñ³ïáõùá ü. ¶ñç·áñû³ý ²ù÷á÷áõù èáõíí³í ¿ ï³ù³û³ï³ýáñ»ý ÷á÷áëíáõ ïáõùý³ïç çýýáñù³óç³ûáí ù³ñïáíû³ý ³õµûáõñç ùá¹»éç ñ³ù³ñ µ³½ù³ïç í³ñï³íý»ñç ëïáõ·ù³ý ëý¹çñá: m ( ¸ 2 ) ñ³í³ý³ï³ý³ûçý µ³ßëáõùý»ñá h³ûïýç »ý, ¨ ûµû»ïïá ï³ù³û³ï³ýáñ»ý áý¹áõýáõù ¿ ¹ñ³ýóçó áñ¨¿ ù»ïá: ²ûë ùá¹»éç ñ³ù³ñ áõëáõùý³ëçñí»é ¿ µáéáñ ñý³ñ³íáñ ½áõû·»ñç ëë³éý»ñç ñ³í³ý³ï³ýáõãûáõýý»ñç óáõóçãý»ñç (ñáõë³éçáõãûáõýý»ñç) ÷áëï³ëí³íáõãûáõýá: ºñáïáõ í³ñï³íý»ñç ¹»åùá áý¹ñ³ï ³é³ýó ñçßáõáõãû³ý ï³åáõõáõ ñ³ù³ñ »ñµ íç׳ïý»ñç ñ³çáñ¹³ï³ýáõãûáõýá ³ýñ³ûï ¿ áý¹áõýáõù áñáßáõçý, ¹çï³ñïí»é ¿ üáõç ¨ þ»ýç ïáõùçó: æëï ýáõûý ùá¹»éç ñ³ù³ñ, »ñµ íç׳ïý»ñç ñ³çáñ¹³ï³ýáõãûáõýá ñ³ûïýç ¿ íç׳ﳷñçý ¹çï³ñïí»é ¿ ²éëí»¹»ç, ð³ñáõãûáõýû³ýç ¨ ²éáû³ýç ïáõùçó: æýãå»ë üáõý ¨ þ»ýá, ù»ýù ýáõûýå»ë ëï³ó»é »ýù ïáõùý³ïç çýýáñù³óç³ûáí ï³ù³û³ï³ýáñ»ý ÷á÷áëíáõ ³õµû»áõñç ñ³ù³ñ ³ñ³·áõãûáõý-ñáõë³éçáõãûáõý ¨ ñáõë³éçáõãûáõý-³ñ³·áõãûáõý ýáõýïóç³ý»ñá: mathematical problems of computer science 49, 49–57, 2018. cloud service for analysis and interactive visualization of weather data in armenia hayk a. grigoryan and rita m. abrahamyan institute for informatics and automation problems of nas ra armenian state hydrometeorological and monitoring service e-mail: hayk.grigoryan.a@gmail.com abstract the lesser caucasus mountains are crossing through the territory of armenia, creating vast differences in altitude, terrain, temperature and precipitation in provinces and towns. even armenia’s lowlands are 500 to 1500m above sea level. armenias highlands extend up to aragats mountain at 4090m where, 75% of the territory is above 1000m, 50% is above 2000m, and 3.4% is above 3000m. this paper presents a cloud service with interactive visualization and analytical capabilities for weather data in armenia by integrating the two existing infrastructures for observational data and numerical weather prediction. the weather data used in the platform consist of near-surface atmospheric elements including air temperature, relative humidity, pressure, wind and precipitation. the visualization and analitycs have been implemented for 2m air temperature. cloud service provides the armenian state hydrometeorological and monitoring service with analytical capabilities to make a comparative analysis between the observation data and the results of a numerical weather prediction model for per station and region for a given period. keywords: cloud service, weather data, observational data, data analysis, numerical weather prediction, wrf, spatial olap. 1. introduction armenia occupies the north-eastern part of armenian plateau and central part of lesser caucasus range (latitude 38.51’ to 41.18’ north, longitude 43.29’ to 46.37’ east), with the area of about 30 000 sq.km. the geographical location of armenia and complex mountainous relief have led to the diversity of natural conditions across the country. armenia is on the northern edge of the subtropical zone, in latitudes characterized by an arid and continental climate. due to the mountainous relief, different climatic zones exist, and the weather may have high spatial gradients. high fluctuations in annual and daily temperatures are typical for the armenian climate. the presence of six climatic zones from dry subtropical to rigorous high mountainous and from everlasting snowcaps to warm humid subtropical forests and humid semi-desert steppes make additional challenges on weather forecasting and climate prediction for the armenian state hydrometeorological and monitoring service (ahms). 49 50 cloud service for analysis and interactive visualization of weather data in armenia the meteorological data, received from 47 meteorological stations, among which only four stations provide historical data and monthly updates to the global climate observing system surface network and three meteorological stations provide synoptic data are being used in the gridded analysis data set, which serves as an input for the global atmospheric models to produce weather forecasts at the global scale. the observation data received from the meteorological stations and data received from a global model serve as inputs and outputs to the high-resolution numerical weather prediction models to produce outputs of temperature, precipitation, and other meteorological elements from the ground to the top of the atmosphere [1, 2]. the high-performance computational (hpc) resources of the armenian e-infrastructure are used to better resolve mesoscale weather events, and hence to give reasonably accurate forecasts in a short range [3]. the limitations in such forecasts lie in the availability of initial conditions at model resolutions. the article aims to present the weather data interactive web-based visualization and analytical platform1. the platform has been developed for the weather data in armenia by integrating the three existing platforms for observational data and numerical weather prediction. the platform provides a way to compare the output of forecasting model with the observation data gathered from different stations for a chosen frame of time. the suggested platform is essential for a wide range of applications, such as urban area management, sustainable development and nature protection, regional and local planning, agriculture, forestry and fisheries, health, civil protection, infrastructure, transport and mobility or tourism. the remainder of this paper is divided into the following sections: section 2 introduces the infrastructure, section 3 represents the discussions and analyses and finally section 4 is the conclusion. 2. cloud service framework the suggested cloud service consists of 5 primary layers (see fig. 1). the base layer provides hpc and data resources, which is particularly important for digital models [4]. the resources of the armenian e-infrastructure are utilized, which is a complex national it infrastructure consisting of both communication and distributed computing infrastructures. fig 1. the framework of the platform. 1cloud service for analysis of numerical weather prediction model accuracy using visualization technique: http://meteo.grid.am h. grigoryan and r. abrahamyan 51 the datasets layer combines two types of data platforms for further analysis: − model output: outputs of weather prediction models; − observation data: meteorological stations observations, as a base to investigate the deviation values with other model outputs. the data management layer provides intelligent tools to transfer raw data to data analytics layer. the integrated rule-oriented data system (irods) provides a middleware between the physical data storage systems and the user interface [5]. as soon as data reaches data analytics layer, it is processed, and only several indexes are left from the enormous amount of starting crude information. finally, the top layer and final destination of the already processed data is the visualization layer, where the result lists are changed to more easy to understand graphs or tables. besides, the advantages of google maps are used to map these indexes with their real location on the map. 2.1 data gathering observational datasets provide the observed meteorological data from different locations acquired from synop (surface synoptic observations) messages issued by official ground weather stations. synop reports are regularly sent every three hours, which consist of groups of numbers describing general weather information of the weather station, such as temperature, barometric pressure and visibility at a weather station, and so on. the numerical weather prediction models are initialized using ncep (national centers for environmental prediction) global forecast system analysis and forecasts at 0.5 deg horizontal resolution [6]. information created during pre-processing and simulations of the model is in the lambert conformal projection, which is appropriate for mid-latitude domains. the model setup (see fig. 2) comprises a parent d01 domain (with a common center located at longitude 44.7, latitude 40.0), partly covering europe, all the caucasus, parts of the central asia and the middle east and the nest domain d02, covering the entire region of armenia. 2.2 numerical weather prediction model the mesoscale weather research and forecasting (wrf) model [9, 10], which is adapted to the territory of armenia, is used in operational weather forecasting. designed to serve both research and operational needs, it has grown to offer a spectrum of options and capabilities for a wide range of applications. the d01 domain with a common center located at longitude 44.7, latitude 40. has horizontal resolution 18-km with 202x202 grid points , the d2 nest with 9770 grid points has 6-km horizontal grid increment . the weather forecasts are performed on a daily basis, using the following 1-way nesting strategy. the model uses vertical 31 eta levels and the geographic data resolution is 30 seconds. the model was initialized with the initial and boundary conditions of global forecast system (gfs) at 0000 utc (local time on 04:00) for 31-day period, namely 1-31 january 2016. the applied version of wrfarw model includes the wrf single moment 6-class scheme for cloud microphysics with ice, snow and graupel processes. sub-grid parameterization of deep and shallow convection is based on the kainfritsch scheme applying mass flux approach with downdrafts and cape removal time scale (kain and fritsch 1993). in this study the melloryamadajanjic (myj) 52 cloud service for analysis and interactive visualization of weather data in armenia fig 2. parent (d01) and nested (d02) domains used in the model. pbl scheme with eta similarity surface layer was used (mellor and yamada 1982). the rapid radiative transfer model (rrtm) scheme was used for longwave radiation, while shortwave radiation processes were represented by the dudhia scheme. due to the limited computing resources, we have considered the inner domain with 6 km resolution, and the range of every run was 24 hours starting always at 0000 utc of each day. observable 2-metres long temperatures from 42 operational stations in armenia are used to study the accuracy of forecasting air temperature by the model wrf for january 2016 it must be noted that the inversion was observed in armenia frequently in january 2016 causing low temperatures in inversion affected regions. the surface inversion , trapped by cold air near the ground, causes fog below the inversion layer . 2.3 data management everyday rapid growth of data and need to analyze these data pushed the development of analytical processing tools. olap (online analytical processing) is a model for accessing multidimensional data in data warehouses [11]. data cubes and olap session are main concepts in olap. a data cube is a collection of facts and dimensions organizing the data of a data warehouse according to different analysis axes and aggregation measures. olap provides a set of operations (such as drill-down and slice-and-dice) that transform one multidimensional query into another, which provide high querying. olap queries are formulated as sequences called olap sessions. for analyzing data with spatial and georeferenced components the spatial olap (solap) technology is used, which allows rapid and easy navigation within spatial databases and that offers many levels of information granularity, many themes, many epochs and many display modes synchronized or not: maps, tables and diagrams [12]. it allows a tight integration of gis and olap systems. a solap system supports three types of spatial dimensions: the non-geometric spatial dimensions, the geometric spatial dimensions and the mixed spatial dimensions. during an olap session, the h. grigoryan and r. abrahamyan 53 user analyzes the results of a query and, contingent upon the particular information, applies an activity to determine a new query that will give a superior comprehension of information. 2.4 user interface and visualization the implemented service consists of 3 main intelligent blocks, and the data flows between these pieces. the first block is the database. postgresql with its postgis extension is used to store data sets with geometrical information. the following block is implementing data analytic logic. the special type of olap approach is used (spatial olap) to process the required data. as a query language for olap, the multidimensional expressions (mdx) [13] is used to interact and perform tasks with multidimensional databases (olap cubes). afterwards, the handled information is exchanged to user interface where user can make different graphs, tables and see the output on the map (see fig. 3). fig 3. web-based visualization and analytical platform consist of the following sections: 1query form, 2 temperature chart, 3 coefficients table the main screen includes many parts, which are marked with red numbers. ⇒ query form user can pick out the station, start and end dates and the period (by default it takes all hours from 0 to 21) for requesting the wished records for plot. after the plot action, the charts and the table will be updated with the corresponding values; ⇒ temperature shows the observation and model temperature information lines correspondingly for each period; ⇒ coefficients table shows the rmse, bias and r correlation coefficients. platform provides the ability to transfer new information to the database by utilizing api endpoints. at present, only the administrator of the platform has access to these activities. for observation data, synop documents (the records are in ascii format) must be used, which will be parsed utilizing a script written in javascript language. for wrf model outputs, which are prroduced in netcdf file format, the python script was made for finding relating values based on the stations information stored in the database (see fig. 4). 54 cloud service for analysis and interactive visualization of weather data in armenia fig. 4. data workflow consists of api endpoint, database, solap and user interface steps. 3. evaluation of results various statistical and object-oriented methods are used in the suggested web-based analytical platform to investigate the characteristics of model-forecast, which is important for giving helpful guidance to end-users. as a case study, the 2 metre temperature has been investigated using the observational and regional high-resolution wrf model data for january, 2016. for the studied period, the rmse, bias and r correlation coefficients are calculated for the observational and modelforecast data for 42 observation points distributed in the territory of armenia (see table 1). table 1: mean estimates of verification of temperature forecasts for several meteo stations per region. h. grigoryan and r. abrahamyan 55 the table shows that the average difference between the forecast data and observations is about 4-50c. for all the stations studied, an acceptable correlation coefficient has been obtained, which allows one to judge the adequacy of the model. the analyses, which have been carried out for 42 stations (some high-altitude stations are not considered), show that the model data for almost all stations are overstated. rmse values are about 1.6-2.50c for the stations located at altitudes above 1500m-2000m, 4.69.80c for the stations below 1000m. it means that the model predicts well the temperature values at a height of 2m for stations located above 1500m and gives unsatisfactory results for stations below 1500m. the worst results are obtained for stations below 1000m, such as stations located in meghri, kapan, ijevan, bagratashen, and the valley of syunik and tavush region. the january 2016 temperature for ararat valley and yerevan is also projected unsatisfactorily. in ararat valley rmse 5.6-6.40c, the worst result was obtained for ararat station. in yerevan, a good result was obtained for yerevan-arabkir station, which is located at an altitude of 1113m, and the worst result for yerevan-zvartnots. the correlation coefficient varies from 0.80-0.97 for mountain and foothill areas, as for valleys it is from 0.7 to 0.25 and from -0.7 to -0.31. from all that has been described above, it can be concluded that the wrf model with the described configuration, for a cold period of time, gives a positive forecast for a variable air temperature of 2 m for the mountain and foothill regions of the republic (shirak, kotayk, gegharkunik, lori, mountain and foothill areas of aragatsotn) and at the same time an unsatisfactory forecast for the valley of syunik, tavush, ararat valley and yerevan. analyzing all the factors that make up the temperature, we see that there were fogs in the valley during the selected period, which was the reason for low temperatures. therefore we can conclude that the model wrf is not predicted by low temperatures of occurrence of the lowland, effect of the fog at surface inversion. such a result does not satisfy, therefore, it is necessary to improve the accuracy of the model data, through a proper tuning, it would be useful to test the impact of resolution given, because the armenian terrain has a rather complex orography and is characterized by several land-category types. 4. conclusion the suggested platform enables to integrate the already available observational and modelforecast informations and use these sources for studies and analyzes in a web-based visualization environment. the interactive comparison charts for 2m air temperature allows to visually analyze and gather the information about model accuracy. it enables to adjust the forecasting results with additional methods by implementing statistical analyses and provides a fairly high result in cases where the model’s sensitivity is low. it is planned to enhance the functionality of the platform by adding a new source of satellite image processing, the land surface temperature, which will give an opportunity to analyze locations where the meteorological stations do not exist. it will include also new visualization tools of various formats, such as to analyze and compare other near-surface atmospheric elements. different nowcasting methodologies based on artificial intelligence will be implemented for the development of a hazardous hydro-meteorological phenomena alarm system. 56 cloud service for analysis and interactive visualization of weather data in armenia references [1] a. gevorgyan, “summertime wind climate in yerevan: valley wind systems”, climate dynamics, vol. 48, no. 5–6, pp. 1827–1840, 2017. [2] a. gevorgyan, h. melkonyan, r. abrahamyan, z. petrosyan, a. shachnazaryan, h. astsatryan, v.sahakyan and yu. shoukourian, a persistent surface inversion event in armenia as simulated by wrf model, in ieee proceedings of the international conference on computer science and information technologies, csit’2015, pp. 105– 110, 2015. [3] h. astsatryan, v. sahakyan, y. shoukourian, p.-h. cros, m. dayde, j. dongarra and p. oster, “strengthening compute and data intensive capacities of armenia”, in ieee proceedings of 14th roedunet international conference networking in education and research, ner’2015, pp. 28–33, 2015. [4] h. astsatryan, yu. shoukourian and v. sahakyan, “the armcluster project: brief introduction”, in proceedings of the international conference on parallel and distributed processing techniques and applications, pp. 1291–1295. 2004. 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[13] m. whitehorn, r. zare and m. pasumansky, fast track to mdx, springer-verlag london, 2006, doi: 10.1007/1-84628-182-2 submitted 09.10.2017, accepted 22.01.2018. h. grigoryan and r. abrahamyan 5 7 ð³û³ëï³ýáõù »õ³ý³ï³ûçý ïíû³éý»ñç ñ³ù³ñ í»ñéáõíáõãû³ý ¨ çýï»ñ³ïïçí íç½áõ³éç½³óç³ûç ³ùå³ûçý í³é³ûáõãûáõý ð. ¶ñç·áñû³ý ¨ è. ²µñ³ñ³ùû³ý ²ù÷á÷áõù öáùñ îáíï³ëû³ý é»éý»ñá ³ýóýáõù »ý ð³û³ëï³ýç ï³ñ³íùáí, ëï»õí»éáí ï»õ³ýùç µ³ñóñáõãû³ý, ç»ñù³ëïç׳ýç ¨ ï»õáõùý»ñç ù»í ï³ñµ»ñáõãûáõýý»ñ: ð³û³ëï³ýç ñáíçïý»ñá ·ïýíáõù »ý íáíç ù³ï³ñ¹³ïçó 500-çó 1500ù µ³ñóñáõãû³ý íñ³: è»éý³ßõã³ý»ñá ï³ñ³ííáõù »ý ùçýã ²ñ³·³í é»éá` 4090ù, áñï»õ ï³ñ³íùç 75 % -á ·»ñ³½³ýóáõù ¿ 1000ù-á, 50 % -á` 2000ù-çó µ³ñóñ ¿, çëï 3.4 % -á` 3000 ù µ³ñóñáõãûáõýçó: ²ßë³ï³ýùç ýå³ï³ïý ¿ ý»ñï³û³óý»é ð³û³ëï³ýáõù »õ³ý³ï³ûçý ïíû³éý»ñç ñ³ù³ñ í»ñéáõíáõãû³ý ¨ çýï»ñ³ïïçí íç½áõ³éç½³óç³ûç ³ùå³ûçý í³é³ûáõãûáõý, áñá µ³õï³ó³í ¿ ó³ù³ù³ûçý ï³û³ýý»ñç ¹çï³ñïáõùý»ñç ¨ »õ³ý³ïç ãí³ûçý ï³ýë³ï»ëù³ý ùá¹»éç »ýã³ï³éáõóí³íùý»ñçó: øß³ïí³í ñ³ù³ï³ñ·áõù û·ï³·áñííáõ »õ³ý³ï³ûçý ïíû³éý»ñá µ³õï³ó³í »ý ù³ï»ñ¨áõûãçý ùáï ùãýáéáñï³ûçý µ³õ³¹ñçãý»ñçó, ý»ñ³éû³é û¹ç ç»ñù³ëïç׳ýá, ñ³ñ³µ»ñ³ï³ý ëáý³íáõãûáõýá, ×ýßáõùá, ù³ùçý ¨ ï»õáõùý»ñá: ìç½áõ³éç½³óç³ý ¨ í»ñéáõíáõãûáõýá çñ³ï³ý³óí»é »ý 2ù µ³ñóñáõãû³ý û¹ç ç»ñù³ëïç׳ýç ñ³ù³ñ: ²ùå³ûçý í³é³ûáõãûáõýá ð³û³ëï³ýç å»ï³ï³ý ñç¹ñáû¹»ñ¨áõã³µ³ý³ï³ý ¨ ùáýçïáñçý·ç í³é³ûáõãû³ýá ñý³ñ³íáñáõãûáõý ¿ ï³éçë áñáß³ïç å³ù³ý³ï³ñ³ïí³íáõù ûáõñ³ù³ýãûáõñ ï³û³ýç ¨ ï³ñ³í³ßñç³ýç ñ³ù³ñ ï³ï³ñ»é ¹çï³ñïáõùý»ñ, çýãå»ë ý³¨ »õ³ý³ïç ï³ýë³ï»ëù³ý ãí³ûçý ùá¹»éç ³ñ¹ûáõýùý»ñç ñ³ù»ù³ï³ï³ý í»ñéáõíáõãûáõýý»ñ: îáëà÷íûé ñåðâèñ äëÿ àíàëèçà è èíòåðàêòèâíîé âèçóàëèçàöèè äàííûõ î ïîãîäå â àðìåíèè à. ãðèãîðÿí è ð. àáðààìÿí àííîòàöèÿ ìàëûå êàâêàçñêèå ãîðû, ïåðåñåêàÿ òåððèòîðèþ àðìåíèè, ñîçäàþò áîëüøèå ïåðåïàäû âûñîò ìåñòíîñòåé, òåìïåðàòóð è îñàäêîâ. ðàâíèíû àðìåíèè íàõîäÿòñÿ íà âûñîòå îò 500 äî 1500ì íàä óðîâíåì ìîðÿ. àðìÿíñêîå íàãîðüå ïðîñòèðàåòñÿ äî ãîðû àðàãàö(âûñîòà 4090 ì), ãäå 75% òåððèòîðèè ïðåâûøàåò 1000ì, 50% ñâûøå 2000ì, à 3,4% ñâûøå 3000ì. öåëü äàííîé ðàáîòû ïðåäñòàâèòü îáëà÷íûé ñåðâèñ äëÿ äàííûõ î ïîãîäå â àðìåíè, ðàçðàáîòàííûé ïóòåì èíòåãðàöèè äâóõ ñóùåñòâóþùèõ èíôðàñòðóêòóð íàáëþäåíèÿ ñ íàçåìíûõ ñòàíöèé è ÷èñëåííàÿ ìîäåëü ïðîãíîçèðîâàíèÿ ïîãîäû. ïîãîäíûå äàííûå, èñïîëüçóåìûå â ðàçðàáîòàííîé ñèñòåìå, ñîñòîÿò èç àòìîñôåðíûõ ýëåìåíòîâ â ïðèçåìíîì ñëîå, âêëþ÷àÿ òåìïåðàòóðó âîçäóõà, îòíîñèòåëüíóþ âëàæíîñòü, äàâëåíèå, âåòåð è îñàäêè. àíàëèç è âèçóàëèçàöèÿ ïðîâîäèëèñü äëÿ òåìïåðàòóðû âîçäóõà íà âûñîòå 2ì. îáëà÷íûé ñåðâèñ ïðåäîñòàâëÿåò âîçìîæíîñòü ãîñóäàðñòâåííîé ñëóæáå ïî ãèäðîìåòåîðîëîãèè è ìîíèòîðèíãó àðìåíèè â îïðåäåëåííûé ïåðèîä âðåìåíè äëÿ êàæäîé ñòàíöèè è ìåñòíîñòè ïðîâåñòè ñðàâíèòåëüíûé àíàëèç íàáëþäàåìûõ äàííûõ ñ ðåçóëüòàòàìè äàííûõ îò ÷èñëåííîé ìîäåëè. hg abstract_hayk_grigoryan_abstract microsoft word sozdanie mat modeli landinga26_05_2005.doc.doc ìàòåìàòè÷åñêèå âîïðîñû êèáåðíåòèêè è âû÷èñëèòåëüíîé òåõíèêè 24, 2005, 125-132. 125 î ñòàòèñòè÷åñêîì ïîäõîäå ê îáíàðóæåíèþ íàðóøåíèé â òåëåôîííûõ ñåòÿõ êàðåí à. ìêðò÷ÿí èíñòèòóò ïðîáëåì èíôîðìàòèêè è àâòîìàòèçàöèè íàí ðà e-mail: kamkrtchyan@ipia.sci.am, kamkrtchyan@yahoo.com àííîòàöèÿ â ñòàòüå ïðåäëàãàåòñÿ ìåòîä ìîäåëèðîâàíèÿ îáðàçîâ îáúåêòîâ, ïîçâîëÿ þùèé îáíàðóæèâàòü íàðóøåíèÿ â òåëåôîííîé ñåòè, ñ ïðèìåíåíèåì ñòàòèñòè÷åñêîãî àíàëèçà. ëèòåðàòóðà [1] l. cao and other. hybrid strategy of analysis and control of telecommunication frauds. attend.it.uts.edu.au/icita05/ cdrom-icita04/papers/62-1.pdf 2004. [2] paul de jager. “introduction to using intelligent techniques for telecommunications fraud detection”. http://www.eurescom.de/~pub/seminars/past/2001/securityfraud/12-jager/ [3] james l. johnson and other. local prosecutors’ experiences fighting telecommunications fraud. www.ndaa-apri.org/pdf/sounds_too_good.pdf american prosecutors research institute, 2004 [4] e. lundin. aspects of employing fraud and intrusion detection systems. technical report no. 2l, department of computer engineering, chalmers university of technology, geoteborg, sweden, 2002. [5] p. gosset and m. hyland. classification, detection and prosecution of fraud on mobile networks. 1998. http://www.esat.kuleuven.ac.be/cosic/aspect/papers/mobsummit.doc [6] e. lundin. combining fraud and intrusion detection meeting new requirements. http://www.ce.chalmers.se/~emilie/ 2004 [7] russell g. smith, stealing telecommunications services, australian institute of criminology, http://www.aic.gov.au/publications/tandi/ti54.pdf 2005 [8] ì. ñêâîðöîâà. ìàòåìàòè÷åñêîå ìîäåëèðîâàíèå. http://archive.1september.ru/mat/2003/14/no14_1.htm 2003. [9] å. àðóò þíÿí è äðóãèå. âåðîÿòíîñòü è ïðèêëàäíàÿ ñòàòèñòèêà. (íà àðìÿíñêîì ÿçûêå), èçäàòåëüñòâî “ãèòóò þí” íàí ðà, åðåâàí 2000. î ñòàòèñòè÷åñêîì ïîäõîäå ê îáíàðóæåíèþ íàðóøåíèé â òåëåôîííûõ ñåòÿõ 126 ð»é³ëáë³ûçý ó³ýó»ñáõù ë³ëïáõùý»ñç µ³ó³ñ³ûïù³ý íç׳ﳷñ³ï³ý ùç ùáï»óù³ý ù³ëçý î. øïñïãû³ý ²ù÷á÷áõù ðá¹í³íáõù ³é³ç³ñïíáõù ¿ ñ»é³ëáë³ûçý ó³ýó»ñç û·ïíáõý»ñç ù³ã»ù³ïçï³ï³ý å³ïï»ñý»ñç íç׳ﳷñ³ï³ý í»ñéáõíáõãû³ý ùççáóáí ùá¹»é³íáñù³ý ùç »õ³ý³ïª ë³ëïáõùý»ñç µ³ó³ñ³ûïù³ý ñ³ù³ñ: vma_toprint_changed.dvi mathematical problems of computer science 23, 2004, 102{118. application of voting m ethods to str ategies analysis and assessment e m m a h . d a n ie lya n institute for informatics and automation problems of nas of ra e-mail emma danielyan@yahoo.com abstract classi¯cation of known voting methods in the frame of the management optimal strategy provision and skill assessment problems is given. methods applicable to these problems are presented with pseudo code and their complexities are calculated. the applicability of voting methods is shown on the scheme with methods separation on two basic classes condorcet consistent and scoring methods. the kendall-wei tournament participants ranking method is analyzed for mentioned problems. refer ences [1 ] t. co r m e n , ch . l e is e r s o n , r . r ive s t introduction to algorithms, ( in r u s s ia n ) , mo s c o w, 2 0 0 1 . [2 ] e . d a n ie lya n , e . p o g o s s ia n " a p p lic a t io n o f v o t in g me t h o d s t o s t r a t e g y a s s e s s m e n t " , p roceedings of the conference on computer science and information technologies (csit'99), pp. 165 169, yerevan, armenia, 1999, [3 ] ma r k d u n lo p " te a c h in g a lg o r it h m s &co m p le xit y: b ig oh r o u g h gu id e " http://www.cs.strath.ac.uk/~mdd/teaching/alg&comp/big oh.html [4 ] j. w . mo o n topics on tournaments, n y , h o lt , r in e h e va r t & w in s t o n , 1 9 6 8 . [5 ] h . mo u lin axioms of cooperative d ecision m aking, is b n -0 -5 2 1 -3 6 0 5 5 -2 , v ir g in ia p o lyt e c h n ic & s t a t e u n ive r s it y, 1 9 8 8 . [6 ] e . m. p o g o s s ia n \ d e ve lo p m e n t o f ma n a g e m e n t s kill a s s e s s m e n t " , p roceedings of the ab se l -98, m aui, honolulu, 1998. [7 ] e . m. p o g o s s ia n \ ma n a g e m e n t s t r a t e g y s e a r c h a n d a s s e s s m e n t p r o g r a m m in g " , p roceedings of the conference on computer science and information technologies (csit'99), pp. 46 66, yerevan, armenia, 1999. [8 ] e . m. p o g o s s ia n \ fo c u s in g ma n a g e m e n t s t r a t e g y p r o vis io n s im u la t io n " p roceedings of the third international conference in computer sciences and information technologies(csit'01), pp. 243 247,yerevan, armenia, 2001. [9 ] l . e . s a d o vs ki, a . l . s a d o vs ki m athematics and sport, ( in r u s s ia n ) , mo s c o w, fiz -ma t l it e r a t u r e , 1 9 8 5 . 1 0 2 e. h. danielyan 1 0 3 [1 0 ] s t e ve n s . s kie n a the algorithm d esign m anual, s p r in g e r -v e r la g n e w-y o r k, in c . 1 9 9 8 . [1 1 ] h e r b e r t s . w ilf \ s e a r c h in g t h e we b wit h e ig e n ve c t o r s " a p r il 1 3 , 2 0 0 1 , http://www.math.scar.utoronto.ca/b24/k endallw ei.pdf. êïñ³ï»·ç³ý»ñç ·ý³ñ³ïáõù ¨ ³ý³éç½ ùí»³ñïáõãû³ý ù»ãá¹ý»ñç ïçñ³éù³ùµ ¾. ð. ¸³ýç»éû³ý ²ù÷á÷áõù ðá¹í³íáõù ¹çï³ñïíáõù ¿ ùç ß³ñù ñ³ûïýç ùí»³ñïáõãû³ý ù»ãá¹ý»ñç ¿ýý»ïïçí ïçñ³éáõãûáõýá ù»ý»çù»ýïç ûåïçù³é ëïñ³ï»·ç³ûç ¨ ù»ý»ç»ñý»ñç ·çï»éçùý»ñç ·ý³ñ³ïù³ý ëý¹çñý»ñáõù: ¶ñ³ýçïáñ»ý óáõûó ¿ ïñí³í ù»ãá¹ý»ñç µ³å³ýáõùá »ñïáõ ñçùý³ï³ý ¹³ë»ñçª ·ý³ñ³ï³ï³ýý»ñç ñ³ßí³éù³ý ¨ ñ³ù³ó³ûý»óí³í îáý¹áñë»ïáí, ýñ³ýó ïçñ³é»éçáõãû³ý ýï³ñ³·ñáõãû³ùµ: ²ßë³ï³ýùáõù ù³ýñ³ù³ëýáñ»ý ¹çï³ñïíáõù ¿ ý³¨ î»ý¹³é-ì»ûç ù»ãá¹á, áñá ãáõûé ¿ ï³éçë ¹³ë³ï³ñ·»é ùñóáõûãç ù³ëý³ïçóý»ñçýª û·ï³·áñí»éáí áõå³ûçý í»ïïáñç ñ³ëï³óáõáõãûáõýá: ²ûë ù»ãá¹á ýáõûýå»ë ³é³ç³ñïíáõù ¿ í»ñáñçßû³é ëý¹çñý»ñç ñ³ù³ñ: øí»³ñïáõãû³ý ³ûý ù»ãá¹ý»ñá, áñáýù ïçñ³é»éç »ý ù»ý»çù»ýïç ûåïçù³é ëïñ³ï»·ç³ç ¨ ù»ý»ç»ñý»ñç ·çï»éçùý»ñç ·ý³ñ³ïù³ý ëý¹çñý»ñáõù ý»ñï³û³óíáõù »ý å먹áïá¹áí: ð³ßí³ñïíáõù ¿ ³û¹ ù»ãá¹ý»ñç µ³ñ¹áõãáõýá: ²é³ýóçý ³õûáõë³ïáí óáõûó ¿ ïñíáõù ùí»³ñïáõãû³ý ù»ãá¹ý»ñç ñ³ù³å³ï³ëë³ýáõãáõýá ùí»³ñïáõãû³ý ï»ëáõãû³ý ñçùý³ï³ý ³ùëçáùý»ñçýª çù³ëï³íáñí³í ëïñ³ï»·ç³ý»ñç ·ý³ñ³ïù³ý ëý¹çñý»ñç ñ³ù³ñ: d:\sbornik\...\tpel.dvi mathematical problems of computer science 32, 45{47, 2009. selection of output e lements by m inimum cost flow algor ithm r u b e n o. a d a m ya n a n d s t e p a n e . ma r ko s ya n yerevan state university ruboam@yahoo.com abstract during the placement stage of vlsi (very large scale of integration) physical design phase it is needed to take into account the external connections of the placing elements. so later, it is possible to get better result in routing stage. thus it is required to map external nets of a circuit to its elements in such a way that the maximum number of nets corresponding to an element is minimal. in this article the problem is solved by reducing it to the problem of ¯nding a minimum cost °ow of a given value in a network. refer ences [1 ] n .a . s h e r wa n i, algorithms for vl si p hysical d esign automation, k lu we r a c a d e m ic p u b lis h e r s , n o r we ll, ma , u s a , 1 9 9 3 . [2 ] à.â. ïåòðîñÿí, ñ.å. ìàðêîñÿí è þ.ã. øóêóðÿí, ìàòåìàòè÷åñêèå âîïðîñû àâòîìàòèçàöèè ïðîýêòèðîâàíèÿ ýâì, àêàäåìèÿ íàóê àðìÿíêîé ññð, 1977. [3 ] e .a .d in it s , \ th e m e t h o d o f s c a lin g a n d t r a n s p o r t a t io n p r o b le m s " , studies in d iscrete m athematics, mo s c o w, p p . 4 6 -5 7 , 1 9 7 3 . [4 ] h .n .ga b o w, \ s c a lin g a lg o r it h m s fo r n e t wo r k p r o b le m s " , 24th annual symposium on f oundations of computer science, n e w y o r k, p p . 2 4 8 -2 5 7 , 1 9 8 3 . [5 ] h .n .ga b o w, \ s c a lin g a lg o r it h m s fo r n e t wo r k p r o b le m s " , j ournal of computer and system science 31, p p .1 4 8 -1 6 8 , 1 9 8 5 . ºéù³ûçý ï³ññ»ñç áýïñáõãûáõýá ýí³½³·áõûý ³ñå»ùáí ñáëù ·ïý»éáõ ³é·áñçãùç ùççáóáí è. ²¹³ùû³ý, ê. ø³ñïáëû³ý ²ù÷á÷áõù ¶»ñù»í çýï»·ñ³é³ûçý ëë»ù³ý»ñç ýç½çï³ï³ý ý³ë³·íù³ý ï»õ³¹ñù³ý ÷áõéáõù ³ýññ³å»ßïáõãûáõý ¿ ³é³ç³ýáõù ñ³ßíç ³éý»é ï»õ³¹ñíáõ ï³ññ»ñç ³ñï³ùçý ï³å»ñá, 4 5 4 6 selection of output elements by minimum cost flow algorithm áñå»ë½ç áõõ»·íù³ý ÷áõéáõù ñý³ñ³íáñ éçýç ëï³ý³é ³í»éç é³í ³ñ¹ûáõýù: àôëïç ³ýññ³å»ßï ¿ ³ñï³ùçý ßõã³ý»ñý ³ñï³å³ïï»ñ»é ëë»ù³ûç ï³ññ»ñçý ³ûýå»ë, áñ ßõã³ý»ñç ³é³í»é³·áõûý ù³ý³ïá, áñá ñ³ù³å³ï³ëë³ýáõù ¿ ù»ï ï³ññá éçýç ùçýçù³é: ðá¹í³íáõù ³û¹ ëý¹ñç éáõíáõùá ñ³ý·»óí³í ¿ ó³ýóáõù ïñí³í ù»íáõãû³ý ýí³½³·áõûý ³ñå»ùáí ñáëù ·ïý»éáõ ëý¹ñçý: mathematical problems of computer science 58, 20–31, 2022. doi:10.51408/1963-0089 udc 519.1 on an extension of the ghouila-houri theorem samvel kh. darbinyan institute for informatics and automation problems of nas ra e-mail: samdarbin@iiap.sci.am abstract let d be a 2-strong digraph of order n ≥ 8 such that for every vertex x ∈ v(d)\{z}, d(x) ≥ n and d(z) ≥ n − 4, where z is a vertex in v(d). we prove that: if d contains a cycle passing through z of length equal to n − 2, then d is hamiltonian. we also give a new sufficient condition for a digraph to be hamiltonian-connected. keywords: digraphs, hamiltonian cycles, hamiltonian-connected, 2-strong. article info: received 21 aprile 2022; received in revised form 16 september 2022; accepted 15 november 2022. acknowledgement: we thank the referees for their valuable comments and suggestions that improved the presentation considerably. 1. introduction in this paper, we consider finite digraphs (directed graphs) without loops and multiple arcs. the order of a digraph d is the number of its vertices. we shall assume that the reader is familiar with the standard terminology on digraphs. terminology and notations not described below follow [1]. every cycle and path is assumed simple and directed. a cycle (path) in a digraph d is called hamiltonian (hamiltonian path) if it includes every vertex of d. a digraph d is hamiltonian if it contains a hamiltonian cycle, and it is hamiltonianconnected if for any pair of ordered vertices x and y there exists a hamiltonian path from x to y. there are numerous sufficient conditions for the existence of a hamiltonian cycle in a digraph (see, [1]–[3]). let us recall the following sufficient conditions for a digraph to be hamiltonian. theorem 1: (ghouila-houri [4]). let d be a strong digraph of order n ≥ 2. if for every vertex x ∈ v(d), d(x) ≥ n, then d is hamiltonian. theorem 2: (meyniel [5]). let d be a strong digraph of order n ≥ 2. if d(x)+d(y) ≥ 2n−1 for all pairs of non-adjacent vertices x and y in d, then d is hamiltonian. nash-williams [6] raised the problem of describing all the extreme digraphs in theorem 1, that is, all digraphs with minimum degree at least |d| − 1, that do not have a hamiltonian 20 s. darbinyan 21 cycle. as a solution to this problem, thomassen [7] proved a structural theorem on the extreme digraphs. an analogous problem for theorem 2 was considered by the author [8]. in [8], we generalize thomassen’s structural theorem (theorem 1, in [7]), characterizing the nonhamiltonian strong digraphs of order n with the degree condition that d(x) + d(y) ≥ 2n − 2 for every pair of non-adjacent distinct vertices x, y. moreover, in [8], it was also proved that if m is the length of a longest cycle in d, then d contains cycles of all lengths k = 2, 3, . . . , m. the following conjecture was suggested by thomassen. conjecture 1: (thomassen [9], see conjecure 1.6.7 in [2]). every 3-strong digraph of order n and with minimum degree at least n + 1 is hamiltonian-connected. in [10], we disprove this conjecture, by proving the following three theorems. theorem 3: every k-strong (k ≥ 1) digraph of order n, which has n − 1 vertices of degrees at least n, is hamiltonian if and only if any (k + 1)-strong digraph of order n + 1 with minimum degree at least n + 2 is hamiltonian-connected. theorem 4: for every n ≥ 8, there is a non-hamiltonian 2-strong digraph d of order n with minimum degree equal to 4 such that d has n − 1 vertices of degrees at least n. theorem 5: for every n ≥ 9, there exists a 3-strong digraph d of order n with minimum degree at least n+1 such that d contains two distinct vertices u, v for which u ↔ v, d+d(u)+ d−d(v) = 6 and there is no (u, v)-hamiltonian path. in view of theorems 4, 5 and conjecture 1, it is natural to pose the following problem. problem: let d be a 2-strong digraph of order n ≥ 9. suppose that n − 1 vertices of d have degrees at least n and a vertex x has degree is at least n − m, where 1 ≤ m ≤ n − 5. find the maximum value of m, for which d is hamiltonian. goldberg, levitskaya and satanovskiy [11] relaxed the conditions of the ghouila-houri theorem. they proved the following theorem. theorem 6: (goldberg et al. [11]). let d be a strong digraph of order n ≥ 2. if for every vertex x ∈ v(d) \ {z}, d(x) ≥ n and d(z) ≥ n − 1, then d is hamiltonian. note that theorem 6 is an immediate consequence of theorem 2. in [11], the authors for any n ≥ 5 presented two examples of non-hamiltonian strong digraphs of order n such that: (i) in the first example, n − 2 vertices have degrees equal to n + 1 and the other two vertices have degrees equal to n − 1. (ii) in the second example, n−1 vertices have degrees at least n and the remaining vertex has degree equal to n − 2. in [12], it was reported that the following theorem holds. theorem 7: (darbinyan [12]). let d be a 2-strong digraph of order n ≥ 9 with minimum degree at least n − 4. if n − 1 vertices of d have degrees at least n, then d is hamiltonian. in this article, we present the first part of the proof of theorem 7, which we formulate as theorem 9. the proof of the last theorem has never been published. it is worth mentioning that the proof presented here differs from the previous handwritten proof and is significantly shorter and more general than the previous one. the second part of the proof (i.e., the complete proof) of theorem 7 we will present in the forthcoming paper, where we also 22 on an extension of the ghouila-houri theorem present two examples of digraphs, which show that the bounds n ≥ 9 and n − 4 in theorem 7 are sharp in a sense. 2. further terminology and notation for the sake of clarity we repeat the most impotent definition. the vertex set and the arc set of a digraph d are denoted by v(d) and a(d), respectively. the order of a digraph d is the number of its vertices. the converse digraph of d is the digraph obtained from d by reversing the direction of all arcs. the arc of a digraph d directed from x to y is denoted by xy or x → y (we also say that x dominates y or y is an out-neighbour of x and x is an in-neighbour of y), and x ↔ y denotes that x → y and y → x (x ↔ y is called 2-cycle). if x → y and y → z, we write x → y → z. if a and b are two disjoint subsets of v(d) such that every vertex of a dominates every vertex of b, then we say that a dominates b, denoted by a → b. we define a(a → b) = {xy ∈ a(d) | x ∈ a, y ∈ b} and a(a, b) = a(a → b) ∪ a(b → a). if x ∈ v(d) and a = {x} we sometimes write x instead of {x}. let n+d(x), n − d(x) denote the set of out-neighbors, respectively the set of in-neighbors of a vertex x in a digraph d. if a ⊆ v(d), then n+d(x, a) = a ∩ n + d(x) and n − d(x, a) = a ∩ n − d(x). the outdegree of x is d+d(x) = |n + d(x)| and d − d(x) = |n − d(x)| is the in-degree of x. similarly, d+d(x, a) = |n + d(x, a)| and d − d(x, a) = |n − d(x, a)|. the degree of the vertex x in d is defined as dd(x) = d + d(x) + d − d(x) (similarly, dd(x, a) = d + d(x, a) + d − d(x, a)). we omit the subscript if the digraph is clear from the context. the subdigraph of d induced by a subset a of v(d) is denoted by d. in particular, d − a = d⟨v(d) \ a⟩. for integers a and b, a ≤ b, by [a, b] we denote the set {xa, xa+1, . . . , xb}. if j < i, then {xi, . . . , xj} = ∅. the path (respectively, the cycle) consisting of the distinct vertices x1, x2, . . . , xm (m ≥ 2) and the arcs xixi+1, i ∈ [1, m − 1] (respectively, xixi+1, i ∈ [1, m − 1], and xmx1), is denoted by x1x2 · · · xm (respectively, x1x2 · · · xmx1). the length of a cycle or a path is the number of its arcs. let d be a digraph and z ∈ v(d). by cm(z) (respectively, c(z)) we denote a cycle in d of length m (respectively, any cycle in d), which contains the vertex z. we say that p = x1x2 · · · xm is a path from x1 to xm or is an (x1, xm)-path. a digraph d is strong (strongly connected) if, for every pair x, y of distinct vertices in d, there exists an (x, y)-path and a (y, x)-path. a digraph d is k-strong (k-strongly connected) if, |v(d)| ≥ ∥ + ∞ and for any set a of at most k − 1 vertices d − a is strong. two distinct vertices x and y are adjacent if xy ∈ or yx ∈ a(d) (or both). the converse digraph of d is the digraph obtained from d by replacing the direction of all arcs. we will use the principle of digraph duality: let d be a digraph, then d contains a subdigraps h if and only if the converse digraph of d contain the converse of subdigraph h. 3. preliminaries in our proofs, we will use the following well-known simple lemma. lemma 1: (häggkvist and thomassen [13]). let d be a digraph of order n ≥ 3 containing a cycle cm of length m, m ∈ [2, n − 1]. let x be a vertex not contained in this cycle. if d(x, v(cm)) ≥ m + 1, then for every k ∈ [2, m + 1], d contains a cycle ck including x. the next lemma is a slight modification of a lemma by bondy and thomassen [14], it is very useful and will be used extensively throughout this paper. s. darbinyan 23 lemma 2:. let d be a digraph of order n ≥ 3 containing a path p := x1x2 . . . xm, m ∈ [2, n − 1]. let x be a vertex not contained in this path. if one of the following condition holds: (i) d(x, v(p)) ≥ m + 2, (ii) d(x, v(p)) ≥ m + 1 and xx1 /∈ a(d) or xmx /∈ a(d), (iii) d(x, v(p)) ≥ m and xx1 /∈ a(d) and xmx /∈ a(d), then there is an i ∈ [1, m − 1] such that xi → x → xi+1, i.e., d contains a path x1x2 . . . xixxi+1 . . . xm of length m (we say that x can be inserted into p). using lemma 2, we can prove the following lemma. lemma 3: let p := x1x2 . . . xm, m ∈ [3, n−1], be a longest (x1, xm)-path in a digraph d of order n. suppose that y ∈ v(d)\v(p) and there is no i ∈ [1, m−2] such that xi → y → xi+2. then the following holds: (i) if yx1 /∈ a(d), x1y ∈ a(d) and d(y, v(p)) ≥ m, then d(y, v(p)) = m and {x1, x2, . . . , xm} → y; (ii) if xmy /∈ a(d), yxm ∈ a(d) and d(y, v(p)) ≥ m, then d(y, v(p)) = m and y → {x1, x2, . . . , xm}; (iii) if d(y, v(p)) ≥ m+1, then d(y, v(p)) = m+1 and there exists an integer q ∈ [1, m] such that {xq, xq+1, . . . , xm} → y → {x1, x2, . . . , xq}. proof. to prove the lemma, it suffices to show that every vertex xi ∈ v(p) is adjacent to y. assume that this is not the case. (i) let y and xt be not adjacent. then t ≥ 2 since x1 → y. since p is a longest (x1, xm)-path, we have that y cannot be inserted into p . using lemma 2(ii) and the assumption that yx1 /∈ a(d), we obtain xmy ∈ a(d), 2 ≤ t ≤ m − 1 and m ≤ d(y, v(p)) = d(y, {x1, . . . , xt−1}) + d(y, {xt+1, . . . , xm}) ≤ t − 1 + (m − t + 1) = m. this means that d(y, {x1, . . . , xt−1}) = t − 1 and d(y, {xt+1, . . . , xm}) = m − t + 1. again using lemma 2, we obtain that xt−1 → y → xt+1, which contradicts the supposition of lemma 3. this contradiction shows that every vertex xi is adjacent to y. in a similar way, one can show that if (ii) or (iii) holds, then every vertex of p also is adjacent to y. lemma 3 is proved. in [10], the author proved the following theorem. theorem 8: (darbinyan [12]). let d be a strong digraph of order n ≥ 3. suppouse that d(x)+d(y) ≥ 2n−1 for all pairs of non-adjacent vertices x, y ∈ v(d)\{z}, where z is some vertex in v(d). then d is hamiltonian or contains a cycle of length n − 1. using theorem 8 and lemmas 1 and 2, it is not difficult to show that the following corollaries are true. corollary 1: let d be a strong digraph of order n ≥ 3 satisfying the condition of theorem 8. then d has a cycle that contains all the vertices of d maybe except z. corollary 2: let d be a strong digraph of order n ≥ 3. suppose that n − 1 vertices of d have degrees at least n. then d is hamiltonian or contains a cycle of length n − 1 (in fact, d has a cycle that contains all the vertices of degrees at least n). in this section, we also will prove the following lemma. we will use this lemma in the second part of the proof of theorem 7. 24 on an extension of the ghouila-houri theorem lemma 4: let d be a digraph of order n ≥ 4 such that for any vertex x ∈ v(d)\{z}, d(x) ≥ n and d(z) ≤ n − 2, where z is some vertex in v(d). suppose that cm(z) = x1x2 . . . xmx1 with m ≤ n−2 is a longest cycle through z. if d⟨v (d)\v (cm(z))⟩ is strong and d contains a cm(z)-bypass p = xiy1y2 . . . ylxj such that |v(cm(z)[xi+1, xj−1])| is smallest possible over all cm(z)-bypasses, then z ∈ v(cm(z)[xi+1, xj−1]). proof. without loss of generality, we assume that xj = x1, xi = xm−k, k ≥ 1, a({y1, . . . , yl}, v(cm(z)[xm−k+1, xm])) = ∅ and k is minimum possible with this property over all cm(z)-bypasses. extending the path cm(z)[x1, xm−k] with the vertices of v(cm(z)[xm−k+1, xm]) as much as possible, we obtain an (x1, xm−k)-path, say r. since cm(z) is a longest cycle through z, some vertices u1, u2, . . . , ud ∈ v(cm(z)[xm−k+1, xm]), 1 ≤ d ≤ k, are not on the obtained extended path r. using lemma 2, we obtain that d(yi, vv (cm(z))) ≤ m − k + 1 and d(ui, v(cm(z))) ≤ m + d − 1. put b := v(d) \ (v(cm(z)) ∪ v(p)). note that |b| = n − m − l. let v be an arbitrary vertex in b. from the minimality of k, we have that d contains no paths of the types ui → v → yj and yj → v → ui, which in turn implies that d+(ui, b) + d−(yj, b) ≤ |b| and d−(ui, b) + d +(yj, b) ≤ |b|. therefore, d(ui, b) + d(yj, b) ≤ 2|b| = 2(n − m − l). thus, we have d(ui) + d(yj) = d(ui, v(cm(z))) + d(yj, v(cm(z))) + d(ui, b) + d(yj, b) + d(yj, {y1, . . . , yl}) ≤ m + d − 1 + m − k + 1 + 2n − 2m − 2l + 2l − 2 = 2n − 2 − (k − d) ≤ 2n − 2. this is possible if ui = z. therefore, d = 1 and z ∈ v(cm(z)[xm−k+1, xm]). lemma 4 is proved. 4. the main result in this section, we prove the following theorem. theorem 9: let d be a 2-strong digraph of order n ≥ 8. suppose that for every x ∈ v(d) \ {z}, d(x) ≥ n and d(z) ≥ n − 4, where z is a vertex in v(d). if d contains a cycle of length n − 2 passing through z (i.e., a cycle cn−2(z)), then d is hamiltonian. before we prove our main result, we will prove the following lemma. lemma 5: let d be a non-hamiltonian 2-strong digraph of order n such that for any vertex x ∈ v(d) \ {z}, d(x) ≥ n and d(z) ≤ n − 2, where z is an arbitrary fixed vertex in v(d). suppose that cm+1(z) = x1x2 . . . xmzx1 with m ∈ [2, n − 3] is a longest cycle in d, d(z, y ) = 0 and d⟨y ⟩ is a strong digraph, where y := v(d) \ v(cm+1(z)). let y1, y2 be two distinct vertices in y . if for each yi ∈ {y1, y2}, d(yi, {x1, x2, . . . , xm}) = m + 1, then n ≥ 6 and d(z) ≤ m − 2. proof. by contradiction, suppose that d(z) ≥ m−1. we denote by p the path x1x2 . . . xm. note that |y | = n − m − 1. since the path p cannot be extended with any vertex y ∈ y , by lemma 2, d(y, v(p)) ≤ m + 1 and n ≤ d(y) = d(y, v(p)) + d(y, y ) ≤ m + 1 + d(y, y ), d(y, y ) ≥ n − m − 1 = |y |. (1) since d is 2-strong and cm+1(z) is a longest cycle, using lemma 2 and d(yi, v(p)) = m + 1 it is not difficult to show that there is an integer l ∈ [2, m − 1] such that {xl, xl+1, . . . , xm} → {y1, y2} → {x1, x2, . . . , xl}. (2) s. darbinyan 25 since d(y, y ) ≥ n − m − 1 = |y |(by (1)), and d⟨y ⟩ is strong, by the ghouila-houri theorem, d⟨y ⟩ is hamiltonian. put e := {x1, x2, . . . , xl−1} and f := {xl+1, xl+2, . . . , xm}. since cm+1(z) is a longest cycle and d⟨y ⟩ is strong, from (2) it follows that a(e → y) = a(y → f) = ∅. (3) note that from |y | ≥ 2, |e| ≥ 1 and |f| ≥ 1 it follows that n ≥ 6. we need to prove the following claims 1-2 bellow. claim 1. (i) if d−(z, e) ≥ 1, then d+(z, f) = 0. (ii) a(e → f) ̸= ∅. proof. (i) by contradiction, suppose that xi ∈ e, xj ∈ f and xi → z → xj. then by (2), y1 → xi+1 and xj−1 → y2. hence, cm+3(z) = x1x2 . . . xizxj . . . xmy1xi+1 . . . xj−1y2x1, a contradiction. (ii) suppose, on the contrary, that a(e → f) = ∅. then using claim 1(i) and (3), we obtain: if d−(z, e) ≥ 1, then d+(z, f) = 0 and a(e ∪ y ∪ {z} → f) = ∅, if d−(z, e) = 0, then a(e ∪ y → f ∪ {z}) = ∅. therefore, d − xl is not strong, which contradicts that d is 2-strong. from now on, we assume that xaxb ∈ a(e → f). note that 1 ≤ a ≤ l − 1 and l + 1 ≤ b ≤ m. we may assume that b is the maximum and a is the minimum with these properties. by (2), we have xb−1 → {y1, y2} → xa+1. (4) since z cannot be inserted into p , using lemma 2(ii) and clam 1(i), we obtain d(z, {x1, x2, . . . , xa}) + d(z, {xb, xb+1, . . . , xm}) ≤ a + m − b + 2. (5) by r(yi, y3−i), where i ∈ [1, 2], we denote a longest (yi, y3−i)-path in d⟨y ⟩. from now on, assume that r(yi, y3−i) = r(y1, y2). claim 2. (i) if i ∈ [a + 1, l − 1], then xiz /∈ a(d). (ii) if j ∈ [l + 1, b − 1], then zxj /∈ a(d). (iii) if i ∈ [a + 1, l] and i − a ≤ 2, then zxi /∈ a(d). (iv) if j ∈ [l, b − 1] and b − j ≤ 2, then xjz /∈ a(d). proof. each of claims (i)-(iv) we prove by contradiction. (i) assume that i ∈ [a + 1, l − 1] and xiz ∈ a(d). then by (2) and (4), we have cm+3(z) = x1x2 . . . xaxb . . . xmy1xi+1 . . . xb−1y2xa+1 . . . xizx1, a contradiction. (ii) assume that j ∈ [l + 1, b − 1] and zxj ∈ a(d). then by (2) and (4), we have cm+3(z) = x1x2 . . . xaxb . . . xmzxj . . . xb−1y1xa+1 . . . xj−1y2x1, a contradiction. (iii) assume that i ∈ [a + 1, l], i − a ≤ 2 and zxi ∈ a(d). then c(z) = x1x2 . . . xaxb . . . xmzxi . . . xb−1r(y1, y2)x1 is a cycle of length at least m + 2, a contradiction. (iv) assume that j ∈ [l, b − 1], b − j ≤ 2 and xjz ∈ a(d). then c(z) = x1x2 . . . xaxb . . . xmr(y1, y2)xa+1 . . . xjzx1 is a cycle of length at least m + 2, a contradiction. claim 2 is proved. 26 on an extension of the ghouila-houri theorem now we will consider the following cases depending on the values of a and b with respect to l. case 1. a ≤ l − 3 and b ≥ l + 3. then by claim 2, d(z, {xa+1, xa+2, xb−2, xb−1}) = 0. therefore, since z cannot be inserted into p , using (5) and lemma 2, we obtain m − 1 ≤ d(z, {x1, x2, . . . , xa, xb, xb+1, . . . , xm}) + d(z, {xa+3, . . . , xb−3}) ≤ a + m − b + 2 + b − 3 − a − 2 + 1 = m − 2, which is a contradiction. case 2. a ≤ l − 3 and b = l + 2. then by claim 2, d(z, {xa+1, xa+2, xl+1}) = 0 and xlz /∈ a(d). therefore, since z cannot be inserted into p , using (5) and lemma 2, we obtain m − 1 ≤ d(z, {x1, x2, . . . , xa, xb, xb+1, . . . , xm}) + d(z, {xa+3, . . . , xl}) ≤ a + m − b + 2 + l − a − 2 = m − (l + 2) + l = m − 2, which is a contradiction. case 3. a ≤ l − 3 and b = l + 1. then by claim 2, d(z, {xa+1, xa+2}) = 0 and xlz /∈ a(d). similar to case 2, we obtain m − 1 ≤ d(z, {x1, x2, . . . , xa, xb, xb+1, . . . , xm}) + d(z, {xa+3, . . . , xl}) ≤ a + m − b + 2 + l − a − 2 = m − b + l = m − (l + 1) = m − 1. this implies that d(z, {xa+3, . . . , xl}) = l − a − 2. hence, by claim 2(i) and xlz /∈ a(d), z → {xa+3, . . . , xl}. from this and (4), we see that the cycle q(z) = x1x2 . . . xaxb . . . xmz xa+3 . . . xlr(y1, y2)x1 has length equal to m − 1 + |v(r(y1, y2))|. since cm+1(z) is a longest cycle and d⟨y ⟩ is hamiltonian, it follows that |v(r(y1, y2))| = |y | = 2. then m = n − 3, y1 ↔ y2, xa+1 ↔ xa+2 and xa+1 (xa+2) is adjacent to each vertex xi ∈ {x1, x2, . . . xm}, as d(xa+1) ≥ n (d(xa+2) ≥ n) and xa+1 (xa+2) cannot be inserted into q(z). we will distinguish two subcases. subcase 3.1. m ≥ l + 2. from the minimality of a and the maximality of b, it follows that a({x1, x2, . . . , xa} → {xb+1, xb+2, . . . , xm}) = ∅. (6) assume that xi → xj with i ∈ [a + 1, l] and j ∈ [l + 2, m]. using (4) and the fact that zxa+3 ∈ a(d), it is not difficult to see that if i ∈ [a + 1, a + 2], then c(z) = x1x2 . . . xa+1(xa+2)xj . . . xmzxa+3 . . . xj−1y1y2x1 is a cycle of length at least m + 2, if i ∈ [a + 3, l − 1], then cm+3(z) = x1x2 . . . xixj . . . xmzxi+1 . . . xj−1y1y2x1, if i = l, then cm+3(z) = x1x2 . . . xaxl+1 . . . xj−1y1y2xa+1 . . . xlxj . . . xmzx1. thus, in all cases, we have a contradiction. we may, therefore, assume that (recall that b = l + 1) a({xa+1, xa+2, . . . , xl} → {xb+1, xb+2, . . . , xm}) = ∅. combining this with (6), we obtain a({x1, x2, . . . , xl} → {xb+1, xb+2, . . . , xm}) = ∅. (7) s. darbinyan 27 assume first that d−(z, e) ≥ 1. then by claim 1(i), d+(z, f) = 0. this together with (3) and (7) implies that a({z, x1, x2, . . . , xl} ∪ y → {xl+2, xl+3, . . . , xm}) = ∅. assume second that d−(z, e) = 0. since xlz /∈ a(d), we obtain a({x1, x2, . . . , xl} ∪ y → {z, xl+2, xl+3, . . . , xm}) = ∅. so, in both cases we have that the subdigraph d − xl+1 is not strong, which contradicts that d is 2-strong. subcase 3.2. b = l + 1 = m. assume that a ≥ 2. as mentioned above, either x1 → xa+1 or xa+1 → x1. therefore, cm+3(z) = x1xa+1 . . . xm−1y1y2x2 . . . xaxmzx1 or cm+2(z) = x1 . . . xaxmzxa+3 . . . xm−1y1y2 xa+1x1. so, in both cases, we have a contradiction. assume next that a = 1. then from d−(z, {x2, x3, . . . , xm−1}) = 0 (by claims 2(i) and 2(iv)) and d−(z) ≥ 2 it follows that x1 → z. we know that z → {xa+3, . . . , xl}. using this, it is not difficult to see that if xi → xm with i ∈ [2, m − 2], then for i = 2, cm+2(z) = x1x2xmzx4 . . . xm−1y1y2x1, and for i ∈ [3, m − 2], cm+3(z) = x1x2 . . . xixmzxi+1 . . . xm−1y1 y2x1, a contradiction. we may, therefore, assume that d−(xm, {x2, x3, . . . , xm−2}) = 0. (8) now we consider the vertex x1. if xj → x1 with j ∈ [2, m − 2], then for j = 2, cm+2(z) = x1xmzx4 . . . xm−1y1y2x2x1, and for j ∈ [3, m − 2], cm+3(z) = x1xmzxj+1 . . . xm−1y1y2x2 . . . xjx1. thus, in both cases, we have a contradiction. we may, therefore, assume that d−(x1, {x2, x3, . . . , xm−2}) = 0. this together with (3), (8) and d−(z, {x2, x3, . . . , xm−1}) = 0 implies that a({x2, x3, . . . , xm−2} → y ∪ {z, x1, xm}) = ∅. this means that d − xm−1 is not strong, which contradicts that d is 2-strong. case 4. a = l − 2. taking into account case 2 and the digraph duality, we may assume that b ≤ l + 2. subcase 4.1. a = l − 2 and b = l + 2. then by claim 2, d(z, {xl−1, xl, xl+1}) = 0. this together with (5) implies that m − 1 ≤ d(z, {x1, x2, . . . , xa, xb, xb+1, . . . , xm}) ≤ a + m − b + 2 = m + l − 2 − l − 2 + 2 = m − 2, a contradiction. subcase 4.2. a = l − 2 and b = l + 1. then by claim 2, d(z, {xl−1, xl}) = 0. assume first that m ≥ l + 2. if there exist i ∈ [l − 1, l] and j ∈ [l + 2, m] such that xi → xj, then c(z) = x1x2 . . . xl−2xl+1 . . . xj−1r(y1, y2)xixj . . . xmzx1 is a cycle of length at least m + 2, a contradiction. we may, therefore, assume that a({xl−1, xl} → {xl+2, xl+3, . . . , xm}) = ∅. this together with (3), the minimality of a and the maximality of b implies that a({x1, x2, . . . , xl} → {xl+2, xl+3, . . . , xm}) = ∅. therefore, if d−(z, e) = 0, then a({x1, x2, . . . , xl}∪y → {z, xl+2, xl+3, . . . , xm}) = ∅, and if d−(z, e) ≥ 1, then d+(z, f) = 0 (claim 1(i)) and a({z, x1, x2, . . . , xl} ∪ y → {xl+2, xl+3, . . . , xm}) = ∅. thus, in both cases, we have that d − xl+1 is not strong, a contradiction. assume next that m = l + 1. then a = l − 2 = m − 3. let a ≥ 2. from the minimality of a it follows that d−(xm, {x1, x2, . . . , xa−1}) = 0. if there exist i ∈ [1, a − 1] and j ∈ [a + 1, a + 2] such that xi → xj, then it is easy to see that c(z) = x1x2 . . . xixj . . . xm−1r(y1, y2)xi+1 . . . xaxmzx1 is a cycle of length at least m+2, a contradiction. we may, therefore, assume that a({x1, x2, . . . , xa−1} → {xa+1, xa+2, xa+3 = xm}) = ∅. 28 on an extension of the ghouila-houri theorem from this we have: if d−(z, {x1, x2, . . . , xa−1) = 0, then a({x1, x2, . . . , xa−1} → y ∪ {z, xa+1, xa+2, xa+3}) = ∅, if d−(z, {x1, x2, . . . , xa−1) ≥ 1, then by claim 1(i), zxm /∈ a(d) and a({x1, x2, . . . , xa−1} ∪ {z} → y ∪ {xa+1, xa+2, xa+3}) = ∅. so, in both cases, we have that d − xa is not strong, which contradicts that d is 2-strong. let now a = 1. then m = 4 = b = l + 1 and d(z, {x2, x3}) = 0. this together with d(z, y ) = 0, d+(z) ≥ 2 and d−(z) ≥ 2 implies that x1 → z → x4, which contradicts claim 1(i). case 5. a = l − 1. taking into account cases 3 and 4, we may assume that b = l + 1. then d(z, {xl}) = 0, and from (3), the minimality of a and the maximality of b it follows that a({x1, x2, . . . , xl−1} → y ∪ {xl+2, xl+3, . . . , xm}) = a({x1, x2, . . . , xl−2} → y ∪ {xl+1, xl+2, . . . , xm}) = ∅. (9) it is not difficult see that: if xl → xj with j ∈ [l + 2, m], then c(z) = x1x2 . . . xl−1xl+1 . . . xj−1r(y1, y2)xlxj . . . xmzx1 is a cycle of length at least m + 3, if xi → xl with i ∈ [1, l − 2], then c(z) = x1x2 . . . xixlr(y1, y2)xi+1 . . . xl−1xl+1 . . . xmzx1 is a cycle of length at least m + 3. so, in both cases we have a contradiction. we may, therefore, assume that d+(xl, {xl+2xl+3, . . . , xm}) = d−(xl, {x1, . . . , xl−2}) = 0. then by (9), a({x1, x2, . . . , xl−2} → {xl, xl+1, . . . , xm}) = a({x1, x2, . . . , xl} → {xl+2, xl+3, . . . , xm}) = ∅. (10) assume that m ≥ l + 2. if d−(z, e) ≥ 1, then d+(z, f) = 0 (claim 1(i)). this together with (3), (10), d(z, {xl}) = 0 and d(z, y ) = 0 implies that a({z, x1, x2, . . . , xl} ∪ y → {xl+2, xl+3, . . . , xm}) = ∅, which in turn implies that d − xl+1 is not strong, a contradiction. we may, therefore, assume that d−(z, e) = 0. now it is not difficult to see that a({x1, x2, . . . , xl} ∪ y → {z, xl+2, xl+3, . . . , xm}) = ∅. this means that d − xl+1 is not strong, a contradiction. assume now that m = l + 1. by the digraph duality, we may assume that a = l − 1 = 1. hence, b = l + 1 = m = 3. then, since d+(z) ≥ 2 and d−(z) ≥ 2, x1 → z → xm, which contradicts claim 1(i). the discussion of case 5 is completed. lemma 5 is proved. now we are ready to prove the main result. for the convenience of the reader, we restate it here. theorem 9: let d be a 2-strong digraph of order n ≥ 8 and z be a fixed vertex in v(d). suppose that for any vertex x ∈ v(d) \ {z}, d(x) ≥ n, d(z) ≥ n − 4, and d contains a cycle of length n − 2 passing through z. then d is hamiltonian. proof. suppose, on the contrary, that d contains a cycle cn−2(z) := x1x2 . . . xn−2x1 but it is not hamiltonian. by theorem 3 (or by theorem 2), d(z) ≤ n − 2. let {y1, y2} = v(d) \ v(cn−2(z)). since z ∈ v(cn−2(z)), we have that d(yi) ≥ n. using lemma 1, it is easy to show that d contains no cn−1(z), d(y1) = d(y2) = n, d(y1, v(cn−2(z))) = s. darbinyan 29 d(y2, v(cn−2(z))) = n − 2 and y1 ↔ y2. if y1 or y2 is adjacent to every vertex xi, i ∈ [1, n − 2], then d contains a cycle c(z) of length at least n − 1, a contradiction. we may, therefore, assume that y1 and some vertex of cn−2(z) are not adjacent, say xn−2. then d(y1, {x1, x2, . . . , xn−3}) = n − 2. since y1 cannot be inserted into x1x2 . . . xn−3, using lemma 2, we obtain that xn−3 → y1 → x1. this together with y1 ↔ y2 implies that d(xn−2, {y1, y2}) = 0 (for otherwise, d contains a cycle of length at least n − 1 through z, which is a contradiction). therefore, d(y2, {x1, x2, . . . , xn−3}) = n − 2, and by lemma 2, xn−3 → y2 → x1. then cn−1 = x1x2 . . . xn−3y1y2x1 is a cycle of length n − 1. we know that cn−1 does not contain the vertex z. therefore, z = xn−2. thus, we have that the conditions of lemma 5 hold. therefore, d(z) ≤ n−5, which contradicts that d(z) ≥ n−4. the theorem is proved. in [15], overbeck-larisch proved the following sufficient condition for a digraph to be hamiltonian-connected. theorem 10: (overbeck-larisch [15]). let d be a 2-strong digraph of order n ≥ 3 such that, for each two non-adjacent distinct vertices x, y we have d(x) + d(y) ≥ 2n + 1. then for each two distinct vertices u, v with d+(u) + d−(v) ≥ n + 1 there is a hamiltonian (u, v)-path. let d be a digraph of order n ≥ 3 and let u and v be two distinct vertices in v(d). follows overbeck-larisch [15], we define a new digraph hd(u, v) as follows: v(hd(u, v)) = v(d−{u, v})∪{z} (z a new vertex) and a(hd(u, v)) = a(d−{u, v})∪{zy | y ∈ n+d−v(u)}∪ {yz | y ∈ n−d−u(v)}. now, using theorem 7, we will prove the following theorem, which is an analogue of the overbeck-larisch theorem. theorem 11: let d be a 3-strong digraph of order n + 1 ≥ 10 with minimum degree at least n + 2. if for two distinct vertices u, v, d+d(u) + d − d(v) ≥ n − 2 or d + d(u) + d − d(v) ≥ n − 4 with uv /∈ a(d), then there is a hamiltonian (u, v)-path in d. proof. let d be a 3-strong digraph of order n + 1 ≥ 10 and let u, v be two distinct vertices in v(d). suppose that d and u, v satisfy the degree conditions of the theorem. now we consider the digraph h := hd(u, v) of order n ≥ 9. by an easy computation, we obtain that the minimum degree of h is at least n − 4, and h has n − 1 vertices of degrees at least n. moreover, we know that h is 2-strong (see [10]). thus, the digraph h satisfies the conditions of theorem 7. therefore, h is hamiltonian, which in turn implies that in d there is a hamiltonian (u, v)-path. 5. conclusion for hamiltonicity of a graph g (undirected graph), there are numerous sufficient conditions in terms of the number k(g) of connectivity, where k(g) ≥ 3 (recall that for a graph g to be hamiltonian, k(g) ≥ 2 is a necessary condition) and the minimum degree δ(g) (or the sum of degrees of some vertices with certain properties), see the survey papers by gould, e.g. [16]. this is not the case for the general digraphs. in [17], the author proved that: for every pair of integers k ≥ 2 and n ≥ 4k + 1 (respectively, n = 4k + 1), there exists a k-strong (n−1)-regular (respectively, with minimum degree at least n−1 and with minimum semi-degrees at least 2k −1 = (n−3)/2) a non-hamiltonian digraph of order n. in [1] (page 30 on an extension of the ghouila-houri theorem 253), it was showed that there is no k such that every k-strong multipartite tournament with a cycle factor has hamiltonian cycle. based on the evidence from theorem 9, we raise the following conjecture, the truth of which in the case k = 0 follows from theorem 9. conjecture 2: let d be a 2-strong digraph of order n and z be a fixed vertex in v(d). suppose that for any vertex x ∈ v(d) \ {z}, d(x) ≥ n + k and d(z) ≥ n − k − 4, where k ≥ 0 is an integer. then d is hamiltonian. references [1] j. bang-jensen and g. gutin, digraphs: theory, algorithms and applications, springer, 2000. [2] j.-c. bermond and c. thomassen, “cycles in digraphs – a survey”, journal of graph theory, vol. 5, no. 1, pp. 1-43, 1981. [3] d. kühn and d. osthus, “a survey on hamilton cycles in directed graphs”, european journal of combinatorics, vol. 33, pp. 750-766, 2012. [4] a. ghouila-houri, “une condition suffisante d’existence d’un circuit hamiltonien”, comptes rendus de i’academie des sciences paris, ser. a-b 251, pp. 495-497, 1960. [5] m. meyniel, “une condition suffisante d’existence d’un circuit hamiltonien dans un graphe oriente”, journal of combinatorial theory, ser. b, vol. 14, pp. 137-147, 1973. [6] c.st.j.a. nash-williams, “hamilton circuits in graphs and digraphs”, the many facets of graph theory, springer verlag lecture notes 110, (springer verlag) pp. 237-243, 1969. [7] c. thomassen, “long cycles in digraphs”, proceedings of london mathematical society, vol. 42, no. 3, pp. 231-251, 1981. [8] s.kh. darbinyan, “cycles of any length in digraph with large semi-degrees”, aakdemy nauk armyan ssr doklady, (arxiv.1911.05998v1) vol. 75, no. 4, pp. 147-152, 1982 . [9] c. thomassen, “long cycles in digraphs with constraints on degrees, survey in combinatorics”, proc. 7th british combinatorial conf., london math. soc. lecture notes, cambridge university press, vol. 38, pp. 211-228, 1979. [10] s.kh. darbinyan, “hamiltonian and strongly hamilton-connected digraphs”, aakdemy nauk armyan ssr doklady, (arxiv.1801.05166v1), vol. 91, no. 1, pp. 3-8, 1990. [11] m.k. goldberg, l.p. levitskaya and l.m. satanovskiy, “on one strengthening of the ghouila-houri theorem”, vichislitelnaya matematika i vichislitelnaya teknika, vol. 2, pp. 56-61, 1971. [12] s.kh. darbinyan, “a sufficient condition for a digraph to be hamiltonian”, aakdemy nauk armyan ssr doklady, vol. 91, no. 2, pp. 57-59, 1990. [13] r. häggkvist and c. thomassen, “on pancyclic digraphs”, journal of combinatorial theory, ser. b, vol. 20, no. 1, pp. 20-40, 1976. [14] j.a. bondy and c. thomassen, “a short proof of meyniel’s theorem”, discrete mathematics, vol. 19, pp. 195-197, 1977. [15] m. overbeck-larisch, “hamiltonian pats in oriented graphs”, journal of combinatorial theory, ser. b, vol. 21, pp. 76-80, 1976. s. darbinyan 3 1 [1 6 ] r .j. go u ld , \ r e s e n t a d va n c e s o n t h e h a m ilt o n ia n p r o b le m : s u r ve y iii" , graphs and combinatorics, vo l. 3 0 , p p . 1 -4 6 , 2 0 1 4 . [1 7 ] s .k h . d a r b in ya n , \ d is p r o o f o f a c o n je c t u r e o f th o m a s s e n " , aakdemy nauk armyan ssr d oklady, vo l. 7 6 , n o . 2 , p p . 5 1 -5 4 , 1 9 8 3 . ¶áõñçé³-ðáõñçç ã»áñ»ùç ùç áý¹é³ûýù³ý ù³ëçý ê³ùí»é ê. ¸³ñµçýû³ý ðð ¶²² æýýáñù³ïçï³ûç ¨ ³íïáù³ï³óù³ý åñáµé»ùý»ñç çýëïçïáõï e-mail: samdarbin@iiap.sci.am ²ù÷á÷áõù ü»ñï³ ³ßë³ï³ýùáõù ³å³óáõóí»é ¿ ñ»ï¨û³é ã»áñ»ùá: â»áñ»ù: ¸çóáõù d-ý 2-áõå»õ ï³å³ïóí³í n-·³·³ã³ýç (n ¸ 8 ) ïáõùýáñáßí³í ·ñ³ý ¿, áñç n ¡ 1 ·³·³ãý»ñç ³ëïç׳ýý»ñá ÷áùñ ã»ý n ãíçó, çëï z ·³·³ãç ³ëïç׳ýá ÷áùñ ã¿ n ¡ 4 ãíçó: ºã» d-ý ý å³ñáõý³ïáõù ¿ n ¡ 2 »ñï³ñáõãû³ùµ óçïé, áñá ³ýóýáõù ¿ ù³ïñçó, ïûáåéçóû³ý ù³ïñçó: îá îäíîì ðàñøèðåíèè òåîðåìû ãóéÿ-óðè ñàìâåë õ. äàðáèíÿí èíñòèòóò ïðîáëåì èíôîðìàòèêè è àâòîìàòèçàöèè íàí ðà e-mail: samdarbin@iiap.sci.am àííîòàöèÿ â íàñòîÿùåé ðàáîòå äîêàçàíà ñëåäóþùàÿ òåîðåìà. òåîðåìà. ïóñòü d åñòü 2-ñèëüíî ñâÿçíûé n ¸ 8 âåðøèííûé îðãðàô, â êîòîðîì n ¡ 1 âåðøèí èìåþò ñòåïåíü íå ìåíüøå ÷åì n , à âåðøèíà z èìååò ñòåïåíü íå ìåíüøå ÷åì n ¡ 4 . åñëè d ñîäåðæèò êîíòóð äëèíû n ¡ 2 , êîòîðèé ñîäåðæèò âåðøèíó z, òî d ñîäåðæèò ãàìèëüòîíîâ êîíòóð. êëþ÷åâûå ñëîâà: îðãðàô, ãàìèëüòîíîâ êîíòóð, 2-ñèëüíî, ãàìèëüòîíîâîñâÿçíûèé. z ·³·³ãáí, ³å³ d å³ñáõý³ïáõù ¿ ñ³ùçéïáýû³ý óçïé: ´³ý³éç µ³é»ñ` ñ³ï³¹³ñó ù³ïñçó, »ñ»ù³ýïûáõý³·í³ûçý ù³ïñçó, ñ»ñùçïû³ý 02_darbinyan_20_31 02 d:\sbornik\...\new.dvi mathematical problems of computer science 29, 2007, 104{106. on simultaneous 2-locally-balanced 2-par tition for t wo for ests with same ver tices h o vik g. ta n a n ya n y a n d r a fa ye l r . k a m a lia n z y russian-armenian state university e-mail: htananyan@yahoo.com, zyerevan state university e-mail: rrkamalian@yahoo.com abstract the existence of a partition of the common set of the vertices of two forests into two subsets, when di®erence of their capacities in the neighbourhood of each vertex of each forest is not greater than 2 is proved, and an example, which shows that improvement of the speci¯ed constant is impossible is brought. refer ences [1 ] s .v . b a likya n , r .r . k a m a lia n , on n p -c o m p le t e n e s s o f t h e p r o b le m o f e xis t e n c e o f l o c a lly-b a la n c e d 2 -p a r t it io n fo r b ip a r t it e gr a p h s g wit h ¢ ( g ) = 3 , r eports of nas r a, applied m athematics , v. 1 0 5 , n 1 , 2 0 0 5 , p p . 2 1 -2 7 . ( in r u s s ia n .) [2 ] s . v . b a likya n , r . r . k a m a lia n , on n p -c o m p le t e n e s s o f t h e p r o b le m o f e xis t e n c e o f l o c a lly-b a la n c e d 2 -p a r t it io n fo r b ip a r t it e gr a p h s g wit h ¢ ( g ) = 4 u n d e r t h e e xt e n d e d d e ¯ n it io n o f t h e n e ig h b o u r h o o d o f a v e r t e x, r eports of nas r a, applied m athematics , v. 1 0 6 , n 3 , 2 0 0 6 , p p . 2 1 8 -2 2 6 . ( in r u s s ia n .) [3 ] f. h a r a r y, gr a p h th e o r y, a d d is o n -w e s le y, r e a d in g , ma , 1 9 6 9 . [4 ] d . d e w e r r a , b a la n c e d s c h e d u le s , inf or j . , 9 ( 3 ) , 1 9 7 1 , p p . 2 3 0 -2 3 7 . ¶³·³ãý»ñç ñ³ùáýïýáõ µ³½ùáõãûáõýý»ñáí »ñïáõ ³ýï³éý»ñç ùç³å³ù³ý³ïû³ 2-éáï³é-ñ³í³ë³ñ³ïßéí³í 2-ïñáñù³ý ù³ëçý ð. ¶. â³ý³ýû³ý ¨ è. è. ø³ù³éû³ý ²ù÷á÷áõù ²å³óáõóí³í ¿, áñ ·³·³ãý»ñç ñ³ùáýïýáõ µ³½ùáõãûáõýý»ñ áõý»óáõ »ñïáõ ³ýï³éý»ñç ñ³ù³ñ ·áûáõãûáõý áõýç ýñ³ýó ·³·³ãý»ñç µ³½ùáõãû³ý ³ûýåçëç ïñáñáõù »ñïáõ »ýã³µ³½ùáõãûáõýý»ñç, áñç ¹»åùáõù ûáõñ³ù³ýãûáõñ ³ýï³éç ûáõñ³ù³ýãûáõñ ·³·³ãç ßñç³ï³ûùáõù ³û¹ »ñïáõ »ýã³µ³½ùáõãûáõýý»ñç ï³ññ»ñç ù³ý³ïý»ñç ï³ñµ»ñáõãûáõýá ãç ·»ñ³½³ýóáõù 2-á, ¨ ³û¹ ñ³ëï³ïáõýá ÷áùñ³óý»é ñý³ñ³íáñ ã¿: 1 0 4 91 mathematical problems of computer science 58, 91–98, 2022. doi: 10.51408/1963-0096 udc 004.7 network management automation through virtualization arusyak d. manasyan institute for informatics and automation problems of nas ra e-mail:armanasyan@iiap.sci.am abstract the study aims to develop methods for automating network management by analyzing its virtual counterpart. the paper substantiates the relevance of this approach, identifies the advantages and disadvantages, highlights the existing problems, and suggests ways to solve them. as a result, the effectiveness of network virtualization was shown by the example of an experimental network. keywords: network, automation, virtualization, sdn (software-defined network), opendaylight (software), openflow (protocol). article info: received 10 december 2021; received in revised form 8 july 2022; accepted 23 august 2022. 1. introduction the more devices are connected to the network, the more inconvenience there will be with the expenses of their utilization. and until the network system is automated, this problem will be constant. organizations will spend a lot of money to buy powerful network devices, but network management will not become easier. that is why a study of network automation and virtualization was carried out, their current applications were discussed and solutions to existing problems were proposed. as network traffic continues to grow, companies increasingly require large-scale network configurations. the move to cloud computing continues as enterprise customers and their applications rely more and more on network efficiency, so networks are expected to be highly reliable with minimal downtime. as the number of devices on the network increases, so does the need for uninterrupted, flexible, fast, and efficient communication between them. to do this, it is necessary to obtain a large number of network devices that will be of a high quality, and have great features, such as a large amount of memory, many interfaces, and powerful processors, and all this is associated with high costs, which is one of the main prerequisites for the emergence automation and virtualization concepts. for service providers, automation is a key strategy to improve network agility and reliability while controlling operating and capital costs. therefore, it is necessary to automate the work network management automation through virtualization 92 with network equipment. automation of daily network tasks and functions, as well as automated monitoring of iterative processes, increases the availability of network services. we can describe the current state of the networking industry as "critical". the marketdominant closed (proprietary) solutions are "boxes" for applications, and the interoperability of solutions from different vendors is best provided at the interface level. networks are extremely complex, making them difficult to scale, manage, and trust. this slows down the further development of networks and programs running in them. therefore, several solutions for network automation have been developed, and we talked about sdn in our research work. software defined networking (sdn) introduces network virtualization capabilities, which makes it easier to build and manage network automation tasks. using sdn, networks can be provisioned at the software layer, abstracting the underlying physical hardware. this takes automation to the next level and significantly accelerates network provisio ning and configuration management. it also enables it to attach network and security services to workloads using a policy-driven approach(see [1]). today, network automation solutions allow us to perform a wide range of tasks, including network planning design, including scenario planning backup management, device testing configuration testing, deployment of deployed physical devices services, as well as virtual device deployment provisioning devices, real-time network data collection systems related to applications, network topology, traffic, services, data analysis, including active artificial intelligence, machine learning analysis, to get an idea of the present and future, network behavior, check configuration compliance, to ensure all network devices and service requirements, software updates, including backing up software if necessary, fixing closed network issues, including troubleshooting, and complex, difficult-to-detect troubleshooting activities, detailed analysis of reports, panels, alarms, warnings, compliance with security requirements, monitoring of the network and its services, service level to maintain customer satisfaction. the purpose of this article is to show the benefits of network virtualization, present the tools necessary for this, and show its effectiveness as a result of the experimental application. 2․analyses and discussion network automation through sdn (see [2]) adds a number of capabilities to conventional automation paradigms, which optimize it resources and require sdn as a networking architecture approach. it enables the control and management of networks using software applications. through sdn networking, the behavior of the entire network and its devices is programmed in a centrally controlled manner through software applications using open apis. sdn improves performance through network virtualization. in sdn[2] software-controlled applications or apis work as a basis of complete network management that may be directing traffic on a network or communicating with underlying hardware infrastructure. so to put it simply, we can say that sdn can create virtual networks or control traditional networks with the help of software to improve security and reduce cost. traditional network refers to the old conventional way of networking, which uses fixed and dedicated hardware devices such as routers and switches to control network traffic. inability to scale, as well as network security and performance are major concerns nowadays in the current growing business situation so sdn is taking control of traditional networks. the traditional network is static and based on hardware network appliances. traditional network architecture was used by many companies until recent years but nowadays due to its drawbacks sdn has been developed and will be used more widely in the coming years(see [3]). a. manasyan 93 table 1. comparison of sdn to traditional network(see [3]). no sdn traditional network 1 virtual networking approach. old conventional networking approach. 2 centralized control. distributed control. 3 programmable network. this network is nonprogrammable. 4 open interface. closed interface. 5 data plane and control plane are decoupled by software. data plane and control plane are mounted on the same plane. 6 it supports automatic configuration so it takes less time. it supports static/manual configuration so it takes more time. 7 it can prioritize and block specific network packets. it leads all packets in the same way with no prioritization support. 8 it is easy to program as per need. it is difficult to program again and replace the existing program as peruse. 9 the cost is low. the cost is high. 10 structural complexity is low. structural complexity is high. 11 extensibility is high. extensibility is low. 12 it is easy to troubleshoot and report as it is centralized and controlled. it is difficult to troubleshoot and report as it is distributed and controlled. 13 its maintenance cost is lower than the traditional network. cost is higher than sdn. as the sdn technology (see [4]) is based on an intelligent controller, it allows you to automatically redistribute traffic. it turned out that the device allows you to centrally change the settings of network equipment in branches, monitor the network status, load and quality of channels online, and solve problems. this ensures the transparency of data transmission networks and reduces the burden on it professionals serving the network. the study also showed that the sdn solution involves the automatic networking of private networks and the transmission of information through all available channels without losing the speed and quality of applications. for example, in the past, only expensive vpn channels were used to transmit audio or video without distortion. now, thanks to sdn, we can only use the internet and lte as a backup(see [5]). in this way, customers can save on telecommunication bill payments and solve vpn reservation issues simply and cheaply. unlike other virtualization technologies, the open-source sdn solution is more promising. sdn[2] already provides companies with many options to choose from openflow, netconf, ovsdb, switches that support the api library, as well as enterprise software that utilizes these protocols. like any other infrastructure, the sdn infrastructure is built on open standards. this open ecosystem accelerates network innovation. although the traditional approach to building a network infrastructure still prevails due to the negative impact of mental inertia and crisis events, sdn already allows you to effectively solve problems in a virtual physical environment. by automating the network, we get the following benefits and services: reduced problems, reduced costs, increased network flexibility, reduced network outages, increased number of strategic employees, advanced analysis, and network management capabilities. 3. methods and applications the article methodology includes the study of epistemological issues, programs (opendaylight), protocols (openflow) in the field of networks, using scientific literature, and research articles. network management automation through virtualization 94 the research aim is to present an example of an automated network as a result of the analysis based on the studied materials. below is the physical experimental network represented by the gns3 simulator, which is fully operational, we will get the virtualized version of the following network, but the initial settings must be done one way or another. this article provides a brief overview of virtual networks and network performance evidence. the physical network shown below is represented by a fully running gns3 simulator. it contains hosts, routers (mikrotik), and a virtual switch openvswitch. fig . 1 ․ network presented with gns3 simulator. here are the settings of one of the devices, almost the same as the rest: /routing ospf instance set [ find default=yes ] router-id=10.255.255.1 /ip address add address=10.0.4.1/24 interface=ether4 network=10.0.4.0 add address=192.168.10.1/24 interface=ether3 network=192.168.10.0 /routing ospf network add area=backbone network=10.0.4.0/24 add area=backbone network=192.168.10.0/24 here are the minimum settings that make the network complete. for network virtualization, as mentioned at the beginning, we implemented an sdn solution. we have demonstrated the use of sdn with the opendaylight software, which is a software platform for sdn. to work with our controller, to connect it to our physical network, we downloaded and activated the following components: opendaylight-user@root>feature:install odl-restconf odl-l2switch-all odl-mdsal-apidocs odl-dlux-all odl-openflowplugin-all a. manasyan 95 they provide a graphical user interface of opendaylight software, as well as the necessary tools and devices. after activating them, immediately after setting the appropriate settings in our physical openvswitch network, we see a virtualized version of our network. to establish a "controller" connection in our physical network, we have previously configured the openvswitch openflow device by giving it the ip address of the controller by typing the following command: ovs-vsctl set-controller br0 tcp: 192.168.18.129:6633, where 192.168.18.129 is the ip address of the controller and it can be different for different devices, 6633 is the connection port and the protocol that controls data transfer over tcp. thanks to this, it was able to communicate with other devices. fig․ 2 ․ example of a virtual network in opendaylight. fig. 2 shows a virtualized version of the physical network in opendaylight. the picture clearly shows all the devices in our network that are connected to the openflow protocol support device, openvswitch. it is thanks to the openflow protocol that our sdn controller sees our entire physical network. openflow is a protocol for managing data processing, which is transmitted over the network through routers and switches using sdn technology. fast packet forwarding (data forwarding) on a classic router or switch and high-level routing decisions (control operations) are made on the same device. the openflow switch separates these two functions. data redirection is performed by the switch itself, while routing decisions are entrusted to a separate controller, usually a standard server. after clicking on the network topology, yang automatically shows us the confrest api url it uses to get this information: fig․ 3․ confrest api url network management automation through virtualization 96 by clicking the send button(fig.3), we can see the topology of our operational network. fig. 4․ operating network topology. in fig. 4, we can see information about our current topology, including the mac (media access control) and ip addresses of our hosts. so, you do not need to enter the device to see them every time, but you can see them from one control panel of sdn. when we send network traffic, all the information about it is mentioned in the flow tables of the sdn: how many packages were sent to us, how many arrived, how many dropped on the way, and what errors we encountered. and all that information we can see in the nodes of fig.5. fig.5. node connector statistics. a. manasyan 97 5. conclusion this paper proposes a solution for network optimization. as a result of the research, we concluded that automation improves the speed of it operations in response to analytical change. the ability to monitor operations, just as needed, provides greater visual control of the network, and transparency of processes within it. network automation improves work efficiency, reduces human error, increases access to network services, and provides better customer service. research has shown that the sdn solution includes the automatic integration of private networks, and the transmission of information over all available channels, without loss of application speed and quality. as a result of the study, it became clear that network automation can be implemented regardless of its type, which facilitates its transition. network virtualization is a more all-encompassing version of virtualization that makes it possible to convert physical network hardware into software that can easily be transitioned to different domains as needed, increasing flexibility and scalability for the network. i came to the conclusion that its use on the network will be of great benefit to network administrators. references [1] how can network automation be improved with sdn. [online]. available: https://www.acadiatech.com/blog/how-can-network-automation-improve-with-sdn/ [1] w. braun and m. menth, "software-defined networking using openflow: protocols applications and architectural design choices", future internet, vol. 6, no. 2, pp. 302-336, 2014, [online]. available: http://www.mdpi.com/1999-5903/6/2/302 [2] s. jena. difference between software defined network and traditional network. [online]. available: https://www.geeksforgeeks.org/difference-between-softwaredefined-network-and-traditional-network/ [3] j. doherty, sdn and nfv simplified: a visual guide to understanding software defined networks and network function virtualization, addison-wesley professional, march 2016. [4] software-defined sd-wan technologies. [online]. available: https://tadviser.com/index.php/article:sd-wan_%28software_defined%29_softwaredefined_wan https://www.tadviser.ru/images/thumb/9/96/idc_sd-wan_market_share_snapshot_blog_ii-1024x876.jpg/840px-idc_sd-wan_market_share_snapshot_blog_ii-1024x876.jpg https://www.tadviser.ru/images/thumb/9/96/idc_sd-wan_market_share_snapshot_blog_ii-1024x876.jpg/840px-idc_sd-wan_market_share_snapshot_blog_ii-1024x876.jpg https://www.acadiatech.com/blog/how-can-network-automation-improve-with-sdn/ http://www.mdpi.com/1999-5903/6/2/302 https://www.geeksforgeeks.org/difference-between-software-defined-network-and-traditional-network/ https://www.geeksforgeeks.org/difference-between-software-defined-network-and-traditional-network/ https://learning.oreilly.com/search/?query=author%3a%22jim%20doherty%22&sort=relevance&highlight=true https://learning.oreilly.com/library/view/sdn-and-nfv/9780134307398/ https://learning.oreilly.com/library/view/sdn-and-nfv/9780134307398/ https://learning.oreilly.com/library/publisher/uuid/1495afeb-554e-4ba9-84fb-f260848d173e https://tadviser.com/index.php/article:sd-wan_%28software_defined%29_software-defined_wan https://tadviser.com/index.php/article:sd-wan_%28software_defined%29_software-defined_wan https://habr.com/ru/company/hpe/blog/255363/ https://habr.com/ru/company/hpe/blog/255363/ network management automation through virtualization 98 ցանցի ղեկավարման ավտոմատացում վիրտուալացման միջոցով արուսյակ դ․ մանասյան հհ գաա ինֆորմատիկայի և ավտոմատացման պրոբլեմների ինստիտուտ e-mail:armanasyan@iiap.sci.am ամփոփում հետազոտության նպատակն էր մշակել ցանցի ղեկավարման ավտոմատացման մեթոդներ՝ վերլուծելով դրանց վիրտուալ անալոգը: աշխատանքում հիմնավորվում է այս մոտեցման արդիականությունը, վեր են հանվում առավելություններն ու թերությունները, ընդգծվում են առկա խնդիրները և առաջարկվում են դրանց լուծման ուղիներ։ արդյունքում ներկայացվել է ցանցերի վիրտուալացման արդյունավետությունը՝ փորձնական ցանցի օրինակով: բանալի բառեր՝ ցանց, ավտոմատացում, վիրտուալացում, sdn (ծրագրակողմնորոշված ցանց), opendaylight (ծրագրային ապահովում), openflow (արձանագրություն)։ автоматизация управления сетью за счет виртуализации арусяк д. манасян институт проблем информатики и автоматизации нан ра e-mail:armanasyan@iiap.sci.am аннотация цель исследования заключалась в разработке методов автоматизации управления сетью путем анализа ее виртуального аналога. в работе обосновывается актуальность такого подхода, выявляются преимущества и недостатки, подчеркиваются существующие проблемы и предлагаются пути их решения. в результате была показана эффективность виртуализации сети на примере пилотной сети. ключевые слова: сеть, автоматизация, виртуализация, sdn (программноопределяемая сеть), opendaylight (программное обеспечение), openflow (протокол). 1. introduction 5. conclusion ամփոփում formalizmodelej.dvi ìàòåìàòè÷åñêèå âîïðîñû êèáåðíåòèêè è âû÷èñëèòåëüíîé òåõíèêè 25, 2006, 80–84. ê ïðèìåíåíèþ òåîðèè ãðàôîâ ïðè èññëåäîâàíèè òåëåêîììóíèêàöèîííûõ ñåòåé. êàðåí à. ìêðò÷ÿí èíñòèòóò ïðîáëåì èíôîðìàòèêè è àâòîìàòèçàöèè íàí ðà e-mails kamkrtchyan@ipia.sci.am, kamkrtchyan@yahoo.com àííîòàöèÿ ñ öåëüþ îïèñàíèÿ è ïîñëåäóþùåãî èññëåäîâàíèÿ ìåòîäàìè òåîðèè ãðàôîâ ìîäåëåé òåëåôîííûõ è äðóãèõ òåëåêîììóíèêàöèîííûõ ñåòåé ñèñòåìàòèçèðóþòñÿ, óòî÷íÿþòñÿ ñîîòâåòñòâóþùèå ïîíÿòèÿ. ðàññìàòðèâàþòñÿ ïðàêòè÷åñêèå ïðèìåðû. refer ences [1 ] mkr t c h ya n k , a p p lic a t io n o f s t a t is t ic a l a p p r o a c h in d e t e c t in g fr a u d c a s e s in t e le p h o n e n e t wo r ks , ma t h e m a t ic a l p r o b le m s o f co m p u t e r s c ie n c e , 2 4 , p p .1 2 5 { 1 3 2 , 2 0 0 5 . [2 ] mkr t c h ya n k , a p p lic a t io n o f s t a t is t ic a l a p p r o a c h ¯ g h t in g fr a u d c a s e s in t e le p h o n e n e t wo r ks , p r o c e e d in g s o f cs it in y e r e va n , s e p t e m b e r 2 2 , 2 0 0 5 . [3] ñêâîðöîâà ì. ìàòåìàòè÷åñêîå ìîäåëèðîâàíèå. 2003. http://archive.1september.ru/mat/2003/14/no14 1.htm [4] àðóòþíÿí å. è äðóãèå. âåðîÿòíîñòü è ïðèêëàäíàÿ ñòàòèñòèêà. (íà àðìÿíñêîì ÿçûêå), èçäàòåëüñòâî ”ãèòóòþí” íàí ðà, åðåâàí 2000. [5] íîñîâ â. êîìáèíàòîðèêà è òåîðèÿ ãðàôîâ. ìîñêâà 1999. [6] îðå î. òåîðèÿ ãðàôîâ. ìîñêâà, íàóêà, 1968. [7] õàððàðè ô. òåîðèÿ ãðàôîâ. ìîñêâà, ìèð, 1973. [8] ×åðíîáàåâ à. ñèñòåìàòè÷åñêèé óêàçàòåëü òåðìèíîâ è îïðåäåëåíèé òåîðèè ãðàôîâ. v.990916. 2004. http://www.carawan.ru/ alex/graphs/graphs terms and definitions.htm [9] êðèñòîôèäåñ í. òåîðèÿ ãðàôîâ. àëãîðèòìè÷åñêèé ïîäõîä. ì., ìèð, 1978. [10] íå÷åïóðåíêî ì.è. ïîïêîâ â.ê. ìàéíàãàøåâ ñ.ì. è äð. àëãîðèòìû è ïðîãðàììû ðåøåíèÿ çàäà÷ íà ãðàôàõ è ñåòÿõ. íîâîñèáèðñê, íàóêà (ñèáèðñêîå îòäåëåíèå), 1990. [11] ôîðä ë.ð. ôàëêåðñîí ä.ð. ïîòîêè â ñåòÿõ. ìîñêâà, ìèð, 1966. [12] êëåéíðîê ë. êîììóíèêàöèîííûå ñåòè. ìîñêâà, íàóêà, 1970. 8 0 ê. à. ìêðò÷ÿí 8 1 [13] ìàòðèöû è îïðåäåëèòåëè. 2006. http://dit.vov.ru/md/md-1.htm [14] ralph p. grimaldi. discrete and combinatorial mathematics. addison–wesley publishing company, 1985. ð»é³ñ³õáñ¹³ïó³ï³ý ó³ýó»ñç ñ»ï³½áïù³ý ñ³ù³ñ ·ñ³ýý»ñç ï»ëáõãû³ý ïçñ³éáõãû³ý í»ñ³µ»ñû³é: î. øïñïãû³ý ²ù÷á÷áõù ¶ñ³ýý»ñç ï»ëáõãû³ý ù»ãá¹ý»ñáí ñ»é³ëáë³ûçý ¨ ³ûé ñ»é³ñ³õáñ¹³ïó³ï³ý ó³ýó»ñç ùá¹»éý»ñç ýï³ñ³·ñù³ý ¨ ñ»ï³·³ ñ»ï³½áïù³ý ýå³ï³ïáí ñ³ù³ï³ñ·íáõù, ×ß·ñïíáõù ¨ ñ³ù³éñíáõù »ý ñ³ù³å³ï³ëë³ý ·³õ³÷³ñý»ñá: ¸çï³ñïíáõù »ý ïçñ³é³ï³ý ûñçý³ïý»ñ: mathematical problems of computer science 51, 98–106, 2019. udc 519.6 on homeomorphism between euclidean subspace and conformally euclidean manifold ashot s. gevorkyan1,2, alek a. aleksanyan1 and suren b. alaverdyan1 1institute for informatics and automation problems of nas ra 2institute of chemical physics after a. b. nalbandyan of nas ra e-mail: gashot@ipia.sci.am abstract the article presents the proof of the homeomorphism between euclidean subspace e6 of the classical three-body system and 6d riemannian manifold m, which allows reducing the dynamical problem to the system of the 6th-order. keywords: system of underdetermined algebraic equations, orientated 3d riemannian manifold, topology of 3d manifolds. 1. introduction as is known, the time evolution of the classical system is uniquely determined by the hamilton equations and is usually reduced to a system of ordinary differential equations of the second order. integrating this system of a differential equation means finding all possible functions of one variable “t” (time), which, when substituted into equations, turns them into an identity. in the case of dynamical systems, as a rule, the system of equations cannot be fully integrated, since the number of integrals of motion often is less than the number of degrees of freedom. in the series of works [1]–[5], using the example of the classical three-body problem, it was shown that the use of riemannian geometry makes it possible to reveal new hidden symmetries of a dynamical system, which makes the integration of the problem more completel. in this paper we examine the question of homeomorphism between 6d euclidean subspace e6 and 6d manifold m. in particular, the question of the decomposition of a manifold in the form m :⇔ m(3) × s3mi is proved, where m(3) denotes the sum of 84 oriented in 9d euclidean space 3d manifolds and s3mi is the group symmetry so(3) at the given point mi ∈ m(3). 98 a. gevorkyan, a. aleksanyan and s. alaverdyan 99 2. on homeomorphism between the euclidean subspace and the conformally euclidean manifold proposition 1: let e6 be a euclidean subspace with metric γµν ({ρ}), on which an orthogonal coordinate system is given: ρ1, ..., ρ6 = {ρ} = ρ1, ρ6 ∈ e6, (1) and, respectively, m is a conformally euclidean manifold, which is determined by the metric tensor gαβ({x}) and the local coordinate system {x}: gαβ({x}) = g({x̄})δαβ, {x} = x1, ..., x6, {x̄} = x1, x3, α, β = 1, 6, (2) where g({x̄}) > 0 is a smooth function belonging to the class c1(r6), then the euclidean subspace e6 is homeomorphic to the manifold m. proof. let us consider a linear infinitesimal element ”ds” in both coordinate systems {ρ} ∈ e6 and {x} ∈ m. equating them, we can write: (ds)2 = γαβ({ρ})dραdρβ = gµν ({x̄})dxµdxν , α, β, µ, ν = 1, 6, (3) from which one can obtain the following system of algebraic equations: γαβ({ρ})ρα,µρβ,ν = gµν ({x̄}) = g({x̄})δµν , (4) where it is necessary to prove that the coefficients ρα,µ({x}) = ∂ρα/∂xµ make sense of derivatives. in this regard, we must prove that the function ρα({x}) is twice differentiable and continuous in its domain of definition and, in addition, satisfy the symmetry condition: ρα,µν ({x}) = ρα,νµ({x}), ∀ µ, ν = 1, 6, (5) (schwartz’s theorem on the symmetry of second derivatives). recall that the set of coefficients ρα,µ({x}) allows us to perform coordinate transformations {ρ} 7→ {x}, which we shall call direct transformations. γαβ({ρ})g−1({x̄}) = xµ, αxν, β δµν , (6) where xµ, α({ρ}) = ∂xµ/∂ρα and γαβ({ρ}) = γαᾱ({ρ})γββ̄({ρ})γᾱβ̄({ρ}). at first we consider the system of equations (4), which is related to direct coordinate transformations. it is easy to see that the system of algebraic equations (4) is underdetermined with respect to the variables ρα,µ({x}), since it consists of 21 equations, while the number of unknown variables is 36. obviously, when these equations are compatible, then the system of equations (4) has an infinite number of real and complex solutions. note that for the classical three-body problem, the real solutions of the system (4) are important, which form a 15-dimensional manifold. since the system of equations (6) is still defined in a rather arbitrary way we can impose additional conditions on it in order to find the minimal dimension of the manifold allowing a separation of the base m(3) from the layer ∪is3mi . similarly, from (3), one can obtain a system of algebraic equations defining inverse transformations: 100 on homeomorphism between euclidean subspace and conformally euclidean manifold let us make new notations: αµ = ρ1,µ, βµ = ρ2,µ, ζµ = ρ3,µ, uµ = ρ4,µ, vµ = ρ5,µ, wµ = ρ6,µ. (7) we also require that the following additional conditions be met: α4 = α5 = α6 = 0, β4 = β5 = β6 = 0, ζ4 = ζ5 = ζ6 = 0, u1 = u2 = u3 = 0, v1 = v2 = v3 = 0, w1 = w2 = w3 = 0. (8) using (7) and conditions (8) from the equation (4) we can obtain two independent systems of algebraic equations: α21 + β 2 1 + γ 33ζ21 = ğ({ρ̄}), α1α2 + β1β2 + γ33ζ1ζ2 = 0, α22 + β 2 2 + γ 33ζ22 = ğ({ρ̄}), α1α3 + β1β3 + γ33ζ1ζ3 = 0, α23 + β 2 3 + γ 33ζ23 = ğ({ρ̄}), α2α3 + β2β3 + γ33ζ2ζ3 = 0, (9) (recall that at obtaining (9) it is assumed that γ11 = γ22 = 1) and, correspondingly: γ44u24 + γ 55v24 + γ 66w24 + 2(γ 45u4v4 + γ 46u4w4 + γ 56v4w4) = ğ({ρ̄}), γ44u25 + γ 55v25 + γ 66w25 + 2(γ 45u5v5 + γ 46u5w5 + γ 56v5w5) = ğ({ρ̄}), γ44u26 + γ 55v26 + γ 66w26 + 2(γ 45u6v6 + γ 46u6w6 + γ 56v6w6) = ğ({ρ̄}), a4u4 + a5v4 + a6w4 = 0, b4u5 + b5v5 + b6w5 = 0, c4u6 + c5v6 + c6w6 = 0. (10) in equations (10), the following notations are made: ai = γ i4u5 + γ i5v5 + γ i6w5, bj = γ j4u6 + γ j5v6 + γ j6w6, ck = γ k4u4 + γ k5v4 + γ k6w4, where i, j, k = 4, 6. it should be noted that the solutions of algebraic systems (9) and (10) form two different 3d manifolds s(3) and r(3), respectively. the manifold s(3) is in a one-to-one mapping on the one hand with the subspace e3 3 {ρ̄} (where e3 ⊂ e6 the internal space in the hyperspherical coordinate system), and on the other hand with the manifold m(3) (see fig. 1). note that this statement follows from the fact that all points of the manifold m(3) and the subspace e3, are pairwise connected through the corresponding derivatives (see (4)), which, as unknown variables, enter the algebraic equations (9), and, in addition, as shown there exist also inverse coordinate transformations (see appendix). now we prove the continuity of these mappings. recall that the unknowns in the equations (9), are in fact functions of coordinates {ρ̄}. performing a shift of coordinates {ρ̄} → {ρ̄} + {δρ̄} in (9), we get the following system of equations: ᾱ21 + β̄ 2 1 + γ̄ 33ζ̄21 = ḡ({ρ̄}), ᾱ1ᾱ2 + β̄1β̄2 + γ̄33ζ̄1ζ̄2 = 0, ᾱ22 + β̄ 2 2 + γ̄ 33ζ̄22 = ḡ({ρ̄}), ᾱ1ᾱ3 + β̄1β̄3 + γ̄33ζ̄1ζ̄3 = 0, ᾱ23 + β̄ 2 3 + γ̄ 33ζ̄23 = ḡ({ρ̄}), ᾱ2ᾱ3 + β̄2β̄3 + γ̄33ζ̄2ζ̄3 = 0, (11) a. gevorkyan, a. aleksanyan and s. alaverdyan 101 fig. 1: in this diagram all spaces are homeomorphic to each other, i.e., e3 ' s(3) ' m(3). where ḡ({ρ̄}) = ğ ( {ρ̄} + {δρ̄} ) , {δρ̄} = (δρ1, δρ2, δρ3). assuming that |δ{ρ̄}| ¿ 1, in the equations (11), we can expand the functions in a taylor series on these small parameters and taking into account the system of equations (9), we get: δρi { 2(α1α1 i + β1β1 i + γ 33ζ1ζ1 i) + γ 33 , i ζ 2 1 − ğ, i({ρ̄}) } = o(|δ{ρ̄}|2), δρi { 2(α2α2 i + β2β2 i + γ 33ζ2ζ2 i) + γ 33 , i ζ 2 2 − ğ, i({ρ̄}) } = o(|δ{ρ̄}|2), δρi { 2(α3α3 i + β3β3 i + γ 33ζ3ζ3,i) + γ 33 , i ζ 2 3 − ğ, i({ρ̄}) } = o(|δ{x̄}|2), δρi { α1α2 i + α2α1 i + β1β2 i + β2β1 i + γ 33(ζ1ζ2 i + ζ2ζ1 i) + γ 33 , i ζ1ζ2 } = o(|δ{ρ̄}|2), δρi { α1α3 i + α3α1 i + β1β3 i + β3β1 i + γ 33(ζ1ζ3 i + ζ3ζ1 i) + γ 33 , i ζ1ζ3 } = o(|δ{ρ̄}|2), δρi { α2α3, i + α3α2 i + β2β3 i + β3β2 i + γ 33(ζ2ζ3 i + ζ3ζ2 i) + γ 33 , i ζ2ζ3 } = o(|δ{ρ̄}|2), (12) where i = 1, 3 and summation is performed by dummy indices. if we require that the expressions with the same increments be equal to zero, then from (12) one can obtain an underdetermined system of algebraic equations, i.e., 18 equations for finding 27 unknowns variables: 2(α1α1 i + β1β1 i + γ 33ζ1ζ1 i) + γ 33 , i ζ 2 1 − ğ, i({ρ̄}) = 0, 2(α2α2 i + β2β2 i + γ 33ζ2ζ2 i) + γ 33 , i ζ 2 2 − ğ, i({ρ̄}) = 0, 2(α3α3 i + β3β3 i + γ 33ζ3ζ3 i) + γ 33 , i ζ 2 3 − ğ, i({ρ̄}) = 0, α2α1 i + α1α2 i + β2β1 i + β1β2 i + γ 33(ζ2ζ1 i + ζ1ζ2 i) + γ 33 , i ζ1ζ2 = 0, α3α1 i + α1α3 i + β3β1 i + β1β3 i + γ 33(ζ3ζ1 i + ζ1ζ3 i) + γ 33 , i ζ1ζ3 = 0, α3α2 i + α2α3 i + β3β2 i + β2β3 i + γ 33(ζ3ζ2 i + ζ2ζ3 i) + γ 33 , i ζ2ζ3 = 0. (13) recall that the set of coefficients {σ} = (σ1, ..., σ9) = [α = (α1, α2, α3), β = (β1, β2, β3), ζ = (ζ1, ζ2, ζ3)] belongs to the manifold s (3). now, we can require that the second derivatives be symmetric σij = σji, where {σ} = [α, β, ζ] and i, j = 1, 3. this, as can be easily seen, allows us to reduce the number of unknown variables and make the system of equations definite, i.e., 18 equations for 18 unknown variables. the system of equations (13) can be written in canonical form: ax = b, a = (dµν ), µ, ν = 1, 18, (14) where a ∈ r18×18 is the basic matrix of the system, b ∈ r18 and x ∈ r18 are columns of free terms and system solutions, respectively. note that, for an arbitrary point {ρ̄i} ∈ e3, 102 on homeomorphism between euclidean subspace and conformally euclidean manifold fig. 2: the form of an oriented manifold generated by a system of equations (9). note that the calculations of the equations system (9) were performed taking into account the following transformations γ33ζ1 → ζ1, γ33ζ2 → ζ2 and γ33ζ3 → ζ3. the first figure shows a general view of a manifold in three-dimensional space, which obviously is a sphere with topological features. the second figure shows the projection of a sphere onto a plane (α2, α3) in the form of a circle, from which one can see a cutting circle in the center. there are obviously six such circular cuts on a sphere. fig. 3: as can be seen, this manifold also has a topology. fig. 4: as can be seen, this manifold also has a topology. a. gevorkyan, a. aleksanyan and s. alaverdyan 103 fig. 5: as can be seen, this manifold also has a topology. the system of equations (9) generates sets of solutions {σ} = [α, β, ζ] that continuously fill a region of e3 space, forming 3d manifold s(3). as for the system of equations (14), it has a solution if the determinant of the basic matrix a is nonzero (see appendix): det(dµν ) 6= 0, µ, ν = 1, 18. on the other hand, the algebraic system (14) does not have a solution when det(dµν ) = 0. in this case at each point {ρ̄i} there exists a countable set w of coefficients {σ} = [α, β, ζ] such that det(dµν ) = 0. it is easy to verify that the measure of this set in comparison with the measure of the s(3) for which det(dµν ) 6= 0, is equal to zero, i.e., w = {0}. in other words, for the case under consideration schwartz’s theorem holds, and σς (where ς = 1, 9) and dµν (see (13)) have the sense of the first and second derivatives, respectively. the same is easily proved for inverse mappings. let us consider the open set ∀ g = ∪αgα, consisting of the union of cards gα arising at continuous mappings f : {ρ̄} 7→ {x̄} using algebraic equations (9). proceeding from the foregoing, it is obvious that the maps can be chosen so that the immediate neighbors have intersections comprising at least one common point, that is a necessary condition for the continuity of the mappings. using the above arguments, we assert that the atlas g can be widened up to g ∼= m(3). now let us discuss the structure of the manifold m(3). it is easy to see that the independent {σ} parameters form 9d space r9, in which the system of algebraic equations (9) generates 3d oriented manifolds. these manifolds can be summed up as sets using a certain order by gluing manifolds having common planes. as a result of this gluing, which similar to the the operation of connected sum of topological manifolds, the 3d manifold m(3) = ∪im(3)i , is formed. the number of submanifolds m (3) i can easily be calculated by the formula cmn = n! m!(n−m)! , where n and m denote the dimension of space r 9 and the dimension of the manifold m(3)i immersed into r9, respectively. as the calculations show (see fig. 2-5), the generated c39 = 84 topological manifolds can be grouped into four incongruent groups of manifolds. it is also necessary to note that all these varieties are oriented in a 9-dimensional space in the sense that they are well-defined 3d submanifolds. thus, all the conditions of the theorem of a homeomorphism between metric spaces e3 and m(3) are satisfied, and therefore we can say that these spaces are homeomorphic or 104 on homeomorphism between euclidean subspace and conformally euclidean manifold topologically equivalent, i.e., f : e3 7→ m(3). as for the system of algebraic equations (10), then at each point of the internal space mi(x 1, x2, x3)i ∈ m(3), it generates 3d manifold r(3) that is a local analogue of the euler angles and, consequently, ∪is3mi ' r(3). the layer r(3), continuously passing through all points of the basis m(3), fills the subspace e6. finally, taking into account the aforesaid, we can conclude that the spaces e6 and m, are homeomorphic too. in addition, the manifold m can be represented in the form of decomposition m ∼= m(3) × s3mi . proposition 1 is proved. 3. conclusion as a. poincaré rightly pointed out, there is no finest geometry, there is a geometry convenient for solving a specific task. usually, when studying complex dynamical systems, coordinate transformations are used to separate variables and reduce the original system. in particular, by coordinate transformations, the three-body problem, which is a system of 18th order, is reduced to the system of 8th order. however, as we have shown, it is possible to make the reduction of a dynamical system more complete if we use the curve (riemannian) geometry. note that in this case it becomes possible to reveal the hidden symmetries of internal motion and, accordingly, to obtain additional integrals of motion. for a three-body system, replacing the geometry allows us to reduce the problem to the 6th order system. the main difficulty arising at the solution of this problem is the generalization of the well-known poincaré theorem on a homomorphism between the 3d sphere with unit radius and 3d compact. in this work, the possibility of such a generalization is strictly proved. 4. appendix as mentioned (see (14)), the vector x consists of 18 independent components. its transposed form looks like this: xt = ( α11, α12, α13, α22, α23, α33, β11, β12, β13, β22, β23, β33, ζ11, ζ12, ζ13, ζ22, ζ23, ζ33 ) . taking into account the form of the vector x, we can write the explicit form of the basic matrix: a =   d11 · · · d181 · · · · · · · · · d118 · · · d1818   , (15) where the superscript indicates the column number, while the subscript indicates the line number. as for the explicit form of elements d νµ = dµν , where µ, ν = 1, 18, then we can find them by multiplying the basic matrix a with the vector x (see equation (14)) and comparing with the system of equations (13). in particular, it is easy to verify that these a. gevorkyan, a. aleksanyan and s. alaverdyan 105 terms are equal: d 11 = d 2 2 = d 3 3 = 2d 2 10 = 2d 4 11 = 2d 5 12 = 2d 3 13 = 2d 5 14 = 2d 6 15 = 2α1, d 24 = d 4 5 = d 5 6 = 2d 1 10 = 2d 2 11 = 2d 3 12 = 2d 3 16 = 2d 5 17 = 2d 6 18 = 2α2, d 37 = d 5 8 = d 6 9 = 2d 1 13 = 2d 2 14 = 2d 3 15 = 2d 2 16 = 2d 4 17 = 2d 5 18 = 2α3, d 17 = d 2 8 = d 3 9 = 2d 8 10 = 2d 10 11 = 2d 11 12 = 2d 9 13 = 2d 11 14 = 2d 12 15 = 2β1, d 84 = d 10 5 = d 6 11 = 2d 7 10 = 2d 8 11 = 2d 9 12 = 2d 9 16 = 2d 11 17 = 2d 12 18 = 2β2, d 97 = d 11 6 = d 6 12 = 2d 7 13 = 2d 8 14 = 2d 9 15 = 2d 8 16 = 2d 10 17 = 2d 11 18 = 2β3, d 113 = d 2 14 = d 3 15 = 2d 17 10 = 2d 16 11 = 2d 17 12 = 2d 15 13 = 2d 17 14 = 2d 18 15 = 2γ 33ζ1, d144 = d 16 5 = d 6 17 = 2d 13 10 = 2d 14 11 = 2d 15 12 = 2d 13 13 = 2d 14 14 = 2d 15 15 = 2γ 33ζ2, d157 = d 17 8 = d 9 18 = 2d 14 16 = 2d 16 17 = 2d 17 18 = 2d 15 16 = 2d 17 17 = 2d 18 18 = 2γ 33ζ3. (16) all elements of the matrix (15) missing in (16) are identically zero. as is known, the algebraic system (13) or (14) does not have a solution in the case when the determinant of the matrix is zero det(a) = det(dµν ) = 0. a class consisting of sets of coefficients {σ} for which the determinant is zero, can be countable and the measure, respectively, will be equal to zero w = {0}. references [1] e. a. ayryan, a. s. gevorkyan and l. a. sevastyanova, “on the motion of a three body system on hypersurface of proper energy”, physics of particles and nuclei letters, vol.10, no. 7, pp. 1-8, 2013. [2] a. s. gevorkyan, “on reduction of the general three-body newtonian problem and the curved geometry”, journal of physics: conference series, 496, 012030, 2014. [3] a. s. gevorkyan, “on the motion of classical three-body system with consideration of quantum fluctuations”, physics of atomic nuclei, vol. 80, no. 2, pp. 358-365, 2017. [4] a. s. gevorkyan, “fundamental irreversibility and times arrow of the classical threebody problem. new approaches and ideas in the study of dynamical systems”. arxiv:1706.09827v2[math-ph] 13 dec 2017. [5] a. s. gevorkyan, “is the hamiltonian mechanics and in general classical mechanics reversible?”, book of abstracts, international conference dedicated to the 120th anniversary of emil artin, yerevan, armenia, may 27-june 2, pp. 58-59, 2018. submitted 04.12.2018, accepted 23.04.2019. 1 0 6 on homeomorphism between euclidean subspace and conformally euclidean manifold ¾íïéç¹û³ý »ýã³ï³ñ³íáõãû³ý ¨ ïáýýáñù-¿íïéç¹û³ý µ³½ù³ó¨áõãû³ý ùçç¨ ñáùçáùáñýç½ùç í»ñ³µ»ñû³é 1 ðð ¶²² æýýáñù³ïçï³ûç ¨ ³íïáù³ï³óù³ý åñáµé»ùý»ñç çýëïçïáõï 2 ðð ¶²² ². ´. ü³éµ³ý¹û³ýç ³ýí³ý ùçùç³ï³ý ýç½çï³ûç çýëïçïáõï e-mail: gashot@ipia.sci.am ²ù÷á÷áõù ðá¹í³íáõù µ»ñí³í ¿ ¹³ë³ï³ý »ñ»ù ù³ñùýç ¿íïéç¹û³ý »ýã³ï³ñ³íáõãû³ý e6 ¨ 6 d èçù³ýç µ³½ù³ó¨áõãû³ý m ùçç¨ ñáùçáùáñýç½ùç ³å³óáõûóá, áñá ãáõûé ¿ ï³éçë èçù³ýç µ³½ù³ó¨áõãûáõý, áõõõáñ¹í³í ïáåáéá·ç³ï³ý µ³½ù³ó¨áõãûáõý, ñáùçáùáñýç½ù µ³½ù³ó¨áõãûáõýý»ñç ùçç¨: î ãîìåîìîðôèçìå ìåæäó åâêëèäîâûì ïîäïðîñòðàíñòâîì è êîíôîðìíî-åâêëèäîâûì ìíîãîîáðàçèåì àøîò ñ. ãåâîðêÿí 1;2, àëåê à. àëåêñàíÿí 1 è ñóðåí á. àëàâåðäÿí1 1èíñòèòóò ïðîáëåì èíôîðìàòèêè è àâòîìàòèçàöèè íàí ðà 2èíñòèòóò õèìè÷åñêîé ôèçèêè èìåíè à. á. íàëáàíäÿíà íàí ðà e-mail: gashot@ipia.sci.am àííîòàöèÿ â ñòàòüå ïðåäñòàâëåíî äîêàçàòåëüñòâî ãîìåîìîðôèçìà ìåæäó åâêëèäîâûì ïîäïðîñòðàíñòâîì e6 êëàññè÷åñêîé ñèñòåìû òðåõ òåë è 6 d ðèìàíîâûì ìíîãîîáðàçèåì m, ÷òî ïîçâîëÿåò ñâåñòè äèíàìè÷åñêóþ çàäà÷ó ê ñèñòåìå 6-ãî ïîðÿäêà. êëþ÷åâûå ñëîâà: ñèñòåìà íåäîîïðåäåëåííûõ àëãåáðàè÷åñêèõ óðàâíåíèé, ðèìàíîâî ìíîãîîáðàçèå, îðèåíòèðîâàííîå òîïîëîãè÷åñêîå ìíîãîîáðàçèå, ãîìåîìîðôèçì ìåæäó ìíîãîîáðàçèÿìè. ²ßáï ê. ¶¨áñ·û³ý1;2, ²é»ք ². ²é»ùë³ýû³ý1 ¨ êáõñ»ý ´. ²é³í»ñ¹û³ý1 ¹çý³ùçï ëý¹çñá ñ³ý·»óý»é 6-ñ¹ ï³ñ·ç ñ³ù³ï³ñ·ç: ´³ý³éç µ³é»ñ՝ ãñëï³ï»óí³í ñ³ýñ³ñ³ßí³ï³ý ñ³í³ë³ñáõùý»ñç ñ³ù³ï³ñ·, 08 08_gevorgyan_98--106 8_gevorgyan_article 8_gevorgyan_abstract d:\user\sbornik_38_pdf\36.dvi mathematical problems of computer science 38, 84{86, 2012. m odeling fi fo queues by dynamic p etr i n ets go h a r r . p e t r o s ya n armenian state pedagogical university after kh. abovyan e-mail: blueayez777@mail.ru p e t r i n e t s ( p n ) a r e a g r a p h ic a l t o o l fo r fo r m a l d e s c r ip t io n o f t h e ° o w o f a c t ivit ie s in c o m p le x s ys t e m s . co m p a r e d t o o t h e r m o r e p o p u la r t e c h n iqu e s o f g r a p h ic a l s ys t e m r e p r e s e n t a t io n ( fo r in s t a n c e , b lo c k d ia g r a m s o r lo g ic a l t r e e s ) , p n a r e p a r t ic u la r ly m a t c h e d fo r r e p r e s e n t a t io n o f lo g ic a l in t e r a c t io n s a m o n g p a r t s o r a c t ivit ie s in a s ys t e m in a n a t u r a l wa y. typ ic a l s it u a t io n s t h a t c a n b e m o d e le d b y p n a r e : s yn c h r o n iz a t io n , s e qu e n t ia lit y, c o n c u r r e n c y a n d c o n ° ic t [1 ], [2 ], [3 ], [5 ]. in c o m p u t e r s c ie n c e , a qu e u e is a p a r t ic u la r kin d o f a b s t r a c t d a t a t yp e o r c o lle c t io n in wh ic h t h e e n t it ie s in t h e c o lle c t io n a r e ke p t in a n o r d e r a n d t h e p r in c ip a l o p e r a t io n s ( o r t h e o n ly o n e ) in t h e c o lle c t io n a r e ( is ) t h e a d d it io n o f e n t it ie s t o t h e r e a r t e r m in a l p o s it io n a n d r e m o va l o f e n t it ie s fr o m t h e fr o n t t e r m in a l p o s it io n . th is m a ke s t h e qu e u e a fir s t -in -fir s t ou t ( fifo) d a t a s t r u c t u r e . in a fifo d a t a s t r u c t u r e , t h e ¯ r s t e le m e n t a d d e d t o t h e qu e u e will b e t h e ¯ r s t o n e t o b e r e m o ve d . th is is e qu iva le n t t o t h e r e qu ir e m e n t t h a t o n c e a n e le m e n t is a d d e d , a ll e le m e n t s t h a t we r e a d d e d b e fo r e h a ve t o b e r e m o ve d b e fo r e a d d it io n o f t h e n e w e le m e n t . a qu e u e is a n e xa m p le o f a lin e a r d a t a s t r u c t u r e [5 ]. de¯nition. a p e t r i n e t is m = ( c; ¹) , wh e r e c = ( p; t; i; o ) is t h e n e t wo r k s t r u c t u r e , a n d ¹ is t h e n e t wo r k c o n d it io n . th e p -p o s it io n s , t -t r a n s it io n s a r e ¯ n it e s e t s ; i : t ! p 1; o : t ! p 1 a r e t h e in p u t a n d o u t p u t fu n c t io n s , r e s p e c t ive ly, wh e r e p 1 a r e a ll p o s s ib le c o lle c t io n s ( r e p e t it ive e le m e n t s ) o f p ; ¹ : p ! n0 is t h e fu n c t io n o f c o n d it io n s , wh e r e n0 = f0 ; 1 ; : : :g is t h e s e t o f in t e g e r s . w e d e t e r m in e ( in a kn o wn m a n n e r ) t h e a llo we d t r a n s it io n s o f p e t r i n e t s a n d t h e t r a n s it io n s fr o m o n e s t a t e t o a n o t h e r , a s we ll t h e s e t o f r e a c h a b le s t a t e s . th e s t a t e o f t h e n e t wo r k is r e p r e s e n t e d b y t h e fo llo win g ve c t o r : ( ¹( p1 ) ; ¹( p2 ) ; :::; ¹( pm ) ) ; if p = fp1; p2; :::; pmg; ¹ is a fu n c t io n , wh ic h c a r r ie s o u t t h e fo llo win g m a p p in g , ¹ : p ! n0, wh e r e n0 is u s e d t o e n c o d e t h e n u m b e r o f t o ke n s in t h e p o s it io n s . b e fo r e s t a r t in g it s wo r k t h e n e t m u s t h a ve a n in it ia l s t a t e , ( ¹0 ( p1 ) ; ¹0 ( p2 ) ; :::; ¹0 ( pm ) ) , wh e r e t h e n u m b e r o f t o ke n s in t h e ir r e s p e c t ive p o s it io n s a r e s h o wn [1 , 4 ]. th e c o m p le xit y o f t h e s t a n d a r d p e t r i n e t m o d e lin g o f t h e t h e fifo qu e u e is 4 m + 6 ( m¡ 1 ) + 2 + 2 = 1 0 m ¡ 2 ( t h e n u m b e r o f p la c e s , t r a n s it io n s , a r c s , fo r a n e t wo r k wit h m e le m e n t s a r e c a lc u la t e d ) . th e d i®e r e n c e b e t we e n d yn a m ic p e t r i n e t s fr o m t h e s t a n d a r d p e t r i n e t s is t h a t t h e s t r u c t u r e o f t h e ¯ r s t c h a n g e s d u r in g it s wo r k, t h a t is , p o s it io n s o r t r a n s it io n s c a n b e a d d e d o r r e m o ve d fr o m t h e n e t , b e s id e s , a n d , c o n s e qu e n t ly, t h e in p u t a n d o u t p u t fu n c t io n s a r e c h a n g e d [4 ]. 8 4 g. petrosyan 8 5 b e lo w is t h e m a t h e m a t ic a l d e ¯ n it io n o f d yn a m ic p e t r i n e t s , wh e r e we h a ve in t r o d u c e d t h e id e a o f d yn a m ic m e m o r y. th is id e a h e lp s u s t o u s e t h e fr e e d m e m o r y s p a c e . fir s t , we d e n o t e t h e s e qu e n c e o f a ll p la c e s a s fo llo ws : ­ = ( p0; p1; p2; :::; pk; :::; p2k ) , a n d t h e s e qu e n c e o f a ll t r a n s it io n s b y t h e fo llo win g ã = ft0; t1; :::; tk; :::; t2k g. l e t 's d e ¯ n e t h e fo llo win g , c = ( p; t; i; o ) , a s t h e c u r r e n t s t r u c t u r e o f t h e n e t wo r k, wh e r e p µ ­ is t h e ¯ n it e s e t o f p o s it io n s a t t h e g ive n m o m e n t , a n d t µ ã is t h e ¯ n it e s e t o f t r a n s it io n s a t t h e g ive n m o m e n t , a n d i : t ! p 1 a n d o : t ! p 1, a r e t h e c u r r e n t in p u t a n d o u t p u t fu n c t io n s , r e s p e c t ive ly. p 1 h a s t h e s a m e m e a n in g a s in t h e p r e vio u s c a s e s . d e n o t e ¹ : ­ ! n¡10 , a s t h e fu n c t io n o f c o n d it io n , wh e r e n¡10 is t h e s e t o f in t e g e r s ( t h e n u m b e r s o f t o ke n s in t h e p o s it io n s ) , in c lu d in g ( -1 ) , t h a t e n c o d e s t h e u n u s e d p o s it io n s . l e t p = fpi1; pi2; :::; pirg is g ive n , ( ¹( pi1 ) ; ¹( pi2 ) ; :::; ¹( pir ) ) b e t h e ve c t o r o f t h e c u r r e n t s t a t e a t t h e g ive n m o m e n t . mo r e o ve r , if p = fpi1; pi2; :::; pirg is t h e n u m b e r o f p o s it io n s in t h e n e t wo r k, t h e n t h e fu n c t io n ¹ p e r fo r m s t h e fo llo win g m a p p in g : ¹ : p ! no; ¹ : ( ­=p ) ! f¡ 1 g: to o b s e r ve t h e p r o c e s s e s in t h e n e t wo r k, yo u ¯ r s t n e e d t o kn o w t h e la ws o f t h e s t r u c t u r a l c h a n g e s . s u p p o s e we h a ve a n m = ( c; ¹ ) d yn a m ic p e t r i n e t , wh ic h c u r r e n t ly h a s c = ( p; t; i; o ) a n d is in t h e ¹ c u r r e n t c o n d it io n . l e t 's s a y t h a t t 2 t t r a n s it io n is a llo we d , if 8p 2 i ( t ) , ¹ ( p ) ¸ ]( p; i ( t ) ) , wh e r e ]( x; a ) h a s t h e s a m e m e a n in g a s in t h e s t a n d a r d p e t r i n e t wo r ks [1 , 4 ]. l e t m = ( c; ¹ ) b e a d yn a m ic n e t wo r k a n d t h e t r a n s it io n t 2 t is a llo we d t o r u n . in t h is c a s e , a s we h a ve n o t e d , n o t o n ly t h e s t a t e o f t h e n e t wo r k will c h a n g e , b u t a ls o t h e s t r u c t u r e . in c o n t r a s t t o t h e p r e vio u s d e ¯ n it io n , we d e ¯ n e t h e fu n c t io n ± [1 , 4 ], wh ic h d e p e n d s o n t h e t h r e e a r g u m e n t s ( c; ¹; t ) , a n d r e t u r n s b a c k t h e n e w s t r u c t u r e a n d s t a t e o f t h e n e t wo r k a ft e r t h e t r a n s it io n t a s a n e w va lu e . in fa c t we c o n c lu d e t h a t , in o p t im iz a t io n s e n s e , t h e p e t r i d yn a m ic n e t s a r e m o r e c o n ve n ie n t fo r m o d e lin g o f s ys t e m s o f s o m e t yp e t h a n t h e s t a n d a r d p e t r i n e t s . in fa c t , ± ( c; ¹; t ) = ( c0; ¹0 ) , wh e r e c = ( p 0; t 0; i0; o0 ) is a n e w s t r u c t u r e . p 0 is t h e s e qu e n c e o f t h e n e w p o s it io n s , t 0 is t h e s e qu e n c e o f t h e n e w t r a n s it io n s , i0 : t 0 ! ( p 0 ) 1 is t h e n e w in p u t fu n c t io n , o0 : t 0 ! ( p 0 ) 1 is t h e n e w o u t p u t fu n c t io n . ¹0 = ( ¹0 ( p0 ) ; ¹ 0 ( p1 ) ; :::; ¹ 0 ( pk ) ; :::; ¹ 0 ( p2k ) ) a n d ¹ 0 : ( ­=p 0 ) ! f¡ 1 g. th e in it ia l s t r u c t u r e o f t h e n e t wo r k is d e n o t e d a s c0, a n d t h e in it ia l c o n d it io n is ¹0. th e n e t wo r k g e n e r a t e s ( c; ¹) p a ir s o f s e t s , a n d t h is s e t is d e n o t e d b y r ( c0; ¹0 ) , wh e r e 1 . ( c0; ¹0 ) 2 r ( c0; ¹0 ) is in it ia l s t a t e , 2 . if ( c0; ¹0 ) 2 r ( c0; ¹0 ) a n d 9 t 2 t 0, s u c h t h a t ± ( c0; ¹0; t ) = ( c0; ¹0 ) 2 r ( c0; ¹0 ) , 3 . th e o t h e r ( c; ¹ ) p a ir s in r ( c0; ¹0 ) d o n o t b e lo n g t o r( c0; ¹0 ) a n d t h e la t t e r c a n b e in ¯ n it e . 8 6 modeling fifo queues by dynamic petri nets r e fe r e n c e s [1 ] j. l . p e t e r s o n ., p etri net theory and the modeling of systems. p r e n t ic e h a ll, e n g le wo o d cli®s , 1 9 8 1 . [2 ] t. mu r a t a , " p e t r i n e t s : p r o p e r t ie s , a n a lys is a n d a p p lic a t io n s " , p roceedings of the ie e e , vo l. 7 7 , n o . 4 , 1 9 8 9 . [3 ] w . r e is in g , g. r o z e n b e r g ( e d s ) . l e c t u r e n o t e s o n p e t r i n e t s . p a r t s i a n d ii / / l e c t u r e n o t e s in co m p u t e r s c ie n c e s . v . 1 4 9 1 1 4 9 2 . s p r in g e r v e r la g , 1 9 9 8 . [4 ] â. å. êîòîâ, ñåòè ïåòðè. ì.: íàóêà, 1984. [5 ] ä. êíóò, èñêóññòâî ïðîãðàììèðîâàíèÿ, ò1-ò3, 2008. formalizmodelej.dvi ìàòåìàòè÷åñêèå âîïðîñû êèáåðíåòèêè è âû÷èñëèòåëüíîé òåõíèêè 26, 2006, 97–100. ìíîãîêîìïîíåíòíûå âåðøèíû ãðàôîâ òåëåêîììóíèêàöèîííûõ ñåòåé. êàðåí à. ìêðò÷ÿí èíñòèòóò ïðîáëåì èíôîðìàòèêè è àâòîìàòèçàöèè íàí ðà e-mails kamkrtchyan@ipia.sci.am, kamkrtchyan@yahoo.com àííîòàöèÿ ñ öåëüþ èññëåäîâàíèÿ ìåòîäàìè òåîðèè ãðàôîâ ìîäåëåé òåëåôîííûõ è äðóãèõ òåëåêîììóíèêàöèîííûõ ñåòåé ââîäÿòñÿ ïîíÿòèÿ ñâÿçàííûå ñ ìíîãîêîìïîíåíòíîñòüþ âåðøèí è îïåðàöèé íàä íèìè. refer ences [1 ] mkr t c h ya n k , a p p lic a t io n o f s t a t is t ic a l a p p r o a c h in d e t e c t in g fr a u d c a s e s in t e le p h o n e n e t wo r ks , ma t h e m a t ic a l p r o b le m s o f co m p u t e r s c ie n c e , 2 4 , p p . 1 2 5 { 1 3 2 , 2 0 0 5 . [2 ] mkr t c h ya n k , a p p lic a t io n o f s t a t is t ic a l a p p r o a c h ¯ g h t in g fr a u d c a s e s in t e le p h o n e n e t wo r ks , p r o c e e d in g s o f cs it in y e r e va n , s e p t e m b e r 2 2 , p p . 2 1 3 { 2 1 8 , 2 0 0 5 . [3] ìêðò÷ÿí ê, ê ïðèìåíåíèþ òåîðèè ãðàôîâ ïðè èññëåäîâàíèè òåëåêîììóíèêàöèîííûõ ñåòåé, ìàòåìàòè÷åñêèå âîïðîñû êèáåðíåòèêè è âû÷èñëèòåëüíîé òåõíèêè, 25, ñòð. 80 – 85, 2006. ð»é³ñ³õáñ¹³ïó³ï³ý ó³ýó»ñç ·ñ³ýý»ñç µ³½ù³µ³õ³¹ñçã ·³·³ãý»ñá î. øïñïãû³ý ²ù÷á÷áõù ¶ñ³ýý»ñç ï»ëáõãû³ý ù»ãá¹ý»ñáí ñ»é³ëáë³ûçý ¨ ³ûé ñ»é³ñ³õáñ¹³ïó³ï³ý ó³ýó»ñç ùá¹»éý»ñç ñ»ï³½áïù³ý ýå³ï³ïáí ý»ñùáõííáõù »ý ·³·³ãý»ñç µ³½ù³µ³õ³¹ñçã éçý»éáõ ¨ ýñ³ýó ùçç¨ ·áñíáõáõãûáõýý»ñç ñ»ï ï³åí³í ·³½³÷³ñý»ñá: ¸çï³ñïíáõù »ý ïçñ³é³ï³ý ûñçý³ïý»ñ: 9 7 начиная с начала 2000 года осуществляется внедрение ghis в здравоохранении, в рамках принятого проекта о реформирование информ mathematical problems of computer science 49, 92--96, 2018. info-communication systems based on e-mail and sms technologies andranik e. mkhitaryan institute for informatics and automation problems of nas ra e-mail: and.mkhitaryan@gmail.com abstract the features of the widespread systems for the exchange of messages in computer and cellular networks, the prerequisites for the creation of combined messaging systems using ip and gsm technologies are considered. the peculiarities of the construction of hybrid e-mail / sms info-communication systems are considered. keywords: mail2sms, e-mail, sms, notification. 1. introduction electronic mail is a system to send/receive text messages between computers by the internet. despite the fast delivery of the letter to the addressee, e-mail is a typical "on demand" system. the reading of the delivered message by the addressee depends on the time when he/she requests it from the e-mail provider. another worldwide system for text message exchange is sms. unlike e-mail, sms is not a postal system: the size of the transmitted message is limited, message delivery is possible only in the coverage area of the cellular network or according to the roaming agreements of the local cellular operator. the undeniable advantage of sms technology is the delivery of the message directly into “the pocket” of a mobile subscriber. the limitations typical of e-mail and sms can be overcome by the use of integrated, "in one bottle," mutually complementary network and sms technologies. at the same time, the problem of "long-range" of a particular sms-service is solved using the internet as a transport for sms messages (websms). 2. solution the problem of "on demand" can be solved by the use of e-mail related sms-systems to promptly notify the recipient of incoming e-mails in his mailbox [1], [2]. these technologies 92 a. mkhitaryan 93 usually use special program selectors, built into the mail server that allocate messages for which a notification is required to be sent to the recipient (according to the permissive list with the corresponding e-mail addresses on the server) and provides necessary data to form an sms notification for the particular sms server, which has a gateway to the cellular network of the region. sms server in the internet environment is equipped with a gsm modem a gateway to a given cellular network, which provides the formation and transmission of sms notifications. in the considered case, the initiative of selecting from among the incoming messages those for which it is desirable to receive an sms notification, is assigned to the recipient. another approach is possible when creating a notification system: in this case, an application for sending an sms notification to the addressee is sent directly by the sender when preparing the letter [3]. in such cases, the message should contain certain signs indicating the need to send the letter to the addressee together with sms notification. bearing in mind that email notification systems are based on the use of sms technologies, it is natural to combine the info-communication resources of e-mail and sms technologies in the form of one service for the user of an e-mail. in such systems, the ability of e-mail account users to send sms notifications to the specified cellular networks is not the only possible way, because in these systems it is easy to integrate "in one package" for an e-mail user: a wide range of services related to the use of sms technologies (transmission of sms notifications via e-mail, group mailings of sms, etc.). using such a hybrid "e-mail + sms" technology allows creating a unified system for the user of webmail to access all the info-communication resources used in this mail system, both e-mail and sms. 3. multiple approaches there are two possible approaches to creating such hybrid systems: a) systems with the selection of incoming mail from the mail server according to certain characteristics and transfer of the necessary data to sms server for configuration and sending sms (figure 1). the criterion for selecting an incoming message containing a request for operations with sms can be marked by the sender in the "subject" section. accordingly, when composing a letter in the subject section, a command line is formed containing a request for the gsm controller to perform a specified specific operation. the selector of mail server selects e-mails containing a command in the subject section from the incoming e-mail stream. from the letter, in turn, the attributes of the letter are highlighted and together with the content of the subject are transferred from the mail server to the gsm controller for further processing in accordance with the command contained in the command line. fig. 1. structure of the first approach. info-communicational systems based on email and sms technologies 94 b) systems that operate with copies of letters with the necessary target designations sent directly to the sms server to create and send an sms to the recipient (figure 2). in this case, the system is presented in the form of an independent e-mail / sms infocommunication resource, not directly connected to webmail services that uses the e-mail as a transport of access to the resource for the user. user access to the system is done directly from the webmail page of the user, without requiring additional access to other resources. during the preparation of the letter, the user needs to send a copy of the letter to the mail address of the system (to request a notification) or send an email directly to the system. such a solution does not require a binding to a specific e-mail provider. these provisions are used in the unimail-asnet project [4]. the server acts as an ordinary e-mail user. for the convenience of webmail users, the mechanisms for accessing and managing sms options are combined with e-mail preparation operations with a minimum number of request commands to the unimail server. all operations with the system are made directly from the webmail user's page when composing the letter. in the highlighted markers of the "subject" field, the command of the message contains the information (phone numbers of the addressees, management of the "black list") for the unimail server to perform the corresponding operations. a copy of the letter should be sent to the e-mail address of the server for further processing. the choice of the mode requested by the user (notification or sending of sms) is made automatically by the system, analyzing the address part of the message copy being processed. commands with descriptions:  *notice to…(phone numbers)* parallel to the letter send sms notification to the addressee/addressees  *sms to… (phone numbers)* send an sms to the addressee/addressees  *add to black list* include in the "black list" of the blocked recipient  *del from black list* exclude the blocked addressee from the "black list" fig. 2. structure of the second approach. a. mkhitaryan 95 4. conclusion besides many undisputable advantages, e-mail is a typical “on demand” info-communication resource. however, another technology – sms, does not have such a problem and allows you to send text messages directly to the addressee’s mobile phone. the use of e-mail/sms hybrid system will help users overcome that disadvantage of e-mail. in the article two different approaches of such systems are presented. both advantages and disadvantages of both models are discussed. the possible design for them and the use cases are described. references [1] д. геворкян, а. нанасян и к. хачатрян, «новые web ресурсы asnet.am», proceedings of international conference of computer science and information technologies, csit-2011, ереван, pp. 311-312, 2011. [2] d. gevorkyan, k. khachatryan, a. nanassian, a. petrosyan, g.petrosyan, v. sahakyan and e. vardanyan, “mail informerselective incoming instant phone notification system”, proceedings of international conference computer science and information technologies, csit, yerevan, armenia, pp. 466-467, 2009. [3] а. нанасян и к. хачатрян «mail2sms.asnet.am – система оповещения о входящих письмах», proceedings of international conference of computer science and information technologies csit-2013. ереван, 2013. [4] а. мхитарян, э. матвеев, а. нанасян, в. саакян и a. petrosyan, “гибридная инфокоммуникационная email/sms система unimail”, proceedings of international conference of computer science and information technologies, csit, yerevan, armenia, pp. 389-391, 2017. submitted 22.11.2017, accepted 12.02.2018. տեղեկատվական հաղորդակցման համակարգեր հիմնված էլ. փոստի և sms-ի տեխնոլոգիաների վրա ա. մխիթարյան ամփոփում աշխատանքում դիտարկվել են համակարգչային և բջջային կապերով տեքստային հաղորդագրություններ ուղարկելու համար նախատեսված մեխանիզմների առանձնահատկությունները, ip և gsm տեխնոլոգիաների համատեղ կիրառմամբ հաղորդագրությունների փոխանցման համակարգի ստեղծման նախադրյալները։ info-communicational systems based on email and sms technologies 96 դիտարկվել են e-mail/sms հիբրիդային տեղեկատվական հաղորդակցման համակարգի ստեղծման առանձնահատկությունները։ инфо-коммуникационные системы на базе технологий e-mail и sms а․ мхитарян аннотация рассмотрены особенности распространенных систем обмена письменными сообщениями в компьютерных и сотовых сетях, предпосылки создания совмещенных систем передачи сообщений, использующих технологии ip и gsm сетей. рассмотрены особенности построения гибридных e-mail/sms инфокоммуникационных систем. d:\sbornik\...\tpel1.dvi mathematical problems of computer science 32, 70{73, 2009. on i nter val t otal color ings of t r ees p e t r o s a . p e t r o s ya n y a n d a n i s . s h a s h ikya n z yinstitute for informatics and automation problems of nas of ra, zdepartment of informatics and applied mathematics, ysu pet petros@ipia.sci.am, anishashikyan@gmail.com abstract an interval total t¡coloring of a graph g is a total coloring of g with colors 1; 2; : : : ; t such that at least one vertex or edge of g is colored by i; i = 1; 2; : : : ; t, and the edges incident to each vertex v together with v are colored by dg(v)+1 consecutive colors, where dg(v) is the degree of a vertex v in g. in this paper we prove that if t (t 6= k1) is a tree and ¢(t ) + 2 · t · m(t ) then t has an interval total t¡coloring, where ¢(t ) is the maximum degree of vertices in t and m(t ) is a parameter which can be e®ectively found for any t . refer ences [1 ] p . a . p e t r o s ya n , \ in t e r va l t o t a l c o lo r in g s o f c o m p le t e b ip a r t it e g r a p h s " , p roceedings of the csit conference, p p . 8 4 -8 5 , 2 0 0 7 . [2 ] p . a . p e t r o s ya n , \ in t e r va l t o t a l c o lo r in g s o f c e r t a in g r a p h s " , m athematical p roblems of computer science, vol. 31, p p . 1 2 2 -1 2 9 , 2 0 0 8 . [3 ] d . b . w e s t , in t r o d u c t io n t o gr a p h th e o r y, p r e n t ic e -h a ll, n e w je r s e y, 1 9 9 6 . [4 ] h . p . y a p , to t a l co lo r in g s o f gr a p h s , l e c t u r e n o t e s in ma t h e m a t ic s 1 6 2 3 , s p r in g e r v e r la g , 1 9 9 6 . ì³é»ñç ùçç³ï³ûù³ûçý éç³ï³ï³ñ ý»ñïáõùý»ñç ù³ëçý ä. ä»ïñáëû³ý, ². þ³ßçïû³ý ²ù÷á÷áõù g ·ñ³ýç éç³ï³ï³ñ ý»ñïáõùá 1 ; 2 ; : : : ; t ·áõûý»ñáí ï³ýí³ý»ýù ùçç³ï³ûù³ûçý éç³ï³ï³ñ 1 ; 2 ; : : : ; t –ý»ñïáõù, »ã» ³ù»ý ùç i ·áõûýáí, i = 1 ; 2 ; : : : ; t, ý»ñïí³í ¿ ³éýí³½ý ù»ï ·³·³ã ï³ù ïáõ ¨ ûáõñ³ù³ýãûáõñ v ·³·³ãçý ïçó ïáõ»ñá ¨ ³û¹ ·³·³ãá ý»ñïí³í »ý dg ( v ) + 1 ñ³çáñ¹³ï³ý ·áõûý»ñáí, áñï»õ dg ( v ) -áí ýß³ý³ïí³í ¿ v ·³·³ãç ³ëïç׳ýá g ·ñ³ýáõù: ²ûë ³ßë³ï³ýùáõù ³å³óáõóí³í ¿, áñ »ã» t (t 6= k1) -ý í³é ¿ ¨ ¢ ( t ) + 2 · t · m ( t ) , ³å³ t -ý áõýç ùçç³ï³ûù³ûçý éç³ï³ï³ñ t –ý»ñïáõù, áñï»õ ¢ ( t ) -ý t -ç ù³ùëçù³é ·³·³ãç ³ëïç׳ý ¿, çëï m ( t ) –ý ³ñ¹ûáõý³í»ï ñ³ßí³ñï»éç å³ñ³ù»ïñ ¿ t –ç ñ³ù³ñ: 7 0 52 mathematical problems of computer science 58, 52–60, 2022. doi: 10.51408/1963-0092 udc 004.891.3 ininfluedeveloping aerial unmanned effective decision makers sedrak v. grigoryan and edward m. pogossian institute for informatics and automation problems of nas ra e-mail: addressforsd@gmail.com, epogossi@aua.am abstract unmanned aerial vehicles (uavs, drones) and similar unmanned units are becoming more and more involved in various spheres, such as agriculture, emergency situations, battles, etc. however, in decision making there are still a lot they can be improved to avoid human direct involvement in those problems. to advance in the problem we develop tools to make uav autonomously effective decision makers, particularly, able to analyze properly given situations and then according to assigned goals select appropriate strategies to achieve the goals. in the following work we aim to provide a solution for a single uav which is able to discover units of interest, and select the target to track, manipulate or hit based on expert specified knowledge, as well as discuss further steps. keywords: object, detection, decision making, combinatorial problems, expert knowledge. article info: received 16 august 2022; accepted 26 september 2022. 1. introduction 1.1. problems of space of uav involvements involvement of programmatic solutions in various types of uav-based environments, such as agriculture, emergency situations, battles, and other types of urgent problems, is important and actual problem. representation of problems can vary from one to another, while given situation for uavbased solutions may stay in scope of the following list: maps, emergencies, opponents, their positions, etc. mailto:addressforsd@gmail.com mailto:epogossi@aua.am s. grigoryan and e. pogossian 53 overall, it is becoming very important to avoid human involvement in these tasks directly to avoid human causalities, to provide descent support and amount of units involved, thus it is important having decision making modules. a non-expensive uav which is able to process the field situation as an image from the top, and make decisions based on the current situation without human involvement is an urgent problem. the advantage of such unit is that it can cost low and has pretty high accuracy and effectiveness. 1.2. programmatic improvements of uav units various tasks can be considered in this space, including: 1.2.1. the tasks of adequate processing of situations. the program has to properly capture and parse the current situation based on retrieved data, mostly from images. this is currently not fully solved, however there are some available solutions for certain types of such tasks, e.g. detecting units of interests, such as emergency areas, e.g. fire sources on the images, etc. such solutions require: a. sufficient preliminary inherited knowledge and ongoing data related to the units on the field to be recognized, particularly the ones to identify the own and opponent units, targeting items, tracking objects, etc. b. proper training and examining the functionality of target models in performing parsing of situations and recognizing there all valuable units (the mistakes might be very costly depending on the problem). 1.2.2. making valuable decisions in situations uav can: a. analyze them to select with respect to (wrt) the goals the most prospective and simultaneously available ones b. select plans of attaining those targets c. analyze compositions of actions, strategies for the perspective plans d. make evaluation of the strategies and perform appropriate strategies to attain the goals. 1.3. to examine our approach, we concentrate on the topic for a battle field strategy games g, which provide good way to track situation from the top (similar to uav images). we consider this as a problem of certain combinatorial rgt class, where the space of solutions is reproducible game trees [1-8]. rgt problems are specified as follows: • there are (a) interacting actors (players, competitors, etc.) performing (b) identified types of actions in the (c) specified types of situations; • there are identified utilities, goals for each actor; • actions for each actor are defined • the scope of solutions at the situations are fully determined by them (i.e., are identified as games with perfect information) actors perform their actions in specified periods of times and do affect situations by actions in time t by transforming them to new situations in time t+1 trying to achieve the best utilities on that situation (goals) by regularities defining these actions. for example, a way to interpret battle field game g as the rgt problem is: 1. the battling sides can be considered as interacting actors 2. military units’ movements, attacks can be considered as actions 3. the battle field area including military units can be considered as the situations 4. different situations can be considered as goals: capture objects, destroy enemy units, push frontline. 5. the analysis of given situations are sufficient for selection of proper strategies ininfluedeveloping aerial unmanned effective decision makers 54 1.4. advances of rgt solvers 1.4.1. there are certain important advances and achievements in cognizers (rgt solvers) [7] development: in it was shown that rgt problems are reducible to each other, particularly, to some standard kernel rgt problem k, say, chess, thus, we get an opportunity to integrate the best-known achievements in solving particular rgt problems into rgt solvers letting us to apply those achievements to any of rgt problem [1]. in rgt solutions, we follow the research lines of botvinnik, pitrat, wilkins and ones successfully started since 1957 in the institute for informatics and automation problems at the academy of sciences of armenia and based on modeling of expert approaches involving: knowledge bases, knowledge-based algorithms of decision making and matching situations to classifiers, as well as algorithms of revealing and modifying knowledge. the advances in rgt [1-10] include the following: 1. solutions for transforming situations for rgt problems, a solution for chess is available. “generals: command and conquer” game is considered as a sample battlefield problem and positive results were achieved for recognition of military units. 2. knowledge presentation and matching algorithms were developed generally for rgt problems and adequacy was experimented for chess, marketing and other rgt problems. 3. planning and decision-making algorithms, igaf and ppit (including tzt) based on botvinnik’s ideas were developed and tested for network intrusion protection problems and chess problems. additionally, partial implementation of ppit algorithms were integrated in general rgt solver and experimented for chess and other rgt problems. various urgent combinatorial problems were investigated as rgt problems including network protection from hacker intrusions [1], single ship defense from various types of attacks [6, 7], chess [2, 4], etc. 1.5. we aim to resolve some of above-mentioned tasks by providing programs for uav, i.e., autonomously effective decision makers, or agents, particularly for type of games g that will allow to process properly situations of g, then according to assigned goals select appropriate strategies for achieving the goals. in the current work we concentrate on the following problems: a. from the input images from uavs detect and classify units of the game g influential for attaining the goals b. from the input situation including already classified influential units select target to hit. 2. units’ detection, classification classification of influential units is performed via the recorded images. popular object detection and image classification methods are now widely based on machine learning solutions, particularly deep convolutional neural networks, e.g. in the following an approach for vehicle detection from aerial images is discussed [11]. in the following we also rely on ml solutions to train a model for influential units’ detection and classification. 2.1. creation of classification dataset based on analysis of available data, we collected influential unit images and videos. from the collected data images were revealed to describe influential units for training the model. we made s. grigoryan and e. pogossian 55 a grouping of some classes of influential for game g units into one class, later to be classified as the same. this allows to have much less classes and possibly higher detection rate. the created dataset mostly consists of aerial photos, because uavs mostly take pictures that way. we have selected 8 groups of influential for game g classes, then created dataset consisting of their aerial images. 2.2. preparation of detection model once we have the images as discussed in above section, we prepare it for training models. in the case of the game g unit’s detection, the model needs both accuracy and speed, but it is more essential to draw accurate detection conclusions. some of the studies reveal that yolo provides better detection and speed combination over other models in various problems, providing real time detection ability [11-16]. based on the available results we use yolov5 as a model to be used for our dataset training and detection. the trained model gave results of accuracy with the values as follows: detection precision about 80%, recall is close to 60-70% and map about 60. the summary is in fig. 1. 3. selection of the target as described in introduction chapter we are relying on the achievements of rgt to provide decision making solutions in such problems, particularly the solutions rely on expert knowledge. here it comes to finding out the knowledge pieces needed in decision making in the game g and specifically for selection a target for uav managing as we concentrate our attention to that specific game g in the current work. the experts’ analysis and descriptions the following nuclear types [2] of knowledge for game g were revealed. for the targeting influential units: 1. the class of the g units as classified in section 2, can be reduced to a value in range of {1-8} for each class having a specific value. fig. 1 metrics of training results. ininfluedeveloping aerial unmanned effective decision makers 56 2. the price of the unit. this is not the actual cost of the unit, but the price of the unit in the battle, describing how much can its damage be. so we assign this a rule of {1-8} range depending on the type of target. 3. other types of expert knowledge also participate in decision making, based on which the decision becomes more accurate. for own uav: 1. it also contains specific types of knowledge, in this case this is related to the managing abilities, the decision realization instruments type and power, which determine the target to be resolved. based on the following nuclear classifiers we construct classes of units that appear as possible targets [2, 9]. in the tasks we only consider decisions relevant to the game g by the uav as our own action. so, the simplified version of goal searching algorithms [2, 10] is applied here. first non-perspective targets are filtered out in the situation. then by unit price the prioritization is applied and, with some additional corrections the target is selected. because the situation is changing, the selected target on each situation can be different. to increase the confidence of correct selection of the targets, in sequential situations the same logic of target selection is applied several times. if examined target is confirmed, the confidence is increasing. with attaining certain confidence in certain time period, the target is locked on. 3. above we discussed the basic approach and some of applied knowledge descriptions for the selection of the targets. the model of detection of influential units and its metrics were provided in section 2. knowledge-based approach adequacy has been discussed in [2, 8, 9]. the performance and the efficiency of programs realizing our uav approach are attributed as follows: a. the program is developed in python programming language to provide easy and fast transitions between various experimenting environments. b. to improve efficiency of the program, when the target is selected, it is only tracked without its recurring detection and matching. fig. 2. the flow of target selection algorithm. s. grigoryan and e. pogossian 57 c. once target is locked on, the program calculates and provides the direction for hitting it. d. the efficiency of the program is experimented with various video inputs with different frame update rate: frames per second (fps), resolution, the program provides close to real time results: for hd and fullhd videos with 20-25 fps the program is able to achieve close to real time performance. e. the prepared program and its performance were tested low power-consuming and gpu enhanced devices, which may be a good fit for uav setup. 4. future works the current solutions demonstrate the positive results of the work, as well as provide background for the future steps. the next steps of the current works are: 1. the accuracy of the detection of game g units affects the whole flow of target selection and situation processing, decision making, thus improvement of detection is one of essential topics, also due to possible fatal problems in actual application mistakes. for this step we go on the following direction: 1.1. enhancement of the dataset with new images, 1.2. enhancement of dataset by machine learning solutions, such as data augmentation, 1.3. applying machine learning techniques to improve quality of the input 1.4. if the amount of data is sufficient, then classify exact types of units instead of grouping them. 2. enlarging the scope of considered situation. this assumes enhancing the knowledge for matching situations, which can help in properly selection targets, provide more than one type of actions for involved other than the given single uav own units, specifying separate targets for own units and the sequence for targets to be hit. the enhancement of knowledge of the experts is an essential part in making decisions and improvement of decision with the increase of expert knowledge is demonstrated in [9], while integration of knowledge-based decision-making algorithms provided in [2, 10] also demonstrated their adequacy. this provides a good background for using the solutions in real uavs. 5. conclusions in the following work an approach to describe battle field problem is discussed, where a way to formalize the problem is given. the following results were achieved: 1. from open sources many photo and video data were analyzed, and images were revealed to create a dataset of g units. the dataset consists of 8 classes, each of them containing a group of units functionally equal to the ones defined by experts, to achieve an acceptable accuracy in detection. 2. yolov5 model was used for training a model to detect the selected classes, and the results of model performance were demonstrated. 3. by close cooperation with experts of that field certain types of knowledge to properly select the target to be hit were revealed. 4. algorithms to select the target based on input images, classified objects on that and the knowledge of the field are developed. 5. experiments were conducted for low power computing units and close to real time processing efficiency is achieved. ininfluedeveloping aerial unmanned effective decision makers 58 relying on the results achieved in this work and achievements described in the field of rgt problems, we plan the next steps of the work as follows: 1. collect more data from available sources, enhance the existing dataset by machine learning tools. this allows to achieve better detection and classification accuracy, as well as makes it possible later to more detailed classification instead of grouping them. 2. enlarge the scope of included problems to consider also agricultural, emergency and other urgent applications, to provide certain types of actions based on decisions it makes using algorithms developed for rgt solvers [2, 9, 10]. 3. enhance knowledge base for the problems based on expert knowledge to enable various types of actions, including ways of more appropriate target selections, target managing sequence selections, etc. references [1] e. pogossian, a. javadyan and e. ivanyan., "effective discovery of intrusion protection strategies" the international workshop on agents and data mining, lecture notes in computer science, st. petersburg, russia, pp. 263-274, 2005. 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[6] d. dionne, e. pogossian, a. grigoryan, j. couture and e. shahbazian, “an optimal sequential optimization approach in application to dynamic weapon allocation in naval warfare”, 11th international fusion 2008 conference in cologne, july 1-3, 2008. [7] e. pogossian, d. dionne, a. grigoryan, j. couture and e. shahbazian, “developing goals directed search models empowering strategies against single ownship air threats”, csit2009: international conference in computer sciences and information technologies, yerevan, armenia, 5 p. 2009. [8] e. pogossian, “towards adequate constructive models of mental systems”, csit2017: international conference in computer science and information technologies, yerevan, armenia, pp. 96-101, 2017. [9] e. pogossian, constructing models of being by cognizing, academy of sciences of yerevan, armenia, 2020. [10] m. buniatyan, developing software for expert knowledge/classifiers-based effective strategies formation and applications, master thesis, iiap nas ra, yerevan, armenia, 2022. s. grigoryan and e. pogossian 59 [11] f. li, s. li, c. zhu, x. lan and h. chang, “cost-effective class-imbalance aware cnn for vehicle localization and categorization in high resolution aerial images”, remote sensing, vol 9, no. 5, pp. 1-29, 2017. [12] k. zhang, ch. wang, x. yu, et al. “research on mine vehicle tracking and detection technology based on yolov5”, systems science and control engineering, vol. 10, no. 1, pp. 347-366, 2022. [13] yolov5 repository [online] available: https://github.com/ultralytics/yolov5 [14] n. sabina, m. aneesa and p. haseena, “object detection using yolo and mobilenet ssd: a comparative study”, internetional journal of engineering research&technology, vol. 11, no. 6, 134-138, 2022. [15] k. r. ahmed and p. smart, “detection using deep learning based on dilated convolution”, sensors, 21, 24, 8406, 2021. doi: 10.3390/s21248406 [16] a. bochkovskiy, ch. wang and m. liao, “yolov4: optimal speed and accuracy of object detection”. arxiv:2004.10934v1, 2020. անօդաչու արդյունավետ որոշումների կայացման ծրագրերի մշակում սեդրակ վ. գրիգորյան և էդվարդ մ. պողոսյան հհ գաա ինֆորմատիկայի և ավտոմատացման պրոբլեմների ինստիտուտ e-mail: addressforsd@gmail.com, epogossi@aua.am ամփոփում անօդաչու թռչող սարքերը (աթս, դրոն), եւ այլ անօդաչու միավորները լայն կիրառում են ստանում են տարատեսակ կարեւոր ոլորտներում, ինչպիսիք են՝ գյուղատնտեսությունը, արտակարգ իրավիճակները, ռազմական խնդիրները եւ այլն, չնայած դրանց որոշումների կայացման եղանակներում դեռ կան լավարկման հնարավորություններ՝ խուսափելու համար մարդկային գործոնի ուղղակի ներգրավվածությունից։ այսպիսի խնդիրներում առաջադիմելու նպատակով մենք մշակում ենք գործիքներ, որոնք հնարավորություն կտան աթսների կողմից ինքնուրույն արդյունավետ որոշումներ կայացնել, մասնավորապես՝ վերլուծելով ստեղծված իրավիճակը, ըստ հասցեագրված նպատակների մշակել ռազմավարություն՝ այդ նպատակներին հասնելու համար։ այս աշխատանքում մենք ձգտում ենք տալ մի լուծում միայնակ աթսի համար լուծում, որը կհայտնաբերի իրավիճակում հետաքրքրություն ներկայացնող միավորները, դրանցից կընտրի թիրախ հետեւելու, խոցելու կամ այլ նպատակի mailto:addressforsd@gmail.com mailto:epogossi@aua.am ininfluedeveloping aerial unmanned effective decision makers 60 համար՝ հիմնված փորձագիտական գիտելիքների վրա։ հաջորդիվ նաեւ բերվում են լուծման հետագա զարգացման քայլերը։ բանալի բառեր՝ օբյեկտների հայտնաբերում, որոշումների կայացում, կոմբինատոր խնդիրներ, փորձագիտական գիտելիքներ: разработка программ принятия эффективных беспилотных решений седрак в. григорян и эдвард м. погосян институт проблем информатики и автоматизации нан ра e-mail: addressforsd@gmail.com, epogossi@aua.am аннотация беспилотные летательные аппараты (бпла, дроны) и подобные беспилотные устройства наряду с возрастающим числом приложений в сельском хозяйстве, управлении при чрезвычайных ситуациях, например боевых и т.д., требуют значительного усовершенствования эффективного принятия решений. нами разрабатывается програмы, позволяющие, в частности, анализировать ситуации, а затем в соответствии с поставленными целями выбирать подходящие стратегии для достижения целей. в работе представлены описание процедуры анализа ситуации для обнаружения целевых обьектов и их отслеживания, анализа версий решений с использованием наличных знаний эксперта, выбора конкретной цели и принятия окончательного решения. kлючевые слова: обнаружение объектов, принятие решений, комбинаторные задачи, экспертные знания. mailto:addressforsd@gmail.com mailto:epogossi@aua.am intfinalsieve.dvi mathematical problems of computer science 23, 2004, 100{101. su±cient condition for the p r oper ty of a gr aph to b e b ipar tite r a fa e l r . k a m a lia n a n d v a h a g n v . mkr t c h ya n institute for informatics and automation problems of nas of ra e-mails rrkamalian@yahoo.com, vahanmkrtchyan2002@yahoo.com abstract a su±cient condition is found to determine when a graph with vertices on the integer sieve of rn is bipartite. this condition can be useful in cases when the description of a graph does not contain an appropriate bipartition of the set of its vertices. refer ences [1 ] h a r a r y f., gr a p h th e o r y , a d d is o n -w e s le y, r e a d in g , ma ,1 9 6 9 . [2 ] k o n ig d ., gr a p h e n u n d ma t r iz e s . m at. f iz. l apok., v.3 8 , p p .1 1 6 -1 1 9 , 1 9 3 1 . ´³í³ñ³ñ å³ûù³ý ·ñ³ýç »ñïïáõù³ýçáõãû³ý ñ³ù³ñ è. è.ø³ù³éû³ý ¨ ì. ì. øïñïãû³ý ²ù÷á÷áõù rn ï³ñ³íáõãû³ý ³ùµáõç³ãçí ó³ýóç íñ³ ïñí³í ·³·³ãý»ñáí ·ñ³ýç ñ³ù³ñ ·ïýí»é ¿ »ñïïáõù³ýçáõãû³ý ñ³ûï³ýçß: ²ûë ñ³ûï³ýçßá ï³ñáõ ¿ û·ï³ï³ñ éçý»é ³ûý ¹»åù»ñáõù, »ñµ ·ñ³ýç ýï³ñ³·ñáõãûáõýá ãç å³ñáõý³ïáõù ·³·³ãý»ñç µ³½ùáõãû³ý ñ³ñù³ñ³í»ï ïñáñáõù: 1 0 0 d:\sbornik\...\article_eng.dvi mathematical problems of computer science 31, 130{141, 2008. on one appr oach to optimization of recur sive function computations a r t a s h e s k . gh a z a r ya n institute for informatics and automation problems of nas of ra. e-mail: artashesg@gmail.com abstract the goal of this work is the theoretical justi¯cation and the development of a new optimizer that synthesizes programs calculating multivariate recursive functions and systems of functions. the current version of the optimizer processes a wide category of multivariate systems of recursive functions using two algorithms { stack recursion optimization and combined total replacement optimization. the results of this work can be used in development of packages, calculating the systems of recursive functions, modeling discrete multivariate systems with complex interconnections, solving boundary-value and ¯eld-value problems, etc. refer ences [1 ] ìàðàíäæÿí ã.á, “îá îäíîì ìåòîäå ñèíòåçà ïðîãðàìì ÷èñëîâûõ ôóíêöèé”, ìàòåìàòè÷åñêèå âîïðîñû êèáåðíåòèêè è âû÷èñëèòåëüíîé òåõíèêè, xvi, 1986. [2 ] ma r a n d jia n h ., general form recursive equations, cs l , p p . 5 0 1 -5 1 1 , 1 9 9 4 . [3 ] ma n n a , z., theory of computation. n y , mc gr o w-h ill 1 9 7 8 . [4 ] b a r r o n d ., r ecursive methods in programming, ge n e r a l e d it o r : s t a n le y gill a s s o c ia t e e d it o r : j. j. flo r e n t in , 1 9 6 9 . [5 ] a h o a .v ., h o p c r o ft j. e . a n d u lm a n j.d ., d ata structures and algoritms. a d d is o n w e s le y, r e a d in g , ma s s a c h u s e t t s . 1 9 8 3 . [6 ] b a r e n d r e g t , h . p ., the lambda calculus. its syntax and semantics, n o r t h -h o lla n d , 1 9 8 4 . [7 ] gh a z a r ya n a ., on one method of °exible numeration, p r o c e e d in g s o f t h e c o n fe r e n c e , cs it, p . 1 5 , 1 9 9 7 . [8 ] k n a s t e r b . une th¶eorµeme sur les fonctions d'ensembles. a n n a le s s o c . p o lo n a is e ma t h ., 6 2 , p p . 1 3 3 { 1 3 4 , 1 9 2 7 . [9 ] r ic e h . g., classes of recursively enumerable sets and their decision problems, tr a n s . a m e r . ma t h . s o c , p p . 3 5 8 { 3 6 6 , 1 9 7 4 . [1 0 ] k le e n e , s . c., introduction to m etamathematics. n e w y o r k to r o n t o , d . v a n n o s t r a n d co ., in c ., 1 9 5 2 . [1 1 ] õàëàòÿí, è. ã., ïàêåò ïðèêëàäíûõ ïðîãðàìì àâòîìàòè÷åñêèé ïðîãðàììíûé ñèíòåç. òåçèñû äîêëàäîâ òðåòüåé ðåñïóáëèêàíñêîé êîíôåðåíöèè àñïèðàíòîâ àðìÿíñêîé ññð, ×àñòü 2, åðåâàí, ññ. 16 -17, 1989. 1 3 0 a. ghazaryan 1 3 1 [1 2 ] a m d a h l g. m., validity of the single-processor approach to achieving large scale computing capabilities. in a fip s co n fe r e n c e p r o c e e d in g s vo l. 3 0 ( a t la n t ic cit y, n .j., a p r . 1 8 -2 0 ) . a fip s p r e s s , r e s t o n , v a ., p p . 4 8 3 -4 8 5 , 1 9 6 7 . è»ïáõñëçí ýáõýïóç³ý»ñç ñ³ßíù³ý ûåïçù³é³óù³ý ùç »õ³ý³ïç í»ñ³µ»ñû³é ². ô³½³ñû³ý ²ù÷á÷áõù ²ûë ³ßë³ï³ýùç ýå³ï³ïý ¿ ýáñ ûåïçù³é³óù³ý ùß³ïáõùý áõ ï»ë³ï³ýáñ»ý ³ñ¹³ñ³óí³í éçý»éá, áñá ëçý㻽áõù ¿ µ³½ù³ã³÷ é»ïáõñëçí ýáõýïóç³ý»ñ ¨ ýáõýïóç³ý»ñç ñ³ù³ï³ñ·»ñ ñ³ßíáõ íñ³·ñ»ñ: úåïçù³é³ñçý ý»ñï³û³óíáõ ï³ñµ»ñ³ïá ùß³ïíáõù ¿ é»ïáõñëçí ýáõýïóç³ý»ñç µ³½ù³ã³÷ ñ³ù³ï³ñ·»ñç é³ûý ¹³ëçª û·ï³·áñí»éáí »ñïáõ ³é·áñçãù. ëï»ï³ûçý é»ïáõñëç³ûç ûåïçù³é³óáõù ¨ éñçí ÷áë³ñçýù³ý ³é·áñçãùç ùç³íáñí³í ûåïçù³é³óáõù: ²ûë ³ßë³ï³ýùç ³ñ¹ûáõýùý»ñá ï³ñ»éç ¿ û·ï³·áñí»é ³ûý íñ³·ñ³ß³ñ»ñç ùß³ïù³ý ù»ç, áñáýù ñ³ßíáõù »ý é»ïáõñëçí ýáõýïóç³ý»ñç ñ³ù³ï³ñ·»ñ, ùá¹»é³íáñáõù »ý µ³ñ¹ ÷áëï³å³ïóáõãûáõýý»ñáí áý¹ñ³ï µ³½ù³ã³÷ ñ³ù³ï³ñ·áñ, éáõíáõù »ý »½ñ³ûçý ¨ ¹³ßï³ûçý ëý¹çñý»ñ ¨ ³ûéý: microsoft word gayane.doc ìàòåìàòè÷åñêèå âîïðîñû êèáåðíåòèêè è âû÷èñëèòåëüíîé òåõíèêè 31, 90--99, 2008. 90 î äëèííûõ êîíòóðàõ â íàïðàâëåííûõ ãðàôàõ, ïðîõîäÿùèõ ÷åðåç äàííó þ âåðøèíó ñàìâåë õ. äàðáèíÿí è èñêàíäàð à. êàðàïåòÿí институт проблем информатики и автоматизации нан ра samdarbin@ipia.sci.am, isko@ipia.sci.am аннотация ïóñòü g åñòü )12( n -âåðøèííûé ( n ≥6) íàïðàâëåííûé ãðàô ñ ìèíèìàëüíûìè ïîëуñòåïеíÿìè, íå ìеíüøèìè 1n . äîêàçûâàåòñÿ, ÷òî ÷åðåç ëþáую âåðøèíу òàêîãî ãðàôà ïðîõîäèò êîíòóð äëèíû .2n литература [1] ф. харари, теория графов, мир, москва, 1973. [2] j. bang-jensen and g. gutin, digraphs. theory. algorithms and applications. springer, 2001. [3] j. moon, topics on tournaments. n. y., 104 p., 1968. [4] m. overbeck-larish, “a theorem on pancyclic oriented graphs”, j. combin. theory ser. b 23, 168-173, 1977. [5] b. jackson, “long paths and cycles in oriented graphs”, j. graph theory, no. 5, pp. 145157, 1981. [6] z. m. song, “pancyclic oriented graphs”, j. graph theory, no. 18, pp. 461-468, 1994. 468, 1994. [7] j. bang-jensen and y. guo, “a note on vertex pancyclic oriented graphs”, odense universitet, [8] g. gutin, “characterizations of vertex pancyclic and pancyclic ordinary complete multipartite digraphs”, discrete math, v. 141, pp. 153-162, 1995. [9] с. х. дарбинян, “îöåíêà äëèí êîíòóðîâ è ïóòåé â ðåãóëÿðíûõ íàïðàâëåííûõ ãðàôàõ”, tanulmanyok , v. 135, pp. 131-144, 1982. [10] с. х. дарбинян, к. м. мосесян, “о панцикличности регулярных орграфов’, дан арм. сср, 1978, т. lxvii, № 4, ñòð. 208-211, 1978. [11] с. х. дарбинян, “о панцикличности направленных графов с большими полустепенями”, дан арм. сср, т. lxxx, № 4, ñòð. 51-54, 1985 ( ñì. òàêæå математические вопросы кибернетики и вычислительной техники, № 14, ñòð. 55-74, 1985 ). [12] ñ. õ. äàðáèíÿí, è. à. êàðàïåòÿí, “î âåðøèííîé ïàíöèêëè÷íîñòè íàïðàâëåííûõ ãðàôîâ ñ áîëüøèìè ïîëóñòåïåíÿìè”, математические вопросы кибернетики и вычислительной техники, № 29, ñòð. 66-84, 2007. [13] ñ. õ. äàðáèíÿí, è. à. êàðàïåòÿí, “î êîíòóðàõ â íàïðàâëåííûõ ãðàôàõ ïðîõîäÿùèõ ÷åðåç äàííóþ âåðøèíó”, математические вопросы кибернетики и вычислительной техники, № 31, ñòð.100--107, 2008. [14] s. darbinyan and i. karapetyan, “on vertex pancyclic oriented graphs, csitconference, pp.154-155, yerevan, armenia, 2005. ñ. äàðáèíÿí è è. êàðàïåòÿí 91 àõõõ áñ¹í³í ·ñ³ ýý»ñáõù ïñí³í ·³·³ãáí ³ýóýáõ »ñï³ñ óçïé»ñç ù³ëçý ê. ¸³ñµçýû³ý ¨ æ. î³ñ³å»ïû³ý ²ù÷á÷áõù ü»ñï³ ³ßë³ï³ýùáõù ³å³óáõóíáõù ¿, áñ »ã» ( 12 n )·³·³ã³ýç )6( n áõõõáõñ¹í³í g ·ñ³ýç ó³ýï³ó³í ·³·³ãç éáï³é ïçë³³ëïç׳ýý»ñá ÷áùñ ã»ý 1n ãíçó, ³å³ g ·ñ³ýç ûáõñ³ù³ýãûáõñ ·³·³ã ·ïýíáõù ¿ n2 »ñï³ñáõãû³ý ïáõùýáñáßí³í óçïéç íñ³: d:\sbornik\...\rgevorgyan3.dvi mathematical problems of computer science 23, 2004, 134{143. a dynamic p r ogr amming appr oach for computing similar ity of the p r otein sequences b ased on continuous functions compar ison r o b e r t k . ge vo r g ya n yerevan state university, dep. of applied mathematics and informatics. e-mail robertg@ysu.am abstract this paper introduces a dynamic programming approach for computing "continuous" similarity of the two protein sequences. the discrete dynamic programming method considers items of each comparable sequence independently; meantime there is a strong interrelation between them. to overcome this disadvantage a "continuous" sequence comparison method is developed. particularly, a certain continuous function is correlated to each comparable protein sequence, and then the comparison is made between those functions. through compressions and expansions the comparable functions are brought to the most similar representation in the meaning of a certain similarity function. by this approach the sequence comparison problem is reduced to a functional maximization problem, which is numerically solved using dynamic programming method. finally some practical results are presented with the application of described method. refer ences [1 ] a le xe e v v .m., tikh o m ir o v v .m. a n d fo m in s .v ., optimal control., n a u ka , 1 9 7 9 . [2 ] a ls t c h u l s .f., glis h w ., mille r w ., mye r s e .w . a n d l ip m a n d .j. b asic local alignment search tool. j. mo l. b io l. 2 1 5 , 4 0 3 -4 1 0 , 1 9 9 0 . [3 ] b a ir o c h a . a n d a p we ile r r . the sw iss-p r ot protein sequence data bank and its supplement tre m b l in 1999. n u c le ic a c id r e s ., 2 7 , 4 9 -5 4 , 1 9 9 9 . [4 ] b a ld i p . a n d b r u n a k s . b ioinformatics, the m achine l earning apprach. mit p r e s s , 2 0 0 1 . [5 ] b a t e m a n a ., b ir n e y e ., ce r r u t i l ., d u r b in r ., e t wille r l ., e d d y s .r ., gr i± t h s -jo n e s s ., h o we k .l ., ma r s h a ll m. a n d s o n n h a m m e r l .l .e . the p fam protein families database. n u c le ic a c id s r e s e a r c h , vo l. 3 0 , n o . 1 , 2 7 6 -2 8 0 , 2 0 0 2 . [6 ] b e llm a n r . d ynamic p rogramming. p r in s t o n u n iv. p r e s s , 1 9 5 7 . 1 3 4 r. k. gevorgyan 1 3 5 [7 ] d u r b in r . e d d y s .r ., k r o g h a ., mit c h is o n g. b iological sequence analysis. ca m b r id g e u n ive r s it y p r e s s , 1 9 9 8 . [8 ] e d d y s .r . p ro¯le hidden m arkov models. b io in fo r m a t ic s , vo l. 1 4 , n o . 9 , 7 5 5 -7 6 3 , 1 9 9 8 . [9 ] ge vo r g ya n r .k ., a continuous m ethod for e valuating d issimilarity of the p rotein sequences. p r o c e e d in g s o f t h e is a a c co n fe r e n c e o n a n a lys is , y e r e va n , a r m e n ia , 2 9 -4 0 , 2 0 0 4 . [1 0 ] gu s ¯ e ld d . algorithms on strings, trees, and sequences. ca m b r id g e u n ive r s it y p r e s s , 1 9 9 7 . [1 1 ] h e ym a n n s ., ga b r ie lya n o. r ., gh a z a r ya n h . a n d o t h e r s . towards a m etrical space of b iological sequences. p r o c e e d in g s o f t h e is a a c co n fe r e n c e o n a n a lys is , y e r e va n , a r m e n ia , 1 -1 8 , 2 0 0 4 . [1 2 ] h o r s t r ., p a r d a lo s p .m. a n d th o a i n .v . introduction to global optimization. k lu we r a c a d e m ic p u b lis h e r s , 1 9 9 5 . [1 3 ] k a n t o r o vic h l .v . a n d r u b in s t e in g.s . on a function space and certain extremum problem. d o kl.a ka d . n a u k s s s r , n 5 , 1 1 5 , 1 0 5 8 -1 0 6 1 , 1 9 5 7 . [1 4 ] l e ve n s t e in v .i. b inary codes capable of correcting insertions and reversals. s o v. p h ys . d o kl., 1 0 :7 0 7 -7 1 0 , 1 9 6 6 . [1 5 ] mikh a le vic h v .s . sequential optimization algorithms and their applications., k ib e r n e t ika , n 1 , 2 , 1 9 6 5 . [1 6 ] mo is e e v n . n . calculus of approximations in the theory of optimal tasks. n a u ka , mo s c o w, 1 9 7 1 . [1 7 ] n e e d e lm a n s .b . a n d w u n s c h c.d . a general method applicable to the search for similarities in the amino acid sequences of two proteins. j. mo l. b io l., 4 8 , 4 4 3 -4 5 3 , 1 9 7 0 . [1 8 ] p e a r s o n w .r . a n d l ip m a n d .j. improved tools for biological sequence comparison. p r o c . n a t . a c a d s c i. u s a , 8 5 , 2 4 4 4 -2 4 4 8 , 1 9 8 8 . [1 9 ] r a b in e r r . a n d ju a n g b .-h . f undamentals of speech recognition. p r e n t ic e h a ll p tr e n g le wo o d cli®s , n e w je r s e y 0 7 6 3 2 , 1 9 9 3 . [2 0 ] s a n ko ® d . a n d k r u s ka l j.b . time w arps, string e dits and m acromolecules. cs l i p u b lic a t io n s , 1 9 9 7 , is b n 1 -5 7 5 8 6 -2 1 7 -4 . [2 1 ] s e t u b a l j.c. a n d me id a n is j. introduction to computational molecular biology. p w s p u b lis h in g c o m p a n y, 1 9 9 7 . [2 2 ] 2 1 . s m it h t.f. a n d w a t e r m a n m.s . identi¯cation of common molecular subsequences. jo u r n a l o f mo le c u la r b io lo g y, 1 4 7 : 1 9 5 -1 9 7 , 1 9 8 1 . [2 3 ] 2 2 . w a s s e r s t e in l .n . m arkov processes over denumerable products of spaces describing large systems of automata. p r o b le m s o f in fo r m a t io n tr a n s is s io n 5 , 4 7 -5 2 , 1 9 6 9 . [2 4 ] 2 3 . w a t e r m a n m. introduction to computational biology. ch a p m a n a n d h a ll, 1 9 9 5 . 1 3 6 a dyn. progr. approach for comp. similarity of the protein seq. based on contin. func. comparison êåçï³ïáõó³ûçý ñ³çáñ¹³ï³ýáõãûáõýý»ñçª ³ýáý¹ñ³ï ýáõýïóç³ý»ñç íñ³ ñçùýí³í ýù³ýáõãû³ý ñ³ßí³ñïáõùá ¹çý³ùçï íñ³·ñ³íáñù³ý ù»ãá¹áí: è. î. ¶¨áñ·û³ý ²ù÷á÷áõù ²ûë ñá¹í³íáõù ý»ñï³û³óí³í ¿ ï»ýë³µ³ý³ï³ý ñ³çáñ¹³ï³ýáõãûáõýý»ñç ñ³ù»ù³ïù³ý ùç ù»ãá¹: ð³ûïýç ¹çëïñ»ï ¹çý³ùçï íñ³·ñ³íáñù³ý ùáãá¹á ûáõñ³ù³ýãûáõñ ñ³ù»ùïíáõ ñ³çáñ¹³ï³ýáõãûáõýý»ñç ³ý¹³ùý»ñá ¹çï³ñïáõù ¿ çñ³ñçó ³ýï³ë, ³ûýçýã ¹ñ³ýó ùçç¨ ï³ý áñáß³ïç ï³å»ñ: ²û¹ ã»ñáõãûáõýá ñ³õã³ñ³ñ»éáõ ñ³ù³ñ ¹çï³ñïíáõù ¿ ñ³çáñ¹³ï³ýáõãûáõýý»ñç ñ³ù»ù³ïù³ý ùç þ³ýáý¹ñ³ïþ ù»ãá¹: ²ûý ¿ª ûáõñ³ù³ýãûáõñ ñ³ù»ù³ïíáõ ñ³çáñ¹³ï³ýáõãû³ýá ñ³ù³å³ï³ëë³ýáõãû³ý ù»ç ¿ ¹ñíáõù ùç ³ýáý¹ñ³ï ýáõýïóç³ ¨ ³å³ ñ³ù»ù³ïáõãûáõýá ï³ï³ñíáõù ¿ ³û¹ ýáõýïóç³ý»ñç ùçç¨: ê»õùáõùý»ñç ¨ ó·áõùý»ñç ùççáóáí ñ³ù»ù³ïíáõ ýáõïóç³ý»ñá µ»ñíáõù »ý ³ù»ý³ýù³ý ï»ëùçª ïñí³í ýù³ýáõãû³ý ýáõýïóç³ûç çù³ëïáí: ²ûë ùáï»óù³ý ùççáóáí ëý¹çñá µ»ñíáõù ¿ ýáõýïóçáý³éç ûåïçùç½³óç³ûç ëý¹ñç, áñá ãí³å»ë éáõí»éáõ ñ³ù³ñ ý»ñï³û³óí³í ¿ ùç ¹çý³ùçï íñ³·ñ³íáñù³ý ù»ãá¹: ²ßë³ï³ýùáõù µ»ñí³í »ý ý³¨ áñáß åñ³ïïçï ³ñ¹ûáõýùý»ñª ý»ñï³û³óí³í ù»ãá¹ç ïçñ³éù³ùµ: microsoft word tpel.doc математические вопросы кибернетики и вычислительной техники 32, 112--115, 2009. 112 об асимптотике коэффициентов представления правильно меняющихся распределений гор п. авагян государственный педагогический университет им. х. абовяна gor_avakyan@yahoo.com аннотация в настоящей работе рассмотрены распределения 1}{ np правильно меняющиеся при n , допускающие асимптотически постоянную медленно меняющуюся компоненту, лог-выпуклые вниз, определяемые последовательностью 1}{ n коэффицентов с заданным основным представлением. на основе асимптотического разложения с двумя членами распределения }{ np находится асимптотическое разложение для }{ n при n . литература 1. э. а. даниелян, г. п. авагян, “об одном представлении правильно меняющихся распределений”, математика в высшей школе, т. 4, с. 17-23, 2008. 2. j. astola, e. danielian, “frequency distributions in biomolecular systems and growing networks”, tampere: ticsp series 31, 2006. 3. g. p. avagyan, “on dstribution’s constant slowly varying component”, proceedings of ysu, pp. 20-23, 2006. 4. э. а. даниелян, г. п. авагян, “об асимптотических разложениях правильно меняющихся распределений”, доклады нан армении, с. 21-31, 2009. 5. и.м., рыжик и. с. градштейн, таблицы интегралов, сумм, рядов и произведений. –м. ленинград: титтл, 1951. կանոնավոր փոփոխվող բաշխումների ներկայացման գործակիցների ասիպտոտիկայի մասին գ. ավագյան ամփոփում ներկայացված աշխատանքում դիտարկվում են 1}{ np կանոնավոր փոփոխվող բաշխումներ, որոնք թույլ են տալիս ասիմպտոտորեն հաստատուն դանդաղ փոփոխվող բաղադրիչ, լոգ-ուռուցիկ են դեպի ներգև և տրված են իրենց հիմնական ներկայացումը որոշող  1}{ n գործակիցների հաջորդականությամբ:  1}{ np -ի, երբ n ասիմպտոտական վերլուծության հիման վրա գտնված է }{ n -երի ասիմպտոտական վերլուծությունը: microsoft word 8_h_ avagyan.doc mathematical problems of computer science 39, 66--71, 2013. 66 a clutter reduction algorithm in non-coherent lfm cw radars hovhannes r. avagyan institute of radiophysics and electronics of nas of ra e-mail: hov.avagyan@yahoo.com abstract the algorithm of clutter reduction in lfm cw radars is presented. similarities in sequential measurements are formalized via the introduced identification function. the ensemble averaging allows significantly to simplify the moving target identification. keywords: algorithm, clutter reduction, fmcw radar, moving target indication. 1. introduction portable cw radars have recently found wide application in military and civil areas taking into account a number of requirements, such as: small dimensions, competitive price, and so on. among the main areas of application for the mentioned radars are surveillance security systems, vehicle collision avoidance systems, etc. [1], [2]. the recently developed applications imposed specific requirements on radars parameters; hence, the parameters of such systems should be improved. the mentioned systems are considered as “intellectual” radio devices and need a serious software support for effective operation. for this purpose, the development of efficient algorithms is vital. as it is known, one of the most important parameters of radar systems is its dynamic range. the total capability of the system can be improved by dynamic range boosting, although it is not always sufficient. sometimes the indirect improvement of system parameters can be more efficient. these include: the suppression of the reflected signal from the local objects (clutter), which will have the same result as the dynamic range boosting. this problem, in traditional coherent-pulse radar systems, is solved by periodical comparison of the received signals of more than two neighboring pulses, and detection of their spectral differences [6]. in cw radar the situation substantially differs, since in this case the spectra are not fully identical and there is only likeness between them. thus, the problem is summarized in detecting similarities of signal spectra received from the neighboring sequential periods. h. avagyan 67 2. system and measurements one of the most effective ways of distance measurement with cw radars is using of a linear frequency modulation (lfm). the system transmits lfm signal with the periodically rising and falling frequency slopes. such measurements during the slopes will allow us to identify the distance and the radial speed of the target. the signal reflected from the target contains a frequency shift, simultaneously due to the signal delay and doppler effect. the received signal is down converting and the result is the beating signal shown in fig. 1. fig. 1. fmcw triangular waveform and beating signal the achieved signal is the sequence of signals with and frequencies corresponding to the lfm upand down-periods. depending on the target movement direction and the period of the transmitted signal, the possible values of the resulting frequency are: thus, it is obvious, that the detected frequencies of the signal reflected from local objects will be the same for rising and falling slopes. meanwhile for the case of the moving object a clutter reduction algorithm in non-coherent lfm cw radars 68 they will differ by the value of twice of the doppler frequency [3], [5]. we have carried out a huge amount of spectral measurements including clutter, human and vehicle targets. the spectra of down converted signals for upand down-periods are shown in fig. 2 (clutter), meanwhile the same for the moving targets is shown in fig. 3. although, the spectra of the signal reflected from the clutter are not strictly identical, but there is a qualitative resemblance between them. fig.2. spectra of clutter (solid – upspectrum, dashed – downspectrum) fig.3. spectra of clutter and moving targets (solid – upspectrum, dashed – downspectrum) taking into account aforementioned considerations, we can conclude that for noncoherent radars it is also possible to suppress the clutter, which will improve the potential of the system if the respective processing algorithm is applied. the obtained results are h. avagyan 69 presented and discussed in [4]. the essence of the tested method is in comparison and suppression of the spectral peaks with the same frequencies. actually, the method is simple enough and realizable, but there is not enough resemblance between the spectral peaks, caused by the system non-coherence and noise existence, that’s why it is hard to get a desirable level of clutter suppression. based on the considerations, that spectral differences are caused by the system non-coherence, it can be concluded that they will have a random character. the method of ensemble averaging of the random processes can be successfully used in this case as well. we have already done some preliminary estimation on that topic, and achieved results proving the reliability of the mentioned considerations. in order to describe properly, we introduce an identification function as the spectrum of point-by-point ratio of two sequentially measured periods. such identification function of spectral components vs. frequency is shown in fig. 4. a) b) fig.4. identification function vs. frequency (a without averaging, b 10 times ensemble averaging) 3. weather conditions and conclusion the recent observations showed, that there is a dependency between the reliability of detection and the weather. weather conditions have substantial influence on the quality of clutter suppression. a clutter reduction algorithm in non-coherent lfm cw radars 70 fig. 5. spectra of land clutter (solid – 1st period spectrum, points – 2nd period spectrum, dashed difference) fig. 6. spectra of land clutter with a layer of 10 cm snow (solid – 1st period spectrum, points – 2nd period spectrum, dashed difference) it is well known that, ideally, the spectrum of the received radar signal must be invariant during lfm sawtooth periods, if there is no moving object in coverage, but really, they differ a little due to fading. fft spectra and the respective difference have been counted during two adjacent periods. the weather dependent results are shown in fig. 5 and fig. 6. it is obvious that snow coating significantly improves the situation and the difference of spectra in two adjacent periods is small enough. probably, it is caused by the reduction of small-scale fading [6]. h. avagyan 71 references [1] i. komarov and s. smolskiy, “fundamentals of short-range fm radar”, artech house radar library, pp. 289, 2003. [2] w. wang, j. cai and y. yang, “a novel method to identify multi-target by transformable periods lfm waveform”, proc. of international conference on communications, circuits and systems, vol. 2, pp. 744-747, 2005. [3] a. muzhikyan, a. hakhoumian, s. martirosyan, v. nikoghosyan, n. poghosyan, t. poghosyan, k. rustamyan, and t. zakaryan, “short-range ku-band hybrid-mode cw-lfm radar”, proc. of 11-th international radar symposium irs, vilnius, lithuania , pp.1-3, 2010. [4] a. hakhoumian, s. martirosyan, a. muzhikyan, v. nikoghosyan, n. poghosyan, t. poghosyan, k. rustamyan and t. zakaryan, “light-weight short-range ku-band cw-lfm radar,” proc. of international conference “the technique of microwave and thz waves and its application in biomedical and radar technologies and in remote sensing”, pp. 87-90, (irphe'2010), 2010. [5] c. c. duarte, b. p. dorta naranjo, a. a. lopez and a. b. del campo, “cwlfm radar for ship detection and identification,” ieee aerospace and electronic systems magazine, vol. 22, no. 2, pp. 22-26, 2007. [6] b. sklar, “rayleigh fading channels in mobile digital communication systems. part i: characterization” , ieee communications magazine, vol. 35, no. 9, pp. 136-146, 1997. submitted 09.01.2013, accepted 20.02.2013. lfm-cw ոչ-կոհերենտ ռադարներում տեղային օբյեկտներից անդրադարձած ազդանշանի ճնշման մի ալգորիթմ հ. ավագյան ամփոփում ներկայացված է lfm-cw ռադարների համար տեղային օբյեկտներից անդրադարձած ազդանշանի ճնշման ալգորիթմ: ներմուծված նույնականացման ֆունկցիայի միջոցով ձևակերպված են հաջորդական չափումների միջև առկա նմանությունները: ցույց է տրված, որ ըստ անսամբլի միջինացումը թույլ է տալիս էականորեն պարզեցնել շարժվող թիրախների հայտնաբերումը: алгоритм подавления фона местных объектов в некогерентных рлс непрерывного излучения с лчм о. авакян аннотация представлен алгоритм для подавления фона местных объектов в некогерентных рлс непрерывного излучения с лчм. с помощью введенной функции идентичности сформулирована степень сходства в последовательных измерениях. показано, что усреднение по ансамблю позволяет значительно упростить процедуру селекции движущихся целей. d:\user\sbornik_38_pdf\28.dvi mathematical problems of computer science 38, 70{71, 2012. m ember ship functions or ®-cuts? algor ithmic (constr uctivist) analysis justi¯es an i nter val appr oach v la d ik k r e in o vic h department of computer science, university of texas at el paso, usa, vladik@utep.edu in h is p io n e e r in g p a p e r s , ig o r za s la vs ky s t a r t e d a n a lg o r it h m ic ( c o n s t r u c t ivis t ) a n a lys is o f fu z z y lo g ic . in t h is p a p e r , we e xt e n d t h is a n a lys is t o fu z z y m a t h e m a t ic s a n d fu z z y d a t a p r o c e s s in g . s p e c i¯ c a lly, we s h o w t h a t t h e t wo m a t h e m a t ic a lly e qu iva le n t r e p r e s e n t a t io n s o f a fu z z y n u m b e r { b y a m e m b e r s h ip fu n c t io n a n d b y ®-c u t s { a r e not a lg o r it h m ic a lly e qu iva le n t , a n d o n ly t h e ®-c u t r e p r e s e n t a t io n e n a b le s u s t o e ± c ie n t ly p r o c e s s fu z z y d a t a . 1 fir s t r e s u lt : two r e p r e s e n t a t io n s a r e n o t e qu iva le n t de¯nition 1. b y a c -m e m b e r s h ip fu n c t io n , we mean a tuple consisting of two real numbers ¢ and ¢ and a function ¹ : h ¢ ; ¢ i ! [0 ; 1 ] for which ¹ ( ¢ ) = ¹ ³ ¢ ´ = 0 , m a x x ¹ ( x) = 1 , and a · b · c implies that ¹( b ) ¸ m in ( ¹ ( a) ; ¹( c) ) . de¯nition 2. w e say that a c-membership function h ¢ ; ¢ ; ¹i is c o m p u t a b le if both real numbers ¢ and ¢ are computable and the function ¹ is computable. de¯nition 3. b y a fa m ily o f ®-c u t s ( o r s im p ly ®-cuts, fo r s h o r t ) c o r r e s p o n d in g t o a c m e m b e r s h ip fu n c t io n ¹, we m e a n a p a ir o f m a p p in g s x : [0 ; 1 ] ! ir a n d x : [0 ; 1 ] ! ir fo r wh ic h , fo r e ve r y ® 2 [0 ; 1 ], we h a ve fx : ¹( x ) ¸ ®g = [x ( ®) ; x( ® ) ]. de¯nition 4. w e say that ®-cuts are c o m p u t a b le if both mapping x and x are computable. p r oposition 1. there exists a computable c-membership function for which the corresponding ®-cuts are not computable. p r oposition 2. there exist computable ®-cuts for which the corresponding c-membership function is not computable. 2 on ly ®-cu t s gu a r a n t e e a lg o r it h m ic fu z z y d a t a p r o c e s s in g s in c e t h e t wo r e p r e s e n t a t io n s o f fu z z y a r e n o t c o m p u t a t io n a lly e qu iva le n t , it is d e s ir a b le t o a n a lyz e wh ic h o f t h e m le a d s t o a n a lg o r it h m ic fu z z y d a t a p r o c e s s in g . h e r e a r e t h e r e s u lt s o f t h is a n a lys is : fu z z y d a t a p r o c e s s in g is c o m p u t a b le fo r ®-c u t s b u t , in g e n e r a l, n o t c o m p u t a b le fo r m e m b e r s h ip fu n c t io n s . 7 0 v. kreinovich 7 1 de¯nition 5. l et ¹1; : : : ; ¹n be membership functions, and let f ( x1; : : : ; xn ) be a function. b y the r e s u lt of applying f to fuzzy sets ¹1; : : : ; ¹n, we mean a membership function de¯ned by the formula ¹( y ) = m a x x1;:::;xn:y=f (x1;:::;xn) m in ( ¹1 ( x1 ) ; : : : ; ¹n ( xn ) ) : p r oposition 3. there exists a computable c-membership function ¹1 ( x1 ) and a computable function f ( x1 ) for which the result ¹ of applying f to ¹1 is not computable. p r oposition 4. there exists an algorithm that, given n computable families of ®-cuts corresponding to the membership functions ¹1; : : : ; ¹n and a computable function f ( x1; : : : ; xn ) , returns computable ®-cuts for the result ¹ of applying f to ¹1; : : : ; ¹n. 3 a u xilia r y r e s u lt : w h y m in a n d n o t a n y ot h e r a n d -op e r a t io n w e wa n t a ll t h e p r o p e r t y t o s a t is fy t h e \ c o n ve xit y" c o n d it io n , t h a t if a · b · c, t h e n ¹ ( b) ¸ m in ( ¹ ( a) ; ¹( c) ) . s o m e t im e s , we kn o w t h a t t h e a c t u a l va lu e x s a t is ¯ e s two p r o p e r t ie s s0 a n d s00 c h a r a c t e r iz e d b y m e m b e r s h ip fu n c t io n s ¹0 ( x ) a n d ¹00 ( x ) ; t h e n , t h e d e g r e e ¹( x ) t o wh ic h a r e a l n u m b e r x is c o n s is t e n t wit h t h is in fo r m a t io n c a n b e d e s c r ib e d a s ¹( x ) = f& ( ¹ 0 ( x) ; ¹00 ( x) ) . it is r e a s o n a b le t o r e qu ir e t h a t t h is c o m b in e d p r o p e r t y s h o u ld a ls o b e \ c o n ve x" ( in t h e a b o ve s e n s e ) . de¯nition 6. a function ¹ : ir ! [0 ; 1 ] is called f-c o n ve x if a · b · c implies that ¹( b ) ¸ m in ( ¹( a ) ; ¹ ( c) ) . de¯nition 7. b y a g e n e r a liz e d a n d -o p e r a t io n , we mean a function f : [0 ; 1 ]£[0 ; 1 ] ! [0 ; 1 ] which satis¯es the following two properties: ² for all a, a0, b, and b0, if a · a0 and b · b0, then f ( a; b ) · f ( a0; b0 ) (monotonicity); ² for all a, we have f ( a; 1 ) = f ( 1 ; a ) = a. p r oposition 5. l et f ( a; b) be a generalized and-operation. then, the following two conditions are equivalent to each other: ² for every two f-convex functions ¹0 ( x ) and ¹00 ( x) , the function ¹( x ) = f ( ¹0 ( x ) ; ¹00 ( x ) ) is also f-convex; ² f ( a; b ) = m in ( a; b ) . r e fe r e n c e s [1 ] b . a . k u s h n e r , l ectures on constructive m athematical analysis, a m e r . ma t h . s o c ., p r o vid e n c e , r h o d e is la n d , 1 9 8 4 . [2 ] h . t. n g u ye n a n d e . a . w a lke r , f irst course in f uzzy l ogic, cr c p r e s s , b o c a r a t o n , flo r id a , 2 0 0 6 . [3 ] i. d . za s la vs ky, \ fu z z y c o n s t r u c t ive lo g ic " , in : studies in constructive mathematics and mathematical logic. p art xi, zap. nauchn. sem. p om i, 2 0 0 8 , v o l. 3 5 8 , p p . 1 3 0 { 1 5 2 ; e n g lis h t r a n s la t io n in j ournal of m athematical sciences, 2 0 0 9 , v o l. 1 5 8 , n o . 5 , p p . 6 7 7 { 6 8 8 . mathematical problems of computer science 59, 45–56, 2023. doi: 10.51408/1963-0101 udc 004.891.3 expert knowledge-based rgt solvers for software testing mane p. buniatyan1, sedrak v. grigoryan2 and emma h. danielyan3 1synopsys armenia, yerevan, armenia 2institute for informatics and automation problems of nas ra,yerevan, armenia 3epam systems inc., yerevan, armenia e-mail: buniatyanmane@gmail.com, addressforsd@gmail.com, emma danielyan@yahoo.com abstract program testing is a way of assessing the quality of software and reducing the risk of software failure in operation [1]. quality issues can cause as financial loss as well as harm to human lives (e.g., when the bug is in medical instruments, cars, etc.). so, it is very hard to underestimate the importance of testing. there are multiple testing techniques, which are split into 3 major categories. one of them includes experience-based techniques. test cases and scenarios used in experience-based testing are derived from the tester’s knowledge and intuition, as well as their experience with similar applications and technologies. these techniques can be helpful in identifying tests that are not identified easily by other more systematic techniques. depending on the tester’s approach and experience, experience-based techniques may achieve widely varying degrees of coverage and effectiveness [1]. we propose a method for automation of experience-based testing via a class of combinatorial problems (rgt class). a solver is developed for the class. it acquires expert knowledge and elaborates effective strategies for rgt problems [2]. the proposed method generates test cases dynamically based on the response of the program. the adequacy of the method is being experimented for ”blender” open-source application, which has python api allowing to experiment with testing and analyze test results. keywords: rgt class, rgt solver, software testing, expert systems. article info: received 25 september 2022; sent for review 11 october 2022; accepted 07 february 2023. acknowledgement: the authors express their deep gratitude to dr. edward pogossian for his contribution and constructive comments to the work. 1. introduction software testing is an approach to assess the quality of software and to reduce the risk of its failure in operation [1]. 45 46 expert knowledge-based rgt solvers for software testing in [1], testing techniques are divided into 3 groups: black-box, white-box and experiencebased techniques. in the case of the last one, test cases are based on the testers’ knowledge and intuition, on experience with similar applications and technologies. these techniques are efficient in identifying tests that are not identified easily by other more systematic techniques as well as when there is a limited testing time or incomplete specifications [1]. according to the world quality report 2021-2022 [3], one of the current trends in quality assurance and software testing is test automation. test automation has the following benefits [1]: • saves time by reducing repetitive manual work • provides greater consistency and repeatability • allows to evaluate the situation more objectively based on static measures, coverage reports, etc. • provides more accurate information about the current state of testing based on gathered statistics, test progress, defect rates and performance. there is a way to automate test case generation, known as the model-based testing (mbt). mbt is a technique for generating a test suite from requirements [4]. instead of individual tests creation, testers create models that allow generating test cases automatically. these methods can be used in regression testing and are especially useful when the system changes frequently. in this case, the test suite can be regenerated easily by adjusting the model instead of readjusting each test case separately. mbt has three important components [4]: • a model (requirement, information, workflow, architectural, behavioral, configuration, deployment, performance, risk, environment, and usage models [5]) • a test-generation algorithm • tools generating a supporting infrastructure (including the expected output). mbt tools are meant to generate test suites by manipulating either with input data or behavior without handling both simultaneously. generated test cases do not provide ways to test the system dynamically (the choice of modules to testing depends on the previous test results). software testing can be considered as a combinatorial problem between a tester and states of a program. hence, testing can be also considered as a representative of reproducible game tree (rgt) class problems. rgt is a class of combinatorial problems, for which the space of solutions is a reproducible game tree. these problems meet the following requirements [6]: • there are interacting actors (players, competitors, etc.) performing identified types of actions in specified moments of time and specified types of situations • there are identified benefits for each actor • there are descriptions of situations in which actors act in and are transformed after actions. m. buniatyan, s. grigoryan and e. danielyan 47 for such problems with a given arbitrary situation x and an actor a, who is going to act in x, we can generate a corresponding game tree gt(x, a) comprising all the games started from x. games represent all possible sequences of legal actions for players and situations that they can create from the given initial, or the root situation x. in our consideration, the games are finite and end with one of the goal situations of the problem [6]. assuming that a plays according to a deterministic program, a strategy, the gt(x, a) represents, in fact, all possible performance trees of the strategies from x. in that sense, the gt(x, a) determines the space of all possible solutions from the situation x. with the given criterion k to evaluate the quality of strategies, we can define the best strategy s*(x, a) and the corresponding best action of a from x [6]. rgt class includes important problems like computer networks intrusion protection, optimal management and marketing strategy elaboration in competitive environments, testing of programs, defense of military units from various types of attacks, communication problems, certain types of teaching, chess and chess-like games [2]. one of the advantages of rgt class is that these problems are reducible to the standard kernel problems k. kmethodology multiplies the achievements for particular problems of this class. distributed development of this methodology is possible. k-methodology enhances the effectiveness of rgt solvers providing answers to urgent rgt questions including the following ones [2]: • measurement of the effectiveness of solvers • analysis and typification of combating knowledge • construction of knowledge-based solvers • regular acquisition of rgt expert knowledge and enhancing the effectiveness of solvers. the validity of k-methodology was proved for certain rgt problems including chess, network intrusion protection, navy defense from attacks, management, marketing etc. [2]. rgt solver is a software that acquires expert knowledge and elaborates effective strategies for rgt problems [2]. it is a universal tool for solving rgt-class problems. strategy searching and game tree. as already mentioned, the space of solutions for rgt problems is a reproducible game tree, and with the given criteria, we can evaluate and choose the best possible actions in given situations for the given actor. as the combinatorial complexity of the mentioned problems is huge, we need to reduce the game tree. otherwise, the computer’s computational resources (memory and storage) will not be enough to solve them. c. shannon suggested reducing the tree by building it until the resources are expired. it is not an effective way because we waste our resources to compute steps that will not improve the current situation. another approach, suggested by m. botvinnik, is to consider only those steps that have potential benefit in the current case, i.e., we should not examine the steps that have no meaning. we can evaluate the possible usefulness of an action with the knowledge (without reviewing the opponent’s answers) and choose the most profitable one. then, by checking the opponent’s potential actions, we can build the game tree and choose the best move in a given situation [7]. the solver builds the game tree, evaluates situations with the knowledge, then chooses the best action using the minimax algorithm. 48 expert knowledge-based rgt solvers for software testing the purpose of this paper: testing of programs can be considered as an rgt problem, and rgt solver can be used for experience-based testing as an expert system when the corresponding knowledge is available. in this work, we aim to provide a definition of testing problems as rgt problems, a way of formulating knowledge, and an approach for proper assessment of tested programs, which also covers the drawbacks of model-based testing approaches (in particular, combining different behaviors and input data, running both functional and non-functional tests at the same time, and generating tests dynamically). thus, the following open questions are addressed: 1. what kind of knowledge are we going to use, who are the actors as well as what are their possible actions? 2. how to evaluate each situation, what kind of goals each actor has, etc.? overall, this leads to proposing a model for representing an experience-based testing as an rgt problem. 2. reduction of program testing to rgt class in rgt problems, it is essential to define the situations, the actors, the actions, and benefits for each of them. let’s define these terms for program testing. the actors in software testing are the system under test (i.e., the program) and the tester. note, that unlike some other problems in the rgt class (e.g., like chess), where the opponent tries to make counteraction, in testing the program just responds to the tester’s actions. the actions are any valid elementary operations that can be performed with the program. while building the ”game” tree, the solver dynamically combines these actions, creates test cases and executes them depending on the response of the program. note, that not all combinations of the elementary operations are meaningful from the perspective of the user (e.g., actions that have no effect or are not connected with each other). that is why we need to find a way to control these combinations. the actions of the program are actually only responses to the tester’s actions. the situations are the current states of the program. we can estimate the current situations with [0;1] numbers, where 0 means that no bugs are found, 1that the program is in a critical state and is not usable. the numbers in-between 0 and 1 are intermediate values, and situations with values closer to 1 are worse than situations with values closer to 0. we suggest the following criteria for evaluating the current state of programs (these criteria can be expanded later): • existence of bugs (difference between expected and observed results): different bugs have different importance; when the main functionalities of the program do not work as expected, the program becomes useless (e.g., if the user is not able to log into a social network, save the result of the accomplished job, do a transfer in the banking system, etc.). • performance degradation: we all would like to have fast, high performing programs, but unfortunately it is not always possible. performance degradation in a part of the program that is used frequently will cause to slowdown the work, but if it is in a part m. buniatyan, s. grigoryan and e. danielyan 49 that can be done without human interaction and/or is performing rarely, then it can be acceptable. • security: this is essential for some programs (e.g., banking system, strategic information storing, transfers, etc.). • crashes and hangovers: this is always bad, and in some cases, they can even cause to a fatal problem, like losing the whole work performed. in most situations, this is not acceptable. we need to take into account the number of problems, as well as their severity and importance, the sequence of actions causing the problem (i.e., how frequently the problem occurs in ”real life”). a bug in a very important functionality is worse than a crash that users might not even encounter, but, on the other hand, having lots of ”minor” issues in the program is also not acceptable. when one of the main functionalities does not meet the requirements mentioned above, the program is in a critical state, and it cannot be delivered to customers. the importance of each functionality is considered as a multiplier for the appropriate criterion. the current state of the program can be measured with the following evaluation function: st = mc ∗ c + mb ∗ b + mp ∗ p + ms ∗ s {1}, where mc, mb, mp, ms ∈ [0; 1], c, b, p, s = {0 | 1}. c, b, p and s are boolean variables, that show the existence of crashes/hangovers, bugs, performance degradations or security problems respectively (1 if the mentioned problems occurred, otherwise 0). mc, mb, mp and ms are multipliers for the occurred problems (they show the importance of the broken functionality). any occurred problem is counted only once, so if, for example, a crash occurs, even if it relates to a security problem or it is a bug (obviously, it is not an expected result) we will consider c = 1, b = 0, s = 0 and p = 0. if the current state of the program is bigger than 1, we consider it as 1. 3. rgt expert knowledge formatting for testing error guessing, exploratory testing and checklist-based testing are representatives of experience-based techniques [1]. considering the characteristics of each of these techniques, we propose the following usage of the solver: by reviewing issues occurred before, the usage of the program and its main functionalities, we create checklists. in the solver, checklists are represented as plans, and the checklists’ actions as goals. based on the coverage reports, the source files responsible for each action can be defined. these connections help to prioritize the created checklists. the user can also define priorities depending on the module he/she is most interested in. checklists lead to the creation of a game tree. each branch in the tree is a test case. it is important to mention that actions in the checklists are general, i.e., many elementary actions can correspond to one action in the checklist. it allows you to combine multiple actions and build a tree. checklists define if it still needs to proceed to the next steps or not in case of a defect occurrences in the current step. multipliers in formula {1} are also given as knowledge for the solver. they show the importance of user action. note, that multipliers should be defined for both elementary 50 expert knowledge-based rgt solvers for software testing and checklist actions. the same elementary action in different situations can have different importance, e.g., if the user tries to save a text file it is more important to save the text than the style. we multiply both multipliers to get one for the action. imagine that in the example below, mb = 0.8 for the elementary action “save” and for the following checklists of actions ”open the program, add text, save”, “open an existing text file, change the style, save”. let’s say we have mb = 1 for the “save” in the checklist1 and mb = 0.6 for the “save” in the checklist2. in this case, if the program is not able to save the text, we will have mb=1*0.8=0.8 and for the second case: mb=0.6*0.8 = 0.48. thus, the first case will be considered worse than the second one. in the case of performance degradation, we need to multiply mp with the coefficient showing how many times the performance was slowed down or how much longer it takes to perform the same action. e.g., if the performance is 2x slower than expected, we need to multiply mp with 2. the testing continues until a. the given time is expired, b. all/chosen checklists are checked or c. if the program gets into a critical state. 4. rgt solver experiments in program testing we have chosen the blender program as a system under test. it is an open-source 3d modeling program with a python interface that can be used for testing. in order to understand how the program testing solver works and how the knowledge and checklists can be represented, let’s study an example. to understand how the knowledge and checklists can be represented, let us review an example. the checklist below checks some of the main functionalities of the program: fig. 1. checklist example to keep it simple, we just added a few basic operations, but this list can be enlarged if needed. the operations in this checklist can be independent, like lines 6 and 7. but if this was a checklist based on the previous failures or a user story, then all steps would depend on each other. this checklist could be used if we had limited testing time and could only check the main operations to make sure that there were no critical issues (like a smoke test). the first line of the checklist (i.e., the comment) represents the name of the checklist and the source file which is associated with the checklist (here, as we don’t know the corresponding m. buniatyan, s. grigoryan and e. danielyan 51 source file, we put x.cpp just to show the structure of the checklist. the source file is not mandatory). if some multipliers are absent in the checklist, we assign 0 to them (e.g., ms=0 for all actions in checklist below, because they could not lead to security problems). the variable nextstep is used to determine whether the next step should be performed or not in case of bug in the current step (e.g., if the user is not able to move the 3d cursor it is still somewhere in the scene and the user can add objects). in line 3 we open the program. if it crashes it is a critical state for the program, thus mc=1. next to mp there is the expected time the operation should take (mp/5s/). if it takes 25 seconds, we multiply mp by 5. as this operation is not repeatable and happens only once, when the work starts, its performance is not very important, but yet the user cannot wait for about 10 minutes to start working. as the performance depends on the users’ computer, the performance parameters are defined for minimum system requirements of the program. in the example above, we just used values based on local resources. in line 4, we need to move the 3d cursor. 3d cursor position defines where the object is being added. it can also be used as a 3d view orientation to define where to move objects, to move the pivot point to the 3d cursor, as the rotation point in the spin tool, etc. so, it is a quite important feature, but in case it does not work users can still find workarounds. note that there is no expected time next to mp for this action. it is because this action should work simultaneously with the click (i.e., should not take noticeable time). like other actions in the checklist, this is one of the basic operations, so crash is unacceptable here, thus mc = 0.9. note that all the multipliers here are conditional and this is just an example. in real world example, probably, multipliers should be chosen more thoroughly. nextstep is 1 here, because even if the 3d cursor cannot be moved, we are still able to add an object. to perform this step using the python api we do the following: fig. 2. elementary operation: move 3d cursor 52 expert knowledge-based rgt solvers for software testing this is an elementary operation for moving the 3d cursor. the first line comment shows the corresponding general operation (in the checklist) and the multiplier. as in this case only 1 elementary operation corresponds to the checklist operation, its multiplier is 1. note that the case is not always the same (the coordinates are randomly generated) and the test also checks if the operation was performed successfully or not. in the 5th line of the checklist, we have the ”add object” operation. many elementary operations correspond to this operation (see fig. 3): there are lots of groups of objects, and each group itself contains various objects. fig. 3. add object the python code below is an example of the “add object” operation. it adds a cube in the current location of the cursor. as all objects can be used for creating different 3d models, and their importance is dependent on what exactly the user tries to create m=1 for all objects. note that if the object is not added then we cannot perform the next action, i.e., we cannot change its geometry. the last command in the checklist is “change geometry”. first of all, the user should switch to the edit mode in order to change the object’s geometry, i.e., move the object’s vertices, edges and faces, and then perform the corresponding operations. for this general action, there are 3 possible elementary actions (move vertices, edges, faces). all of them are important while creating a 3d model, but considering the fact that if a user is not able to move the edge, he/she can choose vertices of the edge and move them together (so that the edge will be moved), or choose all edges/vertices of a face and move it. the most important one in those operations is moving vertices, and then edges, then surfaces. thus, in this case, m. buniatyan, s. grigoryan and e. danielyan 53 fig. 4. elementary operation: add object the multiplier for each operation will be different: fig. 5. elementary operation: change geometry for the given example, the solver moves the 3d cursor to different positions, adds different objects, changes their geometry, and makes sure that these operations work as expected for different objects (i.e., checks that the python tests are passing). to check how the solver behaves if the operation does not work, we can simply use assertnotequal function instead of assertequal (e.g., instead of “assertequal(bpy.context.scene.cursor.location.x, x)” we can write “assertnotequal( bpy.context.scene.cursor.location.x, x)”). the solver will combine different elementary tests together, create test cases and run them. to run tests, we use the following command: in order to use the solver for different programs, we use a configuration file, which defines how to run tests (e.g., paths to test cases, checklists and elementary operations). 54 expert knowledge-based rgt solvers for software testing fig. 6. command for running a test 5. conclusion we propose a new approach for test automation and test results evaluation considering the testing of programs as a rgt-class problem. in this work: 1. tools defining the types of knowledge for testing the target application are described. the described knowledge is being integrated into rgt solver and being used to run test cases, test scenarios with later evaluation of test results. 2. an approach for evaluating the state of the program during the testing is proposed. 3. the adequacy of the proposed approach is being experimented with the open-source blender application. 4. the proposed approach solves drawbacks of the model-based testing approach, namely, allows to generate test cases dynamically. the described solution is generic for the rgt solver and can be used for testing various applications. references [1] k. olsen and m. posthuma and s. ulrich, “ certified tester foundation level syllalbus”, international software testing qualifications board, pp. 56–62, 2019. [2] e. pogossian, constructing models of being by cognizing. yerevan, pp. 150–159, 2020. [3] world quality report, capgemini, sogeti, micro focus, pp 16–37, 2021 [4] d. rakhi, j. ashish, n. karunanithi, j. leaton, c. lott, g. patton and b. horowitz, “model-based testing in practice”, proceedings of the 1999 international conference on software engineering (ieee cat. no.99cb37002), los angeles, ca, usa, 1999, pp. 285-294, doi: 10.1145/302405.302640. [5] i. schieferdecker and a. hoffmann, model-based testing, ieee software 29.1, pp. 14–18, 2012. [6] e. pogossian, v. vahradyan a. grigoryan, on competing agents consistent with expert knowledge, proceedings of second international workshop, ais-adm 2007,autonomous intelligent systems: multi-agents and data mining, st. petersburg, russia, pp. 229–241, 2007. [7] m. botvinnik, computers in chess: solving inexact search problems, springer-verlag, new york, 1983. m. buniatyan, s. grigoryan and e. danielyan 5 5 öáñó³·çï³ï³ý ·çï»éçùý»ñç íñ³ ñçùýí³í rgt solver-ç ïçñ³éáõùá íñ³·ñ³ûçý ³å³ñáíù³ý ã»ëï³íáñù³ý ëý¹ñáõù ø³ý» ä. ´áõýç³ãû³ý1, 껹ñ³ï ì. ¶ñç·áñû³ý 2 ¨ ¾ùù³ ð. ¸³ýç»éû³ý3 1synopsys ð³û³ëï³ý, ºñ¨³ý 2ðð ¶²² æýýáñù³ïçï³ûç ¨ ³íïáù³ï³óù³ý åñáµé»ùý»ñç çýëïçïáõï, ºñ¨³ý, ð³û³ëï³ý 3 epam ð³û³ëï³ý, ºñ¨³ý e-mail: buniatyanmane@gmail.com, addressforsd@gmail.com, emma danielyan@yahoo.com ²ù÷á÷áõù â»ëï³íáñáõùá íñ³·ñç áñ³ïá ·ý³ñ³ï»éáõ ¨ ß³ñ³·áñíù³ý ù»ç íñ³·ñ³ûçý ³å³ñáíù³ý ó³ëáõù³ý éçëï»ñá ýí³½»óý»éáõ ùççáó ¿: ìñ³·ñáõù ëë³éý»ñç ³éï³ûáõãûáõýá ï³ñáõ ¿ µ»ñ»é çýãå»ë ýçý³ýë³ï³ý ïáñáõëïý»ñç, ³ûýå»ë ¿é ù³ñ¹ï³ûçý ½áñ»ñç (ûñçý³ï, µåßï³ï³ý ë³ñù³íáñáõùý»ñç ï³ù ù»ù»ý³ý»ñáõù ³éï³ ëë³éý»ñá): ²ûëåçëáí, µ³ñ¹ ¿ ã»ñ³·ý³ñ³ï»é ã»ëï³íáñù³ý ï³ñ¨áñáõãûáõýá: â»ëï³íáñù³ý ùáï»óáõùý»ñá ï³ñ»éç ¿ µ³å³ý»é 3 ñçùý³ï³ý ëùµ»ñç, áñáýóçó ù»ïá ÷áñóç íñ³ ñçùýí³í (experience-based) ã»ëï³íáñáõùý ¿: ²ûë å³ñ³·³ûáõù ã»ëï»ñá ëï»õííáõù »ý‘ ñçùýí»éáí ã»ëï³íáñáõç ·ç»éçùý»ñç ¨ çýïáõçóç³ç, çýãå»ë ý³¨ ý³ëïçýáõù ýù³ý³ïçå íñ³·ñ»ñç ñ»ï áõý»ó³í ÷áñóç íñ³: öáñóç íñ³ ñçùýí³í ùáï»óáõùý»ñý û·ýáõù »ý µ³ó³ñ³ûï»é ³ûýåçëç ëë³éý»ñ, áñáýù ß³ï µ³ñ¹ ¿ ñ³ûïý³µ»ñ»é ³í»éç ñ³ù³ï³ñ·í³í ùáï»óáõùý»ñáí: ²ûë ³ßë³ï³ýùáõù ù»ýù ³é³ç³ñïáõù »ýù ÷áñóç íñ³ ñçùýí³í ã»ëï³íáñù³ý ³íïáù³ï³óáõù` û·ï³·áñí»éáí ïáùµçý³ïáñ ëý¹çñý»ñç rgt ¹³ëá: rgt ¹³ëç ëý¹çñý»ñç éáõíù³ý ñ³ù³ñ ùß³ïíáõù ¿rgt solver-᪠íñ³·ñ³ûçý ÷³ã»ã, áñá ïáõï³ïáõù ¿ ÷áñó³·çï³ï³ý ·çï»éçùý»ñ ¨ ëï»õíáõù ¿ ³ñ¹ûáõý³í»ï é³½ù³í³ñáõãûáõýý»ñ rgt ¹³ëç ëý¹çñý»ñç éáõíù³ý ñ³ù³ñ: ²é³ç³ñïáõù »ýù rgt solver-ý û·ï³·áñí»é íñ³·ñ»ñç ã»ëï³íáñù³ý ëý¹ñáõù: solver-á ëï»õíáõù ¿ ã»ëï³ûçý çñí³ç׳ïý»ñ` ï³ëí³í íñ³·ñç ³ñó³·³ýùçó/å³ï³ëë³ýçó ¨ ·ý³ñ³ïáõù ¿ ¹ñ³ýù áëï ý³ë³å»ë ë³ñù³ýí³í ã³÷³ýçßý»ñç: ²ûë ùáï»óù³ý ³¹»ïí³ïáõãûáõýá ÷áñó³ñïíáõù ¿ »é³ã³÷ ùá¹»é³íáñù³ý »blender» íñ³·ñç ùççáóáí: ´³ý³éç µ³é»ñ` rgt ¹³ë, rgt solver, íñ³·ñ³ûçý ³å³ñáíù³ý ã»ëï³íáñáõù, ÷áñó³·çï³ï³ý ñ³ù³ï³ñ·»ñ: 5 6 expert knowledge-based rgt solvers for software testing rgt solver íà îñíîâå ýêñïåðòíûõ çíàíèé äëÿ òåñòèðîâàíèÿ ïðîãðàììíîãî îáåñïå÷åíèÿ ìàíå ï. áóíèàòÿí1, ñåäðàê â. ãðèãîðÿí2 è åììà ã. äàíèåëÿí3 1synopsys àðìåíèÿ, åðåâàí 2èíñòèòóò ïðîáëåì èíôîðìàòèêè è àâòîìàòèçàöèè íàí ðà, åðåâàí, àðìåíèÿ 3epam àðìåíèÿ, åðåâàí e-mail: buniatyanmane@gmail.com, addressforsd@gmail.com, emma danielyan@yahoo.com àííîòàöèÿ òåñòèðîâàíèå ïðîãðàìì-ýòî ñïîñîá îöåíêè êà÷åñòâà ïðîãðàììíîãî îáåñïå÷åíèÿ è ñíèæåíèÿ ðèñêà îòêàçà ïðîãðàììíîãî îáåñïå÷åíèÿ â ðàáîòå. î÷åíü òðóäíî íåäîîöåíèòü âàæíîñòü òåñòèðîâàíèÿ: ïðîáëåìû ñ êà÷åñòâîì ïðîãðàìì ìîãóò ïðèâåñòè êàê ê ôèíàíñîâûì ïîòåðÿì, òàê è íàíåñòè óùåðá çäîðîâüþ ëþäåé (íàïðèìåð, êîãäà îøèáêà íàõîäèòñÿ â ìåäèöèíñêèõ ïðèáîðàõ, àâòîìîáèëÿõ è ò. ä.). ìåòîäû òåñòèðîâàíèÿ ìîæíî ïîäðàçäåëèòü íà 3 îñíîâíûå ãðóïïû. îäíà èç íèõ ýòî ìåòîäû, îñíîâàííûå íà îïûòå. çäåñü òåñòîâûå ïðèìåðû ñîçäàþòñÿ íà îñíîâå çíàíèé è èíòóèöèè òåñòèðîâùèêà, à òàêæå íà åãî îïûòå ðàáîòû ñ àíàëîãè÷íûìè ïðèëîæåíèÿìè è òåõíîëîãèÿìè. ýòè ìåòîäû ìîãóò áûòü ïîëåçíû ïðè îïðåäåëåíèè òåñòîâ, êîòîðûå íå ëåãêî èäåíòèôèöèðîâàòü äðóãèìè áîëåå ñèñòåìàòè÷åñêèìè ïîäõîäàìè ê òåñòèðîâàíèþ. â çàâèñèìîñòè îò ïîäõîäà è îïûòà òåñòèðîâùèêà, ýòè ìåòîäû ìîãóò îáåñïå÷èâàòü øèðîêóþ ñòåïåíü ïîêðûòèÿ è ýôôåêòèâíîñòü òåñòèðîâàíèÿ. â äàííîé ñòàòüå ìû ïðåäëàãàåì ìåòîä òåñòèðîâàíèÿ íà îñíîâå îïûòà (àâòîìàòèçàöèÿ òåñòèðîâàíèÿ) ÷åðåç êëàññ êîìáèíàòîðíûõ çàäà÷ (rgt êëàññ). rgt êëàññ âêëþ÷àåò òàêèå âàæíûå çàäà÷è, êàê çàùèòà îò âòîðæåíèé â êîìïüþòåðíûå ñåòè, ðàçðàáîòêà îïòèìàëüíîé ñòðàòåãèè óïðàâëåíèÿ è ìàðêåòèíãà â êîíêóðåíòíîé ñðåäå, òåñòèðîâàíèå ïðîãðàìì, çàùèòà âîèíñêèõ ÷àñòåé îò ðàçëè÷íûõ òèïîâ àòàê, ïðîáëåìû ñî ñâÿçüþ, îòäåëüíûå âèäû îáó÷åíèÿ, øàõìàòû è øàõìàòîïîäîáíûå èãðû. rgt solver ýòî ïðîãðàììà, êîòîðàÿ íàêàïëèâàåò ýêñïåðòíûå çíàíèÿ è ðàçðàáàòûâàåò ýôôåêòèâíûå ñòðàòåãèè äëÿ ðåøåíèÿ çàäà÷ êëàññà rgt. â êà÷åñòâå ýêñïåðòíîé ñèñòåìû äëÿ òåñòèðîâàíèÿ, îñíîâàííîãî íà îïûòå, ïðåäëàãàåòñÿ èñïîëüçîâàòü rgt solver. solver ãåíåðèðóåò òåñòîâûå ñèòóàöèè íà îñíîâå îòâåòà/ðåàêöèè ïðîãðàììû è îöåíèâàåò èõ ïî ðÿäó çàðàíåå îïðåäåëåííûõ êðèòåðèåâ. àäåêâàòíîñòü ìåòîäà ïîêàçàíà íà ïðèìåðå ïðèëîæåíèÿ ñ îòêðûòûì èñõîäíûì êîäîì ”áëåíäåð”. êëþ÷åâûå ñëîâà: rgt êëàññ, rgt solver, òåñòèðîâàíèå ïðîãðàììíîãî îáåñïå÷åíèÿ, çíàíèÿ, ýêñïåðòíûå ñèñòåìû. 05_սեդրակ_59 05 mathematical problems of computer science 39, 31--39, 2013. 31 handwritten signature verification using drt vahe s. khachaturyan institute for informatics and automation problems of nas of ra e-mail: vahe@7smarts.com abstract the purpose of this research is the development of mathematical and algorithmic support, which will improve the accuracy of signature verification. the algorithms compute the distances whilecomparing signatures based on drt and hmm. for acceptance or rejection of the test signature a sliding threshold is used for all the authors, and depending on the author athresholdmethod is used, based on the distances between the test signature and the signatures of control, taking them as signs of the problem of two-class classification, using standard methods of imageclassification. keywords: signature verification, discrete radon transform, hidden markov model. 1. introduction the purpose of our research is to develop a system that automatically classifies handwritten signature images as authentic or fraudulent, with as little misclassifications as possible. at the same time, the processing requirements must be feasible so as to make the adoption of such an automated system economically viable. our work is inspired by, amongst others, the potential financial benefits that the automatic clearing of checks will have for the banking industry. despite theincreasing number of electronic alternatives to paper checks, fraud perpetrated at financial institutions in the united states has become a national epidemic. the national check fraud center report of 2000 [1] states that: “...check fraud and counterfeiting are among the fastest-growing crimes affecting the united states’ financial system, producing estimated annual losses exceeding $10 billion with the number continuing to rise at an alarming rate each year.” since commercial banks pay little attention to verifying signatures on checks —mainly due to the number of checks that are processed daily a system capable of screening casual forgeries should already prove beneficial. in fact, most forged checks contain forgeries of this type. we developed a system that automatically authenticates documents based on the owner’s handwritten signature. it should be noted that our system assumes that the signatures have already been extracted from the documents. methods for extracting signature data from check backgrounds can be found in the following papers [2, 3, 4]. our system will assist commercial banks in the process of screening checks and is not intended to replace the manual screening of checks entirely. those checks the signatures of which do not sufficiently match a model of the handwritten signature verification using drt32 owner’s genuine signature, are provisionally rejected. generally, these rejected checks will constitute a small percentage of the total number of checks processed daily, and only these checks are selected for manual screening. since the introduction of computers, modern society has become increasingly dependent on the electronic storage and transmission of information. in many transactions, the electronic verification of a person’s identity proved beneficial and this inspired the development of a wide range of automatic identification systems. plamondon and srihari [5] note that automatic signature verification systems occupy a very specific niche among other automatic identification systems: “on the one hand, they differ from systems based on the possession of something (key, card, etc.) or the knowledge of something (passwords, personal information, etc.), asthey rely on a specific, well learned gesture. on the other hand, they also differ from systems based on the biometric properties of an individual (finger prints, voice prints, retinal prints, etc.), asthe signature is still the most socially and legally accepted means of personal identification.” although handwritten signatures are by no means the most reliable means of personal identification, the signature verification systems are inexpensive and nonintrusive. handwritten signatures provide a direct link between the writer’s identity and transaction, and are therefore perfect for endorsing transactions. 2. image processing each signature is scanned into a binary image at a resolution of 300 dots per inch, after which a median filtering is applied to remove speckle noise. on the average, a signature image has a width of 400 to 600 pixels and a height of 200 to 400 pixels. the image dimensions are not normalized. subsequently, the drt of each signature is calculated. each column of the drt represents a projection or shadow of the signature at a certain angle. after these projections are processed and normalized, they represent a set of feature vectors (observation sequence) for the signature in question. the drt of an image is calculated as follows. assume that each signature image consists of ψ pixels in total, and that the intensity of the ith pixel is denoted by ii, i = 1, . . . ,ψ. the drt is calculated using β not overlapping beams per angle and θ angles in total. the cumulative intensity of the pixels that lie within the jth beam is denoted by rj , j = 1, . . . , βθ. this is called the jth beam sum. in its discrete form, the radon transform can therefore be expressed as follows: rj=∑ ωψi=1 , = 1,2,… , , (1) fig 1. discrete model for the radon transform withω ≈ 0.9. v. khachaturyan 33 fig 2. (a) a signature and its projections are calculated at angles of 0◦ and 90◦. (b) the drt is displayed as a gray-scale image. this image has θ = 128 columns, where each column represents a projection. here ω indicates the contribution of the ith pixel to the jth beam sum (see figure 1). the value of ω is found through two-dimensional interpolation. each projection therefore contains beam sums that are calculated at a given angle. the accuracy of the drt is determined by θ (the number of angles), β (the number of beams per angle), and the accuracy of the interpolation method. note that the continuous form of the radon transform can be inverted through analytical means. the drt therefore contains almost the same information as the original image and can be efficiently calculated with an algorithm by bracewell [8]. our system calculates the drt at θ angles. these angles are equally distributed between 0◦ and 180◦. a typical signature and its drt are shown in figure 2. the dimension of each projection is subsequently altered from β to d. this is done by first decimating all the zero valued components from each projection. these decimated vectors are then shrunk or expanded to a length of d through interpolation. although almost all the information in the original signature image is contained in the projections at angles that range from 0◦ to 180◦, the projections at angles that range from 180◦ to 360◦ are also included in the observation sequence. these additional projections are added to the observation sequence in order to ensure that the sequence fits the topology of our hmm (see section 3.2). since these projections are simply reflections of the projections already calculated, no additional calculations are necessary. an observation sequence therefore consists of t = 2θ feature vectors, that is, =, ,…, .each vector is subsequently normalized by the variance of the intensity of the entire set of t feature vectors. each signature pattern is therefore represented by an observation sequence that consists of t observations, where each observation is a feature vector of dimension d. the experimental results and computational requirements for various values of d and θ are discussed in section 5, respectively. the drt, as a feature extraction technique, has several advantages. although the drt is not a shift invariant representation of a signature image, the shift and scale invariance is ensured by the subsequent image processing. each signature is a static image and contains no dynamic information. since the feature vectors are obtained by calculating projections at different angles, a simulated time evolution is created from one feature vector to the next, where the angle is the dynamic variable. this enables us to construct an hmm for each signature (see section 3). the drt is calculated at angles that range from 0◦ to 360◦ and each observation sequence is then handwritten signature verification using drt34 modeled by an hmm the states of which are organized in a ring (see section 3.2). this ensures that each set of feature vectors is rotation invariant. our system is also robust with respect to moderate levels of noise. these advantages are now discussed in more detail. noise we explained earlier in this section that the zero-valued components of each projection are decimated before the remaining non-zero components are shrunk or expanded through interpolation. in this way, a feature vector with the required dimension is obtained. the decimation of the zero-valued components ensures that moderate levels of noise (which are represented by a few additional small-valued components within certain projections) are “attached” to the other nonzero components before the decimated vector is shrunk or expanded. since the dimension of the feature vectors are high compared to the number of these additional components, the incorporation of these components has little effect on the overall performance of the system. shift invariance although the drt is not a shift invariant representation of a signature image, the shift invariance is ensured by the subsequent image processing. the zero-valued components of each projection are decimated and the corresponding feature vector is constructed from the remaining components only. rotation invariance the drt is calculated at angles that range from 0◦ to 360◦ and each set of feature vectors is then modeled by an hmm the states of which are organized in a ring (see section 3.2). each signature is therefore represented by a set of feature vectors that is rotation invariant. scale invariance for each projection, the scale invariance has to be achieved in the direction perpendicular to the direction in which the image is scanned, that is, perpendicular to the beams, and in the direction parallel to the beams. the scale invariance perpendicular to the beams is ensured by shrinking or expanding each decimated projection to the required dimension. the scale invariance parallel to the beams is achieved by normalizing the intensity of each feature vector. this is achieved by dividing each feature vector by the variance of the intensity of the entire set of feature vectors. 3. signature modelling we use a first-order continuous observation hmm to model each writer’s signature. for a tutorial on hmms, the reader is referred to a paper by rabiner [9] and the book by deller et al. notation we use the following notation for an hmm λ. (1) we denote the n individual states as = , ,…, (2) and the state at time t as qt . (2) the initial state distribution is denoted by π = {πi}, where= = , = 1,… , . (3) (3) the state transition probability distribution is denoted by a = {ai,j}, where= = = ), = 1, … , , = 1, … , . (4) (4) the probability density function (pdf), which quantifies the similarity between a feature vector x and the state sj, is denoted by, , = 1, … , . (5) v. khachaturyan 35 hmm topology we use an hmm, the states of which are organized in a ring (see figure 3). our model is equivalent to a left-to-right model, but a transition from the last state to the first oneis allowed. since the hmm is constructed in such a way that it is equally likely to enter the model at any state, and the feature vectors are obtained from all the projections, that is, the projections calculated at angles ranging from 0◦ to 360◦, the ring topology of our hmm guarantees that the signatures are rotation invariant. each state in the hmm represents one or more feature vectors that occupy similar positions in a d-dimensional feature space. this implies that the hmm groups certain projections (columns of the drt) together. it is important to note that this segmentation process only takes place after some further image processing has been conducted on the original projections. fig 3. an example of an hmm with a ring topology. this model has ten states with one state skip. training each model is trained using the viterbi reestimation technique. the dissimilarity between an observation sequence x and a model λ can therefore be calculated as follows (see [9]):, = − ln | , (6) in real-world scenarios, each writer can only submit a small number of training samples when he or she is enrolled into the system. since our algorithm uses feature vectors with a high dimension, the reestimated covariance matrix of the pdf for each state is not reliable and may even be singular. a mahalanobis distance measure therefore cannot be found. consequently, these covariance matrices are not reestimated and are initially set to 0.5i, where iis the identity matrix. only the mean vectors are reestimated, which implies that the dissimilarity values are based on theeuclidean distance measure. we assume that training signatures, genuine test signatures, and forgeries are available for only a limited number of writers, that is, for the writers in our database. no forgeries are used in the training process since our system aims to detect only skilled and casual forgeries, and these types of forgeries are not available when our system is implemented. the genuine test signatures and forgeries are used to determine the error rates for our system (see section 5). assuming that there are w writers in our database, the training signatures for each writer are used to construct an hmm, resulting in w models, that is {λ1, λ2, . . . ,λw}. when the training set for the writer w is denoted by ( ), ( ), … , ( ) , where nw is the number of samples in the training set, the dissimilarity between every training sample and the model is used to determine the following statistics for the writer’s signature: handwritten signature verification using drt36 = 1 , λ , = ∑ , λ − . (7) 4. verification when a system aims to detect only random forgeries, the subsets of the other writer’s training sets can be used to model “typical” forgeries. this is called “an impostor validation” and can be achieved through strategies like test normalization. these techniques enable one to construct verifiers that detect random forgeries very accurately (see [6, 7]). since we aim to detect only skilled and casual forgeries, and since the models for these forgeries are generally unobtainable, we are not able to utilize any of these impostor validation techniques. we also do not use any subset of genuine signatures for validation purposes. our verifier is constructed as follows. when a claim is made that the test pattern ( ) belongs to the writer w, the pattern is first matched with the model λw through viterbi alignment. this match is quantified by ( |λ ). the dissimilarity between the test pattern and the model is then calculated as follows (see [9]):λ = − ln ( |λ ) . (8) in order to use a global threshold for all writers, dolfing [6] suggests that every dissimilarity value in (8) is normalized, using the statistics of the claimed writer’s signature, that is, (7):λ = , (9) where λ denotes the normalized dissimilarity between the test pattern and the model of the claimed writer’s signature. this normalization is based on the assumption that the dissimilarity value in (8) is based on the mahalanobis distance measure. for mean vectors the dissimilarity value in (8) is based on the euclidean distance measure. when this is the case, we found that significantly better results are obtained when the standard deviation of the dissimilarities of the training set, that is, σw in (9), is replaced by the mean μw, that is, λ = . (10) a sliding threshold τ, where τ ∈ (−∞,∞), is used to determine the error rates for the test patterns. when λ < τ , that is, λ < 1 + τ , (11) the claim is accepted, otherwise, the claim is rejected. when τ = 0, all the test patterns, for whichλ ≥ , are rejected. this almost always results in an frr close to 100% and an far close to 0%.when τ →∞, all the test patterns, for which λ is finite, are accepted. this always results in an frr of 0% and an far of 100%. this technique is simple to apply in a program coding and further performance can be improved by using a better classification algorithm. v. khachaturyan 37 5. experiments our data set contains 924 signatures from 22 writers. ten training signatures were obtained from each writer during theinitial enrollment session. thirty-two test signatures, that consist of 20 genuine signatures, 6 skilled forgeries, and 6 casual forgeries, were subsequently obtained over a period of two weeks. the 20 genuine test signatures consist of two sets of 10 signatures each. these signatures were supplied by the same writers one week and two weeks after the enrollment session. the forgeries were obtained from 6 forgers. the casual forgeries were obtained first. only the name of the writer was supplied to the forgers and they did not have access to the writer’s signatures. the skilled forgeries were then obtained from the same group of forgers. they were provided with several samples of each writer’s genuine signature and were allowed ample opportunity to practice. each forger submitted 1 casual forgery and 1 skilled forgery for each writer. the writers were instructed to produce each signature within an appropriate rectangular region on a white sheet of paper. the signatures were then digitized with a flatbed scanner at a resolution of 300 dots per inch. the genuine signatures were produced with different pens and the forgeries were produced with the same pens that were used for producing the genuine signatures. these signatures are free of excessive noise, smears, and scratches. fig. 4. the stellenbosch data set. graphs for the frr and the far when d = 512, θ = 128, n = 64, and ℓ = 1. results let ℓ denote the number of allotted forward links in our hmm. figure 4 shows the frr and far as functions of our threshold parameter τ ∈ [−0.1, 1], when d = 512, θ = 128, n = 64, andℓ = 1. the frr and far for a test set that contains only skilled forgeries, and the far for a test set that contains only casual forgeries are plotted on the same system of axes. when, for example, a threshold of τ = 0.16 is selected, equation (11) implies that all the test patterns, for which λ ≥ 1.16 , are rejected;the other patterns are accepted. when only the skilled forgeries are considered, this threshold selection will ensure an eer of approximately 18%. when only the casual forgeries are considered, our algorithm achieves an eer of 4.5%. it is clear that when the dimension of the feature vectors is decreased from d = 512 to d = 256 or even to d = 128, the performance of the system is not significantly compromised. the handwritten signature verification using drt38 performance of our system is generally enhanced when the number of feature vectors, that is, t=2θ, or the number of states in the hmm, that is, n, is increased. the best results are obtained when only one forward link is allowed in the hmm, that is, whenℓ = 1. 6. conclusions the drt is a stable and robust method of feature extraction. the drt creates a simulated time evolution from one feature vector to the next one and enables us to model a signature with an hmm. our system is not sensitive to moderate levels of noise, and the feature vectors are extracted in such a way that rotation, shift, and scale invariance is ensured. our system does not outperform all these systems. these systems do, however, utilize either a technique or features that are fundamentally very different from ours. this implies that it is very likely that a combination of their systems and that of ours will result in a superior merged system, making their approaches complementary to ours. we also expect a significant improvement in our results when local features are incorporated into our algorithm. this is currently being investigated. references [1] national check fraud center, national check fraud center report, 2000. [2] s. djeziri, f. nouboud, and r. plamondon, “extraction of signatures from cheque background based on a filiformity criterion,” ieee trans. image processing, vol. 7, no. 10, pp. 1425– 1438, 1998. [3] a. l. koerich and l. l. lee, “automatic extraction of filledin information from bankchecks based on prior knowledge about layout structure,” in advances in document image analysis: first brazilian symposium, lecture notes in computer science, vol. 1339, pp. 322–333, 1997. [4] j. e. b. santos, f. bortolozzi, and r. sabourin, “a simple methodology to bankcheck segmentation,” in advances in document image analysis: first brazilian symposium, lecture notes in computer science, vol. 1339, pp. 334–343, 1997. [5] r. plamondon and s. n. srihari, “on-line and off-line handwriting recognition: a comprehensive survey,” ieee trans. on pattern analysis and machine intelligence, vol. 22, no. 1, pp. 63–84, 2000. [6] r. sabourin, g. genest, and f. prˆeteux, “off-line signature verification by local granulometric size distributions,” ieee trans. on pattern analysis and machine intelligence, vol. 19, no. 9, pp. 976–988, 1997. [7] a. el-yacoubi, e. j. r. justino, r. sabourin, and f. bortolozzi, “off-line signature verification using hmms and cross-validation,” in ieee international workshop on neural networks for signal processing, pp. 859–868, 2000. [8] r. n. bracewell, two-dimensional imaging, prentice-hall, englewood cliffs, nj, usa, 1995. [9] l. r. rabiner, “a tutorial on hidden markov models and selected applications in speech recognition,” proceedings of the ieee, vol. 77, no. 2, pp. 257–286, 1989. submitted 10.12.2012, accepted 15.02.2013. v. khachaturyan 39 ձեռագիր ստորագրություններ իստուգումը drt-ի օգտագործմամբ վ. խաչատուրյան ամփոփում տվյալ աշխատանքի նպատակն է մաթեմատիկական և ալգորիթմային ապահովման մշակումը, որը թույլ կտա բարձրացնել ստորագրության ստուգման ճշգրտությունը: ալգորիթմները հաշվարկում են հեռավորությունները ստորագրությունների համեմատման ժամանակ՝ օգտագործելով drt-ն և hmm-ը: փորձարկվող ստորագրության ընդունման կամ մերժման համար օգտագործվում է սահող շեմքի մեթոդը բոլոր հեղինակների համար և կախված հեղինակից` շեմքի մեթոդը՝ օգտագործելով փորձարկվող ստորագրության և ստուգողական ստորագրությունների միջև եղած հեռավորությունները՝ դասակարգելով դրանք երկու դասերի՝ պատկերների դասակարգման ստանդարտ մեթոդների միջոցով: проверка рукописных подписей с использованием drt в. хачатурян аннотация целью данного исследования является разработка математического и алгоритмического обеспечения, которая позволит повысить точность проверки подписи. алгоритмы вычисляют расстояния при сравнении подписей с помощью drt и hmm. для принятия или отказа тестовой подписи используется метод скользящего порога для всех авторов и зависящий от автора – метод порога, с использованием расстояний между тестовой подписью и контрольными подписями, классифицируя их по двум классам, с использованием стандартных методов классификации изображений. d:\user\sbornik_38_pdf\22.dvi mathematical problems of computer science 38, 53{55, 2012. on m edial-like functional e quations a m ir e h s a n i department of mathematics mahshahr branch, islamic azad university mahshahr, iran. mahshahr branch, islamic azad university, mahshahr, iran. a.ehsani@mahshahriau.ac.ir l e t a b e a n o n e m p t y s e t , n a n d m b e p o s it ive in t e g e r s a n d f : an ! am b e a n a r b it r a r y fu n c t io n . th e n ( a; f ) is c a lle d [n; m]-g r o u p o id . th e n-a r y o p e r a t io n s , f1; : : : ; fm, a r e d e ¯ n e d b y t h e fo llo win g : f ( x1; : : : ; xn ) = ( y1; : : : ; ym ) , yi = fi ( x1; : : : ; xn ) ; fo r e ve r y 1 · i · m, a r e c a lle d t h e c o m p o n e n t o p e r a t io n s o f f a n d t h e y a r e d e n o t e d b y f = ( f1; : : : ; fm ) [1 1 , 1 2 , 1 3 ]. th e [n; m]-g r o u p o id is p r o p e r i® n; m; jqj ¸ 2 . th e [n; m]-g r o u p o id ( a; f ) is c a lle d [n; m]-qu a s ig r o u p ( o r m u lt iqu a s ig r o u p [2 , 3 , 1 4 ]) i® fo r e ve r y in je c t io n , á : nn ! nn+m, wh e r e nn = f1 ; : : : ; ng, a n d e ve r y ( a1; : : : ; an ) 2 qn t h e r e e xis t s a u n iqu e ( b1; : : : ; bn+m ) 2 qn+m s u c h t h a t : f ( b1; : : : ; bn ) = ( bn+1; : : : ; bn+m ) a n d bá(i) = ai; fo r i = 1 ; : : : ; n. it is c le a r t h a t q( f ) is a n [n; 1 ]-qu a s ig r o u p i® q( f ) is a n n-qu a s ig r o u p [1 ]. q( f ) is a [1 ; m]qu a s ig r o u p i® t h e r e e xis t p e r m u t a t io n s , f1; : : : ; fm, o f q s u c h t h a t f ( x ) = ( f1 ( x ) ; : : : ; fm ( x) ) . it is a ls o c le a r t h a t a ll c o m p o n e n t s o f a m u lt iqu a s ig r o u p a r e b in a r y qu a s ig r o u p o p e r a t io n s . if t h e c o m p o n e n t o p e r a t io n s o f t h e [n; m]-qu a s ig r o u p a r e b in a r y o p e r a t io n s , i.e . n = 2 , t h e n we s a y t h a t t h e [n; m]-qu a s ig r o u p is a b in a r y m u lt iqu a s ig r o u p . l e t u s c o n s id e r t h e fo llo win g h yp e r id e n t it ie s [7 , 8 , 9 ]: g ( f ( x; y ) ; f ( u; v ) ) = f ( g ( x; u ) ; g ( y; v ) ) ; ( me d ia lit y) g ( f ( x; y ) ; f ( u; v ) ) = f ( g ( v; y ) ; g ( u; x) ) ; ( p a r a m e d ia lit y) g ( f ( x; y ) ; f ( u; v ) ) = g ( f ( x; u ) ; f ( y; v ) ) ; ( co -m e d ia lit y) g ( f ( x; y ) ; f ( u; v ) ) = g ( f ( v; y ) ; f ( u; x) ) ; ( co -p a r a m e d ia lit y) f ( x; x ) = x: ( id e m p o t e n c y) th e b in a r y a lg e b r a , ( a; f ) , is c a lle d : ² m e d ia l, if it s a t is ¯ e s t h e id e n t it y ( 1 .1 ) , ² p a r a m e d ia l, if it s a t is ¯ e s t h e id e n t it y ( 1 .2 ) , 5 3 5 4 on medial-like functional equations ² c o -m e d ia l, if it s a t is ¯ e s t h e id e n t it y ( 1 .3 ) , ² c o -p a r a m e d ia l, if it s a t is ¯ e s t h e id e n t it y ( 1 .4 ) , ² id e m p o t e n t , if it s a t is ¯ e s t h e id e n t it y ( 1 .5 ) , fo r e ve r y f; g 2 f . th e b in a r y a lg e b r a , ( a; f ) , is c a lle d m o d e , if it is m e d ia l a n d id e m p o t e n t . de¯nition 1 the binary multiquasigroup ( a; f ) with f = ( f1; : : : ; fm ) is called: ² medial, if the binary algebra, ( a; f1; : : : ; fm ) , is medial, ² paramedial, if the binary algebra, ( a; f1; : : : ; fm ) , is paramedial, ² co-medial, if the binary algebra, ( a; f1; : : : ; fm ) , is co-medial, ² co-paramedial, if the binary algebra, ( a; f1; : : : ; fm ) , is co-paramedial, ² idempotent, if the binary algebra, ( a; f1; : : : ; fm ) , is idempotent, ² mode, if the binary algebra, ( a; f1; : : : ; fm ) , is a mode. th e n e xt c h a r a c t e r iz a t io n o f b in a r y m e d ia l m u lt iqu a s ig r o u p s fo llo ws fr o m [6 , 1 0 ]. t heor em 1 l et ( q; f ) be a binary multiquasigroup, where f = ( f1; : : : ; fm ) . if ( q; f ) is a binary medial multiquasigroup, then there exists an abelian group, ( q; +) , such that: fi ( x; y ) = ®ix + ¯iy + ci; where ®i; ¯i are automorphisms of the group ( q; +) , and ci 2 q is a ¯xed element and: ®i¯j = ¯j®i; ®i®j = ®j®i; ¯i¯j = ¯j¯i, for i; j = 1 ; : : : ; m. the group, ( q; +) , is unique up to isomorphisms. m oreover, if ( q; f ) is a mode, then fi ( x; y ) = ®ix + ¯iy; where ®i; ¯i are automorphisms of both the group, ( q; +) , and of the algebra, ( q; f1; : : : ; fm ) . in t h is p a p e r we c h a r a c t e r iz e t h e b in a r y p a r a m e d ia l, c o -m e d ia l a n d c o -p a r a m e d ia l m u lt iqu a s ig r o u p s ( c f. [4 , 5 ]) . r e fe r e n c e s [1 ] b e lo u s o v, v . d . ( 1 9 7 2 ) . n-a r y qu a s ig r o u p s . in : shtiinca, k is h in e v. [2 ] ·c u p o n a , g., u ·sa n , j., s t o ja ko vi¶c, z. ( 1 9 8 0 ) . mu lt iqu a s ig r o u p s a n d s o m e r e la t e d s t r u c t u r e s . p rilosi m anu. i n o . 2 , 5 -1 2 . [3 ] ·c u p o n a , g., s t o ja ko vi¶c, z., u ·sa n , j. ( 1 9 8 1 ) . on ¯ n it e m u lt iqu a s ig r o u p s . p ubl. inst. m ath. v o l. 2 9 , n o . 4 3 , 5 3 -5 9 . [4 ] e h s a n i, a ., mo vs is ya n , y u . m. ( 2 0 1 2 ) . l inear representation of medial-like algebras. co m m u n ic a t io n s in a lg e b r a . a. ehsani 5 5 [5 ] e h s a n i, a ., mo vs is ya n , y u . m. ( 2 0 1 2 ) . b inary m ultiquasigroups with m edial-l ike e quations. ir a n ia n jo u r n a l o f ma t h e m a t ic a l s c ie n c e s a n d in fo r m a t ic s . [6 ] mo vs is ya n , y u . m. ( 1 9 9 9 ) . generalization of toyoda theoram. p r o c e e d in g o f t h e l o o p s '9 9 , p r a g u e . [7 ] mo vs is ya n , y u . m. ( 1 9 9 0 ) . h yp e r id e n t it ie s a n d h yp e r va r ie t ie s in a lg e b r a s . in : yerevan state university p ress. y e r e va n . ( r u s s ia n ) . [8 ] mo vs is ya n , y u . m. ( 1 9 9 8 ) . h yp e r id e n t it ie s in a lg e b r a s a n d va r ie t ie s . uspekhi m ath. nauk. 5 3 : 6 1 -1 1 4 . e n g lis h t r a n s la t io n in : r u s s ia n ma t h . s u r ve ys 5 3 ( 1 9 9 8 ) , n o 1 , 5 7 1 0 8 . [9 ] mo vs is ya n , y u . m. ( 1 9 8 6 ) . in t r o d u c t io n t o t h e t h e o r y o f a lg e b r a s wit h h yp e r id e n t it ie s . in : yerevan state university p ress. y e r e va n . ( r u s s ia n ) . [1 0 ] mo vs is ya n , y u . m., n a z a r i, e . ( 2 0 1 1 ) . tr a n s it ive mo d e s . d emonstratio m athematica. x l iv : n o . 3 , 5 1 1 -5 2 2 . [1 1 ] p o lo n ijo , m. ( 1 9 8 1 ) . s t r u c t u r e t h e o r e m fo r cn+1-s ys t e m s . glasnik m at. v o l. 1 6 , n o . 3 8 , 2 1 1 -2 1 7 . [1 2 ] p o lo n ijo , m. ( 1 9 8 2 ) . on cn+1-s ys t e m s a n d [n; m]-g r o u p o id s . algebraic conference, b eograd. [1 3 ] s t o ja ko vi¶c, z. ( 1 9 8 2 ) . on b is ym e t r ic [n; m]-g r o u p o id s . r eview of research f aculty of science-university of novi sad. v o l. 1 2 , 3 9 9 -4 0 5 . [1 4 ] s t o ja ko vi¶c, z., p a u n i¶c, d j. ( 1 9 8 2 ) . id e n t it ie s o n m u lt iqu a s ig r o u p s . p roc. symp. on n-ary structures. 1 9 5 -2 0 0 . d:\user\sbornik_38_pdf\24.dvi mathematical problems of computer science 38, 59{60, 2012. p auly m atr ix and t r ansfor mation oper ator s for dir ac system t. n . h a r u t yu n ya n , h . h . a z iz ya n yerevan state university, e-mail: hartigr@yahoo.co.uk armenian state agrarian university, e-mail: hermineazizyan@mail.ru l e t ¾1 = ã 0 i ¡i 0 ! ; ¾2 = ã 1 0 0 ¡ 1 ! ; ¾3 = ã 0 1 1 0 ! a r e we ll kn o wn p a u ly m a t r ix a n d e = ã 1 0 0 1 ! . it is kn o wn t h a t t h e s o lu t io n y = '( x; ¸; ®) o f ca u s c h y p r o b le m f¾1 1 i d dx + ¾2p ( x ) + ¾3q ( x ) gy = ¸y; ¸ 2 c y ( 0 ) = ã sin® ¡cos® ! ; c a n b e r e p r e s e n t e d in t h e fo r m ã '0 ( x; ¸; ® ) = ã sin( ¸x + ®) ¡cos( ¸x + ® ) !! '( x; ¸; ® ) = '0 ( x; ¸; ®) + z x 0 k ( x; t ) '0 ( t; ¸; ® ) dt = ( e + k ) '0: op e r a t o r e + k is c a lle d t h e t r a n s fo r m a t io n o p e r a t o r . u n d e r d i®e r e n t c o n d it io n s o n s c a la r fu n c t io n s p a n d q t h is o p e r a t o r a n d h is ke r n e l k ( x; t) wa s in ve s t ig a t e d in d i®e r e n t p a p e r s ( s e e [1 ]-[6 ]) . t heor ema. l et p; q 2 l1loc ( 0 ; 1) : then the kernel k ( x; t ) and the kernel h ( x; t ) of inverse operator '0 ( x; )̧ = '( x; ¸ ) + r x 0 h ( x; t ) '( t; ¸ ) dt can be represented in the form k ( x; t ) = a¾1 + b¾2 + c¾3 + d ¢ e h ( x; t) = ~a¾1 + ~b¾2 + ~c¾3 + ~d ¢ e; where the functions (of two variables ( x; t ) )) a; b; c; d and ~a; ~b; ~c; ~d are represented by functions p and q: 5 9 6 0 pauly matrix and transformation operators for dirac system r e fe r e n c e s [1 ] ga s ym o v m. g., l e vit a n b . m., d e t e r m in a t io n o f a d i®e r e n t ia l e qu a t io n b y t wo o f it s s p e c t r a , u s p . ma t . n a u k v.1 9 , n 2 , 1 9 6 4 , p p .3 -6 3 [2 ] ma r c h e n ko v . a ., s t u r m -l io u ville o p e r a t o r s a n d t h e ir a p p lic a t io n s , n a u ko va d u m ka , k ie v, 1 9 7 7 . [3 ] me lik-a d a m ya n f. e ., on t h e c a n o n ic d i®e r e n t ia l o p e r a t o r s in h ilb e r t s p a c e , iz ve s t . a n a r m . s s r , ma t h e m a t ic s , v.x ii, n 1 , p p .1 0 -3 0 . [4 ] l e vit a n b . m., s a r g s ya n i. s ., s t u r m -l io u ville a n d d ir a c o p e r a t o r s , n a u ka , mo s c o w, 1 9 8 8 . [5 ] h a r u t yu n ya n t. n ., tr a n s fo r m a t io n s o p e r a t o r s fo r c a n o n ic d ir a c s ys t e m , d i®e r e n t ia ln ie u r a vn e n iya , v.4 4 , n 8 , 2 0 0 8 , p p . 1 0 1 1 -1 0 2 1 . [6 ] a lb e ve r io s ., h r in iv r ., mikit u k y a ., in ve r s e s p e c t r a l p r o b le m s fo r d ir a c o p e r a t o r s wit h s u m m a b le p o t e n t ia ls , r u s . j. o f ma t h . p h ys . v.1 2 , n 4 , 2 0 0 5 , p p .4 0 6 -4 2 3 . d:\sbornik\...\paper.dvi mathematical problems of computer science 25, 2006, 85{91. a novel approach for yield o ptimization v a z g e n s . k a r a p e t ya n ponte solutions inc., russian-armenian (slavonic) state university, e-mail vazgen.karapetyan@pontesolutions.com abstract in this paper a novel approach for yield optimization is presented. while there are di®erent types of defects or other issues a®ecting the yield, only the "extra conductive material" type (potentially causing shorts circuits) is studied. the approach tends to improve the yield of the ic layout by reducing the area which is sensitive to random defects, the so-called critical area. refer ences [1 ] r . b a ld ic k, a . k a h n g , a . k e n n in g s , a n d i. ma r ko v. " e ± c ie n t op t im iz a t io n b y mo d ifyin g t h e ob je c t ive fu n c t io n " . ie e e transactions on circuits and systems, v o l 4 8 , n o 8 :9 4 7 { 9 5 7 , 2 0 0 1 . [2 ] c. b a m ji a n d r . v a r a d a r a ja n . "l eaf cell and hierarchical compaction techniques". k lu we r a c a d e m ic p u b lis h e r s , 1 9 9 7 . [3 ] a . v . fe r r is -p r a b h u . "introduction to semiconductor d evice yield m odeling". a r t e c h h o u s e p u b lis h e r s , 1 9 9 2 . [4 ] i. k o r e n . " d e fe c t to le r a n c e in v l s i cir c u it s : te c h n iqu e s a n d y ie ld a n a lis ys " . p roceedings of the ie e e , v o l 8 6 , n o 9 :1 8 1 7 { 1 8 3 6 , s e p t e m b e r , 1 9 9 8 . [5 ] ch . w e b e r , v . s a n ka r a n , k .w . to b in , a n d g. s c h e r . " qu a n t ifyin g t h e v a lu e o f own e r s h ip o f y ie ld a n a lys is te c h n o lo g ie s " . ie e e transactions on semiconductor m anufacturing, v o l 1 5 , n o 4 :4 1 1 { 4 1 9 , n o ve m b e r , 1 9 9 8 . äçï³ýç ³ñï³¹ñ³ýùç ûåïçù³é³óù³ý ùç ýáñ ùáï»óù³ý ù³ëçý ì. î³ñ³å»ïû³ý ²ù÷á÷áõù ²ßë³ï³ýùáõù ý»ñï³û³óí³í ¿ ¶øæê ³ñï³¹ñ³ï³ý ëáï³ýç ýí³½»óù³ý ùç ùáï»óù³ý ýï³ñ³·ñáõãûáõý: ¸çï³ñïíáõù »ý ñ³ïáõï ïçåç å³ï³ñ³ï³ý 8 5 8 6 a novel approach for yield optimization ¹»ý»ïïý»ñá ¨ ¹ñ³ýóáí å³ûù³ý³íáñí³í ”ï³ñ× ùç³óáõù” ï»ë³ïç ë³÷³ýáõùý»ñá: àõëáõùý³ëçñíáõù ¿ ¹ñ³ýó å³ï׳éá ¨ ³é³ç³ñïíáõù ¿ ¶øæê ï»õ³¹ñáõãûáõýá ÷áë»éáõ ùççáóáí åçï³ýç ³ñï³¹ñ³ýùç ã³÷ç ù»í³óù³ý ùç ùáï»óáõù: d:\user\sbornik_38_pdf\41.dvi mathematical problems of computer science 38, 95{96, 2012. on the h ier ar chies of some p r opositional systems for classical and n on-classical logics ¤ a n a h it ch u b a r ya n department of informatics and applied mathematics, yerevan state university, armenia achubaryan@ysu.am th e fa m ilie s o f e lim in a t io n s ys t e m s wit h fu ll s u b s t it u t io n r u le a n d wit h r e s t r ic t e d s u b s t it u t io n r u le s a r e in t r o d u c e d fo r cla s s ic a l, in t u it io n is t ic a n d min im a l ( io h a n s s o n 's ) p r o p o s it io n a l lo g ic s ( cp l , ip l ,mp l ) , a n d t h e e ± c ie n c ie s o f in t r o d u c e d s ys t e m s a r e c o m p a r e d fo r e ve r y m e n t io n e d lo g ic . w e u s e t h e n o t io n s o f d e t e r m in a t ive c o n ju n c t a n d d e t e r m in a t ive n o r m a l fo r m s , in t r o d u c e d fo r cp l in [1 ], fo r ip l a n d mp l in [2 ]. l e t ' b e a p r o p o s it io n a l fo r m u la , p = fp1; p2; : : : ; png b e t h e s e t o f a ll va r ia b le s o f ', a n d p 0 = fpi1; pi2 ; : : : ; pimg ( 1 · m · n) b e s o m e s u b s e t o f p . de¯nition 1. given ¾ = f¾1; : : : ; ¾mg ½ em (unit b oolean cube), the conjunct k¾ = fp¾1i1 ; p ¾2 i2 ; : : : ; p¾mim g is called ' ¡ 1 -determinative (' ¡ 0 -determinative) if assigning ¾j ( 1 · j · m) to each pij we obtain the value of ' (1 or 0) independently of the values of the remaining variables. de¯nition 2. d nf d = fk1; k2; : : : ; krg is called determinative d nf (dd nf ) for ' if ' = d and every conjunct ki ( 1 · i · r ) is 1-determinative for '. th e in ve s t ig a t e d s ys t e m s a r e t h e fo llo win g s ys t e m s e c, e i a n d e m a n d t h e ir g e n e r a liz a t io n s . th e a xio m s o f ec a r e n o t ¯ xe d , b u t fo r e ve r y fo r m u la ' e a c h c o n ju n c t fr o m s o m e d d n f o f ' c a n b e c o n s id e r e d a s a n a xio m . th e elimination rule ( "-r u le ) in fe r s k0 [ k00 fr o m c o n ju n c t s k0 [fpg a n d k0 [fpg, wh e r e k0 a n d k00 a r e c o n ju n c t s a n d p is a va r ia b le . d n f d = fk1; k2; : : : ; klg is c a lle d fu ll ( t a u t o lo g y) if u s in g "-r u le c a n b e p r o ve d t h e e m p t y c o n ju n c t io n ( ;) fr o m t h e a xio m s fk1; k2; : : : ; klg. th e a n a lo g ie s o f t h e d e t e r m in a t ive c o n ju n c t s a n d d d n f fo r ip l a n d mp l ( i-d d n f a n d m-d d n f a c c o r d in g ly ) a r e c o n s t r u c t e d in [2 ]. n o t e t h a t t h e lit e r a ls in t h e la t t e r c o n ju n c t s a r e o n ly va r ia b le s wit h n e g a t io n o r wit h d o u b le n e g a t io n s . b y a n a lo g y t h e c o r r e s p o n d in g p r o o f s ys t e m ei ( em ) c a n b e c o n s t r u c t e d fo r ip l ( mp l ) . a s a xio m is c o n s id e r e d e ve r y i-d e t e r m in a t ive ( m -d e t e r m in a t ive ) c o n ju n c t fr o m s o m e i-d e t e r m in a t ive ( m-d e t e r m in a t ive ) dnf . ¤this work is supported by grant 11-1b023 of ssc of goverment of armenia. 9 5 9 6 on the hierarchies of some propositional systems for classical and non-classical logics fo r ei ( em ) we t a ke t h e fo llo win g in fe r e n c e r u le k0 [ p k00 [ p k0 [ k00 i² ¡ rule ã k0 [ ( p ¾ ? ) ¾ ? k00 [ ( p ¾ ? ) k0 [ k00 m² ¡ rule ! ; wh e r e k0 a n d k00 a r e c o n ju n c t s a n d p is a va r ia b le . l e t u s in t r o d u c e t h e s u b s t it u t io n r u le fo r t h e s e t o f c o n ju n c t s c a s fo llo win g c s ( c ) ap ; wh e r e s ( c ) ap d e n o t e s t h e s e t o f r e s u lt s o f s u b s t it u t io n o f fo r m u la a in s t e a d o f va r ia b le p e ve r ywh e r e in t h e c o n ju n c t s o f t h e s e t c , a n d t h e r e fo r e we h a ve t h e generalized e lim in a t io n r u le fo r a fo r m u la a c1 [ fag c2 [ fag c1 [ c2 ; wh e r e a is a lit e r a l o r o n a n y s t e p s u b s t it u t e d fo r m u la : b y sec we d e n o t e t h e s ys t e m e c wit h s u b s t it u t io n r u le a n d g e n e r a liz e d e lim in a t io n r u le . if t h e n u m b e r o f c o n n e c t ive s o f s u b s t it u t e d fo r m u la s is b o u n d e d b y `, t h e n t h e c o r r e s p o n d in g s ys t e m is d e n o t e d b y s`ec. th e s ys t e m s sei, sem, s`ei, s`em a r e d e ¯ n e d b y a n a lo g y o n t h e b a s e o f t h e s ys t e m s e i a n d e m, u s in g t h e c o r r e s p o n d in g g e n e r a liz e d e lim in a t io n r u le s . w e d e ¯ n e t h e c o m p le xit y t o b e t h e s iz e o f a p r o o f ( = t h e t o t a l n u m b e r o f s ym b o ls ) . th e m in im a l c o m p le xit y o f a fo r m u la ' ( o r it s r e p r e s e n t a t io n ) in a p r o o f s ys t e m © we d e n o t e b y l©' . to c o m p a r e t h e e ± c ie n c ie s o f in t r o d u c e d s ys t e m s we u s e t h e we ll-kn o wn n o t io n s o f p s im u la t io n , p -e qu iva le n c e a n d e xp o n e n t ia l s p e e d -u p fr o m [3 ]. w e u s e a ls o t h e we ll-kn o wn n o t io n s o f fr e g e s ys t e m s fc, fi a n d fm fo r cp l , ip l a n d mp l a c c o r d in g ly ( s e e fo r e xa m p le in [2 ]) . m ain t heor em 1 . fo r e ve r y l > 0 t h e s ys t e m s`+1ec ( s`+1ei, s`+1em ) h a s e xp o n e n t ia l s p e e d -u p o ve r t h e s ys t e m s`ec ( s`ei, s`em ) in t r e e fo r m . 2 . th e s ys t e m s sec ( sei, sem ) a n d fc ( fi, fm) a r e p -e qu iva le n t . r e fe r e n c e s [1 ] a n . ch u b a r ya n , a r m . ch u b a r ya n , a n e w c o n c e p t io n o f e qu a lit y o f t a u t o lo g ie s , l &p s , v o l.v , n o 1 , tr ie s t , it a ly, 2 0 0 7 , 3 -8 . [2 ] a n . ch u b a r ya n , a r m . ch u b a r ya n , h . n a lb a n d ya n , s . s a ya d ya n , a h ie r a r c h y o f r e s o lu t io n s ys t e m s wit h r e s t r ic t e d s u b s t it u t io n r u le s , co m p u t e r te c h n o lg y a n d a p p lic a t io n 3 , d a vid p u b lis h in g , u s a , 2 0 1 2 , 3 3 0 -3 3 6 . [3 ] s . co o k, r . r e c kh o w, th e r e la t ive e ± c ie n c y o f p r o p o s it io n a l p r o o fs s ys t e m s , jo u r n a l o f s ym b o lic l o g ic , 4 4 , 1 9 7 9 , 3 6 -5 0 . article_eng12.dvi mathematical problems of computer science 32, 86{95, 2009. m axwell e lectr odynamics subjected to quantum vacuum fluctuations a s h o t s . ge vo r kya a n d a r a ks ya a . ge vo r kya n 1 institute for informatics and automation problems of nas of ra. 1yerevan state university g ashot@sci.am abstract the propagation of electromagnetic waves in vacuum is considered taking into account quantum °uctuations in the limits of maxwell-langevin (ml) equations. for a model of "white noise" °uctuations, using ml equations, the second order partial di®erential equation is found which describes the quantum distribution of virtual photons in vacuum. it is proved that in order to satisfy observed facts, the lamb shift etc, the virtual photons should be quantized in nonperturbed vacuum. for a model of the reverse harmonic quantum oscillator, the quantum distribution of photons is obtained precisely. it is shown, that the quantized virtual photons having negative energies, in toto (approximately 85 percent) are condensed on the energy level absolute value of which is minimal. it is proved that the extension of maxwell electrodynamics with inclusion of vacuum quantum ¯eld °uctuations may be constructed on 6d space-time continuum with 2d compacti¯ed subspace. the problem of propagation of various types electromagnetic waves in vacuum is investigated. their in°uence on the refraction index of vacuum is studied. refer ences [1 ] w . e . l a m b , jr ., a n d r . c. r e t h e r fo r d , " fin e s t r u c t u r e o f t h e h yd r o g e n a t o m b y a mic r o wa ve me t h o d " , p hys. r ev., vo l. 7 2 , 2 4 1 ( 1 9 4 7 ) . [2 ] p . w . milo n n i, r . j. co o k a n d m. e . go g g in , " r a d ia t io n p r e s s u r e fr o m t h e v a c u u m : p h ys ic a l in t e r p r e t a t io n o f t h e ca s im ir fo r c e " , p hys. r ev., a vo l. 3 8 , 1 6 2 1 ( 1 9 8 8 ) . [3 ] r . l . fo r wa r d , " e xt r a c t in g e le c t r ic a l e n e r g y fr o m t h e v a c u u m b y co h e s io n o f ch a r g e d fo lia t e d co n d u c t o r s " , p hys. r ev., b vo l. 3 0 , 1 7 0 0 ( 1 9 8 4 ) . [4 ] a . d . s a kh a r o v, " v a c u u m qu a n t u m flu c t u a t io n s in cu r ve d s p a c e a n d t h e th e o r y o f gr a vit a t io n , d o kl. a ka d . n a u k. s s s r ( sov. p hys. d okl., vo l. 1 2 , 1 0 4 0 ( 1 9 6 8 ) . s e e a ls o d is c u s s io n in c. w . mis n e r , k . s . th o r n e a n d j. a . w h e e le r , gravitation ( fr e e m a n , s a n fr a n c is c o , 1 9 7 3 ) , p . 4 2 6 . [5 ] t. w . ma r s h a ll, " r a n d o m e le c t r o d yn a m ic s " , p r o c . r o y. s o c ., a vo l. 2 7 6 , p . 4 7 5 ( 1 9 6 3 ) . [6 ] t. h . b o ye r , " r a n d o m e le c t r o d yn a m ic s : th e t h e o r y o f c la s s ic a l e le c t r o d yn a m ic s wit h c la s s ic a l e le c t r o m a g n e t ic z e r o -p o in t r a d ia t io n " . p h ys . r e v., vo l. 1 1 , p . 7 9 0 -8 0 8 ( 1 9 7 5 ) . [7 ] b . h a is c h , a . r u e d a , a n d h . e . p u t h o ®, " in e r t ia a s a z e r o -p o in t -̄ e ld l o r e n t z fo r c e " , p h ys . r e v., a vo l. 4 9 , p . 6 7 8 -6 9 4 ( 1 9 9 4 ) . 8 6 a. gevorkyan and ar. gevorkyan 8 7 [8 ] l . d e la p e n a , a n d a . m. ce t t o , " th e qu a n t u m d ic e : a n in t r o d u c t io n t o s t o c h a s t ic e le c t r o d yn a m ic s " , d o r d r e c h t : k lu we r ( 1 9 9 6 ) . l . d e la p e n a , a n d a . m. ce t t o , ( " co n t r ib u t io n fr o m s t o c h a s t ic e le c t r o d yn a m ic s t o t h e u n d e r s t a n d in g o f qu a n t u m m e c h a n ic s " . a r x iv: ja n 2 0 0 5 qu a n t -p h / 0 5 0 1 0 1 1 4 ja n 2 0 0 5 . [9 ] t. h . b o ye r , " a b r ie f s u r ve y o f s t o c h a s t ic e le c t r o d yn a m ic s ," in fo u n d a t io n s o f r a d ia t io n th e o r y a n d qu a n t u m e le c t r o d yn a m ic s , e d it e d b y a . o. b a r u t ( p le n u m , n e w y o r k, 1 9 8 0 ) s e e a ls o t h e ve r y r e a d a b le a c c o u n t " th e cla s s ic a l v a c u u m ," in s c ie n t i¯ c a m e r ic a n , p . 7 0 ( a u g u s t 1 9 8 5 ) [1 0 ] je a n -l u c ca m b ie r , " in e r t ia l ma s s fr o m s t o c h a s t ic e le c t r o d yn a m ic s " in : ma r c g. millis ( e t a l.) : f rontiers of p ropulsion science. p . 4 2 3 -4 5 4 , a m e r ic a n in s t . o f a e r o n a u t ic s a n d a s t r o n a u t ic s , r e s t o n ( 2 0 0 9 ) . [1 1 ] a . s . ge vo r kya n , " e xa c t ly s o lva b le m o d e ls o f s t o c h a s t ic qu a n t u m m e c h a n ic s wit h in t h e fr a m e wo r k o f l a n g e vin -s c h r e o d in g e r t yp e e qu a t io n " , analysis and applications. e d s . b y b a r s e g ia n g. a n d b e g e h r h ., n a to s c ie n c e p u b lic a t io n s , p p . 4 1 5 -4 4 2 , k lu we r , ( 2 0 0 4 ) . ø³ùëí»éç ¿é»ïïñá¹çý³ùçï³ï³ý ñ³ßíç ³éýí³í ùí³ýï³ûçý ø³ùëí»éçí³ïáõùç ýéáõïïáõ³óç³ý»ñá ². ¶¨áñ·û³ý ¨ ²ñ. ¶¨áñ·û³ý ²ù÷á÷áõù øí³ýï³ûçý ¹³ßïç ï»ëáõãû³ý ßñç³ý³ïý»ñáõù í³ïáõùá çñ»ýçó ý»ñï³û³óýáõù ¿ ï»õ, áñï»õ ³éï³ »ý ¿ý»ñ·»ïçï ³ù»ý ï»ë³ï ù³ëýçïý»ñ ¨ ¹³ßï»ñç ãéãéáõùý»ñ: ²ûé ëáëù»ñáí, í³ïáõùá µýáõã³·ñíáõù ¿ ýç½çï³ï³ý ã³÷»ñáí ¨ ï³éáõóí³íùáí, áñá ï³½ùáõù¿ µ³ó³ë³ï³ý ¿ý»ñ·»ïçï ùçç³í³ûñá, áñá ﻽»ñùáõù ã³÷³ýóáõù ¿ ³ù»ýáõñ»ù: ºã» ùí³ýï³ûçý ¹³ßïç ï»ëáõãûáõýá ëáïáñáùý»ñç ùççáóáí ï³ñáõ³ý³ñ ×ß·ñçï ýï³ñ³·ñ»ñ »ñ¨áõûãý»ñá, ³å³ í³ïáõùç ñ³ïïáõãûáõýý»ñá ýù³ý ïéçý»çý ùí³ýï³ûçý ý»ñ¹³ßý³ï ï³ï³ý³ïç ñ³ïïáõãûáõýý»ñçý: ø»ýù ³é³ççý ³ý·³ù ³ûë ëý¹çñá ¹çï³ñï»é »ýù ø³ùëí»é–è³ý娻ý ïçåç å³ï³ñ³ï³ý ¹çý»ñ»ýóç³é ñ³í³ë³ñáõùý»ñç ßñç³ý³ïý»ñáõù: ¸³ ù»½ ñý³ñ³íáñáõãûáõý ¿ ïí»é í³ïáõùáõùª ùí³ýï³ûçý µ³ßëù³ý ñ³ù³ñ ½³ñ·³óý»é ï³ýáý³íáñ ï»ëáõãûáõý ³é³ýó û·ï³·áñí»éáõ ëáïáñáõùý»ñç ù»ãá¹ý»ñá: ì»ñççý ñ³ý·³ù³ýùá ñý³ñ³íáñáõãûáõý ¿ ïí»é ù³ýñ³ù³ëýáñ»ý ñ»ï³½áï»é í³ïáõùç ¿é»ïïñáù³·ýçë³ï³ý µ³õ³¹ñç íç׳ﳷñáõãûáõýá ¨ ï³éáõóí³íùá: ø³ëý³íáñ³å»ë óáõûó ¿ ïñí³í, áñ ùí³ýï³ûçý í³ïáõùç ý»ñ³éáõùá ù³ùëí»éç ¿é»ïïñá¹çý³ùçï³ûç áõñí³·íáõù ýï³ñ³·ñíáõù ¿ »ñïáõ éñ³óáõóçã ã³÷»ñáí, áñáýù ïáùå³ïïýçï³óí³í »ý: ²ñï³ùçý ¹³ßïáõù ·ïýíáõ í³ïáõùç ùí³ýï³ûçý µ³ßëù³ý ñ³ù³ñ ëï³óí³í ¿ üáïï»ñ– äé³ýïç ïçåç ñ³í³ë³ñáõù ¨ ï³éáõóí³í »ý í³ïáõùç µ»ïù³ý óáõóçãý»ñá: òáõûó ¿ ïñí³í, áñ ýñ³ýù ï³ñáõ »ý ÷áëí»é ³ñï³ùçý ¹³ßïç ³½¹»óáõãû³ý ï³ï: d:\user\sbornik_38_pdf\33.dvi mathematical problems of computer science 38, 77{79, 2012. on the algebr as with h yper identities of the var iety of de m or gan algebr as y u . m. mo vs is ya n , v . a . a s la n ya n yerevan state university, alex manoogian 1, yerevan 0025, armenia e-mail: yurimovsisyan@yahoo.com, vahagn.aslanyan@gmail.com 1 in t r o d u c t io n in p a p e r [1 ] t h e a lg e b r a s wit h h yp e r id e n t it ie s o f t h e va r ie t y o f b o o le a n a lg e b r a s a r e c h a r a c t e r iz e d . in t h is p a p e r t h e a lg e b r a s wit h h yp e r id e n t it ie s o f t h e va r ie t y o f d e mo r g a n a lg e b r a s a r e c h a r a c t e r iz e d . fo r t h e s e a lg e b r a s wit h t wo b in a r y o p e r a t io n s we p r o ve a s t r u c t u r e r e s u lt . a s a c o n s e qu e n c e , we o b t a in t h e n e w ¯ n it e b a s e o f t h e h yp e r id e n t it ie s o f t h e va r ie t y o f d e mo r g a n a lg e b r a s , h a vin g fu n c t io n a l a n d o b je c t ive r a n ks n o t e xc e e d in g t h r e e . a n a lg e b r a q( +; ¢;0 ) wit h t wo b in a r y a n d o n e u n a r y o p e r a t io n s is c a lle d a d e mo r g a n a lg e b r a if q ( +; ¢ ) is a d is t r ib u t ive la t t ic e a n d q ( +; ¢;0 ) s a t is ¯ e s t h e fo llo win g id e n t it ie s : ( x + y ) 0 = x0 ¢ y0; x00 = x; wh e r e x00 = ( x0 ) 0. th e s t a n d a r d fu z z y a lg e b r a f = ( ( 0 ; 1 ) ; max ( x; y ) ; min( x; y ) ; 1 ¡ x ) is a n e xa m p le o f a d e mo r g a n a lg e b r a . d e mo r g a n a lg e b r a s we r e c o n s id e r e d b y j.a .k a lm a n [2 ]( a s i-la t t ic e s ) , g.c.mo is il [3 ], h .r a s io wa a n d a .b ia lyn ic ki-b ir u la [4 ], y u .m.mo vs is ya n [5 ], j. b e r m a n a n d w . b lo k [6 ] a n d o t h e r s . th e y a ls o r e la t e d t o c o n s t r u c t ive lo g ic wit h s t r o n g n e g a t io n ( a .a .ma r ko v [7 ], d .n e ls o n [8 ], n .n .v o r o b e v [9 ], i.d .za s la vs ky [1 0 ]) . e xc e p t in m a t h e m a t ic a l lo g ic a n d a lg e b r a , d e mo r g a n a lg e b r a s ( a n d d e mo r g a n b is e m ila t t ic e s ) h a ve a p p lic a t io n s in m u lt i-va lu e d s im u la t io n s o f d ig it a l c ir c u it s t o o ( [1 1 , 1 2 ]) . th e h yp e r id e n t it ie s o f t h e va r ie t y o f d e mo r g a n a lg e b r a s a r e c h a r a c t e r iz e d in [1 3 ]. de¯nition 1.1 a t -algebra a = ( q; §) , where t = f1 ; 2 g, is called d e m organ quasilattice if it satis¯es all hyperidentities of the variety of d e m organ algebras. fo r e xa m p le , t h e s u p e r p r o d u c t ( [1 4 , 1 5 , 1 6 , 1 7 ]) o f t h e t wo d e mo r g a n a lg e b r a s ( d e mo r g a n qu a s ila t t ic e s ) is a d e mo r g a n qu a s ila t t ic e . 7 7 7 8 on the algebras with hyperidentities of the variety of de morgan algebras 2 ma in r e s u lt b e lo w we d e ¯ n e t h e c o n c e p t o f d e mo r g a n s u m a n a lo g o u s t o b o o le a n s u m in t r o d u c e d in [1 ]. de¯nition 2.1 l et a = ( q; ­ [ ff g) be an algebra with a unary operation f . l et ( qi;­ ) ; i 2 i be subalgebras of the algebra a, and ai = ( qi; ­ [ ffig) be algebras with a unary operation fi. the algebra a is called d e m organ sum of algebras ai, if the following conditions hold true: 1 ) qi \ qj = â for all i; j 2 i; i 6= j; 2 ) q = s i2i qi; 3 ) two binary operations +; ¢ and a unary operation ¹ can be de¯ned on i such that i ( +; ¢; ¹ ) is a d e m organ algebra; 4 ) if i; j 2 i and i · j (here " · " is the order of the lattice i ( +; ¢ ) ), then there exists an isomorphism ( 'i;j; ~" ) : ai ! aj ; where ~" ( fi ) = fj; ~" ( a ) = a for any a 2 ­. m oreover, 'i;i is the identical mapping of the set qi, and for all i · j · k we have 'i;j ¢ 'j;k = 'i;k; 5 ) f or every i 2 i there exists an isomorphism ( hi;i; ~" ) : ai ! ai; such that h¡1 i;i = hi;i and hi;i ¢ 'i;k = 'i;k for all k ¸ i + i; k 2 i; 6 ) f or any n-ary operation a 2 ­ ( n ¸ 2 ) and for any x1; : : : ; xn 2 q we have: a ( x1; : : : ; xn ) = a ( 'i1;i0 ( x1 ) ; : : : ; 'in;i0 ( xn ) ) ; where xj 2 qij , ij 2 i; j = 1 ; n, i0 = i1 + : : : + in; 7 ) f or any x 2 q we have: f ( x ) = hi;i ( fi ( x ) ) ; where x 2 qi. t heor em 2.1 an algebra a = ( q; f+; ¢;¹ g) with two binary operations +; ¢ and one unary operation ¹ is a d e m organ quasilattice i® it is a d e m organ algebra or d e m organ sum of d e m organ algebras. cor ollar y 2.1 the variety of d e m organ algebras has a ¯nite base of hyperidentities having functional and objective ranks not exceeding three. r e fe r e n c e s [1 ] y u .m. mo vs is ya n , algebras with hyperidentities of the variety of b oolean algebras. iz ve s t iya r o s s iys ko y a ka d e m ii n a u k: s e r iya ma t e m a t ic h e s ka ya 6 0 , 1 2 7 -1 6 8 , 1 9 9 6 . e n g lis h t r a n s la t io n in r u s s ia n a c a d e m y o f s c ie n c e iz ve s t iya ma t e m a t is c h e s ka ya , 6 0 , 1 2 1 9 1 2 6 0 , 1 9 9 6 . [2 ] j. a . k a lm a n , l a t t ic e s wit h in vo lu t io n , trans. amer. m ath. soc. 8 7 , ( 1 9 5 8 ) , 4 8 5 -4 9 1 . [3 ] g.c.mo is il, r e c h e r c h e s s u r l'a lg e b r e d e la lo g iqu e , annales scienti¯ques de l'universite de j assy, 2 2 , ( 1 9 3 5 ) , 1 -1 1 7 . yu. movsisyan, v. aslanyan 7 9 [4 ] a .b ia lyn ic ki-b ir u la , h .r a s io wa , on t h e r e p r e s e n t a t io n o f qu a s i-b o o le a n a lg e b r a s , b ull. acad. p olon. sci., ser. m ath. astronom. p hys., 5 , ( 1 9 5 7 ) , 2 5 9 -2 6 1 . [5 ] y u .m.mo vs is ya n , b in a r y r e p r e s e n t a t io n s o f a lg e b r a s wit h a t m o s t t wo b in a r y o p e r a t io n s . a ca yle y t h e o r e m fo r d is t r ib u t ive la t t ic e s , international j ournal of algebra and computation, v o l.1 9 , 1 ( 2 0 0 9 ) , 9 7 -1 0 6 . [6 ] j. b e r m a n , w . b lo k, s t ip u la t io n s , m u lt i-va lu e d lo g ic a n d d e mo r g a n a lg e b r a s , m ultivalued l ogic 7 ( 5 -6 ) ( 2 0 0 1 ) , 3 9 1 -4 1 6 . [7 ] a .a .ma r ko v, co n s t r u c t ive l o g ic ( in r u s s ia n ) ,uspekhi m at. nauk, 5 ( 1 9 5 0 ) ,1 8 7 -1 8 8 . [8 ] d .n e ls o n , co n s t r u c t ib le fa ls it y, j . symbolic l ogic, 1 4 ( 1 9 5 9 ) , 1 6 -2 6 . [9 ] n .n .v o r o b e v, a c o n s t r u c t ive p r o p o s it io n a l c a lc u lu s wit h s t r o n g n e g a t io n ( in r u s s ia n ) , d okl. akad. nauk ssr , 8 5 ( 1 9 5 2 ) , 4 6 5 -4 6 8 . [1 0 ] i.d .za s la vs ky, symmetric constructive l ogic ( in r u s s ia n ) , p u b lis h in g h o u s e o f a c a d e m y o f s c ie n c e s o f a r m e n ia n s s r ( 1 9 7 8 ) . [1 1 ] j.a . b r z o z o ws ki, d e mo r g a n b is e m ila t t ic e s , p roceedings of the 30th ie e e international symposium on m ultiple-valued l ogic ( is mv l 2 0 0 0 ) , ma y 2 3 -2 5 , ( 2 0 0 0 ) , p .1 7 3 . [1 2 ] j.a . b r z o z o ws ki, p a r t ia lly o r d e r e d s t r u c t u r e s fo r h a z a r d d e t e c t io n , special session: the m any l ives of l attice theory, j oint m athematics m eetings, s a n d ie g o , ca , ja n u a r y 6 -9 , ( 2 0 0 2 ) . [1 3 ] y u .m.mo vs is ya n , v .a .a s la n ya n , hyperidentities of d e m organ algebras, l o g ic jo u r n a l o f t h e igp l .( d o i:1 0 .1 0 9 3 / jig p a l/ jz r 0 5 3 ) [1 4 ] y u .m. mo vs is ya n , introduction to the theory of algebras with hyperidentities ( in r u s s ia n ) , y e r e va n s t a t e u n ive r s it y p r e s s , y e r e va n , 1 9 8 6 . [1 5 ] y u .m. mo vs is ya n , h yp e r id e n t it ie s in a lg e b r a s a n d va r ie t ie s , uspekhi m atematicheskikh nauk, v o l.5 3 , n o . 1 ( 3 1 9 ) , ( 1 9 9 8 ) , 6 1 { 1 1 4 . e n g lis h t r a n s la t io n in r u s s ia n ma t h e m a t ic a l s u r ve ys 5 3 , 1 , ( 1 9 9 8 ) , 5 7 { 1 0 8 . [1 6 ] y u .m.mo vs is ya n , b ila t t ic e s a n d h yp e r id e n t it ie s , p roceedings of the steklov institute of m athematics, v o l. 2 7 4 , ( 2 0 1 1 ) , p p . 1 7 4 -1 9 2 . [1 7 ] y u .m.mo vs is ya n , a .b .r o m a n o ws ka , j.d .h s m it h , s u p e r p r o d u c t s , h yp e r id e n t it ie s , a n d a lg e b r a ic s t r u c t u r e s o f lo g ic p r o g r a m m in g , comb. m ath. and comb. comp., 2 0 0 6 , v.5 8 , p .1 0 1 -1 1 1 . d:\sbornik\...\article_eng.dvi mathematical problems of computer science 30, 31{35, 2008. on e xistence of 2-par tition of a t r ee, which obeys the given p r ior ity s u r e n v . b a likya n y, r a fa ye l r . k a m a lia n z y yerevan state university e-mail: suren.balikyan@gmail.com z russian-armenian state university e-mail: rrkamalian@yahoo.com abstract a necessary and su±cient condition is obtained for the problem of such partitioning of the set of vertices of a tree g into two disjoint sets v1 and v2, which, for a given function p : v (g) ! f¡1; 0; 1g with some special restriction, satis¯es the condition j¸(v) \ v1j ¡ j¸(v) \ v2j = p(v) ¢ (jfvg \ v1j ¡ jfvg \ v2j) for any vertex v of g, where ¸(v) is the set of all vertices of g adjacent to v. refer ences [1 ] s . v . b a likya n , r . r . k a m a lia n , " on n p -c o m p le t e n e s s o f t h e p r o b le m o f e xis t e n c e o f lo c a lly-b a la n c e d 2 -p a r t it io n fo r b ip a r t it e g r a p h s g wit h ¢ ( g) = 3 " , r eports of nas r a, applied m athematics, vo l. 1 0 5 , n u m . 1 , p p . 2 1 { 2 7 , 2 0 0 5 . [2 ] s . v . b a likya n , r . r . k a m a lia n , " on n p -c o m p le t e n e s s o f t h e p r o b le m o f e xis t e n c e o f lo c a lly-b a la n c e d 2 -p a r t it io n fo r b ip a r t it e g r a p h s g wit h ¢ ( g ) = 4 u n d e r t h e e xt e n d e d d e ¯ n it io n o f t h e n e ig h b o u r h o o d o f a ve r t e x" , r eports of nas r a, applied m athematics, vo l. 1 0 6 , n u m . 3 , p p . 2 1 8 { 2 2 6 , 2 0 0 6 . [3 ] s . v . b a likya n , " on lo c a lly-b a la n c e d 2 -p a r t it io n s o f s o m e b ip a r t it e g r a p h s " , abstracts of papers of 15th international conference "m athe m atics. com p uting. e d ucation.", vo l. 1 5 , p . 7 , d u b n a , r u s s ia , ja n u a r y 2 8 fe b r u a r y 0 2 2 0 0 8 . [4 ] f. h a r a r y, graph theory, a d d is o n -w e s le y, r e a d in g , ma , 1 9 6 9 . [5 ] c. b e r g e , graphs and hypergraphs, e ls e vie r s c ie n c e l t d , 1 9 8 5 . 3 1 3 2 on existence of 2-partition of a tree, which obeys the given priority ì³éç ³ûýåçëç 2-ïñáñù³ý ·áûáõãû³ý ù³ëçý, áñá »ýã³ñïíáõù ¿ ïñí³í ý³ë³å³ïíáõãû³ýá ê. ´³éçïû³ý, è. ø³ù³éû³ý ²ù÷á÷áõù êï³óí³í ¿ ³ýññ³å»ßï ¨ µ³í³ñ³ñ å³ûù³ý g í³éç ·³·³ãý»ñç µ³½ùáõãû³ý v1 ¨ v2 ãñ³ïíáõ »ýã³µ³½ùáõãûáõýý»ñç ³ûýåçëç ïñáñù³ý ·áûáõãûáõýá å³ñ½»éáõ ñ³ù³ñ, áñ ïñí³í ñ³ïáõï ë³ñù³ý³÷³ïáõùý»ñáí ýáõýïóç³ûç ñ³ù³ñ µ³í³ñ³ñíç ñ»ï¨û³é å³ûù³ýá p : v ( g ) ! f¡ 1 ; 0 ; 1 g í³éç ûáõñ³ù³ýãûáõñ v ·³·³ãç ñ³ù³ñ j (̧ v ) \v1j¡j¸ ( v ) \ v2j = p( v ) ¢ ( jfvg \ v1j ¡ jfvg \ v2j ) , áñï»õ ¸ ( v ) -áí ýß³ý³ïí³í ¿ v-çý ïçó ·³·³ãý»ñç µ³½ùáõãûáõýá: d:\sbornik\...\19davit\rev.dvi mathematical problems of computer science 31, 167{168, 2008. t he subsystem of i nfor ming on cr itical situations t hr ough cor r espondence in acm cluster d. petrosyan, d. gevorgyan and k. khachaturyan institue for informatics and automation problems of nas of ra on t h e b a s e o f m o b ile c o m m u n ic a t io n o p p o r t u n it ie s t h e a cm s ys t e m a llo ws t o b e we llin fo r m e d a b o u t t h e e xt r e m e s it u a t io n s o c c u r r in g in t h e wo r kin g t e r r it o r y o f clu s t e r t h r o u g h c a llin g t h e p h o n e n u m b e r o f t h e c o r r e s p o n d in g t e c h n ic a l s t a ® g ive n b e fo r e h a n d , b e s id e s it m a n a g e s a n d p r o t e c t s t h e wo r kin g t e r r it o r y o f clu s t e r fr o m ille g a l e n t r a n c e , a ls o a llo win g t o fo r m a fr e qu e n c y-t im e m o n it o r in g o f p e o p le e n t e r in g t h a t t e r r it o r y. a lo n g wit h t h e b a s ic wo r kin g p r o g r a m s , p r e ve n t io n fr o m ille g a l e n t r a n c e a n d ¯ r e p r o t e c t io n s ys t e m s ( via in fo r m a t io n fr o m t h e r m a l, m a g n e t ic -c o n t a c t , in fr a -r e d , o p t ic a l t r a n s m it t e r s ) in clu s t e r , it s t e c h n ic a l p a r a m e t e r s o b t a in in g is r e g u la r ly p e r fo r m e d , wh e r e t h e t e m p e r a t u r e , wo r kin g s t a t e a n d o t h e r p a r a m e t e r s o f t h e e le c t r o n ic e qu ip m e n t s a va ila b le in clu s t e r a r e e s t im a t e d . a n d , o f c o u r s e , it is d e s ir a b le a n d n e c e s s a r y t o g o ve r n t h o s e p a r a m e t e r s ( fo r c e r t a in g o a ls , p a r t ic u la r ly p r o vid in g fo r a n a lys is , s t a t is t ic s , s e c u r it y) , t o c o n t r o l a n d p r e ve n t o r a r r a n g e t h e e xt r e m e s it u a t io n s a r is in g in t h a t e qu ip m e n t s if n e c e s s a r y. th o u g h t h e r e a r e m e c h a n is m s in clu s t e r d e s ig n e d t o d is c o ve r a n e xt r e m e s it u a t io n a b o u t a n y e qu ip m e n t , it is a c t u a l t o g e t e m e r g e n c y in fo r m a t io n im m e d ia t e ly wh e n b e in g o u t o f t h e wo r kin g t e r r it o r y o f clu s t e r . th e a cm s ys t e m h a s b e e n e s t a b lis h e d fo r t h is g o a ls . it is b a s e d o n t h e m o b ile c o m m u n ic a t io n o p p o r t u n it ie s a n d a llo ws t o in fo r m a b o u t e xt r e m e s it u a t io n s . a cm p r o g r a m a llo ws " h is " u s e r s t o e n t e r t h e wo r kin g t e r r it o r y o f clu s t e r a ft e r t u r n in g o ® t h e p r o t e c t io n fr o m ille g a l e n t r a n c e s ig n a l s ys t e m o f t h e wo r kin g t e r r it o r y o f clu s t e r , p r o vid in g t h a t t h e p r o t e c t io n s ig n a l s ys t e m is t u r n e d o n wh e n g o in g o u t o f t h e t e r r it o r y. a n d t h e s ys t e m " r e c o g n iz e s " it s u s e r s a c c o r d in g t o t h e p a s s wo r d im p o r t e d d u r in g t h e r e g is t r a t io n in t h e u s e r s ' s ys t e m . u p o n d ia lin g t h e s p e c ia l p h o n e n u m b e r o f t h e s ys t e m a n d e n t e r in g t h e m a in m e n u o f a cm p r o g r a m , t h e s ys t e m d e m a n d s t o im p o r t t h e p a s s wo r d o f t h a t u s e r b e fo r e h a n d , a n d t h e n it c a n im p le m e n t t h e fo llo win g a c t io n s , 1 . t u r n o ® t h e p r o t e c t io n fr o m ille g a l e n t r a n c e s ig n a l s ys t e m o f t h e wo r kin g t e r r it o r y o f clu s t e r , 2 . t u r n o n t h e p r o t e c t io n s ig n a l s ys t e m o f t h e wo r kin g t e r r it o r y o f clu s t e r fr o m ille g a l e n t r a n c e , 3 . o b t a in d e ¯ n it e c u r r e n t p a r a m e t e r s a b o u t s o m e p h ys ic a l e qu ip m e n t s o f clu s t e r ( t h e a m o u n t o f wo r kin g p r o c e s s o r s , e t c .) b e s id e s g ivin g t h e e n t r a n c e r ig h t t o t h e u s e r s , t h e s ys t e m r e g is t e r s t h e u s e r s o f it s s e r vic e s a c c o r d in g t o t im e ( e .g . wh e n t h a t u s e r e n t e r e d t h e wo r kin g t e r r it o r y o r wh e n le ft ) . a s it wa s m e n t io n e d a b o ve m o n it o r in g o f t e c h n ic a l p a r a m e t e r s , in c lu d in g t e m p e r a t u r e , wo r kin g s t a t e a n d o t h e r p a r a m e t e r s o f t h e e le c t r o n ic e qu ip m e n t s o f clu s t e r is r e g u la r ly p e r fo r m e d , in t h e p r o c e s s o f wh ic h t h e p a r a m e t e r s a r e e s t im a t e d . in c a s e e xt r e m e s it u a t io n s a r is e in clu s t e r 1 6 7 1 6 8 the subsystem of informing on critical situations through correspondence in acm cluster a n d o t h e r r o o m s a d ja c e n t t o it t h e n p r e s e n c e o f c r it ic a l p a r a m e t e r s s p e c ī c t o t h e s t r u c t u r e o f e le c t r o n ic e qu ip m e n t s , b e in g d a m a g e d o r in e xt r e m e s it u a t io n is ¯ xe d in t h e m o n it o r in g s u b s ys t e m . a ll t h e t e c h n ic a l p a r a m e t e r s d e r ive d fr o m clu s t e r a r e a n a lyz e d in t h e o u t lin e d s ys t e m a n d a r r a n g e d in a s p e c ia l b a s e . th e p a r a m e t e r s r e c e ive d in t h e c o n d it io n s o f n o r m a l wo r kin g s it u a t io n fr o m e a c h e le c t r o n ic e qu ip m e n t ( a s n u m e r ic a l in d ic a t o r s ) , a r e in a d e ¯ n it e kn o wn in t e r va l, t h e n in c a s e o f p a r a m e t e r va lu e s o u t o f t h o s e in t e r va ls ( e .g . a n y e qu ip m e n t is o ve r h e a t e d ) , t h e n u m b e r o f t h e e qu ip m e n t a n d t h e n u m e r ic a l in d ic a t o r o f t h e p a r a m e t e r s o u t o f it s in t e r va l a r e d e d u c e d . th e p a r a m e t e r s a r e r e c e ive d b y t h e s ys t e m wit h t h e h e lp o f t h e t e c h n o lo g y b a s e d o n ip d is t a n t c o m m u n ic a t io n , a n d t h e n t h e s ys t e m a u t o m a t ic a lly c a lls t h e p h o n e n u m b e r o f t h e c o r r e s p o n d in g t e c h n ic a l s t a ® g ive n b e fo r e h a n d a n d in fo r m s o n s p e c ī c a lly wh ic h p r o c e s s o r is u n d e r c r it ic a l s it u a t io n . refer ences [1 ] âàðäàíÿí, ä. ãåâîðêÿí, à. íàíàñÿí, â. ñààêÿí, ä. òàäåâîñÿí, ê.õà÷àòðÿí, àêì (àðìêëàñòåðìîíèòîð) ñèñòåìà ìîíèòîðèíãà òåõíîëîãè÷åñêèõ ïàðàìåòðîâ, âûñîêîïðîèçâîäèòåëüíîé ñèñòåìû ”àðìêëàñòåð”. csit 2007 ²îø îé³ëï»ñáõù ïñçïçï³ï³ý çñ³íç׳ïý»ñç ù³ëçý ñ»é³ï³ ï»õ»ï³óù³ý »ýã³ñ³ù³ï³ñ·á ¸.ä»ïñáëû³ý, ¸. ¶¨áñ·û³ý, î. ê³ã³ïñû³ý ²ù÷á÷áõù ²îø ñ³ù³ï³ñ·á‘ ñ»ýí»éáí µçç³ûçý ï³åç ñý³ñ³íáñáõãûáõýý»ñç íñ³, ãáõûé ¿ ï³éçë çñ³½»ïí»é ïé³ëï»ñç ³ßë³ï³ýù³ûçý ï³ñ³íùáõù å³ï³ñ³í >ùëïñ»ù³é çñ³íç׳ïý»ñç ù³ëçý, ½³ý·³ñ³ñ»éáí ý³ë³å»ë ïñí³í ñ³ù³å³ï³ëë³ý ï»ëýçï³ï³ý ï³½ùç ñ»é³ëáë³ñ³ñçý, µ³óç ¹ñ³ýçó ³ûý ï³é³í³ñáõù ¨ å³ñå³ýáõù ¿ ïé³ëï»ñç ³ßë³ï³ýù³ûçý ï³ñ³íùá ³åûñçýç ùáõïù ·áñí»éáõó, ãáõûé ï³éáí ý³¨ ï³½ù»é ³û¹ ï³ñ³íù ùïýáõ ³ýó³ýó ñ³×³ë³ï³ý-å³ù³ý³ï³ûçý ùáýçïáñçý·: microsoft word tigran_hakobhin.doc mathematical problems of computer science 30, 71--75, 2008. 71 stop rule for image hierarchical segmentation algorithm david asatryan, grigor sazhumyan institute for informatics and automaation problems of nan ra e-mail dasat@ipia.sci.am abstract in this paper we consider an important problem of stopping the hierarchical segmentation procedure when the appropriate segmentation is achieved. this problem arises at every segmentation procedure, which uses a searching algorithm for selection of acceptable decision. we propose an algorithm for stopping the hierarchical segmentation procedure. stop-rule is based on the segmentation homogeneity measure, and uses a ratio of special sum of squares. the first sum equals to summarized variance of pixel intensity relative to the centers of intervals, being determined by thresholds, the second one expresses variance of mean values of the segments relative to the same centers. examples of segmentation results to demonstrate the features and properties of proposed technique are considered. references 1. d. martin, c. fowlkes, d. tal, and j. malik, “a database of human segmented natural images and its application to evaluating segmentation algorithms and measuring ecological statistics”, proc. iccv'01, vol. ii. vancouver, canada, pp. 416-423, 2001. 2. y. j. zhang, “a survey on evaluation methods for image segmentation”, pattern recognition 29(8), pp. 1335-1346, 1996. 3. q. luo, and t. m. khoshgoftaar, “unsupervised multiscale color image segmentation based on mdl principle”. ieee transactions on image processing, vol. 15, no 9, pp. 2755-2761, 2006. 4. nasa. “recursive hierarchical segmentation (rhseg) pre-processing software”, [http://techtransfer.gsfc.nasa.gov/rhseg 21.06.2006]. 5. д. г. асатрян, г. с. сажумян, “об одном методе пороговой локальной сегментации изображения”, mathematical problems of computer science, vol. 26, pp. 15-20, 2006. 6. д. г. асатрян, г. с. сажумян, “метод когерентной сегментации и его приложение к восстановлению поврежденных изображений”, вестник гиуа, сер. моделирование, оптимизация, управление, вып. 9, т. 2, с. 15-21, 2006. 7. d. g. asatryan, g. s. sazhumyan, and h. s. shahverdyan, “technique for coherent segmentation of image and applications”, mathematical problems of computer science, vol. 28, pp. 88-93, 2007. 72 stop rule for image hierarchical segmentation algorithm î³ý·³éç ï³ýáý` å³ïï»ñç ñç»ñ³ñëçï ñ³ïí³í³íáñù³ý ù»ãá¹ç ñ³ù³ñ ¸. ²ë³ïñû³ý, ¶. ê³åáõùû³ý, ²ù÷á÷áõù ü³ëïçýáõù ùß³ïí³í` å³ïï»ñç ñç»ñ³ñëçï ïáñ»ñ»ýï ñ³ïí³í³íáñù³ý ù»ãá¹ç ñ³ù³ñ ³é³ç³ñïí»é ¿ ï³ý·³éç ï³ýáý, ñ»ýí³í ù³é³ïáõëçý»ñç ·áõù³ñý»ñç »ñïáõ ³ñï³ñ³ûïáõãûáõýý»ñç ñ³ñ³µ»ñáõãû³ý íñ³: ¸ñ³ýóçó ³é³ççýý ³ñï³ñ³ûïáõù ¿ ëï³óí³í ë»·ù»ýïý»ñç ÷çùë»éý»ñç å³ûí³éáõãû³ý ù³é³ïáõë³ûçý ß»õáõùá ïçñ³éí³í ß»ù»ñç ³é³ç³óñ³í ùçç³ï³ûù»ñç ï»ýïñáýý»ñç ýï³ïù³ùµ, çëï »ñïñáñ¹á` ýáõûý ë»·ù»ýïý»ñç ùçççýý»ñç ù³é³ïáõë³ûçý ß»õáõùá ýáõûý ï»ýïñáýý»ñç ýï³ïù³ùµ: üï³ñ³·ñí»é »ý ³é³ç³ñïí³í ï³ýáýç ïçñ³éáõãû³ý ³ñ¹ûáõý³í»ïáõãûáõýá óáõó³¹ñáõ ñ³ù³å³ï³ëë³ý å³ïï»ñý»ñ ¨ ãí³ûçý ³ñ¹ûáõýùý»ñ: d:\user\sbornik_38_pdf\2.dvi ìàòåìàòè÷åñêèå âîïðîñû êèáåðíåòèêè è âû÷èñëèòåëüíîé òåõíèêè 38, 8{9, 2012. î ìèíèìàëüíûõ è ìàêñèìàëüíî òóïèêîâûõ nðàñïîçíàþùèõ ñèñòåìàõ íàòóðàëüíûõ ÷èñåë ñ.ì.âàðäàíÿí institute for informatics and automation problems of nas of ra e-mail:seyranv@ipia.sci.am íèæå ðàññìàòðèâàþòñÿ n-ðàñïîçíàþùèå ñèñòåìû â êëàññå äâóõýëåìåíòíûõ ïîäìíîæåñòâ îòíîñèòåëüíî îïåðàöèé ïåðåñå÷åíèÿ è äîïîëíåíèÿ. àíîëîãè÷íûå çàäàè ðàññìîòðåíû â [1-3]. ïðèâåäåì îñíîâíûå îïðåëåäåíèÿ ïîíÿòèé, èñïîëçóåìûõ íèæå. ðàññìîòðèì êîíå÷íîå ìíîæåñòâî [n] = f1 ; 2 ; ¢ ¢ ¢ ; ng, n ¸ 3 . ×åðåç r[n] îáîçíà÷èì ìíîæåñòâî ïîäìíîæåñòâ ìíîæåñòâà [n]. ïóñòü n¤ = fa1; a2; ¢ ¢ ¢ ; akg ÿâëÿåòñÿ ïîäìíîæåñòâîì ìíîæåñâà r[n]. îïðåäåëåíèå 1. áóäåì ãîâîðèòü, ÷òî ñèñòåìà n¤ ðàñïîçíàåò ýëåìåíò i 2 [n], åñëè ñ ïîìîùüþ îïåðàöèé ïåðåñå÷åíèÿ è äîïîëíåíèÿ (ïî îòíîøåíèþ ê k[n]) èç ìíîæåñòâ a1; a2; ¢ ¢ ¢ ; ak ìîæíî ïîëó÷èòü fig. îïðåäåëåíèå 2. áóäåì ãîâîðèòü ÷òî n¤ ÿâëÿåòñÿ n-ðàñïîçíàþùèé ñèñòåìîé, åñëè ñèñòåìà n¤ ðàñïîçíàåò êàæäûé ýëåìåíò i 2 [n]. îïðåäåëåíèå 3. n-ðàñïîçíàþùàÿ ñèñòåìà n¤ íàçûâàåòñÿ òóïèêîâûé, åñëè ëþáîå ñîáñòâåííîå ïîäìíîæåñòâî ìíîæåñòâà n¤ íå ÿâëÿåòñÿ n-ðàñïîçíàþùåé ñèñòåìîé. îïðåäåëåíèå 4. n-ðàñïîçíàþùàÿ ñèñòåìà n¤ íàçûâàåòñÿ ìèíèìàëüíîé, åñëè íå ñóøåñòâóåò n-ðàñïîçíàþùåé ñèñòåìû ñ ìîùíîñòüþ, ìåíøåé, ÷åì jn¤j. îïðåäåëåíèå 5. äâà ïîäìíîæåñòâà ìíîæåñòâà r[n] íàçûâàþòñÿ èçîìîðôíûìè, åñëè ñóøåñòâóåò âçàèìíî-îäíîçíà÷íîå îòîáðàæåíèå ìíîæåñòâà [n] íà [n], ïåðåâîäÿùåå îäíî èç íèõ â äðóãîå. îïðåäåëåíèå 6. ñèñòåìó n¤ íàçûâàåì ìàêñèìàëüíî òóïèêîâîé, åñëè íå ñóùåñâóåò òóïèêîâîé n-ðàñïîçíàþùåé ñèñòåìû, ó êîòîðîé êîëè÷åñòâî ìíîæåñòâ áîëüøå ÷åì ó ñèñòåìû n¤. íèæå ðàññìàòðèâàþòñÿ òîëüêî òàêèå n-ðàñïîçíàþùèå ñèñòåìû n¤, äëÿ êîòîðûõ jaij = 2 ïðè ëþáîì i. â [4,5] óñòàíàâëèâàþòñÿ ìîùíîñòè ìàêñèìàëüíî òóïèêîâûõ, à òàêæå ìèíèìàëüíûõ ðàñïîçíàþùèõ ñèñòåì. òåîðåìà 1. ïðè n ´ 0 ( m o d 3 ) è n ¸ 9 ÷èñëî ìèíèìàëüíûõ ïîïàðíî íå èçîìîðôíûõ n-ðàñïîçíàþùèõ ñèñòåì ðàâíî 7.(ïðè n = 6 , ýòî ÷èñëî ðàâíî 4, à ïðè n = 3 îíî ðàâíî 1). òåîðåìà 2. ïðè n ´ 1 ( m o d 3 ) è n ¸ 4 ñóøåñòâóåò åäèíñòâåííàÿ ìèíèìàëíàÿ n-ðàñïîçíàþùàÿ ñèñòåìà ñ òî÷íîñòüþ äî èçîìîðôèçìà. òåîðåìà 3. ïðè n ´ 2 ( m o d 3 ) è n ¸ 5 ÷èñëî ìèíèìàëüíûõ ïîïîðíî íå èçîìîðôíûõ n-ðàñïîçíàþùèõ ñèñòåì ðàâíî 2. òåîðåìà 4. ïðè n ¸ 3 , ÷èñëî ïîïàðíî íå èçîìîðôíûõ ìàêñèìàëíî òóïèêîâûõ n-ðàñïîçíàþùèõ ñèñòåì ðàâíî h n 2 i (ãäå h n 2 i åñòü öåëàÿ ÷àñòü n 2 ). 8 ñ. âàðäàíÿí 9 ñïèñîê ëèòåðàòóðû [1] ï. ýðäåù, äæ. ñïåíñåð, âåðîÿòíîñòíûå ìåòîäû â êîìáèíàòîðèêå. ìîñêâà, ìèð 1976. [2] ñ.ì. âàðäàíÿí, îá îäíîé çàäà÷å ðàñïîçíàâàíèÿ ìíîæåñòâ, äàí àðì. ññð, òîì 72, ñ. 141-143, 1981. [3] s. m. vardanyan, recognizing sets (systems), proceedings of the international conference ”computer science and information technologies” csit05, pp 161 162, yerevan, armenia 2005. [4] s. m. vardanyan, on the powers of dead-end recognizing systems in the class of two-element sets concerning operations of intersection and complement, mathematical problems of computer sciences, vol.35, pp. 104 108, 2011. [5] ñ. ì. âàðäàíÿí, î ìèíèìàëüíîñòè íåêîòîðûõ ðàñïîçíàþùèõ ñèñòåì â êëàññå äâóõýëåìåíòíûõ ïîäìíîæåñòâ îòíîñèòåëíî îïåðàöèé ïåðåñå÷åíèÿ è äîïîëíåíèÿ, äíàí ðà, òîì 112, n1, ñ.57 62, 2012. 61 mathematical problems of computer science 58, 61–66, 2022. doi: 10.51408/1963-0093 udc 510.64 proof complexity of hard-determinable balanced tautologies in frege systems anahit a. chubarya yerevan state university e-mail: achubaryan@ysu.am abstract hard-determinable property and balanced property of tautologies are specified as important properties in the study of proof complexities formerly. in this paper harddeterminable and balanced properties are studied together. it is shown that some sequences of hard determinable balanced tautologies have polynomially bounded frege proofs. keywords: hard-determinable tautologies, balanced tautologies, frege systems, proof complexity characteristics. article info: received 29 june 2022; accepted 29 september 2022. 1. introduction one of the most fundamental problems in proof complexity theory is to find an efficient proof system for classical propositional logic (cpl). there is a widespread understanding that polynomial time computability is the correct mathematical model of feasible computation. according to the opinion, a truly "effective" system should have a polynomial size 𝑝(𝑛) proof for every tautology of size 𝑛. in [1] cook and reckhow named such a system a supersystem. they showed that 𝑁𝑃 = 𝑐𝑜𝑁𝑃 iff there exists a supersystem. it is well known that many systems are not super. this question about the frege system, the most natural calculi for propositional logic, is still open. in many papers, some specific sets of tautologies are introduced, and it is shown that the question about polynomial bounded sizes for frege proofs of all tautologies is reduced to an analogous question for a set of specific tautologies. in particular the hard-determinable tautologies and balanced tautologies are introduced in [2,3] as such sets of specific tautologies. in this paper, the hard-determinable and balanced properties are studied together and it is shown that some mailto:achubaryan@ysu.am proof complexity of hard-determinable balanced tautologies in frege systems 62 sequences of hard-determinable balanced tautologies have polynomial bounded frege proofs. using the notions and results of this paper and the results of [3-4] the above-mentioned statement of cook and reckhow can be rephrased as follows: 𝑁𝑃 = 𝑐𝑜𝑁𝑃 iff in some frege system of cpl the proofs for all hard-determinable balanced formulas are polynomially bounded. 2. preliminaries to prove our main result, we recall some notions and notation. we will use the current concepts of the unit boolean cube (𝐸𝑛), a propositional formula, a tautology, a proof system for cpl, and proof complexity. the particular choice of a language for presenting propositional formulas is immaterial in this consideration. however, because of some technical reasons we assume that the language contains propositional variables, denoted by small latin letters with indices. logical connectives ¬, &, ∨, ⊃, and parentheses ( , ). note that some parentheses can be omitted in generally accepted cases. 2.1. hard-determinable and balanced tautologies following the usual terminology we call the variables and negated variables literals. the conjunct 𝐾 (clause) can be represented simply as a set of literals (no conjunct contains a variable and its negation simultaneously). in [3] the following notion is introduced. we call each of the following trivial identities for a propositional formula ψ a replacement-rule: 0&ψ = 0, ψ&0 = 0, 1&ψ = ψ, ψ&1 = ψ, ψ&ψ = ψ, ψ&¬ψ = 0, ¬ψ&ψ = 0, 0∨ ψ =ψ, ψ∨ 0=ψ, 1∨ψ =1, ψ∨1 =1, ψ∨ψ = ψ, ψ∨¬ ψ =1, ¬ψ∨ψ=1, 0⊃ψ=1, ψ⊃0=¬ψ, 1⊃ψ =ψ, ψ⊃1=1, ψ⊃ψ =1, ψ⊃¬ψ = ¬ψ, ¬ψ⊃ ψ = ψ, ¬0 = 1, ¬1 = 0, ¬¬ψ = ψ. application of a replacement rule to certain word consists in replacing some its subwords, having the form of the left-hand side of one of the above identities by the corresponding right-hand side. let 𝜑 be a propositional formula, let 𝑃 = {𝑝1, 𝑝2, … , 𝑝𝑛} be the set of the variables of 𝜑, and let 𝑃′ = {𝑝𝑖1 , 𝑝𝑖2 , … , 𝑝𝑖𝑚 } (1 ≤ 𝑚 ≤ 𝑛) be some subset of 𝑃. definition 1: given 𝜎 = {𝜎1, 𝜎2, … , 𝜎𝑚} ∈ 𝐸 𝑚, the conjunct 𝐾𝜎 = {𝑝𝑖1 𝜎1 , 𝑝𝑖2 𝜎2 , … , 𝑝𝑖𝑚 𝜎𝑚 } is called 𝜑determinative if assigning 𝜎1 (1 ≤ 𝑗 ≤ 𝑚) to each 𝑝𝑖𝑗 and successively using replacement rules we obtain the value of 𝜑 (0 or 1) independently of the values of the remaining variables. definition 2: we call the minimal possible number of variables in a 𝜑-determinative conjunct the determinative size of 𝜑 and denote it by ds(𝜑). by | 𝜑| we denote the size of the formula 𝜑, defined as the number of all logical signs entries in it. it is obvious that the full size of the formula, which is understood to be the number of all symbols is bounded by some linear function in |𝜑 |. definition 3: for sufficiently large 𝑛 the tautologies 𝜑𝑛 are called hard-determinable if there is some constant c such that 𝑙𝑜𝑔|𝜑𝑛|𝑑𝑠(𝜑𝑛) → 𝑐 for 𝑛 → ∞. definition 4: a formula 𝜑 is balanced if every propositional variable occurring in 𝜑 occurs exactly twice, once positive and once negative. a. chubaryan 63 example 1. the tautologies 𝜑𝑛 = 𝑝1 ⊃ (𝑝1 ⊃ (𝑝2 ⊃ (¬𝑝2 ⊃ (… ⊃ (𝑝𝑛 ⊃ 𝑝𝑛 ) … )))) are balanced. it is not difficult to see that 𝑑𝑠(𝜑𝑛) = 1, hence 𝜑𝑛 are not hard-determinable. example 2. the tautologies 𝑄𝐻𝑄𝑛 = 𝑉0≤𝑖≤𝑛 &1≤𝑗≤𝑛 [𝑉1≤𝑘≤𝑖 �̅�𝑖,𝑗,𝑘 ∨ 𝑉𝑖<𝑘≤𝑛 𝑞𝑘,𝑗,𝑖+1](𝑛 ≥ 1), are balanced. put 𝑄𝑖,𝑗 = 𝑉1≤𝑘≤𝑖 �̅�𝑖,𝑗,𝑘 ∨ 𝑉𝑖<𝑘≤𝑛𝑞𝑘,𝑗,𝑖+1(𝑛 ≥ 1, 0 ≤ 𝑖 ≤ 𝑛, 1 ≤ 𝑗 ≤ 𝑛), then 𝑄𝐻𝑄𝑛 = 𝑉0≤𝑖≤𝑛(𝑄𝑖1&𝑄𝑖2& … &𝑄𝑖𝑗 & … &𝑄𝑖(𝑛−1)&𝑄𝑖𝑛) and therefore 𝑑𝑠(𝑄𝐻𝑄𝑛). it is not difficult to see, that |𝑄𝐻𝑄𝑛| = 3𝑛2(𝑛+1) 2 − 1 |, hence 𝑄𝐻𝑄𝑛 are hard-determinable as well. 2.2. proof systems and proof complexities let us recall some notions from [1]. a frege system 𝓕 uses a denumerable set of propositional variables, a finite, complete set of propositional connectives; 𝓕 has a finite set of inference rules defined by a figure of the form 𝐴1𝐴2… 𝐴𝑚 𝐵 (the rules of inference with zero hypotheses are the schemes of axioms); 𝓕 must be sound and complete, i.e. for each rule of inference 𝐴1𝐴2… 𝐴𝑚 𝐵 every truth-value assignment, satisfying 𝐴1𝐴2 … 𝐴𝑚, also satisfies 𝐵, and 𝓕 must prove every tautology. in the theory of proof complexity two main characteristics of the proof are: 𝑙 – complexity to be the size of a proof (= the sum of all formulae sizes) and 𝑡 – complexity to be its length (= the total number of lines). the minimal 𝑙 – complexity (𝑡 – complexity) of a formula 𝜑 in a proof system φ we denote by 𝑙𝜑 φ(𝑡𝜑 φ). the polynomial equivalence (𝑝 − 𝑙 --equivalence, 𝑝 − 𝑡 --equivalence) of two proof systems by some proof complexity measure means that the transformation of any proof in one system into a proof in another system can be performed with no more than polynomial increase of proof complexity measure. it is well known that any two frege systems are 𝑝 − 𝑙 -equivalent (𝑝 − 𝑡 -equivalent). let 𝑀 be some set of tautologies. definition 5: we call the ф-proofs of tautologies from the set 𝑀 𝑡 -polynomially (𝑙 – polynomially) bounded if there is a polynomial 𝑝() such that 𝑡𝜑 𝛷 ≤ 𝑝(|𝜑|)(𝑙𝜑 𝛷 ≤ 𝑝(|𝜑|)) for all 𝜑 from 𝑀. 2.3. former results it was previously proven that a) tautologies without hard-determinability condition have 𝑡 -polynomially (𝑙 polynomially) bounded proofs in all systems of cpl [4], b) hard-determinability condition is sufficient (but not necessary) to obtain exponential lower bounds for both proof complexities of tautologies in “weak” proof systems of cpl (cutfree sequent, resolution, cutting planes etc.) [4], c) hard-determinability condition is not sufficient for exponential lower bounds of proof complexities in frege systems: for some examples of hard-determinable formulas the 𝑡 polynomially (𝑙 polynomially) bounded frege-proofs are given in [2]. some proof systems of cpl (calculus of structures with deep inference rules), where the author considers only formulas in negation normal form, are studied in [3], where among the rest of the results it is proved that proof complexity of hard-determinable balanced tautologies in frege systems 64 a) the set of above mentioned balanced formulas 𝑄𝐻𝑄𝑛 have polynomially bounded proofs in one of the studied system 𝑠𝐾𝑆, b) the relations between the proof complexities in the system 𝑠𝐾𝑆 and the frege systems are unknown for the present. 3. main result let 𝐹 be some frege system with inference rule modus ponens. theorem1: the 𝐹 -proofs of tautologies 𝑄𝐻𝑄𝑛 (𝑛 ≥ 1) are 𝑡-polynomially (𝑡-polynomially) bounded. to prove, we use the method of [2] for description of some polynomially bounded proof of 𝑄𝐻𝑄𝑛 direct in 𝐹 by reducing it to 𝐹 -proofs of well-known tautologies 𝑃𝐻𝑃𝑛 = &0≤𝑖≤𝑛𝑉1≤𝑗≤𝑛 𝑝𝑖𝑗 ⊃ 𝑉0≤𝑖<𝑘≤𝑛 𝑉1≤𝑗≤𝑛(𝑝𝑖𝑗 &𝑝𝑘𝑗 )(𝑛 ≥ 1) presenting the pigeonhole principle . it is proved in [5] that the set of these formulas is tpolynomially (𝑙polynomially) bounded. the following two auxiliary statements will be of use: lemma 1: given arbitrary formulas 𝛼, 𝛽, 𝛾, 𝛼𝑖, 𝛽𝑖, 𝛼𝑖𝑗 and 𝛽𝑖𝑗, the 𝐹-proofs of the following tautologies are 𝑡-polynomially (𝑙-polynomially) bounded: 1) α ∨ α¯, 2) (α ⊃ β) ⊃ ((β ⊃ γ) ⊃ (α ⊃ γ)), 3) (β¯ ⊃ α) ⊃ (¯α ⊃ β), 4) α1 ⊃ (α2 ⊃ (... ⊃ (αk ⊃ α1 &α2 &···&αk)...)) (k ≥ 2), 5) α ∨ α¯ ⊃ β1 ∨···∨ βk∨α ∨ βk+1 ∨··· ∨ βk+r ∨ α¯ ∨ βk+r+1 ∨··· ∨ βk+r+t (k ≥ 1, r ≥ 1, t ≥ 1), 6) ¬(𝑉1≤𝑖≤𝑘 &1≤𝑗≤𝑚𝛼𝑖𝑗 ) ⊃ &1≤𝑖≤𝑘 𝑉1≤𝑗≤𝑚�̅�𝑖𝑗 (𝑘 ≥ 1, 𝑚 ≥ 1) 7) &1≤𝑖≤𝑘 (𝛽1𝑖 ⋁𝛽2𝑖 ) ⊃ ¬(𝑉1≤𝑖≤𝑘 (�̅�1𝑖&�̅�2𝑖 )) (𝑘 ≥ 1). the proof is obvious. lemma 2: let 𝑄𝑖𝑗 and 𝑄𝑘𝑗 (0 ≤ 𝑖˂𝑘 ≤ 𝑛, 1 ≤ 𝑗 ≤ 𝑛) be the above denoted subformulas of 𝑄𝐻𝑄𝑛, then 𝐹-proofs of the formulas 𝑄𝑖𝑗 ∨ 𝑄𝑘𝑗 be 𝑡-polynomially (𝑙-polynomially) bounded. the proof follows from the fact of existence of some 𝑠 and 𝑚 (1 ≤ 𝑠 ≤ 𝑛, 1 ≤ 𝑚 ≤ 𝑛) such that 𝑄𝑖𝑗 contains 𝑞𝑠𝑗𝑚 and 𝑄𝑘𝑗 contains ¬𝑞𝑠𝑗𝑚, and also from 1) and 5) of lemma 1. from 6) of lemma 1 we infer for the formula 𝑄𝑛 = 𝑉0≤𝑖≤𝑛 &1≤𝑗≤𝑛 𝑄𝑖𝑗 . condition 1: the f-proofs of the formulas ¬𝑄𝐻𝑄𝑛 ⊃ &0≤𝑖≤𝑛𝑉1≤𝑗≤𝑛¬𝑄𝑖𝑗 are 𝑡-polynomially (𝑙-polynomially) bounded. put 𝑃𝐻𝑃𝑛 ’ = &0≤𝑖≤𝑛 𝑉1≤𝑗≤𝑛¬𝑄𝑖𝑗 ⊃ 𝑉0≤𝑖<𝑘≤𝑛𝑉1≤𝑗≤𝑛¬(𝑄𝑖𝑗 &¬𝑄𝑘𝑗 ) (1) a. chubaryan 65 the formulas (1) are obtained from the 𝑃𝐻𝑃𝑛 by the corresponding substitutions. hence, condition 2: the 𝐹-proofs of the formulas (1) are 𝑡-polynomially (𝑙-polynomially) bounded. let 𝐴𝑛 = 𝑉0≤𝑖<𝑘≤𝑛 𝑉1≤𝑗≤𝑛(¬𝑄𝑖𝑗 & ¬𝑄𝑘𝑗). using conditions (1), (2), and item 2) of lemma 1, we obtain condition 3: the 𝐹-proofs of the formulas ¬ 𝑄𝐻𝑄𝑛 ⊃ 𝐴𝑛 are 𝑡-polynomially (𝑙-polynomially) bounded. from lemma 2 and item 4) of lemma 1 we have condition 4: the f-proofs of the formulas 𝐵𝑛 = &0≤𝑖<𝑘≤𝑛 &1≤𝑗≤𝑛 (𝑄𝑖𝑗 ⋁𝑄𝑘𝑗 ) are 𝑡-polynomially (𝑙-polynomially) bounded, and from item 7) of lemma 1 it follows that the 𝐹proofs of the formulas ¬𝐴𝑛,𝑚 are 𝑡-polynomially (𝑙-polynomially) bounded as well. from the conditions (3), (4), and item 3) of lemma 1 we have a t-polynomial (l-polynomial) bound for the f-proofs of 𝑄𝑛 . corollary1: there are hard-determinable balanced formulas the f-proofs of which are tpolynomially (l-polynomially) bounded. 4. conclusion using the polynomial equivalence of different frege systems [1], the above mentioned result of cook and reckhow can be rephrased as follows: 𝑁𝑃 = 𝑐𝑜𝑁𝑃 iff in some frege system of cpl the proofs for all hard-determinable balanced formulas are polynomially bounded. references [1] s. a. cook and a. r. reckhow, “the relative efficiency of propositional proof systems,” j. symbolic logic, vol. 44, pp. 36–50, 1979. [2] s. r. aleksanyan and a. a. chubaryan, “the polynomial bounds of proof complexity in frege systems”, siberian mathematical journal, springer verlag, vol. 50, no. 2, pp. 243249, 2009. [3] l. sraßburger, “extension without cut”, annals of pure and applied logic, vol.163, pp. 19952007, 2012. [4] a. a. chubaryan, “relative efficiency of a proof system in classical propositional logic,” izv. nan armenii mat., vol. 37, no. 5, pp. 71–84, 2002. [5] s. r. buss, “polynomial size proofs of the propositional pigeonhole principle,” journal symbolic logic, vol. 52, pp. 916–927, 1987. proof complexity of hard-determinable balanced tautologies in frege systems 66 դժվար-որոշելի բալանսավորված նույնաբանությունների արտածումների բարդությունները ֆրեգեի համակարգերում անահիտ ա. չուբարյան երևանի պետական համալսարան e-mail: achubaryan@ysu.am ամփոփում նախկինում նույնաբանությունների դժվար-որոշելիության հատկությունը և բալանսավորված լինելու հատկությունը առանձնացվել էին որպես կարևոր հատկություններ արտածումների բարդությունների ուսումնասիրություններում: այս հոդվածոմ դժվար-որոշելիության և բալանսավորված լինելու հատկությունները ուսումնասիրվում են համատեղ: ապացուցվել է, որ դժվար-որոշելի բալանսավորված նույնաբանությունների մեկ դասի համար արտածումները ֆրեգեի համակարգերում բազմանդամորեն սահմանափակ են: բանալի բառեր՝ դժվար-որոշելի նույնաբանություններ, բալանսավորված նույնաբանություններ, ֆրեգեի համակարգեր, արտածման բարդությունների բնութագրիչներ: сложности выводов трудно-определяемых балансированных формул в системах фреге аанаит а. чубарян ереванский государственный университет e-mail: achubaryan@ysu.am аннотация ранее свойство трудно-определяемости и свойство балансированности тавтологий были выдлены как важные свойства в исследованиях сложностей выводов. в настоящей статье свойства трудно-определяемости и балансированности изучаются совместно. доказана полиномиальная ограниченность выводов в системах фреге для некоторого класса трудно-определяемых балансированных формул. ключевые слова: трудно-определяемые тавтологии, балансированные тавтологии, системы фреге, характеристики сложностей выводов. mailto:achubaryan@ysu.am mailto:achubaryan@ysu.am references hnalbandyan-dsm.dvi mathematical problems of computer science 23, 2004, 119{126. on system with distr ibuted shar ed m emor y h o vh a n n e s z. n a lb a n d ya n state engineering university of armenia e-mail hovign@web.am abstract distributed shared memory systems combine the scalability of loosely coupled multiple computer systems with the ease of usability of tightly coupled multiprocessors, providing with transparent replication and caching of data. this paper introduces distributed system for parallel computing { dspc, that provides distributed shared memory on top of network of workstations. programming model, memory organization, cache-coherence protocol and adaptive techniques are discussed in the paper. an evaluation with some well-known dsm benchmarks was done to present the overall performance of the dspc system. refer ences [1 ] d . b a ile y, l . d a g u m , e . b a r s z c z a n d h . s im o n . n a s p a r a lle l b e n c h m a r k r e s u lt s . in supercomputing, p a g e s 3 8 6 { 3 9 3 , 1 9 9 2 . [2 ] a . gh a z a r ya n . on s ys t e m fo r d is t r ib u t e d p a r a lle l c o m p u t a t io n s d s p c . in p roc. of the int'l conference on computer science and information technologies (csit 2001), a u g u s t 2 0 0 1 . [3 ] l . ift o d e , j. p . s in g h , a n d k . l i. s c o p e c o n s is t e n c y: a b r id g e b e t we e n r e le a s e c o n s is t e n c y a n d e n t r y c o n s is t e n c y. in p roc. of the 8th acm annual symp. on p arallel algorithms and architectures (sp aa'96), p a g e s 2 7 7 { 2 8 7 , ju n e 1 9 9 6 . [4 ] h . l u , s . d wa r ka d a s , a . co x, a n d w . zwa e n e p o e l. qu a n t ifyin g t h e p e r fo r m a n c e d i®e r e n c e s b e t we e n p vm a n d t r e a d m a r ks . j ournal of p arallel and d istributed computing, 4 3 , n o . 2 :6 5 { 7 8 , ju n e 1 9 9 7 . [5 ] h . n a lb a n d ya n . on p a r a lle l p r o c e s s in g wit h d is t r ib u t e d s h a r e d m e m o r y. in p roc. of the int'l conference on computer science and information technologies (csit 2003), p a g e s 3 1 7 { 3 2 0 , s e p t e m b e r 2 0 0 3 . [6 ] s . w o o , m. oh a r a , e . to r r ie , j.p . s in g h , a n d a . gu p t a . th e s p la s h -2 p r o g r a m s : ch a r a c h t e r iz a t io n a n d m e t h o d o lo g ic a l c o n s id e r a t io n s . in p roc. of the 22th annual int'l sump. on computer architecture (isca'95), p a g e s 2 4 { 3 6 , ju n e 1 9 9 5 . 1 1 9 1 2 0 on system with distributed shared memory ´³ßëí³í áý¹ñ³ýáõñ ñçßáõáãûáõýáí ñ³ù³ï³ñ·ç ù³ëçý ð. ¼. ü³éµ³ý¹û³ý ²ù÷á÷áõù ²ßë³ï³ýùáõù ¹çï³ñïíáõù ¿ ½áõ·³ñ»é ñ³ßí³ñïý»ñç ñ³ù³ñ ëï»õíí³í ýáñ µ³ßëí³í áý¹ñ³ýáõñ ñçßáõáõãûáõýáí ñ³ù³ï³ñ·ç ýï³ñ³·ñáõãûáõýá: øýý³ñïí³í »ý ñ³ù³ï³ñ·ç ï³éáõóí³íùá, ñçßáõáõãû³ý ùá¹»éá, ïíû³éý»ñç ñ³ù³å³ñ÷³ïáõãû³ý ³ñó³ý³·ñáõãûáõýá: æñ³·áñíí»é »ý ùç ß³ñù ëý¹çñý»ñ ñ³ù³ï³ñ·ç ³ñï³¹ñáõ³ï³ýáõãûáõýá ·ý³ñ³ï»éáõ ýå³ï³ïáí: êï³óí³í ³ñ¹ûáõýùý»ñá íï³ûáõù »ý ñ³ù³ï³ñ·ç µ³ñóñ ³ñ³·³·áñíáõãû³ý ù³ëçý: microsoft word aslanyan1.doc îçµ»éý»ïçï³ûç ¨ ñ³ßíáõ³ï³ý ï»ëýçï³ûç ù³ã»ù³ïçï³ï³ý ñ³ñó»ñ 30, 111--122, 2008. 111 ê³ñù³ý³÷³ï áõéáõóçïáõãû³ý ïáùá·ñ³ýç³ ¨ é³·ñ³ýåû³ý ùáï³ñïáõùý»ñç ù³ëçý è¨áý ²ëé³ýû³ý, ð³ëùçï ê³ñ³ïû³ý, ²ñïûáù ðáíë»÷û³ý ðð ¶²² æýýáñù³ïçï³ûç ¨ ³íïáù³ï³óù³ý åñáµé»ùý»ñç çýëïçïáõï lasl@sci.am hasmik@ipia.sci.am artyom.hovsepyan@gmail.com ²ù÷á÷áõù êáõûý ñá¹í³íáõù ¹çï³ñïíáõù ¿ ¹çëïñ»ï ïáùá·ñ³ýç³ûç áý¹ñ³ýáõñ ¹³ëç ùç ù³ëý³íáñ ëý¹çñ, áñç éáõíù³ý ñ³ù³ñ ³é³ç³ñïíáõù ¿ è³·ñ³ýåû³ý é»é³ùë³óç³ûç íñ³ ñçùýí³í ùç ùáï³íáñ ù»ãá¹ ñ³ù³å³ï³ëë³ý íñ³·ñ³ûçý çñ³ï³ý³óù³ùµ: ¶ñ³ï³ýáõãûáõý 1. r. j. gardner, p. gritzmann, d. prangenberg, on the computational complexity of reconstructing lattice sets from their x-rays. technical report (970-05012), techn. univ. munchen, fak. f. math, 1997. 2. g. j. woeginger. the reconstruction of polyominoes from their orthogonal projections. inform. process. lett., 77, pp 225-229, 2001. 3. e. barcucci, a. del lungo, m. nivat, and r. pinzani. reconstructing convex polyominoes from horizontal and vertical projections. theoret. comput. sci., 155, pp. 321-347, 1996. 4. g. dahl and t. flatberg. lagrangian decomposition for reconstructing hv-convex (0, 1) matrices, report 303, university of oslo, pp. 1-13, 2002. 5. m. guignard and s. kim. lagrangian decomposition: a model yielding stronger lagrangian bounds. math. prog., 39, pp215-228, 1987. 6. à. à. ñààêÿí, ãðàäèåíòíûå àëãîðèòìû ñèíòåçà (0,1)-ìàòðèö ñ ðàçëè÷íûìè ñòðîêàìè. äàí àðì ññð, lxxxiii, 5, ñòð. 207-209. 1986. 7. h. j. ryser. combinatorial properties of matrices of zeros and ones. canad. j. math., 9:pp 371-377, 1957. 8. d. gale. a theorem on flows in networks. pacific j. math., 7, pp 1073-1082, 1957. 112 ê³ñù³ý³÷³ï áõéáõóçïáõãû³ý ïáùá·ñ³ýç³ ¨ é³·ñ³ýåû³ý ùáï³ñïáõùý»ñç ù³ëçý on constrained convexity tomography and lagrangean approximations l. aslanyan, h. sahakyan, a. hovsepyan abstract in this paper one particular problem of general type of discrete tomography problems is considered and an approximate algorithm for its solution based on lagrangean relaxation is introduced. a program’s implementation is given as well. microsoft word kanal.doc îçµ»éý»ïçï³ûç ¨ ñ³ßíáõ³ï³ý ï»ëýçï³ûç ù³ã»ù³ïçï³ï³ý ñ³ñó»ñ 25, 2006, 12–17. 12 î³åáõõ³ûçý áõõ»·íù³ý ýáñ ³é·áñçãùç ù³ëçý ì³ñ¹³ý ². ø³ýáõïû³ý ðð¶²² æýýáñù³ïçï³ûç ¨ ³íïáù³ï³óù³ý åñáµé»ùý»ñç çýëïçïáõï e-mail vardanm2003@yahoo.com ²ù÷á÷áõù ²ßë³ï³ýùáõù ³é³ç³ñïí³í ¿ æê ý³ë³·íù³ý ï³åáõõ³ûçý áõõ»·íù³ý ùç ³é·áñçãù, áñáõù û·ï³·áñííáõù »ý áã-ù³ýñ»ã»ýû³ý é³ñ»ñ: àõõ»·íù³ý ³é·áñçãùá ñçùýí³í ¿ ½áõ·³ñ»é åõåç³ï³ûçý ï»ë³ï³íáñù³ý ¨ ï»õ³÷áëáõãûáõýý»ñç íñ³: ü³ëý³ï³ý ³ñ¹ûáõýùý»ñá óáõûó »ý ï³éçë, áñ ï³åáõõ³ûçý áõõ»·íù³ý ³ûë ¹³ëá µýáõã³·ñíáõù ¿ ³í»éç ÷áùñ é³ûýáõãû³ùµ, ù³ý ø³ýñ»ã»ýû³ý ùá¹»éç û·ï³·áñíù³ý ¹»åùáõù: ¶ñ³ï³ýáõãûáõý [1] t. ohtsuki, layout design and verification, 1986 [2] â.à. ñåëþòèí, ìàøèííîå êîíñòðóèðîâàíèå ýëåêòðîííûõ óñòðîéñòâ. ìîñêâà, “ñîâåòñêîå ðàäèî”, 1977. [3] a. frank. disjoint paths in a rectilinear grid. in combinatorica, 2(4), pages 361-371, 1982. [4] k. mehlhom, ep. preparata, and m. sarrafzadeh. channel routing in knock-knee mode: simplified algorithms and proof, in algorithmica, 1(2), pages 213-221,1986. [5] ronald l. rivest , charles m. fiduccia, a “greedy” channel router, proceedings of the 19th conference on design automation, p.418-424, january 1982. [6] m. sarrafzadeh. channel-routing problem in the knock-knee mode is np-complete. in ieee trans. on cad, cad-6(4), pages 503-506, 1987. [7] t. yoshimura and e.s. kuh. efficient algorithms for channel routing. in ieee trans. on cad of lntegrated circuits and systems, v. cad.l, pages 25-35, 1982. on new algorithm for channel routing vardan a. manukyan abstract we present new channel routing algorithms that consider the characteristic of net crossings. the routing strategy is based on parallel bubble sorting technique. nonmanhattan wires as well as overlapping wires are introduced. preliminary results show that a class of channel routing problems can be routed in height less than the manhattan density. microsoft word article.doc ìàòåìàòè÷åñêèå âîïðîñû êèáåðíåòèêè è âû÷èñëèòåëüíîé òåõíèêè 29, 2007, 107–116. 107 ðàçðàáîòêà ñòðàòåãèé â íàðäàõ* ñ èñïîëüçîâàíèåì èíäèâèäóàëèçèðîâàííûõ ýêñïåðòíûõ çíàíèé геворг карапетян институт проблем информатики и автоматизации нан ра e-mail: gevorgk@gmail.com аннотация ýôôåêòèâíîñòü èãðîâûõ àëãîðèòìîâ îïðåäåëÿåòñÿ ñïîñîáíîñòüþ âûáîðà îïòèìàëüíûõ ñòðàòåãèé. ýôôåêòèâíîñòü æå ëþáîé ñòðàòåãèè âî ìíîãîì çàâèñèò îò ìåòîäà ïðåäñòàâëåíèÿ çíàíèé, íà îñíîâå êîòîðûõ ñòðàòåãèÿ îïðåäåëÿåò íåîáõîäèìûå äåéñòâèÿ. â [7, 2], ïîêàçàíî, ÷òî èíäèâèäóàëüíîå ïðåäñòàâëåíèå çíàíèé ðàçëè÷íî, òàê êàê êîíêðåòíûå ïîíÿòèÿ, êîòîðûìè îïèñûâàþòñÿ ýêñïåðòíûå çíàíèÿ âûèãðûøíîñòè, êàê ïðàâèëî, ó ðàçíûõ èãðîêîâ ïðåäñòàâëÿþòñÿ ðàçíûìè ãèïîòåçàìè. â äàííîé ðàáîòå ðàññìîòðåíû âîïðîñû ïîñòðîåíèÿ ïëàíîâ è ñòðàòåãèé ñ èñïîëüçîâàíèåì èíäèâèäóàëèçèðîâàííûõ ýêñïåðòíûõ çíàíèé è ïðèâåäåíû ïðèìåðû èõ èñïîëüçîâàíèÿ â àëãîðèòìàõ èãðû â íàðäû*. ëèòåðàòóðà [1] pogossian e., vahradyan v., grigoryan a. on competing agents consistent with expert knowledge. lecture notes in computer science, ais-adm-07: the intern.workshop on autonomous intelligent systems agents and data mining, june6-7, 2007, st. petersburg. [2] pogossian e., karapetyan g., vahradyan v., experiments in simulation of conceptualchess knowledge, csit 2005, yerevan, 5p. [3] äæèäæÿí ð, êàðàïåòÿí ã., ðàçðàáîòêà àëãîðèòìîâ è ìåòîäîâ îáó÷åíèÿ èãðû â íàðäû áåç èãðàëüíûõ êîñòåé, íàó÷íàÿ êîíôåðåíöèÿ ãèóà, ñáîðíèê ìàòåðèàëîâ, òîì 1, åðåâàí 2004 [4] botvinnik m., about solving approximate problems, sov. radio, moscow, 1979 [5] furenkranz j., machine learning in games: a survey, nova scientific, 2001 [6] pogossian e., adaptation of combinatorial algorithms, academy of sciences of armenia, yerevan, 1983 [7] pogossian e.: specifying personalized expertise. international association for development of the information society (iadis): international conference cognition and exploratory learning in digital age (celda 2006), 8-10 dec., barcelona, spain (2006) 151-159 [8] ïîãîñÿí ý., âàãðàäÿí â., ãðèãîðÿí à., ýêñïåðèìåíòû ñîãëàñîâàíèÿ çíàíèé ýêñïåðòîâ ñ ïðèíÿòèåì ðåøåíèé â ýòþäàõ ðåòè è íîäàðåèøâèëè (ñì. íàñòîÿùèé ñáîðíèê). [9] pirat j., a chess combination program which uses plans, ai, v. 8, 1972 [10] g.f. luger. “artificial intelligence: structures and strategies for complex problem solving” , 4-th ed., addison-wesley, 2003 [11] zermelo e. uber eine anwendung der mengenlehre auf die theorie des ðàçðàáîòêà ñòðàòåãèé â íàðäàõ* ñ èñïîëüçîâàíèåì èíäèâèäóàëèçèðîâàííûõ ýêñïåðòíûõ çíàíèé 108 shachspiels. proceedings of the fifth international conference of mathematicians, cambridge, cambridge university press (1912) 501-504 [12] pogossian e.: focusing management strategy provision simulation. proceedings of the csit2001, 3d inter. conf. in comp. sci. and inf. technologies, yerevan (2001) 37-42 [13] wilkins d. using knowledge to control tree searching. ai, v.18, 1-51 [14] baghdasaryan t., danielyan e, pogossian e.: supply chain management strategy provision by game tree dynamic analysis international conference: management of small and medium business: information technologies (sbm2006), sevastopol, sept. 3-8, (2006) 37-41 [15] pogossian e., javadyan a., ivanyan e.: effective discovery of intrusion protection strategies. the intern. workshop on agents and data mining, st. petersburg, russia, lecture notes in computer science, vol. 3505 (2005) 263-274 [16] turing a.m.: computing machinery and intelligence. mind 49 (1950)[reprinted in minds and machines. a. anderson (ed.), engelwood cliffs nj, prentice hall (1964) 433-460 [17] pogossian e., hambartsumyan m., harutunyan y.: a repository of units of chess vocabulary ordered by complexity of their interpretations. national academy of sciences of armenia, ipia, (research reports 1974-1980) (in russian) 1-55 [18] djidjian r. getting ready for great discoveries.yerevan state university, 2004,pp 231 [19] flavell j. 1962. the develometal psycology of jean piaget, d.vannostrand company inc., princeton, new jersey [20] winograd t., flores f. 1986. understanding computers and cognition (a new foundation for design). publishers, chapter 2, pp. 11–59, huntington, ny [21] pylyshyn z. 2004. seeing and visualizing: it’s not what you think, an essay on vision and visual imagination, http://ruccs.rutgers.edu/faculty/pylyshyn.html [22] kosslyn s. 1980, image and mind. cambridge, ma harvard university press ü³ñ¹çç ñ³ù³ñ ÷áñó³·»ïç ³ýñ³ï³ï³ý³óí³í ·çï»éçùý»ñç û·ï³·áñíù³ùµ é³½ù³í³ñáõãûáõýý»ñç ùß³ïáõù ¶. î³ñ³å»ïû³ý ամփոփում ê³õ³ûçý ³é·áñçãùý»ñç ¿ý»ïïçíáõãûáõýá å³ûù³ý³íáñíáõù ¿ ýñ³ýó ûåïçù³é é³½ù³í³ñáõãû³ý áýïñù³ý ñý³ñ³íáñáõãû³ùµ: æëï ï³ù³û³ï³ý é³½ù³í³ñáõãû³ý ¿ý»ïïçíáõãûáõýá ñçùý³ï³ýáõù å³ûù³ý³íáñí³í ¿ ·çï»éçùý»ñç ý»ñï³û³óù³ý ù»ãá¹áí, áñç ñçù³ý íñ³ ïíû³é é³½ù³í³ñáõãûáõýá áýïñáõù ¿ ³ýññ³å»ßï ·áñíáõáõãûáõýý»ñá: ð»ï³½áïáõãû³ý ýå³ï³ïý ¿ ùß³ï»é ïáùµçý³ïáñ ë³õ»ñáõù é³½ù³í³ñáõãûáõýý»ñç ó¨³íáñù³ý ³é·áñçãùý»ñ` ñçùýí³í ïáýïñ»ï ÷áñó³·»ïý»ñç ·çï»éçùý»ñçý ñ³ù³å³ï³ëë³ý ³ýñ³ï³ï³ý³óí³í åé³ýý»ñç íñ³ ¨ µ»ñí³í »ý ¹ñ³ýó ïçñ³éù³ý ûñçý³ïý»ñ ý³ñ¹çç ³é·áñçãù»ñáõù: ´»ñí³í »ý ý³ñ¹çáõù ·çï»éçùý»ñç ý»ñï³û³óù³ý ¨ û·ï³·áñíù³ý ùá¹»éý»ñ, ý»ñï³û³óí³í »ý ïíû³é ùá¹»éý»ñç ¹»åùáõù ³é³ç³óáõ ³é·áñçãù»ñç ùß³ïù³ý åñáµé»ùý»ñá ¨ ¹ñ³ýó éáõíáõùý»ñá: ü»ñï³û³óí»é ¿ ý³ñ¹çç ³é·áñçãù»ñç ùß³ïù³ý ûñçý³ï ppit ùáï»óù³ý ùççáóáí ¨ óáõûó ¿ ïñí»é ³û¹ ùáï»óù³ý ³ñ¹ûáõý³í»ïáõãûáõýá ³é³ç³ó³í åñáµé»ùý»ñç éáõíù³ý å³ñ³·³ûáõù: microsoft word 37.doc mathematical problems of computer science 38, 87--88, 2012. 87 on models of meaning processing edward pogossian cognitive algorithms and models laboratory of the institute for informatics and automation problems of the academy of sciences of armenia, state engineering university of armenia epogossi@aua.am 1. what follows, is an explanation of my understanding of certain meanings tended to provide a theoretically consistent specification of what i mean by “meanings” generally. 2. we, a community, have meanings, and we present them via explanations. explanations of a meaning m are either the activated m itself, or the units of communications (counits) corresponding to m, or are models of durables. whatever causing in us prints are our realities (recall “thing in itself” by kant), while the totality of realities is our universe. we do store prints, reveal there regularities (regs) (say, rules by a.a. markov) and compose meanings as assembles of regs. durables are realities that cause prints with certain regs that are quasi stable in time. classes of prints matching to regs of meanings imply corresponding classes of durables. realities r’ are the models of realities r if the meanings mr and mr’ have equal parts, and r and r’ are equal if have equal meanings. 3. totalities of counits of some types comprise languages of those types while syntaxes of languages are parts of their meanings that present common constituents of counits of languages required for correct communications. since we acquire meanings and corresponding counits in certain languages of our communities, the structure of meanings obey to the syntax of those languages. thus, meanings, say in english, have to be structured by have, be, do, time, aspects, voice, mood and other syntax categories and the completeness of explanations depends on their presence in meanings and their presentations in explanations. 4. we do process meanings to promote our utilities. by meanings we model strategies promotion and estimate their consequences to choose strategies with the most perspective impacts to our utilities. the better meanings present our realities, i.e. the more adequate they are, the more effective the modeling can be. the scale we use to evaluate meanings is induced by the explanations we are able to put in correspondence to the meanings, which includes the following increasing degrees: be activated for a person, have language explanations, have model explanations, be a specification, be a theoretically consistent specification, have theoretically consistent models reproducible by certain communities. 5. in the presentation we are going to refine the above and consequent categories and provide the results of experiments in chess and other applications. 88 on models of meaning processing references 1. brutyan ch., zaslavski i., mkrtchyan l. on methods of automated synthesis of positional strategies in games , problemi kibernetiki, moscow, 1967. 2. e.pogossian. on modeling cognition. proceedings of csit , 2011,pp194-197 3. g. atkinson chess and machine intuition. ablex publ. corporation, new jersey, 1993. 4. mandler j. the foundations of mind: origins of conceptual thought. oxford univ., 2004. 5. t.winograd, f.flores. understanding computers and cognition. a new foundation for design publishers, huntington, ny, 1986. 6. j.flavell. the developmental psychology of jean piaget, d.vannostrand comp. inc., princeton, n.j., 1962 7. z. pylyshyn seeing and visualizing: it’s not what you think, an essay on vision and visual imagination, http://ruccs.rutgers.edu/faculty/pylyshyn.htm,l2004. 8. j.searle is the brain’s mind a computer program? scientific american 262, pp26-31, 1990. 9. 11–59, huntington, ny, 1986. 10. r. feynman. the meaning of it all. addison wesley, massachusetts, 1998 11. e. pogossian, v. vahradyan and a. grigoryan. on competing agents consistent with expert knowledge, lecture notes in computer science, ais-adm-07: the international workshop on autonomous intelligent systems agents and data mining, st. petersburg, russia, 229-241pp, june 6-7, 2007. 12. e.pogossian. on measures of performance of functions of human mind. 6th international conference in computer sci. and inf. technologies, csit2007, yerevan, 2007, 149-154 13. e.pogossian. specifying personalized expertise. international association for development of the information society (iadis): international conference cognition and exploratory learning in digital age (celda 2006), 8-10 dec., barcelona, spain (2006) 151-159 14. e pogossian. on a transparent presentation of written english syntax. 5th international cognitive linguistics conference, vrije universiteit, amsterdam, july ,209-14 15. e.pogossian. on measurable models of promotion of negentroping strategies by cognition, new trends in information technology, sofia, 161-169,2010 16. e.pogossian. adaptation of combinatorial algorithms. acad.emy of sci. of armenia, y., 293, 1983 (in russian) mathematical problems of computer science 54, 18–33, 2020. udc 519.2 a neyman-pearson proper way to universal testing of multiple hypotheses formed by groups of distributions evgueni a. haroutunian and aram o. yesayan institute for informatics and automation problems of nas ra e-mail: eghishe@sci.am, armfrance@yahoo.fr abstract the asymptotically optimal neyman-pearson procedures of detection for models characterized by m discrete probability distributions arranged into k, 2 ≤ k ≤ m groups considered as hypotheses are investigated. the sequence of tests based on a growing number of observations is logarithmically asymptotically optimal (lao) when a certain part of the given error probability exponents (reliabilities) provides positives values for all other reliabilities. lao tests sequences for some models of objects, including cases, when rejection of decision may be permitted, and when part, or all given error probabilities decrease subexponentially with an increase in the of number of experiments, are desined. for all reliabilities of such tests single-letter characterizations are obtained. a simple case with three distributions and two hypotheses is considered. keywords: statistical hypotheses testing, families of hypotheses, optimal detection, test with no match detection, neyman-pearson approach, neyman-pearson lemma, principle of maximum of kullback-leibler distance, error exponent. 1. introduction this paper is devoted to the generalization of neyman-pearson criterion for some specific universal hypotheses testing problem pointed out in the title. in [8] and in the following papers [9], [10], cox formulated a number of divers examples of problems for two families of hypotheses testing and developed a general modification of the neyman-pearson maximumlikelihood ratio procedures for solving such problems. in a series of papers and in disseration of f. harmosi-nejad and all [27], two stage procedures were investigated for certain models of problems of hypotheses testing. the first stage in these actions executes detecting between families of distributions, and the second stage performs detection of certain distribution in the selected family. investigation of the present paper can be considered as a more detailed analysis of this first-stage problems. the asymptotically optimal testing of two hypotheses was investigated by hoeffding in [32], also the concept of universal hypotheses testing was introduced there. 18 e. haroutunian and a. yesayan 19 the hypotheses testing problems for two hypotheses were also studied by borovkov [6], levy [35], van trees [40], csiszár and longo [12], tusnady [39], longo and sgarro [36]. neyman-pearson criterion of multiple hypotheses testing for discrete random variables was explored in [25]. in publications [1], [24] and [26], many hypotheses logarithmically asymptotically optimal (lao) testing for the models consisting of many independent objects was investigated. following birgé [3], we called the sequence of tests logarithmically asymptotically optimal (lao), when for given values of some reliabilities (error probability exponents) the test ensures the best values for the rest of them. haroutunian [18]-[20] investigated the problem of multiple hypotheses testing at the suggestion of r. l. dobrushin. construction of lao tests sequence is realized applying “kullback-leibler balls” around the hypothetic distributions in the space of distributions as sets for detection of corresponding hypotheses. this concept, introduced in [16]-[17] and applied in [20]-[31] and in the present paper, conforms to the idea of “r-divergent sequences” defined in [17] and used in other works. hypotheses testing with no-match decision was considered by gutman [16]. in papers [28]-[30] the results of researches of characteristics of lao hypotheses testing with possibility of rejection of decision for some models with one or multiple objects, with side information are presented. our study is based on information theoretic methods including the method of types. applications of methods of information theory in mathematical statistics, in particular in hypotheses testing, are exposed in the monographs by csiszár and körner [11], blahut [5], cover and thomas [7], csiszár and shields [13], poor [37], kullback [33] haroutunian and all [31] , in paper of blahut [4]. the structure of this paper is as follows. section 2 contains definitions, notations and problem argument. in central section 3 the construction of desired lao tests is exposed for model with groups of distributions and with possibility of rejection of decision. in section 4, the theorem of the section 3 is reformulated for the case without rejection option. section 5 is devoted to the models with some reliabilities equal to 0. in final section 6, the testing for simplest model with three distributions and two hypotheses is discussed. conclusion also contains some open problems. 2. problem presentation let p(x) be the space of all probability distributions (pds) on a finite alphabet x . let x be a random variable (rv) taking values in the set x with one of m possible pds gm ∈ p(x), m = 1,m. let x = (x1,x2, ...,xn ), xn ∈ x , n = 1,n, be a vector of results of n independent observations of the rv x. then the pd gnm(x) = n∏ n=1 gm(xn) and gnm ∈p(xn ) the m different pds are arranged into k, 2 ≤ k ≤ m different groups b1, b2, ..., bk, which we consider as k hypotheses (suppositions) hk concerning the distributions of the studied object. we consider also an empty “group” bk+1. these groups are mutually disjoint, contain |b1|, |b2|, ..., |bk+1| pds such that k∑ k=1 |bk| = m, |bk+1| = 0. 20 a neyman-pearson proper way to universal testing of multiple hypotheses formed by groups of distr. when |bk| > 1, the hypothesis hk is composite [6], [14], [35]. in applications, the groups may be formed with some different values of parameters of a certain pd. we study the hypothesis testing problem, which is that to decide, based on the observed sample x, where this vector has originated from a source with a pd from a series bk, k = 1,k, or to accept bk+1, that is to reject to make any judgement. the procedure is the universal test (do not specializing individual pds in groups), we denote it by φn [34]. this problem may also be considered as specific task of detection for multiple composite hypotheses, also having the possibility to refuse any decision. the test φn can be defined by partition of the space xn into k + 1 disjoint subsets an1 ,an2 , ...,ank+1, where ank , k = 1,k, contains all vectors x for which the test adopts the hypothesis hk, and ank+1 includes all vectors x for which the test refuses to take a certain answer. we denote by φ the infinite sequence of tests φn. let αl|k(φn ) for l 6= k, l = 1,k, k = 1,k be the probability of the erroneous acceptance of the hypothesis hl by the test φn provided that the hypothesis hk is true, we define (see [6], [35]): αnl|k = αl|k(φn ) 4 = max gm∈bk gnm(a n l ). (1) when we decline any decision, but the hypothesis hk is true, we consider the following probability of error: αnk+1|k = αk+1|k(φn ) 4 = max gm∈bk gnm(a n k+1), k = 1,k. (2) the probability of not accepting the true hypothesis hk, we define in the following way: αnk|k = αk|k(φn ) 4 = ∑ l 6=k, l=1,k+1 αnl|k = max gm∈bk gnm(ank ), k = 1,k. (3) note that our approach differs from the approaches in [6], where only αnk|k are studied, and in [38] where the αnk|k are not considered. we study the corresponding reliabilities (error probabilities exponents) el|k of the tests sequence φ: el|k = el|k(φ) 4 = lim n→∞ ( − 1 n log αnl|k ) , k = 1,k, l = 1,k + 1. (4) all reliabilities are arranged in (k + 1) ×k matrix. for instance, at k = 3 the matrix of reliabilities has the following form e(φ) =   e1|1 e2|1 e3|1 e4|1e1|2 e2|2 e3|2 e4|2 e1|3 e2|3 e3|3 e4|3   . definitions (3) and (4) imply that ek|k = min l 6=k, l=1,k+1 el|k, k = 1,k. (5) we call the tests sequence φ∗ logarithmically asymptotically optimal (lao) for this model if for given positive values of certain k elements of the reliabilities matrix e(φ∗) the e. haroutunian and a. yesayan 21 procedure φ∗ provides maximal values for all other elements of it [3]. this criterion can be considered as a proper specification of the neyman-pearson approach to the universal test of multiple hypotheses in the sense of optimality of reliabilities. in certain publications, the lao approach is referred to as the “exponential rate optimal” (ero) [32], [16], [39]. in opposition to the criterion adopted by gutman [16], we recognize the asymmetry in the importance of different hypotheses and consider unequal requirements to error probabilities, or reliabilities of their detection. we use the following notions and notations: shannon entropy of pd p on alphabet x : h(p) = − ∑ x p(x) log p(x), divergence, kullback-leibler information, relative entropy, or“distance” of two pds p1 and p2 on x : d(p1||p2) = ∑ x p1(x) log p1(x) p2(x) , a new notion introduced in [22], divergence of three pds p1,p2,p3 on x : d(p1||p2||p3) 4 = ∑ x p1(x) log p2(x) p3(x) = d(p1||p2) −d(p1||p3). as was noted in introduction, our study applies the method of types, developed in information theory [7, 11, 13, 31]. the basic notion in this method is the notion of the type qx of the vector x ∈xn , which is equivalent to the statistical notion of the empirical distribution of the sample x: qx = {qx(x) = n(x/x)/n,x ∈x}, where n(x/x) is the number of repetitions of the element x in the sample x. we denote by q(xn ) the set of all possible types on xn . it is clear that q(xn ) ⊂p(x). we will denote divergence by dn (q||p) when q ∈ q(xn ) and p ∈ p(x). note that dn (q||p) → d(q||p), when n →∞. let t nq (x) be the family of all vectors x of the type q. for q /∈ q(xn ) , we have t nq (x) = ∅. we will use the following estimates [11], [31]: | q(xn ) |≤ (n + 1)|x|, (6) (n + 1)−|x| exp{nh(q)}≤ |t nq (x)| ≤ exp{nh(q)}. (7) we will denote for brevity: for q ∈p(x), d(q||bk) 4 = min gm∈bk d(q||gm), (8) and for q ∈q(xn ), dn (q||bk) 4 = min gm∈bk dn (q||gm), (9) for rl ⊂p(x), d(rl||bk) 4 = min q∈rl d(q||bk), (10) and for rnl ⊂q(x n ), dn (rnl ||bk) 4 = min q∈rn l dn (q||bk). (11) in the following sections, we present ways of optimal tests construction for the considered models and investigate the corresponding error probabilities and reliabilities. 22 a neyman-pearson proper way to universal testing of multiple hypotheses formed by groups of distr. 3. testing for model with rejection option to construct the desired lao test corresponding to preliminary given strictly positive numbers e1|1,e2|2, ...,ek|k we define the following subsets of distributions: rnk 4 = {q ∈q(xn ) : dn (q||bk) ≤ ek|k}, k = 1,k, (12) rnk+1 4 = {q ∈q(xn ) : dn (q||bk) > ek|k, k = 1,k}, (13) rk 4 = {q ∈p(x) : d(q||bk) ≤ ek|k}, k = 1,k, (14) rk+1 4 = {q ∈p(x) : d(q||bk) > ek|k, k = 1,k}, (15) it is clear that rnk ⊂rk, k = 1,k + 1. (16) define also the following values of reliabilities: e∗k|k = e ∗ k|k(ek|k) 4 = ek|k, k = 1,k, (17) e∗l|k = e ∗ l|k(el|l) 4 = d(rl||bk), k = 1,k, k 6= l, l = 1,k, (18) e∗k+1|k = e ∗ k+1|k(e1|1,e2|2, ...,ek|k) 4 = d(rk+1||bk) = e∗k|k, k = 1,k. (19) theorem 1: if all distributions gm, m = 1,m, are different in the sense that d(gm′||gm) > 0, m′ 6= m, and the strictly positive numbers e1|1,e2|2, ...,ek|k are such that the following inequalities hold e∗1|1 < min l=2,k d(rl||b1) (20) e∗k|k < min( min l=1,k−1 e∗l|k(el|l), min l=k+1,k d(rl||bk)) (20′) e∗k|k < min l=1,k−1 e∗l|k(el|l) (20 ′′) then there exists an lao sequence of tests, all elements of the reliability matrix e∗ = {e∗l|k} of which are defined in (17)-(19) and are strictly positive. when at least one of the inequalities in (20) is violated, then at least one element of the matrix of reliabilities e∗ is equal to 0. more than that, if we try to detect with such el|l which for some l ∈ [1; k + 1] and k ∈ [1; k] is greater than d(rl||bk), then the test for all n = 1, 2, ... will make an error with the probability 1. proof: having a collection of numbers satisfying the conditions (20) we pass to the proof of the positive statement of the theorem, that is, to the construction of the test. consider a sequence of tests φ∗, which is defined by partition of sample space xn on the following k + 1 subsets: an∗k = ⋃ q∈rn k t nq (x), k = 1,k, (21) an∗k+1 = x n − k⋃ k=1 an∗k , n = 1, 2, .... e. haroutunian and a. yesayan 23 this an∗k ⊂ xn , because when q /∈ q(xn ), then t nq (x) is empty. let us prove that the collection of sets in (21) determines a test, namely, each x belongs to one and only to one subset an∗k , an∗k ⋂ an∗r = ∅, r 6= k, and k+1∑ k=1 an∗k = x n. really, for k = 2,k, r = 1,k − 1, for each k > r let us consider arbitrary x ∈an∗k . we see that in accordance with (21) and (12) there exists t nqx (x) ⊂a n∗ k , such that d n (qx||bk) ≤ ek|k. as k > r from conditions (20) it follows that er|r < e ∗ k|r(ek|k). from definition (18) and inequality dn (qx||bk) ≤ ek|k we obtain er|r < e∗k|r(ek|k) = min q∈rn k d(q||br) ≤ dn (qx||br). hence qx /∈rnr and from (21) it follows that x /∈an∗r . we can verify that an∗k+1 ⋂ an∗k = ∅, k = 1,k, because if x ∈ an∗k+1, then by (15) for type qx the inequality d n (qx||bk) > ek|k is true for k = 1,k. according to the definition (21) of an∗k , k = 1,k, we see that x /∈an∗k . the sample x from t nq (x) ⊂q(xn ) has the following probability: gnm(x) = n∏ n=1 gm(xn) = ∏ x gm(x) n(x/x) = ∏ x gm(x) nqx(x) = exp{n ∑ x (−qx(x) log qx(x) gm(x) + qx(x) log qx(x))} = exp{−n[dn (qx||gm) + h(qx)]}. (22) now for k = 1,k, using (3), (21), (6), (7), (12) and (22) we can upper estimate αk|k(φ ∗ n ) as follows: αk|k(φ ∗ n ) = max gm∈bk gnm(an∗k ) = max gm∈bk gnm( ⋃ q/∈rn k t nq (x)) ≤ (n + 1)|x| max gm∈bk max q:dn (q||bk)>ek|k gnm(t n q (x)) ≤ (n + 1)|x| max q:dn (q||bk)>ek|k exp{−ndn (q||bk)} ≤ exp{−n[ inf q:dn (q||bk)>ek|k dn (q||bk) −on (1)]} = exp{−n(ek|k −on (1))}, (23) where on (1) → 0 with n →∞. from here e∗k|k ≥ ek|k, k = 1,k. now let us prove the lower inequalities for l = 1,k, k = 1,k, l 6= k. from (1), (21), (7) and (22) we obtain αk|k(φ ∗ n ) = max gm∈bk gnm(an∗k ) = max gm∈bk gnm( ⋃ q/∈rn k t nq (x)) ≥ max gm∈bk max q/∈rn k gnm(t n q (x)) ≥ (n + 1)−|x| max gm∈bk max q:dn (q||bk)>ek|k exp{−ndn (q||gm)} = exp{−n( min gm∈bk inf q:dn (q||bk)>ek|k dn (q||gm) + o(1))} 24 a neyman-pearson proper way to universal testing of multiple hypotheses formed by groups of distr. = exp{−n( inf q:dn (q||bk)>ek|k dn (q||bk) + o(1))} = exp{−n(ek|k + o(1))}. (24) (23) and (24) give us (17). we can obtain similar upper estimates e∗l|k ≥ el|k for l = 1,k, k = 1,k, l 6= k. according to (1), (6), (7) and (10) we have αl|k(φ ∗ n ) = max gm∈bk gnm(a n∗ l ) = max gm∈bk gnm( ⋃ q∈rn l t nq (x)) ≤ (n + 1)|x| max gm∈bk max q∈rn l exp{−ndn (q||gm)} = exp{−n(dn (rl||bk) −on (1))}. (25) again for l = 1,k,k = 1,k,l 6= k, we lower estimate αl|k(φ ∗ n ) = max gm∈bk gnm(a n∗ l ) = max gm∈bk gnm( ⋃ q∈rn l t nq (x)) ≥ (n + 1)−|x| max gm∈bk max q∈rn l exp{−nd(q||gm)} = exp{−n(d(rl||bk) + on (1))} = exp{−n(el|k + on (1))}. (26) according to the definition (4), the reliability el|k(φ ∗) of the test sequence φ∗ is the limit inferiour lim n→∞ (− 1 n log αl|k(φ ∗ n )), taking into account (25), (26) and the continuity of the functional dn (q||gl), we obtain that lim n→∞ (− 1 n log αl|k(φ ∗ n )) exists and (18) is correct. similarly we can obtain upper and lower bounds for αk+1|k(φ ∗ n ), k = 1,k. applying the analogous resoning we get (19). the proof of the first part of the theorem will be accomplished if we demonstrate that the sequence of tests φ∗ is lao, that is, for every other sequence of tests φ∗∗ with the same reliabilities e1|1, ...,ek|k for all l = 1,k + 1, l 6= k, k = 1,k, inequalities el|k(φ∗∗) ≤ el|k(φ ∗) hold. suppose the contrary is the case, that is there exists sequence of tests φ∗∗ defined by the sets dn1 , ...,dnk+1 such that el|k(φ ∗∗) > el|k(φ ∗) for some l ∈ [1,k + 1], k ∈ [1,k], l 6= k. (27) for tests φ∗ and φ∗∗ the space xn is decomposed into subsets an∗l and, respectively, into dnl , l = 1,k + 1, such that for l = 1,k and n large enough. max gm∈bl gnm(a n∗ l ) = max gm∈bl gnm(d n l ) = 1 − exp{−nel|l} (28) and an∗l are constructed in (21) with sets of types t nq (x) including almost all vectors x having positive probability max gm∈bl gm. from here for the set an∗l −dnl ⊂dn∗l we have lim n→∞ max gm∈bl gnm(a n∗ l −d n l ) ≤ lim n→∞ exp(−nel|l) = 0. (29) by (4), (1) and (27) el|k(φ ∗∗) = lim n→∞ ( − 1 n log max gm∈bk gnm(d n l ) ) e. haroutunian and a. yesayan 25 ≥ lim n→∞ ( − 1 n log max gm∈bk gnm(a n l ) ) . that is for n large enough max gm∈bk gnm(d n l ) ≤ max gm∈bk gnm(a n∗ l ), and hence max gm∈bk gnm(a n∗ l −d n l ) > 0, which contradicts (29). so, we must conclude that (27) is not possible. for the proof of the second part of theorem 1, it is enough to note that if one of the conditions (20) is violated, then from (12)-(19) it follows that at least one of the elements el|k is equal to 0. in case when el|l > d(rl||bk) from (3), (21), (12), (22), (7) we have for all n αnl|l = max gm∈bl gnm(a n l ) = max gm∈bl gnm( ⋃ q∈rn k t nq (x)) ≥ max gm∈bl max q∈rn l gnm(t n q (x)) = exp{−n min gm∈bl min q∈rn l dn (q||gm)} = exp{−n × 0} = 1, because for gm ∈bnl we have min gm∈bl min q∈rn l dn (q||gm) = 0. theorem 1 is proved. 4. case without rejection of decision consider also the standard case when the decision is obligatory. again we have m possible pds gm ∈p(x), m = 1,m, which are placed in k groups b1, b2, ...,bk, which we envisage as hypotheses hk, k = 1,k. the unknown hypothesis must be detected on the base of sample x = (x1,x2, ...,xn ). the test φn can be designed by dividing the sample space xn into k subsets an1 ,an2 , ...,ank as acceptance regions for the hypotheses of the same number. the test is characterized by error probabilities. αnl|k = αl|k(φn ) 4 = max gm∈bk gnm(a n l ), l 6= k, l,k = 1,k. αnk|k = αk|k(φn ) 4 = ∑ l 6=k, l=1,k αnl|k, k = 1,k. the reliabilities are difined as in (4) el|k = el|k(φ) 4 = lim n→∞ ( − 1 n log αnl|k ) , k, l = 1,k. we shape the lao sequence of tests φ for preliminary given positive numbers e1|1,e2|2, ...,ek−1|k−1 by the following regions of pds rk 4 = {q ∈p(x),d(q||bk) ≤ ek|k}, k = 1,k − 1, rk 4 = {q ∈p(x),d(q||bk) > ek|k, k = 1,k − 1}. 26 a neyman-pearson proper way to universal testing of multiple hypotheses formed by groups of distr. let the corresponding reliabilities be as in (17)-(19) e∗k|k = e ∗ k|k(ek|k) 4 = ek|k, k = 1,k − 1, (30) e∗l|k = e ∗ l|k(el|l) 4 = d(rl||bk), k = 1,k, k 6= l, l = 1,k − 1, (31) e∗k|k = e ∗ k|k(e1|1,e2|2, ...,ek−1|k−1) 4 = d(rk||bk), k = 1,k. (32) theorem 2: if all pds gm, m = 1,m, are different, d(gm′||gm) > 0, m′ 6= m, and the strictly positive numbers e1|1,e2|2, ...,ek−1|k−1 are such that the following inequlities hold e∗1|1 < min l=2,k−1 d(rl||b1)) (33) e∗k|k < min( min l=1,k−1 e∗l|k(e ∗ l|l), min l=k+1,k−1 d(rl||bk)) k = 2,k − 2, (33′) e∗k−1|k−1 < min l=1,k−2 e∗l|k−1(e ∗ l|l) (33 ′′) then there exists an lao sequence of tests, all elements of the reliabilities matrix e of which are defined in (30)-(32) and are strictly positive. when one of the inequalities in (33) is violated then at least one element of the matrix of reliabilities e∗ is equal to 0. 5. some or all given reliabilities are equal to zero the well-known stein lemma [11] also called chernoff-stein lemma in [7] provides the estimate of the error probability for the case of two hypotheses. it asserts that when the error probability αn1|1 is postulated as a constant, then the error probability α n 2|2 goes to 0 as exp{−nd(p1||p2)} as the number of observations n tends to infinity. in this section, we present a generalization of stein lemma in two directions. first, a more general model, when section 2 consists of m pds grouped in k hypotheses and the test has to detect an unknown hypothesis or reject any decision. and secondly, it is known that some error probabilities αnl|l, or all k of them, tend to 0 when n goes to infinity as a function δnl|l, such that lim n→∞ ( − 1 n log δnl|l ) = 0. (34) in practice, δnl|l can be constants or polynomials by n. the following theorem is a generalization of a result from [23] as an addition to theorem 1. theorem 3: when all distributions gm, m = 1,m are different in the sense that d(gm′||gm) > 0, m′ 6= m and given numbers ek|k, k = 1,k partly or all of them are equal to 0 and verify condition (34), then there exists an lao test sequence φ∗, the elements of reliabilities matrix e(φ∗) = {e∗l|k} of which are defined by (14), (15), (17)-(19), if the conditions (20) hold. but in the case when the given el|l is equal to zero, the formula (18) changes as follows: e∗l|k = e ∗ l|k(0) = d(bl||bk), k = 1,k, k 6= l. for the proof, it is enough to replace (12) by the following expressions: rnl 4 = {q ∈q(xn ) : dn (q||bl) ≤− 1 n log δnl|l}. e. haroutunian and a. yesayan 27 6. case of three distributions and two hypotheses in this section, we discuss a number of questions concerning the most simple model amongst the considered in this paper. at first we represent a generalization of the fundamental result of neyman-pearson lemma for the noted case of 3 pds and two groups. there are given three distributions g1, g2, g3 for a random variable x. these distributions are divided into two groups (hypotheses) such that the first hypothesis h1 is the first distribution and the second hypothesis is the group of two other pds h1 = (g1), h2 = (g2, g3). (35) the statistician must accept or reject the first hypothesis on the base of the sample x. theorem 4: (neyman-pearson lemma) for a threshold t > 1, consider test ψ∗n defined by the region of acceptance an∗ for hypothesis h1: an∗ = {x : gn1 (x) max(g2 n (x); g3 n (x)) > t}, (36) and acceptance region an∗ for h2. the corresponding error probabilities are αn∗1|1 (t) = α n∗ 2|1 (t) = g n 1 (an∗) αn∗2|2 (t) = α n∗ 1|2 (t) = max(g n 2 (a n∗ 1 ); g n 3 (a n∗)). let an ⊂ xn be the decision region for h1 of the another test φn with error probabilities αn1|1 and α n 2|2. if α n 1|1 ≤ α n∗ 1|1 , then α n 2|2 ≥ α n∗ 2|2. proof: the numbers n and t are fixed, we can do not note them during proof. let ψan∗ and ψan be indicator functions of the regions. it is not difficult to verify that for all x ∈xn , (ψan∗(x) − ψan (x))(gn1 (x) − t max(g2 n (x); g3 n (x))) ≥ 0. then ∑ x∈xn (ψan∗(x)g n 1 (x) − tψan∗(x) max(g2 n (x); g3 n (x)) −ψan (x)gn1 (x) + tψan (x) max(g2 n (x); g3 n (x))) = ∑ x∈an∗ (gn1 (x) − t max(g n 2 (x); g n 3 (x))) − ∑ x∈an (gn1 (x) − t max(g n 2 (x); g n 3 (x))) = (1 −α∗1|1) − tα ∗ 2|2 − (1 −α1|1) + tα2|2 = (α1|1 −α∗1|1) + t(α2|2 −α ∗ 2|2) ≥ 0. so from α1|1 ≤ α∗1|1 it follows that α2|2 ≥ α ∗ 2|2. now we reformulate theorem 2 for the model given in (35). theorem 5: if pds g1, g2, g3 are different, the strictly positive number e1|1 is such that e1|1 ≤ min(d(g2||g1), d(g3||g1)), 28 a neyman-pearson proper way to universal testing of multiple hypotheses formed by groups of distr. for r1 4 = {q ∈p(x), d(q||g1) ≤ e1|1}, r2 4 = {q ∈p(x), d(q||g1) > e1|1}, we consider e∗1|1 = e1|1 = e ∗ 2|1 e∗2|2 = e ∗ 1|2 = min q∈r1 min(d(q||g2),d(q||g3)) then for the hypotheses in (35) there exists an lao sequence of tests with strictly positive reliabilities given above and with regions of decision for hypotheses hk an∗k = ⋃ q∈rk t nq (x), k = 1, 2. in the paper [20] of 1990 haroutunian noted that “the principle of maximum of likelihood is equivalent to the principle of maximum of kullback-leibler distance” and “the desired tests sequence is constructed by means of distances between the sample distribution and the hypothetical distributions”. it is worth to note that this assertion is something in common with the following note in cover and thomas monograph of 1991 (p. 307) [7] concerning the test of two hypotheses (the next one with our adopted exposition). ”in the above theorem (the neyman-pearson lemma), we have shown that the optimum test is a likelihood ratio test. we can rewrite the log-likelihood ratio as the difference between the relative entropy distances of the sample type to each of the two distributions. hence the likelihood ratio test (in our notation) gn1 (x) g2 n (x) > t > 1 is equivalent to d(qx||gn2 ) −d(qx||g n 1 ) > log t n or (with our new notation of divergence of three pds) d(qx||gn1 ||g n 2 ) > log t n . it remains to add that for the case of simple model in (36) the likelihood ratio test is equivalent to the following condition specifying the region of detection of the first hypothesis in (36) min[d(qx||gn1 ||g n 2 ),d(qx||g n 1 ||g n 3 )] > log t n . 7. conclusion here we offer some concluding remarks and open problems. in this paper, we have discussed error exponents trade-off of neyman-pearson suitable strategy of hypotheses testing for models with m known discrete probability distributions joined in k (2 ≤ k ≤ m) clusters, considered as hypotheses. we presented a single letter characterization of the error exponents of all possible pairs of hypotheses of tests for some cases. after a detailed proof of the point in question for a e. haroutunian and a. yesayan 29 general case with the possibility of decision rejection, analogical results are announced the case without a rejection option, the case when all or a part of the given reliabilities are equal to zero, and finally, for a particular case of three disributions and two hypotheses. the reasonings at the end of the previous section confirm the optimality of the tests considered in the paper based on the use of distances between the sample and hypotheses. for further works it is deserving exploration of characteristics of testing for generalization and enlargement of models studied in this paper. interesting is the case with multiple objects [cf. 21, 24, 26, 27]. significant are arbitrarily varying models with a sequence of states known to the decision maker [cf. 12] and also the case when states are not known to the statistician [cf. 2, 25]. important is the problem of hypothesis identification [cf. 1, 18, 23]. bayesian framework of the problem, and sources with other than independent issues, for instance with markov dependence must also be investigated [cf. 16]. acknowledgement the authors are thankful to the reviewers for their helpful comments. references [1] r. f. ahlswede and e. a. haroutunian, “on logarithmically asymptotically optimal testing of hypotheses and identification”, lecture notes in computer science, volume 4123, ”general theory of information transfer and combinatorics”, springer, pp. 462– 478, 2006. 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[40] h. van trees, detection, estimation and modulation theory, pt.1 new york, wiley, 1968. submitted 28.07.2020, accepted 26.11.2020. 3 2 a neyman-pearson proper way to universal testing of multiple hypotheses formed by groups of distr. ´³ßëáõùý»ñç ëùµ»ñáí ï³½ùí³í µ³½ù³ãçí í³ñï³íý»ñç ü»ûù³ýç-äçñëáýç ñ³ùáý¹ñ³ýáõñ ï»ëï³íáñù³ý ñ³ïáõï áõõç ºí·»ýç ². ð³ñáõãûáõýû³ý ¨ ²ñ³ù ú. ºë³û³ý ðð ¶²² æýýáñù³ïçï³ûç ¨ ³íïáù³ï³óù³ý åñáµé»ùý»ñç çýëïçïáõï e-mail: eghishe@sci.am, armfrance@yahoo.fr ²ù÷á÷áõù ð»ï³½áïí»é »ý ü»ûù³ýç-äçñëáýç ³ëçùåïáïáñ»ý ûåïçù³é ñ³ûïý³µ»ñù³ý áýã³ó³ï³ñ·»ñá ³ûý ùá¹»éý»ñç ñ³ù³ñ, áñáýù µýáõã³·ñíáõù »ý m ¹çëïñ»ï ñ³í³ý³ï³ýáõãûáõýý»ñç µ³ßëáõùý»ñáí, áñáýù ëùµ³íáñí³í »ý áëï k ¹³ë»ñç, 2 · k · m , áñáýù ¹çï³ñïíáõù »ý áñå»ë í³ñï³íý»ñ: ¸çï³ñïáõùý»ñç ù³ý³ïç ³×ç íñ³ ñïåöèàëüíûé ïóòü íåéìàíà-ïèðñîíà ê óíèâåðñàëüíîé ïðîâåðêå ìíîãèõ ãèïîòåç, ñôîðìèðîâàííûõ ãðóïïàìè ðàñïðåäåëåíèé åâãåíèé à. àðóòóíÿí è àðàì î. åñàÿí èíñòèòóò ïðîáëåì èíôîðìàòèêè è àâòîìàòèçàöèè íàí ðà e-mail: eghishe@sci.am, armfrance@yahoo.fr àííîòàöèÿ èññëåäóþòñÿ àñèìïòîòè÷åñêè îïòèìàëüíûå ïðîöåäóðû íåéìàíà-ïèðñîíà îáíàðóæåíèÿ äëÿ ìîäåëåé, õàðàêòåðèçóþùèõñÿ m äèñêðåòíûìè ðàñïðåäåëåíèÿìè âåðîÿòíîñòåé, ñãðóïïèðîâàííûìè â k ãðóïï, 2 · k · m, ðàññìàòðèâàåìûõ êàê ãèïîòåçû. ïîñëåäîâàòåëüíîñòü òåñòîâ, ïðè âîçðàñòàíèè ÷èñëà íàáëþäåíèé, ÿâëÿåòñÿ ëîãàðèôìè÷åñêè àñèìïòîòè÷åñêè îïòèìàëüíîé (lao), êîãäà îïðåäåëåííàÿ ´³ý³éç µ³é»ñ` íç׳ﳷñ³ï³ý í³ñï³íý»ñç ëïáõ·áõù, í³ñï³íý»ñç áýï³ýçùý»ñ, ûåïçù³é ñ³ûïý³µ»ñáõù, ü»ûù³ý-äçñëáýç ùáï»óáõù, ü»ûù³ý-äçñëáýç é»ùù³, îáõéµ³ïçèáûµé»ñç ñ»é³íáñáõãû³ý ù³ùëçùáõùç ëï½µáõýù, ëë³éç óáõóçã: ñçùýí³í ï»ëï»ñç ñ³çáñ¹³ï³ýáõãûáõýá éá·³ñçãùáñ»ý ³ëçùåïáïçïáñ»ý ûåïçù³é (è²ú) ¿, »ñµ ïñí³í óáõóçãý»ñç (ñáõë³éçáõãûáõýý»ñç) áñáß³ïç ù³ëá ³å³ñáíáõù ¿ùý³ó³í µáéáñ ñáõë³éçáõãûáõýý»ñç ñ³ù³ñ ¹ñ³ï³ý ³ñå»ùý»ñ áõý»ý³éá: î³éáõóí»é »ý ûµû»ïïý»ñç áñáß ùá¹»éý»ñç ñ³ù³ñ è²ú ï»ëï»ñç ³ûý ¹»åù»ñá, »ñµ ãáõûé³ïñíáõù ¿ ññ³å³ñí»é áñáßáõù áý¹áõý»éáõó ¨ »ñµ ëë³éý»ñç ñ³í³ý³ï³ýáõãûáõýý»ñç ùç ù³ëá ï³ù µáéáñá ÷áñó»ñç ãíç ³×ç ñ»ï ù»ïï»õ ýí³½áõù »ý »ýã³óáõóã³ûçý ûñ»ýùáí: êï³óí»é »ý ³û¹åçëç ï»ëï»ñç µáéáñ ñáõë³éçáõãûáõýý»ñç ùç³ï³é µýáõã³·ñáõùý»ñá: ¸çï³ñïí³í ¿ ùç ñ³ë³ñ³ï ¹»åù, »ñµ µ³ßëáõùý»ñç ãçíá »ñ»ù ¿, çëï í³ñï³íý»ñç ãçíá` »ñïáõ: e. haroutunian and a. yesayan 3 3 ÷àñòü çàäàííûõ ýêñïîíåíò (íàäåæíîñòåé) îáåñïå÷èâàåò ïîëîæèòåëüíûå çíà÷åíèÿ äëÿ âñåõ äðóãèõ íàäåæíîñòåé. ñêîíñòðóèðîâàíû lao ïîñëåäîâàòåëüíîñòè òåñòîâ äëÿ íåêîòîðûõ ìîäåëåé, â òîì ÷èñëå â ñëó÷àÿõ, êîãäà ðàçðåøåí îòêàç îò ïðèíÿòèÿ ðåøåíèÿ è êîãäà ÷àñòü èëè âñå çàäàííûå âåðîÿòíîñòè îøèáîê óáûâàþò ñóáýêñïîíåíöèàëüíî ñ ðîñòîì êîëè÷åñòâà ýêñïåðèìåíòîâ. ïîëó÷åíû îäíîáóêâåííûå õàðàêòåðèñòèêè äëÿ âñåõ íàäåæíîñòåé òàêèõ òåñòîâ. ðàññìîòðåí ïðîñòîé ñëó÷àé ñ òðåìÿ ðàñïðåäåëåíèÿìè è äâóìÿ ãèïîòåçàìè. êëþ÷åâûå ñëîâà: ïðîâåðêà ñòàòèñòè÷åñêèõ ãèïîòåç, ñåìåéñòâà ãèïîòåç, îïòèìàëüíîå îáíàðóæåíèå, ïîäõîä íåéìàíà-ïèðñîíà, ëåììà íåéìàíà-ïèðñîíà, ïðèíöèï ìàêñèìóìà ðàññòîÿíèÿ êóëüáàêà-ëåéáëåðà, ïîêàçàòåëü îøèáêè. 02_haroutunian_54_18_33 haroutunian_new microsoft word gayane.doc ìàòåìàòè÷åñêèå âîïðîñû êèáåðíåòèêè è âû÷èñëèòåëüíîé òåõíèêè 31, 100--107, 2008. 100 î êîíòóðàõ â íàïðàâëåííûõ ãðàôàõ ïðîõîäÿùèõ ÷åðåç äàííó þ âåðøèíó ñàìâåë õ. äàðáèíÿí è èñêàíäàð à. êàðàïåòÿí институт проблем информатики и автоматизации нан ра samdarbin@ipia.sci.am, isko@ipia.sci.am аннотация ïóñòü g åñòü )12( n -âåðøèííûé ( n ≥6) íàïðàâëåííûé ãðàô ñ ìèíèìàëüíûìè ïîëуñòåïеíÿìè, íå ìеíüøèìè 1n . äîêàçûâàåòñÿ, ÷òî ÷åðåç ëþáую âåðøèíу òàêîãî ãðàôà ïðîõîäèò êîíòóð äëèíû .12 n литература [1] ф. харари, теория графов, мир, москва, 1973. [2] j. bang-jensen and g. gutin, digraphs. theory. algorithms and applications. springer, 2001. [3] b. jackson, “long paths and cycles in oriented graphs”, j. graph theory, no. 5, pp. 145157 ,1981. [4] z. m. song, “pancyclic oriented graphs”, j. graph theory, no. 18, pp. 461 468 , 1994. [5] j. bang-jensen and y. guo, “a note on vertex pancyclic oriented graphs”, odense universitet, preprint 20 , 1997. [6] g. gutin, “characterizations of vertex pancyclic and pancyclic ordinary complete multipartite digraphs”, discrete math, v. 141, pp. 153-162, 1995. [7] с. х. дарбинян, “îöåíêà äëèí êîíòóðîâ è ïóòåé â ðåãóëÿðíûõ íàïðàâëåííûõ ãðàôàõ”, tanulmanyok , v. 135 , pp. 131-144, 1982. [8] с. х. дарбинян, к. м. мосесян, “о панцикличности регулярных орграфов“, дан арм. сср, 1978, т. lxvii, № 4, ñòð. 208-211, 1978. [9] с. х. дарбинян, “о панцикличности направленных графов с большими полустепенями”, дан арм. сср, т. lxxx, № 4, ñòð. 51-54, 1985 (ñì. òàêæå математические вопросы кибернетики и вычислительной техники, № 14, ñòð. 55-74, 1985). [10] ñ. õ. äàðáèíÿí, è. à. êàðàïåòÿí, “î âåðøèííîé ïàíöèêëè÷íîñòè íàïðàâëåííûõ ãðàôîâ ñ áîëüøèìè ïîëóñòåïåíÿìè”, математические вопросы кибернетики и вычислительной техники, № 29, ñòð. 66-84, 2007. [11] s. darbinyan and i. karapetyan, “on vertex pancyclic oriented graphs, csit conference, pp.154-155, yerevan, armenia, 2005. ñ. äàðáèíÿí è è. êàðàïåòÿí 101 àõõõ áñ¹í³í ·ñ³ ýý»ñáõù ïñí³í ·³·³ãáí ³ýóýáõ óçïé»ñç ù³ëçý ê. ¸³ñµçýû³ý ¨ æ. î³ñ³å»ïû³ý ²ù÷á÷áõù ü»ñï³ ³ßë³ï³ýùáõù ³å³óáõóíáõù ¿, áñ »ã» 12 n -·³·³ã³ýç )6( n áõõõáõñ¹í³í g ·ñ³ýç ó³ýï³ó³í ·³·³ãç éáï³é ïçë³³ëïç׳ýý»ñá ÷áùñ ã»ý 1n ãíçó, ³å³ g ·ñ³ýç ûáõñ³ù³ýãûáõñ ·³·³ã ·ïýíáõù ¿ 12 n »ñï³ñáõãû³ý ïáõùýáñáßí³í óçïéç íñ³: mathematical problems of computer science 49, 58–65, 2018. usage of neural networks for asm research hayk e. nahapetyan institute for informatics and automation problems of nas ra e-mail: hayknahapetyan@yahoo.com abstract purpose of this paper is to describe the possible usage of artificial neural networks for abelian sandpile model research. for developing neural networks, neuroph studio has been chosen, and abelian sandpile model has been considered on 2-dimensional grid. keywords: neural networks, ca, asm, neuroph studio, limiting shape. 1. introduction the structure and function of neural networks (nn) are based on our current understanding of the biological nervous system. nns are built on a large number of simple and adaptable processing units (pu) which are interconnected in such a way that they can store experiential knowledge through learning from examples and, like biological systems, have the ability to take in hazy information from the outside world and process it without an explicit set of rules. this approach (parallel and distributed) is in contrast to the traditional computing approach which processes information sequentially according to a set of exact rules. also, their structure and function provide a typical example of the applications of systems perspective concept which puts much emphasis on, in addition to the individual elements and their operations, the relationships among the elements and how they influence each other within the system (wu, 1992). perhaps due to some of the difficulties that have been experienced with the traditional expert system applications, and because of the rapid development and introduction of nn system development tools, nns have created a substantial amount of interest in the manufacturing arena, with systems and techiques being developed for organization, operational, as well as machine-level applications. the concept of self-organized criticality was first introduced by bak, tang and wiesenfeld in 1987 [1], and gave rise to growing interest in the study of self-organizing systems. bak et al. argued that in many natural phenomena, the dissipative dynamics of the system is such that it drives the system to a critical state, thereby leading to ubiquitous power law behaviors. this mechanism has been invoked to understand the power law distributions observed in turbulent fluids, earthquakes, distribution of visible matter in the universe, solar flares and surface roughening of growing interfaces. the sandpile models, being a class of cellular automata, are among the simplest theoretical models, which exhibit self-organized criticality. a special subclass of interest consists of so called abelian sandpile models (asm). the abelian property means that the final stable state of the ca is independent of the order 58 h. nahapetyan 59 in which the updates of cells are carried out. this property plays a key role during the numerical, as well as analytical studies of the asm [2]–[5]. in this paper we describe the usage of neural networks in the research of abelian sandpile model on 2-dimensional grid. the problem statement is considered in the ”problem of interest” section. 2. basic structure of nns and neuroph studio as mentioned above, artificial neural networks consist of processing units (pu), which are the building bricks of nns. pus usually take the form shown in fig.1, and emulate (assumingly) the function of a neuron in the brain. basically, pus are logic processing devices endowed with a fundamental function over the sum of their weighted inputs and a certain threshold value. mathematically, this is expressed as: yi = fi ( n ∑ j=1 wijxj − si ) = fi(ai), (1) where yi, wij, xj and si stand for the pu output signal, the weight of the j to i interconnection, input value from puj and the threshold value of pui, respectively. the input to pui can be either the output from other pus or directly from outside the nn, i.e., input to the nn. the output from pui can be used either as an input to the subsequent pus or as an output from the nn. the value of wij determines how strongly the output of puj influences the activity of pui. the magnitude of a weight can be changed over time. during a training operation, it is mainly through this mechanism that the pu is made adaptive to new information put to it, and the learning process is accomplished. as will become clear later, the total weight matrix w of an nn encompasses and reflects the nn ′s knowledge and skills that it has learnt through previous training, and is therefore referred to as its long-term memory. fig 1. the processing unit. the threshold si acts as a filter for incoming signals. the term inside the brackets in equation 1, ai, is known as the activation of pui, which provides temporary and local information around it. this is therefore referred to as the pu ′s short-term memory. the value of ai is transformed by the pu ′s output function, fi, to determine the magnitude of its current output signal. a number of activation functions have been used to 60 usage of neural networks for asm research construct nns, of which the step function is the simplest and the most straightforward (fig. 2a). with the step activation function, a pu produces an output signal of either ′1′ or ′0′ depending on whether or not the level of its activation is above a certain threshold value. that is: yi = { 1 if ai > 0 0 otherwise where ai = n ∑ j=1 wijxj − si however, in order to filter out the noise and hence enhance the ability of achieving a true steady state of operation, for some nns a sigmoid activation is usually used in practice, expressed in the form of: fig. 2. activation functions: (a) step activation function; (b) sigmoid activation function. h. nahapetyan 61 yi = fi(ai) = 1 1 + e−cai , where, ai = n ∑ j=1 wijxj − si. here c is a constant which determines the degree of ’uncertainty’ introduced into pui activity. the general shape of this function is as shown in fig. 2b. some of the advantages offered by this type of function will become clear later in the text (1/c is also known as the ’temperature’). in some cases its value can be set at an artificially high level initially to ’shake’ the nn so that it has a better chance of achieving its true stable state. this is then gradually reduced to allow it to cool down to the ideal state, i.e. step function with zero degree of temperature. this is known as simulated annealing). neuroph studio is lightweight java neural network framework to develop common neural network architectures. it contains a well-designed, open-source java library with a small number of basic classes, which correspond to basic nn concepts. it also has a nice gui as neural network editor to quickly create java neural network components. for creating and testing neural networks over cellular automata, neuroph studio has been chosen. 3. sandpile model consider an undirected graph g = (v, e) described with the set of vertices v = {v1, v2, . . . , vn} and the set of edges e. each vertex vi ∈ v is assigned a variable hi, which takes integer values and represents the height of the sand at that vertex. hmaxi denotes the maximal allowed height for the vertex vi in the graph g. for a d-dimensional lattice, we take hmaxi = 2d + 1. ct denotes the set of heights hi, which determines the configuration of the system at a given discrete time t . a configuration is called stable, if all heights satisfy hi < h max i . the vertex vi is called closed, if h max i = deg(vi), where deg(vi) indicates the degree of vi. the dynamics of the system is defined by the following rules. consider a stable configuration ct at a given time t . we add a grain of sand to a random vertex vi ∈ v by setting hi to hi + 1 (we assume that the vertex is chosen randomly with a uniform distribution on the set v ). this new configuration, if stable, defines ct+1. if hi ≥ h max i , then the vi becomes unstable and topples losing h max i grains of sand, while all neighbors of vi receive one grain. note that if the vertex is open, then the system loses grains. during the toppling of the closed vertices, the number of grains is conserved. note also that toppling of a vertex may cause some of its neighboring vertices to become unstable. in this case, those vertices also topple according to the same toppling rule. once all unstable vertices are toppled, a new stable configuration ct+1 is obtained. if the finite connected graph g has at least one open vertex, then all vertices become stable after a finite number of topplings. moreover, the new stable configuration is independent of the toppling order. let âi be an operator, which acts on sandpile configurations and adds a grain to vertex i. it can be easily shown that âiâj = âjâi. this is the reason why the sandpile model is called abelian. 62 usage of neural networks for asm research 4. problem of interest the research problem concerns the abelian sandpile model over a 2 dimensional square lattice of size (n ∗ n), where n stands for the number of nodes on each lattice line. consider n− > ∞. ri,j will be called the distance between vi and vj, or in other words, the minimum count of edges, which is needed to pass between vi and vj. ci,j will be the min count of grains, that by toppling that much grains on vi, and letting the model to become stable, we can make sure that at least one grain has reached the node vj, in oder words hj ≥ 1. let’s consider a 2-dimensional lattice, where hi = 0, ∀i, 0 ≤ i ≤ n 2. let’s choose any vi node on the lattice. the problem is to find a formula describing connection between ri,j and ci,j. there are articles [6], [7] regarding this problem. it should be noted that an exact formula describing c0,j dependency on j does not exist, also all the results obtained up to date are interpreted via approximation formulas only. in order to obtain correct results for c0,j , wherej > 0, a software program has been created which simulates the sandpile model and produces the data for neural network learning. 4.1 results in this subsection we give a comparative analysis of the results obtained by the applications of newly created software package and the ones that gave neural networks. as an example of a neural network, a so-called ”multilayer perceptron” with one input neuron and one output neuron, has been chosen. bias neurons have been used in nn structures, where the sigmoid type has been chosen for transfer function, meanwhile, the learning rule is developed based on back-propagation methodology. in listed examples, the difference between neural networks structure are hidden neurons count only. there are 3 cases regarding the neural networks architecture depending on neurons’ count. the first case concerns the presence of a lot of neurons, when the nn will memorize all input values while training and will produce the results of tests without thinking. in the second case, a very little number of neurons are under consideration, and the nn will produce rather wrong results and, therefore, will be not that smart. the third case, named ”the golden mean”, has a big range regarding the neurons’ count, and it is not that hard to find out the structure corresponding to this case. in this example, the results of tests, given below in figures 4 and 5, illustrate that the total mean square error is less for nn with 10 neurons comparing with the test results with 14 ones. the bigger the training set is, the more neurons are needed. fig. 3. neural network state after training h. nahapetyan 63 fig. 4. test results with 10 neurons fig. 5. test results with 14 neurons fig. 6. attributes 64 usage of neural networks for asm research 5. conclusion in this paper, neural networks’ usage for solving asm problems has been discussed. the goal of this work was to find out neural network structure corresponding to the problem such as the one described in “problem of interest” section. comparative analyzes between actual results and the ones that gave nn has been discussed, and in case of acceptable oversight asm simulation could be changed via neural networks described in this research in order of minimizing time consumptions. also the comparative analysis of different neural networks’ architectures has been conducted. perspectives on work are to use cluster systems for getting results of c0,j for bigger j and compare with results of already known solutions/formulas and with the ones from asm simulation. 6. acknowledgement the author is grateful to dr. s. poghosyan and dr. y. alaverdyan for important discussions and critical remarks at all stages of the work. this work was supported by the state committee of science mes ra, in the frames of the research project no. 16yr-1b008. references [1] p. bak, c. tang and k. wiesenfeld,“self-organized criticality: an explanation of the 1/f noise”,phys. rev. lett., vol.59, no. 4, pp. 3811–7384, 1987. [2] v. s. poghosyan, s. y. grigorev, v. b. priezzhev and p. ruelle, “pair correlations in the sandpile model: a check of logarithmic conformal field theory”, phys. lett. b, vol. 659, pp. 76817772, 2008. [3] su. s. poghosyan, v. s. poghosyan, v. b. priezzhev and p. ruelle, “numerical study of correspondence between the dissipative and fixed-energy abelian sandpile models”, phys.rev. e, 84, 066119, 2011. [4] v. s. poghosyan, s. s. poghosyan and h. e. nahapetyan, “the investigation of models of self-organized systems by parallel programming methods based on the example of an abelian sandpile model”, proc. csit conference 2013, yerevan armenia, sept. 23-27, pp. 260-262, 2013. [5] h. nahapetyan, j.-pierre jessel, s.poghosyan and y. shoukourian,“a multi user and multi purpose ca simulator17”,phys. rev. lett., vol.59, no. 4, pp. 38117384, 1987. proc. csit conference 2017, yerevan armenia, sept. 23-27, pp. 260-262. [6] l. levine and y. peres, “asymptotics for rotor-router aggregation and the divisible sandpile”, potential anal (2009) 30: 1. https://doi.org/10.1007/s11118-008-9104-6 [7] a. fey and f. redig, “limiting shapes for deterministic centrally seeded growth models”, j. stat. phys., vol. 130, no. 3, 57917597, 2008. submitted 06.10.2017, accepted 22.12.2017. h. nahapetyan 6 5 ü»ûñáý³ûçý ó³ýó»ñç û·ï³·áñíáõùá ³í³½³ïáõûïç ³µ»éû³ý ùá¹»éç ñ»ï³½áïáõãû³ý ñ³ù³ñ ð. ü³ñ³å»ïû³ý ²ù÷á÷áõù ²ûë ñá¹í³íç ýå³ï³ïý ¿ ýï³ñ³·ñ»é ³ññ»ëï³ï³ý ý»ûñáý³ûçý ó³ýó»ñç ñý³ñ³íáñ û·ï³·áñíáõùá ³í³½³ïáõûïç ³µ»éû³ý ùá¹»éç ñ»ï³½áïáõãû³ý ñ³ù³ñ: ü»ûñáý³ûçý ó³ýó»ñç ùß³ïù³ý ñ³ù³ñ áýïñí»é ¿ neuroph studio-ý, çëï ³í³½³ïáõûïç ³µ»éû³ý ùá¹»éá ¹çï³ñïí»é ¿ »ñïã³÷ ù³é³ïáõë³ûçý ó³ýóç íñ³: èñïîëüçîâàíèå íåéðîííûõ ñåòåé äëÿ èññëåäîâàíèÿ ìîäåëüè ïåñ÷àíîé êó÷è ã. íàãàïåòÿí àííîòàöèÿ öåëü ýòîé ñòàòüè îïèñàòü âîçìîæíîå èñïîëüçîâàíèå èñêóññòâåííûõ íåéðîííûõ ñåòåé äëÿ èññëåäîâàíèé ìîäåëüè ïåñ÷àíîé êó÷è. äëÿ ðàçðàáîòêè íåéðîííûõ ñåòåé áûëà âûáðàíà neuroph studio, à ìîäåëü ïåñ÷àíîé êó÷è áûëà ðàññìîòðåíà íà äâóìåðíîé ñåòêå. 07__ abstract_hayk d:\sbornik\...\untitled12.dvi ìàòåìàòè÷åñêèå âîïðîñû êèáåðíåòèêè è âû÷èñëèòåëüíîé òåõíèêè 23, 2004, 144–149. òåñòèðîâàíèå äëèòåëüíîñòè ïåðèîäîâ â ðÿäàõ áèðæåâûõ öåí ¤ åâãåíèé à. àðóòþíÿí, èðèíà à. ñàôàðÿí è ìóøåã ñ. ñààêÿí èíñòèòóò ïðîáëåì èíôîðìàòèêè è àâòîìàòèçàöèè íàí ðà e-mails evhar@ipia.sci.am, safari@ipia.sci.am, m.sahakianalumni@lse.ac.uk àííîòàöèÿ ðåøàåòñÿ çàäà÷à òåñòèðîâàíèÿ ïîñòîÿíñòâà ôóíêöèè ðèñêà ðÿäîâ äèíàìèêè öåí àêöèé ïðîòèâ àëüòåðíàòèâû ìîíîòîííîãî ðîñòà ðèñêà. ïðåäëîæåíà íîâàÿ ðàíãîâàÿ ñòàòèñòèêà, ÿâëÿþùàÿñÿ ìîäèôèêàöèåé ñòàòèñòèêè ëîã-ðàíãîâûõ ìåòîê. îáîñíîâàíî åå ïðèìåíåíèå äëÿ íåñòàáèëüíûõ ðûíêîâ. refer ences [1 ] ge n o n -ca t a lo t v ., je a n t h e a u t. a n d l a r e d o c. " p a r a m e t e r e s t im a t io n fo r d is c r e t e ly o b s e r ve d s t o c h a s t ic vo la t ilit y m o d e ls " . b ernoulli, vo l. 4 , n o . 5 , p p . 8 5 5 -8 7 2 , 1 9 9 9 . [2 ] k o ko s z ka l ., l e ip u s r ." ch a n g e -p o in t e s t im a t io n in a r ch m o d e ls " . vo l. 6 , n o . 3 , p p . 5 1 3 -5 3 9 , 2 0 0 0 . [3 ] ch e n j., gu p t a a . k . " te s t in g a n d lo c a t in g va r ia n c e c h a n g e p o in t s wit h a p p lic a t io n t o s t o c k p r ic e s " . j . amer. statist. assoc. , vo l. 9 2 , p p . 7 3 9 -7 4 7 , 1 9 9 9 . [4 ] co m p t e f., r e n a u lt e . " l o n g m e m o r y in c o n t in u o u s -t im e s t o c h a s t ic vo la t ilit y m o d e l " .m athematical f inance, vo l. 8 , n o . 4 , p p . 2 9 1 -3 2 3 , 1 9 9 8 . [5 ] êîêñ ä. ð., îóêñ ä. ”àíàëèç äàííûõ òèïà âðåìåíè æèçíè”. ì., ôèíàíñû è ñòàòèñòèêà, 1988. [6] heckman j., singer b. ”econometric duration analysis”. j ournal of econometrics 24, pp. 63-132 , 1984. 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[14] ýìáðåõòñ ï., êëþïïåëüáåðã ê. ”íåêîòîðûå àñïåêòû ñòðàõîâîé ìàòåìàòèêè”. òåîð. âåðîÿòíîñòåé è åå ïðèì., òîì 38, âûï. 3, ññ. 374-416, 1993. ´áñë³û³ï³ý ·ý»ñç ß³ñù»ñáõù å³ñµ»ñáõãûáõýý»ñç »ñï³ñáõãû³ý ï»ëï³íáñáõù º. ². ð³ñáõãûáõýû³ý, æ. ². ê³ý³ñû³ý ¨ ø. ê. ê³ñ³ïû³ý ²ù÷á÷áõù èáõííáõù ¿ ³ñå»ãõã»ñç ·ý»ñç ¹çý³ùçï³ûç ß³ñù»ñç éçëïç ýáõýïóç³ûç ñ³ëï³ïáõý éçý»éáõ í³ñï³íç ëïáõ·ù³ý ëý¹çñá áý¹¹»ù éçëïç ùçáýã³ó ³×ç: ²é³ç³ñïí³í ¿ é³ý·³ûçý ýáñ íç׳ï³ýç, áñá éá·-é³ý·³ûçý ýßçãý»ñç íç׳ï³ýáõ ùá¹çýçï³óç³ý ¿: ðçùý³íáñí³í ¿ ¹ñ³ ïçñ³éáõãûáõýá áã ï³ûáõý ßáõï³ý»ñç ¹»åùáõù: d:\sbornik\...\rev.dvi mathematical problems of computer science 31, 142{149, 2008. n ew m athematical appr oach for i nvestigation of statistical p r oper ties of random e nvir onment of 1d quantum n-p ar ticles system i n e xter nal field ashot s. gevorkyan y and arax a. gevorkyan z y institue for informatics and automation problems of nas of ra z yerevan state university e-mail: g ashot@sci.am, g arax@ipia.sci.am abstract the investigation of 1d quantum n-particles system (ps) with relaxation in the random environment under the in°uence of external ¯eld is conducted within the limits of the stochastic di®erential equation (sde) of langevin-schräodinger (l-sch) type. using l-sch equation the 2d second order non-stationary partial di®erential equation is found, which describes the quantum distribution in the environment, depending on energy of nonperturbed 1d quantum n-ps and on the external ¯eld's parameters. it is shown that the average value of interaction potential between 1d disordered quantum n -ps and on the external ¯eld, has the ultraviolet divergence. this problem is solved by renormalization of equation for the function of quantum distribution. it is shown that it has a sense of dimensional renormalization which is characteristic for the quantum ¯eld theory. critical properties of environment are investigated in detail. the possibility of ¯rst-order phase transition in environment depending on amplitude of an external ¯eld is shown. refer ences [1 ] a . s . ge vo r kya n a n d ch in -k u n h u , on a m a t h e m a t ic a l a p p r o a c h fo r t h e in ve s t ig a t io n o f s o m e s t a t is t ic a l p h e n o m e n a o f a d is e r d o r e d 3 d s p in s ys t e m in t h e e xt e r n a l ¯ e ld . p r o c e e d in g s o f t h e is a a c co n f. o n a n a lys is , y e r e va n , a r m e n ia , e d s . b y g. a . b a r s e g ia n e t a l., 1 6 5 -1 7 8 , 2 0 0 4 . [2 ] a . v . b o g d a n o v, a . s . ge vo r kya n , a .g. gr ig o r ya n , a ms / ip s t u d ie s in a d va n c e d ma t h e m a t ic s , 13, 8 1 , 1 9 9 9 . [3 ] i. m. l ifs h it s , s . a . gr e d e s ku l a n d l . a . p a s t u r , in t r o d u c t io n t o t h e t h e o r y o f d is o r d e r e d s ys t e m s . mo s c o w, n a u ka , ( in r u s s ia n ) 1 9 8 2 . [4 ] a . s . ge vo r kya n , e xa c t ly s o lva b le m o d e ls o f s t o c h a s t ic qu a n t u m m e c h a n ic s wit h in t h e fr a m e wo r k o f l a n g e vin -s c h r e o d in g e r t yp e e qu a t io n , a n a lys is a n d a p p lic a t io n s . p r o c e e d in g o f t h e n a to a d va n c e d r e s e a r c h wo r ks h o p , y e r e va n 2 0 0 2 , e d s . b y g. a . b a r s e g ia n a n d h . b e g e h r , n a to s c ie n c e p u b lic a t io n s , 4 1 5 -4 4 2 , k lu we r , 2 0 0 4 . [5 ] v . i. k lya t s kin , s t a t is t ic a l d e s c r ip t io n o f d yn a m ic a l s ys t e m s wit h ° u c t u a t in g p a r a m e t e r s . mo s c o w, n a u ka , ( in r u s s ia n ) 1 9 7 5 . 1 4 2 a. gevorkyan and ar. gevorkyan 1 4 3 [6 ] a . n . v a s il'e v, th e qu a n t u m -̄ e ld r e n o r m g r o u p in th e o r y o f cr it ic a l b e h a vio u r a n d o f s t o c h a s t ic d yn a m ic s . p u b lis h in g h o u s e p in f, s t . p e t e r s b u r g ( in r u s s ia n ) 1 9 9 8 . [7 ] m. v . fe d o r yu k, me t h o d o f s a d d le p o in t s , p u b lis h e r " n a u ka " ( in r u s s ia n ) 1 9 7 7 . 1d ùí³ýï³ûçý n-ù³ëýçïý»ñç ñ³ù³ï³ñ·ç å³ï³ñ³ï³ý ßñç³ï³ûùç íç׳ﳷñ³ï³ý ñ³ïïáõãûáõýý»ñá ³ñï³ùçý ¹³ßïáõù áõëáõùý³ëçñ»éáõ ýáñ ù³ã»ù³ïçï³ï³ý å³ïï»ñ³óáõù ². ¶¨áñ·û³ý ¨ ²ñ. ¶¨áñ·û³ý ²ù÷á÷áõù 1d ùí³ýï³ûçý n-ù³ëýçïý»ñç ñ³ù³ï³ñ·ç (øð) å³ï³ñ³ï³ý ßñç³ï³ûùç é»é³ïë³óç³ý ³ñï³ùçý ¹³ßïáõù ýï³ñ³·ñí³í ¿ è³ý娻ý-þñ»¹çý·»ñç (è-þñ) ïçåç å³ï³ñ³ï³ý ¹çý»ñ»ýóç³é ñ³í³ë³ñù³ý ßñç³ý³ïý»ñáõù: ú·ï³·áñí»éáí è-þñ ñ³í³ë³ñáõùá‘ ëï³óí³í ¿ »ñïñáñ¹ ï³ñ·ç 2d áã ëï³óçáý³ñ ù³ëý³ïç ³í³ýóç³éý»ñáí ¹çý»ñ»ýóç³é ñ³í³ë³ñáõù 1d ùí³ýï³ûçý n-øð ßñç³ï³ûùç µ³ßëù³ý‘ ï³ëí³í ñ³ù³ï³ñ·ç ãëáïáñí³í ¿ý»ñ·ç³ûçó ¨ ³ñï³ùçý ¹³ßïç å³ñ³ù»ïñ»ñçó: òáõûó ¿ ïñí³í, áñ n-øð ¨ ³ñï³ùçý ¹³ßïç ùçç¨ ÷áë³½¹»óáõãû³ý åáï»ýóç³éç ùçççý ù»íáõãûáõýá áõýç áõéïñ³ù³ýáõ߳ﳷáõûý ï³ññ³ùçïáõù: ²ûë åñáµé»ùá éáõíí»é ¿ ùí³ýï³ûçý µ³ßëù³ý ñ³í³ë³ñù³ý é»ýáñù³éç½³óç³ûç ù»ãá¹áí: òáõûó ¿ ïñí³í, áñ é»ýáñù³éç½³óç³ý áõýç ï³ñ³í³ã³÷³ûçý çù³ëï, áñá ñ³ïáõï ¿ ùí³ýï³ûçý ¹³ßïç ï»ëáõãûáõýý»ñçý: ø³ýñ³ù³ëýáñ»ý áõëáõùý³ëçñí³í ¿ ßñç³ï³ûùç íç׳ﳷñáõãû³ý ïñçïçï³ï³ý ñ³ïïáõãûáõýý”ñá ¨ óáõûó ¿ ïñí³í, áñ ³ûý ï³ëí³í ³ñï³ùçý ¹³ßïç ³ùåéçïáõ¹çó áõýç ³é³ççý ï³ñ·ç ÷áõé³ûçý ³ýóáõù: d:\user\sbornik_38_pdf\29.dvi mathematical problems of computer science 38, 72, 2012. e xtensions of m ar kov's constr uctive continuum and unifor m continuity of constr uctive functions b o r is a . k u s h n e r department of mathematics university of pittsburgh at johnstown johnstown w e c o n s id e r e ve r ywh e r e d e ¯ n e d c o n s t r u c t ive fu n c t io n s ( c .f.) o n t h e c lo s e d u n it c o n s t r u c t ive in t e r va l. a s is we ll-kn o wn b y t h e fa m o u s za s la vs ky-ts e it in th e o r e m s u c h a c .f. c a n b e e ®e c t ive ly n o n u n ifo r m ly c o n t in u o u s . in t h is c a s e it c a n n o t b e e xt e n d e d t o a c la s s ic a l c o n t in u o u s fu n c t io n . in r e a lit y, in e ve r y kn o wn c o u n t e r -e xa m p le a s in g u la r it y c o u ld b e d is c o ve r e d a lr e a d y o n t h e le ve l o f p s e u d o n u m b e r s . l e t u s r e c a ll t h a t a p s e u d o n u m b e r is a r e c u r s ive s e qu e n c e o f r a t io n a ls t h a t is a ca u c h y s e qu e n c e c la s s ic a lly. p s e u d o n u m b e r s c a n b e c o n s id e r e d a s á0 ( ¢ 2 ) -c o m p u t a b le n u m b e r s a s we ll. l e t d b e t h e s e t o f a ll c o n s t r u c t ive r e a l n u m b e r s ( ma r ko v's co n t in u u m in t h e t it le ) , d1 t h e s e t o f a ll p s e u d o n u m b e r s . a c .f. f is s a id t o b e 1 -c o m p le t e if it c a n b e e xt e n d e d t o a c o m p u t a b le ( a n d s o c o n t in u o u s ) fu n c t io n o ve r d1. t heor em 1 there is a 1-complete c.f. that is e®ectively nonuniformly continuous. th is r e s u lt is r a t h e r p r e c is e a s a c .f. c o n t in u o u s ly e xt e n d ib le t o á00-c o m p u t a b le n u m b e r s is u n ifo r m ly c o n t in u o u s c la s s ic a lly. a s is we ll kn o wn e ve r y c .f. c a n b e c o m p u t e d o n d b y a k le e n e o p e r a t o r ( p a r t ia l-r e c u r s ive o p e r a t o r ) . th e fo llo win g r e s u lt t o g e t h e r wit h th e o r e m 1 s h o ws t h a t t h e r e is a n e s s e n t ia l d i®e r e n c e b e t we e n ma r ko v's a n d k le e n e 's co m p u t a b ilit y o ve r d1. t heor em 2 a c.f. f is constructively uniformly continuous i® there is a 1-complete k leene operator that computes f. r e fe r e n c e s [1 ] b .a . k u s h n e r , s o m e e xt e n s io n s o f ma r ko v's co n s t r u c t ive co n t in u u m a n d t h e ir a p p lic a t io n s t o t h e th e o r y o f co n s t r u c t ive fu n c t io n s , th e l .e .j.b r o u we r ce n t e n a r y s ym p o s iu m , n o r t h -h o lla n d p u b l. co ., a m s t e r d a m , 1 9 8 2 , p p . 2 6 1 { 2 7 3 [2 ] b .a . k u s h n e r , l e c t u r e s o n co n s t r u c t ive ma t h e m a t ic a l a n a lys is . ( tr a n s la t io n fr o m t h e r u s s ia n ) , a ms , p r o vid e n c e , r h o d e is la n d , 1 9 8 4 7 2 mathematical problems of computer science 49, 97–102, 2018. existence of maximum entropy problem solution in a general n-dimensional case ruben a. gevorgyan and narek d. margaryan yerevan state university ruben−gevorgyan@yahoo.com, narek−margaryan@outlook.com abstract in the following paper, we will define conditions, which need to be satisfied in order for the maximum entropy problem applied in european call options to have a solution in a general n-dimensional case. we will also find a minimum right boundary for the price range in order to have at least one risk neutral measure satisfying the option pricing formula. the results significantly reduce the computational time of optimization algorithms used in maximum entropy problem. keywords: entropy, boundary, distribution, options. 1. introduction the maximum entropy methodology has recently started to become a quite popular tool with a huge potential of application in different fields [1, 2, 3, 4]. the core of the theory is based on shannon’s classical definition of information entropy [5], which is a crucial foundation in information theory. the maximum entropy approach has been broadly studied for its application in finance and financial extrapolation [6], and there have been significant contributions to its development since then, including the application of legendre transforms [7], partially finite convex programming [8], the employment of risk neutral moments [9], as well as the application of the problem as a non-parametric approach in american options pricing [10]. by theory, in the discrete case, the price of a european call option should be equal to the mathematical expectation of future pay-offs’ discounted value, thus lying in their convex hull. in reality, actual market prices may be biased from the theoretical ones [11] and lie out of the convex hull. we will concentrate on the derivation of conditions for the existence of solution which will not only reduce the computational time but will also result in an automated distribution recovery process [12, 13, 14] and will later allow us to develop algorithmic trading strategies that train on huge data sets. 2. outline of the problem consider having european call options for n different strike prices. let us denote the vector of strike prices with k and the vector of future states with x. x and k needn’t be of the 97 98 existence of maximum entropy problem solution in a general n-dimensional case same dimension. maximum entropy methodology seeks a risk neutral probability measure p, such that ap = b, (1) n∑ i=1 pi = 1, pi ≥ 0, (2) s(p) = n∑ i=1 piln(pi) is maximal, (3) where b is the vector of current option prices’ future values for each strike and a is (x1 − k1) + (x2 − k1)+ . . . (xn − k1)+ ... ... ... ... (x1 − kn)+ (x2 − kn)+ . . . (xn − kn)+   , (4) where (x)+ = max(x, 0). the probability vector p and the vector of future states x have the same dimension, in fact pi is the probability mass assigned to the future state xi. the distribution of future states will change as we change the state vector x. the question that interests us is what kind of state vector should be considered in order for a probability measure satisfying (1), (2) to exist in the first place. it is obvious that the greater the number of a’s linearly independent columns is, the bigger will their convex hull be, and so the more p vectors may exist satisfying (1), (2). so first of all we will consider the state vector (k1, . . . , kn, kn + t) for some arbitrary t. matrix a will now have the form below.  0 k2 − k1 . . . kn − k1 kn − k1 + t 0 0 . . . kn − k2 kn − k2 + t ... ... ... ... ... 0 0 . . . kn − kn−1 kn − kn−1 + t 0 0 . . . 0 t   . (5) we will denote a’s columns by a0, a1, . . . , an. let α(t) denote the angle between an and i, where i is the unit vector (1, . . . , 1). it is easy to show that limt→∞ cos α(t) = 1, so in order to see if any t exists, s.t. (1), (2) are satisfied, we will consider i instead of an, assuming that the angle between b and i isn’t 0 (this assumption holds throughout the text). let’s consider the following n + 1 hyperplane vector pairs (we denote hyperplanes by hp(·)).  hp(a1, a2, . . . , an−1, i), a0 hp(a0, a2, . . . , an−1, i), a1 ... hp(a0, a1, . . . , an−2, i), an−1 hp(a0, a1, . . . , an−2, an−1), i . (6) for each hyperplane above, we will denote by ni its normal “pointing” in the direction of the associated vector ai (note that a0 = (0, . . . , 0))  ⟨n0 − a1, a0 − a1⟩ ≥ 0 ⟨n1, a1⟩ ≥ 0 ... ⟨nn, an⟩ ≥ 0 , (7) r. gevorgyan and n. margaryan 99 where ⟨·, ·⟩ denotes the scalar (dot) product. 3. existence of solution the following proposition is obvious. proposition 1. there exists a finite t, s.t. (1), (2) are satisfied if and only if the following inequalities take place.   ⟨n0 − a1, b − a1⟩ ≥ 0 ⟨n1, b⟩ ≥ 0 ... ⟨nn, b⟩ ≥ 0 . (8) we now proceed to finding a minimal value for t, s.t. conditions (1) and (2) are satisfied. b represents the vector of prices and, thus its components are non-negative. assume that (8) takes place. if the last component of b, bn is 0, then the minimal value of t for which b ∈ conv(a0, a1, . . . , an−1, an) is 0 (conv(·) denotes the convex hull). in case bn is greater than 0, we will use the following lemmas (note that an = an−1 + ti). lemma 1. ∃µ > 0, s.t. ∀t for which b ∈ conv(a0, . . . , an−1, an), t ≥ µ > 0. proof. assume the opposite, then ∀ϵ > 0 ∃t0 < ϵ, s.t. ∃γ0, . . . , γn, γi ≥ 0, ∑n i=0 γi = 1, for which γ0a0 + . . . + γnan = b. let r = inf q∈conv(a0,...,an−1) ρ(b, q), ϵ = r ρ(b, i) , where ρ is the euclidean distance. for the ϵ above there exists 0 < t0 < ϵ, s.t. γ0a0+. . .+ γnan = b ⇔ γ0a0 + . . . + (γn−1 + γn)an−1 + t0γni = b ⇒ ρ(γ0a0 + . . . + (γn−1 + γn)an−1, b) = ρ(b − t0γni, b) ≤ t0ρ(i, b) < r, resulting in a contradiction. lemma 2. if for some t0 b ∈ conv(a0, . . . , an), then this also holds for any t > t0. proof. let t > t0, γ ′ n−1 = γn−1 + γn t−t0 t , γ ′ n = γn t0 t , then γ ′ n−1 + γ ′ n = γn−1 + γn and γ0a0 + . . . + γ ′ n−1an−1 + γ ′ nan = γ0a0 + . . . + γnan = b we now know that the set t of all possible t’s for which b ∈ conv(a0, . . . , an) is bounded from below by a positive number and unbounded from above. the next lemma proves that for t = inf t b is again in the convex hull conv(a0, . . . , an). lemma 3. let t be the set of all t’s, s.t. b ∈ conv(a0, . . . , an), then t = inf t ∈ t. proof. let’s assume the opposite. as t is the infimum of t, then for ∀ϵ > 0 ∃t0 ∈ t, s.t. 0 < t0 − t < ϵ. let r = inf q∈conv(a0,...,an−1,an−1+ti) ρ(b, q), 100 existence of maximum entropy problem solution in a general n-dimensional case ϵ = r ρ(b, i) . by the assumption there exists a t0 < t+ϵ, s.t. b = γ0a0 +. . .+γnan. let a ′ n = an−1 +ti, then ρ(γ0a0 + . . . + γn−1an−1 + γna ′ n, b) = ρ(b + (t − t0)i, b) ≤ (t0 − t)ρ(i, b) < r, resulting in a contradiction. based on the lemmas we may now formulate the main theorem of the article. theorem 1. if condition (8) is satisfied, the angle between b and i isn’t 0, then b ∈ conv(a0, . . . , an), where an = an−1 + ti and γn−1 = 0 in the linear representation of b by vectors a0, . . . an. the minimal value of t, t is given by t = bn(kn − kn−1) bn−1 − bn . (9) proof. we only need to show that γn−1 = 0. assume it’s not, then b = γ0a0 + . . . + γnan = γ0a0 + . . . + γn−1an−1 + γn(an−1 + ti) = γ0a0 + . . . + (γn + γn−1)an−1 + tγn γn + γn−1 (γn + γn−1)i = γ0a0 + . . . + (γn + γn−1)(an−1 + tγn γn + γn−1 i). as γn > 0, then tγn γn + γn−1 < t, which is a contradiction. having known that an−1 doesn’t ”participate” in the linear representation of b, we only need to find the value of t, s.t. b ∈ hp(a0, . . . , an−2, an). for that we will find the normal n of the hyperplane and solve ⟨n, b⟩ = 0 for t. we find n by observing the determinant of the following matrix based on the vectors from the hyperplane.  k2 − k1 0 . . . . . . 0 k3 − k1 k3 − k2 0 . . . 0 ... kn−1 − k1 kn−1 − k2 . . . 0 0 kn − k1 + t kn − k2 + t . . . kn − kn−1 + t t e1 e2 . . . en−1 en   . (10) the determinant is (−1)2n−1t(k2 − k1) . . . (kn−1 − kn−2)en−1+ (−1)2n(kn − kn−1 + t)(k2 − k1) . . . (kn−1 − kn−2)en. so n = (0, . . . , 0, −t, kn − kn−1 + t), and therefore ⟨n, b⟩ = 0 ⇔ t = bn(kn − kn−1) bn−1 − bn . r. gevorgyan and n. margaryan 101 4. conclusion as a result we obtained a way of checking whether a solution to the maximum entropy problem applied in european call options exists, before starting the optimization. if (8) takes place, then in order for the solution to exist, the right bound of future states vector must be greater than or equal to the value of t described in the theorem above. checking the existence of solution prevents the user from unknowingly proceeding to the stage of entropy maximization over an empty set of discrete probability distributions, which would yield unpredictable results. references [1] y. alhassid, n. agmon and r. d. levine, “an upper bound for the entropy and its applications to the maximal entropy problem”, chem. phys. lett., vol. 53, pp. 22, 1978. 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[8] j. borwein, r. choksi and p. marechal, “probability distributions of assets inferred from option prices via the principle of maximum entropy”, siam j. optim., vol. 14, no 2, pp. 464-478, 2003. [9] l. s. rompolis, “a new method of employing the principle of maximum entropy to retrieve the risk neutral density”, http://web.xrh.unipi.gr/attachments/seminars/2008 [10] yu. xishen and li yang, “pricing american options using a nonparametric entropy approach”, hindawi publishing corporation, pp. 16, article id 369795, 2014. [11] m.rubinstein, “implied binomial trees.” finance working paper, vol. 49, no 3, pp. 771-818, 1994. [12] n. d. margaryan “assessment of asset price distributions using maximum entropy method”, proc. of engineering academy of armenia, vol. 14, no 1, pp. 57-61, 2017. [13] n. d. margaryan “an algorithmic approach to solving the maximum entropy problem”, proc. of engineering academy of armenia, vol. 14, no 3, pp. 371-374, 2017. [14] n. d. margaryan “a boundary for the existence of solution to the maximum entropy problem applied in european call options”, proc. of the yerevan state university, vol. 52, no 1, pp. 3-7, 2018. submitted 18.10.2017, accepted 12.02.2018. 1 0 2 existence of maximum entropy problem solution in a general n-dimensional case àý¹ñ³ýáõñ n-ã³÷³ýç ¹»åùáõù ³é³í»é³·áõûý ¿ýïñáåç³ûç ëý¹ñç éáõíù³ý ·áûáõãûáõýá è. ¶¨áñ·û³ý ¨ ü. ø³ñ·³ñû³ý ²ù÷á÷áõù ð»ï¨û³é ³ßë³ï³ýùáõù ïë³ñù³ý»ýù å³ûù³ýý»ñ, áñáýó µ³í³ñ³ñí³íáõãûáõýý ³ýññ³å»ßï ¿ áý¹ñ³ýáõñ n-ã³÷³ýç ¹»åùáõù ºíñáå³ï³ý ûåóçáýý»ñáõù ïçñ³éíáõ ³é³í»é³·áõûý ¿ýïñáåç³ûç ëý¹ñç éáõíù³ý ·áûáõãû³ý ñ³ù³ñ: ü³¨ ï·ïý»ýù ·ý³ûçý ùçç³ï³ûùç ýí³½³·áõûý ³ç³ïáõùû³ý ë³ñù³ýá` ûåóçáýý»ñç ·ý³·áû³óù³ý µ³ý³ó¨çý µ³í³ñ³ñáõ ³éýí³½ý ù»ï éçëïçó 㻽áù ã³÷ç ·áûáõãû³ý ñ³ù³ñ: êï³óí³í ³ñ¹ûáõýùý»ñá ½·³éçáñ»ý ýí³½»óýáõù »ý ³é³í»é³·áõûý ¿ýïñáåç³ûç ëý¹ñç ù»ç û·ï³·áñííáõ ûåïçùç½³óçáý ³é·áñçãùý»ñç ñ³ßí³ñï³ûçý å³ù³ý³ïá: ñóùåñòâîâàíèå ðåøåíèÿ ïðîáëåìû ìàêñèìàëüíîé ýíòðîïèè â îáùåì n-ìåðíîì ñëó÷àå ð. ãåâîðãÿí è í. ìàðãàðÿí àííîòàöèÿ â ñëåäóþùåé ñòàòüå ìû îïðåäåëèì óñëîâèÿ, âûïîëíåíèå êîòîðûõ íåîáõîäèìî äëÿ ñóùåñòâîâàíèÿ ðåøåíèÿ ïðîáëåìû ìàêñèìàëüíîé ýíòðîïèè, ïðèìåíÿåìîé â åâðîïåéñêèõ îïöèîíàõ, â îáùåì n-ìåðíîì ñëó÷àå. ìû òàêæå íàéäåì ìèíèìàëüíóþ ïðàâóþ ãðàíèöó äëÿ öåíîâîãî äèàïàçîíà, êîòîðàÿ íåîáõîäèìà äëÿ ñóùåñòâîâàíèÿ õîòÿ áû îäíîé ðèñê-íåéòðàëüíîé ìåðû óäîâëåòâîðÿþùåé ôîðìóëå öåíîîáðàçîâàíèÿ îïöèîíîâ. ïîëó÷åííûå ðåçóëüòàòû çíà÷èòåëüíî óìåíüøàþò âû÷èñëèòåëüíîå âðåìÿ îïòèìèçàöèîííûõ àëãîðèòìîâ, èñïîëüçóåìûõ â çàäà÷å ìàêñèìàëüíîé ýíòðîïèè. article abstract_n microsoft word aslanyan1.doc mathematical problems of computer science 30, 76--86, 2008. 76 agent interaction protocols in an intelligent agent server system levon h. aslanyan, david a. karapetyan institute for informatics and automation problems of nas of ra e-mail: lasl@sci.am, david@dm-lab.sci.am url: http://dm-lab.sci.am abstract we study interaction protocols of software agents in an intelligent agent server system, which employs software agents in regard to different applied problems. four models of agent interaction protocols are proposed. the protocols are evaluated for utility, implementation and applicability. the logical level of system is designed and implemented algorithmically. the test application is the intrusion detection problem. references 1. l. aslanyan, k. margaryan, h. sahakyan, data analysis algorithms in network protection systems, iii international conference on “digital information processing and control in extreme situations”, minsk., may 28-30 2002, isbn: 985-6453-80-1, pp. 221-225. 2. d. milojicic, m. breugst, masif the omg mobile agent system interoperability facility. mobile agents second international workshop, ma '98 (stuttgart, germany, september 1998). 3. d. a. karapetyan, intelligent agent server (netint) system, mathematical problems of computer science vol. 25, pp. 64-70, 2006. 4. c. a. r. hoare, communicating sequential processes, prentice hall, 1985. 5. g. lowe, breaking and fixing the needham-schroeder public-key protocol using fdr, proceedings of tacas ’96, springer lncs 1055, 1996. l. aslanyan, d. karapetyan 77 բանական ագենտ սերվերների համակարգում միջագենտային համագործակցության արձանագրություններ լ. ասլանյան, դ. կարապետյան ամփոփում աշխատանքում ուսումնասիրվում են ծրագրային ագենտների վրա հիմնված բանական ագենտ սերվերների համակարգում միջագենտային համագործակցության ընթացակարգի մասին: ներկայացված են միջագենտային համագործակցության ընթացակարգերի 4 մոդել: ընթացակարգերը գնահատված են ըստ արդյունավետության, իրականացման և հնարավոր կիրառությունների: համակարգի տրամաբանական մակարդակը նախագծված և իրականացված է ալգորիթմորեն: տրված է ներխուժման հայտնաբերման խնդրի տեստային լուծման տարբերակ: d:\sbornik\...\chubaryan4.dvi mathematical problems of computer science 30, 36{39, 2008. compar ison of the complexities in fr ege p r oofs with di®er ent substitution rules a n a h it a . ch u b a r ya n , a r m in e a . ch u b a r ya n , s o n a r . a le ks a n ya n department of informatics and applied mathematics, yerevan state university department of applied mathematics, state engineering university of armenia e-mails: achubaryan@ysu.am, chubarm@ysu.am, sonush@rambler.ru abstract we compare the proof complexities in frege systems with multiple substitution rule and with constant bounded substitution rule. we prove that any two constant bounded substitution frege systems are polynomially equivalent both by size and by steps. frege system with multiple substitution rule and frege system with constant bounded substitution rule are also polynomially equivalent by size, but the ¯rst system has exponential speed-up over the second system by steps. refer ences [1 ] s . a . co o k, a . r . r e c kh o w, th e r e la t ive e ± c ie n c y o f p r o p o s it io n a l p r o o f s ys t e m s " , j ournal of symbolic l ogic, vo l. 4 4 , p p . 3 6 { 5 0 , 1 9 7 9 . [2 ] g. ce jt in , a . ch u b a r ya n , on s o m e b o u n d s t o t h e le n g t h s o f lo g ic a l p r o o fs in c la s s ic a l p r o p o s it io n a l c a lc u lu s " , ( in r u s s ia n ) , trudy vycisl.centra an arm ssr i yerevan univ., vo l. 8 , p p . 5 7 { 6 4 , 1 9 7 5 . [3 ] a . a . ch u b a r ya n , th e c o m p le xit y in fr e g e p r o o fs wit h s u b s t it u t io n " , m athem. p roblems of computer science, nan, armenia, vo l. 2 2 , p p . 7 { 1 1 , 2 0 0 1 . ²ñï³íáõùý»ñç µ³ñ¹áõãûáõýý»ñç ñ³ù»ù³ïáõùá ï³ñµ»ñ ï»õ³¹ñù³ý ï³ýáýý»ñáí üñ»·»ç ñ³ù³ï³ñ·áõù ², âáõµ³ñû³ý, ². âáõµ³ñû³ý, ê. ²é»ùë³ýû³ý ²ù÷á÷áõù ²ßë³ï³ýùáõù ñ³ù»ù³ïíáõù »ý áëï ³ñï³íáõùý»ñç »ñïáõ µ³ñ¹áõãû³ý µýáõã³·ñçãý»ñç (»ñï³ñáõãûáõý ¨ ù³ûé»ñç ù³ý³ï) üñ»·»ç ñ³ù³ï³ñ·ç µ³½ù³ïç ï»õ³¹ñù³ý ¨ ë³ñù³ý³÷³ï ï»õ³¹ñù³ý ï³ýáýýáí »ñïáõ áý¹é³ûýáõùý»ñ: ²å³óáõóí³í ¿, áñ áëï ³ñï³íù³ý »ñï³ñáõãû³ý µ³½ù³ïç ¨ ë³ñù³ý³÷³ï ï»õ³¹ñù³ý ï³ýáýý»ñáí üñ»·»ç ñ³ù³ï³ñ·»ñá µ³½ù³ý¹³ùáñ»ý ñ³ù³ñå»ù »ý, ë³ï³ûý áëï ù³ûé»ñç ù³ý³ïç µ³½ù³ïç ï»õ³¹ñù³ý ï³ýáýáí üñ»·»ç ñ³ù³ï³ñ·ý áõýç óáõóã³ûçý ³ñ³·³óáõù ë³ñù³ý³÷³ï ï»õ³¹ñù³ùµ üñ»·»ç ñ³ù³ï³ñ·»ñç ýï³ïù³ùµ: 3 6 microsoft word tigran_hakobhin.doc mathematical problems of computer science 31, 169--177, 2008. 169 anomalies dynamic analysis and correction software edward pogossian1,2 and arthur grigoryan1 1institute for informatics and automation problems of nan ra, 2state engineering university of armenia, e-mail: epogossi@aua.am abstract the research is aimed to develop an effective anomalies dynamic analysis and correction software (adacs) for grid armenia. a software is developed that in addition to predetermined and fixed forms of protection of variety of servers generates game trees of possible anomalies and elaborates recommendations to avoid them by analyzing possible strategies throughout the game trees and searching the best correction strategies. experiments on correction of anomalies in overfilling the memory of the cluster of ipia are processed. references 1. h. v. astsatryan, yu. h. shoukourian and v. g. sahakyan, “the armcluster1 project: creation of high performance computation cluster and databases in armenia proceedings of conference”, computer science and information technologies, pp. 376-379, 2001. 2. r. butler, d. engert, i. foster, c. kesselman, s. tuecke, j. volmer, and v. welch, “design and deployment of a national-scale authentication infrastructure”, ieeecomputer, 33(12):60-66. 2000. 3. g.a. bolcer and g. kaiser, “swap: leveraging the web to manage workflow”, ieee internet computing,:85-88. 1999. 4. i. foster, c. kesselman and s. tuecke, “the anatomy of the grid enabling scalable virtual organizations”, intl journal, supercomputer applications 2001. 5. i. foster, c. kesselman, g. tsudik, and s. tuecke, “a security architecture for computational grids”, in acm conference on computers and security, 83-91, 1998. 6. k. ilgun, r.a. kemmerer and p.a. porras, “state transition analysis: a rule-based intrusion detection system”, ieee trans. software eng. vol. 21, no. 3, mar. 1995. 7. f. martinelli, p. mori and a. vaccarelli, “improving grid services security with fine grain policies”, instituto de informatica e telematica, pisa, italy, 2003. 8. e. pogossian, a. javadyan and e. ivanyan, “effective discovery of intrusion protection strategies”, lecture notes in computer science, ais-adm-05: the international workshop on autonomous intelligent systems -agents and data mining st. petersburg, russia http://space.iias.spb.su/ais05/, lnai 3505, pp.263-276, june 6-8, 2005. 9. e. pogossian and a. javadyan, “a game model for effective counteraction against computer attacks in intrusion detection systems”, nato asi 2003, data fusion for situation monitoring, incident detection, alert and response management, tsahkadzor, armenia, pp.30, august 19-30, 2003. 10. e. pogossian, v. vahradyan and a. grigoryan, “on competing agents consistent with expert knowledge”, lecture notes in computer science, ais-adm-07: the international workshop on autonomous intelligent systems agents and data mining, st. petersburg, russia, pp. 229-241, 2007.  a-1451 grid armenia istc project for the institute for informatics and automation problems of nan ra 170 anomalies dynamic analysis and correction software 11. e. pogossian, “combinatorial game models for security systems. nato arw on security and embedded system”, porto rio, patras, greece, aug. 8-18, 2005. 12. m. botvinnik, computers in chess: solving inexact search problems. springer series in symbolic computation, with appendixes, springer-verlag: ny, 1984. þ»õáõùý»ñç ¹çý³ùçï í»ñéáõíáõãû³ý ¨ í»ñ³óù³ý ñ³ù³ï³ñ· ¾. äáõáëû³ý ¨ ². ¶ñç·áñû³ý ²ù÷á÷áõù grid ùçç³í³ûñáõù ß»õáõùý»ñçó å³ßïå³ýáõãû³ý åñáµé»ùç ñ³ù³ñ ¹çï³ñïíáõù »ý ñ³ñó»ñ, ï³åí³í áñáßáõùý»ñç áý¹áõýù³ý í³é»ñç ¹çý³ùçï í»ñéáõíáõãû³ý ñ»ï,: êï»õíí³í ¿ íñ³·çñ, ³û¹åçëç í³é»ñç ¹çý³ùçï ·»ý»ñ³óç³ûç ñ³ù³ñ, ¨ ýï³ñ³·ñí³í »ý ëï³óí³í ¿ùëå»ñ»ù»ýï³é ÷áñó³ñïù³ý ³ñ¹ûáõýùý»ñá` ñçßáõáõãû³ý ·»ññ³·»óù³ý åñáµé»ùç ûñçý³ïç ñ³ù³ñ: d:\user\sbornik_38_pdf\3.dvi ìàòåìàòè÷åñêèå âîïðîñû êèáåðíåòèêè è âû÷èñëèòåëüíîé òåõíèêè 38, 10{11, 2012. àñèììåòðèÿ è ÿäðî êîäèðîâàííîãî ìåòðè÷åñêîãî ïðîñòðàíñòâà þ. ã. ãðèãîðüÿí åâðîïåéñêàÿ îáðàçîâàòåëüíàÿ ðåãèîíàëüíàÿ àêàäåìèÿ e-mail: yurgrig@yahoo.com â ðàáîòå ðàññìàòðèâàþòñÿ âîïðîñû, ñâÿçàííûå ñ ïîíÿòèÿìè àñèììåòðèè è ÿäðà äèñêðåòíîãî ìåòðè÷åñêîãî ïðîñòðàíñòâà @( d ) [1,2,3], èìåþùèìè âàæíîå çíà÷åíèå äëÿ ïîñëåäóþùåãî ðàçâèòèÿ äàííîãî íàïðàâëåíèÿ. îïðåäåëâíèå 1. ìíîæåñòâî äåéñòâèòåëüíûõ ÷èñåë < áåç íóëÿ íàçûâàåòñÿ àñèììåòðè÷åñêèì ìíîæåñòâîì, åñëè ñóùåñòâóåò õîòÿ-áû îäèí ýëåìåíò a 2 < òàêîé, ÷òî ¡a 62 <. íàïðèìåð, <( d ) = fdg[ ( jdj; 1) ÿâëÿåòñÿ àñèììåòðè÷åñêèì ìíîæåñòâîì ñ ôèêñèðîâàííûì d < 0 . çàôèêñèðóåì öåëîå ÷èñëî d · ¡ 2 è ðàññìîòðèì áåñêîíå÷íîå ìíîæåñòâî öåëûõ ÷èñåë: @ ( d ) = fd; ¡d + 1 g [ fd2 ¡ d + i; i = 0 ; 1 ; 2 ; : : :g ( 1 ) â [1,2] ïîêàçàíî, ÷òî ìíîæåñòâî @( d ) äëÿ êàæäîãî ôèêñèðîâàííîãî d0 · ¡2 îáðàçóåò àñèìåòðè÷åñêîå ìåòðè÷åñêîå ïðîñòðàíñòâî ñ ìåòðèêîé r ( x; y ) = ( p x + y; x 6= y 0 ; x = y ( 2 ) ñòðóêòóðà ìíîæåñòâà @( d ) ïðè d = ¡ 2 ïðèâåäåíà íà ðèñ. 1. -. . . . . . . . . . . . . . -4 -3 -2 -1 0 1 2 3 4 5 6 7 ¡1 1 ðèñ. 1 ­ ­ ­ ­ ­ ­ ­ ­ ­­ ­ ­­ ­ ­­ ­ ­­ ­ ­­ ­ ­­ ­ ­­ ­ ­­ ­ ­­ ­ ­­ ­ ­­ ­ ­­ ­ ­­ ­ ­­ ­ ­­ â ïðîñòðàíñòâå @( d ) ïðè ôèêñèðîâàííîì d0 ââîäÿòñÿ ïîíÿòèÿ òðåõ äèñêðåòíûõ îáúåêòîâ "; "; p ñî ñâîèìè îïðåäåëÿþùèìè òî÷êàìè: "fd0; k; a; bg, "fd0; k; a; b; cg, pfd0; k; a; b; c; e; f g, óäîâëåòâîðÿþùèìè ñèñòåìàì óðàâíåíèé (3) k + d0 = 1 a + d0 = b + k ad0 + bk = 0 3 75 " kc + d0c + kd0 = 0 3 7775 " k2 + d20 + c 2 = e2 a2 + b2 + c2 = f 2 3 777777775 p ( 3 ) 1 0 þ. ãðèãîðüÿí 1 1 " ¾ " ¾ p ( 4 ) îïðåäåëåíèå 2. ìíîæåñòâî pfd0; k; a; b; c; e; f g ïðè ôèêñèðîâàííîì d0 · ¡2 íàçûâàåòñÿ ÿäðîì ïðîñòðàíñòâà @ (1), åñëè ñóùåñòâóåò ñîâåðøåííîå ÷èñëî h 2 @ ( d0 ) , óäîâëåòâîðÿþùåå óðàâíåíèþ: d20 + k 2 + a2 + b2 + c2 + e2 + f 2 = h2 ( 5 ) ïîêàçàíî, ÷òî äèîôàíòîâî óðàâíåíèå (5) èìååò ðåøåíèå â öåëûõ ÷èñëàõ, ÷òî ñîãëàñíî îïðåäåëåíèþ 2 îáåñïå÷èâàåò ñóùåñòâîâàíèå ÿäðà äèñêðåòíîãî àñèììåòðè÷åñêîãî ïðîñòðàíñòâà @( d ) . ñïèñîê ëèòåðàòóðû [ 1] grigoryan yu.g. “principles of inhomogeneous geometry”, j. algebra, geometry and their appl. seminar proc. – 2004 – 3-4, – p. 40-53. [2] ãðèãîðüÿí þ.ã. “ïðîñòðàíñòâî äèñêðåòíûõ ãåîìåòðèé”, êèáåðíåòèêà è ñèñòåìíûé àíàëèç, – 2006 – n5 – ñ. 22-32. [3] ãðèãîðüÿí þ.ã. “íåêëàññè÷åñêèå ñâîéñòâà ïðîñòðàíñòâà äèñêðåòíûõ ãåîìåòðèé”, êèáåðíåòèêà è ñèñòåìíûé àíàëèç, –2009 – n5 – ñ. 51-59. начиная с начала 2000 года осуществляется внедрение ghis в здравоохранении, в рамках принятого проекта о реформирование информ mathematical problems of computer science 48, 57-63, 2017. automation script for wrf model data assimilation system hayk a. grigoryan institute for informatics and automation problems of nas ra e-mail: hayk-grigoryan@ipia.sci.am abstract climatology and weather forecasting are important since they help to determine future climate expectation. they are important to most aspects of day to day life, including aviation, boating, other modes of transportation, farming, tourism, health care, sports, etc. many software models exist for modeling and forecasting the weather. the weather prediction is very complex and needs in a lot of observational data. in the weather forecasting domain armenia as a developing country gets much of the forecasting data from outside resources, and moreover, not all observation stations in armenia provide daily observation data and they are not integrated into the global forecast systems. that’s why the data assimilation techniques exist for making small corrections to a short-range forecast. in this paper an automation script is introduced which in easy and user-friendly way generates a little_r format ascii files from station observations synop code, which is necessary for data assimilation wrfda system in wrf model. keywords: data assimilation, automation, big data analysis, wrfda, synop, little_r 1. introduction in recent years the area of the weather forecast is growing very fast and in parallel with the computer technologies the different models are getting more accessible [1]. numerical weather prediction (nwp) based on current weather predicts the weather using different mathematical models of the atmosphere and oceans. nwp focuses on taking current observations of weather and processing these data with computer models to forecast the ongoing state of weather. as the numerical computer models processing the initial data it is important to know the current state of the weather. data assimilation is a process which produces outputs of precipitation, temperature and a lots of other meteorological elements from the oceans to the top of atmosphere as an input serving on the current weather observations. a number of global and regional forecast models are 57 https://en.wikipedia.org/wiki/mathematical_model https://en.wikipedia.org/wiki/mathematical_model automation script for wrf model data assimilation system 58 run in different countries worldwide, using current weather observations relayed from radiosondes, weather satellites and other observing systems as inputs [2]. the mesoscale wrf model is specialized to the territory of armenia and used for exploitative weather forecasting. one of the programs of the model is the wrfda data assimilation system, used for making small corrections to a short-range forecast. 2. tools and definitions 2.1. weather research forecast (wrf) the weather research and forecasting (wrf) model is an nwp system and a set of software and tools, designed to bestead both atmospheric research and operational forecasting applications [3]. wrf features two dynamical (computational) cores, a data assimilation system, and a software architecture allowing for parallel computation and system expansibility. the model used in a wide range of meteorological applications across scales ranging from tens meters to thousands of kilometers. it includes the following features:  meteorological studies  real-time nwp  idealized  simulations  data assimilation  earth system model coupling  model training and educational support in national centers for environmental prediction (ncep) and other national meteorological centers wrf is currently in operational use. in many laboratories, universities and private companies wrf model used in real-time forecasting configurations. 2.2. data assimilation – wrfda system data assimilation is the technique by which observations are combined with an nwp product (the first guess or background forecast) and their respective error statistics to make small corrections to a short-range forecast (background), which is assumed to be good, to produce a model analysis. variational (var) data assimilation achieves this through the iterative minimization of a prescribed cost (or penalty) function. differences between the analysis and observations/first guess are penalized (damped) according to their perceived error. the wrfda system is designed to be a flexible, state-of-the-art atmospheric data assimilation system that is portable and efficient on available parallel computing platforms (fig. 1). h. grigoryan 59 fig 1. wrfda system in wrf model. wrfda is suitable for use in a broad range of applications, across scales ranging from kilometers for regional and mesoscale modeling to thousands of kilometers for global scale modeling [4]. the obsproc program of the system reads observations in little_r format. it ingests multiple types of observations that are converted to little_r format and concatenated to one file, process the observation data and output the ascii file(s) suitable for wrfda needs − 3dvar, fgat (first guess at appropriate time), 4dvar. the current script is mainly used for the 3dvar system the basic goal of which is to seek an optimal estimate of the true atmospheric state at the time of analysis. 2.3. little_r little_r is an ascii-based observation file format. because of many possible formats of raw observation data files such as ascii, bufr, prebufr, madis and hdf, little_r is designed to be an intermediate format so that wrfda might be able to assimilate as many observation types as possible in a universal manner [5]. the report-based file format allows to all manner of observation types to easily combine together into an easy-to-read and edit text file. it consists of reports. these reports are composed of sections: a header, the data itself, and three tail integers. header is a single line located in the first line of each observation and contains information about their observation location, type and other fields. data section comes after header section and it may contain multiple data records which usually correspond to the multiple vertical levels data for a single observation depending on the observation type. in case of fm-12 type of observation in armenia the data section contains one row. for all observation types, after data automation script for wrf model data assimilation system 60 record(s), the single line ending record comes and for obsproc program the height and pressure fields must have a fixed flag value “-777777.00000”. the tail integers are used by obsproc program to decide which observation to keep if two identical observation types are at identical places and times. as the little_r is not used only for obsproc/wrfda, but also for the other programs, there are many unused or missing fields which must be filled with missing flag value (for most of fields it’s “-888888.00000”) to indicate that that value does not exist. 2.4. synop code synop (surface synoptic observations) is a numerical code called fm-12 by wmo used for reporting weather observations made by manned and automated weather stations [6]. a report consists of groups of numbers describing general weather information, such as the temperature, barometric pressure and visibility at a weather station. synop reports are typically sent every three hours. 3. automation script the script is written in python programming language. the structure is designed to be similar to the wrf model programs: properties are provided via configuration file called “namelist”, which is used for main executable program. it contains 2 sections: input and output. the input section contains “stations_file” (path to the stations csv file) and “synop_file” (path to the file containing the synop code) parameters. stations csv file must contain the following required fields: [station wmo number], [name], [elevation], [latitude], [longitude] (see fig 3). fig 3. stations csv file. synop code files must contain sections, each one has a date time in the first row and is followed by code information row for each station (see fig 4). h. grigoryan 61 fig 4. synop code file. output section contains “prefix” parameter, which is used in the name of the generated little_r files. for using the program at first configuration parameters in “namelist” file must be set with the corresponding values and after that running the main file run.py with python will generate little_r file(s). in the case of armenia the daily report of land station records contains 8 groups of information. each group contains observation of fix time and the interval between those times is 3 hour. each little_r file may contain information of many stations observations but for the same date time. so for each daily report the script generates 8 little_r format files. 4. conclusion the provided automation script makes simplify the work of the climatologists by using a user friendly approach obtain the observation data in little_r format, which are ready to be used as an input for wrf model wrfda system. in the future it’s planned to implement this script through a web-based platform of the weather forecast system. references [1] t. ringelband, p. schäfer and a. moser, “probabilistic ampacity forecasting for overhead lines using weather forecast ensembles”, springer-verlag, electrical engineering, vol. 95, no. 2, pp 99–107, 2013. [2] t. semmler, t. jung and m. a. kasper, “using nwp to assess the influence of the arctic atmosphere on midlatitude weather and climate”, science press, advances in atmospheric sciences, vol. 35, no. 1, pp 5–13, 2018. [3] g. emmanouil, d. vlachogiannis, a. sfetsos, s. karozis and a. tasopoulou, “a study of an extreme hot weather event in greece with the wrf-arw atmospheric model”, https://link.springer.com/journal/202 https://link.springer.com/journal/202 https://link.springer.com/journal/202/95/2/page/1 https://link.springer.com/journal/376 https://link.springer.com/journal/376 https://link.springer.com/journal/376/35/1/page/1 automation script for wrf model data assimilation system 62 in: karacostas t., bais a., nastos p. (eds) perspectives on atmospheric sciences. springer atmospheric sciences. springer, cham 2017. [4] r. b. zaripov, y. v. martynova and v. n. krupchatnikov, “atmosphere data assimilation system for the siberian region with the wrf-arw model and threedimensional variational analysis wrf 3d-var”, allerton press, russian meteorology and hydrology, vol. 41, no. 11–12, pp 808–815, 2016. [5] (2017) the ieee website. [online]. available: http://www2.mmm.ucar.edu/wrf/users/wrfda/onlinetutorial/help/littler.html [6] (2017) the ieee website. [online]. available: http://weather.unisys.com/wxp/appendices/formats/synop.html submitted 05.08.2017, accepted 04. 12.2017. ավտոմատացման սցենար wrf մոդելի տվյալների ասիմիլացիայի համակարգի համար հ. գրիգոյան ամփոփում կլիմայագիտությունը և եղանակային կանխատեսումը կարևոր են, քանի որ օգնում են որոշել ապագա կլիմայի սպասումները: նրանք կարևոր են առօրյա կյանքի բազմաթիվ ոլորտներում, այդ թվում `ավիացիայի, նավագնացության, տրանսպորտի, գյուղատնտեսության, զբոսաշրջության, առողջապահության, սպորտի և այլն: առկա են շատ ծրագրային մոդելներ եղանակի մոդելավորման և կանխատեսման համար: եղանակի կանխատեսումը շատ բարդ է և պահանջում է շատ դիտողական տվյալներ: հայաստանը, որպես զարգացող երկիր եղանակային կանխատեսման ոլորտում, կանխատեսման տվյալների մեծ մասն օգտագործում է արտաքին ռեսուրսներից: դիտորդական կայանների միայն շատ քիչ մասն է տրամադրում ամենօրյա դիտարկման տվյալները այդ համաշխարհային կանխատեսման համակարգերին: նմանատիպ դեպքերի համար գոյություն ունեն տվյալների ասիմիլացիայի համակարգեր, որոնք նախատեսված են կարճաժամկետ հեռանկարում փոքր ուղղումներ կատարելու համար: այս հոդվածում ներկայացված է ավտոմատացման սցենար, որը հեշտ և հարմար օգտագործման ձևով ստեղծում է little_r ֆորմատով ascii ֆայլեր՝ օգտագործելով կայանների դիտարկումների synop կոդը, որն անհրաժեշտ է wrf մոդելի տվյալների ասիմիլացիայի համար wrfda համակարգում: https://link.springer.com/journal/11983 https://link.springer.com/journal/11983 https://link.springer.com/journal/11983/41/11/page/1 h. grigoryan 63 скрипт автоматизации для системы ассимиляции данных модели wrf а. григорян аннотация климатология и прогнозирование погоды важны, поскольку они помогают определить будущие климатические ожидания. они важны для большинства аспектов повседневной жизни, включая авиацию, плавание, другие виды транспорта, сельское хозяйство, туризм, здравоохранение, спорт и т. д. многие модели программного обеспечения существуют для моделирования и прогнозирования погоды. прогноз погоды очень сложный и нуждается во множестве наблюдательных данных. в области прогнозирования погоды армения, как развивающаяся страна, получает большую часть данных прогнозирования из внешних источников, и, кроме того, не все наблюдательные станции в армении предоставляют ежедневные данные наблюдений, и они не интегрируют их в глобальные системы прогнозов. вот почему существуют методы ассимиляции данных для внесения небольших поправок в краткосрочный прогноз. в этой статье представлен скрипт автоматизации, который легко и удобно создает файлы ascii формата little_r из кода наблюдений станции synop, что необходимо для системы ассимиляции данных wrfda в wrf-модели. on_lower_bound2.dvi mathematical problems of computer science 23, 2004, 127{129. on lower b ound for w (k2 n) r a fa e l r . k a m a lia n a n d p e t r o s a . p e t r o s ya n institute for informatics and automation problems of nas of ra e-mails rrkamalian@yahoo.com, pet petros@yahoo.com abstract the lower bound w (k2n) ¸ 3n ¡ 2 is proved for the greatest possible number of colors in an interval edge coloring of the complete graph k2n. refer ences [1 ] f. h a r a r y, gr a p h th e o r y, a d d is o n -w e s le y, r e a d in g , ma ,1 9 6 9 . [2 ] a .s . a s r a t ia n , r .r . k a m a lia n , in t e r va l c o lo r in g s o f e d g e s o f a m u lt ig r a p h , appl. m ath.5 , 2 5 -3 4 ,1 9 8 7 . [3 ] s .v . s e va s t ia n o v, on in t e r va l c o lo u r a b ilit y o f e d g e s o f a b ip a r t it e g r a p h , m eth. of d iscr. anal. in s o lu t io n o f e xt e r n a l p r o b le m s . th e in s t it u t e o f ma t h e m a t ic s o f t h e s ib e r ia n b r a n c h o f t h e a c a d e m y o f s c ie n c e s o f u s s r . n o vo s ib ir s k, n 5 0 , 6 1 -7 2 , 1 9 9 0 . [4 ] s . co o k, th e c o m p le xit y o f t h e o r e m -p r o vin g p r o c e d u r e s . in p roc.3rd acm symp. o n th e o r y o f co m p u t in g , 1 5 1 -1 5 8 , 1 9 7 1 . [5 ] r .m. k a r p , r e d u c ib ilit y a m o n g co m b in a t o r ia l p r o b le m s , in \ co m p le xit y o f co m p u t e r co m p u t a t io n s " ( r .e . mille r a n d j.w . th a t c h e r , e d s .) , p p . 8 5 -1 0 3 , n e w y o r k, p le n u m , 1 9 7 2 . [6 ] r .r . k a m a lia n , in t e r va l e d g e co lo r in g s o f gr a p h s , d o c t o r a l d is s e r t a t io n , n o vo s ib ir s k, 1 9 9 0 . [7 ] k . gia r o , m. k u b a le , m. ma la ¯ e js ki, co n s e c u t ive c o lo r in g s o f t h e e d g e s o f g e n e r a l g r a p h s , d iscr. m ath. 2 3 6 , 1 3 1 -1 4 3 ,2 0 0 1 . [8 ] a .a . zyko v, th e o r y o f ¯ n it e g r a p h s , n o vo s ib ir s k, n a u ka , 1 9 6 9 . [9 ] a .s . a s r a t ia n , r .r . k a m a lia n , in ve s t ig a t io n o n in t e r va l e d g e c o lo r in g s o f g r a p h s , j . combin. theory ser. b 6 2 , 3 4 -4 3 ,1 9 9 4 . [1 0 ] v .g. v iz in g , th e c h r o m a t ic in d e x o f a m u lt ig r a p h , k ibernetika 3 , 2 9 -3 9 , 1 9 6 5 . [1 1 ] i. h o lye r , th e np -c o m p le t e n e s s o f e d g e c o lo r in g , siam j . comput. 1 0 , n 4 , 7 1 8 -7 2 0 , 1 9 8 1 . 1 2 7 1 2 8 on lower bound for w (k2n) êïáñçý ·ý³ñ³ï³ï³ý w ( k2n ) -ç ñ³ù³ñ è. è. ø³ù³éû³ý, ä. ². ä»ïñáëû³ý ²ù÷á÷áõù êï³óí³í ¿ w ( k2n ) ¸ 3 n ¡ 2 ³ýñ³í³ë³ñáõãûáõýá, áñý ³å³ñáíáõù ¿ ëïáñçý ·ý³ñ³ï³ï³ý k2n éñçí ·ñ³ýç ùçç³ï³ûù³ûçý ïáõ³ûçý ý»ñïù³ý ù»ç û·ï³·áñííáõ ·áõûý»ñç ³é³í»é³·áõûý ñý³ñ³íáñ ãíç ñ³ù³ñ: d:\sbornik\...\noic.dvi mathematical problems of computer science 25, 2006, 92{100. on lao t esting of m ultiple h ypotheses for p air of objects e vg u e n i a . h a r o u t u n ia n a n d p a r a n d z e m m. h a ko b ya n institue for informatics and automation problems of nas of ra e-mail evhar@ipia.sci.am, par h@ipia.sci.am abstract many hypotheses testing for a model consisting of two independent by functioning objects is considered. it is known that m(¸ 2) probability distributions are given and objects independently of other follows to one of them. the matrix of asymptotic interdependencies (reliability{reliability functions) of all possible pairs of the error probability exponents (reliabilities) in optimal testing for this model is studied. this problem was introduced (and solved for the case with two given probability distributions) by ahlswede and haroutunian. the situation with three hypotheses was examined by haroutunian and hakobyan. refer ences [1 ] r . f. a h ls we d e a n d e . a . h a r o u t u n ia n , " te s t in g o f h yp o t h e s e s a n d id e n t i¯ c a t io n " , e lectronic notes on d iscrete m athematics, vol. 21, p p . 1 8 5 { 1 8 9 , 2 0 0 5 . [2 ] r . f. a h ls we d e a n d e . a . h a r o u t u n ia n , " on s t a t is t ic a l h yp o t h e s e s op t im a l te s t in g a n d id e n t ī c a t io n " . m athematical p roblems of computer science 24, p p . 1 6 { 3 3 , 2 0 0 5 . [3 ] e . a . h a r o u t u n ia n , " r e lia b ilit y in mu lt ip le h yp o t h e s e s te s t in g a n d id e n t i¯ c a t io n p r o b le m s " . p r o c e e d in g s o f t h e n a to a s i, y e r e va n , 2 0 0 3 . n a to s c ie n c e s e r ie s iii: co m p u t e r a n d s ys t e m s s c ie n c e s { vo l. 1 9 8 , p p . 1 8 9 { 2 0 1 . ios p r e s s , 2 0 0 5 . [4 ] r . e . b e c h h o fe r , j. k ie fe r , a n d m. s o b e l, s e qu e n t ia l id e n t ī c a t io n a n d r a n kin g p r o c e d u r e s . th e u n ive r s it y o f ch ic a g o p r e s s , ch ic a g o , 1 9 6 8 . [5 ] r . f. a h ls we d e a n d i. w e g e n e r , s e a r c h p r o b le m s . w ile y, n e w y o r k, 1 9 8 7 . [6 ] e . a . h a r o u t u n ia n , " l o g a r it h m ic a lly a s ym p t o t ic a lly o p t im a l t e s t in g o f m u lt ip le s t a t is t ic a l h yp o t h e s e s " , p roblems of control and information theory, vo l. 1 9 ( 5 -6 ) , p p . 4 1 3 { 4 2 1 , 1 9 9 0 . [7 ] e . a . h a r o u t u n ia n a n d p . m. h a ko b ya n , " on lo g a r it h m ic a lly a s ym p t o t ic a lly o p t im a l h yp o t h e s is t e s t in g o f t h r e e d is t r ib u t io n s fo r p a ir o f in d e p e n d e n c e o b je c t s " , m athematical p roblems of computer science vol. 24, p p . 7 6 { 8 1 , 2 0 0 5 . 9 2 e. a. haroutunian and p. m. hakobyan 9 3 [8 ] i. cs is z ¶a r a n d j. k äo r n e r , information theory: coding theorems for d iscrete m emoryless systems, a c a d e m ic p r e s s , n e w y o r k, 1 9 8 1 . [9 ] e . a . h a r o u t u n ia n , " a s ym p t o t ic a lly o p t im a l t e s t in g o f m a n y s t a t is t ic a l h yp o t h e s e s c o n c e r n in g ma r ko v c h a in " , 5th intern. vilnius conference on p robability theory and m athem. statistics, vo l. 1 , ( a -l ) , p p . 2 0 2 { 2 0 3 , 1 9 8 9 . [1 0 ] e . tu n c e l, " on e r r o r e xp o n e n t s in h yp o t h e s is t e s t in g " . ie e e trans. on it, vo l. 5 1 , n o . 8 , p p . 2 9 4 5 { 2 9 5 0 , 2 0 0 5 . [1 1 ] l . b ir g ¶ e , " v it e s s e s m a xim a ls d e d ¶ e c r o is s a n c e d e s e r r e u r s e t t e s t s o p t im a u x a s s o c ie ¶ s " . z. w a h r s c h . ve r w. ge b ie t e , vo l. 5 5 , p p . 2 6 1 { 2 7 3 , 1 9 8 1 . ºñïáõ ûµû»ïïý»ñç ½áõû·ç ýï³ïù³ùµ µ³½ù³ïç í³ñï³íý»ñç è²ú ëïáõ·ù³ý ù³ëçý º. ². ð³ñáõãûáõýû³ý ¨ ö. ø. ð³ïáµû³ý ²ù÷á÷áõù èáõíí³í ¿ »ñïáõ ³ýï³ë ûµû»ïïý»ñçó ï³½ùí³í ùá¹»éç ñ³ù³ñ µ³½ù³ïç í³ñï³íý»ñç ëïáõ·ù³ý ëý¹çñá: m ( ¸ 2 ) ñ³í³ý³ï³ý³ûçý µ³ßëáõùý»ñá h³ûïýç »ý, ¨ ûµû»ïïý»ñçó ûáõñ³ù³ýãûáõñá ³ýï³ëáñ»ý áý¹áõýáõù ¿ ¹ñ³ýóçó ù»ïá: ²ûë ùá¹»éç ñ³ù³ñ áõëáõùý³ëçñí»é ¿ µáéáñ ñý³ñ³íáñ ½áõû·»ñç ëë³éý»ñç ñ³í³ý³ï³ýáõãûáõýý»ñç óáõóçãý»ñç (ñáõë³éçáõãûáõýý»ñç) ÷áëï³ëí³íáõãûáõýá: ²ûë ëý¹çñá ³é³ç³¹ñ»é »ý (¨ éáõí»é »ñïáõ ñ³í³ý³ï³ý³ûçý µ³ßëáõùý»ñç ¹»åùç ñ³ù³ñ) ð³ñáõãûáõýû³ýá ¨ ²éëí»¹»ý: mathematical problems of computer science 59, 7–15, 2023. doi:10.51408/1963-0097 udc 519.1 a note on large cycles in graphs around conjectures of bondy and jung zhora g. nikoghosyan institute for informatics and automation problems of nas ra, yerevan, armenia e-mail: zhora@iiap.sci.am abstract new sufficient conditions are derived for generalized cycles (including hamilton and dominating cycles as special cases) in an arbitrary k-connected (k = 1, 2, ...) graph, which prove the truth of bondy’s (1980) famous conjecture for some variants significantly improving the result expected by the given hypothesis. similarly, new lower bounds for the circumference (the length of a longest cycle) are established for the reverse hypothesis proposed by jung (2001) combined inspiring new improved versions of the original conjectures of bondy and jung. keywords: hamilton cycle, dominating cycle, longest cycle, large cycle. article info: received 27 january 2021; sent for review 14 february 2022; received in revised form 11 january 2023; accepted 7 march 2023. 1. introduction we consider only finite undirected graphs without loops or multiple edges. the set of vertices of a graph g is denoted by v (g); the set of edges by e(g). for a subset s of v (g), we denote by g−s the maximum subgraph of g with the vertex set v (g)−s. for a subgraph h of g, we use g − h, short for g − v (h). a good reference for any undefined terms is [3]. let α and δ be the independence number and the minimum degree of a graph g, respectively. we define σk by the minimum degree sum of any k independent vertices if α ≥ k; if α < k, we set σk = +∞. in particular, we have σ1 = δ. a simple cycle (or just a cycle) q of order t (the number of vertices) is a sequence v1v2...vtv1 of distinct vertices v1, ..., vt with vivi+1 ∈ e(g) for each i ∈ {1, ..., t}, where vt+1 = v1. when t = 1, the cycle v1 coincides with the vertex v1. so, by this standard definition, all vertices and edges in a graph can be considered as cycles of orders 1 and 2, respectively. such an extension of the cycle definition allows to avoid unnecessary repetition ”let g be a graph of order n ≥ 3” in a large number of results. further, a simple path (or just a path) of order t is a sequence v1v2...vt of distinct vertices v1, ..., vt with vivi+1 ∈ e(g) for each i ∈ {1, ..., t − 1}. a graph g is hamiltonian if g contains a hamilton cycle, i.e., a cycle of order |v (g)|. 7 8 a note on large cycles in graphs around conjectures of bondy and jung now let q be an arbitrary cycle in g. we say that q is a dominating cycle in g if v (g − q) is an independent set of vertices. the first type of generalized cycles, including hamilton and dominating cycles as special cases, was introduced by bondy [4]. for a positive integer λ, q is said to be a dλ-cycle if |h| ≤ λ − 1 for every component h of g − q. alternatively, q is a dλ-cycle of g if and only if every connected subgraph of order λ of g has at least one vertex with q in common. thus, a dλ-cycle dominates all connected subgraphs of order λ. by this definition, q is a hamilton cycle if and only if q is a d1-cycle. analogously, q is a dominating cycle if and only if q is a d2-cycle. we now present another two types of more interesting generalized cycles that form the main topic of this paper. for a positive integer λ, the cycle q is called a pdλ-cycle (pd path dominating) if each path of order at least λ in g has at least one vertex with q in common. similarly, we call the cycle q a cdλ-cycle (cd cycle dominating; introduced in [13]) if each cycle of order at least λ has at least one vertex with q in common. in fact, a pdλ-cycle dominates all paths of order λ in g; and a cdλ-cycle dominates all cycles of order λ in g. in terms of pdλ and cdλ-cycles, q is a hamilton cycle if and only if either q is a pd1-cycle or a cd1-cycle. further, q is a dominating cycle if and only if either q is a pd2-cycle or a cd2-cycle. throughout the paper, we consider a graph g on n vertices with minimum degree δ and connectivity κ. further, let c be a longest cycle in g with c = |c|, and let p and c denote the orders of a longest path and a longest cycle in g − c, respectively. in particular, c is a hamilton cycle if and only if p ≤ 0 or c ≤ 0. similarly, c is a dominating cycle if and only if p ≤ 1 or c ≤ 1. in 1980, bondy [4] conjectured a common generalization of some well-known degree-sum conditions for pdλ-cycles (called (σ, p)-version) including hamilton cycles (pd1-cycles) and dominating cycles (pd2-cycles) as special cases. conjecture 1. (bondy [4],1980): (σ, p)-version let c be a longest cycle in a λ-connected (1 ≤ λ ≤ δ) graph g of order n. if σλ+1 ≥ n + λ(λ − 1), then p ≤ λ − 1. parts of conjecture 1 were proved for λ = 1, 2, 3. (a) κ ≥ 1, σ2 ≥ n =⇒ p ≤ 0 (ore[15], 1960), (b) κ ≥ 2, σ3 ≥ n + 2 =⇒ p ≤ 1 (bondy[4], 1980), (c) κ ≥ 3, σ4 ≥ n + 6 =⇒ p ≤ 2 (zou[17], 1987). for the general case, conjecture 1 is still open. the long cycles analogue (the so called reverse version) of bondy’s conjecture (conjecture 1) can be formulated as follows. conjecture 2. (reverse, σ, p)-version let c be a longest cycle in a λ-connected (1 ≤ λ ≤ δ) graph g. if p ≥ λ − 1, then c ≥ σλ − λ(λ − 2). parts of conjecture 2 were proved for λ = 1, 2, 3, 4. (d) κ ≥ 1, p ≥ 0 =⇒ c ≥ σ1 + 1 (dirac[6], 1952), zh. nikoghosyan 9 (e) κ ≥ 2, p ≥ 1 =⇒ c ≥ σ2 (bondy[2], 1971; bermond[1], 1976; linial[11], 1976), (f) κ ≥ 3, p ≥ 2 =⇒ c ≥ σ3 − 3 (fraisse, jung[8], 1989), (g) κ ≥ 4, p ≥ 3 =⇒ c ≥ σ4 − 8 (chiba, tsugaki, y amashita[5], 2014). note that the initial motivations of conjecture 1 and conjecture 2 come from their minimal degree versions the most popular and much studied versions, which also remain unsolved. conjecture 3. (bondy [4],1980): (δ, p)-version let c be a longest cycle in a λ-connected (1 ≤ λ ≤ δ) graph g of order n. if δ ≥ n+2 λ+1 +λ−2, then p ≤ λ − 1. conjecture 4. (jung [10], 2001): (reverse, δ, p)-version let c be a longest cycle in a λ-connected (1 ≤ λ ≤ δ) graph g. if p ≥ λ − 1, then c ≥ λ(δ − λ + 2). parts of conjecture 3 were proved for λ = 1, 2, 3. (h) κ ≥ 1, δ ≥ n 2 =⇒ p ≤ 0 (dirac[6], 1952), (i) κ ≥ 2, δ ≥ n+2 3 =⇒ p ≤ 1 (nash − williams[12], 1971), (j) κ ≥ 3, δ ≥ n+6 4 =⇒ p ≤ 2 (fan[7], 1987). parts of conjecture 4 were proved for λ = 1, 2, 3, 4. (k) κ ≥ 1, p ≥ 0 =⇒ c ≥ δ + 1 (dirac[6], 1952), (l) κ ≥ 2, p ≥ 1 =⇒ c ≥ 2δ (dirac[6], 1952), (m) κ ≥ 3, p ≥ 2 =⇒ c ≥ 3δ − 3 (v oss, zuluaga[16], 1977), (n) κ ≥ 4, p ≥ 3 =⇒ c ≥ 4δ − 8 (jung[9], 1990). note that cdλ-cycles are more suitable for research than pdλ-cycles since cycles in g − c are more symmetrical than paths in view of the connections between g − c and cdλ-cycles. this is the main reason why some minimum degree versions of conjectures 1 and 2 have been solved just for cdλ-cycles. according to the above arguments, it is natural to consider the exact analogues of bondy’s generalized conjecture (conjecture 1) and its reverse version (conjecture 2) for cdλ-cycles, which we call (σ, c) and (reverse, σ, c)-versions, respectively. conjecture 5. (σ, c)-version let c be a longest cycle in a λ-connected (1 ≤ λ ≤ δ) graph g of order n. if σλ+1 ≥ n + λ(λ − 1), then c ≤ λ − 1. conjecture 6. (reverse, σ, c)-version let c be a longest cycle in a λ-connected (1 ≤ λ ≤ δ) graph. if c ≥ λ − 1, then c ≥ σλ − λ(λ − 2). in 2009, the author proved [14] the validity of minimum degree versions of conjectures 5 and 6. 10 a note on large cycles in graphs around conjectures of bondy and jung theorem 1. ([14], 2009): (δ, c)-version let c be a longest cycle in a λ-connected (1 ≤ λ ≤ δ graph g of order n. if δ ≥ n+2 λ+1 + λ − 2, then c ≤ λ − 1. theorem 2. ([14], 2009): (reverse, δ, c)-version let c be a longest cycle in a λ-connected (1 ≤ λ ≤ δ) graph. if c ≥ λ−1, then c ≥ λ(δ−λ+2). actually, in [14], a significantly stronger result than theorem 1 was proved showing that the conclusion c ≤ λ − 1 in theorem 1 can be strengthened to c ≤ min{λ − 1, δ − λ}, called c-improvement. theorem 3. ([14], 2009): (δ, c)-version, c-improvement let c be a longest cycle in a λ-connected (1 ≤ λ ≤ δ) graph g of order n. if δ ≥ n+2 λ+1 +λ−2, then c ≤ min{λ − 1, δ − λ}. analogously, the condition c ≥ λ − 1 in theorem 2 was weakened [14] to c ≥ min{λ − 1, δ − λ + 1}. theorem 4. ([14], 2009): (reverse, δ, c)-version, c-improvement let c be a longest cycle in a λ-connected (1 ≤ λ ≤ δ) graph g. if c ≥ min{λ − 1, δ − λ + 1}, then c ≥ λ(δ − λ + 2). in this paper, we present new analogous further improvements of theorems 1, 2, 3, 4 inspiring new conjectures in forms of improvements of the initial generalized conjectures of bondy and jung. 2. results first, we prove that the connectivity condition κ ≥ λ in theorem 1 can be weakened to κ ≥ min{λ, δ − λ + 1}. theorem 5. (δ, c)-version, κ-improvement let c be a longest cycle in a graph g of order n and λ a positive integer with 1 ≤ λ ≤ δ. if κ ≥ min{λ, δ − λ + 1} and δ ≥ n+2 λ+1 + λ − 2, then c ≤ λ − 1. analogously, we prove that the connectivity condition κ ≥ λ in theorem 2 can be weakened to κ ≥ min{λ, δ − λ + 2}. theorem 6. (reverse, δ, c)-version, κ-improvement let c be a longest cycle in a graph g and λ a positive integer with 1 ≤ λ ≤ δ. if κ ≥ min{λ, δ − λ + 2} and c ≥ λ − 1, then c ≥ λ(δ − λ + 2). next, we prove that the conclusion c ≤ λ − 1 in theorem 5 can be strengthened to c ≤ min{λ − 1, δ − λ}. theorem 7. (δ, c)-version, (c, κ)-improvement let c be a longest cycle in a graph g of order n and λ a positive integer with 1 ≤ λ ≤ δ. if κ ≥ min{λ, δ − λ + 1} and δ ≥ n+2 λ+1 + λ − 2, then c ≤ min{λ − 1, δ − λ}. finally, we prove that the condition c ≥ λ − 1 in theorem 6 can be weakened to c ≥ min{λ − 1, δ − λ + 1}. theorem 8. (reverse, δ, c)-version, (c, κ)-improvement let c be a longest cycle in a graph g and λ a positive integer with 1 ≤ λ ≤ δ. if κ ≥ min{λ, δ − λ + 2} and c ≥ min{λ − 1, δ − λ + 1}, then c ≥ λ(δ − λ + 2). zh. nikoghosyan 11 3. generalized improvements of conjectures of bondy and jung motivated by theorems 5, 6, 7, 8 (minimum degree versions) with conjectures 1 and 2, in this section we propose their exact analogs in terms of degree sums as generalized improvements of bondy and jung conjectures. conjecture 7. (σ, c)-version, (c, κ)-improvement let c be a longest cycle in a graph g of order n and λ a positive integer. if κ ≥ min{λ, δ − λ + 1} and σλ+1 ≥ n + λ(λ − 1), then c ≤ min{λ − 1, δ − λ}. conjecture 8. (reverse, σ, c)-version, (c, κ)-improvement let c be a longest cycle in a graph g and λ a positive integer. if κ ≥ min{λ, δ − λ + 2} and c ≥ min{λ − 1, δ − λ + 1}, then c ≥ σλ − λ(λ − 2). conjecture 9. (σ, p)-version, (p, κ)-improvement let c be a longest cycle in a graph g of order n and λ a positive integer. if κ ≥ min{λ, δ − λ + 1} and σλ+1 ≥ n + λ(λ − 1), then p ≤ min{λ − 1, δ − λ}. conjecture 10. (reverse, σ, p)-version, (p, κ)-improvement let c be a longest cycle in a graph g and λ a positive integer. if κ ≥ min{λ, δ − λ + 2} and p ≥ min{λ − 1, δ − λ + 1}, then c ≥ σλ − λ(λ − 2). 4. proofs proof of theorem 7. we shall prove that c ≤ min{λ − 1, δ − λ} under the conditions κ ≥ min{λ, δ − λ + 1}, δ ≥ n + 2 λ + 1 + λ − 2 for each 1 ≤ λ ≤ δ. if min{λ, δ − λ + 1} = λ, that is λ ≤ ⌊δ+1 2 ⌋, then we shall prove that c ≤ λ − 1 under the conditions κ ≥ λ, δ ≥ n + 2 λ + 1 + λ − 2. but the latter follows from theorem 1 for all λ = 1, 2, ..., ⌊δ+1 2 ⌋ immediately. now let min{λ, δ − λ + 1} = δ − λ + 1, that is λ ≥ ⌊δ+2 2 ⌋. to conclude the proof, it remains to show that κ ≥ δ − λ + 1, δ ≥ n + 2 λ + 1 + λ − 2 ⇒ c ≤ δ − λ ( λ = δ, δ − 1, ..., ⌊ δ + 2 2 ⌋) . (1) put δ − λ + 1 = µ. acording to this notation, (1) is equivalent to κ ≥ µ, δ ≥ n + 2 δ − µ + 2 + δ − µ − 1 ⇒ c ≤ µ − 1 ( µ = 1, 2, ..., ⌊ δ + 1 2 ⌋) . (2) in (2), the inequality δ ≥ n + 2 δ − µ + 2 + δ − µ − 1 12 a note on large cycles in graphs around conjectures of bondy and jung is equivalent to δ ≥ n + 2 µ + 1 + µ − 2, implying that (2) is equivalent to κ ≥ µ, δ ≥ n + 2 µ + 1 + µ − 2 ⇒ c ≤ µ − 1 ( µ = 1, 2, ..., ⌊ δ + 1 2 ⌋) . (3) observing that (3) follows from theorem 1 immediately, we obtain (1) ≡ (2) ≡ (3) ⇐ ”theorem 1”. theorem 7 is proved. proof of theorem 5. let g be a graph with κ ≥ min{λ, δ − λ + 1}, δ ≥ n + 2 λ + 1 + λ − 2 for each 1 ≤ λ ≤ δ. we shall prove that c ≤ λ−1. observing that min{λ−1, δ −λ} ≤ λ−1, we can weaken the conclusion c ≤ min{λ − 1, δ − λ} in theorem 7 to c ≤ λ − 1 and the result follows immediatly. proof of theorem 8. let g be a graph with κ ≥ min{λ, δ − λ + 2}, c ≥ min{λ − 1, δ − λ + 1} for each 1 ≤ λ ≤ δ. we shall prove that c ≥ λ(δ − λ + 2). if λ = 1, then the result follows from the fact that each graph has a cycle of length at least δ + 1 [6]. let λ ≥ 2. further, if min{λ, δ−λ+2} = λ, then we are done by theorem 2. now let min{λ, δ−λ+2} = δ−λ+2, that is λ ≥ ⌊δ+3 2 ⌋. then it remains to prove that κ ≥ δ − λ + 2, c ≥ δ − λ + 1 ⇒ c ≥ λ(δ − λ + 2) ( λ = δ, δ − 1, ..., ⌊ δ + 3 2 ⌋) . (4) put δ − λ + 2 = µ. by this notation, the statement (4) is equivalent to κ ≥ µ c ≥ µ − 1 ⇒ c ≥ µ(δ − µ + 2) ( µ = 2, 3, ..., ⌊ δ + 2 2 ⌋) , (5) which follows from theorem 2 immediately. so, (4) ≡ (5) ⇐ ”theorem 2”. theorem 8 is proved. proof of theorem 6. let g be a graph with κ ≥ min{λ, δ − λ + 2}, c ≥ λ − 1 for each 1 ≤ λ ≤ δ. we shall prove that c ≥ λ(δ−λ+2). observing that min{λ−1, δ−λ+1} ≤ λ − 1, we can strengthen the condition c ≥ min{λ − 1, δ − λ + 1} in theorem 8 to c ≥ λ − 1 and the result follows immediately. theorem 6 is proved. . zh. nikoghosyan 13 5. conclusion in 2009 [14], a minimum degree sufficient condition for large cycles in graphs is established showing that the famous conjecture of bondy principally is improvable. in the same paper, a lower bound for the length of a longest cycle (the circumference) is derived showing that the conjecture of jung (reverse version of bondys conjecture) principally is improvable as well. in this note, two new analogous sufficient conditions for large cycles and two new lower bounds for the circumference are derived inspiring four new improved versions of bondys and jungs conjectures. references [1] j.c. bermond, “on hamiltonian walks”, congressus numerantium, vol.15, pp. 41-50, 1976. 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[9] h.a. jung and h.a. jung, “longest cycles in graphs with moderate connectivity”, topics in combinatorics and graph theory, essays in honour of gerhard ringel, physicaverlag, heidelberg, pp. 765778, 1990. [10] h.a. jung, “degree bounds for long paths and cycles in k-connected graphs”, computational discrete mathematics, lecture notes in comput. sci., springer, berlin, vol. 2122, pp. 56-60, 2001. [11] n. linial, “a lower bound on the circumference of a graph”, discrete math., vol. 15, pp. 297-300, 1976. [12] c.st.j.a. nash-williams, “edge-disjoint hamiltonian cycles in graphs with vertices of large valency”, studies in pure mathematics, academic press, san diego, london, pp. 157-183, 1971. [13] zh. g. nikoghosyan, “cycle-extensions and long cycles in graphs”, transactions of the institute for informatics and automation problems (iiap) of nas of ra, mathematical problems of computer science, vol. 21, pp. 121-128, 2000. 1 4 a note on large cycles in graphs around conjectures of bondy and jung [1 4 ] zh .g. n iko g h o s ya n , \ d ir a c -t yp e g e n e r a liz a t io n s c o n c e r n in g la r g e c yc le s in g r a p h s " , d iscrete m athematics, vo l. 3 0 9 , p p . 1 9 2 5 -1 9 3 0 , 2 0 0 9 . [1 5 ] o. or e , \ a n o t e o n h a m ilt o n ia n c ir c u it s " , amer. m ath. m onthly, vo l. 6 7 , p . 5 5 , 1 9 6 0 . [1 6 ] h .-j. v o s s a n d c. zu lu a g a , \ ma xim a le g e r a d e u n d u n g e r a d e k r e is e in gr a p h e n " , i, w is s . z. te c h n . h o c h s c h u le ilm e n a u , vo l. 4 , p p . 5 7 -7 0 , 1 9 7 7 . [1 7 ] y . zo u , \ a g e n e r a liz a t io n o f a t h e o r e m o f ju n g " , j . nanjing normal univ. nat. sci., vo l. 2 , p p . 8 -1 1 , 1 9 8 7 . ²ïý³ñï ·ñ³ýý»ñáõù ù»í óçïé»ñç ù³ëçý ´áý¹çç ¨ úáõý·ç í³ñï³íý»ñç ßáõñç äáñ³ ¶. üçïáõáëû³ý ðð ¶²² æýýáñù³ïçï³ûç ¨ ³íïáù³ï³óù³ý åñáµé»ùý»ñç çýëïçïáõï, ºñ¨³ý, ð³û³ëï³ý e-mail: zhora@iiap.sci.am ²ù÷á÷áõù êï³óí»é »ý ýáñ µ³í³ñ³ñ å³ûù³ýý»ñ ·ñ³ýç áý¹ñ³ýñ³óí³í óçïé»ñç ñ³ù³ñ (áý¹·ñï»éáí ð³ùçéãáýû³ý ¨ ¹áùçý³ýï óçïé»ñá áñå»ë ù³ëý³íáñ ¹»åù»ñ) ï³ù³û³ï³ý k-ï³å³ïóí³í ( k = 1 ; 2 ; ::: ) ·ñ³ýáõù, áñáýù ³å³óáõóáõù »ý ´áý¹çç (1980) ñ³ûïýç í³ñï³íç ×ßù³ñï³óçáõãûáõýá áñáß ï³ñµ»ñ³ïý»ñç ¹»åùáõù, çýãç ßýáññçí ½·³éçáñ»ý é³í³óíáõù ¿ ïíû³é í³ñï³íáí ³ïýï³éíáõ ³ñ¹ûáõýùá: ð³ù³ýù³ýáñ»ý, ³ù»ý³»ñï³ñ óçïéç »ñï³ñáõãû³ý ñ³ù³ñ ëï³óí»é »ý ýáñ ëïáñçý ·ý³ñ³ï³ï³ýý»ñ ñ³ï³¹³ñó í³ñï³íç ñ³ù³ñ, áñý ³é³ç ¿ ù³ß»é úáõý·á 2001-çý: êï³óí³í ³ñ¹ûáõýùý»ñá µ³í³ñ³ñ ñçùù»ñ »ý ï³éçë ³é³ç ù³ß»éáõ ýáñ é³í³óí³í ï³ñµ»ñ³ïý»ñ ´áý¹çç ¨ úáõý·ç ý³ëý³ï³ý í³ñï³íý»ñç ÷áë³ñ»ý: çàìåòêà î áîëüøèõ öèêëàõ â ãðàôàõ âîêðóã ãèïîòåç áîíäè è þíãà æîðà ã. íèêîãîñÿí èíñòèòóò ïðîáëåì èíôîðìàòèêè è àâòîìàòèçàöèè íàí ðà, åðåâàí, àðìåíèÿ e-mail: zhora@iiap.sci.am àííîòàöèÿ ïîëó÷åíû íîâûå äîñòàòî÷íûå óñëîâèÿ äëÿ îáîáùåííûõ öèêëîâ (âêëþ÷àÿ ãàìèëüòîíîâûå è äîìèíàíòíûå öèêëû êàê ÷àñòíûå ñëó÷àè) â ïðîèçâîëüíîì k´³ý³éç µ³é»ñ` ð³ùçéãáýç óçïé, ¹áùçý³ýï óçïé, ³ù»ý³»ñï³ñ óçïé, ù»í óçïé: zh. nikoghosyan 1 5 ñâÿçíîì ãðàôå ( k = 1 ; 2 ; ::: ) , äîêàçûâàþùèå ñïðàâåäëèâîñòü èçâåñòíîé ãèïîòåçû áîíäè (1980) äëÿ íåêîòîðûõ âàðèàíòîâ, çíà÷èòåëüíî óëó÷øèâ îæèäàåìûé ïî äàííîé ãèïîòåçå ðåçóëüòàò. àíàëîãè÷íî, ïîëó÷åíû íîâûå íèæíèå îöåíêè äëÿ äëèíû äëèííåéøåãî öèêëà ãðàôà äëÿ îáðàòíîé ãèïîòåçû, ïðåäëîæåííîé þíãîì (2001). ïîëó÷åííûå ðåçóëüòàòû â ñî÷åòàíèè äàþò îñíîâàíèÿ âûäâèæåíèÿ íîâûõ óëó÷øåííûõ âàðèàíòîâ äëÿ èñõîäíûõ ãèïîòåç áîíäè è þíãà. êëþ÷åâûå ñëîâà: ãàìèëüòîíîâ öèêë, äîìèíàíòíûé öèêë, äëèííåéøèé öèêë, áîëüøîé öèêë. 01_nikoghosyan_59 01 microsoft word tpelu.doc mathematical problems of computer science 31, 108--115, 2008. 108 on interval-separable subsets of vertices of a complete graph hakob z. arakelyan1 and rafayel r. kamalian2 1department of informatics and applied mathematics, ysu, 2institute for informatics and automation problems of nas of ra, e-mail: arak_hakob@yahoo.com, rrkamalian@yahoo.com abstract a subset r of the set of vertices of a graph g is called interval-separable iff there exists a proper edge coloring of g in which colors of edges incident with any vertex x of g form an interval of integers iff x r . all interval-separable subsets of the set of vertices of the complete graph are found. references [1] f. harary, graph theory, addison-wesley, reading, ma, 1969. [2] v.g. vizing, the chromatic index of a multigraph, kibernetika 3, pp. 29-39, 1965. [3] a.s. asratian, r.r. kamalian, interval colorings of edges of a multigraph, appl. math 5, yerevan state university, pp 25-34, 1987. [4] r.r. kamalian, interval edge-colorings of graphs, doctoral dissertation, the institute of mathematics of the siberian branch of the academy of sciences of ussr, novosibirsk, 103p, 1990. [5] r.r. kamalian, p.a. petrosyan, on lower bound for w(k2n), mathematical problems of computer science, vol.23, pp 127-129, yerevan, 2004. [6] p.a. petrosyan, interval color–feasible sequences for some classes of graphs, phd thesis, institute for informatics and automation problems of nas of ra, yerevan, 130 p, 2006. [7] r.r. kamalian, interval colorings of complete bipartite graphs and trees, preprint of the computing centre of the academy of sciences of armenia, 11p, 1989. èñçí ·ñ³ýç ·³·³ãý»ñç µ³½ùáõãû³ý ùçç³ï³ûù³ûýáñ»ý ³é³ýóý³óíáõ »ýã³µ³½ùáõãûáõýý»ñç ù³ëçý ð. ²é³ù»éû³ý è. ø³ù³éû³ý ²ù÷á÷áõù g ·ñ³ýç ·³·³ãý»ñç µ³½ùáõãû³ý r »ýã³µ³½ùáõãûáõýá ïáãíáõù ¿ ùçç³ï³ûù³ûýáñ»ý ³é³ýóý³óíáõ ³ûý ¨ ùç³ûý ³ûý å³ù³ý³ï, »ñµ ·áûáõãûáõý áõýç g ·ñ³ýç ³ûýåçëç ×çßï ïáõ³ûçý ý»ñïáõù, áñ ï³ù³û³ï³ý x ·³·³ãçý ïçó ïáõ»ñç ·áõûý»ñá ï³½ùáõù »ý µý³ï³ý ãí»ñç µ³½ùáõãû³ý ù»ç ùçç³ï³ûù ³ûý ¨ ùç³ûý ³ûý ¹»åùáõù, »ñµ x r : ¶ïýí³í »ý éñçí ·ñ³ýç ·³·³ãý»ñç µ³½ùáõãû³ý µáéáñ ùçç³ï³ûù³ûýáñ»ý ³é³ýóý³óíáõ »ýã³µ³½ùáõãûáõýý»ñá: üï³ñ³·ñí»é »ý ³é³ç³ñïí³í ï³ýáýç ïçñ³éáõãû³ý ³ñ¹ûáõý³í»ïáõãûáõýá óáõó³¹ñáõ ñ³ù³å³ï³ëë³ý å³ïï»ñý»ñ ¨ ãí³ûçý ³ñ¹ûáõýùý»ñ: начиная с начала 2000 года осуществляется внедрение ghis в здравоохранении, в рамках принятого проекта о реформирование информ mathematical problems of computer science 46, 81--86, 2016. increasing the visibility of scientific data in armenia using persistent identifiers hayk a. grigoryan institute for informatics and automation problems of nas ra e-mail: hayk-grigoryan@ipia.sci.am abstract during the development of computer technologies, the scientific research becomes more data intensive and collective than in the past. data practices of researchers such as data sharing, discovery, reuse and preservation can be useful for other researchers in the same domain. data sharing allows the verification of results and extends scientific research from previous results. many scientific fields such as biology, astronomy, weather forecast, etc., produce a vast amount of data, these data need to be shared and increase its accessibility, because sharing data has an important role for today’s science communities. in this paper, we introduce a deployed infrastructure to enable data-sharing using metadata which increases the accessibility of this data. keywords: persistent identifier, scientific data, data sharing, metadata. 1. introduction data form the basis for good scientific decisions, management, and use of resources and informed decision-making. in addition, “science is becoming data-intensive and collaborative” [1]. due to developments in computational simulation and modeling and communication technologies, the amount of data collected, analyzed and stored has increased enormously [2] and the science confronts with big data and complex data structures that traditional data processing applications are insufficient to operate with them. digital data are not only the outputs of research but provide inputs to new hypotheses, enabling new scientific conceptions and driving innovation [3]. data sharing becomes more important as science becomes more data intensive and amount of data daily increasing. because of the huge size of scientific data, it’s good practice to share not only the data but also the metadata (data about data). 81 increasing the visibility of scientific data in armenia using persistent identifiers 82 metadata recapitulates basic information about data, which can make easier to find and work with specific instances of data. metadata is structured information that declares, locates, explains, or otherwise makes it easier to retrieve, use, or manage an information resource. it can describe a different kind of resources such as single, collection or even a part of larger resources. as stated in [4] “metadata is a key to ensuring that resources will survive and continue to be accessible into the future”. author, date created and file size are one example of basic document metadata. the ability to filter through that metadata helps someone to find the specific document. in a phase where data-intensive science is moving towards automated processes, the usage of persistent identifiers (pid) for any type of digital object is crucial [5]. currently, existing solutions provide good services for dealing with data sharing but usually they had some complex structures and the scientist who does not have knowledge of working with that kind of services may confront with difficulties of using that. also, that services provide a metadata which may not match with our scientific rules. our interface is simple to use and as it targets to the armenian scientific communities, it is more flexible and can be changed depending on the community needs. 2. persistent identifier a persistent identifier is a long-lived reference to a digital resource. it has two components: a unique identifier which ensures the provenance of a digital resource; and a service. when the resource location changes, the server locates the resource in the course of time and guarantee that the identifier resolves to the current location. the aim of persistent identifiers is to solve the problem of the persistence of accessing cited resource, particularly in the scientific data. it can be used also for scientific data which is stored in the web network. frequently, web addresses fail to take users to the expected referenced resource because of the technical problems with server or event more often by human-created failures. organizations shift journals to new publishers, reconstruct their websites, or moving forward without using the older content, leading to broken links. if the referenced resource is essential for medical, legal or scientific reasons this can be frustrating for users [6]. 2.1 data sharing the data lifecycle cannot be considered independently from research lifecycle. starting from ideas and finalizing with publications the researchers confront with the search process which is a crucial part of this lifecycle (figure 1) [7]. h. grigoryan 83 fig. 1. joint information systems committee (jisc), stages of the research and data lifecycle. 3. related works in the area of data sharing already exist working services providing the scientists to share and save their data on the server which can be accessible for other communities. epic can be considered as one of the famous providers of such services which was founded in 2009 by a consortium of european partners. it’s providing the pid services based on the handling system [8] for the european research. handling systems are used for the allocation and resolution of persistent identifiers. for the scientific research community epic provides 4 services: pid service, pid resolution, pid replication and global handle mirror server. the epic api provides a software stack for a pid service [9]. there is another service provided by eudat which offers common data services, supporting multiple research communities as well as individuals, through a geographically distributed, resilient network of 35 european organizations. these shared services and storage resources are distributed throughout 15 european nations, and data is stored beside some of the most powerful supercomputers in europe. covering both access and deposit, from informal data sharing to long-term archiving, and addressing identification, discoverability and computability of both long-tail and big data, eudat’s services address the full life cycle of research data [10]. the existing services resolve the problem with data sharing and allowing scientific communities to share their data within the network. however, they suggest some constraint rules for metadata and all data stored on their servers. our deployed infrastructure described in this paper will allow to be more flexible for storing and sharing data and will construct local data sharing rules. as our platform is based on the pid service provided by epic, it will allow our local scientific communities easily register their data and share them within the european research. http://journals.plos.org/plosone/article/figure/image?size=medium&id=info:doi/10.1371/journal.pone.0021101.g001 increasing the visibility of scientific data in armenia using persistent identifiers 84 4. deployed infrastructure the schematic representation of the deployed infrastructure is presented below (figure 2). fig. 2. the model of infrastructure. the model of infrastructure can be divided into 3 sections. section 1: registered communities can create metadata for their data and provide a reference for their resources. metadata will be saved on our server and will generate the url for that. section 2: using the api provided by epic, the platform will request for pid generation passing the url which was generated in section 1. epic api allows to construct a generated pid by providing a custom suffix and prefix to the guid . as it’s working with one account that the epic provides to us the suffix of the generated pid will have a structure: -- section 3: infrastructure will assign that generated pid with the metadata in our server and communities which are using our system can edit their metadata without touching the already existed persistent identifier. they can also delete their metadata which automatically will delete the pid from the epic system. we were also providing the search functionality within epic persistent identifiers and within our server data. the infrastructure consists of two sides: back-end and front-end. for programming backend side the node js programming language with express library was used. for storing the created metadata used the document based on mongodb database engine. the front-end part is constructed with angularjs technique which is based on the javascript script language. h. grigoryan 85 5. conclusion in this paper, we have presented our infrastructure for data sharing which has a simple structure and provides an easy way to store and access data by scientists. the primary benefit of the work is to increase the visibility of the scientific community in armenia by making their scientific output data more visible, trusted and accessible, and this in its turn will increase the productivity and collaboration between armenian research communities and international communities. for future work the infrastructure can be improved by adding local replica storages which can be used for faster disaster recovery. references [1] (2010) national science foundation. press release 10-077 scientists seeking nsf funding will soon be required to submit data management plans. online. [available]: http://www.nsf.gov/news/news_summ.jsp?cntn_id=116928. accessed 2010 oct 2. [2] (2009) national academies of science, committee on ensuring the utility and integrity of research data in a digital age ensuring the integrity, accessibility, and stewardship of research data in the digital age. online. [available]: http://www.nap.edu/catalog.php?record_id=12615. accessed 2010 oct 5. [3] (2010) national science foundation, office of cyber infrastructure directorate for computer & information science & engineering (2008) sustainable digital data preservation and access network partners (datanet) program solicitation nsf 07601. online.[available]:http://www.nsf.gov/funding/pgm_summ.jsp?pims_id=503141. accessed 2010 sep 22. [4] (2013) niso press “understanding metadata”, online. [available]: http://www.niso.org/publications/press/understandingmetadata.pdf [5] (2015) gary berg-cross, raphael ritz, peter wittenburg “rda dft core terms and model”, december 02, 2015, online. [available]: http://hdl.handle.net/11304/5d760a3e-991d-11e5-9bb4-2b0aad496318 [6] (2010) juhahakala “persistent identifiers – an overview”, twr technology watch review, online.[available]:http://www.metadaten-twr.org/2010/10/13/persistentidentifiers-an-overview/ [7] (2011) carol tenopir, suzie allard, kimberly douglass, arsev umur aydinoglu, lei wu, eleanor read, maribeth manoff, mike frame , “data sharing by scientists: practices and perceptions”, online. [available]: http://dx.doi.org/10.1371/journal.pone.0021101.g001 [8] (2016) the ieee website. [online]. available: http://www.handle.net/ [9] (2016) the ieee website. [online]. available: http://www.pidconsortium.eu/ [10] (2016) the ieee website. [online]. available: https://eudat.eu/ submitted 04.08.2016, accepted 12.11.2016. http://www.nsf.gov/news/news_summ.jsp?cntn_id=116928 http://www.nap.edu/catalog.php?record_id=12615 http://www.nsf.gov/funding/pgm_summ.jsp?pims_id=503141 http://www.niso.org/publications/press/understandingmetadata.pdf https://b2share.eudat.eu/search?f=author&p=gary%20berg-cross%2c%20raphael%20ritz%2c%20peter%20wittenburg&ln=en http://hdl.handle.net/11304/5d760a3e-991d-11e5-9bb4-2b0aad496318 http://www.metadaten-twr.org/2010/10/13/persistent-identifiers-an-overview/ http://www.metadaten-twr.org/2010/10/13/persistent-identifiers-an-overview/ http://dx.doi.org/10.1371/journal.pone.0021101.g001 http://www.handle.net/ http://www.pidconsortium.eu/ increasing the visibility of scientific data in armenia using persistent identifiers 86 հայաստանում գիտական տվյալների տեսանելիության բարձրացումը մշտական նույնացուցիչների միջոցով հ. գրիգոյան ամփոփում համակարգչային տեխնոլոգիաների զարգացման ընթացքում գիտական հետազոտությունների տվյալները դարձել են ավելի ինտենսիվ և հավաքական: հետազոտողների տվյալների օգտագործման մեթոդները, ինչպիսիք են՝ տվյալների փոխանակում, բացահայտում, վերակազմակերպման օգտագործում և պահպանություն, կարող են օգտակար լինել միևնույն ոլորտի այլ հետազոտողների համար: տվյալների տարածումը արդյունքների ստուգման հնարավորություն է տալիս և ընդլայնում է առկա գիտական հետազոտությունների արդյունքները: շատ գիտական ոլորտներում, ինչպիսիք են՝ կենսաբանություն, աստղագիտություն, եղանակի կանխատեսում և այլն, արտադրվում են հսկայան քանակությամբ տվյալներ, որոնք պետք է լինեն ընդհանուր և բարձր հասանելիության, քանի որ տվյալների տարածումն ունի շատ մեծ դեր ժամանակակից գիտական հասարակությունում: այս հոդվածում մենք ներկայացրել ենք տեղակայված ենթակառուցվածք, որը մետատվյալների օգտագործմամբ տվյալների փոխանակման հնարավորություն է տալիս, որը մեծացնում է այդ տվյալների հասանելիությունը: повышение видимости научных данных в армении с использованием постоянных идентификаторов а. григорян аннотация при разработке компьютерных технологий, данные научных исследований становятся более коллективными и интенсивными, чем в прошлом. методы использования данных исследователей, таких как обмен данными, открытие, повторное использование и сохранение, могут быть полезными для других исследователей в той же сфере. обмен данными позволяет проверку результатов и расширяет научные исследования от предыдущих результатов. многие научные направления, такие как биология, астрономия, прогноз погоды и т.д. производят огромное количество данных, эти данные должны быть общими и повысить его доступность, поскольку обмен данными играет важную роль для современных научных сообществ. в этой статье мы представляем инфраструктуру, позволяющую обмен данными с использованием метаданных, что увеличивает доступность этих данных. microsoft word ob odnom.doc ìàòåìàòè÷åñêèå âîïðîñû êèáåðíåòèêè è âû÷èñëèòåëüíîé òåõíèêè 23, 2004, 150–153. 150 об одном методе прогнозирования динамики ценных бумаг мушег с. саакян èíñòèòóò ïðîáëåì èíôîðìàòèêè è àâòîìàòèçàöèè íàí ðà e-mail m.sahakian-alumni@lse.ac.uk àííîòàöèÿ в статье приводится метод прогнозирования динамики ценных бумаг и результаты испытаний программной реализации этого метода. . литература 1. c. lee giles, b. g. horne and t. lin. learning a class of large finite state machines with a recurrent neural network. neural networks 8(9), 1359-1365, 1995. 2. êîêñ ä. ð., îêóñ ä. àíàëèç äàííûõ òèïà âðåìåíè æèçíè. ì. ôèíàíñû è àòàòèñòèêà. 1988. 3. ãîëÿíäèíà í. ý. ìàòîä “ãóñåíèöà” – ssa: àíàëèç âðåìåííûõ ðÿäîâ. ó÷åáíîå ïîñîáèå ñïá: èçä-âî ñ-ïåòåðáóðãñêîãî óíèâåðñèòåòà. 2004. 78 ñ. ²ñå»ãõã»ñç ¹çý³ùçï³ûç ï³ýë³·áõß³ïù³ý ùç ù»ãá¹ç ù³ëçý ø. ê. ê³ñ³ïû³ý ²ù÷á÷áõù ðá¹í³íáõù ý»ñï³û³óí³í »ý ³ñå»ãõã»ñç ¹çý³ùçï³ûç ï³ýë³·áõß³ïù³ý ùç ù»ãá¹ ¨ ¹ñ³ íñ³·ñ³ûçý çñ³·áñíù³ý ÷áñó³ñïáõùý»ñç ³ñ¹ûáõýùý»ñá: начиная с начала 2000 года осуществляется внедрение ghis в здравоохранении, в рамках принятого проекта о реформирование информ mathematical problems of computer science 48, 98-104, 2017. consideration of congestion situations in telecommunication networks and methods of their processing as a special type of emergency situations hamlet h. harutyunyan and hmayak a. avanesyan armenian state pedagogical university after kh. abovyan national polytechnic university of armenia e-mail: hamhar@lycos.com, hmayak.avanesyan@gmail.com abstract this article discusses the methods of struggle with congestion situations in telecommunication networks, considering them as a special type of emergency situations, which frequently occur in computing systems. keywords: telecommunication network, congestion situation, emergency, network traffic, intensity of traffic, data transmission. 1. introduction the problem of struggle with congestion situations is one of the most difficult and important problems of telecommunication networks. many scientific works have been devoted to this problem. currently, the main directions of research are conducted generally in the following areas:  increase the performance of switching centers (nodes);  increase the bandwidth of communication channels;  improvement of data transfer protocols;  creation of effective traffic processing algorithms;  development of methods for preventing, detecting, processing and eliminating of congestion situations with minimal data loss. in this article, the congestion situation is considered as an emergency in switching centers. in accordance with this idea, the methods of struggle with congestion situations are considered as a special case in general methodology of emergency situations processing. such approach to congestion situations extends the methodology of their research and processing, using the existing methodologies of emergency management. 98 mailto:hamhar@lycos.com h. harutyunyan and h. avanesyan 99 2. congestions as a special type of emergency situations the concept of congestion situation is defined in telecommunication networks literature in different ways [1,2,4]. mostly it is associated with network traffic increase. here is a brief description of the process of occurrence of congestion situation. the congestion occurs when the current load of system exceeds its maximum load capacity. while the current system load is below the acceptable value, the intensity of the incoming traffic is equal or almost equal to the intensity of the outgoing traffic in the switching center. as soon as the current load approaches and exceeds the system load capacity, the intensity of the incoming traffic becomes more than the intensity of the outgoing traffic. a queue of packets waiting to be processed accumulates in the switching center and the delay of their processing begins to increase. the more the load increases, the longer the queue becomes and, hence, the longer the delay period becomes. with the continuation of this process, the queue length grows to the limit, after that, new incoming packets are not queued but discarded. without having received a confirmation from the recipient, the sender starts to resend the packets, which in turn, further increases the load of the receiving center. in this situation, the bandwidth capability of the switching center begins to fall and the outgoing traffic decreases sharply. in the end, it tends to zero, which brings about the switching center dysfunction. this situation in the scientific literature is called congestion [1,5]. the negative consequences of congestion situations are:  telecommunication network characteristics deterioration;  transmission of data delays and losses;  bandwidth reduction of switching centers;  partial or complete dysfunction of the switching center. in this context, the state of congestion resembles the classical definition of the computing systems emergency state, when due to hardware or software errors, data losses and partial or complete system faults occur [1]. despite the similarities, different types of emergencies are different by nature, and there is no universal method for their detection and processing. therefore, for every type of emergency, special methods are required to develop appropriate for their nature, conditions of occurrence and impact on the functioning of the system. the general emergency management strategy involves the following procedures: • prevention of occurrence of emergency situations; • control and detection of emergency situations; • processing of detected emergency situations and restoring the normal functioning mode of the system; • elimination of consequences of emergency conditions. the strategy of struggle with congestions also includes the same procedures mentioned above. however, despite the general methodology, congestion as an emergency situation has a completely different nature. unlike the classic concept of emergency situations, congestion can occur in a state of full operational capability of hardware and software systems and complete correctness of information. the congestion situation occurs when the rate of incoming traffic exceeds a certain threshold value corresponding to the switching center maximum value throughput. the congestion situation can be eliminated itself, when the incoming traffic becomes less than the specified threshold value. in that case the congestion situation is like a “transient fault” in hardware. the concept of congestion is closely related to the data processing in real time mode, when there is a limitation for packets transmission time, and violation of limits is considered as an error. the absence of time limits formally means that the tasks queue length can be unlimited and the system can cope with the processing of traffic of any intensity. in this case, the congestion situation does not occur, because there is no violation of the time limit for data processing. of course, such consideration of congestion situations in telecommunication networks and methods 100 statement of question is purely theoretical. in practice, the queue length cannot be infinite, and, besides, any service must be completed within a reasonable time corresponding to the service specificity. if the time is exceeded, the interest of users to this service decreases. it is important to note that the congestion situation can occur not only with the increase in incoming traffic, but also with a decrease in the bandwidth capability of the switching center due to hardware and software failures. those failures can be caused by system characteristics deterioration. in this case, the system bandwidth can be decreased and become insufficient for processing tasks in proper time. that leads to a congestion situation. it should be noted that, due to failures in hardware and communication channels, the delays of processing packets in the queues sometimes can exceed the permissible value. however, that is not a signal of congestion. the signal of violation of the permissible delay time is a necessary but insufficient condition for fixing of congestion situation. therefore, in such cases the danger of congestion occurs when the intensity of failures exceeds a preset threshold. such a situation can arise due to the problems in the communication channels or congestion in the receiving centers. the definition of congestion situation due to the conditions of processing tasks in real time is presented in [3]. 3. the “sensitivity” of switching centers to the congestion situation telecommunication network has complex structures, consisting of many switching centers, which provide data transfer through the network. in packet switching mode switching centers have different “sensitivity” to the congestion situation. the probability of congestions depends on many factors, some of which are given below. the efficiency of use of system and network resources. the switching centers as component parts of the common route usually have different loads. it depends on their bandwidth, number of routes, crossing in the switching center and intensity of the flow on them. the probability of congestion at any time depends on the difference between the bandwidth and the current load of the switching center (loading reserve). сr = (𝐶𝐶 − 𝐶𝐶𝑡𝑡), where сr is the loading reserve, 𝐶𝐶 – the switching center bandwidth, 𝐶𝐶t the current load. the less is this difference, the less free resources are left, and the greater is the probability of congestion. it is obvious, that in case of increasing traffic on the route, the congestion situation will occur in the switching center, which has less сr. the rank of the switching center. the switching centers depending on their position in a network topology, have different roles in the functioning of the network. the role of any switching center determines its “importance” in ensuring network bandwidth. in networks terminology the “importance” of switching nodes is characterized by the parameter "rank" of the node, which is determined by the number of communication channels connected to this node. we can use the above-mentioned term "rank" for determination of switching centers “importance” in the topology of the telecommunication network. in this case, the parameter "rank" of the switching center can be determined by the number of possible routes passing through it. the network topology can be represented as a graph. the graph vertices represent switching centers, and edges to communication channels. in the terminology of graph theory the parameter "rank" is the degree of the vertex of the graph. obviously, under the same bandwidth capability switching centers, the probability of congestion will be more in a switching center with a higher rank. on the other hand, for equal values of the rank of switching centers, the probability of congestion is greater at the center with a lower bandwidth capability. an example of a graph of a network topology is shown in fig. 2.1. h. harutyunyan and h. avanesyan 101 fig. 2.1. example of a graph topology of a telecommunication network. in this example, switching centers have different "ranks": the rank of the switching center c is equal to 2, k is 6. it is obvious that the switching center number k is more significant in the network topology than the switching center number c. the reliability of the switching center. the switching centers belong to the class of complex systems that are able to provide system functioning in case of fault of some components. the faults in hardware in such systems lead to the degradation of a system or functional characteristics. a fault in any component can lead to the state of partial or complete fault of system depending on the possibility of continuation of execution of system target functions. for switching center, a partial fault may lead to reduction of system and network resources, reducing the performance of the computer system or bandwidth of communication channels, the partial destruction of routes. in this connection, the switching center can get into a state of congestion even the intensity of incoming traffic is acceptable for a full configuration of system. sustainability of communication channels to errors. increased intensity of failures in communication channels leads to increasing of number of received packets with errors and to increasing of number of resending of packets. in the end, it may lead to increased queue lengths in the buffer file and to the congestion of the sender center. the probability of overloading depends on the faulttolerance of the channels. consideration of congestion situations in telecommunication networks and methods 102 4. the influence of switching center congestion situation to the quality of functioning of the entire network congestion delays the flows on all routes passing through the switching center. it means that the local congestion in any switching center may cause congestion in other centers, which are connected to that center with common routs. this process can cover other switching centers and spread to levels of the subnet and the entire network. the magnitude of the effect of congestion in any switching center on the performance of the whole network is determined by two factors: 1. the level of decreasing of network bandwidth. 2. the number of destroyed routes. congestion situation leads to an afunctional state of switching center for a while, which, in turn, leads to the change in the topology of network. to determine the impact of faults of a switching center on the network bandwidth, we introduce a "parameter" weight for any switching center. we suggest the following method of calculating the parameter "weight" for switching centers. 1. to estimate the network bandwidth at the maximum allowed traffic over all possible routes (c). 2. to disconnect i-th switching center and to estimate the network bandwidth (сi) again. the "weight" of the i-th center is denoted by pi. 𝑃𝑃𝑖𝑖 = 1 − 𝐶𝐶𝑖𝑖 𝐶𝐶 , 𝑃𝑃𝑖𝑖 can vary in the range 0≤ pi ≤ 1; if disabling the i-th switching center does not change the network bandwidth, then pi=0 (сi=c). it can be in case, when there are reserves in the network topology and the output of the switching center from a working configuration does not affect the network bandwidth. if disabling i-th center pi=1 (сi=0), the network stops functioning, it means that a fault of i-th center leads to a complete fault of the whole network. the more the pi value, the greater the magnitude of the impact of i-th center on the functioning of the whole network. the fault of the switching center can reduce the number of possible routes in the network, increase the load on other routes, change the network connectedness and stop transfer data by some routes. when evaluating the impact of a switching center fault on the performance of the network, it should be taken into account not only the reducing number of possible routes, but the ‘importance” of the destroyed routes. it is important to distinguish congestion due to lack of own resources and congestion due to receiving centers. criteria for detection and a strategy of processing of both types of congestion situations are totally different. the impact of congestion in a switching center to others also depends on the duration of congestion situation. we introduce the notion of "permissible duration of congestion situation”. this is a time interval, after which the sending center fixes the fact of congestion state in receiving center. if during that time the congestion situation is eliminated, the functioning mode will be restored. the congestion duration depends on the behavior of the incoming traffic. when the incoming traffic intensity becomes lower than the system bandwidth, the overload is eliminated h. harutyunyan and h. avanesyan 103 by itself. taking into account the self-similar nature of network traffic, it can be assumed, that in such a short time as permissible duration interval, the probability of maintaining high intensity is higher than the probability of its decrease [6]. the permissible duration can be determined by the parameter of permissible delays of processing of packets in switching center, which for different traffic categories is different. the strategy of processing of different types of emergency situations is discussed in scientific literature. however, the congestion situation is considered usually in the context of resources planning and flows management issues, but not within the framework of the emergency management strategy. all these works are mainly aimed at improving the efficiency of planning and managing the resources of telecommunication networks and also improving the data transmission procedures. it, of course, increases the efficiency of system and its stability to overload. however, with all this, the probability of congestion situation will still remain. therefore, in addition to the conventional approach to congestion, it is advisable to consider the congestion as a special type of emergency situation in telecommunication networks and to apply the whole methodology of dealing with emergency situations. references [1] a. s. tanenbaum, david j. wetherall, computer networks, fifth edition, prentice hall press upper saddle river, nj, usa 2010. [2] i. villy, teletraffic engineering and network planning, technical university of denmark, ørsteds plads, denmark, 2015. [3] h. harutyunyan and h. avanesyan, “causes and conditions of occurrence of congestion situations in telecommunication networks”, computer science and information technologies. int. conference, september 25 – 29, yerevan, armenia, pp. 441--442, 2017. [4] w.stallings, foundations of modern networking, by pearson education, inc usa, 2016 [5] о. и. шелухин и др., самоподобие и фракталы , физико-математическая литература, москва, 2008. submitted 09.10.2017, accepted 06.12.2017. հեռահաղորդակցական ցանցերում գերբեռնվածքային իրավիճակները որպես հատուկ տիպի վթարային իրավիճակ և դրանց մշակման մեթոդները հ. հարությունյան և հ. ավանեսյան ամփոփում գերբեռնվածքային իրավիճակների դեմ պայքարի խնդիրը հանդիսանում է հեռահաղորդակցության ցանցերի առավել բարդ և կարևոր խնդիրներից մեկը։ .այդ խնդրի ուսումնասիրմանը նվիրված են բազմաթիվ գիտական աշխատանքներ: ներկայումս հետազոտությունները հիմնականում տարվում են հետևյալ ուղղություններով՝ consideration of congestion situations in telecommunication networks and methods 104  հեռահաղորդակցական կենտրոնների (հանգույցների) արտադրողականության բարձրացում  կապուղիների թողունակության բարձրացում  տվյալների փոխանցման արձանագրությունների կատարելագործում  երթուղիների կառավարման արդյունավետ ալգորիթմների մշակում  գերբեռնվածություն իրավիճակների կանխարգելման, հայտնաբերման, մշակման և վերացման նոր մեթոդների մշակում։ հոդվածում հեռահաղորդակցական կենտրոններում գերբնակվածության իրավիճակները դիտարկվում են որպես հատուկ տիպի վթարային իրավիճակներ և դրա հետ կապված՝ գերբնակվածության դեմ պայքարի մեթոդները դիտարկվում են վթարային իրավիճակների դեմ գոյություն ունեցող մեթոդաբանության համատեքստում; նման մոտեցումը ընդլայնում է գերբեռնվածության իրավիճակների ուսումնասիրության և մշակման հնարավորությունները։ перегрузка как аварийная ситуация и методы ее обработки г. арутюнян и а. аванесыан аннотация борьба с перегрузками является одной из сложных проблем телекоммуникационных сетей. решению этой проблемы посвящено много работ. в настоящее время исследования ведутся в направлениях увеличения производительности коммутационных центров, пропускной способности каналов связи, совершенствования протоколов передачи данных, создания эффективных алгоритмов управления трафиком, разработки методов предотвращения, обнаружения и устранения перегрузки с возможно минимальными потерями. все эти исследования имеют цель поддерживать баланс между входящими и исходящими потоками коммутационного центра. в данной работе перегрузка рассматривается как аварийное состояние в телекоммуникационных сетях. общая стратегия управления аварийными ситуациями предполагает использование следующих процедур: • предотвращения возникновения аварийной ситуации; • контроля и обнаружения возникновения аварийной ситуации; • обработки аварийной ситуации и восстановление нормального режима работы системы; • устранения последствия аварийного состояния. в соответствии с этой постановкой, методы борьбы с перегрузками в работе рассматриваются как методы борьбы со специальным видом аварийных ситуаций. рассмотрение состояния перегрузки как особый вид аварийных ситуаций расширяет методологию исследования и борьбы с перегрузками, используя существующие методологии борьбы с аварийными ситуациями. начиная с начала 2000 года осуществляется внедрение ghis в здравоохранении, в рамках принятого проекта о реформирование информ mathematical problems of computer science 46, 50--54 , 2016. selection of methods to provide end-to-end email traffic security arthur s. petrosyan and gurgen s. petrosyan institute for informatics and automation problems of nas ra e-mail: arthur@sci.am, gurgen@sci.am abstract the goal of the research described in this paper is to select methods of securing email traffic. while it is known that email service is not secure by default, both end users and service providers can implement available securing mechanisms to ensure end-to-end email traffic security as much as possible. latest developments and research in this area, like smtp mta strict transport security (sts) and smtp tls reporting are presented. this paper includes a best practice configuration of protection methods for both mail user agent (mua) and mail transfer agent (mta). recommendations given are oriented for the members of academic scientific research computer network of armenia (asnet-am) in regard to secure use of asnet-am email service. keywords: email, security, smtp, mta, strict transport security, sts, tls, mail user agent, mua, mail transfer agent, mta. 1. introduction despite so many other ways of communication today, email is still one of the widely used and popular ways to exchange information. since email is not an online service and is based on a store-and-forward model (picture 1), implementing end-to-end email traffic security can be a complex task, depending on several parties to support and implement specific configuration requirements. this includes configuration of both email clients, called mail user agent (mua) and email servers, called mail transfer agent (mta). generally speaking email security can mean two measures: 1. connection security 2. data security 50 a. petrosyan and g. petrosyan 51 this paper is focused on the measures of connection security in view of latest developments in this field. implementing connection security for email means encryption of email traffic during network transfer. email message always originates at mua and is then transferred to mta for further delivery via other mtas to the appropriate user’s mailbox, from which the recipient can fetch it using its own mua. so end-to-end email traffic security could be achieved only in case all parties use the secure methods of communications. end-to-end email communication presented in picture 1 can be divided into two main parts: mua-mta communication and mta-mta communication. fig. 1. email store-and-forward model. 2. mua-mta communication mua-mta connection security means encryption of data during network transfer. for email it can be implemented in two ways: ssl/tls and starttls. both options provide the same level of connection security but there is some important difference. the "ssl/tls" method means: "always encrypt connection or don't connect at all". the "starttls" method means: "encrypt connection if both ends support tls, otherwise connect without encryption". so, starttls can be treated as less secure, because not only can it failback to insecure data transfer without notification, but because it's also subject to man-inthe-middle (mitm) attack [1]. starttls as an extension of the smtp, imap and pop3 protocols (smtps, imaps, pop3s) enables establishing an encrypted connection with the support of the ssl/tls protocol without separate special network port for encrypted communication. although separate ports are registered for the smtps, imaps and pop3s protocols, the use of standard port enables the usage of both protected and unprotected communication [2]. but that’s the issue for users, who use starttls, because by using it, they choose to get email service work at any price, even sacrificing the connection security for just having their email work. and that will surely happen if the mta doesn’t provide ssl at least at that time. on the other hand, if users configure mua to use ssl/tls method and specify only separate selection of methods to provide end-to-end email traffic security 52 special network port for encrypted communication for smtp and imap/pop3, then the user can be sure, that at least mua-mta connection is encrypted and secure. asnet-am memebers are urged not to use starttls, but use ssl/tls instead for mua-mta connection [3]. 3. mta-mta communication in case of mua-mta communication the decision to use encrypted communication can be freely made by the end user (the owner of mua). but in case of mta-mta communication, which is the next step of forwarding the email message, it is out of the end user control. that part of the chain is to be properly configured by the administrator of mta. unfortunately, today most of mtas are accepting non-encrypted connections from other mtas for backward compatibility. it means that currently we can’t be sure that our email traffic passes the internet securely. as described above for mua-mta connection regarding starttls method is also true for mta-mta connection. here also use of starttls can be treated as not reliable and vulnerable to man-in-the-middle (mitm) and encryption downgrade attacks. thus, starttls for mta-mta connection does not guarantee either message confidentiality or proof of server authenticity. a brief description of the security issue with starttls mechanism follows. when a starttls-enabled mta wants to establish an smtp session with another mta, it first initially asks the remote mta if it supports ssl or not. and that process is not encrypted. so if an attacker intercepts this unencrypted communication and alters the handshaking process to trick the original mta into believing that the remote mta doesn't support encrypted communication, it can trick original mta to use non-ssl communication, i.e., perform encryption downgrade, even in case the real remote mta can talk ssl. latest developments and research in this area are trying to improve the situation. for example, the new smtp mta strict transport security (sts) mechanism is now being actively developed by google, yahoo!, microsoft, linkedin and other big companies as an internet-draft document [4]. smtp mta sts has been designed to enhance the email communication security. this new proposal has been recently submitted to the internet engineering task force (ietf). the primary goal of smtp sts is to prevent mitm attacks that have compromised past efforts like starttls at making smtp a more secure protocol. the use of smtp mta sts would force mta-mta communication to be always encrypted. smtp mta sts mechanism will enable administrators of mtas to:  declare mtas ability to receive tls-secured connections  declare particular methods for certificate validation  request that sending mta report upon  and/or refuse to deliver messages that cannot be delivered securely. smtp mta sts can protect mta-mta communication against mitm attacks. it is designed to rely on certificate validation process via tls identity checking. the new email security standard will check if recipient mta supports smtp mta sts and has valid and upto-date encryption certificate published in its dns zone. if it does successful encrypted mtamta communication will take place and email traffic will securely pass on. otherwise, the connection will be dropped and notification about the reason will be generated. of course, smtp mta sts is an attempt to improve the situation where starttls fails. but since the smtp mta sts mechanism is only a draft proposal right now, we need to wait a. petrosyan and g. petrosyan 53 for it to become usable. but even before that almost any mta can be configured to strictly use ssl. for example, postfix mta has an approptiate option ‘smtpd_tls_security_level’ [5], which can be turned on and set to the value ‘encrypt’. this way administrator of mta can enforce the use of tls, so that the postfix mta accepts no mail without encryption, by setting "smtpd_tls_security_level = encrypt". unfortunately, this will bring many problems in a real mta, because many mtas are not able to talk ssl today. so much of email traffic will just be dropped. that is why it is currently not recommended to have such configuration in case of a publicly-referenced mta [8]. in postfix mta default configuration this option is off by default and should only seldom be used. example: /etc/postfix/main.cf: smtpd_tls_security_level = encrypt 4. conclusion according to the investigations presented above it becomes clear, that currently there is no way to achieve total end-to-end security of email traffic. for mua-mta communication part email traffic security currently can be obtained, but it mostly depends on the mua correct configuration. best practice configuration discussed above is important to be used, i.e., using strict transport security measures both for incoming and outgoing email traffic, but avoiding the use of starttls mechanism, to be sure encryption always takes place. asnet-am members are strongly recommended to use only "ssl/tls" method, when configuring muas. for mtamta communication part email traffic security currently is in the state of development until the smtp mta sts mechanism becomes a standard and will be implemented at least by the major parties managing the email traffic in the internet. it can be expected then to provide proper endto-end email traffic security. references [1] rfc4949 man-in-the-middle (mitm) attack. https://tools.ietf.org/html/rfc4949 [2] rfc7435 opportunistic security: some protection most of the time. [online]. available: https://tools.ietf.org/html/rfc7435 [3] a. petrosyan, e. prokhorenko and m. khachatryan, “securing e-mail service in asnetam network”, proceedings of the conference csit’2015, yerevan, pp. 249-250, 2015. [4] smtp mta strict transport security. internet-draft, [online]. available: https://tools.ietf.org/html/draft-ietf-uta-mta-sts-01 [5] enabling tls in the postfix smtp server, . [online]. available: http://www.postfix.org/tls_readme.html#server_cert_key submitted 04.07.2016, accepted 12.11.2016. https://tools.ietf.org/html/rfc7435 https://tools.ietf.org/html/draft-ietf-uta-mta-sts-01 http://www.postfix.org/tls_readme.html%23server_cert_key selection of methods to provide end-to-end email traffic security 54 էլ. փոստի տվյալների ամբողջական հոսքի անվտանգության մեթոդների ընտրություն ա. պետրոսյան և գ. պետրոսյան ամփոփում այս հոդվածում նկարագրված հետազոտության նպատակն է փնտրել այնպիսի մեթոդներ, որոնք կապահովեն էլ. փոստի տվյալների փոխանակման ամբողջական անվտանգությունը: ինչպես հայտնի է էլ. փոստի ծառայությունն ի սկզբանե անվտանգ չէ: այդ պատճառով վերջնական օգտագործողները և ծառայությունների մատակարարները կարող են կիրառել հնարավորինս շատ հասանելի էլ. փոստի տվյալների ամբողջական հոսքի անվտանգության ապահովման մեխանիզմներ: հոդվածում ներկայացված են այդ բնագավառում վերջին մշակումները և հետազոտությունները, ինչպիսիք են՝ smtp mta strict transport security (sts) և smtp tls reporting: այս հոդվածում ներառված են պաշտպանության մեթոդների լավագույն կարգավորումների առաջարկությունները՝ ինչպես mail user agent (mua)-ների, այնպես էլ mail transfer agent (mta)-ների համար: առաջարկությունները հիմնականում նախատեսված են հայաստանի ակադեմիական գիտահետազոտական կոմպյուտերային ցանցի (asnet-am) անդամների կողմից էլ. փոստի ծառայությունից անվտանգ օգտվելու համար: выбор методов обеспечения безопасности трафика электронной почты «из-конца-в-конец» а. петросян и г. петросян аннотация описанные в статье исследования имеют цель поиска способов обеспечения безопасного трафика электронной почты «из-конца-в-конец». как известно, служба электронной почты не является безопасной по умолчанию, поэтому необходимо, чтобы как конечные пользователи, так и провайдеры почтовых услуг реализовывали как можно больше доступных механизмов защиты обеспечения безопасности трафика электронной почты «из-конца-в-конец». в статье представлены последние разработки и исследования в этой области, такие как smtp mta strict transport security (sts) и smtp tls reporting . статья содержит рекомендации по выбору наилучшей конфигурации методов защиты как mail user agent (mua), так и mail transfer agent (mta). рекомендации в основном ориентированы на членов академической научно-исследовательской компьютерной сети армении (asnet-am) для безопасного использования службы электронной почты сети asnet-am. microsoft word gayane.doc ìàòåìàòè÷åñêèå âîïðîñû êèáåðíåòèêè è âû÷èñëèòåëüíîé òåõíèêè 31, 150—157, 2008. 150 применение математической статистики в моделировании геохимических систем на примере северо-восточного побережья бассейна озера севан гаяне а. минасян институт проблем информатики и автоматизации нан ра e-mail mgaya@sci.am аннотация проанализированы распределения содержаний химических элементов в геологических образованиях бассейна озера севан на основе построения статистических моделей геохимических данных. рассматривается содержание групп химических элементов: ni, co и cr, ti, v, в породах северо-восточного побережья оз. севан. построены и изучены модели нормального распределения содержаний кобальта, логнормального распределения содержаний хрома, корреляция содержаний титана, ванадия и никеля, кобальта в ультраосновных породах и модели стандартного отклонененя элементов в породах. получено, что распределение элементов подчиняется логнормальному закону; имеется корреляционная связь между парами ni-co, v-ti, и ее отсутствие для cr с четырьмя другими изученными элементами; cr, co и ni дают наибольшие концентрации в дунитах, а ti и v – в габбро и кварцевых диоритах. литература 1. а. а. ярошевский. применение математики в геохимии: некоторые типы задач и методы решения. московский государственный университет им. м.в. ломоносова. http://journal.issep.rssi.ru/page.php?year=1996&number=7&page=67 2. дж.девис. статистика и анализ геологических данных. москва, издательство “мир”, 1977. 3. элементарные понятия статистики. http://www.statsoft.ru/home/textbook/esc.html 4. с. а. айвазян, и. с. енюков, л. д. мешалкин. прикладная статистика. основы моделирования и первичная обработка данных. москва. финансы и статистика, 1983. 5. º. ð³ñáõãûáõýû³ý, î. ô³½³ýãû³ý, ü. ø»ëñáåû³ý, ¸. ²ë³ïñû³ý, ø. ð³ñáõãûáõýû³ý, ø. ê³ñ³ïû³ý, ð. þ³ñáõùû³ý: ð³í³ý³ï³ýáõãûáõý ¨ ïçñ³é³ï³ý íç׳ﳷñáõãûáõý, ºñ¨³ý, 2000: 6. а.а.беус, с.в.григорян, м.т.ойзерман, п.г.чолакян, а.а.стояновский. руководство по предварительной математической обработке геохимической информации при поисковых работах. москва, издательство “недра”, 1965, 120 с. г. минасян 151 ø³ã»ù³ïçï³ï³ý íç׳ﳷñáõãû³ý ïçñ³éáõùá »ñïñ³ùçùç³ï³ý ñ³ù³ï³ñ·ç ùá¹»é³íáñù³ý ñ³ù³ñ ꨳý³ é×ç ³í³½³ýç ñûáõëçë-³ñ¨»éû³ý ³÷ç ûñçý³ïáí ¶. øçý³ëû³ý ²ù÷á÷áõù î³ï³ñí³í ¿ ꨳý³ é×ç ³í³½³ýç »ñïñ³µ³ý³ï³ý ³é³ç³óáõùý»ñáõù ùçùç³ï³ý ï³ññ»ñç å³ñáõý³ïáõãûáõýý»ñç µ³ßëù³ý í»ñéáõíáõãûáõý` »ñïñ³ùçùç³ï³ý ïíû³éý»ñç íç׳ﳷñ³ï³ý ùá¹»éý»ñç ï³éáõóù³ý ùççáóáí: ¸çï³ñïíáõù ¿ ꨳý³ é×ç ñûáõëçë-³ñ¨»éû³ý ³÷ç ³å³ñý»ñáõù ùçùç³ï³ý ï³ññ»ñç å³ñáõý³ïáõãûáõýý»ñç ñ»ï¨û³é ëáõùµá` ni, co ¨ cr, ti, v: î³éáõóí³í ¨ áõëáõùý³ëçñí³í »ý ñ»ï¨û³é ùá¹»éý»ñá` áõéïñ³ñçùù³ûçý ³å³ñý»ñáõù co -ç å³ñáõý³ïáõãû³ý ýáñù³é µ³ßëáõùá, cr ç å³ñáõý³ïáõãû³ý éá·ýáñù³é µ³ßëáõùá, ti, v ¨ ni, co å³ñáõý³ïáõãûáõýý»ñç ïáé»éû³óç³ý ¨ ³å³ñý»ñáõù ï³ññ»ñç ï³ýáý³ó¨ ß»õáõùý»ñá: êï³óí»é ¿, áñ µ³ßëáõùá »ýã³ñïíáõù ¿ éá·ýáñù³é µ³ßëù³ý ûñ»ýùçý, ·áûáõãûáõý áõýç ïáé»éû³óçáý ï³å ni-co, vti ½áõû·»ñç ùçç¨ ¨ ³ûý µ³ó³ï³ûáõù ¿ cr -ç ñ³ù³ñ áõëáõùý³ëçñíáõ ãáñë ï³ññ»ñç ¹»åùáõù, cr-á, co-ý ¨ ni-ý ï³éçë »ý áñáß ïáõï³ïáõùý»ñ ¹áõýçïý»ñáõù, çëï ti-ý ¨ v-ý - ·³µµñáý»ñáõù ¨ ùí³ñó³ûçý ¹çáñçïý»ñáõù: microsoft word 9_grigor_zargaryan.doc mathematical problems of computer science 39, 72--80, 2013. 72 verification environments for usb controller grigor y. zargaryan institute for informatics and automation problems of nas of ra e-mail: grigorzargaryan@gmail.com abstract the complexity of electronic devices with the everyday growing requirements is constantly increasing. sowtfare/hardware (sw/hw) integration, validation and reducing time to market have become one of the major bottlenecks in the design and verification flow. this paper presents the main ways of design and verification flows with their advantages and disadvantages for usb 3.0 controller. it discusses the design flow with the combination of simulation and prototype-based design and presents а simulation-based verification and also two types of field-programmable gate array (fpga) based verification environments with their advantages and disadvantages. the design done with this flow will enable system on a chip (soc) designers to develop a high-quality usb 3.0 silicon solution to meet the growing market demands in a timely manner. keywords: universal serial bus, verification, simulation-based verification, fpga-based verification. 1. introduction as electronic devices have already combined a lot of different functions, the market requires more and more new functionalities. however, it poses problematic issues such as a long development time and a hard design verification due to the increasing chip complexity. another challenge also rises to verify the complex design efficiently and timely under the situation the time-to-market is decreasing exponentially [1]. the dependencies of hardware and software result in an intricate relationship between the different company types. fig. 1 shows the research results on failure types on the failing first silicone provided by collett international [2]. verification techniques can be classified into a simulation-based method and an emulation-based method [3]. in the simulation-based method, even if it has an advantage of being able to verify the design exactly and minutely, an excessively long simulation time is required. by using register transfer level (rtl) hw models simulation, the verification times have increased to the level when they now take up to 70% of the device design time [4, 5]. in the emulation-based method, since it needs a certain emulation system such as a fpga board, the board development time is added to the design verification time [6]. it is necessary to make a useful environment and to establish an efficient verification methodology for fpga-based verification. when rtl is largely verified and stable, the software development ramps up. it is split between os support and porting, a low-level software development and a high-level application software g. zargaryan 73 development. all the software development efforts consume 40% of the total cost for 27 months design[7]. when amortizing development and production cost onto expected sales, this project reaches break even after 34 months, i.e. seven months after the product launch but almost three years after the starting product development. the challenges in this example are that we have to predict nearly three years in advance what is going to be sold in high-volumes in order to specify our chip. how can this almost intolerable situation be made easier? the answer is to “start software sooner”. if software development and validation started seven months earlier and subsequently the time to break even would have been reduced by five months. additional revenue gain could be expected over the production volume due to submit to market design earlier than other similar products. so it is extremely important to decide most appropriate development and verification flow to ensure the product success on the market and deliver high-quality, verified designs. figure 1. failure types on the first silicone. 2. simulation-based verification the simulation-based verification model is one of the most popular and effective ways for functional verification. the simulation-based design requires more than just “a simulation tool”. depending on the complexity of the system, the design process may require many tools, many ways to link the tools together. the simulation environment that supports the design process should be flexible enough to disaggregate a complex system into any number of smaller pieces and conversely to aggregate independent objects into a complete representation of the system. it should, as much as possible, reduce the degree of complexity for the concept designer. this means that a graphical interface is essential, that energy couplings between the system components should be automatically and transparently handled, that the existing models should be reusable and that the rapid iteration of the design cycle and incremental refinement of the system should be supported on a group-wise basis [8]. several approaches have been developed to reduce the simulation time and to increase the verification quality. one of the methods to reduce the device design time is the transaction level modeling (tlm) [4]. it can be used for hw/sw modeling, co-design and co-verification. another reason to use tlm model is the availability to start sw development at an early stage. verification environments for usb controller 74 2.1 simulation-based verification environment the verification environment is needed, which will allow us to run simulations and fix issues. usb host/device controller verification environment consists of a device under test (dut), usb verification ip (vip) and pcie vip (fig. 2). it is possible to use other vip instead of pcie vip, such as axi/ahb/amba depends on what bus will be used next to the controller. if the design is implemented on the fpga or on the asic, then dut can be the top for this design, including usb controller and other necessary modules. verification runs as follows: each transaction is a combination of a request and response. vip starts the transaction by sending a request and waiting for an appropriate response. simulation tests pass when all the required requests are sent and the responses are received.. vip generating test vectors using systemc, which allows to short verification time. figure 3 provides a part of systemc code which compares the data transferred between the device and the host using usb simulation environment. after running simulation the .vpd file can be used for debugging. the dve tool provides graphical user interface which allows performing debugging in effective ways. figure 4 shows ssrxp, ssrxn, sstxp, sstxn and ssclk states of the signals after performing simulation. the ssrxp, ssrxn, sstxp and sstxn signals are external connections and responsible for date transfer on super speed mode. figure 2. usb verification environment. figure 3. system c code performing comparision of transferred data. pcie vip dut usb vip $display(" data comparison(tx and rx buffers) after transfer is done... @", $time); first = 0; for(j=0; j < dut_number_of_trbs; j=j+1) begin for(i=0; i < dut_bytes_per_trb/4; i=i+1) begin tmp_data32 = start_data + i*step; k = (j*1280) + i; tmp_data2 = {32'h0, mem[3+4*k], mem[2+4*k], mem[1+4*k], mem[4*k]}; if(first == 0) $display("first location-a: rx buffer data= %h; tx buffer data= %h", tmp_data2[31:0], tmp_data32, $time); first = 1; if(tmp_data2[31:0] !== tmp_data32) begin fail = 1; $display("data mismatch; expected= %h; received= %h @%t", tmp_data32, tmp_data2[31:0], $time); end end end end if(fail==0) print_banner(" comparison passed "); else print_banner(" comparison failed "); end g. zargaryan 75 figure 4. simulation diagrams 3. fpga-based verification fpga-based prototyping accelerates the creation of an asic prototype with high-speed hardware prototyping systems including a software flow for the conversion of asic rtl into one or more fpga ics. fpga-based prototypes provide cycle-accurate, high-performance execution and real world interface connectivity prior to tape-out of test chips. in the effort to reduce time-to-market (ttm) engineering organizations continue to seek ways to develop hardware and software in parallel. advanced asic prototyping techniques enable a more parallel development methodology. and firms, which have achieved more concurrent engineering practices, have not only reduced the time to product introduction, but additionally reduced product support & maintenance effort during the product’s time in market due to higher quality. the sooner the real software development begins, the more feasible it will be to make progress on the integration & test, and validation phases prior to the tape-out milestone (fig. 5). from the aspects of debugging and control capabilities, the virtual platforms or any simulation allow much easier ways than fpgas. but on the other hand, fpga allows much debugging and control capabilities than the actual silicon provides when available. to allow debugging on fpga boards before running synthesis it is necessary to define which signals will be used for debugging. also additional tools are required to grab debugging signals from fpga platform. if some additional wires or signal are needed for debugging which are not defined before the synthesis, it will be necessary to define and rerun the synthesis again. also it is recommended to start fpag implementation after rtl verification has stabilized due to the efforts of mapping the rtl to fpga-based prototype. for the same reason it is not useful for hardware/software co-development. prototyping provides powerful methods for validating the design of hardware and software in models. fpga-base prototyping is specifically useful during the hardware and software integration. nowadays fpga technologies allow high density, high speed, broad bandwidth, low-voltage, low-power and low cost. there are built-in ip cores, which can extend the application area and shorten the cycle of r&d. more functional cores like networking, audio, video and image can be integrated into a single fpga chip. verification environments for usb controller 76 …wit h advanced asic prototyping methods time in market higher productivity higher quality earlier ttm product support & maintenance hw sw integration & test time to market d ev el op m en t e ffo rt time in market time to market d ev el op m en t e ffo rt hw sw integration & test product support & maintena & test traditional flow figure 5. terms reduce time-to-market 3.1 fpga implementation the fpga-based verification environment consists of pc, pcie cards, fpga board and usb physical layer (phy) (fig. 6). an appropriate os will be loaded on the pc with the controller driver and high level applications. the pcie bus will make a connection between pc and fpga board. fpga board is connected to usb phy with a parallel interface, such as pipe3, ulpi and utmi. usb phy layer contains an analog receiver, transceivers and convert sequential data into parallel. this type of environment will allow easy software debugging. figures 7 and 8 present host’s and device’s logical components. host includes the following: usb host controller, aggregate usb system software (usb driver, host controller driver and host software), client. device includes the following: usb bus interface, usb logical device, function[9]. on the market there are few companies that provide tracers for packet level debugging. there are pcie and usb tracers which can be used for more effective debugging. figure 9 shows an example of usb 3.0 trace recorded on this type of environment. figure 6. usb controller implementation pc fpga usb phy g. zargaryan 77 figure 7. usb host’s logical components figure 8. usb device’s logical components figure 9. usb 3.0 trace example. 4. fpga-based embedded systems embedded system often refers to the non-pc systems which combine hardware and software design. in general, it contains embedded micro-processor (8-bit, 16-bit or 32 bit), storage and peripherals, embedded os (real-time and multi-task) and applications (fig. 10). embedded systems have some characteristics which differ from other computing systems [10].  small system kernel.  specific-functioned.  real-time os. in terms of embedded hardware, its core component is the embedded microprocessor. at present there are over 1,000 kinds of embedded processors in the world and the popular architectures are more than thirty, in which intel mcs-8051 is ever the overwhelming majority. in recent years the small volume, high performance and low power consumption become dominant factors of embedded system design considerations. the professional intellectual property (ip) core providers like arm, mips corps. offer high-quality embedded cores to semiconductor manufacturers, by which all kinds of chips on different devices applied to diverse areas, are widely produced. client sw usb system sw usb host controller host function usb logical device usb bus interface physical device verification environments for usb controller 78 jta g programmable logic on-chip ram flash embedded processor cache uart external bus interface fpga serial port external devices target board sw code hw binary file 4.1 fpga-based embedded environment embedded design flow combine embedded sw flow and fpga hw flow. the hardware design flow consists of standard fpga design steps such as design entry, simulation, synthesis and implementation. the software design flow consists of c code, c/c++ compilation to linker and debugger. generated hw binary file for fpga configuration and sw code written into the board through jtag. this type of setup verification environment also requires usb phy, and consists of target board and usb phy (fig. 11). figure 10. design diagram of fpga-based embedded system. figure 11. fpga-based embedded system implementation. 5. conclusion simulation-based verification can be used to start developing usb controller. developing systemc level models and parallel workaround on hw and sw can allow shorter time to market. systemc models usage allows fast simulation time. after rtl has stabilized fpga-based or fpgabased embedded system can be created. prototyping can help most in the following sw validation and integration tasks: os configuration & installing, kernel space debugging, on-chip debugging, user space debugging, unit testing, system testing, field diagnostics and lab diagnostics. fpga-based prototypes in particular provide the most help in the highlighted area enabling: physical-layer interface compatibility checking at-speed debug regression testing multi-core integration in-field tests and finally prototyping usb controller allows testing sw and hw with real world usb phy and with real world usb devices. testing with real world devices and real speed will allow silicone success on first tape out. it is very hard to imagine more useful environment than the real world testing environment. real world testing at usb 3.0 speeds helps to verify the architecture, such as memory management and interoperability tests for usb 3.0 standard compliance. finally, the fpga validation platform can also be used for usb implementers forum certification of a prototype design and windows hardware certification kit by microsoft. new driver stacks are required to handle the faster usb 3.0 speeds, and simply extending usb 2.0 architectures to support usb 3.0. universality of the usb protocol requires that hosts are tested with hundreds of usb 2.0 devices and all available usb 3.0 devices. fpga-based embedded system usb phy g. zargaryan 79 real world prototype can be present in many technical exhibitions. also it can be given to customers to try if it meets their needs, to try different configurations of rtl, different modes, etc. references [1] e. jimenez, “challenges in system on chip verification”, international workshop on microprocessor test and verification, pp.52-60, 2006. [2] source: collett international research, inc. [3] c. pixley, et al., “functional verification 2003: technology, tools and methodology”, international conference on asic, vol.1, pp.1-5, 2003. [4] s. swan, “systemc transaction level modelsand rtl verification”, proc. 43rd acm/ieee design automation conference, pp. 90-92, 2006. [5] s. tasiran and k. keutzer, "coverage metrics for functional validation of hardware designs", ieee design & test of computers, vol. 18, no. 4, pp. 36-45, 2001. [6] y. lin, et al., “versatile pc/fpga-based verification/fast prototyping platform with multimedia applications”, ieee transactions on instrumentation and measurement, vol.2, pp.1490-1495, 2007. [7] a.doug, l. austin, fpga-based prototyping methodology manual, published by synopsys, inc., mountain view, ca, usa, 2011. [8] r. a. dougal, “design tools for electric ship systems”, ieee electric ship technologies symposium, pp. 8-11, philadelphia, pa, july 2005. [9] usb 2.0 specification, april 27, 2000, www.usb.org. [10] f. vahid and t. givargis, embedded system design – a unified hardware/softwareiintroduction, john wiley & sons, inc., pp. 1.1-1.2, 2002. submitted 20.12.2012, accepted 21.02.2013. ð³ù³åçï³ýç ñ³çáñ¹³ï³ý ¹áõç õ»ï³í³ñáõ ñ³ý·áõûóç ëïáõ·ù³ý ùçç³í³ûñ»ñá ¶. ¼³ñ·³ñû³ý ²ù÷á÷áõù ¾é»ïïñáý³ûçý ë³ñù³íáñáõùý»ñç µ³ñ¹áõãûáõýá ¨ ýñ³ýó íñ³ ¹ñí³í å³ñ³ýçý»ñá ûñ»óûñ ³×áõù »ý: ²å³ñ³ï³-íñ³·ñ³ûçý ñ³ù³¹ñáõùá, ëïáõ·áõùá ¨ ßáõï³ ¹áõñë ·³éáõ ññ³ï³åáõãûáõýá ý³ë³·íù³ý ÷áõéáõù ¹³ñó»é »ý ï³ñ¨áñ³·áõûý å³ñ³ýçý»ñ: ²ßë³ï³ýùáõù ý»ñï³û³óí³í »ý ñ³ù³åçï³ýç ñ³çáñ¹³ï³ý ¹áõç (ðð¸) õ»ï³í³ñáõ ñ³ý·áõûóç ý³ë³·íù³ý ¨ ëïáõ·ù³ý ÷áõé»ñի ñçùý³ï³ý áõõçý»ñá` çñ»ýó ³é³í»éáõãûáõýý»ñáí áõ ã»ñáõãûáõýý»ñáí: î³ëí³í ý³ë³·íù³ý ¨ ëïáõ·ù³ý ³éï³ íç׳ïçó` ï³ñµ»ñ ùá¹»éý»ñ áõý»ý ï³ñµ»ñ արդյունավետություն: ð³ßíç ³éý»éáí ëïáñ¨ ý»ñï³û³óí³íá` ï³ñ»éç ¿ ë»õù å³ùï»ïý»ñáõù ý³ë³·í»é µ³ñóñ³ï³ñ· ðð¸-ç õ»ï³í³ñáõ ñ³ý·áõûó: verification environments for usb controller 80 среда проверки управляющего узла универсальной последовательной шины г. заргарян аннотация сложность электронного оборудования и предьявляемые к нему требования растут с каждым днем. аппаратно-программное сопоставление, проверка и актуальность выхода на рынок стали важнейшими составляющими на этапе его проектирования. в работе представлены основные пути этапа проектирования и проверки управляющего узла универсальной последовательной шины (упш) со всеми своими преимуществами и недостатками. в зависимости от способа проектирования и проверки, различные модели имеют различную эффективность. с учетом нижеизложенного можно в сжатые сроки проектировать высококачественные управляющие узлы упш. microsoft word 5_david beybutyan.doc mathematical problems of computer science 39, 40---47, 2013. 40 proposing novel schema for downstream using multiple olts in epon system davit v. beybutyan institute for informatics and automation problems of nas of ra e-mail: beybutyandavid@gmail.com abstract the paper proposes a new schema for downstream using multiple olt mechanisms in epon system. in the proposed mechanisms a new dba algorithm is used, so-called cdba-controlled dynamic bandwidth allocation, this controls the assigned bandwidth for each olt. we proposed an architecture which also comforts the system reliability; in case of network failure another olt will be able to recover the system in a short period of time. it also allows the unused bandwidth of one olt to be used by the other olt. in the future, companies can provide their services without deploying another pon. keywords: multimedia services, passive optical network, multi-optical line terminal, epon, controlled dynamic bandwidth allocation. 1. introduction network operators have already started the deployment of the new multimedia services such as video-on-demand (vod), high-definition television (hdtv), peer-to-peer (p2p) live video streaming and iptv. by the predictions of cisco, these markets will grow exponentially in the next few years [1]. the development of aforementioned multimedia services will represent 80 percent of the global consumer traffic by 2015 [2]. the financial investments that are needed for updating the infrastructures to support those services put more pressure on the operators who need to recover the investment within a very short period of time. on the other hand, clients are more demanding and require better services at lower costs. therefore, competition among operators is becoming increasingly strong. for each new service that is being introduced, one of the main questions is: what is the best way of serving both the client and the operator needs? the biggest issues of the companies in the future that will provide video-on-demand, p2p and other on demand services will be the cost of the equipment and the provision of high quality services at low costs for the users. in vod services, large varieties of contents are available to the customers who are not supposed to access the same content in the exact same instant and broadcasting information is no longer a solution. contents are requested on demand, and user experience and network performance are affected by both the user traffic profiles variability and the video server location. great number of users requesting videos and video servers far from customer’s access network would result in long waiting delays and a bad experience for a user, which might not find this new service very interesting. d. beybutyan 41 network operators are facing the challenge of carrying the large data-centric traffic with tighter timing and quality-of-services (qos) requirements for expanding upon the existing network infrastructures [3]. in the access network domain, ethernet passive optical networks (epons) are regarded as one of the best solutions for the access networks due to its simplicity, high data rate, and low-cost [4]. an epon consists of an optical line terminal (olt) located in the central office (co), a feeder fiber, passive splitter (ps) and optical network units (onus). the olt connects a group of related onus over point-to-multipoint topologies to deliver broadband packets, and it also reduces costs relative to maintenance power. in addition, the optical liner terminal (olt) has the entire channel bandwidth to broadcast the control data packets and messages to each optical network unit (onu) because of the directional properties of the splitter. in epon system, the data sent from olt toward onus are split at a passive splitter and simultaneously broadcasted to each of onus. it should be noted that each onu receives all downstream traffic designated to not only itself but also every other onu. in another way, we can say that the data received by non-destination onus is discarded. in the upstream transmissions from the onus to the olt, a polling technique is used as a multiple access technology to allow the onus to share the same optical line. the onus are assigned by time slots, and they are allowed to transmit data only during the assigned time slots. despite all the advantages of epon, there are still many challenges in providing multimedia services in the epons. it might be about how the network operators can guarantee the quality of service (qos) while achieving sufficient profits and revenues. more video content will be sent over unicasts, which will increase the bandwidth and due to the evolution from hdtv formats towards super hd and ultra hd and 3d formats the demands of the bandwidth are even higher [2]. providing higher capacities than the existing pons will not solve all the existing problems. there must be a proper infrastructure to provide the protection to this new high capacity network. so far, to our knowledge not so many papers discuss the downstream in epon systems: mostly papers are concentrated in the upstream but, however, the present-day users care more about the downstream because of growing on demand for services, and users need more and more traffic. in this paper, we propose a new architecture of pon that has multiple olts. all olts are connected to each other with a cable, and they all are also connected to the core network thus assuring the network confidence. in case of network failure, another olt will be able to recover the system in a short period of time. the proposed architecture can be used to solve another important problem of the network. nowadays, only a one vendor can provide services to the users connected to the pon network [5, 6, 7]. it means that the users have no freedom to choose one from multiple service providers and services such as vod, ip television, hdtv and ip telephony are differed from vendor to vendor. moreover, the proposed architecture can make new possibilities for the users who like online gaming and sharing files. using this network, the companies which are going to provide their own services over the same pon system, as well as the users can trek from one service provider to another according to their preference. the proposed architecture can also support not only localtraffic redirection but also provide better qos, particularly for p2p video-streaming. figure 1 shows the proposed architecture that can efficiently support p2p multimedia services. the main purpose of this paper is to design an architecture that will be reliable for the next generation optical networks, and the internet providers can easily provide p2p vod services. the providers can deliver their own services in the same pon system using the proposed network architecture. the rest of the paper is organized as follows: section ii introduces the related works with the existing bandwidth allocation algorithms for single-olt pon system. section iii introduces new dba schemes for downstream transmission. in section iv, the system performance is evaluated by simulation results. section v discusses the lost or wasted bandwidth and section vι concludes our paper. proposing novel schema for downstream using multiple olts in epon system 42 fig. 1. proposed architecture 2. related works in pon systems an important factor is the dynamic bandwidth allocation; each onu’s upstream bandwidth is decided by assigning time slots specified by the olt in a unit time [8]. in general, the bandwidth allocation algorithms have a main impact in minimizing latency, meeting quality of service guarantees, improving the fairness and requirement of buffer size in upstream direction. bandwidth allocation algorithms can be classified into two main groups; the first one is called dynamic bandwidth allocation (dba) and the second one fixed bandwidth allocation (fba) algorithms. the performance of epon by using a fixed bandwidth allocation algorithm is studied well, where all traffics are considered to a single class [9]. as it is shown in figure 2 the scheme is very simple and continuously grants the maximum window to all onus. as a matter of fact the cycle time tcycle is constant for all kinds of traffic loads. fig. 2. fixed bandwidth allocation scheme the main disadvantage of this algorithm is that the light loaded onus will under-utilize their allocated bandwidth leading to enlarged delay to other onus and finally deteriorate the throughput and bandwidth utilization of the system. dba is suitable for burst traffic such as voip and ftths and they can provide flexible bandwidth sharing of allocation among the users. so far different types of dba algorithms have been developed to improve the bandwidth utilization and to adopt the current demand of vast traffic. figure 3 shows the ls bandwidth allocation scheme. meanwhile the granted window is based on the requested window, the cycle time tcycle is variable. as it is shown in the figure, the cycle time for the first cycle is tcycle1= tmax, because every onu requests for maximum bandwidth w [max]. on the other hand, cycle time in the second cycle is tcycle2= tmaxts, here, ts is the cycle time saving due to light-loaded onus. the main advantage of this scheme is that it reduces the d. beybutyan 43 bandwidth wastage by granting smaller bandwidth to the light-loaded onus. nevertheless, one limitation of this algorithm is that it makes tcycle too small, which will result in lower bandwidth utilization because of constant guard time for every two successive onus. fig. 3. dynamic bandwidth allocation scheme these dba algorithms are proposed only for the single-olt pon and should require a guard time between every two consecutive onus to avoid data overlapping. due to this guard time, some bandwidth wastage problem is observed. 3. bandwidth adjustment approach among multiple olts as it is already known in a single-olt system, a number of onus share the upstream bandwidth while a single olt utilizes complete downstream bandwidth. thus, the bandwidth allocation algorithms have evolved only for allocating the upstream bandwidth among onus. multiolt pon architecture is a rather complex architecture than the current single-olt pon. in our proposed multi-olt passive optical network architecture several olts need to share the downstream bandwidth. since the proposed architecture already has multi-olts, thus in this state the downstream bandwidth allocation scheme is needed, too. the present fixed bandwidth allocation (fba) is not suitable for flexibly maintaining bandwidth among the multiple olts. the fba is suitable only for architectures with a single olt with several onus. fba will always provide a fixed percentage of bandwidth, which means a huge waste of capacity when traffic generations of the olts are far different from the estimated values. hence, in this paper a new dba is proposed, socalled controlled dynamic bandwidth allocation (cdba) scheme for the downstream transmission. we named it controlled dynamic bandwidth allocation because it guarantees the assigned bandwidth for each olt and it also allows the unused bandwidth of one olt to be used by the other olts. hence, it can increase the efficiency of the proposed architecture without losing the unused bandwidth. the amount of delivery is limited so that the assigned bandwidth at each olt is controlled. figure 4 shows the algorithm of cdba. proposing novel schema for downstream using multiple olts in epon system 44 fig. 4. algorithm of controlled dba 4. performance evaluation table ι. system parameters parameter value number of olts in the system 2 number of onus in the system 16 or 32 downstream/upstream link capacity 1gbps olt-onu distance 10-20 km buffer size 10 mb maximum transmission cycle time 2ms guard time 5µs computation time of dba 10µs control message length 0.517µs the system model is set up in opnet simulator with 2 olts and 16 or 32 onus. table i shows the system parameters used in the study. the distance from olt to onu is 10-20 km, the downstream and upstream link capacities are equally 1 gbps and each onu has 10 mb of buffer size. in the simulation the maximum transmission cycle time is 2 ms with 5µs of guard time. fig. 5. downstream delay with assigned bandwidth ratio of 30%-70% d. beybutyan 45 fig. 6. downstream throughput with assigned bandwidth ratio of 30%-70%. figure 5 and figure 6, accordingly, show the downstream delay and downstream throughput with the assigned bandwidth ratio. in the downstream performance the oltm through fba with 50% indicates an average delay of oltm. we can clearly see that the cdba facilitates olts and also guarantees the quality of service for oltm. moreover, in our proposed architecture we can also save the bandwidth in the feeder fiber. 5. lost bandwidth an imprecise bandwidth allocation results in wasteful resource allocation. the imprecise bandwidth allocation means that the olt is assigning too much or too little bandwidth in terms of the requested bandwidth to onus. when the system traffic load is too high and is greater than 50%, the lost bandwidth is reduced because there is no more available bandwidth which can be granted to the onus. figure 7 shows the lost bandwidth ratio versus the total number of the traffic loads. the simulations are completed for the proposed cdba and dba_vod. fig. 7. lost bandwidth ratio with traffic load the simulation results show that in the proposed architecture the lost bandwidth is reduced, when the traffic load is greater than 50%. additionally, it is scalable to large-size mesh access networks and achieves higher network survivability, faster recovery and reliability. proposing novel schema for downstream using multiple olts in epon system 46 6. conclusion the paper proposes a novel scheme for downstream using multiple olts in epon systems. in future the internet providers can use the proposed architecture for providing their own services without deploying another pon. using this network, the internet providers which are going to deliver their own services over the same pon system, as well as the users can trek from one service provider to another according to their preference. it provides protection facilities as well. moreover, they can deliver video-on-demand, p2p and other on demand services: their biggest issues which are the cost of the equipment and providing high quality services at low costs for the users can be finally solved. in the paper new dba schemes are introduced for downstream transmission. the new dba algorithm is one of the key factors to let the providers deliver their individual services in p2p vod systems. the simulation results show that the proposed multi-olt epon system saves the unused bandwidth in case of network failure, and if olt fails or feeder fiber damages, another olt will be able to recover the system in a short time: it can also accommodate 3% more traffic load than the single-olt pon without suffering any congestion and is cost-effective. at last the cdba and dba_vod ratio is quite good, which is the main aspect of saving the lost bandwidth while providing multimedia services through multiolt epon systems. acknowledgement i want to express my gratitude to professor i. s. hwang, as well as to a. nikoukar and a. t. liem for useful discussions and very important comments. i’d also like to express my special thanks to professor e. pogossian for his valuable comments and very important advice. references [1] cisco, “cisco visual networking index: forecast and methodology, 2010-2015,” white paper, 2011. [2] p. chanclou, a. cui, f. geilhardt, h. nakamura, and d. nesset, “network operator requirements for the next generation of optical access networks,” ieee network, vol. 26, no. 2, pp. 8-14, 2012. [3] i.tomkos, l. kazovsky, and k. i. kitayama, “next-generation optical access networks: dynamic bandwidth allocation, resource use optimization, and qos improvements,” ieee network, vol. 26, no. 2, pp. 4-6, 2012. [4] s. choi and j. park, “sla-aware dynamic bandwidth allocation for qos in epons,” ieee/osa journal of optical communications and networking, vol. 2, no. 9, pp. 773-781, 2010. [5] f. effenberger and j. kani, “trends in standardization of optical access networks in itu-t,” ieice transaction on communication, vol. e93-b, no.2, pp. 255-262, 2010. [6] f. effenberger, d. cleary, o. haran, g. kramer, r. d. li, m. oron, and t. pfeiffer, “an introduction to pon technologies,” ieee communications magazine, vol. 45, no. 3, pp. 517525, 2007. [7] d. payne and r. davey, “the future of fiber access systems,” bt technology journal, vol. 20, no. 4, pp. 104-114, 2002. [8] masaki tanaka, takashi nishitani, hiroaki mukai, seiji kozaki, and hideaki yamanaka, “adaptive dynamic bandwidth allocation scheme for multiple-service in 10g-epon system,” proc. of ieee icc, 2011. [9] g. kramer and b. mukherjee, “ethernet pon: design and analysis of an optical access network,” photonic network communication, vol. 3, no. 3, pp 307-319, 2001. submitted 12.12.2012, accepted 14.02.2013. d. beybutyan 47 հոսքի ուղղությամբ նոր սխեմայի առաջարկություն epon-ում բազմաթիվ olt-երի օգտագործմամբ դ. բեյբության ամփոփում հոդվածում հոսքի ուղղության համար առաջարկված է նոր սխեմա epon համակարգում` բազմակի olt մեխանիզմներ օգտագործելու միջոցով: առաջարկված մեխանիզմներում օգտագործվել է նոր դինամիկ թողունակության բաշխման (դթբ) ալգորիթմ, այսպես կոչված` կառավարվող դինամիկ թողունակության բաշխում (կդթբ), որը վերահսկում է հանձնարարված թողունակությունը յուրաքանչյուր օպտիկական գծի տերմինալում (օգտ): մենք առաջարկել ենք այնպիսի կառուցվածք, որը նաև ապահովում է համակարգի հուսալիությունը. ցանցի խափանման դեպքում մեկ այլ օգտ-ն ի վիճակի կլինի կարճ ժամանակահատվածում վերականգնել համակարգը: այն թույլ է տալիս նաև մեկ օգտ-ի չօգտագործած թողունակությունը օգտագործել մեկ այլ օգտ-ով: ապագայում ընկերությունները կկարողանան առաջարկել ծառայություններ` առանց լրացուցիչ պասիվ օպտիկական ցանցի տեղադրման: предложение новой схемы для потока передачи с использованием множественных терминалов оптических линий в пассивной оптической ethernet сети д. бейбутян аннотация в статье предложена новая схема для потока передачи с использованием механизмов множественных терминалов оптических линий (тол). в предложенных механизмах использован новый алгоритм распределения динамической пропускной способности (рдпс), так называемая контролируемая рдпс, которая контролирует заданную пропускную способность для каждого тол. мы предложили архитектуру, которая также обеспечивает надежность системы: в случае обрыва сети другой тол будет в состоянии восстановить систему за короткий промежуток времени. она также позволяет использовать невостребованную пропускную способность одного тол другим. в будущем компании могут предложить свои услуги без установления дополнительных пассивных оптических сетей (пос). начиная с начала 2000 года осуществляется внедрение ghis в здравоохранении, в рамках принятого проекта о реформирование информ mathematical problems of computer science 45, 27--34, 2016. determination of received unknown signal modulation type using higher order statistics or spectral correlation method eduard r. sivolenko institute of radiophysics and electronics of nas ra e-mail: e_sivolenko@yahoo.com abstract nowadays modulated signals are being used everywhere. so, modulation type recognition becomes more important for communication systems. in this paper we suggest a new method to discriminate between two modulated signals: amplitude modulated (am) and frequency modulated (fm). the simulation results are given below. the method is based on higher order statistics. in this paper we use bispectrum and triple autocorrelation function for signal modulation type recognition. keywords: hos, bispectral estimating, cumulants, phase coupling, object detection, automatic modulation type recognition, carrier suppressed am, fm. 1. introduction the technological world is moving forward rapidly. a lot of smart communication systems are developed. one of the main problems in communication is the received signal modulation type automatic recognition. received signal modulation type automatic recognition plays an important role in various applications of signal analysis. the modulation type is considered as a signal signature. it is very important for receiver to be able to detect the received signal modulation type with less or without any knowledge about the transmitted signal. modulation type automatic recognition is very important not only for civil but also for military applications such as useful information finding, interference identification and spectrum control, electronic warfare, control of communication quality, etc. different methods have been developed for the received signal modulation type automatic recognition. a lot of algorithms have been evolved for analogue and digital modulations [1, 2]. there are a lot of different detection algorithms which are based on the first and second order statistics such as autocorrelation function and 27 determination of received unknown signal modulation type using higher order statistics 28 fourier transform [3]. autocorrelation function and fourier transform are more effective for linear signals. besides, the first and second order statistics are providing information only about signal amplitude. meanwhile, many of the signals are highly non-gaussian, nonlinear processes. so there is a need of additional information frequencies and phases of components in power spectrum. that additional information can be provided by higher order statistics. higher order statistics can suppress gaussian noise and provide with useful information about the received signal. bispectrum, which is also known as third-order statistics, is known as a powerful tool for coupled phase’s information providing. as it is known the power spectrum can be found from the second ordered autocorrelation function using the wiener-khinchin theory [4]. by the same way the bispectrum can be found from the triple autocorrelation function using the wienerkhinchin theory. we use the bispectrum and third-order statistics for signal modulation type automatic recognition. in this paper deep analyses were done for two types of modulated signals, i.e., for amplitude and frequency modulation. these two modulations are the most interesting because in certain cases we have the same picks in specific frequencies in power spectrum. but power spectrum can provide information only about frequencies. it is very difficult to recognize which frequencies belong to the same modulation type. from this point of view it is very important to find additional information about the received signal for recognition modulation type automatically. this information can be provided by bispectrum. it can provide information about coupled phases which cannot do power spectrum because it is phase-blinded [5]. we use bispectrum to retain the phase information for the received signal modulation recognition. 2. bispectrum processing as it is mentioned above bispectrum is a very powerful tool. phase’s relationship finding is the main motivation for using bispectrum estimation in modulation type recognition tasks. besides, bispectrum is usually used for extraction of useful signal from noise. first we will consider the bispectrum properties for a real-valued stationary discrete process �x(m) (i)� with finite sample number 𝑖𝑖 = 1, … , 𝑁𝑁 − 1 and with finite set of 𝑚𝑚 = 1,2, … , 𝑀𝑀 independent realizations x(m) (i) [6]. autocorrelation function can be written as 𝑅𝑅𝑥𝑥(𝑘𝑘) = 〈�[𝑥𝑥(𝑚𝑚) (𝑖𝑖) − 𝐸𝐸] 𝑁𝑁−1 𝑖𝑖=0 [𝑥𝑥(𝑚𝑚) (𝑖𝑖 + 𝑘𝑘) − 𝐸𝐸]〉∞, (1) where 𝑘𝑘 = −𝑁𝑁 + 1, … , 𝑁𝑁 − 1 is the shift index, 〈… 〉∞ denotes the ensemble averaging for infinite realization number, i.e., for 𝑀𝑀 → ∞ ; 𝐸𝐸 = 〈1 𝑁𝑁 ∑ 𝑥𝑥(𝑚𝑚) (𝑖𝑖)𝑁𝑁−1𝑖𝑖=0 〉∞ is the mean value; 𝑅𝑅𝑥𝑥(0) = 𝜎𝜎𝑥𝑥2 = 〈∑ [𝑥𝑥(𝑚𝑚) (𝑖𝑖) − 𝐸𝐸]𝑁𝑁−1𝑖𝑖=0 2 〉∞ is the variance. autocorrelation function is a function of one variable. spectral density px(p) is defined from wiener-khinchin theorem using direct fourier transfer: 𝑃𝑃𝑥𝑥(𝑝𝑝) = � 𝑅𝑅𝑥𝑥(𝑘𝑘)exp (−𝑗𝑗2𝜋𝜋𝑘𝑘𝑝𝑝) 𝑘𝑘=+∞ 𝑘𝑘=−∞ , (2) or by 𝑃𝑃𝑥𝑥(𝑝𝑝) = 〈𝑋𝑋(𝑚𝑚)(𝑝𝑝)𝑋𝑋∗(𝑚𝑚)(𝑝𝑝)〉∞, (3) where 𝑝𝑝 = −𝑁𝑁 + 1, … , 𝑁𝑁 − 1 is the frequency sample index; e. sivolenko 29 𝑋𝑋(𝑚𝑚)(𝑝𝑝) = ∑ 𝑥𝑥(𝑚𝑚)(𝑖𝑖)exp (−𝑗𝑗2𝜋𝜋𝑖𝑖𝑝𝑝)𝑁𝑁−1𝑖𝑖=0 is fourier transform for m-th realization; * denotes complex conjugation. in equation (3) due to multiplication of the complex conjugated functions the phase information is lost. in opposite to autocorrelation function and spectral density, 𝑅𝑅𝑥𝑥(𝑘𝑘, 𝑙𝑙) triple autocorrelation function and �̇�𝐵𝑥𝑥(𝑝𝑝, 𝑞𝑞) bispectrum are functions of two variables. 𝑅𝑅𝑥𝑥(𝑘𝑘, 𝑙𝑙) triple autocorrelation function is set as 𝑅𝑅𝑥𝑥(𝑘𝑘, 𝑙𝑙) = 〈�[𝑥𝑥(𝑚𝑚) (𝑖𝑖) − 𝐸𝐸] 𝑁𝑁−1 𝑖𝑖=0 [𝑥𝑥(𝑚𝑚) (𝑖𝑖 + 𝑘𝑘) − 𝐸𝐸][𝑥𝑥(𝑚𝑚) (𝑖𝑖 + 𝑙𝑙) − 𝐸𝐸]〉∞, (4) where 𝑘𝑘 = −𝑁𝑁 + 1, … , 𝑁𝑁 − 1 and 𝑙𝑙 = −𝑁𝑁 + 1, … , 𝑁𝑁 − 1 are the independent shift indices. unlike the spectral density, �̇�𝐵𝑥𝑥(𝑝𝑝, 𝑞𝑞) bispectrum is a complex-valued function of two independent frequencies p and q. it can be written as 2-d discrete fourier transform of triple autocorrelation function �̇�𝐵𝑥𝑥(𝑝𝑝, 𝑞𝑞) = � � 𝑅𝑅𝑥𝑥(𝑘𝑘, 𝑙𝑙) exp[−𝑗𝑗2𝜋𝜋(𝑘𝑘𝑝𝑝 + 𝑙𝑙𝑞𝑞)] , 𝑁𝑁−1 𝑙𝑙=−𝑁𝑁+1 𝑁𝑁−1 𝑘𝑘=−𝑁𝑁+1 (5) or as �̇�𝐵𝑥𝑥(𝑝𝑝, 𝑞𝑞) = 〈�̇�𝑋 (𝑚𝑚)(𝑝𝑝)�̇�𝑋(𝑚𝑚)(𝑞𝑞)�̇�𝑋∗(𝑚𝑚)(𝑝𝑝 + 𝑞𝑞)〉∞ = 〈�̇�𝑋(𝑚𝑚)(𝑝𝑝)�̇�𝑋(𝑚𝑚)(𝑞𝑞)�̇�𝑋(𝑚𝑚)(−𝑝𝑝 − 𝑞𝑞)〉∞ (6) where �̇�𝐵𝑥𝑥(𝑝𝑝, 𝑞𝑞) = ��̇�𝐵𝑥𝑥(𝑝𝑝, 𝑞𝑞)� exp[𝑗𝑗𝛾𝛾𝑥𝑥(𝑝𝑝, 𝑞𝑞)], ��̇�𝐵𝑥𝑥(𝑝𝑝, 𝑞𝑞)� and 𝛾𝛾𝑥𝑥(𝑝𝑝, 𝑞𝑞) are the magnitude bispectrum (bimagnitude) and phase bispectrum (biphase), respectively 𝑝𝑝 = −𝑁𝑁 + 1, … , 𝑁𝑁 − 1 and 𝑞𝑞 = −𝑁𝑁 + 1, … , 𝑁𝑁 − 1 are the frequency indices. from (3) power spectrum is the ensemble averaging of the multiplication of two complex conjugated functions of one variable. meanwhile, from (6) bispectrum is an ensemble averaging of three complex-valued functions corresponding to different frequency values. so, spectral density is providing information only about amplitude, meanwhile bispectrum can provide information about phase and amplitude. we use bispectral estimating for one of the main properties: coupled phase information retention [7]. 3. modulation type recognition method this method relies on one of the main properties of bispectrum estimation: coupled phase information retention. as it was mentioned above this method is being used for recognition amplitude and frequency modulations. as it is known the carrier suppressed amplitude modulated signal has two picks in power spectrum. time domain and power spectrum of carrier suppressed amplitude modulated signal are shown in figure 1. determination of received unknown signal modulation type using higher order statistics 30 fig.1. time domain and power spectrum of carrier suppressed amplitude modulated signal. time domain and power spectrum of frequency modulated signal with clearly mentioned five picks are shown in figure 2. fig.2. time domain and power spectrum of frequency modulated signal. carrier suppressed amplitude modulated and frequency modulated signals in this simulation have the same carrier. time domain and power spectrum of the signal, which is the amount of carrier suppressed amplitude modulated and frequency modulated signals, are shown in figure 3. fig.3. time domain and power spectrum of the amount of carrier suppressed amplitude modulated and frequency modulated signals. e. sivolenko 31 it is very difficult for the receiver to recognize the modulation type of the received signal using only power spectrum. only amplitude information is very little to separate amplitude and frequency modulation. so, it is very important in such cases to have information about phases. this information can be provided by third-order statistics especially by bispectral estimating. bispectrum estimating results are shown in figure 4. (a) (b) fig. 4. bispectrum estimation of the amount of carrier suppressed amplitude modulated and frequency modulated signals: default view (a) and top view (b). as it is shown in figure 4 there are lots of picks on bispectrum graph. the picks of amplitude modulated signal are higher than the frequency modulated signal picks. besides, the number of frequency modulated signal picks is more than the number of amplitude modulated signal picks. so, there is a possibility to separate those two signals using bispectrum estimation. the separation was done step by step. received signal filtering is the main part of modulation type recognition. the first step is disclosing the useful information from noise. received signal with noises and disclosed useful signal are shown in figure 5. (a) determination of received unknown signal modulation type using higher order statistics 32 (b) fig. 5. carrier suppressed amplitude modulation and frequency modulation with the same carrier: top and default views with noise (a) and without noise (b). as it is shown in figure 5 there are a lot of useful pics after deleting noise. some of these picks belong to amplitude modulation and another part belongs to frequency modulation. so, it is very important to separate these modulation types. in this case we use phase coupled phenomena for separation. fig. 6. carrier suppressed amplitude modulated signal disclosing using bispectrum estimating. figure 6 illustrate the disclosed carrier suppressed amplitude modulated signal picks which were found using phase coupled phenomena. after finding amplitude modulated signal picks it is very easy to disclose frequency modulated signal picks. those picks are shown in figure 7. fig. 7. frequency modulated signal disclosing using bispectrum estimating. e. sivolenko 33 4. conclusion bispectrum estimation provides more information about signal than power spectrum. for automatic modulation type recognition it is more effective using bispectrum estimation than power spectrum. the main aim of the paper is to explain the modulation type recognition method based on higher order statistics: especially bispectrum and triple autocorrelation function. it has been found that different modulations have different bispectrum graphs. finally, a received signal modulation type automatic recognition new method was presented. it has been shown how bispectrum based estimation techniques can be used in communication systems. the studies indicate the feasibility of bispectra application for solving the task of modulation recognition based on the results of measurements. references [1] m. ali khan, m. muhammad khan and m. saad khan, “automatic modulation recognition of communication signals”, asian journal of natural & applied sciences, vol. 2, no.1, pp. 17-22, oyama, japan 2013. [2] d. le guen and a. mansour, “automatic recognition algorithm for digitally modulated signals”, iasted international conference “signal processing, pattern recognition & applications”, crete, greece, june 25-28, pp. 3237, 2002. [3] s. v. vaseghi, advanced digital signal processing and noise reduction, second edition, uk by john wiley & sons, ltd 2000. [4] и. в. шиховцев и в. п. якубов, “статистическая радиофизика”, с. 26-28, новосибирск, 2011. [5] w. kicinski and a. szczepanski “quadratic phase coupling phenomenon and its properties”, akademia marynarki wojennej, pp. 81-103 ,gdynia, poland. [6] а. в. тоцкий, я. астола, к. о. егиазарян, а. а. зеленский, и. в. курбатов и в. в. лукин, <<восстановление сигналов по оценкам биспектров в присутствии гауссовых и негауссовых помех>>, успехи современной радиоэлектроники, 2002, n 11, с. 44-58 [7] a. a. hakhoumian and e.r. sivolenko “pedestrian detection using higher order statistics (hos) or polyspectral analyses”, proceedings of the international conference on “microwave and thz technologies and applications”, aghveran, armenia, pp. 68-71, 2014. submitted 07.09.2015, accepted 20.01.2016 determination of received unknown signal modulation type using higher order statistics 34 գրանցված անհայտ ազդանշանի մոդուլյացիայի տեսակի որոշումը բարձր կարգի վիճակագրությամբ կամ սպեկտրալ կոռելյացիայի միջոցով է. սիվոլենկո ամփոփում այսօր մոդուլացված ազդանշաններ օգտագործվում են գրեթե ամենուր: այդ իսկ պատճառով գրանցված անհայտ ազդանշանի մոդուլյացիա տեսակի որոշելը շատ կարևոր դեր է խաղում հեռահաղորդակցական համակարգերի համար: այս աշխատանքում առաջարկվում է ալգորիթմ, որը հնարավորություն է տալիս իրարից տարանջատել երկու տեսակի ՝ ամպլիտուդային (am) և հաճախամոդուլված (fm) ազդանշանները: ալգորիթմի աշխատանքի արդյունքները ներկայացված են աշխատանքում: մեթոդը հիմնված է բարձր կարգի վիճակագրության վրա: այս աշխատանքում մոդուլյացիայի տեսակները տարանջատելու համար մենք օգտվում ենք բիսպեկտրումից և եռակի ավտոկոռելյացիոն ֆունկցիայից: определение типа модуляции неизвестного сигнала методом статистики высших порядков или корреляции спектра э. сиволенко аннотация на сегодня модулированные сигналы используются везде. а это значит, что распознавание типа модуляции самая главная задача телекоммуникационных систем. в работе представлен алгоритм определения типа модуляции для двух типов модулированных сигналов: амплитудной (am) и частотной (fm) модуляции. метод основан на статистике высших порядков. для распознавания типа модуляции сигналов мы используем биспектрум и тройную автокорреляционную функцию. результаты алгоритма представлены в работе. microsoft word tmail1.doc mathematical problems of computer science 30, 40--46, 2008. 40 custom software installation and availability assurance of computing elements in lhc grid 1 karen mkoyan yerevan physics institute karen@yerphi.am abstract: this paper describes the process of custom software installation and validation in the lhc grid. it also provides set of scripts developed by author for availability assurance and overall monitoring of computing elements prior to installation, and a script for the installation and validation process itself. references [1] tank and spark http://grid-deployment.web.cern.ch/grid deployment/eis/docs/internal/chep04/sw_installation.pdf [2] lightweight middleware for grid computing http://glite.web.cern.ch/glite/ [3] experiment software installation in lcg-2 http://grid-deployment.web.cern.ch/grid-deployment/eis/docs/expswinstall/sw-install.pdf [4] “custom installer” script http://grid-web0.desy.de/ceat/custom-installer.tgz [5] service availability monitor (sam) http://sam-docs.web.cern.ch/sam-docs/ [6] site functional tests http://goc.grid.sinica.edu.tw/gocwiki/site_functional_tests [7] grid operations centre http://grid-it.cnaf.infn.it/index.php?id=853&type=1 [8] standalone version of sam client http://wiki.egee-see.org/index.php/see-grid_standalone_sam [9] slightly modified version of standalone sam client http://grid-web0.desy.de/ceat/lightweight-ssam_source.tgz [10] ceat at desy http://grid-web0.desy.de/ceat/stats/ 1 this work has been supported by intas, young scientists fellowship grant ref. 05-110-4812 k. mkoyan 41 ð³ïáõï íñ³·ñ³ûçý ÷³ã»ãý»ñç ï»õ³¹ñáõùá ¨ ñ³ßíçã ë³ñù»ñç ýáõýïóçáý³éáõãû³ý ñ³í³ëïç³óáõùá lhc ·ñç¹áõù î. øïáû³ý ²ù÷á÷áõù ðá¹í³íáõù ýï³ñ³·ñí³í ¿ ñ³ïáõï íñ³·ñ³ûçý ÷³ã»ãý»ñç ï»õ³¹ñù³ý áýã³óùá lhc ·ñç¹áõù: øß³ïí³í »ý íñ³·ñ³ûçý ÷³ã»ãý»ñ ï»õ³¹ñù³ý áýã³óùç ³íïáù³ï³óù³ý ¨ ñ³ßíçã ë³ñù»ñç ýáõýïóçáý³éáõãû³ý ñ³í³ëïç³óù³ý ¨ í»ñ³ñëïù³ý ñ³ù³ñ: microsoft word 4.doc математические вопросы кибернетики и вычислительной техники 38, 12, 2012. 12 о стратификации множества отображений групп самвел далалян ереванский государственный университет пусть a –группа с (мультипликативной) бинарной операцией xy, m – группа с бинарной операцией [a,.b]. упорядоченную пару из элементов a и b будем обозначать ab, а множество отображений из m в a – через s(m, a).. определение. для произвольного   s(m, a) элемент (a, b) группы a. однозначно определяемый соотношением ([a, b]) = (a)(b)(a, b), назовем правым сомножителем отклонения отображения  от гомоморфности в (или для) ab. отображение  = : mm  a, ab  (a, b) назовем правым индикатором отклонения отображения  от гомоморфности. образ im  = {(a, b), a, b  m} : = f назовем правой системой факторов (= сомножителей), порожденную ею подгруппу h группы a – правой группой, а порожденную f (эквивалентно, h) нормальную подгруппу n группы a – правым нормальным делителем, ассоциированными с правым индикатором отклонения отображения  от гомоморфизма. очевидно, отображение  является гомоморфизмом тогда и только тогда, когда f = h = n = e – {e}, где e – нейтральный элемент группы a. . пусть f – некоторое подмножество группы a. отображение   s(m, a) назовем fморфизмом, если f = f. класс всех f-морфизмов из группы m в группу a обозначим через [f] . множество всех классов [f] задает стратификацию (разбиения на классы попарно непересекающихся подмножеств) множества s(m, a) аналогично определяются h-морфизмы и n-морфизмы из группы m в группу a и подмножества [h], [n]  s(m, a) для произвольных подгруппы h и, соответственно, нормального делителя n группы a, а также соответствующие стратификации. очевидно, для любого   s(m, a) имеем [f]  [h]  [ n]. поэтому каждая следующая из полученных стратификаций доминирует над предыдущей. в работе дается внутреннее и внешнее описание стратов этих стратификаций. microsoft word 09_andranik's article_74--78.doc mathematical problems of computer science 49, 74--78, 2018. 74 unimail info-communicational software andranik e. mkhitaryan, aram s. nanassian and eduard z. matveev institute for informatics and automation problems of nas ra e-mail: and.mkhitaryan@gmail.com, ananas@sci.am, edo.matveev1996@mail.ru abstract in this paper, the design and features of the independent infocommunicational software resource are presented based on e-mail and sms technologies, named unimail. the components of the server and their responsibilities are also considered. peculiarities of the unimail’s architecture and principles of workflow are discussed. acceptable commands and configuration details are described for users. keywords: unimail, info-communication, mail2sms, e-mail, sms, notification. 1. introduction e-mail and sms are world-spread info-communicational technologies. e-mail provides an exchange mechanism of text messages between computers via the internet. despite the fast delivery of the message to the addressee, email is a typical "on demand" system. another mechanism to send/receive text messages is sms. however, it has several disadvantages that are listed below: • the size of the transmitted message is limited, • message delivery is possible only in the coverage area of the cellular network or according to the roaming agreements of the local cellular operator. despite its disadvantages, sms technology has an undeniable advantage the messages are being delivered directly to the “pocket” of the addressee (mobile subscriber). the most important disadvantages of e-mail and sms can be solved by combined use of both technologies [1][2]. the limitation of the size of message as in case of sms will be overcome by e-mail and the problem of “on demand” of e-mail will be overcome by sms [3]. 2. unimail software unimail is an independent software product designed and implemented by the institute for informatics and automation problems of nas ra. with the use of both sms and e-mail technologies, unimail gives possibility to the e-mail users to send an informative sms notification in parallel to the letter to the addressee. the initiator of sending notification for this a. mkhitaryan, a. nanassian and e. matveev 75 system is the sender of the e-mail message, which makes it unique and different from the existing instances. the structure of unimail (figure 1) consists of three general components. the first part is responsible for sending and receiving e-mails. unimail has its own mail server (smtp), and there is a separated block, which provides functionality to communicate with that server. the second section is responsible for working with gsm network. as a part of unimail there also is a gsm modem. the independent block provides the ability to control gsm modem. in general, unimail uses the modem to send sms messages/notifications. the third section – the general one, is the core of unimail software, which is responsible for the main logic. the components of unimail core are described in figure 2. it consists of seven different subsections. the received e-mails at first are being filtered by the following steps:  acceptable and correct command exists in the subject of the e-mail  addressee has not blocked the services of unimail  addressee’s black/white lists allow the sender to notify him by unimail  the specified phone number (in commands where phone number is required) is a valid number from the current regional cellular network. fig. 2. main components of unimail software. fig. 1. structure of unimail server. unimail info-communicational software 76 by design, unimail should work server side. as an optimal solution, it is being run as a service to avoid using additional resources. it is repetitively requesting for new e-mails from the mail server and is storing them for further processing. in case there is a new e-mail, it will go through the mail filter. usually there are lots of spam and useless e-mails in mailbox. there are several requirements that each mail should satisfy to pass the filtering process. first, it should contain a command in the subject. in case the subject does not contain any command, or the command is specified incorrectly, the mail will be ignored by unimail. for the monitoring of unimail the informative logging mechanism on standard ‘/var/log/unimail’ directory is implemented. figure 2 presents the software in action by logs. in the logs the current operations of the server are presented. for this specific case the user has sent an e-mail message to two addressees and unimail notifying them by sms. this is a usual lifecycle of the server. it filters all the inbox messages, then finds out if there is an e-mail satisfying the requirements. in the subject of that e-mail there is a command that notifies the addressees about the e-mail sent. unimail sends notifications one by one and checks the results. 3. basic functions of unimail-asnet to use any service of unimail, the users should send (to / cc / bcc) a copy of the e-mail to the address of the server unimail@unimail.asnet.am. to send an sms notification in parallel to an e-mail to the addressee, in the subject there should be specified a list of phone numbers with asterisks (subject: **). similar markers could also be used on the body of e-mail to append the mentioned part to the sms message (body: **). the selected fragment should not exceed 110 characters, including spaces. if there is no selected fragment, a standard notification will be sent by sms. as a result, the addressee will receive an sms in format: “unimail: you got email from :
” in case there is no e-mail address specified in to, cc or bcc fields of the received e-mail, except for the address of the server, then just the sms message will be sent to the specified phone numbers instead of the notification. to summarize, for this specific case, unimail allows to send sms via e-mail. the initiator of notifying about the e-mail is the sender. however, the receiver should have an opportunity to control the list of people who are permitted to send him sms notifications. to be able to control the lists, the user should tie his phone number with a personal email address. for this configuration, unimail requires a simple registration:  to: unimail@unimail.asnet.am fig. 3. output log file of unimail. a. mkhitaryan, a. nanassian and e. matveev 77  subject: *reg: * after sending the e-mail, an sms containing a randomly generated 4 or 5-digit number will be sent to the specified phone number. the user should send that number back to the server from the same e-mail address:  to: unimail@unimail.asnet.am  subject: ** once the registration is complete, the user’s e-mail address and phone number will be paired. the next operations are allowed for the registered users only. unimail supports two well-known systems of controlling permitted lists, i.e., black and white lists. any registered user can choose the appropriate type of control. by default, the mode being chosen is “a black list” mode. to change the mode the user should send an e-mail to the address of the unimail server with subject:  for switching to “white list” mode: o *use white list *  for switching to “black list” mode: o *use black list * to include or exclude someone in/out of the black or white lists, the user should send an e-mail to the same e-mail address with the following indicators:  subject: o *add white/black list * o *del white/black list *  body: o * / , / , …* the registered user can also block/unblock all the services provided by unimail by sending an email with subject:  * block sms *  * unblock sms * 4. conclusion unimail is an info-communication autonomous software resource, which allows you to send sms notifications in parallel to the e-mail. it uses a unique approach when the initiator of notification is the sender, who considers his e-mail to be important enough. at the same time, the receiver can easily control the list of people who are permitted to send him notifications. receivers can even turn off all the services provided by unimail. to sum up, unimail provides a complete package to control notification system of both senders and receivers. it operates with any e-mail provider and can notify anyone from the regional cellular network. references [1] д. геворкян, а. нанасян, к. хачатрян, «новые web ресурсы asnet.am», proceedings of international conference of computer science and information technօlogies, csit-2011, ереван, pp. 311-312, 2011. unimail info-communicational software 78 [2] d. gevorkyan, k. khachatryan, a. nanassian, a. petrosyan, g.petrosyan, v. sahakyan and e. vardanyan, “mail informerselective incoming instant phone notification system”, proceedings of international conference computer science and information technologies, csit, yerevan, armenia, pp. 466-467, 2009. [3] а. нанасян и к. хачатрян «mail2sms.asnet.am – система оповещения о входящих письмах», proceedings of international conference of computer science and information technologies csit-2013. ереван, 2013. [4] а. мхитарян, э. матвеев, а. нанасян, в. саакян и a. petrosyan, “гибридная инфокоммуникационная email/sms система unimail”, proceedings of international conference of computer science and information technologies, csit, yerevan, armenia, pp. 389-391, 2017. submitted 22.11.2017, accepted 12.02.2018. unimail-ի տեղեկատվական հաղորդակցման համակարգի ծրագրային ապահովում ա. մխիթարյան, ա. նանասյան և է. մատվեև ամփոփում աշխատանքում ներկայացված են sms և email տեխնոլոգիաների վրա հիմնված տեղեկատվական հաղորդակցման անկախ ծրագրային ռեսուրսը՝ unimail անվամբ, unimail սերվերի առանձին մասերը և դրանց պարտականությունները։ դիտարկվել են unimail-ի կառուցվածքային առանձնահատկությունները և աշխատանքի սկզբունքները։ նկարագրված է հրամանների ցանկը և կարգավորումների մանրամասները օգտատերերի համար։ программная поддержка инфокоммуникационной системы unimail а. мхитарян, а. нанасян и э. матвеев аннотация инфокоммуникационная система unimail независимый аппаратнопрограммный ресурс на основе технологий электронной почты и sms. приведены функциональные компоненты сервера unimail. обсуждаются особенности архитектуры и принципы работы, базовые команды системы. mathematical problems of computer science 46, 117–125, 2016. the parallel simulation method for d-dimensional abelian sandpile automata hayk e. nahapetyan, suren s. poghosyan, vahagn s. poghosyan and yuri h. shoukourian institute for informatics and automation problems of nas ra e-mail: povahagn@gmail.com, shouk@sci.am abstract in this paper, the star-packing problem introduced in [1] for a square lattice is generalized for d-dimensional lattice ld, d ∈ n. the problem is to pack the lattice ld with star graphs s2d. using the solution of this problem, a parallel algorithm for the simulation of d-dimensional cellular automata is developed. as an example of cellular automata, the relaxation process of unstable states of abelian sandpile model is considered. appropriate software packages have been developed using openmp and cuda technologies. the parallel simulation results, carried out for 3-dimensional lattices of different sizes, are presented. keywords: abelian sandpile model, dense packing problem, parallel algorithm, cellular automata. 1. introduction cellular automata (ca) are discrete models studied in computability theory, mathematics, physics, complexity science, theoretical biology and microstructure modeling. modern multicore computers with shared memory and multiprocessor clusters with distributed memory make it possible to efficiently parallelize simulation algorithms for ca. in the work [2], a cluster-based parallel algorithm and a package to simulate two-dimensional ca is introduced. the concept of self-organized criticality was first introduced by bak, tang and wiesenfeld in 1987 [4], and gave rise to growing interest in the study of self-organizing systems. bak et al. argued that in many natural phenomena, the dissipative dynamics of the system is such that it drives the system to a critical state, thereby leading to ubiquitous power law behaviors. this mechanism has been invoked to understand the power law distributions observed in turbulent fluids, earthquakes, distribution of visible matter in the universe, solar flares and surface roughening of growing interfaces. the sandpile models, being a class of cellular automata, are among the simplest theoretical models which show self-organized criticality. a special subclass of interest consists of so called abelian sandpile models (asm). the abelian property means that the final stable state of the ca is independent of the order in which the updates of cells are carried out. this property plays a key role during the numerical, as well as analytical studies of the asm [5, 6, 7, 8, 9, 10, 11, 12, 13, 14]. in numerous works, in the absence of analytical relations for 117 118 the parallel simulation method for d-dimensional abelian sandpile automata various physical characterizers, different methods of statistical analysis have been applied for calculation or verification of hypothetical and analytical expressions for sandpile models. the luck of such systems, also the demand for accurate calculations of physical observables would dramatically increase the acceptable sizes of lattice as well as the time constraints for calculations. an approach of efficient parallel simulation of sandpile model on multicore computers with shared memory, which is based on the idea that there are few active cells (unstable vertices) at each moment of time, is given in [3]. in general, when the initial unstable configuration contains many unstable vertices, this approach does not give good results for big lattices. in what follows we present a more efficient parallel algorithm for the simulation of d-dimensional ca implemented on systems with shared memory, which is based on the star-packing of the d-dimensional lattices. a solution to the star-packing problem is given in [1] for the 2-dimensional square lattice. in this paper we generalize the solution to the case of general d dimensions. below a cellular automaton for implementing the abelian sandpile model on a d-dimensional lattice is considered aiming at the development of parallel simulation. 2. the star-packing problem consider a d-dimensional lattice ld of linear size n with periodic boundary conditions, which is a graph with the vertex set v = v (ld) = {v = (x1, x2, . . . , xd) : xi = 0, 1, 2, . . . , n − 1; i = 1, 2, . . . , d} (1) and the set of undirected edges e = e(ld) defined by the following rule. each vertex v = (x1, x2, . . . , xd) ∈ v is connected with 2d vertices given by adjacency list adj(v) = {(x1, x2, . . . , (xk ± 1) mod n, . . . , xd) : k = 1, 2, . . . , d}. (2) the number of vertices is |v | = nd, and the number of edges is |e| = d nd. two different vertices v1 and v2 are called independent, if they are not adjacent and they do not have any common adjacent vertex: v1 /∈ adj(v2), v2 /∈ adj(v1) and adj(v1) ∩ adj(v2) = ∅. (3) the star-packing problem is to pack the lattice ld with non-overlapping star graphs s2d. consider a star packing with a set of central vertices v ′ ⊂ v of stars. the following two rules define the set v ′: 1. full coverage condition: ( ∪ v∈v ′ adj(v) )∪ v ′ = v. 2. non-overlapping condition (independence of stars): any two different vertices v1, v2 ∈ v ′ are independent. it is obvious, that a star packing exists if |v | is dividable by 2d + 1. given a vertex v = (x1, x2, . . . , xd) ∈ v and a vector r⃗ = (p1, p2, . . . , pd) ∈ z. define the sum v + r⃗ to be a vertex in v with coordinates v + r⃗ ≡ ((x1 + p1) mod n, (x2 + p2) mod n, . . . , (xd + pd) mod n) ∈ v. (4) h. nahapetyan, su. poghosyan, v. poghosyan and yu. shoukourian 119 equivalently, the sum of a vector r⃗ and an arbitrary subset v ′ ⊆ v to be the following subset of v v ′ + r⃗ ≡ {v + r⃗ : v ∈ v ′} ⊆ v. (5) note that, given a star packing defined by a set v ′, the set v ′ + r⃗ also defines a star packing, where r⃗ ∈ z. theorem: consider a d-dimensional lattice ld of linear size n with periodic boundary conditions. assume that n is dividable by 2d + 1. then a star packing of ld exists, and the set of central vertices of stars is v ′ = {v = (x1, x2, . . . , xd) : (x1 + 2x2 + 3x3 + . . . + d xd) mod (2d + 1) = 0, v ∈ v }. (6) proof: given a vector r⃗ = (p1, p2, . . . , pd) ∈ z, define a function q(r⃗) = p1 + 2p2 + 3p3 + . . . + d pd. (7) let us introduce the d-dimensional basis vectors e⃗k = (0, 0, . . . , 0︸ ︷︷ ︸ k-1 , 1, 0, 0, . . . , 0︸ ︷︷ ︸ d-k ), k = 1, 2, 3, . . . , d, (8) and the null vector e⃗0 = (0, 0 . . . , 0). first, we prove the full coverage condition. given an arbitrary vertex v ∈ v \ v ′, it is necessary to prove that there exists a vertex u ∈ adj(v), which is a central vertex of some star, i.e., u ∈ v ′. in other words, ∀r⃗ = (p1, p2, . . . , pd) ∈ z, 0 ≤ pi ≤ n − 1, (9) one of the following 3 conditions holds 1. q(r⃗) mod (2d + 1) = 0. 2. ∃k = 1, 2, . . . , d; q(r⃗ − e⃗k) mod (2d + 1) = 0. 3. ∃k = 1, 2, . . . , d; q(r⃗ + e⃗k) mod (2d + 1) = 0. assume that q(r⃗) has the following form: q(r⃗) = (2d + 1)s + t, 0 ≤ t < 2d + 1, s ≥ 0. (10) let the value of k be k =   0, if t = 0; ⇒ q(r⃗) mod (2d + 1) = 0 t, if t < d; ⇒ q(r⃗ − e⃗k) mod (2d + 1) = 0 2d + 1 − t, if t > d; ⇒ q(r⃗ + e⃗k) mod (2d + 1) = 0 , (11) which proves the coverage condition. now we prove the non-overlapping condition. it states, ∀ r⃗ = (p1, p2, . . . , pd) ∈ z, 0 ≤ pi ≤ n − 1, i = 1, 2, . . . , d, which satisfies the condition q(r⃗) mod (2d + 1) = 0, (12) and ∀∆⃗r = (∆p1, ∆p2, . . . , ∆pd) with 0 < |∆p1| + |∆p2| + . . . + |∆pd| ≤ 2, (13) 120 the parallel simulation method for d-dimensional abelian sandpile automata we have q(r⃗ + ∆⃗r) mod (2d + 1) ̸= 0. (14) since the q function is linear, we have to show that q(∆⃗r) mod (2d + 1) ̸= 0. then, it is sufficient to prove the inequality 0 < |q(∆⃗r)| < 2d + 1. note that there are 3 possible cases, which satisfy the condition (13): 1. ∃k0 = 1, 2, . . . , d, for which ∆pk0 = ±2 and ∆ps = 0, ∀s ̸= k0. then 0 < |q(∆⃗r)| = 2k0 < 2d + 1. (15) 2. ∃k0 = 1, 2, . . . , d, for which ∆pk0 = ±1 and ∆ps = 0, ∀s ̸= k0. then 0 < |q(∆⃗r)| = k0 < 2d + 1. (16) 3. ∃k1, k2 = 1, 2, . . . , d; k1 ̸= k2, for which ∆pk1 = ±1, ∆pk2 = ±1 and ∆ps = 0, ∀s ̸= k1, k2. then 0 < |q(∆⃗r)| = |k1∆pk1 + k2∆pk2| ≤ k1 + k2 < 2d + 1. (17) 2 fig.1 and fig.2 illustrate the star packing of the 2 and 3-dimensional lattices, respectively. fig. 1. a star-packing of a 2-dimensional lattice. h. nahapetyan, su. poghosyan, v. poghosyan and yu. shoukourian 121 fig. 2. a star-packing of a 3-dimensional lattice. the layer z=4 is depicted separately. 3. abelian sandpile model consider an undirected graph g = (v, e) with vertex set v = {v1, v2, . . . , vn} and edge set e. a random variable hi, which takes integer values, is attached to each vertex vi ∈ v , representing the height of the sand at that vertex. hmaxi denotes the maximum allowed height of vertex vi of the graph g. for d-dimensional lattice we take h max i = 2d−1. ct denotes the collection of heights hi, which defines a configuration of the system at a given discrete time t . a configuration is called stable, if all heights satisfy hi ≤ hmaxi . vertex vi is called closed, if hmaxi = deg (vi) − 1, where deg (vi) indicates the number of adjacent vertices of vi. vertex vi is called open, if h max i ≥ deg (vi). the dynamics of the system is defined by the following rules. consider a stable configuration ct at a given time t . we add a grain of sand at a random vertex vi ∈ v by setting hi to hi + 1 (we assume that the vertex is chosen randomly with a uniform distribution on the set v ). this new configuration, if stable, defines ct+1. if hi > h max i , vi becomes unstable and topples losing h max i +1 grains of sand, while all neighbors of vi receive one grain. note that if the vertex is open, the system loses grains. during the toppling of the closed vertices, the number of grains is conserved. note also that toppling of a vertex may cause some of its neighboring vertices to become unstable. in this case those vertices also topple according to the same toppling rule. once all unstable vertices have been toppled, a new stable configuration ct+1 is obtained. if the finite connected graph g has at least one open vertex, then all vertices become stable after finite number of topplings. moreover, the new stable configuration is independent of the toppling order. therefore, the dynamics is well defined. let âi be an operator, which acts on sandpile configurations and adds a grain at vertex i. it can easily be shown that âiâj = âjâi. this is the reason why the sandpile model is called abelian. 122 the parallel simulation method for d-dimensional abelian sandpile automata 4. simulation results in this section we present the simulation results of the implemented software with built-in parallelization algorithms. the abelian sandpile model on the finite d-dimensional lattice of linear size n is considered. the number of nodes is n = nd. the value of maximal height at all vertices vi ∈ v is hmaxi = 2d − 1, i = 1, 2, . . . , n. to perform parallel simulation, the concurrently toppled unstable vertices should meet the requirement of being independent. partitioning the set of vertices into the sets of independent vertices is carried out by starpacking discussed above. let twait be the time needed for a cell to finish its job, and h denotes the initial number of grains at each node. in order to obtain a parallel software environment, the openmp and cuda technologies have been used, which was implemented on the cpu i7 2670qm by intel and geforce gt. simulation results include 2 types of cellular automata simulating sandpile model. difference is that for the second system an additional job is introduced during each toppling, which takes twait constant time to finish. four types of implementations have been analyzed and compared: 1. standard − topplings are implemented sequentially on cpu. there is a loop around all nodes, and once an unstable node is found, it becomes stable. 2. no omp − topplings are implemented sequentially on cpu. by a parallelization algorithm, we partition the set of nodes into 7 sections for a 3-dimensional lattice. there are two types of loops, one type over all groups, and the second type over all vertices for each group. 3. omp − topplings are implemented parallel for each group on cpu, also there is one loop over all groups. 4. gpu − topplings are implemented in parallel for each group on gpu, also there is one loop over all groups. the simulations have been done on the same cpu and gpu architecture. in figs. 3-7 the simulation results are presented. 140 210 280 n 5 10 15 20 25 30 computation time (ms) standard no-omp omp gpu fig. 3. sandpile on a 3-dimensional lattice. grains are added randomly with uniform distribution. total amount of added grains is 10 × n3. h. nahapetyan, su. poghosyan, v. poghosyan and yu. shoukourian 123 n=7,h=6 n=7,h=12 n=14,h=6 n=14,h=12 100 200 300 400 computation time (ms) standard no-omp omp gpu fig. 4. sandpile on a 3-dimensional lattice with initial height h=6, h=12. twait = 5ms. n=7,h=6 n=7,h=12 n=14,h=6 n=14,h=12 200 400 600 800 computation time (ms) standard no-omp omp gpu fig. 5. sandpile on a 3-dimensional lattice with initial height h=6, h=12. twait = 10ms. n=21,h=6 n=21,h=12 n=42,h=6 n=42,h=12 10000 20000 30000 40000 50000 computation time (ms) standard no-omp omp gpu fig. 6. sandpile on a 3-dimensional lattice with start height h=6, h=12. twait = 3ms. 5. conclusion in this paper, a parallel algorithm for simulating d-dimensional ca has been described. the algorithm is based on the idea of star-packing of d-dimensional lattices. delmas and perennes in their paper [15] found a star packing for 3-dimensional lattice of linear size n = 7i with integer i. we found an explicit star packing of d-dimensional lattice, which meets weaker restriction on n, namely, n mod (2d + 1) = 0. as the results show, the algorithm developed gains about 8× speedup with openmp, and more than 50× on cuda (depending on gpu architecture and cellular automata, it can be 100× and more faster than non-paralyzed 124 the parallel simulation method for d-dimensional abelian sandpile automata performance). our next aim is to use algorithms for computing different observables of the sandpile model on d-dimensional lattices, meanwhile analytical calculations have been obtained for 2-dimension. references [1] v. s. poghosyan, s. s. poghosyan and h. e. nahapetyan, “the investigation of models of self-organized systems by parallel programming methods based on the example of an abelian sandpile model”, proc. csit conference 2013, yerevan armenia, sept. 23-27, pp. 260–262, 2013. [2] e. davtyan, h. karapetyan and k. shahbazyan, “software tool for cluster-based modeling of 2d cellular automata”, proc. csit conferance, yerevan armenia, 28 sept. 2 oct., pp. 404–407, 2013. [3] s. frehmel, “the sandpile model: parallelization of efficient algorithms for systems with shared memory”, acri, lecture notes in computer science 6350, pp. 35–45, 2010. [4] p. bak, c. tang and k. wiesenfeld, “self-organized criticality: an explanation of the 1/f noise”, phys. rev. lett., vol.59, no. 4, pp. 381–384, 1987. [5] d. dhar, “self-organized critical state of sandpile automaton models”, phys. rev. lett., vol. 64, no. 14, pp. 1613–1616, 1990. [6] d. dhar, “theoretical studies of self-organized criticality”, physica a, vol. 369, no. 1, pp. 29–70, 2006. [7] p. grassberger and s. s. manna, “some more sandpiles”, j. phys. france, vol. 51, pp. 1077–1098, 1990. [8] v. s. poghosyan, s. y. grigorev, v. b. priezzhev and p. ruelle, “pair correlations in the sandpile model: a check of logarithmic conformal field theory”, phys. lett. b, vol. 659, pp. 768–772, 2008. [9] su. s. poghosyan, v. s. poghosyan, v. b. priezzhev and p. ruelle, “numerical study of correspondence between the dissipative and fixed-energy abelian sandpile models”, phys.rev. e, 84, 066119, 2011. [10] a. fey, l. levine, and d.b. wilson, “driving sandpiles to criticality and beyond”, phys. rev. lett., 104, 145703, 2010. [11] a. fey, l. levine, and d. b. wilson, “approach to criticality in sandpiles”, phys. rev. e 82, 031121 (2010). [12] v. s. poghosyan, s. y. grigorev, v. b. priezzhev and p. ruelle, “logarithmic two-point correlators in the abelian sandpile model”, j. stat. mech., vol. 2010, no. 07, p07025, 2010. [13] v. s. poghosyan and v. b. priezzhev, “the problem of predecessors on spanning trees”, acta polytechnica, vol. 51, no. 1, pp. 59–62, 2011. [14] s. n. majumdar and d. dhar, “height correlations in the abelian sandpile model”, j. phys. a: math. gen., vol. 24, no. 7, l357–l362, 1991. [15] o.delmas, s. perennes, “circuit-switched gossiping in 3-dimensional torus networks”, proc. euro-par, parallel-processing, pp. 370–373, 1996. submitted 10.06.2016, accepted 25.10.2016. h. nahapetyan, s. poghosyan, v. poghosyan and yu. shoukourian 1 2 5 ²í³½³ïáõûïç d-ã³÷³ýç ²µ»éû³ý ³íïáù³ïç ½áõ·³ñ»é ùá¹»é³íáñù³ý ù»ãá¹ ð. ü³ñ³å»ïû³ý, ê. äáõáëû³ý, ì. äáõáëû³ý ¨ úáõ. þáõùáõñû³ý ²ù÷á÷áõù ²ûë ñá¹í³íáõù µ»ñí³í ¿ ù³é³ïáõëç ó³ýó»ñç ñ³ù³ñ [1]-áõù ý»ñùáõíí³í ³ëïõ³ûçý í³íïáõûãç ëý¹ñç áý¹ñ³ýñ³óáõùá l d, d 2 n d -ã³÷³ýç ó³ýóç ñ³ù³ñ: êý¹çñá ï³û³ýáõù ¿ s2d ·ñ³ýáí l d ó³ýóç í³íïù³ý ù»ç: îíû³é ëý¹ñç éáõíù³ý ñçù³ý íñ³ ùß³ïí»é ¿ d-ã³÷³ýç µçç³ûçý ³íïáù³ïç ùá¹»é³íáñù³ý ½áõ·³ñ»é³óí³í ³é·áñçãù: àñå»ë µçç³ûçý ³íïáù³ïç ûñçý³ï ¿ ¹çï³ñïí»é ²µ»éû³ý ³í³½³ïáõûïç ³ýï³ûáõý íç׳ïý»ñç é»é³ùë³óçáý åñáó»ëá: ð³ù³å³ï³ëë³ý íñ³·ñ³ûçý ÷³ã»ãý»ñá ý³ë³·íí»é »ý openmp ¨ cuda ï»ëýáéá·ç³ý»ñç ïçñ³éù³ùµ: ´»ñí³í »ý ï³ñµ»ñ ã³÷»ñç »é³ã³÷ ó³ýó»ñç ½áõ·³ñ»é³óí³í ùá¹»é³íáñù³ý ³ñ¹ûáõýùý»ñá: ìåòîä ïàðàëëåëüíîé ñèìóëÿöèè d-ìåðíûõ àâòîìàòîâ àáåëåâîé ïåñî÷íîé ãîðêè à. íàãàïåòÿí, ñ. ïîãîñÿí, â. ïîãîñÿí è þ. øóêóðÿí àííîòàöèÿ â ýòîé ñòàòüå ïðèâåäåíî îáîáùåíèå ïðîáëåìû çâåçäíîãî ïîêðûòèÿ, âûäâèíóòîé â [1] äëÿ êâàäðàòíûõ ðåøåòîê. îáîáùåíèå äîñòèãíóòî äëÿ d-ìåðíîé ðåøåòêè l d, d 2 n. ïðîáëåìà ñîñòîèò â ïîêðûòèè l d ðåøåòêè çâåçäíûì ãðàôîì s2d. íà îñíîâàíèè ðåøåíèÿ äàííîé ïðîáëåìû, ðàçðàáîòàíà ïàðàëëåëèçîâàííàÿ ïðîãðàììà ñèìóëÿöèè d-ìåðíîãî êëåòî÷íîãî àâòîìàòà. â êà÷åñòâå ïðèìåðà êëåòî÷íîãî àâòîìàòà ðàññìîòðåí ïðîöåññ ðåëàêñàöèè íåñòàáèëüíûõ ñîñòîÿíèé àáåëåâîé ìîäåëè ïåñî÷íîé ãîðêè. ñîîòâåòñòâóþùèå ïðîãðàììíûå ïàêåòû ðàçðàáîòàíû ñ èñïîëüçîâàíèåì òåõíîëîãèé openmp è cuda. ïðåäñòàâëåíû ðåçóëüòàòû ïàðàëëåëüíîé ñèìóëÿöèè, ïðîâåäåííîé äëÿ 3-ìåðíîé ðåøåòêè ðàçëè÷íûõ ðàçìåðîâ. mpcs.pdf (p.1-8) vahagn_abstract.pdf (p.9) microsoft word 18_tadevos_baghdasaryan.doc mathematical problems of computer science 39, 135--145, 2013. 135 adapting rgt solver interface to management strategy search problems tadevos s. baghdasaryan institute for informatics and automation problems of nas of ra e-mail: tbs_@mail.ru abstract we develop a unified solver for the class of problems where the space of possible solutions can be specified by reproducible game trees (rgt) [11]. in this paper we adapt the interface of rgt solver for marketing and supply chain management (scm) problems of the class presented by valuewar and trading agents competition models [1, 2, 4]. particularly, we describe: the ways for construction of operating entities, actions and strategy plans for the valuewar tool, the ways for construction of operating entities, actions, moves and strategy plans for the tac scm game. keywords: combating and competing games, expert systems, optimal strategies, valuewar, tac. 1. introduction 1.1. definition of rgt class in the variety of problems we identify the class where the space of possible solutions can be specified by reproducible combinatorial game trees (rgt) and develop a unified software, rgt solver, for elaborating optimal strategies for any input specified problem of the class [11]. as it was demonstrated in [11], rgt is a spacious class of problems with only a few following requirements to belong to: there are (a) interacting actors (players, competitors, etc. performing (b) identified types of actions in the (c) specified moments of time and (d) specified types of situations; there are identified benefits for each of the actors; the situations the actors act in and transformed after the actions, can be specified by certain rules, regularities. we do solve games of rgt class with meanings we do specify by states, situations, actors, actions of players, evaluators of situations and regularities of transformation of situations [11]. many security and competition problems belong to rgt class since those problems always interact, and rgt requirements include the most common of them. specifically, these are network intrusion protection (ip), management in oligopoly competitions and chess-like combinatorial problems, various security problems. the unified rgt specification of problems makes possible to design a unified solver for the problems of the class [9, 10]. the solver of rgt problems is a adapting rgt solver interface to the management strategy search problem 136 package aimed to acquire strategic expert knowledge to become comparable with a human in solving hard combinatorial competing and combating problems. 1.2. rgt interpretation of management strategy search problem according to [11] market competitions by valuewar [1-3] and scm by trading agents competitions [4] (tac scm) will be the following: states are determined by the set of parameters of current competition and scenarios of competition situations are determined by the states and actions of competition actors: a company competing against a few others actions are changes of the product price and quality in valuewar case or operational moves in tac game evaluators: algorithms calculating for input situations regularities, or transformation rules, in valuewar case are determined by general microand macroeconomics laws, which are applied to the situations. for tac case these regularities are implemented on the tac server side. the shell of rgt solvers is developed to provide the user with friendly java environment for unified solution of any rgt problem. the aims of the paper are the following: to adapt marketing (by vw) and scm (by tac) problems into the framework of rgt solver to develop ways for the construction of marketing and scm strategies. in the paper we present first the adaptation of valuewar followed by tac scm. valuewar (vw) integration includes: presentation of its composing nucleous entities presentation of actions and strategy plans definitions of vw situations embedding vw interface into rgt solver tac scm integration includes: presentation of scm nucleous and composite entities presentation of moves (actions) and strategy plans definition of situation in tac scm embedding tac scm interface into rgt solver. 2. adapting rgt solver to valuewar 2.1. valuewar so far the existing framework of rgt solver was implemented for the chess problem. but it is being modified to be more flexible in order to include other problems of the rgt class as well, in particular case the marketing and supply chain management problems presented by valuewar tool and tac scm model [9, 12, 13]. valuewar is a tool for the marketing strategy analysis, which presents a model of oligopoly competition of a few companies competing within a specified market. every company competes for one of the predefined goals. each of them is assigned to one of the various strategy plans that describe qualitative changes of basic competition parameters price and quality of items (services) they produce [2]. running the simulation a number of times (every time with a different strategy plan) and having the needed results, it is possible to separate the most optimal strategy plan for the given competitor playing within the selected type of market [5]. t. baghdasaryan 137 2.2. valuewar nucleous entities as an equivalent of figure in chess application of rgt solver, in valuewar we represent operating parameters as price (p) and quality (q). valuewar operates with strategy plans that represent the qualitive changes of the operating parameters mentioned above. in the frames of the solver we describe each parameter as a nucleous element with a particular name, type and value range. namely, for the price it would look in the following way [10]: price: quality: name “price” value range >=0 name “quality” value range in [0,100] 2.3. strategy plans for valuewar another equivalent is the definition of action, which is represented as move in chess and as a concept of strategy plan in valuewar. an sp describes the manner of changing p and q operational parameters. a typical example of sp would be, for instance, “decrease p, increase q”. at this point the actions are increase and decrease. each action will change the operating parameter in a predefined manner and by some delta. we describe action objects with the solver and further add their implementations within each reality. in our case we should apply actions over price and quality, so the same action is implemented twice both for the price object and quality object. depending on the application the agent selects and implements the corresponding version, passing the object as a parameter. in other words, in real world we can create some set of verbs. and having some real objects we define how to apply a particular verb to a particular object. it is possible that some action cannot be applied to the given object. such a case brings to nonsense unless we describe an appropriate implementation for that case. here we define some set of actions (verbs) and some set of objects (realities). so, for each object we have to create an implementation of the same action and associations between the object’s instance of that verb and the original verb. decrease price price reality price > 0 set of actions applicable to price increase implementation decrease implementation maximize implementation keep_unchanged implementation ..... implementations decrease quality quality reality quality in [1..100] set of actions applicable to quality increase implementation decrease implementation maximize implementation keep_unchanged implementation ..... implementations action applied to quality action applied to price fig. 1. application of the same action to different entities: price and quality. price price > 0 quality quality > 0 adapting rgt solver interface to the management strategy search problem 138 at the association level when met a term like “action over an object” (“increase price”) the agent starts to find a link between the “action” and its corresponding instance, related to the given “object”. the presence of such a link means that the action is applicable to that object and the agent will implement that term, otherwise the term is interpreted as nonsense. thus, for example, “increase” action will work differently for p and q (assuming that it is implemented for both of them). so, having such a set of actions, we create a set of various strategy plans and control the way of changing the operational parameters (p and q). 2.4. situation of valuewar a situation in valuewar is considered as a current market state. all parameters that form the market model with their values in couple with parameters of the participating parties (competitors) represent a set that we will call a situation (state). in its turn, every single move of a competitor (in accordance with its strategy plan) affects the market and changes its state, in other words, the competitor’s move changes the situation. in order to run the created strategy plans (performing strategy plan simulation) in the frames of the rgt solver we created a simple market model, based on all the parameters present in the basic valuewar market model. also, the market model may be switched to perform different predefined market types. in initial valuewar there are about 70 parameters forming the model, including such factors as demand, supply, population, per-capita-income, sales, total sales, market share and others. each of the forming factors is in interconnection with one or more others and is calculated by its corresponding formulas. the detailed description of the model is out of the scope of this paper and in fact, there may be different variations of it. actually, any market model is acceptable for valuewar framework problem, the only requirement is that the model environment should develop and change, affected by two single driving parameters price and quality. 2.5. embedding valuewar interface into rgt solver the valuewar simulator is embedded into rgt solver framework as a separate tab, which contains the main valuewar gameboard and subwindows for displaying various parameters. the gameboard is composed of two axes, forming a square of [p, q] positions field. the competitors are presented on the board as smaller squares of predefined color and their positions on the board reflect their values of p and q. as the simulation is in process, their positions may change in accordance to their strategy plans and achieved [p,q] values. sp competitor parameters sp competitor parameters competitor parameters sp competitor parameters market parameters sp fig. 3. affection of competitors’ strategy plans on a valuewar market model. figure 2. set of valuewar strategy plans. strategy plan action for the price action for the quality strategy plan action for the price action for the quality strategy plan action for the price action for the quality t. baghdasaryan 139 before the simulation starts, the user selects a type of a market and strategy plans for each of competing parties. then the user may choose the simulation to proceed automatically or step-by-step. the simulation is splitted into n periods. during a single period each party makes one pq-move and after that the market changes are calculated. every party has its separate window that reflects its related market parameters. in addition, market has its own window for reflecting its specific factors’ values. the user can change the strategy for any competitor at any period of the simulation, as well as the market type itself. in both cases the simulation continues with the new changes, starting from the next period. 3. adapting rgt solver to tac scm 3.1. tac scm comparing to the valuewar, tac problem is more complex not only because of more parameters present, but also that in tac there is higher level of uncertainty. there are two separate parts of a game – the first onedealing with customers and the second onedealing with suppliers [4]. in the tac scm game several agents compete in the market of personal computers assembling pcs of 16 configurations. configurations depend on types of each 4 main supply components used: cpu, ram, motherboard and hdd. every main component is being produced by 2 supplier brands and available in 2 qualitative options: high and low (speed or capacity). besides, there are limitations in compatibility of cpus with mboards: an mboard of one brand must be equipped only by a cpu of the corresponding brand. the actors in tac scm are: customers, which compose the demand of different types of pcs. on the first d day of every tac cycle they send numbers of rfqs (requests for quotes) to agents. an rfq includes: a configuration of requested pc, quantity, desired price, delivery date and penalty for any delay. agents (which have assembly factories (with the given daily production capacity), bank accounts and warehouse), which receive rfq bundles, analyze them and send back to customers as offer bundles (an offer includes: proposed configuration, quantity, price and delivery date). as a result, on the d+1 second day of the cycle customers may order some of the offers to the agent. suppliers, which produce the supply components. there are 2 suppliers for each main component (cpu, hdd, ram and mboard). each supplier has its daily production capacity and may produce 2 qualities of the same supply component. suppliers receive rfq bundles figure 4. the valuewar gameboard. -p +q -q -p+q +p+q -p-q +p-q p1= x11 p2= x12 p3= x13 p4= x14 p1= x21 p2= x22 p3= x23 p4= x24 p1= x31 p2= x32 p3= x33 p4= x34 p1= x41 p2= x42 p3= x43 p4= x44 c4 c1 c2 c3 adapting rgt solver interface to the management strategy search problem 140 from agents on day d+1, just after getting orders from customers. an agent rfq includes: requested supply component, quantity, desired price and delivery date. on the next d+2 day suppliers send back offer bundles with available quantities and delivery dates of the requested supply to the agent. then the agent may order some of those offers from the supplier on the same day. when an agent gets all the necessary supplies needed for any particular customer’s order, it starts the assembling itself on its factory. the assembling may last a number of days, taking into account: number of production cycles required for a single pc of the type, quantity of ordered pcs of the type and daily production capacity of the factory, in cycles. after the order is completed it is delivered to the customers and the agent’s bank account is updated accordingly, subtracting penalties for any delay (if any). the minimal full tac cycle lasts for 6 days. the game duration is t periods (days). every next day a new main tac cycle starts, meanwhile the deals started in previous cycles are still being proceeded. after the last period the game is over and the agent with the biggest bank account is considered as a winner. in valuewar we had two driving parameters, p and q – let’s say two dimensions. in tac they are much more: 4 types of components x 2 brands for each type x 2 qualities of each component, also various dates, etc – so, due to this “multidimentional factor” there is no explicit gameboard applicable. 3.2. nucleous and composite entities of tac scm the main base nucleous entities for tac are: date, price, quantity, speed, base component type, base component quality, base component brand, factory capacity, factory utilization, bank account, etc. more complex (composite) entities are: assembled product, particular supply component, bank, factory, agent, customer, supplier, etc. [10]. let's take an example for component. there are 4 types of components: cpu, hdd, mboard and ram. each of them is manufactured by 2 component-specific brands, and each brand produces 2 qualities of that component. for cpu component there are 2 manufacturers, pintel and imd, which manufacture 2 qualities of cpu 5ghz and 2ghz. for hdd component there are 2 manufacturers (watergate and mintor) producing 2 qualities of them (300gb and 500gb). for ram component mec and quinmax produce 2 qualities of them (1gb and 2gb). finally, for motherboard component basus and macrostar produce motherboards for pintel and imd (there is a limitation that pintel cpus will work only on pintel motherboards and imd cpus will work only on imd motherboards). first of all we should describe the basic nucleous elements with solver (examples below describe brands, capacity, component type, mboard type, price, date, utilization, production capacity entities): then based on them, we can describe some specific elements (examples for capacities for different components, production-specific brands, types of motherboards): fig. 5. example tac nucleous elements. component type name component type value in [cpu, ram, hdd, mboard] component quality name speed value in [2ghz, 5ghz] brands name brand value in [pintel, imd, basus, macrostar, mec, queenmax, w atergate, mintor] component quality name capacity value in [1gb, 2gb, 300gb, 500gb] price name price value >= 0 qualtity name quantity value >= 0 utilization % name utilization value in[1, 100] production capacity name production capacity value > 0 date name date value >= 0 t. baghdasaryan 141 further, more complex and final entities will be described (particular supply components, like "imd cpu 2ghz with base price of $1000", "mintor hdd 500gb with base price of $300" or final entities like "cpu supplier #1 branded 'pintel', with production capacity of 2000 cycles"): following this path, we can describe all the entities present in the tac problem, which will be equivalent to "figure" concept in a chess problem. below in figure 8 some examples are given: fig. 8. examples of tac entities. fig.7. a complex tac entity “queenmax 2gb ram” as a supply component fig. 6. example of specific tac nucleous elements. component quality name capacity value in [1gb, 2gb, 300gb, 500gb] component capacity name ram capacity value = 15 capacity in [1gb, 2gb] component capacity name hdd capacity value = 16 capacity in [300gb, 500gb] brands name brand value in [pintel, imd, basus, macrostar, mec, queenmax, w atergate, mintor] component brand name hdd brand value =11 brand in [w atergate, mintor] component brand name ram brand value =12 brand in [mec, queenmax] component brand name cpu brand value =13 brand in [pintel, imd] component brand name mboard brand value =14 brand in [basus, macrosta] component type name mboard type value in [pintel board, imd board] component type =mboard component type name component type value in [cpu, hdd, mboard, ram] price name price value >= 0 supply components name supply value queenmax 2gb component type = ram ram capacity = 2gb ram brand = queenmax price = 200 component capacity name ram capacity value =15 capacity in [1gb, 2gb] component brand name ram brand value = 12 brand in [mec, queenmax] component type name component type value in [cpu, ram, hdd, mboard] factory name factory value =1 production capacity = 2000 utilization = 50 factory name factory value =1 production capacity = 2000 utilization = 50 supplier name supplier value cpu supplier #1 component type = cpu production capacity = 2000 brand = pintel component name basus pintel board value =200 component type =mboard mboard type =pintel board mboard brand =basus price =250 supply component name supply value pintel 5ghz component type =cpu speed =5ghz cpu brand =pintel price =1500 supply component name supply value imd 2ghz component type =cpu speed =2ghz cpu brand =imd price =1000 component name mintor 500gb value =401 component type =hdd hdd capacity =500gb hdd brand =mintor price =300 supply component name supply value queenmax 2gb ram component type = ram ram capacity = 2gb ram brand = queenmax price = 200 adapting rgt solver interface to the management strategy search problem 142 3.3. defining moves and strategy plans for tac scm as in valuewar, strategy plans for tac specify the qualitative changes of some operational parameter(s). in the frames of tac problem, we can separate two concepts from each other: move and strategy plan (sp). we call a statement as “move” which contains entities with all their parameters specified with exact values (equivalent to “figure move” in chess). in contrast, we call a statement as “sp” in which at least one parameter is not specified with its value. again, an sp describes qualitative action(s) over tac entity (entities), whereas a tac move describes an exact (quantitative) action. in case of move the agent directly executes the statement. in case of sp (with missing values of one or more parameters) the agent may generate several moves from the same sp by substituting the missing values with some others from their acceptable value range. in order to select the most optimal move from the generated set it applies a game subtree for that appeared intermediate problem with possible usage of the existing knowledge base. a typical example of tac move would be the following statement: “buy n quantity of queenmax 2gb ram by d date” [7]. here we have 3 predefined entities: “quantity”, “queenmax 2gb ram” supply component and “date”. all three entities are populated with their exact values: quantity=n, date=d, supply=“queenmax 2gb ram”. replacing at least a single value from any of parameters will force to interpret the given statement as a strategy plan as shown in examples: note, that in complex entities not just any parameter may be missed. if in the example above we replace the value of “component type” (“ram”), then the statement loses its meaning in tac context: in “buy n quantity of queenmax 2gb by d date” the agent will not understand the meaning of “queenmax 2gb” – that is ram, hdd, mboard or cpu. otherwise, it will require more complex searching mechanisms in the agent implementation, which is currently out of scopeof this project. therefore, a typical example of sp would be the following statements: “buy n quantity of queenmax by d date” (as queenmax produces rams of different capacity); “buy quantity of ram by d date”(quantity may be any number within its predefined range); supply component supply queenmax 2gb ram component type = ram ram capacity ram brand = queenmax price = 200 qualtity quantity date date d fig.9. a single missing parameter in quantity (above) or “supply component” (below) causes the statement to be treated as an sp. supply component supply queenmax 2gb ram component type = ram ram capacity ram brand = queenmax price = 200 qualtity quantity n date date d fig.8. entities with fully populated values from the move’s statement. supply component supply queenmax 2gb ram component type = ram ram capacity = 2gb ram brand = queenmax price = 200 qualtity quantity n date date d t. baghdasaryan 143 “buy n quantity of 2gb ram in advance” (date is specified to be any coming number of days within the acceptable range); etc. also in the example above there is an action – “buy”. as in valuewar, here in tac we also should describe action objects with a further addition of their implementations into each entity it may be applied to. in the case above the “buy” action should be described in general and its appropriate implementation should exist for “queennmax 2gb ram” entity. that action may also take “quality” and “date” entities as mandatory (the first) or optional (the second) parameters. if no “buy” action is defined for “queennmax 2gb ram” entity, then that statement loses its meaning and that move or sp will take no effect. note, that “buy” is not defined either for entities “quality” or “date”, so the separate statements “buy n quantity” or “buy d date” are senseless and will not work. generally, in real world we can create some vocabulary of verbs. having some real objects we define how to apply a particular verb to a particular object. it is possible that some action cannot be applied to the given object. such a case brings to nonsense unless we describe an appropriate implementation for that case. here we define some set of actions (verbs) and some set of objects (realities). so, for each object we have to create an implementation of the same action and associations between the object’s instance of that verb and the original verb. at the association level when met a term like “action over an object” (“increase price”) the agent starts to search a link between the “action” and its corresponding instance within the “object”’s action set. the presence of such a link means that the given action is applicable to that object so the agent will implement that term; otherwise the term is treated as nonsense. thus, for example, “increase” action will work differently for p and q (assuming that it is implemented for both of them). 3.4. defining situation of tac scm we do not consider the tac market model because it is implemented on the server side, and according to the rules is inaccessible, so we have to deal only on the tac agent’s side. that is why we will define a tac situation by including only the states of agent-related parameters. all the agent’s entities with their current conditions and values compose a situation for the tac agent at a specific period of time. it includes: factory utilization, warehouse of supplies with quantities of any present components, bank account, orders from customers, orders to suppliers, current date, start date, etc. so there is not any entity associated to a situation, it is represented as parameters set from other entities. having defined the concept of situation, we can use it when forming a knowledge base. the knowledge base, in turn, consists of a set of knowledge elements. and a knowledge element, in general, has the following structure: an initial situation, the applied action (or sp) and the resulting situation. moves (or strategy plans) in action may consider situations directly whenever they include any conditional content. fig. 10. implementation of “buy queenmax 2gb ram” action. action name buy value supply component supply queenmax 2gb ram component type = ram ram capacity = 2gb ram brand = queenmax price = 200 buy implementation module mandatory parameters optional parameters adapting rgt solver interface to the management strategy search problem 144 3.5. embedding tac scm interface into rgt solver the tac agent template has its own interface; we just duplicate its data monitors into our interface to track the game progress. for that we integrate a data transfer module into the rgt solver and the tac agent’s template in order to transfer the ongoing data between each other. the tac solver module handles all the objects defined for the tac by the solver interface (namely: entities, actions, moves, strategies), so it is actually considered as the agent’s “brain”. the module may be integrated either in the agent or in the rgt solver. on the first case the agent will transfer only the information to be displayed on the solver’s tac gameboard tab. on the second case at every game event the agent will pass the incoming information from the server to the solver to be handled. after that the solver processes the data and passes the results back to the agent for further sending to the tac server. conclusion the rgt solver framework with two integrated management-focused problems (the valuewar strategy assessment tool and the tac scm game) is represented. the ways for constructing objects, actions and strategies for the given management problems are described. particularly: the ways for construction of operating entities, actions and strategy plans for valuewar tool; the ways for construction of operating entities, actions, moves and strategy plans for tac scm game. also, we embedded the corresponding interfaces of each valuewar and tac scm problems into separate tabs within the main solver framework. references [1] m. chussil and d. reibstein, “putting the lessons before the test. in wharton on to analyse & develop competitive strategies”, john wesley & sons, pp 343-368, 1997 [2] m. chussil, d. reibstein, strategy analysis with value war, the scipress, 1994. [3] e. pogossian, "focusing management strategy provision simulation", csit2001 3d international conference in computer science and information technologies, 5p., 2001. [4] tac scm web page: http://tac.sics.se/page.php?id=1 [5] t. baghdasaryan, e. danielyan and e. pogossian, “testing oligopoly strategy plans by their on the job performance simulation”, international conference csit2005, 8p, 2005. [6] t. baghdasaryan, e. danielyan and e. pogossian, "supply chain management strategy provision by game tree dynamic analysis", fifth international conference sbm2006, 2p., 2006. [7] t. baghdasaryan, “scripting language and design tool for trading agents strategy plans”, csit2007 6th international conference in computer science and information technologies, 5p., 2007. [8] t. baghdasaryan, a. grigoryan and z. naghashyan, "development of a scripting language interpreter for acquisition of expert knowledge in a regular way", csit2009 7th international conference in computer science and information technologies, 5p., 2009. [9] z naghashyan and e. pogossian, "developing java software for representation, acquisition and management of strategy knowledge", mathematical problems of computer sciences, pp.187-195, 2010. [10] k. khachatryan and v. vagradyan, “graphical language interpreter unified for ssrgt problems and relevant complex knowledge”, csit2011 8th international conference in computer science and information technologies, pp. 178-182, 2011. [11] e. pogossian, “effectiveness enhancing knowledge based strategies for ssrgt class of defense problems”, proceedings of nato advanced study institute (asi), 2011. t. baghdasaryan 145 [12] k. khachatyan and s. grigoryan “java programs for presentation and acquisition of meanings in ssrgt games”, proceedings of seua annual conference, 7p., 2012. [13] k. khachatryan and s. grigoryan, “java programs for matching situations to the meanings of ssrgt games”, proceedings of seua annual conference, 5p., 2012 submitted 30.11.2012, accepted 02.02.2013. rgt solver համակարգի ինտերֆեյսի համակերպումը կառավարման ռազմավարությունների փնտրման խնդիրներին թ. բաղդասարյան ամփոփում մշակվում է rgt solver ունիֆիկացված ծրագրային համակարգ այն դասի խնդիրների լուծման համար, որում հնարավոր լուծումների տիրույթը ներկայացվում է վերարտադրվող խաղային ծառերի միջոցով: նկարագրվում է rgt solver համակարգի ինտերֆեյսի համակերպումը տվյալ դասին պատկանող մարքեթինգի և մատակարարման շղթայի կառավարման խնդիրներին, որոնք ներկայացված են համապատասխանաբար մարքեթինգի ռազմավարությունների վերլուծման valuewar գործիքային փաթեթով և մրցակցող առևտրային գործակալներ (tac scm) խաղով: մասնավորապես, նկարագրվում է հետևյալը. valuewar թաղանթի համար բաղադրիչների, գործողությունների և ռազմավարական պլանների նկարագրման եղանակները, tac scm մոդելի համար բաղադրիչների, գործողությունների, քայլերի և ռազմավարական պլանների նկարագրման եղանակները: адаптация решателя rgt-игр к задачам поиска стратегий менеджмента т. багдасарян аннотация мы разрабатываем унифицированный механизм (rgt solver) для решения класса задач, в которых пространство возможных решений представляется воспроизводимыми игровыми деревьями (rgt). в рамках статьи мы описываем адаптацию интерфейса rgt solver к задачам маркетинга и управления цепью снабжения, относящихся к данному классу и представленными, соответственно, моделями valuewar и конкурирующего торгового агента (tac scm). в частности, мы описываем: способы описания компонент, действий и стратегических планов для оболочки valuewar; способы описания компонент, действий, ходов и стратегических планов для модели tac scm. начиная с начала 2000 года осуществляется внедрение ghis в здравоохранении, в рамках принятого проекта о реформирование информ mathematical problems of computer science 45, 122--126, 2016. symbiosis of email and sms arman h. harutyunyan1, sona h. gharagyozyan2 1yerevan state university 2institute for informatics and automation problems of nas ra e-mail:a_harutyunyan@ipia.sci.am, sona@ipia.sci.am abstract the specific features of the means of written communication through computer and mobile networks, as well as the prehistory of the establishment of hybrid communication system using ip and gsm network technologies have been examined. keywords: symbiosis, sms, email, predictive. 1. introduction the new means of written communicationemail and sms, appearing at the end of the twentieth century, pressed the traditional postal service keeping it mainly as a means to send official documents. in place of the past epistolary genre with its inherent personality came concise information emails. nevertheless, the speed of delivery of messages to the addressee provides undeniable advantages to the electronic means of written communication. at the same time, integrated systems using "one package," technology of email and sms, which are complementary to each other are of particular interest. without going into a detailed assessment of the advantages and disadvantages of these systems, we have pointed out only the specific functional features of email and sms technologies. 2. problems and solutions originally designed for the exchange of written communications between computers, email is considered as a system “on demand” despite of its rapid delivery to the addressee. the fact of reading delivered message depends on the time the addressee opens his mailbox. in contrast to email, sms is not only a mailing system: the volume of transmitted message is limited, the 122 mailto:a_harutyunyan@ipia.sci.am mailto:sona@ipia.sci.am a. harutyunyan, s. gharagyozyan 123 delivery of the message is only possible in the zone of mobile network coverage or according to the current roaming agreement of mobile operator. the message delivery to the “pocket” of the mobile subscriber is an indisputable advantage of sms technology. accordingly, either the message is delivered “on demand” almost anywhere in the world (email), or directly into the hands of the addressee, but for the above mentioned limitations (sms). the integrated solution to the “distance” problem of particular sms service is solved by using ip-communications, as a main environment followed by the locking of current mobile network and web technologies providing the possibility sending sms to a specified, local mobile network from computers from virtually any region of the world. another above discussed problem: problem “on demand” is associated with the integration of sms technology in email system in the form of smsnotification systems on receipt of emails in the subscriber's mailbox. such postal sms-informants analyze the incoming correspondence of the subscriber separating the letters with return addresses listed by the subscriber to the "white list" to which the subscriber receives smsnotification, in case of receiving correspondence [1-3].. in other systems of email-informants, the initiative of sending notification is given to the sender. the problems to create a "hybrid" system of internet/sms is connected to the specifics of displaying web "pictures" on the display of a mobile phone and with the difficulties inputting alphabetic information. if a modern smartphone with an informant and virtual keyboard is being used while working with the system, no special problems will arise. however, the possibility of a user-friendly operation in a system with other common models of mobile phones leads to the need for a special service software. mobile subscribers use a wide variety of models of mobile phones: from budget phone with a small screen and a set of keys, providing only the basic functions of mobile communication: a telephone communication and exchange of sms to multifunctional smartphones with display of 5-6" touchscreen, virtual keyboard, multi-core processors and with the possibility to use internet. there often exists the possibility of connecting to the internet in common models of mobile phones with screens of about 2" and button dialing keyboard. however, the phenomenon of screen with small format and resolution, as well as of touch-tone, makes it extremely inconvenient use these phones working with network resources. however, not having a wide range of additional functions specific smartphones (often redundant), and they are today in demand (the share of push-button telephones in russian mobile networks, for example, is more than 30%. in the first quarter of 2015 there have been sold more than 2.7 million of these mobile phones [4,5]) corresponding to the basic purpose of mobile phones and having the indubitable advantage of the "pocket" device, due to the size and the weight (unlike the majority of smartphones, which are considered as "wearable" devices). these circumstances determine the feasibility of studies and the development of new solutions providing information on mobile devices of a similar class, the development of adaptive interactive "comfort" systems graphical interaction of mobile phone users with communication and information resources of mobile and ip networks (it is evident that this interaction is provided by gsm / ip servers). these dialogue systems should be extremely concise, limitative as far as possible, the participation of the user answers "yes-no" considering the complexity of a set of queries and text on a mobile phone keypad. problems of writing text messages are solved using predictive typing systems anticipating in the already typed letters of the options of the current and the next word (or phrases) while user is typing using the built-in phone dictionary. such systems for mobile phones (designed mainly for typing sms) have been known since 1999 with the development and encapsulation of the t9 system in mobile phone software and similar functions of itap and ezi systems [6,7,8]. being developed by tegic communications, t9 is used in mobile phones by most of the major manufacturers (nokia, symbiosis of email and sms 124 samsung, sony ericsson, etc.). t9 is considered as the most popular system of predictive typing. unlike т9, itap attempts to predict short phrases as well, analyzing not only typed letters of the current word, but also the previous text. basic dictionaries of these predictive typing systems include 35-60 thousands of words and expressions, and support most of the european and some asian languages. unlike the mentioned predictive systems when typing the text of sms there can be offered an algorithm of serial-word letter by letter for the possible continuation of the typing word, using funded words database individual for each user according to the most frequently used words by the user while typing sms: "verbal image" of the lexicon used by the user stored on the sms server. this database is formed by accumulated words containing in the previous sms sent by the user. depending on the frequency of the repetition of words corresponding “weight” is attributed to each word determined by the number of repetitions of the word accumulated in the sms. such predictive system is installed not in the phone but in a centralized server. the system can be used with any mobile telephone having access to the internet, respectively, without requiring additional resources (memory, cpu) for its functioning. the system generates prediction not by semantic analysis of words and sentences comparing them with the information from the database but by the coincidence of images: putting in comparison the literal consistent of the typing string to the "images" of the vocabulary fund used by the user while sending sms accumulated in the individual database. the analysis on the coincidence of characters ("images of words") allows to use prediction system without being tied to a particular language. the database with which the system operates is located on a server in the form of individual lists for each registered user containing a sequence of characters (letters) forming a word. databases are automatically replenished with words from sms sent by the user. the use of automatically generated personal database "images of the words" increases the probability of a correct prediction at the initial stages of typing allowing, at the same time, significantly to reduce the size of the lexicon of the dictionary. while typing, the first two letters of the typing word are taken as a reference (base) letters of the word; further the letters set is compared with the dictionary with the issuance tips with the options of word completion. when typing messages a user can use both latin alphabet and cyrillic alphabet. accordingly, while sending sms there is a need to convert in cyrillic in the similar-sounding words, written in latin letters (operation of transliteration) for reliable message playback at any gadget. in asnet computer network users can access a number of services of infocommunication applications sharing web and sms technologies (www. asnet.am/sms applications),developed in the years from 2007 to 2013 at the institute for informatics and automation problems of national academy of sciences of the republic of armenia(iiap nas ra) [9,10]. the majority of the above mentioned services used as a basis for the development of info-communication mss system of iiap for mobile phones, providing the possibility to comfortably work with smsoip and emails. such system provides: 1. formation and sending of sms via ip network using web technologies 2. formation and sending of email with sms notification (mobile network of the region where the server is running) containing user-selected fragment of the letter. 3. the use of graphical dialog interface, automatically adapting to the class of the gadget served at the moment. 4. when accessing the system by a phone with a small screen and keyboard, automatic switching to the dialogue version, minimizing data volume, user typing in the formation a. harutyunyan, s. gharagyozyan 125 of sms/email at the same time the formation of the address part of email, sms in single request in a natural language 5. the mechanism of "prediction" typed by the user during the formation of the message. 6. the use of the centralized individual savings with no database on the server system (user requisites, tel. book, address book, dictionary of sms: "verbal images”, etc.). references [1] [online]. available: https://help.mail.ru/mail-help/settings/notifications [2] [online]. available: http://iglous.ru/besplatnyj-sposob-poluchat-sms-uvedomleniya-opochte/ [3] d. gevorkyan, k. khachatryan, a. nanassian, a. petrosyan, g. petrosyan, v. sahakyan and e.vardanyan “mail informerselective incoming e-mail instant phone notification system”, proceedings of international conference computer science and information technologies, yerevan, pp. 466-468, 2009. [4] [online]. available: http://secretmag.ru/news/2015/06/24/knopochnie-telefoni/ [5] [online]. available: http://sia.ru/?action=show_news&id=305085§ion=484 [6] [online]. available: http://www.ixbt.com/mobile/review/prtxtsms.shtml [7] [online]. available: http://www.genon.ru/getanswer.aspx?qid=29112f32-777b-4d02b3ab-bbec2afb72ef [8] [online]. available: http://solo-project.com/articles/category/12/message/1241/ [9] d. gevorkyan, а. nanassian and k. khachatryan “new web resources asnet.am”, computer science and information technologies, proceedings of international conference, yerevan, pp. 311-313, 2011. [10] а. nanassian and k. khachatryan “mail2sms.asnet.am – the alert system of incoming”, proceedings of international conference computer science and information technologies, pp. 459-462, 2013. submitted 04.11.2015, accepted 22.02.2016 էլ-փոստի և sms հաղորդագրության համակցում (symbiosis) ա. հարությունյան և ս. ղարագյոզյան ամփոփում դիտարկվել են համակարգչում և բջջային ցանցում հաղորդագրությունների փոխանակման տարածված համակարգերը, ստեղծելով տվյալների փոխանակման նախադրյալներ, որոնք օգտագործում են ip և gsm ցանցերը։ symbiosis of email and sms 126 симбиоз email и sms а. арутюнян и с. карагезян аннотация рассмотрены особенности распространенных систем обмена письменными сообщениями в компьютерных и сотовых сетях, предпосылки создания гибридных систем передачи сообщений, использующих технологии ip и gsm сетей. рассмотрены проблемы доступа к ресурсам подобных систем с мобильных гаджетов, пути их решений. microsoft word informstochnor.doc ìàòåìàòè÷åñêèå âîïðîñû êèáåðíåòèêè è âû÷èñëèòåëüíîé òåõíèêè 30, 87--91, 2008. 87 оптимальная передача информации через пороговую систему со стохастическим резонансом арсен а. ахумян1, вардан ж. товмасян2, оганес c. ароян2 1 институт радиофизики и электроники нан армении, аштарак 2 ереванский государственный университет, армения e-mail: arsen@irphe.am, tvardan@mail.ru, hharoyan@ysu.am аннотация рассмотрена зависимость скорости передачи информации через нелинейный канал со стохастическим резонатором от уровня шума. выведена связь межу спектром сигнала и спектральной полосой шума, при котором соотношение сигнал/шум на выходе канала, а также скорость передачи информации имеют максимальное значение. литература [1] r. benzi, a. sutera and a. vulpiani, the mechanism of stochastic resonance, j. phys. a 14, pp 453–457, 1981. [2] b. mcnamara and k. wiesenfeld, theory of stochastic resonance, phys. rev. a 39, pp 4854–4869, 1989. [3] z. gingl, l. b. kiss and f. moss, non-dynamical stochastic resonance: theory and experiments with white and arbitrarily coloured noise, europhys. lett. 29, pp 191–196, 1995. [4] f. chapeau-blondeau and x. godivier, theory of stochastic resonance in signal transmission by static nonlinear systems, phys. rev. e 55, pp 1478–1495, 1997. [5] l. gammaitoni, p. hänggi, p. jung and f. marchesoni, stochastic resonance, revs. of modern physics 70, pp 223–287, 1998. [6] c. e. shannon and w. weaver, the mathematical theory of communication, the university of illinois press, 1949. [7] в. с. анищенко, а. б. нейман, ф. мосс, л. шиманский-гайер, стохастический резонанс как индуцированный шумом эффект увеличения степени порядка, уфн, том 169, №1, pp 7–38, 1999. оптимальная передача информации через пороговую систему со стохастическим резонансом 88 æýýáñù³óç³ûç ûåïçù³é ñ³õáñ¹áõùá ëïáë³ëïçï 黽áý³ýëáí ß»ù³ûçý ñ³ù³ï³ñ·áõù ². ð³ëáõùû³ý, ì. âáíù³ëû³ý, ð. ð³ñáû³ý ²ù÷á÷áõù ¸çï³ñïí³í ¿ çýýáñù³óç³ûç ñ³õáñ¹ù³ý ³ñ³·áõãû³ý ï³ëí³íáõãûáõýá ³õùáõïý»ñç ù³ï³ñ¹³ïçó ëïáë³ëïçï 黽áý³ïáñáí ï³åáõõáõù£ ¸áõñë ¿ µ»ñí³í ³½¹³ýß³ýç ¨ ³õùáõïç ëå»ïïñ³é é³ûýáõãûáõýý»ñç ùçç¨ ï³åá, áñç ¹»åùáõù »éù³ûçý ³½¹³ýß³ý ³õùáõï ñ³ñ³µ»ñáõãûáõýá ¨ çýýáñù³óç³ûç ñ³õáñ¹ù³ý ³ñ³·áõãûáõýá ëï³óíáõù »ý ³é³í»é³·áõûý£ d:\sbornik\...\hodvac1.dvi mathematical problems of computer science 24, 2005, 5{10. data flow analysis b y linear p r ogr amming m odel a r m e n h . a la ve r d ya n , v ilik m. k a r a kh a n ya n a n d v a r d a n r . to n o ya n institue for informatics and automation problems of nas of ra e-mail armen am@yahoo.com abstract the paper presents the general discussion of the data °ow algorithms structuring problem on one hand, and, as an example of that particular problem class, the analysis of the linear programming problem when the objective function coe±cients vary depending on the data °ow. the problem is in reconstruction of current result with such an approach, which is the most plain from the full solution of the problem by the dynamic data set °ow. refer ences [1 ] l . a s la n ya n , j. ca s t e lla n o s , f. min g o , h . s a h a kya n , v . r ya z a n o v, a lg o r it h m s fo r d a t a flo ws , in t e r n a t io n a l jo u r n a l in fo r m a t io n th e o r ie s a n d a p p lic a t io n s , is s n 1 3 1 0 -0 5 1 3 3 , v o lu m e 1 0 , n u m b e r 3 ( 2 0 0 3 ) , p p . 2 7 9 -2 8 2 . [2 ] b a r b a r a g. r yd e r a n d ma r vin c. p a u ll, in c r e m e n t a l d a t a -° o w a n a lys is a lg o r it h m s , a cm tr a n s a c t io n s o n p r o g r a m m in g l a n g u a g e s a n d s ys t e m s ( top l a s ) , v o lu m e 1 0 is s u e 1 ( 1 9 8 8 ) , p p . 1 -5 0 . [3] â. ê. ëåîíòüåâ, óñòîé÷èâîñòü â ëèíåéíûõ äèñêðåòíûõ çàäà÷àõ, ”ïðîáëåìû êèáåðíåòèêè”, áûï. 35, 1979. [4 ] a . a la ve r d ya n , l in e a r p r o g r a m m in g w it h ch a n g in g co e ± c ie n t s of ob je c t ive fu n c t io n , in tr a n s a c t io n s o f t h e in s t it u t e fo r in fo r m a t ic s a n d a u t o m a t io n p r o b le m s o f n a s r a , ma t h e m a t ic a l p r o b le m s o f co m p u t e r s c ie n c e , 2 0 0 5 , y e r e va n . [5 ] ch . p a p a d im it r iu , k . s t e ig lit z , co m b in a t o r ia l o p t im iz a t io n : a lg o r it h m s a n d co m p le xit y, p r e n t ic e -h a ll in c , 1 9 8 2 . [6] þ. êóçíåöîâ, â. êóçóáîâ, à. âîëîùåíêî, ìàòåìàòè÷åñêîå ïðîãðàììèðîâàíèå, âûñøàÿ øêîëà, ìîñêâà, 1980. [7] á. ìóðòàô, ñîâðåìåííîå ëèíåéíîå ïðîãðàììèðîâàíèå, òåîðèÿ è ïðàêòèêà, ìîñêâà, ìèð, 1984. 5 6 data flow analysis by linear programming model îíû³éý»ñç ñáëù»ñç í»ñéáõíáõùá áëï ·í³ûçý íñ³·ñ³íáñù³ý ùá¹»éç ². ð. ²é³í»ñ¹û³ý, ì. ø. î³ñ³ë³ýû³ý, ì. è. îáýáû³ý ²ù÷á÷áõù ðá¹í³íá ýíçñí³í ¿ ïíû³éý»ñç ñáëù»ñç ³é·áñçãùý»ñç ï³éáõóù³ý áý¹ñ³ýáõñ ëý¹ñç ùýý³ñïù³ýá ùç ïáõùçó, ¨ áñå»ë ³ûë ¹³ëç ù³ëý³íáñ ëý¹ñç ûñçý³ï‘ ·í³ûçý íñ³·ñ³íáñù³ý ëý¹ñç áõëáõùý³ëçñù³ýá, »ñµ ñáëù³ûçý ïíû³éý»ñçó ï³ëí³í ÷á÷áëíáõù »ý ýå³ï³ï³ûçý ýáõýïóç³ûç ·áñí³ïçóý»ñá: êý¹çñá áýã³óçï ³ñ¹ûáõýùç í»ñ³ï³éáõóù³ý ù»ç ¿ ³ûýåçëç ùç ùáï»óù³ùµ, áñá ³é³í»é å³ñ½ ¿ ëý¹ñç éç³ï³ï³ñ éáõíáõùçó áëï ñáëùç ¹çý³ùçï ïíû³éý»ñç µ³½ùáõãû³ý: mathematical problems of computer science 59, 57–68, 2023. doi: 10.51408/1963-0102 udc 004.934 making speaker diarization system noise tolerant davit s. karamyan1,2, grigor a. kirakosyan2,3 and saten a. harutyunyan2 1russian-armenian university, yerevan, armenia 2krisp.ai, yerevan 3institute of mathematics of nas ra, yerevan, armenia e-mail: {dkaramyan, gkirakosyan, sharutyunyan }@krisp.ai abstract the goal of speaker diarization is to identify and separate different speakers in a multi-speaker audio recording. however, noise in the recording can interfere with the accuracy of these systems. in this paper, we explore methods such as multi-condition training, consistency regularization, and teacher-student techniques to improve the resilience of speaker embedding extractors to noise. we test the effectiveness of these methods on speaker verification and speaker diarization tasks and demonstrate that they lead to improved performance in the presence of noise and reverberation. to test the speaker verification and diarization system under noisy and reverberant conditions, we created augmented versions of the voxceleb1 cleaned test and voxconverse dev datasets by adding noise and echo with different snr values. our results show that, on average, we can achieve a 19.1% relative improvement in speaker recognition using the teacher-student method and a 17% relative improvement in speaker diarization using consistency regularization compared to a multi-condition trained baseline. keywords: speaker recognition, speaker diarization, noise robustness, teacherstudent, consistency regularization. article info: received 9 january 2023; send to review 30 january 2023, received in revised form 11 april 2023; accepted 17 april 2023. acknowledgement: this research was supported by krisp.ai. 1. introduction and related work speaker recognition (sr) is a broad field of study that addresses two major tasks: speaker identification and speaker verification. speaker identification is the task of identifying a person, whereas speaker verification is the task of determining whether the speaker is who they claim to be. in this study, we focus on far-field, text-independent speaker recognition, where the speaker’s identity is determined by the speaking style rather than the content of the speech. typically, such speaker recognition systems operate on unconstrained speech utterances that are converted into a fixed-length vector known as speaker embedding. many speech0-processing tasks use speaker embedding such as speaker diarization (sd) [1, 2], automatic speech recognition (asr) [3], and speech synthesis [4, 5]. 57 58 making speaker diarization system noise tolerant in recent years, deep neural networks have actively been employed for speaker embedding extractors since d-vector [6] was proposed. subsequently, the x-vector [7] was widely used because of the superior performance achieved by employing statistical pooling and time delay neural network (tdnn). other architectures such as resnet-based convolutional neural networks and cnns with cross-convolutional layers [8, 9] were employed for capturing the traits of speech. in addition, to deal with variable-length inputs, transformer [10], cnn-lstm [11] and a slew of variants of tdnn [12] were applied for dnn-based speaker embedding extractors. finally, to reduce the computational complexity and make the models smaller, [13, 14] employed 1d depth-wise separable convolutions for the speaker recognition task. metric learning techniques have been successful in speaker recognition tasks. these methods aim to create speaker embeddings with small distances between embeddings of the same speaker and large distances between embeddings of different speakers since unsupervised clustering will be applied to embeddings later in the speaker diarization pipeline. the triplet loss was proposed in [15] which required a careful selection of a triplet because the effectiveness of the performance depended on the contrast between negative and query samples. the prototypical loss was proposed in [16], where many negative samples were used and the euclidean distance between the centroid of all negative samples and the query embedding was maximized. in the generalized end-to-end loss [17], every utterance in the mini-batch functions as a query as opposed to just one in the prototypical loss. the angular prototypical (ap) loss [18] used only one utterance from each class as the query like the prototypical loss, but with a cosine similarity-based metric. the primary use case for speaker embeddings is speaker diarization. speaker diarization is the process of dividing an input audio stream into homogeneous segments according to the speaker’s identity. a typical speaker diarization system usually consists of several steps: (1) speech segmentation, where the input audio is segmented into short sections that are assumed to have a single speaker, and the non-speech sections are filtered out by voice activity detection (vad), (2) speaker embedding extractor, where speaker embeddings are extracted from segmented sections, (3) clustering, where the extracted audio embeddings are grouped [1] into clusters based on the number of speakers present in the audio recording, and optionally, (4) resegmentation step is performed to further refine clustering results. in real-world environment, noise causes significant degradations to the performance of speaker diarization systems, and is, hence, a major problem requiring special attention. the goal of noise-tolerant speaker diarization is to achieve improved performance in noisy environments. a recent work [19] tackles this problem using the auto-encoder architecture as a dimensionality reduction module. they extract two low-dimensional codes from speaker embeddings, representing the speaker identity and irrelevant noise information, then remove the noise factors. to our knowledge, there hasn’t been a lot of research done in this particular area. asr systems also suffer deterioration due to audio noise, and this has been the subject of extensive research [20, 21, 22], some of which inspired us. in this paper, we explore several approaches, borrowed from unsupervised domain adaptation, to make the speaker recognition models noise tolerant. in particular, we apply teacherstudent and consistency regularization techniques on speaker recognition and diarization tasks and compare them with multi-condition training when various noise augmentations are used. we were inspired by the significant results of this work for teacher-student [22], where clean and noisy audios are fed to the teacher and the student, respectively, to enforce similarity between the output distributions. consistency regularization is a commonly-used d. karamyan, g. kirakosyan and s. harutyunyan 59 technique amongst a variety of tasks in machine learning. this work [20] applies it in a manner similar to that mentioned previously, only here clean and noisy inputs are both fed to the student model. in the paragraphs that follow, we’ll discuss in detail how we apply these concepts to obtain noise-robust speaker recognition and diarization. 2. improving noise robustness of speaker diarization system there are several ways to improve the performance of speaker diarization systems in noisy and reverberant environments. for instance, work in [1] proposed the sequence of refinement operations to smooth and denoise data in the similarity space. in this work, we will focus only on the speaker embedding extraction part, and we are going to use unsupervised domain adaptation techniques to make the model noise tolerant. given a training dataset consisting of pairs (xi, yi) where xi represents an audio signal and yi represents the speaker id. our goal is to learn a parametrized function fθ, which should be able to compress any given audio into a d-dimensional vector, also known as a speaker embedding. moreover, if two audio signals are spoken by the same speaker, then the cosine similarity between their corresponding embeddings should be higher. conversely, if the two audios are spoken by different speakers, the cosine similarity between their embeddings should be lower. the additive angular margin (aam) loss, as proposed in [23], is a prevalent method for training speaker embedding extractors. the aim of the aam loss is to minimize the angle between speaker embeddings belonging to the same speaker while simultaneously maximizing the angle between speaker embeddings belonging to different speakers. 2.1 consistency regularization the core idea behind consistency regularization (cr) is to make sure that the network produces similar embeddings for the augmented versions of the same unlabeled utterance [20, 24, 25]. it is enforced by an additional regularization term in the loss function: lcr = 1 n n∑ i=1 |fθ(a(xi)) − fθ(a(xi))|22, where fθ is an embedding extractor with parameters θ, n represents the total number of training examples within the dataset. by a(x) we denote a stochastic operation that augments the audio in such a way that the speaker identity remains the same. so the difference is most likely non-zero. the final form of loss is a weighted combination of laam and lcr as shown below: l = (1 − α)laam + αlcr, where α is a hyperparameter taking values between 0 and 1. 2.2 teacher-student one critical problem with lcr loss is that it is not stable because of unstable target. to mitigate unstable target problem, the teacher-student model was proposed in [26], where two separate models were used: a student network with θ parameters and a teacher with 60 making speaker diarization system noise tolerant θ′ parameters. on unlabeled examples, the teacher network provides the learning target for the student network: lts = 1 n n∑ i=1 |fstudentθ (a(xi)) − f teacher θ′ (a(xi))| 2 2. student is trained as usual. teacher model is not trained via back-propagation. instead, its weights are updated at each iteration using the weights from the student network. again, the final loss is a weighted combination of laam and lts as shown below: l = (1 − α)laam + αlts. 2.3 knowledge distillation if the teacher model is already trained, it is desirable that its weights remain constant. this training setup is known as ”knowledge distillation”, where the student model is trained to mimic a pre-trained, larger model [27]. 3. experiments 3.1 model architecture in all experiments, we will use the speakernet [13] architecture as the backbone model. speakernet models are made up of 1d depth-wise separable convolutional layers. on top of the model, a statistical pooling layer is used to obtain a fixed-length vector. the proposed variation of speakernet (speakernet-m) has fewer parameters (5m) when compared to sota and shows very similar performance on voxceleb1 [28] trial files when compared to sota systems. the model provides embeddings of size 256 for a given audio sample. in teacher-student experiments, both the teacher and the student have the same architecture. 3.2 datasets the voxceleb1 [28] and voxceleb2 [29] datasets are widely recognized benchmarks in the field of speaker recognition. these datasets have pre-defined development and test sets, which allow for an objective and consistent evaluation of speaker recognition models. we trained our speaker recognition models using only the development part, which consisted of 7205 distinct speakers. for evaluation of speaker embeddings quality, we use voxceleb1 cleaned test trial file. the test trial file contains a list of audio pairs, and the model’s performance is evaluated based on its ability to correctly determine whether the two recordings belong to the same speaker or not. to evaluate speaker diarization, we use the voxconverse [30] development set. the dataset statistics are shown in table 1. 3.3 metrics the equal error rate (eer) metric is used to evaluate the speaker verification. this is the rate used to determine the threshold value for a system when its false acceptance rate and d. karamyan, g. kirakosyan and s. harutyunyan 61 table 1: statistics of datasets used for training speakernet. dataset # speakers duration (h) # utterances voxceleb1 1211 340.4 148642 voxceleb2 5994 2359.77 1,092,009 false rejection rate are equal. we calculate eer on voxceleb1 cleaned test trial file under original, noisy and echo conditions. for diarization evaluation purposes, we used diarization error rate (der). this is the sum of three error terms: false alarm (fa), missed detection (ms) and speaker confusion error rate (cer). similar to the previous works [12, 14], we use collar 0.25 sec and ignore overlap speech regions for confusion error rate calculation. we test the diarization system in original, noisy, and echo scenarios, just like we do for speaker verification. both eer and der are calculated using the cosine similarity back-end. 3.4 experiment setup 3.4.1 input features our audio pre-processing procedure is identical to the one described in the speakernet paper [13]. for each frame window of 20 ms, shifted by 10 ms, 64-dimensional acoustic features were calculated from the speech recordings. each utterance fed to the encoder has a size t × 64, where t is the number of frames in a given audio sample. we crop speech segments into random chunks from 3 to 8 seconds. with larger chunks, the model converges faster. 3.4.2 clean teacher our first baseline is a clean teacher trained on voxceleb1 and voxceleb2 datasets with additive angular margin loss. we set the aam loss hyperparameters to s = 30 and m = 0.2, as it was shown in [13, 14], these values give the best results. to avoid overfitting, we added specaugment [31] to the training pipeline, which randomly masks blocks of frequency and time channels. 3.4.3 noisy teacher our second baseline is a noisy teacher trained with the same objective as a clean baseline, and with the additional augmentation steps described below: • no augment: leave the utterance unchanged • rir augment: reverberate an input audio using an impulse response from rirs dataset [32] • noise augment: add noise from musan [33] dataset with signal-to-noise (snr) values randomly chosen from 0-50db • rir-noise augment: apply noise and echo perturbations to the same audio at the same time 62 making speaker diarization system noise tolerant • speed augment: speed perturbation with 0.95x and 1.05x speeds rir, noise, and rir-noise augmentations all have a probability of 0.25 and are mutually exclusive. speed augmentation is applied independently with a probability of 0.1. 3.4.4 consistency regularization we add an extra mean squared loss between embeddings for the augmented and nonaugmented versions of the same utterance to the aam loss during training. we set the α hyperparameter in the final loss to 0.1. 3.4.5 teacher-student in order to supervise the student model, we choose our clean-teacher baseline as the teacher. we did not update teacher weights during the training and no perturbations were applied to the input of the teacher model. the flow chart of teacher-student training is presented in fig. 1. during the training procedure, in addition to the aam loss, the mean squared loss between the student and teacher-produced embeddings is minimized. we set the α hyperparameter in the final loss to 0.1. fig.1. flow chart of teacher-student learning for improving noise robustness of sr. 3.4.6 optimization all models are trained for 200 epochs with an sgd optimizer, with an initial learning rate (lr) of 0.08 using a cosine annealing lr scheduler on 4 a100 gpus. 3.5 evaluations 3.5.1 speaker verification all the experiment findings are displayed in table 2. the results of the original speakernet and the pre-trained checkpoint1 publicly released by nvidia are also provided for comparison. 1https: //catalog.ngc.nvidia.com/orgs/nvidia/teams/nemo/models/speakerverification_speakernet d. karamyan, g. kirakosyan and s. harutyunyan 63 the pre-trained checkpoint was trained solely with noise augmentation using the abovementioned datasets. in order to examine the speaker verification system under noisy and reverberant conditions, we created augmented versions of voxceleb1 clean test trials by injecting noise and echo with different snr values. table 2: comparison of different speaker verification models under noise and reverb conditions. the results are reported in equal error rates. the more aggressively noise has been applied, the lower the snr values were. a noise level of 0 db indicates that the sound and the noise have the same energy. model orig 0db 5db 10db rir speakernet [13] 2.14 speakernet (nvidia) 1.92 9.75 5.43 3.61 16.5 clean teacher 1.87 12.9 6.94 4.21 16.5 noisy teacher 2.6 9.35 5.84 4.23 12.74 consistency reg. 1.76 8.05 4.40 3.13 12.26 teacher-student 1.73 9.16 4.79 3.26 9.18 table 2 showcases the effectiveness of the methods applied. we can see that training the speakernet model with data augmentation (noisy teacher) improves the results in the noisy/reverberant environment with a small deterioration of eer on the original (not perturbed) audios. the teacher-student method achieves the lowest eer scores in original and reverberant cases (rir), whereas the consistency regularization method shows the best results for noisy audios. using the teacher-student method, we were able to improve the eer by an average of 19.1% compared to the multi-condition trained model. with consistency regularization, we were able to improve the eer by an average of 14.8% compared to the multi-condition trained model. 3.5.2 speaker diarization we employ our trained speakernet models for speaker diarization task to see which model has the smallest performance degradation in noisy conditions. we found that the optimal sliding window size and shift for speech segmentation are 1.5 and 0.5 seconds, respectively. in addition, diarization experiments are based on oracle vad to evaluate the vad-independent performance. the affinity matrix a is constructed using the cosine similarity between segment embeddings. we further apply the following sequence of refinement operations to the affinity matrix a: • row-wise thresholding: for each row, keep top-12 largest elements and set the rest to 0 • symmetrization: y = 1 2 (a + at ) • diffusion: y = aat we use the spectral clustering method [34] to obtain speaker labels. to get a full picture, we present the diarization results for both known (oracle) and unknown numbers of speakers. in the latter case, we utilize the maximal eigen-gap approach to determine the number of speakers [1]. 64 making speaker diarization system noise tolerant table 3: comparison of speaker diarization systems with various speaker embedding extractors under noise and reverberant conditions. the results are reported in diarization error rate (der). model known #speakers unknown #speakers 0db 5db 10db rir orig avg 0db 5db 10db rir orig avg clean teacher 12.13 4.48 1.96 2.44 1.26 4.45 15.44 7.59 2.74 4.48 1.78 6.40 noisy teacher 9.20 4.49 3.13 3.12 1.57 4.30 13.09 7.94 4.18 4.14 1.95 6.26 consistency reg. 9.50 3.46 2.0 2.50 1.45 3.78 13.40 4.90 2.57 3.45 1.67 5.20 teacher-student 9.84 3.41 2.11 2.43 1.36 3.83 13.99 6.17 3.09 3.52 1.61 5.67 in order to assess the performance of the speaker diarization system under noisy and reverberant conditions, we modified the voxconverse dev dataset by adding noise and echo at various signal-to-noise ratios. the results, shown in table 3, indicate that the teacher-student and consistency regularization methods generally outperform the multi-condition baseline model for both scenarios involving known and unknown numbers of speakers. in particular, when the number of speakers is unknown, we observed approximately 17% and 9.5% relative performance improvements for the consistency regularization and teacher-student methods, respectively, compared to the multi-condition baseline. however, it is worth noting that in certain specific scenarios, the baseline models may outperform the models with the overall best average performance. 4. conclusions in this research, we explore ways to increase the accuracy of speaker recognition and speaker diarization in noisy and reverberant environments, such as multi-condition, teacher-student, and consistency regularization. the key component of the methods used is the additional regularization term between embeddings for augmented and non-augmented versions of the same utterance. through the use of teacher-student and consistency regularization, we were able to improve the performance of speakernet on speaker recognition and diarization tasks in noisy and reverberant situations. references [1] q. wang, c. downey, l. wan, p. mansfield and i. moreno, “speaker diarization with lstm”, 2018 ieee international conference on acoustics, speech and signal processing (icassp). pp. 5239-5243, 2018. 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[3 4 ] u .v o n l u xb u r g , \ a t u t o r ia l o n s p e c t r a l c lu s t e r in g " , statistics and computing, vo l. 1 7 , p p . 3 9 5 -4 1 6 , 2 0 0 7 . ²õùï³¹çù³óïáõýáõãû³ý ³å³ñáíáõùá ëáëý³ïý»ñç ¹ç³ñç½³óç³ûç ñ³ù³ï³ñ·áõù 1ð³û-èáõë³ï³ý ñ³ù³éë³ñ³ý, ºñ¨³ý, ð³û³ëï³ý 2krisp.ai, ºñ¨³ý, ð³û³ëï³ý 3ðð ¶²² ù³ã»ù³ïçï³ûç çýëïçïáõï,ºñ¨³ý, ð³û³ëï³ý e -m a il: f d ka r a m ya n , s h a r u t yu n ya n , g kir a ko s ya n g@kr is p .a i ²ù÷á÷áõù ¸³íçã ê. ø³ñ³ùû³ý1;2, ¶ñç·áñ ². îçñ³ïáëû³ý2;3, ê³ã»ý ². ð³ñáõãûáõýû³ý2 êáëý³ïý»ñç ¹ç³ñç½³óç³ûç ýå³ï³ïá ³áõ¹çá ó³ûý³·ñáõãû³ý ù»ç ï³ñµ»ñ ëáëý³ïý»ñç ñ³ûïý³µ»ñáõùý áõ ³é³ýóý³óáõùý ¿: ²ûýáõ³ù»ý³ûýçí, ýáý³ûçý ³õùáõïá ï³ñáõ ¿ ³½¹»é ³ûë ñ³ù³ï³ñ·»ñç ×ß·ñïáõãû³ý íñ³: ²ûë ñá¹í³íáõù áõëáõùý³ëçñí»é »ý ³ûýåçëç ù»ãá¹ý»ñ, çýãåçëçù »ý` ï³ñµ»ñ ³áõ·ù»ýï³óç³ý»ñáí áõëáõóáõùá, ï³ûáõýáõãû³ý ï³ñ·³íáñáõùá (consistency regularization) ¨ áõëáõóçã-³ß³ï»ñï ù»ãá¹á` ëáëý³ïý»ñç ó³ûý³ûçý ñ³ïï³ýçßý»ñ ¹áõñë µ»ñáõ ùá¹»éç ï³ûáõýáõãûáõýá ³õùáõïç ýï³ïù³ùµ µ³ñóñ³óý»éáõ ñ³ù³ñ: üßí³í ù»ãá¹ý»ñç ³ñ¹ûáõý³í»ïáõãûáõýá ëïáõ·í»é ¿ ëáëý³ïý»ñç ýáõûý³ï³ý³óù³ý ¨ ¹ç³ñç½³óç³ûç ëý¹çñý»ñáõù ¨ óáõûó ¿ ïñí»é, áñ ¹ñ³ýù ñ³ý·»óýáõù »ý ï³ûáõýáõãû³ý µ³ñ»é³íù³ýá` ³õùáõïç ¨ ³ñó³·³ýùç ³éï³ûáõãû³ý ¹»åùáõù: êáëý³ïý»ñç ýáõûý³ï³ý³óù³ý ¨ ¹ç³ñç½³óç³ûç ñ³ù³ï³ñ·»ñá ³õùáõïç ¨ ³ñó³·³ýùç å³ûù³ýý»ñáõù ÷áñó³ñï»éáõ ñ³ù³ñ ëï»õíí»é »ý voxceleb1 ¨ voxconverse dev ïíû³éý»ñç ñ³í³ù³íáõý»ñç áý¹é³ûýí³í ï³ñµ»ñ³ïý»ñá` ³í»é³óý»éáí ï³ñµ»ñ snr ³ñå»ùý»ñáí ýáý³ûçý ³õùáõï ¨ ³ñó³·³ýù: êï³óí³í ³ñ¹ûáõýùý»ñá óáõûó »ý ï³éçë, áñ ùçççý ñ³ßíáí ï³ñ»éç ¿ ñ³ëý»é ëáëý³ïý»ñç ýáõûý³ï³ý³óù³ý ×ß·ñïáõãû³ý ñ³ñ³µ»ñ³ï³ý µ³ñ»é³íù³ýá` 1 9 ; 1 % -áí` û·ï³·áñí»éáí áõëáõóçã-³ß³ï»ñï ù»ãá¹á ¨ ëáëý³ïý»ñç ¹ç³ñç½³óç³ûç ×ß·ñïáõãû³ý ñ³ñ³µ»ñ³ï³ý µ³ñ»é³íù³ýá` 1 7 % áí` û·ï³·áñí»éáí ï³ûáõýáõãû³ý ï³ñ·³íáñù³ý ù»ãá¹á` ñ³ù»ù³ï³í ï³ñµ»ñ ³áõ·ù»ýï³óç³ý»ñáí í³ñå»óí³í ùá¹»éç ñ»ï: ´³ý³éç µ³é»ñ` ëáëý³ïý»ñç ýáõûý³ï³ý³óáõù, ëáëý³ïý»ñç ¹ç³ñç½³óç³, ³õùï³¹çù³óïáõýáõãûáõý, áõëáõóçã-³ß³ï»ñï, ï³ûáõýáõãû³ý ï³ñ·³íáñáõù: 6 8 making speaker diarization system noise tolerant îáåñïå÷åíèå øóìîóñòîé÷èâîñòè ñèñòåìû äèàðèçàöèè äèêòîðîâ äàâèä ñ. êàðàìÿí1;2, ãðèãîð à. êèðàêîñÿí2;3, ñàòåí à. àðóòþíÿí2 1ðîññèéñêî-àðìÿíñêèé óíèâåðñèòåò, åðåâàí, àðìåíèÿ 2krisp.ai, åðåâàí, àðìåíèÿ 3èíñòèòóò ìàòåìàòèêè íàí ðà, åðåâàí, àðìåíèÿ e-mail: dkaramyan, sharutyunyan, gkirakosyan@krisp.aig àííîòàöèÿ öåëüþ ñèñòåìû äèàðèçàöèè äèêòîðîâ ÿâëÿåòñÿ èäåíòèôèöèðîâàíèå è ðàçäåëåíèåðàçíûõ äèêòîðîâ â àóäèîçàïèñè. îäíàêî øóì â çàïèñè ìîæåò ïîâëèÿòü íà òî÷íîñòü ýòèõ ñèñòåì. â ýòîé ñòàòüå ìû èññëåäóåì òàêèå ìåòîäû, êàê îáó÷åíèå ñ ðàçëè÷íûìè àóãìåíòàöèÿìè, ðåãóëÿðèçàöèÿ ñîãëàñîâàííîñòè (consistency regularization) è ìåòîä ”ó÷èòåëü-ó÷åíèê”, ÷òîáû ïîâûñèòü óñòîé÷èâîñòü ýêñòðàêòîðîâ ðå÷åâûõ õàðàêòåðèñòèê ê øóìó. ìû ïðîâåðÿåì ýôôåêòèâíîñòü ýòèõ ìåòîäîâ â çàäà÷àõ ðàñïîçíàâàíèÿ äèêòîðîâ ïî ãîëîñó è äèàðèçàöèè äèêòîðîâ è äåìîíñòðèðóåì, ÷òî îíè ïðèâîäÿò ê óëó÷øåíèþ óñòîé÷èâîñòè ïðè íàëè÷èè øóìà è ðåâåðáåðàöèè. ×òîáû ïðîâåðèòü ñèñòåìó ðàñïîçíàâàíèÿ è äèàðèçàöèè äèêòîðîâ â óñëîâèÿõ øóìà è ðåâåðáåðàöèè, ìû ñîçäàëè ðàñøèðåííûå âåðñèè voxceleb1 è íàáîðîâ äàííûõ voxconverse dev, äîáàâèâ øóì è ýõî ñ ðàçíûìè çíà÷åíèÿìè snr. íàøè ðåçóëüòàòû ïîêàçûâàþò, ÷òî â ñðåäíåì ìû ìîæåì äîáèòüñÿ îòíîñèòåëüíîãî óëó÷øåíèÿ ðàñïîçíàâàíèÿ äèêòîðîâ íà 1 9 ; 1 % ñ èñïîëüçîâàíèåì ìåòîäà ”ó÷èòåëü-ó÷åíèê” è îòíîñèòåëüíîãî óëó÷øåíèÿ äèàðèçàöèè äèêòîðîâ íà 1 7 % ñ èñïîëüçîâàíèåì ìåòîäà ðåãóëÿðèçàöèè ñîãëàñîâàííîñòè ïî ñðàâíåíèþ ñ áàçîâîé ìîäåëüþ, îáó÷åííîé ñ ïîìîùüþ ðàçëè÷íûõ àóãìåíòàöèé. êëþ÷åâûå ñëîâà:ðàñïîçíàâàíèå ïî ãîëîñó, äèàðèçàöèÿ äèêòîðîâ, óñòîé÷èâîñòü ê øóìó, ó÷èòåëü-ó÷åíèê, ðåãóëÿðèçàöèÿ ñîãëàñîâàííîñòè. 06_karamyan_davit_59 (1) 06 d:\sbornik\...\sav.dvi mathematical problems of computer science 30, 18{24, 2008. e ±cient m ar ch-like algor ithm for detection of all t wo-oper ation dynamic faults fr om subclass sav h . a ve t is ya n y, g. h a r u t u n ya n z, v .a . v a r d a n ia n z y russian-armenian state university e-mail: hamazasp avetisyan@yahoo.com zvirage logic e-mail: fgurgen.harutyunyan, valery.vardaniang@viragelogic.com abstract this paper introduces an e±cient march-like algorithm for detection of the well known class sav of dynamic faults. sav is the subclass of all two-operation dynamic functional fault models that are sensitized by means of applying two consecutive operations, one applied on the aggressor cell and the second operation applied on the victim cell. earlier, only subclasses saa and svv were considered by a few authors when both sensitizing operations were applied either on the aggressor or victim cell, and march algorithms were developed by them. subclasses sav and sva were not considered due to their complexity. a larger class of march-like algorithms has to be considered for detection of those subclasses since march algorithms cannot detect them. it is shown that 392n operations are su±cient for detection of faults from sav. refer ences [1 ] a . j. va n d e go o r , " te s t in g s e m ic o n d u c t o r m e m o r ie s : th e o r y a n d p r a c t ic e " , j ohn w iley and sons, 1 9 9 1 . 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[8 ] a . b e n s o , a . b o s io , s . d i ca r lo , g. d i n a t a le , p . p r in e t t o , " ma r c h a b , ma r c h a b 1 : n e w ma r c h t e s t s fo r u n lin ke d d yn a m ic m e m o r y fa u lt s " , itc, 2 0 0 5 . [9 ] a . b e n s o , a . b o s io , s . d i ca r lo , g. d i n a t a le , p . p r in e t t o , " a u t o m a t ic m a r c h t e s t g e n e r a t io n fo r s t a t ic a n d d yn a m ic fa u lt s , in s r a ms " , p roc. e ts 2005, tallinn, p p . 1 2 2 -1 2 7 , 2 0 0 5 . [1 0 ] g. h a r u t u n ya n , v . a . v a r d a n ia n , y . zo r ia n , " min im a l m a r c h t e s t s fo r d yn a m ic fa u lt s in r a n d o m a c c e s s m e m o r ie s " , j ournal of e lectronic testing: theory and applications, vo l. 2 3 , n u m b e r 1 , p p . 5 5 -7 4 , 2 0 0 7 . [1 1 ] g. h a r u t u n ya n , v .a . v a r d a n ia n , y . zo r ia n , " min im a l m a r c h t e s t s fo r d yn a m ic fa u lt s in r a n d o m a c c e s s m e m o r ie s " , in p roc. of ie e e e uropean test symposium, p p . 4 3 -4 8 , 2 0 0 6 . [1 2 ] a .j. va n d e go o r , i. s c h a n s t r a , " a d d r e s s a n d d a t a s c r a m b lin g : ca u s e s a n d im p a c t o n m e m o r y t e s t s " , p roc. ie e e w orkshop d e l ta, p p . 1 2 8 -1 3 6 , 2 0 0 2 . [1 3 ] j.-f. l i, k .-l . ch e n g , c.-t. h u a n g , a n d c.-w . w u , " ma r c h b a s e d r a m d ia g n o s t ic a lg o r it h m s fo r s t u c k-a t a n d c o u p lin g fa u lt s " , p roc. ie e e itc, p p . 7 5 8 -7 6 7 , 2 0 0 1 . [1 4 ] v . a . v a r d a n ia n , y . zo r ia n , " a m a r c h -b a s e d fa u lt lo c a t io n a lg o r it h m fo r s t a t ic r a n d o m a c c e s s m e m o r ie s " , p roc. ie e e int. w orkshop m td t, p p . 6 2 -6 7 , 2 0 0 2 . ²ñ¹ûáõý³í»ï ù³ñß³ïçå ³é·áñçãù ¹çý³ùçï ³ýë³ñùáõãûáõýý»ñç »ýã³¹³ëç µáéáñ ³ýë³ñùáõãûáõýý»ñç ñ³ûïý³µ»ñù³ý ñ³ù³ñ ð. ²í»ïçëû³ý, ¶. ð³ñáõãûáõýû³ý, ì. ì³ñ¹³ýû³ý ²ù÷á÷áõù ²ûë ñá¹í³íáõù ý»ñï³û³óíáõù ¿ ³ñ¹ûáõý³í»ï ù³ñß³ïçå ³é·áñçãù, áñá ï³ñáõ³ýáõù ¿ ñ³ûïý³µ»ñ»é ¹³ëç µáéáñ ¹çý³ùçï ³ýë³ñùáõãûáõýý»ñá. ¹³ëá »ñïáõ ·áñíáõõáõãû³ùµ ½·³ûáõý³óíáõ ¹çý³ùçï ³ýë³ñùáõãûáõýý»ñç »ýã³¹³ë ¿, ³ûëçýùý ³ýë³ñùáõãûáõýý»ñ, áñáýù ½·³ûáõý³óíáõù »ý ñçßáõáõãû³ý µççç ýï³ïù³ùµ ñ³çáñ¹³ï³ý »ñïáõ ·áñíáõáõãûáõý ï³ï³ñ»éçë, ³é³ççý ·áñíáõáõãûáõýá ³·ñ»ëáñ ¿, çëï »ñïñáñ¹á ½áñ ¿ µççç ýï³ïù³ùµ: ü³ëïçýáõù ¹çï³ñïí³í »ý »õ»é ¨ »ýã³¹³ë»ñá, áñáýó ñ³ù³ñ ñ»õçý³ïý»ñá ý»ñï³û³óñ»é ¿çý ù³ñß ³é·áñçãù»ñ: ºí ¹³ë»ñá áõëáõùý³ëçñí³í ã»ý »õ»é, ù³ýç áñ ýñ³ýó ñ³ù³ñ ñý³ñ³íáñ ã¿ ï³éáõó»é ù³ñß ³ý·áñçãù: ê³ï³ûý ³ûë ¹³ë»ñç ñ³ù³ñ ñý³ñ³íáñ ¿ ï³éáõó»é ù³ñß³ïçå ³ý·áñçãù»ñ: ðá¹í³íáõù ý»ñï³û³óí³í ¿ ù³ñß³ïçå ³é·áñçãù, áñá ï³ï³ñ»éáí ·áñíáõáõãûáõý ñ³ûïý³µ»ñáõù ¿ ¹³ëç µáéáñ ³ýë³ñùáõãûáõýý»ñá: microsoft word 10_zaslavski.doc mathematical problems of computer science 39, 81--87, 2013. 81 on the comparative complexity of primitive recursive arithmetical and string functions1 igor d. zaslavsky and mikayel h. khachatryan institute for informatics and automation problems of nas ra e-mail: zaslav@ipia.sci.am, mikayel.khachatur@gmail.com abstract formal languages la and lw are introduced as in [1] for the representation of primitive recursive arithmetical and string functions. shannon functions shaw and shwa describing the relations between the complexities of functions representations in these languages are defined as in [1]. a new proof of the upper bounds for shaw is presented; it is based on a new method giving in some cases new possibilities for applications in comparison with the methods considered in [1]. keywords: string function, arithmetical function, term, alphabetic enumeration, shannon function, primitive recursive function. investigations described in this paper may be considered as the continuation of those presented in [1]. let us recall definitions of some notions given in [1]. we suppose that an alphabet 1 2{ , ,..., },pa a a a where 1,p  is fixed. the set of all strings in this alphabet (including the empty string  ) is denoted by a*; the set of all k-tuples 1 2( , ,..., ),kq q q where * iq a for 1 ,i k  will be denoted by *( ) .ka the set of all non-negative integers {0, 1, 2, ... } will be denoted by n; the set of all k-tuples 1 2( , ,..., ),kx x x where ix n for 1 ,i k  will be denoted by .kn k-dimensional string function in a is defined ( [1], [2] ) as a mapping of *( )ka into a*; kdimensional arithmetical function is defined as a mapping of ( )kn into n. primitive recursive string functions in a as well as primitive recursive arithmetical functions are defined in a usual way as in [1] and [2]. the alphabetic enumeration of the set a* is defined as in [1] and [2]; let us recall that this enumeration defines a one-to-one correspondence between the sets a* and n. the non-negative integer, corresponding to a string q in the alphabetic enumeration is denoted by ( ).q the string in a* corresponding to the number n in this enumeration is denoted by ( )p n or .n the length of a string q is denoted by .q all these notations are used in [1]. the alphabetic enumeration of strings gives also a one-to-one correspondence between ndimensional string functions in a, and n-dimensional arithmetical functions. 1 this work is supported by the grant 11-1b 189 of the government of the republic of armenia. on the comparative complexity of primitive recursive arithmetical and string functions 82 namely, we say ( [1], [2] ) that an n-dimensional arithmetical function f represents an ndimensional string function f, if 1 2 1 2( , , ... , ) ( , , ... , )n nf x x x f x x x    for all 1 2, ,..., nx x x in n. in this case we say also that f and f correspond to one another. the mentioned correspondence gives also a one-to-one correspondence between primitive recursive string functions in a and primitive recursive arithmetical functions ( [1], [2] ). in [1] the formal languages la and lw are introduced for the representation of primitive recursive arithmetical functions and primitive recursive string functions. the formal expressions in these languages are said to be terms; by t la and r lw we denote the statements “t is a term in la”, “r is a term in lw”. in the definition of la the symbols s and r are used for the operators of superposition and primitive recursion of arithmetical functions; in the definition of lw the symbols s and r are used for the operators of superposition and alphabetic primitive recursion of string functions ( [1], [2] ). special notations for some modifications of the mentioned operators ( sbl, sbr, sel, ser, sb, se in la; sbl, sbr, sel, ser, sb, se in lw) are also included in la and lw ([1]). we shall consider below special cases of the implementation of the modifications sb and se of the operator s (see [1]); these cases are described in the following points (1), (2), (3). let us note that all the terms considered in (1), (2), (3) are terms in the language lw. (1) if f and g are terms expressing correspondingly a v  dimensional function f (where 2v  ) and a one-dimensional function g, then the term ( , )f g se expresses the v  dimensional function h such that 1 2 1 2 1( , ,..., ) ( , ,..., , ( ))v v vh q q q f q q q g q for all values of the variables 1 2, , ... , .vq q q (2) if f and g are terms expressing correspondingly a 2-dimensional function f and a k-dimensional function g (where 1k  ), then the term ( , )f g sb expresses the (k+1)dimensional function h such that 1 2 1 1 2 1( , ,..., ) ( ( , ,..., ), )k k kh q q q f g q q q q  for all values of the variables 1 2 1, , ... , .kq q q  (3) if 1 2, ,f g g   are terms expressing correspondingly a v  dimensional function f (where 2v  ) and one-dimensional functions 1g and 2 ,g then the term 1 2( , , )f g g  sb expresses the ( 1)v  dimensional function h such that 1 2 1 1 1 2 1 2 1( , ,..., ) ( ( ), ( ), ,..., )v vh q q q f g q g q q q  for all values of the variables 1 2 1, , ... , .vq q q  as it will be seen below, it is convenient to represent the list of variables for the function h in the following form: 3 4, , ,... , .vr q q q using this list, we can write the expression for h as follows: 3 4 1 2 3 4( , , ,... , ) ( ( ), ( ), , ,..., ).v vh r q q q f g r g r q q q in [1] shannon functions ( )awsh n and ( )wash n are introduced; these functions describe the relations between the lengths of terms expressing arithmetical functions (in la) and string functions (in lw) when the considered functions correspond to one another. namely, if t lw , then by ( )la t we denote the set of all terms in la expressing the arithmetical function corresponding to the string function expressed by t. similarly, if r la , then by ( )lw r we denote the set of all terms in lw expressing the string function corresponding to the arithmetical function expressed by r. now we can give (see [1]) the definitions of ( )awsh n and ( )wash n as follows: i. zaslavski and m. khachatryan 83  ( )( )&( )( ) max minwa r la tt lw t nsh n r  ;  ( )( )&( )( ) max min .aw t lw rr la r nsh n t  in [1] the following statement is established (see the main theorem in [1]): there are upper and lower bounss for ( )awsh n and ( )wash n such that each of them has the form ,cn d where c and d are some constants. we shall consider the function ( ).awsh n there are some defects in the proof of the upper bouns for this function in [1]; their removal requires essential changes in the proof. below we give another proof of the mentioned bouns based on a method which is different from those used in [1]. namely, we shall give a new proof of the following theorem. theorem. there are constants c and d such that for any non-negative integer n   .awsh n cn d  we shall use three lemmas in the proof given in [1] (similar statements are proved also in [2]). by ( )n we denote the function such that (0) ,   1 1 1( ) ... n n a a a   times for any positive integer n. lemma 1. there are constants c and d such that for any term t la expressing a function 1 2( , ,..., ),mx x x a term lw expressing some function 1 2( , ,..., )mq q q can be constructed such that the following conditions are satisfied: 1. 1 2 1 2( ( ), ( ),..., ( )) ( ( , ,..., )),m mx x x x x x      for any 1 2, ,..., mx x x in n. 2. ' '.c t d   lemma 2. there is a primitive recursive string function g such that   g m m  for any .m n lemma 3. the one-dimensional string function  ( ) ( )q q   is primitive recursive. proof of theorem. let t be any term in la expressing some function 1 2( , , ... , ).mx x x as it is proved in [1], the following inequality holds: .m t the string function corresponding to  let us denote by 1 2( , , ... , ).mq q q we shall construct a term  in lw having the length mentioned in theorem and expressing the function . using lemma 1 we construct a term φ in la such that ,c t d    where c and d are constants (fixed in lemma 1), and φ expresses a function  satisfying the condition 1 2 1 2( ( ), ( ),..., ( )) ( ( , ,..., ))m mx x x x x x      for any 1 2, ,..., mx x x in n. using lemmas 1 and 2 we obtain the following equalities               1 2 1 2 1 2 1 2 1 2 ( , ... ) ( ( ), ( )... ( )) ( ( ), ( )... ( )) ( ) , ( ) ... ( ) ( ), ( )... ( ) , m m m m m q q q q q q g q q q g q q q g q q q                             by g and  we denote the terms in lw expressing the functions g and . let us consider the well-known primitive recursive arithmetical functions c, l, r, defining a one-to-one correspondence between 2n and n. such functions we define by the following equalities: on the comparative complexity of primitive recursive arithmetical and string functions 84 ( )( 1) ( , ) , 2 x y x y c x y x      ( ( ), ( )) , ( ( , )) , ( ( , )) . c l z r z z l c x y x r c x y y    we consider also the following functions (where 2, 2n k n   ): 1 2 1 2 3 ( 1) times ( 1) times 1 ( ) times ( , ,..., ) (... ( ( , ), ),..., ); ( ) ( (... ( )...)); ( ) ( ( (... ( )...))). n n n n n n n k nk c x x x c c c x x x x c z l l l z c z r l l l z          obviously, for any 1 2, ,..., ,nx x x z in n and for 1 ,k n  the following equalities hold: 1 2 1 2 ( ( ), ( ),..., ( )) ; ( ( , ,..., )) . n n n nn n nk n k c c z c z c z z c c x x x x   using lemma 1 we construct string functions , ,    , such that for any x, y, z in n ( ( ), ( )) ( ( , )); ( ( )) ( ( )); ( ( )) ( ( )). x y c x y z l z z r z              let us note a peculiarity of these functions. if some strings 1 2, , q q q in a do not contain other letters except 1.a then the following equalities hold: 1 2 1 1 2 2( ( ), ( )) , ( ( , )) , ( ( , )) .q q q q q q q q q         however, in general such equalities are not valid. let us consider also the following string functions (where 2, 2n k n   ) 1 2 1 2 3 ( 1) times 1 ( ) times ( , ,..., ) (... ( ( , ), ),..., ); ( ) ( (... ( )...)); ( ) ( ( (... ( )...))). n n n n n n k nk q q q q q q q q q q q                     the terms in lw expressing the functions 1, , , , , n n nk      (where 2, 2n k n   ) we denote, correspondingly, by 1, , , , , . n n nk         if some strings 1 2, , ... , nq q q q in a do not contain other letters except 1.a then the following equalities hold (where 2, 2n k n   ): 1 2 1 1 2 1 1 2 ( ( ), ( ),..., ( )) , ( ( , ,..., )) ; ( ( , ,..., )) . n n n nn n n n n nk n k q q q q q q q q q q q q            in general such equalities are not valid. now in the case, when 2m  , let us construct the term m as follows: ( 1) times ( 2) times ( ( ,..., ( ( , ) , ))..., )) , ). m m m                 se sb se sb i. zaslavski and m. khachatryan 85 here the group of symbols ( ( ,se sb is repeated (m-1) times; after this the group ) is repeated once; after this the group , )) is repeated (m-2) times; finally, the group , ) is repeated once. it is easily seen that the length of the term m does not exceed 10 10 ,c m d where 10c and 10d are some constants. let us consider some subterms of the term m as well as functions expressed by them. it is easily seen that the following statements are valid. the term ( , ) sb expresses the function  1 2( ), .q q  the term 2 ( ( , ), )     se sb expresses the function  1 2( ), ( ) ,q q   that is, the function  2 1 2( ), ( ) .q q   the term ( , ( ( , ), ))     sb se sb expresses the function   1 2 3( ), ( ) , .q q q    the term 3 ( ( , ( ( , ), )), )         se sb se sb expresses the function   1 2 3( ), ( ) , ( ) ,q q q     that is, the function  3 1 2 3( ), ( ), ( ) .q q q    using similar considerations, we conclude that the term m expresses the function 1 2 3(... ( ( ( ), ( )), ( )),..., ( )),mq q q q       that is, the function 1 2 3( ( ), ( ), ( ),..., ( )). m mq q q q     further, let us construct the term m (where 1m  ) as follows: ( 1) times( 1) times ( ( ( , , ), , )..., , ). mm m                sb ...sb sb it is easily seen that the length of the term m does not exceed 11 11,c m d   where 11c and 11d are some constants. using the inequalities c t d    and m t we conclude that the length m does not exceed 12 12 ,c t d where 12c and 12d are some constants. let us consider some subterms of the term m , as well as functions expressed by them. it is easily seen that the following statements are valid. as it is said above, the term  expresses the function  depending on m variables. the function  we denote also by 0 . the term 1 is defined as the term which is equal to . the term 2 ( , , )     sb expresses some function 1 depending on ( 1)m  variables; the list of variables for this function we denote by 1 3, ,..., .mr q q using such notations we can represent the equality describing the function 1 1 3( , ,..., )mr q q as follows: 1 1 3 1 1 3( , ,..., ) ( ( ), ( ), ,..., ),m mr q q r r q q    that is 1 1 3 21 1 22 1 3( , ,..., ) ( ( ), ( ), ,..., ).m mr q q r r q q    the term 3 ( ( , , ), , )        sb sb expresses the function 2 2 4 5( , , ,..., )mr q q q depending on ( 2)m  variables; the equality describing this function can be represented as follows: 2 2 4 5 2 2 2 4 5( , , ,..., ) ( ( ( )), ( ( )), ( ), , ,..., ),m mr q q q r r r q q q       that is 2 2 4 5 31 2 32 2 33 2 4 5( , , ,..., ) ( ( ), ( ), ( ), , ,..., ).m mr q q q r r r q q q     on the comparative complexity of primitive recursive arithmetical and string functions 86 the term 4 ( ( ( , , ), , ), , )            sb sb sb expresses the function 3 3 5 6( , , ,..., )mr q q q depending on ( 3)m  variables; the equality describing this function can be represented as follows: 3 3 5 6 3 3 3 3 5 6( , , ,..., ) ( ( ( ( ))), ( ( ( ))), ( ( )), ( ), , ,..., ),m mr q q q r r r r q q q           that is 3 3 5 6 41 2 42 2 43 2 44 2 5 6( , , ,..., ) ( ( ), ( ), ( ), ( ), , ,..., ).m mr q q q r r r r q q q      using similar considerations, we conclude that the term m expresses the function ( 1)m  depending on one variable (we shall denote this variable by ( 1)mr  ). the equality describing this function can be represented as follows:  ( 1) ( 1) ( 1) ( 1) ( 1) ( 1) ( 2)( 1) ( 2) ( ) ( ( (... ( )...)) , ( ( (... ( )...))),..., ( )),m m m m m m mm m r r r r                    that is ( 1) ( 1) 1 ( 1) 2 ( 1) ( 1)( ) ( ( ), ( ),..., ( )).m m m m m m mm mr r r r         now let us construct the term ( , ).m m s this term expresses the function 1 1 2 2 1 2 1 2 ( ( ( ( ), ( ),..., ( ))), ( ( ( ), ( ),..., ( ))),... ..., ( ( ( ), ( ),..., ( )))). m m m m m m m mm m q q q q q q q q q                 but the strings 1 2( ), ( ),..., ( )mq q q   do not contain other letters except 1.a so, we can conclude that the function expressed by ( , ),m m s is equal to 1 2( ( ), ( ),..., ( )).mq q q    hence the term ( , ( , ))m mg   s s expresses the function 1 2( ( ( ), ( ),..., ( ))),mg q q q    that is, the function 1 2( , ,..., ).mq q q clearly, 13 13 ,c t d   where 13c and 13d are some constants. so, the statement of theorem is proved for 2.m  the cases 1m  and 0m  are considered in a similar way. this completes the proof of theorem. note. applying usual methods of the recursive functions theory, we can obtain essentially more simple and more natural expressions for the term  than those considered above, for example 1 2( , ( , ( , ), ( , ),..., , ( , ))), m m m mg i i i    s s s s s      where any term mki for 1 k m  expresses the function 1 2( , ,..., ) . m k m ki q q q q however, such expressions do not give the required bounds of . for this aim special methods should be used. one of such methods is implemented above. i. zaslavski and m. khachatryan 87 references [1] m. h. khachatryan. “on the representation of arithmetical and string functions in formal languages,” transactions of iiap of nas of ra, mathematical problems of computer science, vol. 27, pp. 37-53, 2006. [2] a. i. maltsev. algorithms and recursive functions. 2nd edition, moskow, nauka , 1986 (in russian). submitted 28.11.2012, accepted 30.01.2013. պարզագույն անդրադրարձ (ռեկուրսիվ) թվաբանական և բառային ֆունկցիաների համեմատական բարդության մասին ի. զասլավսկի և մ. խաչատրյան ամփոփում դիտարկվում են [1]-ում սահմանված պարզագույն անդրադարձ (ռեկուրսիվ) թվաբանական և բառային ֆունկցիաների ներկայացման la և lw ձևային լեզուները։ շենոնի awsh և wash ֆունկցիաները, որոնք բնութագրում են թվաբանական և բառային ֆունկցիաների ներկայացումների բարդությունների միջև եղած կապերը նշված լեզուներում, սահմանվում են, ինչպես [1]-ում։ մի նոր մեթոդով տրվում է awsh ֆունկցիայի վերին գնահատականի ապացույցը։ այդ մեթոդը որոշ դեպքերում ապահովում է կիրառությունների ավելի լայն հնարավորություններ, քան` [1]-ում դիտարկվող մեթոդները։ о сравнительной сложности примитивно рекурсивных арифметических и словарных функций и. д заславский и м. хачатрян аннотация рассматриваются формальные языки la и lw, введенные в [1] для представления примитивно рекурсивных арифметических и словарных функций. функции шеннона awsh и wash , выражающие соотношения между сложностями представления арифметических и словарных функций в этих языках, определяются так же, как в [1]. дается новое доказательство верхней оценки для awsh , основанное на методе, дающем в ряде случаев новые возможности для приложений по сравнению с методами, рассматриваемыми в [1]. d:\sbornik\...\article_eng.dvi mathematical problems of computer science 31, 5{15, 2008. on a class of i r r educible p olynomials over fp me ls ik k . k yu r e g ya n , e d it a y u . h a r u t yu n ya n a n d mika ye l g. e vo ya n y institute for informatics and automation problems of nas of ra yfaculty of informatics and applied mathematics, yerevan state university email: edita@ipia.sci.am, email:michael.ipm@gmail.com abstract the paper presents some results regarding constructive theory of synthesis of irreducible polynomials of degree pt over fp from the given primitive elements in fp, where p is an odd prime and t is an integer whose prime factors all divide p ¡ 1. keywor ds p r im it ive p o lyn o m ia l, e xp lic it ly c o n s t r u c t e d s e qu e n c e s , p e r io d , lin e a r o p e r a t o r , p r im it ive e le m e n t refer ences [1 ] a lb e r t a .a ., f undamental concepts of higher algebra, u n ive r s it y o f ch ic a g o p r e s s , ch ic a g o , 1 9 5 6 . [2 ] p e t e r s o n w . w ., w e ld o n e . i. e rror-correcting codes, 2 n d e d ., m.i.t. p r e s s , ca m b r id g e , ma s s ., 1 9 7 2 . [3 ] v a r s h a m o v r .r . on a m e t h o d o f c o n s t r u c t in g ir r e d u c ib le p o lyn o m ia ls o ve r ¯ n it e ¯ e ld s . d okladi akademii nauk of armenia, vo l. 7 9 , n o .1 , 1 9 8 4 , p p . 2 6 { 2 8 ( in r u s s ia n ) . fp ¹³ßïç íñ³ ³ýí»ñí³ý»éç µ³½ù³ý¹³ùý»ñç ùç ¹³ëç ù³ëçý ø. îûáõñ»õû³ý, ø. ¾íáû³ý, ¾. ð³ñáõãûáõýû³ý ²ù÷á÷áõù ðá¹í³íáõù ý»ñï³û³óí³í »ý ùç ß³ñù ³ñ¹ûáõýùý»ñ, áñáýù ³éýãíáõù »ý ³ýí»ñ³í»éç µ³½ù³ý¹³ùý»ñç ëçý㻽ç ïáýëïñáõïïçí ï»ëáõãû³ýá, ù³ëý³íáñ³å»ë fp ¹³ßïç íñ³ pt ³ëïç׳ýç ³ýí»ñ³í»éç µ³½ù³ý¹³ùý»ñç ï³éáõóù³ýá û·ï³·áñí»éáí fp ¹³ßïç åñçùçïçí ¾é»ù»ýïý»ñá, áñï»õ p-ý å³ñ½ ï»ýï ãçí ¿, çëï t-ý ³ùµáõç ãçí ¿, áñç µáéáñ å³ñ½ µ³½ù³å³ïïçãý»ñá µ³å³ýáõù »ý p ¡ 1 ãçíá : 5 mathematical problems of computer science 47, 37–49, 2017. statistical tests for mixmax pseudorandom number generator narek h. martirosyan, gevorg a. karyan and norayr z. akopov yerevan physics institute, alikhanian brothers street 2, yerevan, armenia e-mail: narek.h.martirosyan@gmail.com, narek@yerphi.am abstract the pseudo-random number generators (prngs) are key tools in monte carlo simulations. more recently, the mixmax prng has been included in root and class library for high energy physics (clhep) software packages and claims to be a state of the art generator due to its long period, high performance and good statistical properties. in this paper the various statistical tests for mixmax are performed. the results compared with those obtained from other prngs, e.g., mersenne twister, ranlux, lcg reveal better qualities for mixmax in generating random numbers. the mersenne twister is by far the most widely used prng in many software packages including packages in high energy physics (hep), however, the results show that mixmax is not inferior to mersenne twister. keywords: mixmax, statistical tests, mcmc. 1. introduction in recent years, there is a growing interest on prngs in different branches of physics and not only. a good prng is important to have guaranteed results of monte carlo(mc) methods. there are many software packages for mc simulations where prngs are the central components. among these packages one can mention the geant4/clhep[1], a widely used simulation toolkit in hep for modeling the passage of elementary particles through matter, also used for medical and space science simulations. prngs are also crucial in markov chain monte carlo (mcmc) methods which are used for sampling from desired probability distribution by constructing markov chain on state space whose stationary distribution is of interest[2, 3, 4]. uniform prngs play a central role in constructing such markov chains. most of mcmc algorithms are developed within random walk models. a widely used example of random walk monte carlo method is metropolishastings algorithm[3, 4, 5, 6] which is also included in the list of the top 10 algorithms[7]. mcmc methods are mainly used for sampling from large dimensional spaces and computing multidimensional integrals. for example, in statistical mechanics, one needs to compute thermal averages of quantities, such as the total energy, magnetization, etc. by performing multidimensional integration or summation over configuration space. however,, the total number of configurations can be very large, e.g., in 3-dimensional ising model the number of spin configurations with particles at n3 lattice sites is 2n 3 . in thermodynamic 37 38 statistical tests for mixmax pseudorandom number generator equilibrium the probabilities of occurring each configuration is represented by boltzmann distribution. thereby having samples drawn from boltzmann distribution one can compute expectation values of thermodynamic quantities. the necessity to have large amounts of simulated data imposes a strict requirements on prngs, such as statistical properties of generated numbers, swiftness in number generation, replicability, lengthiness of generated random cycle and independence of produced random numbers. to address these challenges the renewed version of mixmax prng[8, 9] based on anosov c-systems and kolmogorov k-systems has been introduced in [10, 11, 12]. the mixmax is matrix-recursive prng and it has been shown that the properties of the mixmax generator is improved with increasing the size n of mixmax matrix[12]. the period of mixmax is also increased with increasing n and it can reach up to 1057824, note that the period of commonly used version of mersenne twister based on mersenne prime has the period of 219937 − 1. while having a long period, however, statistical properties and time characteristics of prngs are crucial to consider a generator ”good” or ”bad”. in this paper we will present the results of the statistical tests performed with the matrix size of n = 256 which is considered to be a default dimension of mixmax matrix with flexibility to be further increased. 2. visual demonstration we can reveal the defect of uniform prngs simply plotting random points in highdimensional euclidean space, if these points form a lattice structure then to a first approximation we can say that prng has defects in generating random points since the space is not filled uniformly. the fig.1 shows the comparison of mixmax with the linear congruential generator(lcg), which is known to be defective prng. in contrast to lcg, mixmax does not form a lattice structure. we obtain these figures by generating two u(0, 1) random number sequences and assigning a point in two-dimensional space. fig. 1. random points in two-dimensional space generated by lcg(left) and mixmax(right). 3. statistical testing with testu01 most of prng algorithms produce numbers uniformly distributed in the interval of [0, 1], hence, prngs should pass statistical tests of uniformity. many empirical statistical testing n. martirosyan, g. karyan and n. akopov 39 packages implement tests for these purposes, some of which are [13, 16, 14, 15]. most of the statistical tests implemented in these packages are discussed in knuth’s book [17], e.g., the package [14] implements mainly the tests of knuth. currently, one of the well-known tools for statistical testing is testu01 software library which provides implementations of the empirical statistical tests for uniform prngs. it contains more than 160 different empirical tests and offers several batteries of tests including the most powerful one, i.e., the ’big crush’. when a specific statistical test is applied to random numbers produced by prng the p-value of the test is printed as a measure of deviation from null-hypothesis, which in our case is a uniform distribution of random numbers. in comparison with other libraries testu01 is more flexible and efficient, and it can deal with larger sample sizes and has a wider range of statistical tests than the other libraries. in table 1, the outcome of testu01 bigcrush suite applied on mixmax, mersenne twister and lcg is stored by using 64-bit computer with intel core i3 −4150 processor of clock speed 3.50 × 4 ghz. table 1: testu01 bigcrush suite results. prng total cpu time bigcrush failed test’(s) p value mixmax 2h 43m 51s all tests were passed mersenne twister 3h 19m 27s 3 0.9990, 1 − 10−15 lcg 3h 30m 33s 22 < 10−300 as we can see from the table the mixmax passes the same test suite faster than mersenne twister and does not fail any test. testu01 test suite has been applied to ranlux prng with its modifications ranlux24, ranlux48. it is observed that ranlux, though having good statistical properties is very slow at generating random numbers. comparing with mixmax, ranlux24 is 10 times slower and ranlux48 is 17 times slower. this fact makes it not convenient for the use in generation of large amount of random numbers. 4. kolmogorov-smirnov tests kolmogorov-smirnov (k-s) test is one of the powerful tools that can be used to examine the statistical features of prngs. though one-dimensional (1d) k-s test is already implemented in testu01, we perform ks test independently for various parameters of sample size (n) and extract the distribution of k-s test statistic. the idea behind the test is to calculate the maximum distance between the expected cumulative distribution function(cdf) f(x), f(x) = pr(x ≤ x) and measured or empirical cumulative distribution function(ecdf) fn(x) of n data points fn(x) = 1 n (number of xi ≤ x). (1) the null hypothesis h0 is whether the sample of n random numbers comes from the expected distribution f(x) or not. if data comes from f(x), then the strong law of large numbers provides fn(x) → f(x), as n → ∞. the latter is strengthened by the glivenko-canteli (g-c) theorem[18], which states that under h0 hypothesis pr( lim n→∞ supx |fn(x) − f(x)| = 0) = 1. (2) 40 statistical tests for mixmax pseudorandom number generator hence the difference between cdfs can be used as a measure of agreement between a data and a given distribution. there are several statistical tests based on (g-c) theorem known as cramer-von mises tests [19, 20]. the key feature of tests based on g-c theorem is that their distributions are independent of the hypothesized models under h0 when data sample is large. the one dimenisonal(1d) k-s test is defined as follows: dn = supx |fn(x) − f(x)|. (3) under null hypothesis the distribution of √ n · dn converges to kolmogorov distribution for sufficiently large n when f(x) is continuous[21] lim n→∞ pr( √ n · dn ≤ x) ≡ k(x) = 1 − 2 ∞∑ i=1 (−1)i−1e−2i 2x2. (4) it is of interest to note that for small values of n kolmogorov distribution is not adequate, but there is a way to compute p − value for randomly produced dn[21]. in fig. 4. the normalized histogram of √ n·dn data points for mixmax and mersenne twister is presented and compared with probability density function (pdf) of kolmogorov distribution: f(x) = k′(x) f(x) = 8x ∞∑ i=1 (−1)i−1i2e−2i 2x2. (5) the histograms in fig.(2–7) are normalized to unity dividing each bin entry by the product of sample size and bin width (n · width). visual comparison shows that under h0 the distribution of √ n · dn follows the pdf of kolmogorov distribution. due to fast convergence of the series of partial sums in eq.(4,5) it suffices to take a limited number of terms in the sum, e.g., the first 100 terms are enough. 0.0 0.5 1.0 1.5 2.0 2.5 0.0 0.5 1.0 1.5 x fh x l mersenne mixmax fig. 2. distribution of √ n · dn for mixmax and mersenne twister. the size of a samples is n = 108 and the number of different replicas is 104. the black curve is the pdf of kolmogorov distribution. two-level tests can be used on k-s test to give evidence of visual coincidence on fig. 4.. for this purpose the chi-square test(next section) is applied. it has been checked that both distributions agree with theoretical expectation (black curve) in 95% confidence level (cl). n. martirosyan, g. karyan and n. akopov 41 in multidimensional k-s test one has d-dimensional data (d ≥ 2) and to test h0 it is needed to compare d-variate ecdf with the hypothetical d-variate cdf. the complication in multidimensional case is caused by the ambiguity in definition of the cdf since there are 2d − 1 independent ways of defining cdfs. there have been proposed different ways to calculate the multidimensional k-s statistic [22],[23]. in [22] four quadrants around all combinations (xi, xj) of data points are considered and d is taken as the maximum of 4 differences between cdfs over all quadrants. therefore, this idea makes the test statistic independent of ordering the data. the number of all pairs (xi, xj) for n points is equal to n2, therefore, to calculate dn one needs to compute the differences between cdfs in 3n 2 quadrants(the probability for fourth quadrant is found from normalization). this method suffers from the computing time when n is large. in [23], it is proposed to consider only the observed points rather than all combinations, thereby reducing the computational time by computing the differences for 3n quadrants only. it is possible to compute dn with computationally higher efficiency introducing a binning technique applied to a continuous multidimensional data, i.e., discretizing the data space. the idea of binning technique is discussed in [24]. in [25] the algorithm for 2d k-s test is presented when only one cdf from all the possible configurations is taken into account. as a result of it, the procedure used to compute d evaluating the difference of cdfs is reduced to a small number of data points. in our studies we have extended the standard definition of one dimensional cumulative distribution to its two dimensional “analogue” and computed k-s statistic using the algorithm presented in [25](see fig. 4.). 0.0 0.5 1.0 1.5 2.0 2.5 0.0 0.5 1.0 1.5 x fh x l mersenne mixmax fig. 3. distribution of √ n · dn for mixmax and mersenne twister for 2-dimensional case. the size of a samples is n = 103, note that n is the total number of random points in 2-dimensional space, i.e., 106 numbers are generated by prngs, the number of different replicas is 104. the black curve is the pdf of kolmogorov distribution. the shape of histogram shows a clear shift from the kolmogorov distribution. mixmax and mersenne twister that have been used in these studies give almost the same distributions for √ n · dn which seems to be independent from the dimension of the kolmogorov-smirnov test. 42 statistical tests for mixmax pseudorandom number generator three-dimensional extension of k-s test is presented in [26], where 8 cdfs are considered, and using mc techniques the table of critical values are also presented in this paper. fig. 4. distribution of √ n · dn for the 1st and 31st projections of mixmax rng in comparison with the theoretical expectation (black curve). the sample size is n = 108 and for each projection 104 different replicas are generated. to detect possible non-uniformities in the multidimensional random sequences of mixmax prng, an arbitrary selected projection has been checked via kolmogorov-smirnov test and the results are compared with those of the first projection. in fig. 4., the probability density distributions of kolmogorov-smirnov statistic are presented for the 1st and the 31st projections and the comparison is provided with the expected distribution. 5. chi-square tests the chi-square χ2 test is one of the famous statistical tests which is found in many applications when one deals with grouped or binned data. the chi-square test is applied to categorical sample distributions unlike the k-s test which using each random point compares the continuous sample distributions with the hypothesized ones. the χ2 statistic has the following form [17, 27]: χ2 = k∑ i=1 (oi − ei)2 ei , (6) where oi is the observed number of data points in ith bin and ei = npi is the expected number of data points falling into ith bin, here pi is the probability that observation falls into ith bin. to apply the test to prngs [0,1] interval is divided into k bins and the χ2 statistic is computed noting that pi = 1/k. if h0 is true then statistics defined in (6) computed for random samples follows chi-squared distribution with ν = (k − 1) degrees of freedom gν(y) = 2− ν 2 e− y 2 y ν 2 −1 γ(ν 2 ) , (7) n. martirosyan, g. karyan and n. akopov 43 where γ(ν) is a gamma function. it is useful to introduce a new random variable x = y ν and consider the pdf of x denoted as fν(x). this enables to get rid of small numbers in pdfs when ν is big. since 1 = ∫ fν(x)dx = ∫ gν(y)dy it follows that fν(x) = νgν(νx) = 2− ν 2 ν ν 2 e− x 2ν x ν 2 −1 γ(ν 2 ) . (8) the new random variable introduced in (8) is called a reduced chi-square. the distribution of the reduced chi-square for mixmax and mersenne twister is shown in fig. 5. in comparison with the theoretical expectation. 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 0.0 0.5 1.0 1.5 2.0 x f ν hx l ν= 49 mersenne mixmax fig. 5. comparing denisty histogram of reduced chi-square for mixmax and mersenne twister with the pdf in (8). the size of a samples is n = 106 and the number of different replicas is 104. 6. serial tests the serial test also known as a chi-square test of independence is a multidimensional analogue of the chi-square test which checks the independence between two or more random variables [17, 28]. when the serial test is applied to prngs one divides the random sequence into groups of non-overlapping d-tuples (xid, xid+, ..., xid+k−1), where i = 1, 2, ..., n d , hence the elements of d-tuple are considered as realizations of d random variables and the relationship between them is of interest . if xis are u(0, 1) random variables then k-tuples are uniformly distributed in [0, 1]d. to check this each dimension of unit hypercube is divided into k bins and the data of d-tuples is binned into [0, 1]d. now chi-square statistic (6) is applied to this data comparing the number of observations falling in each sub-hypercube with theoretical expectation: ei,j,...,d = npi,j,...,d, where the joint probability pi,j,...,d of d-dimensional data point to fall into (i, j, ..., d) sub-hypercube is the product of probabilities of each individual coordinate to fall into an appropriate bin, which is the condition of independence. pi,j,...,d = d∏ n=1 pn = ( 1 k )d . (9) 44 statistical tests for mixmax pseudorandom number generator unlike the non-overlapping tuples, the overlapping d-tuples of random sequence fall on neighboring parallel planes. the largest distance between the adjacent parallel hyperplanes is called a spectral test statistic[17, 29, 30, 31, 32, 33]. if the largest distance is small then it implies that overlapping d-tuples are more uniformly distributed in unit hypercube, therefore, prng is considered good. fig. 6. shows d = 3 and d = 5 dimensional cases of serial test, where each dimension is divided into k = 10 bins and the histogram of the reduced chi-square test statistics is compared with the distribution of (8) with appropriate degrees of freedom. the reduced chi-square distribution reveals no significant distinction between mixmax and mersenne twister. note that the serial test here is applied to a single random stream and measures the correlations between adjacent random tuples. this test can be also applied to different streams to check the independence between them. 7. parallel streams of mixmax all tests described in previous sections use one stream generated by prng. however,, in multiprocessor stochastic computations it is important to have uniformly distributed and statistically independent simultaneous random streams partitioned across the processors[34, 35, 36, 37, 38]. different parallelisation approaches of prngs have been studied in literature[39, 40, 41, 38]. one trivial technique for parallelisation is to take random seeds on each processor, but since every prng has a finite number of states, one should be careful in order to avoid possible overlapping between different streams. mixmax has very large state space, therefore, even taking random seeds on each processor does not affect the independence between multiple streams. another approach is to take a single sequence and partition it into different processors. mixmax provides a skipping-ahead algorithm which enables to skip forward by large amount of numbers in sequence, this technique guarantees the non-collision of partitioned streams[39]. the check for randomness of each individual stream can be done via the standard chi-square or k-s tests. the test of independence of multiple streams can be done via parallel version of serial test simply forming d-tuples of random numbers taken from each of d streams at a time. however,, it is not practical to test empirically all random streams when the period is very large, different techniques for testing parallel streams can be found in [38]. the serial test up to dimensions d = 7 has been performed and it is observed that multiple streams of mixmax are statistically independent which is also guaranteed by underlying theory of mixmax. one can analyze the independence and uniformity of parallel streams using the fact that the sum of n independent u(0, 1) random variables follow the irwinhall distribution of order n[42]. under h0 if each stream of prng is generated from a uniform distribution then the random sequence resulted from the element-wise addition of multiple streams has irwinhall distribution. the irwinhall distribution has the following form: fn(x) = 1 2(n − 1)! n∑ i=1 (−1)i ( n i ) (x − i)n−1sgn(x − i), (10) where sgn(x − i) is a sign function. when n = 2, then (10) reduces to the well known n. martirosyan, g. karyan and n. akopov 45 0.8 0.9 1.0 1.1 1.2 0 2 4 6 8 x f ν hx l ν= 999 mersenne mixmax 0.98 0.99 1.00 1.01 1.02 0 20 40 60 80 x f ν hx l ν= 99999 mersenne mixmax fig. 6. comparing denisty histogram of reduced chi-square with the test distribution (8) for 3-dimensional(up) and 5-dimensional(down) cases. triangular distribution f2(x) = { x, 0 ≤ x < 1, 2 − x, 1 ≤ x ≤ 2. (11) in fig. 7. visual comparison of the data with irwinhall distribution is shown, where up to 15 random streams are taken to form the sum. to check visual consistency of the histogram and model prediction in fig.(7.) a chi square test is applied for comparison with irwin-hall distribution. table 2 represents p-values of chi square statistic. 46 statistical tests for mixmax pseudorandom number generator n = 2 n = 5 n = 10 n = 15 0 2 4 6 8 10 0.0 0.2 0.4 0.6 0.8 1.0 x f n hx l fig. 7. the distribution of irwin-hall of order (2,5,10,15) compared with density histogram of data. table 2: p-values from chi-square test. test n = 2 n = 5 n = 10 n = 15 p-value 0.76 0.97 0.39 0.37 8. conclusion this paper presents the study of newly released mixmax prng. the various statistical tests including a very high quality testu01 have been used to check the quality of mixmax compared with other generators, mainly with mersenne twister. the results show that mixmax is not inferior to mersenne twister and even better in the sense of speed and period. acknowledgement the authors would like to thank george savvidy for very useful discussions and comments. this project has received funding from the european union’s horizon 2020 research and innovation programme under the marie sḱlodowska-curie grant agreement no 644121. references [1] s. agostinelli et al., “geant4: a simulation toolkit”, nucl. instrum. meth. a ,vol. 506, no. 250, 2003. 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[42] n.l. johnson, s. kotz and n. balakrishnan, continuous univariate distributions, new york: wiley series in probability and mathematical statistics, 1995. submitted 25.10.2016, accepted 24.01.2017. n. martirosyan, g. karyan and n. akopov 4 9 êï³ïçëïçï³ï³ý ã»ëï»ñ å먹á-å³ï³ñ³ï³ý ãí»ñç ·»ý»ñ³ïáñ mixmax-ç ñ³ù³ñ ü. ø³ñïçñáëû³ý, ¶. î³ñû³ý ¨ ü. ²ïáåáí ²ù÷á÷áõù ä먹á-å³ï³ñ³ï³ý ãí»ñç ·»ý»ñ³ïáñý»ñá ³é³ýóù³ûçý ·áñíçù »ý ñ³ý¹çë³ýáõù øáýï» î³ééá ù»ãá¹áí ùá¹»é³íáñù³ý ñ³ù³ñ: ì»ñç»ñë ýáñ ”mixmax” ·»ý»ñ³ïáñá áý¹·ñïí»ó root ¨ class library for high energy physics (clhep) íñ³·ñ³ûçý ÷³ã»ãý»ñç ù»ç »ñï³ñ å³ñµ»ñáõãû³ý, ³ñ³·³·áñíáõãû³ý ¨ é³í ëï³ïçëïçï ñ³ïïáõãûáõýý»ñç ßýáññçí: ü»ñï³û³óíáõ ñá¹í³íáõù áõëáõùý³ëçñí»é »ý “mixmax”ç ëï³ïçëïçï ñ³ïïáõãûáõýý»ñá ï³ñµ»ñ ëï³ïçëïçï³ï³ý ã»ëï»ñáí: ²ñ¹ûáõýùý»ñá ñ³ù»ù³ïí»é »ý ï³ñµ»ñ ·»ý»ñ³ïáñý»ñçó ëï³óí³í ³ñ¹ûáõýùý»ñç ñ»ï, ûñçý³ï` “mersenne twister”-ç, “ranlux”-ç ¨ “lcg”-ç: òáõûó ¿ ïñí»é áñ “mixmax”-ý áõýç ³í»éç é³í ñ³ïïáõãûáõýý»ñ å³ï³ñ³ï³ý ãí»ñç ·»ý»ñ³óù³ý ñ³ù³ñ: “mersenne twister” ·»ý»ñ³ïáñá ñ³ý¹çë³ýáõù ¿ ³ù»ý³ï³ñ³íí³í ·»ý»ñ³ïáñá ß³ï íñ³·ñ³ûçý ÷³ã»ãý»ñáõù, ë³ï³ûý ý»ñï³û³óí³í ³ñ¹ûáõýùý»ñá óáõûó »ý ï³éçë, áñ “mixmax”-á áã ùç³ûý ãç ½ççáõù “mersenne twister”-çý, ³ûé ý³¨ ·»ñ³½³ýóáõù ¿ ýñ³ý: còàòèñòè÷åñêèå òåñòû äëÿ ãåíåðàòîðà ïñåâäîñëó÷àéíûõ ÷èñåë mixmax í. ìàðòèðîñÿí, ã. êàðÿí è í. àêîïîâ àííîòàöèÿ ãåíåðàòîðû ïñåâäî-ñëó÷àéíûõ ÷èñåë (ãïñ×) ÿâëÿþòñÿ êëþ÷åâûìè èíñòðóìåíòàìè ìîäåëèðîâàíèÿ ïî ìåòîäó ìîíòå êàðëî. íåäàâíî íîâûé ãïñ× ”mixmax” áûë âêëþ÷åí â ïàêåòû ïðîãðàììíîãî îáåñïå÷åíèÿ root è class library for high energy physics (clhep) ââèäó òîãî, ÷òî ýòîò ãåíåðàòîð ïðèçíàí îäíèì èç ëó÷øèõ ñóùåñòâóþùèõ ãïñ× ïî äëèííîìó ïåðèîäó, âûñîêîé ïðîèçâîäèòåëüíîñòè, à òàêæå îòëè÷íûì ñòàòèñòè÷åñêèì ñâîéñòâàì. â ïðåäëàãàåìîé ñòàòüå ïðèâîäÿòñÿ ðåçóëüòàòû ðàçëè÷íûõ ñòàòèñòè÷åñêèõ òåñòîâ äëÿ ïðîâåðêè ãåíåðàòîðà ”mixmax”. ïðîâåäåíî ñðàâíåíèå õàðàêòåðèñòèê ”mixmax” ñ äðóãèìè ãïñ×, íàïðèìåð ñ ”mersenne twister”, ”ranlux” è ”lcg”, ïîêàçàíî, ÷òî ”mixmax” îáëàäàåò ëó÷øèìè ñâîéñòâàìè äëÿ ãåíåðàöèè ñëó÷àéíûõ ÷èñåë. ãåíåðàòîð ”mersenne twister” ÿâëÿåòñÿ îäíèì èç ñàìûõ èñïîëüçóåìûõ âî ìíîãèõ ïðèëîæåíèÿõ, âêëþ÷àÿ ïàêåòû äëÿ ôèçèêè âûñîêèõ ýíåðãèé, îäíàêî ïðèâåäåííûå ðåçóëüòàòû ïîêàçûâàþò, ÷òî ”mixmax” íè â ÷åì íå óñòóïàåò è äàæå ïðåâîñõîäèò ”mersenne twister”. narek_47_mpcs abstract d:\sbornik\...\article_eng.dvi mathematical problems of computer science 31, 116{121, 2008. n ecessar y and su±cient condition for e xistence of locally-balanced 2-par tition of a t r ee under the e xtended de¯nition of a n eighbour hood of a ver tex s u r e n v . b a likya n y a n d r a fa ye l r . k a m a lia n z y yerevan state university e-mail: suren.balikyan@gmail.com z russian-armenian state university e-mail: rrkamalian@yahoo.com abstract a necessary and su±cient condition is obtained for the problem of partitioning of the set of vertices of a tree g into two disjoint sets v1 and v2 such that it satis¯es the condition jj¸(v) \ v1j ¡ j¸(v) \ v2jj · 1 for any vertex v of g, where ¸(v) is the set of all vertices of g the distance of which from v does not exceed 1. refer ences [1 ] s .v . b a likya n , r .r . k a m a lia n , " on n p -c o m p le t e n e s s o f t h e p r o b le m o f e xis t e n c e o f l o c a lly-b a la n c e d 2 -p a r t it io n fo r b ip a r t it e gr a p h s g wit h ¢ ( g ) = 3 " , r eports of nas r a, vo l. 1 0 5 , n u m . 1 , p p . 2 1 { 2 7 , 2 0 0 5 . [2 ] s .v . b a likya n , r .r . k a m a lia n , " on n p -c o m p le t e n e s s o f t h e p r o b le m o f e xis t e n c e o f l o c a lly-b a la n c e d 2 -p a r t it io n fo r b ip a r t it e gr a p h s g wit h ¢ ( g ) = 4 u n d e r t h e e xt e n d e d d e ¯ n it io n o f t h e n e ig h b o u r h o o d o f a v e r t e x" , r eports of nas r a, vo l. 1 0 6 , n u m . 3 , p p . 2 1 8 { 2 2 6 , 2 0 0 6 . [3 ] s .v . b a likya n , " on l o c a lly-b a la n c e d 2 -p a r t it io n s o f s o m e b ip a r t it e gr a p h s " , abstracts of papers of 15th international conference "m athematics. computing. e ducation.", vo l. 1 5 , p . 7 , d u b n a , r u s s ia , ja n u a r y 2 8 fe b r u a r y 0 2 2 0 0 8 . [4 ] f. h a r a r y, graph theory, a d d is o n -w e s le y, r e a d in g , ma , 1 9 6 9 . [5 ] c. b e r g e , graphs and hypergraphs, e ls e vie r s c ie n c e l t d , 1 9 8 5 . 1 1 6 s. balikyan and r. kamalian 1 1 7 ì³éáõù éáï³é-ñ³í³ë³ñ³ïßéí³í 2-ïñáñù³ý ·áûáõãû³ý ñ³ù³ñ ³ýññ³å»ßï ¨ µ³í³ñ³ñ å³ûù³ý ·³·³ãç ßñç³ï³ûùç áý¹é³ûýí³í ë³ñù³ýù³ý ¹»åùáõù ê. ´³éçïû³ý, è. ø³ù³éû³ý ²ù÷á÷áõù êï³óí³í ¿ ³ýññ³å»ßï ¨ µ³í³ñ³ñ å³ûù³ý í³éç ·³·³ãý»ñç µ³½ùáõãû³ý v1 ¨ v2 ãñ³ïíáõ »ýã³µ³½ùáõãûáõýý»ñç ³ûýåçëç ïñáñù³ý ·áûáõãûáõýá å³ñ½»éáõ ñ³ù³ñ, áñ í³éç ûáõñ³ù³ýãûáõñ v ·³·³ãç ñ³ù³ñ ï»õç áõý»ý³ jj (̧ v ) \ v1j ¡ j¸ ( v ) \ v2jj · 1 ³ýñ³í³ë³ñáõãûáõýá, áñï»õ (v)-áí ýß³ý³ïí³í ¿ ³ûý ·³·³ãý»ñç µ³½ùáõãûáõýá, áñáýó ñ»é³íáñáõãûáõýá v-çó ãç ·»ñ³½³ýóáõù 1-çý: microsoft word tpel.doc математические вопросы кибернетики и вычислительной техники 31, 158—162, 2008. 158 квадратурная формула золотого сечения с.б. аллахвердян институт проблем информатики и автоматизации нан ра и ергу e-mail: souren@ipia.sci.am резюме получены формулы апpроксимирующие корни многочлена лежандра с помощью золотого сечения (фибоначчи). сравнены результаты квадратурной формулы золотого сечения с известными формуйами чебышева и лежандра. литература [1] н. м. воробьев, числа фибоначчи, москва, наука, 1992. [2] д. каханер, к. моулер, с. нэш. численные методы и программное обеспечение (пер. с англ.). м.: мир, 2001. [3] r. fischer, fibonacci applications and, strategies for traders, new york: wiley, 1993, p. 13. ”àëï» ñ³ïù³ý” ù³é³ïáõë³ûçý µ³ý³ó¨ ê. ²éé³ëí»ñ¹û³ý ²ù÷á÷áõù êï³óí³í »ý µ³ý³ó¨»ñ, áñáýù ùáï³ñïáõù »ý è»å³ý¹ñç µ³½ù³ý¹³ùç ³ñù³ïý»ñá ”áëï» ñ³ïù³ý” (üçµáý³ããçç ãí»ñç) ùççáóáí: ´»ñí³í »ý ³ñ¹ûáõýùç` ”áëï» ñ³ïù³ý” ù³é³ïáõë³ûçý µ³ý³ó¨ç ñ³ù»ù³ïáõùá ⻵çß¨ç ¨ è»å³ý¹ñç ñ³ûïýç ù³é³ïáõë³ûçý µý³ý³ó¨»ñç ñ»ï: article_with_style.dvi mathematical problems of computer science 31, 73{78, 2008. application of ldt to m any h ypotheses optimal t esting for m ar kov chain l e a d e r n a va e i payame noor university (pnu), iran abstract the problem of many (l > 2) hypotheses testing on distributions of a ¯nite state markov chain is studied. we apply large deviations techniques (ldt). it is proved that this method of investigation in solving the problem of logarithmically asymptotically optimal (lao) hypotheses testing is easier than the procedure that was introduced by haroutunian. the matrix of exponents e = feljmg; m; l = 1; l, of error probabilities of the lao test eljm(á) = lim n !1 ¡ 1 n log ®ljm(án ); where ® n ljm(án ) for l 6= m is the probability to accept the hypothesis l, when the hypothesis m is true, is determined. refer ences [1 ] b la h u t r . e . \ p r in c ip le a n d p r a c t ic e o f in fo r m a t io n th e o r y" , r e a d in g , ma , a d d is o n we s le y, 1 9 8 7 . [2 ] cs is z ¶a r i. a n d s h ie ld s p . \ in fo r m a t io n th e o r y a n d s t a t is t ic s " , fu n d e m e n t a ls a n d tr e n d s in co m m u n ic a t io n s a n d in fo r m a t io n th e o r y, vo l. 1 , n o . 4 , 2 0 0 4 . [3 ] cs is z ¶a r i. a n d k äo r n e r j. \ in fo r m a t io n th e o r y: co d in g th e o r e m fo r d is c r e t e me m o r yle s s s ys t e m s " , a c a d e m ic p r e s s , n e wy o r k, 1 9 8 1 . [4 ] cs is z ¶a r i. \ me t h o d o f t yp e s " , ie e e tr a n s . in fo r m . th e o r y, vo l. 4 4 . n o . 6 . p p . 2 5 0 5 -2 5 2 3 , 1 9 9 8 . [5 ] d e m b o a . a n d ze it o u n i o. \ l a r g e d e via t io n s te c h n iqu e s a n d a p p lic a t io n s " , jo n s a n d b a r t le t . p u b lis h e r s , l o n d o n , 1 9 9 3 . [6 ] gu t m a n m. \ a s ym p t o t ic a lly o p t im a l c la s s ī c a t io n fo r m u lt ip le t e s t wit h e m p ir ic a lly o b s e r ve d s t a t is t ic s " , ie e e tr a n s . in fo r m . th e o r y, vo l. 3 5 , n o . 2 . p p . 4 0 1 -4 0 8 , 1 9 8 9 . [7 ] h a r o u t u n ia n e . a . \ on a s ym p t o t ic a lly o p t im a l t e s t in g o f h yp o t h e s e s c o n c e r n in g ma r ko v c h a in " , ( in r u s s ia n ) . iz ve s t ia a c a d . n a u k a r m e n ia n s s r . s e r ia ma t h e m . vo l. 2 2 , n o . 1 . p p . 7 6 -8 0 , 1 9 8 8 . [8 ] h a r o u t u n ia n e . a , h a r o u t u n ia n m. e a n d h a r u t yu n ya n a . n .\ r e lia b ilit y cr it e r ia in in fo r m a t io n th e o r y a n d in s t a t is t ic a l h yp o t h e s is te s t in g " , fo u n d a t io n s a n d tr e n d s in co m m u n ic a t io n s a n d in fo r m a t io n th e o r y, vo l. 4 , n o . 2 -3 , 2 0 0 7 . [9 ] k u llb a c k s . \ in fo r m a t io n th e o r y a n d s t a t is t ic s " , w ile y, n e w y o r k, 1 9 5 9 . [1 0 ] n a t a r a ja n s .\ l a r g e d e via t io n s , h yp o t h e s e s t e s t in g , a n d s o u r c e c o d in g fo r ¯ n it e ma r ko v c h a in " , ie e e tr a n s . in fo r m . th e o r y, vo l. 3 1 , n o . 3 , p p . 3 6 0 -3 6 5 , 1 9 8 5 . [1 1 ] n a va e i l .\ on m a n y h yp o t h e s e s l a o t e s t in g via t h e t h e o r y o f la r g e d e via t io n s " , fa r e a s t jo u r n a l o f ma t h e m a t ic a l s c ie n c e s , vo l. 2 5 , n o . 2 , p p . 3 3 5 -3 4 4 , 2 0 0 7 . 7 3 7 4 application of ldt to many hypotheses optimal testing for markov chain øþî ïçñ³éáõãûáõýá ø³ñïáíç ßõã³ý»ñç ñ³ù³ñ µ³½ù³ïç í³ñï³íý»ñç ûåïçù³é ï»ëï³íáñù³ýá è. ü³í³ûç ²ù÷á÷áõù àõëáõùý³ëçñí³í ¿ í»ñç³íáñ íç׳ïý»ñáí ø³ñïáíç ßõã³ûç ýï³ïù³ùµ µ³½ù³ïç í³ñï³íý»ñç ëïáõ·ù³ý ëý¹çñá: îçñ³éíáõù ¿ ù»í ß»õáõùý»ñç ï»ëýçï³ý (øþî): ²å³óáõóí»é ¿, áñ í³ñï³íý»ñç éá·³ñçãùáñ»ý ³ëçùåïáïáñ»ý ûåïçù³é (è²ú) ï»ëï³íáñù³ý ñ»ï³½áïù³ý ³ûë »õ³ý³ïá ³í»éç ñ»ßï ¿, ù³ý ð³ñáõãûáõýû³ýç ïáõùçó ý»ñùáõíí³íá: начиная с начала 2000 года осуществляется внедрение ghis в здравоохранении, в рамках принятого проекта о реформирование информ mathematical problems of computer science 46, 55--58, 2016. about some queueing models for computational grid systems vladimir g. sahakyan, yuri h. shoukourian and hrachya v. astsatryan institute for informatics and automation problems of nas ra e-mail: vladimir.sahakyan@sci.am, shouk@sci.am, hrach@sci.am abstract in this paper parametric models of the queues are proposed in grid systems. the models take into account the restrictions on the waiting and interval of execution time of the tasks. keywords: grid computing, queueing systems, resource allocation systems. the optimal usage of cpu times in multi-core computational grid infrastructures depends on several factors, such as the job scheduling, the possibility of dynamic allocation of computational resources, the possibility of job migration to the different phases of implementation in optimal requested environment, performing a stop of the job with the possibility of continuing, etc. the use of grid infrastructures to handle the information stream requires a flexible approach to the allocation of resources as well as for the timely performance of jobs. the job receiving process in the queue for further executing in a grid infrastructure plays an important role in the organization of the whole process. at this stage the incoming job service script is generated based on a maintenance data bursts. this requires synchronization of distributed processes to manage the resources of a grid system. a set of gram (grid resource allocation and management) services allows to manage the resources and jobs, job queues handling, ensuring their execution on the required computational resources. in addition to gram services, the schedulers play special roles. the adoption of the job queue for a service applies to the scheduler responsible for ensuring its timely implementation. nowadays the usage of modern schedulers, such as maui, condor, pbs (portable batch system) has opportunities to perform the planning tasks on specified priorities [1,2,3]. however, in complex cases (for instance, to run the job at a specified time and in specified resources, or to manage multi-stream queues) we need to implement a flexible approach for scheduling. 55 mailto:vladimir.sahakyan@sci.am mailto:shouk@sci.am mailto:hrach@sci.am about some queueing models for computational grid systems 56 the optimal run of the programs in multi-core environments involve selection of service discipline, delivering at least a "loss function", which characterizes the quality of the functioning of the process. as such a function can be considered some functional depending on the average waiting time, utilization of time, the number of denial of service and others. many authors have shown that the discipline of the relative priority is optimal for linear functional without service interruption. if prevent the implementation of programs interrupted, sometimes absolute priority disciplines are better. however, in systems with many devices maintenance procedure can be disrupted, while retaining the properties of known subjects. for example, in case of a fifo discipline, if a job is received after the service does not affect the incoming job before him, it may be served. this can be done using batchfill mechanism (check available periods for low-priority tasks). often it is required to provide manage the job not later or earlier, or in a specified period of time after entering the system. for a description of these conditions we consider the classification of the job queues on systems diagrams defining the service priorities. schemes depending on the order of receipt  mfifo (modified fifo) – received to the job system, it is queued in order of receipt and can serve, if it does not affect the start time of service to all previously received assignments;  mlifo (modified lifo) received to the job system, it is queued in order of receipt and can serve, if it does not affect the start of the service later he received jobs from the queue.  mrs (modified random service) received to the job system, it is queued and can cater for random selection depending on specific task parameters. schemes based on the level of required resources -ju (job up): received to the job system, it is queued in order of required resources for the service ([service time] * [number of required cores]) and can serve, if it does not affect the start of the maintenance of all available jobs in the queue ahead of its assignments. previously, all will go on maintenance task requires the most resources. -jd (job down): received to the job system, it is queued in order of increasing required for maintenance resources ([service time] * [number of required cores]) and can serve, if it does not affect the start of the maintenance of all available jobs in line behind its assignments. previously, all will go on maintenance task requires the least resources. schemes based on the number of processors required -pu (processor up): received to the job system, it is queued in order of processors required for service and can serve if it does not affect the start of the maintenance of all available jobs in the queue ahead of its assignments. previously, all will go on maintenance task requires the most resources. -pd (p down): received to the job system, it is queued in order of increasing required maintenance for the processor and can serve, if it does not affect the start of service available queued jobs behind it. previously, all will go on maintenance task requires the least resources. schemes, with a restriction on the waiting time -wtr (waiting time restriction): received to the job system, it is queued in order acceptable waiting time, i.e., the higher priority admission service has a job with less latency. -itr (interval time restriction): received to the job system, it is queued and must start the handling after a specified time, but no later than the specified timeout. the first goes on a mission handling shall start the service before anyone else. all queuing schemes permitted job handling from the queue, if its handling does not affect the maintenance of higher-priority jobs. v. sahakyan, yu. shoukourian and h. astsatryan 57 schemes are mutually exclusive and only one of the circuits can be selected and when queuing. however, restrictions on the waiting time can be in all schemes. in systems allowing job migrations, these schemes can be described as a service processes without interruption (relative priority) and with interruption (absolute priority). the job interruption may take place in case of entering higher priority jobs. we assume that the job under the new entry to the service will be continued (not started first). setting with interrupted service it is returned to the queue with the adjusted design parameters (service time, waiting time, etc.). for a more detailed formulation let’s consider a computational consisting of m (m ≥ 1) processors. when using systems based on virtual clusters, limiting the number of processors is arbitrary and depends on the performance of the basic configuration. we assume that the number of jobs that can be in the queue is not limited. the grounds of refusal of service can only serve as the impossibility of his service with the user-defined constraints (time, number of processors, etc.). let job stream be supplied. each job can be characterized by parameters (ν, β, ω, γ), where ν the number of processors required for the job, β the time required to run, ω permissible total residence time job in the queue, γ the time from the admission to the system after it is allowed to start the service. fig. 1. the example of task an acceptable execution. if γ = 0, the value of this parameter can be omitted. in systems with no limit on the waiting time the value of the parameter ω is also lowered. for jobs with a restriction on the waiting time for admission to the system is checked on the ability to perform a task and then either taken in turn to perform, or is refused by the implementation. the time required for maintenance is in some sense arbitrary, which means maximum allowable. in reality, it is random and may be less than the predetermined one. therefore, the order may vary both in case of receiving the jobs and finishing the maintenance of jobs. service denial gets jobs, if at the time of admission to the system it finds that it cannot serve the specified parameters. for example, to start services at a specified time. the system is considered at discrete points in time, the job proceeds to the queue or completion of service. we call these moments 0-momentums. in each 0-momentum it carries out queue recalculation and a new order of the queue in the computational system. for various schemes a multithreading queuing model is proposed. between the streams are set priorities, and inside threads can be used as one of the described schemes of service. about some queueing models for computational grid systems 58 for the above mentioned schemes, new algorithms have been developed to schedule jobs with mixed priorities. note that the algorithms can be used both in homogeneous and in lines with mixed schemas with a restriction on waiting time. references [1] i. foster, computational grid, morgan-kufman, 1999. [2] i. foster, c. kesseman and s. tuecke, “the anatomy of the grid”, international journal of hpc applications, vol. 15, no. 3, pp. 25-37, 2001. [3] f. xhafa, “batch mode scheduling in grid systems”, int.j. web and grid services, vol.3, no. 1, pp.19-37, 2007. submitted 04.07.2016, accepted 20.10.2016. հերթերի որոշ մոդելների մասին հաշվողական գրիդ– համակարգերում վ.սահակյան, յու.շուքուրյան և հ. ասցատրյան ամփոփում հոդվածում առաջարկվում է հաշվողական գրիդ–համակարգի համար հերթի պարամետրիկ մոդելը։ մոդելը հաշվի է առնում առաջադրանքի հերթում սպասման և կատարման թույլատրելի ժամանակները։ о некоторых моделях очередей в вычислительных гридсистемах в. саакян, ю. шукурян и г. асцатрян аннотация в статье предложена параметрическая модель очереди для вычислительной гридсистемы. модель учитывает ограничения на допустимые времена ожиданий и промежутков выполнения заданий. microsoft word pa;yan_14.doc математические вопросы кибернетики и вычислительной техники 34, 2010. 39 о необходимости восстановления специальности для подготовки программистов в бакалавриате технических вузов армении (ра) а. х. палян с 1992 по 2007 годы в государственном инженерном университете армении (гиуа) на факультете компьютерных систем и информатики велось обучение бакалавров по следующим специальностям в области информационных технологий (ит):  вычислительные машины, комплексы, системы и сети (вмксс)  программное обеспечение вычислительной техники и асу (повтас)  автоматизированные системы обработки информации и управления (асоиу). по этим же специальностям велась обучение в технических вузах российской федерации (рф). в 2001 году авторитетная организация acm (сша) объявила рекомендуемый перечень специальностей бакалавриата в области ит, который был расширен в 2005 году. перечень опубликован в документе computing curricula 2005 (http://www.acm.org/education/curric_vols/cc2005-march06final.pdf). в настоящее время в части технических вузов он включает:  computer engineering (вмксс)  software engineering (повтас)  information systems (асоиу)  information technology (асоиу). в скобках указано примерное соответствие вышеупомянутым специальностям. данный перечень принят в большинстве вузов сша в качестве стандарта де-факто . после 2001 года в рф ряд ученых, например, профессор мгу в. сухомлин, выступали за введение перечня специальностей acm в бакалавриате вузов рф. в настоящее время на сайте минобразования и науки рф опубликован перечень специальностей бакалавриата ( http://mon.gov.ru/dok/fgos/7198/). в данном перечне направление “информатика и вычислительная техника” в части технических вузов включает специальности :  информатика и вычислительная техника (computer engineering)  программная инженерия (software engineering)  информационные системы и технологии (information systems, information technology). очевидно, что в рф приняли за основу перечень специальностей acm, примерное соответствие которым указано в скобках. в ра в связи с переходом на кредитную систему обучения в 2007 году минобразования и науки ра ввело новый перечень специальностей для бакалавриата технических вузов (гиуа и других вузов):  информатика и вычислительная техника пстроение ассоциативных правил путем цепного раздробления n-мерного единичного куба цепями 40  информационные системы  информационные технологии. в этом перечене отсутствует специальность для подготовки программистов, что по некоторым сведениям является следствием принятого минобразования и науки ра курса на укрупнение специальностей. таким образом, специальность для подготовки программистов, существовавшая долгие годы в технических вузах ра под названием повтас, а в настоящее время в рф и сша под названием “программная инженерия (software engineering)”, оказалась утраченной. вследствие этого в гиуа бывшие специальности повтас и вмксс были включены в новую специальность “информатика и вычислительная техника” в статусе специализаций. программистская специальность в статусе специализации выглядит очень странно. дело в том, что программирование включает широкий набор дисциплин и в соответствии с мировой практикой включает ряд специализаций, как например, программирование баз данних, сетевое программирование, программирование операционных систем и т. д. снижение статуса подготовки программистов до уровня специализации, несомненно, скажется на качестве их обучения. студент, специализирующийся по программированию, должен изучать также предметы, относящиеся к общей специальности, в частности, к вмксс. общеизвестно, что программисты составляют большинство специалистов, работающих в области ит ра. снижение уровня подготовки программистов в гиуа и других вузах приведет к снижению их общего потенциала, что отрицательно скажется на состоянии приоритетной для ра области информационных технологий . учитывая вышеизложенное, представляется целесообразным восстановление специальности для подготовки программистов в бакалавриате технических вузов ра под названием “программная инженерия (software engineering)”. для сохранения общего числа специальностей можно было бы объединить “информационные системы” и “информационные технологии” в одну специальность так, как сделано в рф, достигая тем самым полного соответствия перечней ра и рф. палян арменак христафорович докт. техн. наук, проф. (гиуа) 4_rafayel_23--32_new.dvi mathematical problems of computer science 48, 23{32, 2017. about complexity of fft algor ithms for length of q £ 2p r a fa ye l v . b a r s e g h ya n institute for informatics and automation problems of nas ra e-mail: rafayelbarseghyan@ipia.sci.am abstract the paper presents logarithmic formula which allows to compute the exact number of necessary operations for computing the discrete fourier transform (dft) of an arbitrary q £ 2p length, where q is an odd integer. keywords: fast fourier transform (fft), split-radix algorithm, computational complexity. 1 . in t r o d u c t io n th e d is c r e t e fo u r ie r t r a n s fo r m ( d ft) h a s a wid e r a n g e o f a p p lic a t io n s in m a n y ¯ e ld s o f s c ie n c e a n d e n g in e e r in g [1 ],[2 ]. th e m a in r e a s o n fo r it s p o p u la r it y is t h e e xis t e n c e o f va r io u s a lg o r it h m s wh ic h a llo ws t o s ig n i¯ c a n t ly r e d u c e t h e c o m p u t a t io n a l c o m p le xit y. th e s e a lg o r it h m s a r e g e n e r a lly kn o wn a s fa s t fo u r ie r t r a n s fo r m s ( fft) . fa s t a lg o r it h m fo r e ± c ie n t c o m p u t a t io n o f d ft wa s ¯ r s t in t r o d u c e d b y cu le y a n d tu ke y b y t h e ir h is t o r ic a l p a p e r in 1 9 6 5 [3 ]. fft a lg o r it h m s a llo w t o c o m p u t e d ft o f s iz e n wit h o ( n lg n ) o p e r a t io n s in c o n t r a s t t o d ir e c t fo r m c o m p u t a t io n wh ic h r e qu ir e s o ( n 2 ) o p e r a t io n s . th e r e a r e a n u m b e r o f fft a lg o r it h m s , b u t t h e m o s t p o p u la r m e t h o d s a r e b a s e d o n ¯ xe d -r a d ix a n d s p lit -r a d ix a p p r o a c h e s . s p lit -r a d ix a lg o r it h m s h a ve b e e n c o n s id e r e d t o b e t h e m o s t c o m p u t a t io n a lly e ± c ie n t a n d s t r u c t u r a lly r e g u la r . s p lit -r a d ix a lg o r it h m wa s ¯ r s t in t r o d u c e d b y y a vn e [4 ] in 1 9 6 9 a n d la t e r b y va r io u s a u t h o r s [8 ]. s p lit -r a d ix a lg o r it h m a llo ws t o c o m p u t e d ft o f n = 2 p wit h 4 n lg n ¡ 6 n + 8 a r it h m e t ic o p e r a t io n s . in r e c e n t ye a r s b y va r io u s a u t h o r s [5 ], a n e w m o d i¯ c a t io n o f s p lit r a d ix a lg o r it h m wa s d e ve lo p e d wh ic h a llo ws t o p e r fo r m d ft o f n = 2 p wit h 34 9 n lg n ¡ 124 27 n ¡ 2 lg n ¡ 2 9 ( ¡ 1 ) lg n lg n + 16 27 ( ¡1 ) lg n + 8 a r it h m e t ic o p e r a t io n s . fo r a p p lic a t io n s , wh ic h n e e d t o p e r fo r m d ft o f s iz e s n 6= 2 p, u s u a lly s p e c ia lis t s u s e t h e z e r o p a d d in g t e c h n iqu e . it m e a n s t h a t t h e in p u t s e qu e n c e is ¯ lle d wit h z e r o e s u n t il it b e c o m e s a p o we r o f t wo le n g t h fo r p e r fo r m in g a n y a va ila b le fft a lg o r it h m . s u c h m e t h o d s ig n i¯ c a n t ly d e c r e a s e s t h e r e qu ir e d n u m b e r o f a r it h m e t ic o p e r a t io n s . d ft fo r in p u t s e qu e n c e s wh ic h h a s le n g t h n o n -p o we r -o f-2 is r e qu ir e d in m a n y p r a c t ic a l a p p lic a t io n s , it is a n im p o r t a n t p r o b le m . a lg o r it h m fo r c o m p u t in g d ft fo r s iz e s q £ 2 p, wh e r e q is a n o d d in t e g e r , wa s in t r o d u c e d b y b i a n d ch e n in 1 9 9 8 [6 ]. a lg o r it h m h a s a 2 =4 s p lit -r a d ix s t r u c t u r e a n d in c a s e o f q = 1 h a s t h e s a m e c o m p le xit y a s t h e c o n ve n t io n a l s p lit -r a d ix fft a lg o r it h m . a ft e r t h a t , in 2 0 0 4 b y b o u g u e z e l a n d e t . a l. [1 0 ] a n e w im p r o ve d a lg o r it h m fo r q£ 2 p le n g t h d ft wa s p r e s e n t e d . 2 3 2 4 about complexity of fft algorithms for length of q £ 2p a lg o r it h m is b a s e d o n 2 = 8 s p lit -r a d ix fft a lg o r it h m s c h e m e a n d im p r o ve s s u c h im p o r t a n t fa c t o r s a s d a t a t r a n s fe r , a d d r e s s g e n e r a t io n , t wid d le fa c t o r c o m p u t a t io n a n d a c c e s s t o t h e lo o ku p t a b le , b u t it d id n o t r e d u c e n u m b e r o f a r it h m e t ic o p e r a t io n s . in 2 0 1 0 b i a n d ch e n [7 ] p u b lis h e d a n e w p a p e r wh e r e t h e y p r e s e n t e d a u n i¯ e d m e t h o d fo r g e n e r a t io n o f 2 =2 a ( wh e r e a is a n in t e g e r a n d a > 1 ) s p lit -r a d ix a lg o r it h m s fo r q £ 2 p le n g t h d fts . in t h is p a p e r a g e n e r a l lo g a r it h m ic fo r m u la is d e r ive d fo r c a lc u la t in g n u m b e r o f a r it h m e t ic o p e r a t io n s fo r 2 = 4 a n d 2 =8 s p lit -r a d ix a lg o r it h m s fo r q £ 2 p le n g t h d fts [1 4 ]. fo r a ll q < 2 0 s p e c ia l c a s e s , fo r m u la s a r e d e ve lo p e d fo r c o u n t in g e xa c t n u m b e r o f a r it h m e t ic o p e r a t io n s a n d s o m e c a s e s a r e s h o wn , wh e r e c o m p u t a t io n a l e ®e c t ive n e s s is in ve r s e ly p r o p o r t io n a l t o t h e le n g t h o f t h e d ft-s iz e . 2 . ge n e r a l a lg o r it h m l e t x = fx0; x1; : : : ; xn¡1gt b e a c o m p le x va lu e d c o lu m n -ve c t o r o f le n g t h n, wh e r e n = q £ 2 p a n d q is a n o d d in t e g e r . th e d ft o f t h is ve c t o r a r e d e ¯ n e d a s x[k] = n ¡1x k=0 x[n]w nkn ; ( 1 ) wh e r e 0 · k · n ¡ 1 ; w nn = e xp ( ¡j 2¼n n) = c o s ( 2¼ n n) ¡ j s in ( 2¼ n n) ; j = p ¡ 1 : b e lo w t h e a lg o r it h m fr o m [7 ] is p r e s e n t e d . e ve n in d e xe s o f t h e t r a n s fo r m a r e c o m p u t e d b y x[2 k] = n=2¡1x n=0 ( x[n] + x[n + n=2 ]]) w nkn=2; ( 2 ) wh e r e x[2 k] is a n n=2 d ft. th e o d d in d e xe s a r e d e ¯ n e d b y x[2 ak + l] = n ¡1x n=0 x[n]w n(2ak+l) n ; ( 3 ) wh e r e 0 · k · ( n=2 a ) ¡ 1 , a n d a is a n in t e g e r ( a > 1 ) a n d l h a s a s e le c t e d o d d va lu e s s o t h a t 2 ak + l g e n e r a t e s n=2 o d d in t e g e r s t h a t c a n b e u n iqu e ly m a t c h e d t o a ll t h e o d d in d e x va lu e s b e t we e n 0 a n d n. w it h s o m e m a n ip u la t io n s b a s e d o n t h e p e r io d ic a n d s ym m e t r ic p r o p e r t ie s o f w n(2ak+l) n , ( 3 ) c a n b e r e p r e s e n t e d a s x[2 ak + l] = (n=2a)¡1x n=0 x0[n]w nln w nk n=2a + (n=2a)¡1x n=0 x0[n + n=2 a]w (n+n=2a)l n w nk n=2a + : : : + (n=2a)¡1x n=0 x0[n + ( a ¡ 1 ) n 2 a ]w (n+ ( a¡1) n 2a )l n w nk n=2a; ( 4 ) wh e r e n = 0 ; 1 ; 2 : : : ; n= 2 ¡ 1 a n d x0[n] = x[n] ¡ x[n + n=2 ]: ( 5 ) it c a n b e o b s e r ve d t h a t ( 4 ) is a le n g t h -n=2 a d ft t h e in p u t s e qu e n c e o f wh ic h is t h e r e s u lt o f t h e c o m p u t a t io n in s id e t h e b r a c ke t s o f ( 4 ) fo r n = 0 ; 1 ; 2 : : : ; n= 2 ¡ 1 . in s u m m a r y, t h e e ve n in d e xe d o u t p u t s o f ( 1 ) a r e o b t a in e d fr o m o n e le n g t h -n= 2 d ft d e ¯ n e d in ( 2 ) b a s e d o n r. barseghyan 2 5 t h e r a d ix-2 d e c o m p o s it io n , a n d t h e o d d in d e xe d o u t p u t s a r e o b t a in e d fr o m a le n g t h -n= 2 a d fts , b a s e d o n t h e r a d ix-2 a d e c o m p o s it io n . th e c o m p le xit y o f a lg o r it h m c a n b e c o m p u t e d b y t h e fo llo win g e xp r e s s io n s : c£n = c £ n=2 + ac £ n=2a + n 2a c£ + 2 n ¡ c£t ; c+n = c + n=2 + ac + n=2a + n 2a c+ + 3 n ¡ c+t ; ( 6 ) wh e r e b y c£ a n d c+ d e n o t e t h e n u m b e r o f r e a l m u lt ip lic a t io n s a n d a d d it io n s t h a t a r e u s e d fo r e a c h o f t h e in n e r s u m s d e ¯ n e d in ( 4 ) , a n d c£t a n d c + t is t h e n u m b e r o f r e a l m u lt ip lic a t io n s a n d a d d it io n s s a ve d fr o m a ll t h e t r ivia l t wid d le fa c t o r s w nln in ( 4 ) . 3 . 2 / 4 s p lit -r a d ix a lg o r it h m fo r a = 2 t h e a lg o r it h m b e c o m e s a m o d i¯ e d ve r s io n o f c o n ve n t io n a l 2 = 4 s p lit -r a d ix a lg o r it h m . in s e r t in g a = 2 in t o ( 4 ) we g e t x[4 k + l] = n=4¡1x n=0 w nln ( 1x n=0 x0[n + i n 4 ]w iln=4 ) w nk n=4 = = n=4¡1x n=0 w nln ( x 0[n] + ( ¡j ) lx0[n + n 4 ]) w nkn=4; ( 7 ) fr o m ( 7 ) we c a n s e e t h a t it b e c o m e s a c o n ve n t io n a l s p lit -r a d ix a lg o r it h m wh ic h is r e p o r t e d in [4 ],[8 ],[9 ]. to c o ve r a ll o d d in d e xe s we s e t l = f¡ 1 ; 1 g. in t h is c a s e , we h a ve a n a r it h m e t ic c o m p u t a t io n a l g a in o n ly in c a s e s o f n = 0 a n d n = n=8 ( w 0n a n d w ln=8 n t wid d le fa c t o r s b e c o m e t r ivia l) . n o w it is e a s y t o s e e t h a t t h e n u m b e r o f a r it h m e t ic o p e r a t io n s a r e c+n = c + n=2 + 2 c + n=4 + 4 n ¡ 4 q c£n = c £ n=2 + 2 c£ n=4 + 2 n ¡ 1 2 q: ( 8 ) u s in g t h e t h e o r y o f d i®e r e n c e e qu a t io n s a n d ma xim a [1 2 ] c o m p u t e r a lg e b r a s ys t e m , we g e t t h e n u m b e r o f a r it h m e t ic o p e r a t io n s r e qu ir e d fo r c o m p u t a t io n o f ( 7 ) in lo g a r it h m ic fo r m c+n = ¡ 2p(28q¡3c+2q¡3c + q ) 9 + (¡1)p(10q¡3c+2q+6c + q ) 9 + p2 p+3q 3 + 2 q; c£n = ¡ 2p(44q¡3c£2q¡3c £ q ) 9 ¡ (¡1) p (10q+3c £ 2q ¡6c £ q ) 9 + p2 p+2q 3 + 6 q; ( 9 ) wh e r e cq a n d c2q d e n o t e t h e c o m p le xit ie s o f q a n d 2 q le n g t h d fts , r e s p e c t ive ly. u s in g m e t h o d s fr o m [6 ] fo r c o m p u t in g 2 q-le n g t h d ft we o b t a in c+n = 2 c + q + 4 q; c£n = 2 c £ q ; ( 1 0 ) ¯ n a lly s u b s t it u t in g ( 1 0 ) in t o ( 9 ) a n d 2 p = n q we g e t c+n = 8 3 pq 2 p ¡ 2p 9 ( 1 6 q ¡ 9 c+q ) ¡ 29 q ( ¡ 1 ) p + 2 q; c£n = 4 3 pq 2 p ¡ 2p 9 ( 4 4 q ¡ 9 c£q ) ¡ 109 q ( ¡1 ) p + 6 q; ( 1 1 ) 2 6 about complexity of fft algorithms for length of q £ 2p 4 q-le n g t h d ft c a n b e c o m p u t e d b y c+4q = 4 c + q + 1 6 q c£4q = 4 c £ q u s in g it a n d ( 8 ) , ¯ n a lly we g e t t h e fo r m u la s wh ic h s h o w t h e n u m b e r o f r e a l a r it h m e t ic o p e r a t io n s o f q £ 2 p c+n = 8 3 pq 2 p ¡ 2p 9 ( 1 6 q ¡ 9 c+q ) ¡ 29 q ( ¡1 ) p + 2 q =; = 8 3 n lo g 2 ³ n q ´ ¡ n ³ 16 9 ¡ 1 q c+q ´ ¡ 2 9 q ( ¡ 1 ) log2 ( n q ) + 2 q; c£n = 4 3 pq 2 p ¡ 2p 9 ( 3 8 q ¡ 9 c£q ) + 29 q ( ¡1 ) p + 6 q =; = 4 3 n lo g 2 ³ n q ´ ¡ n ³ 38 9 ¡ 1 q c+q ´ + 2 9 q ( ¡ 1 ) log2 ( n q ) + 6 q: ( 1 2 ) if q = 1 a n d , t h e r e fo r e c+1 = 0 a n d c + 1 = 0 fr o m ( 1 2 ) we c a n g e t c+n = 8 3 n lo g 2 n ¡ 169 n ¡ 2 9 ( ¡1 ) log2 n + 2 ; c£n = 8 3 n lo g 2 n ¡ 389 n + 2 9 ( ¡1 ) log2 n + 6 : ( 1 3 ) ( 1 3 ) is t h e s a m e a s t h e n u m b e r o f r e a l a r it h m e t ic o p e r a t io n s c o u n t r e qu ir e d b y c o n ve n t io n a l s p lit -r a d ix a lg o r it h m . d o in g s o m e o p t im iz a t io n fr o m [6 ], we g e t t h e fo llo win g r e c u r r e n t e xp r e s s io n s fo r c o m p u t in g 8 q le n g t h d fts . c+n = c + 4q + 4 c + q + 3 6 q = 8 c + q + 5 2 q; c£n = c £ 4q + 2 c £ q + 2 c £ sq = 6 c £ q + 2 c £ sq; ( 1 4 ) wh e r e c£sq d e n o t e s t h e n u m b e r o f a r it h m e t ic o p e r a t io n s r e qu ir e d b y s c a le d d ft [6 ]. u s in g ( 1 4 ) , we g e t a n im p r o ve m e n t in t h e n u m b e r o f a r it h m e t ic o p e r a t io n s c+n = 8 3 pq 2 p ¡ 2p 9 ( 1 6 q ¡ 9 c+q ) ¡ 29 q ( ¡ 1 ) p + 2 q =; = 8 3 n lo g 2 ³ n q ´ ¡ n ³ 16 9 ¡ 1 q c+q ´ ¡ 2 9 q ( ¡ 1 ) log2 ( n q ) + 2 q; c£n = 4 3 pq 2 p ¡ 2p 18 ( 8 2 q ¡ 3 ( 5 c£q + c£sq ) ) + 29 ( 7 q + 3 ( c £ q ¡ c£sq ) ) ( ¡1 ) p + 6 q =; = 4 3 n lo g 2 ³ n q ´ ¡ n 18 ³ 8 2 ¡ 3 q ( 5 c£q + c £ sq ) ´ + 2 9 ( 7 q + 3 ( c£q ¡ c£sq ) ) ( ¡ 1 ) log2 ( nq ) + 6 q: ( 1 5 ) 3 .1 co m p le xit y o f 2 =4 s p lit -r a d ix a lg o r it h m fo r q < 2 0 in [6 ] a m e t h o d fo r c o m p u t in g 3 -p o in t d ft wit h o p t im iz a t io n in c a s e o f 2 4 -p o in t is p r e s e n t e d . u s in g t h a t r e s u lt we c a n g e t a c o m p le t e e xp r e s s io n wh ic h d e s c r ib e s t h e n u m b e r o f a r it h m e t ic o p e r a t io n s r e qu ir e d b y 3 £ 2 p-le n g t h d ft. c+n ( 3 £ 2 p ) = 8 p2 p + 203 2 p ¡ 2 3 ( ¡ 1 ) p + 6 = = 8 3 nlo g 2 ( n 3 ) + 20 9 n ¡ 2 3 ( ¡ 1 ) log2( n3 ) + 6 ; c£n ( 3 £ 2 p ) = 4 p 2 p ¡ 1 1 2 p + 2 ( ¡1 ) p + 1 8 = = 4 3 nlo g 2 ( n 3 ) ¡ 11 3 n + 2 ( ¡ 1 ) log2( n3 ) + 1 8 : ( 1 6 ) r. barseghyan 2 7 fo r q = 9 c+n ( 9 £ 2 p ) = 2 4 p 2 p + 6 8 2 p ¡ 2 ( ¡ 1 ) p + 1 8 = = 8 3 nlo g 2 ³ n 9 ´ ¡ 68 81 n ¡ 2 3 ( ¡ 1 ) log2 ( n 9 ) + 1 8 ; c£n ( 9 £ 2 p ) = 1 2 p 2 p ¡ 2 4 2 p + 1 0 ( ¡ 1 ) p + 5 4 = = 4 3 nlo g 2 ³ n 9 ´ ¡ 8 3 n + 1 0 ( ¡ 1 ) log2 ( n 9 ) + 5 4 : ( 1 7 ) fo r q = 1 5 c+n ( 1 5 £ 2 p ) = 4 0 p 2 p + 4243 2 p ¡ 10 3 ( ¡ 1 ) p + 3 0 = = 8 3 nlo g 2 ³ n 15 ´ ¡ 424 45 n ¡ 10 3 ( ¡ 1 ) log2 ( n 15 ) + 3 0 ; c£n ( 1 5 £ 2 p ) = 2 0 p 2 p ¡ 1093 2 p + 46 3 ( ¡ 1 ) p + 9 0 = = 4 3 nlo g 2 ³ n 15 ´ ¡ 109 45 n ¡ 2 3 ( ¡1 ) log2 ( n 15 ) + 9 0 : ( 1 8 ) fo r c o m p u t in g q = 7 le n g t h d ft, we u s e t h e m e t h o d p r e s e n t e d in [1 3 ]. th a t m e t h o d a llo ws t o c o m p u t e d ft wit h c+7 = 7 2 ; c £ 7 = 1 6 . fo r g e t t in g a fu ll lo g a r it h m ic e xp r e s s io n fo r 7 £ 2 p, we u s e ( 1 2 ) . c+n ( 7 £ 2 p ) = 73 2 3+pp + 67 9 2 3+p ¡ 14 9 ( ¡ 1 ) p + 1 4 = = 8 3 n lo g 2 ³ n 7 ´ + 536 63 n ¡ 14 9 ( ¡ 1 ) log2 ( n 7 ) + 1 4 ; c£n ( 7 £ 2 p ) = 73 2 2+pp ¡ 61 18 2 p + 14 9 ( ¡ 1 ) p + 4 2 = = 4 3 n lo g 2 ³ n 7 ´ ¡ 61 126 n + 14 9 ( ¡ 1 ) log2 ( n 7 ) + 4 2 : ( 1 9 ) in c a s e o f q = 1 1 fr o m [1 3 ], we h a ve c+11 = 1 6 8 ; c £ 11 = 4 0 c+n ( 1 1 £ 2 p ) = 113 2 3+pp + 167 9 2 3+p ¡ 22 9 ( ¡ 1 ) p + 2 2 = = 8 3 nlo g 2 ³ n 11 ´ + 1336 99 n ¡ 22 9 ( ¡ 1 ) log2 ( n 11 ) + 2 2 ; c£n ( 1 1 £ 2 p ) = 113 2 2+pp ¡ 29 9 2 1+p + 22 9 ( ¡ 1 ) p + 6 6 = = 4 3 nlo g 2 ³ n 11 ´ ¡ 58 99 n + 22 9 ( ¡ 1 ) log2 ( n 7 ) + 6 6 : ( 2 0 ) in c a s e o f q = 1 3 fr o m [1 3 ], we h a ve c+13 = 1 8 8 ; c £ 13 = 4 0 c+n ( 1 3 £ 2 p ) = 133 2 3+pp + 371 9 2 2+p ¡ 26 9 ( ¡ 1 ) p + 2 6 = = 8 3 nlo g 2 ³ n 13 ´ + 1484 117 n ¡ 26 9 ( ¡ 1 ) log2 ( n 13 ) + 2 6 ; c£n ( 1 3 £ 2 p ) = 133 2 2+pp ¡ 67 9 2 1+p + 26 9 ( ¡ 1 ) p + 7 8 = 4 3 nlo g 2 ³ n 7 ´ ¡ 134 117 n + 26 9 ( ¡1 ) log2 ( n 7 ) + 7 8 : ( 2 1 ) 2 8 about complexity of fft algorithms for length of q £ 2p in c a s e o f q = 1 7 fr o m [1 3 ], we h a ve c+17 = 2 7 4 ; c £ 13 = 8 2 c+n ( 1 7 £ 2 p ) = 173 2 3+pp + 1097 9 2 1+p ¡ 34 9 ( ¡ 1 ) p + 3 4 = = 8 3 nlo g 2 ³ n 17 ´ + 2194 153 n ¡ 34 9 ( ¡ 1 ) log2 ( n 17 ) + 3 4 ; c£n ( 1 7 £ 2 p ) = 173 2 2+pp + 23 9 2 2+p + 34 9 ( ¡1 ) p + 1 0 2 = = 4 3 nlo g 2 ³ n 17 ´ ¡ 92 153 n + 34 9 ( ¡ 1 ) log2 ( n 17 ) + 1 0 2 : ( 2 2 ) in c a s e o f q = 1 9 fr o m [1 3 ], we h a ve c+19 = 4 0 4 ; c £ 19 = 7 6 c+n ( 1 9 £ 2 p ) = 193 2 3+pp + 833 9 2 2+p ¡ 38 9 ( ¡1 ) p + 3 8 = = 8 3 nlo g 2 ³ n 19 ´ + 3332 171 n ¡ 38 9 ( ¡ 1 ) log2 ( n 19 ) + 3 8 ; c£n ( 1 9 £ 2 p ) = 193 2 2+pp ¡ 19 9 2 1+p + 38 9 ( ¡ 1 ) p + 1 1 4 = = 4 3 nlo g 2 ³ n 19 ´ ¡ 38 171 n + 38 9 ( ¡1 ) log2 ( n 19 ) + 1 1 4 : ( 2 3 ) 4 . 2 / 8 s p lit -r a d ix a lg o r it h m in c a s e o f a = 4 , t h e a lg o r it h m b e c o m e s 2 =8 s p lit -r a d ix a lg o r it h m . x[8 k + l] = n=4¡1x n=0 w nln ( 3x n=0 x0[n + i n 8 ]w il8 ) w nk n=8: ( 2 4 ) th e t o t a l n u m b e r s o f r e a l m u lt ip lic a t io n s a n d r e a l a d d it io n s n e e d e d b y t h e a lg o r it h m a r e c+n = c + n=2 + 4 c+ n=8 + 11 2 n ¡ c+t c£n = c £ n=2 + 4 c£ n=8 + 5 2 n ¡ c£t : ( 2 5 ) b e lo w t h e n u m b e r o f a r it h m e t ic o p e r a t io n s in lo g a r it h m ic fo r m a r e p r e s e n t e d c+n = 11 4 pq 2 p ¡ 55 16 q 2 p ¡ 1 8 2 p ³ c+t ¡ ( 2 c+q + c+2qc+4q ) ´ + 1 4 c+t + +( ¡ 1 ) p 2 p=2 [7 ( 1 2 c+q ¡ 2 ( c+2q + c+4q + c+t ) + 5 5 q ) c o s ( p a r c t a n ( p 7 ) ) + p 7 ( 4 c+q ¡ 2 ( 1 1 c+2q ¡ 5 c+4q ¡ c+t ) ¡ 9 9 q ) s in ( p a r c t a n ( p 7 ) ) ] c£n = 5 4 pq 2 p ¡ 25 16 q 2 p ¡ 1 8 2 p ³ c£t ¡ ( 2 c£q + c£2qc£4q ) ´ + 1 4 c£t + +( ¡ 1 ) p 2 p=2 [7 ( 2 ( c£t ¡ 6 c£q + c£2q + c£4q ) ¡ 2 5 q ) c o s ( p a r c t a n ( p 7 ) ) + p 7 ( c£t + 4 c £ q ¡ 2 2 c£2q + 5 c£4q ¡ 4 5 q ) s in ( p a r c t a n ( p 7 ) ) ]: ( 2 6 ) wh e r e b y cq,c2q a n d c4q d e n o t e n u m b e r o f a r it h m e t ic o p e r a t io n s r e qu ir e d fo r c o m p u t a t io n s o f q, 2 q a n d 4 q le n g t h d fts , r e s p e c t ive ly. fo r c o m p u t in g 2 q a n d 4 q le n g t h d ft, we c a n u s e r. barseghyan 2 9 m e t h o d s d e s c r ib e d in t h e p r e vio u s s e c t io n , wh ic h a llo ws t o r e wr it e ( 2 6 ) c+n = 11 4 pq 2 p ¡ 15 16 q 2 p ¡ 1 8 2 p ³ c+t ¡ 8 c+q ´ + 1 4 c+t + +( ¡ 1 ) p 2 p=2 [7 ( 1 5 q ¡ 2 c+t ) c o s ( p a r c t a n ( p 7 ) ) + p 7 ( 2 c+t ¡ 2 7 q ) s in ( p a r c t a n ( p 7 ) ) ] c£n = 5 4 pq 2 p ¡ 25 16 q 2 p ¡ 1 8 2 p ³ c£t ¡ 8 c£q ´ + 1 4 c£t + +( ¡ 1 ) p 2 p=2 [7 ( 2 5 q ¡ 2 c£t ) c o s ( p a r c t a n ( p 7 ) ) + p 7 ( 2 c£t ¡ 4 5 q ) s in ( p a r c t a n ( p 7 ) ) ]: ( 2 7 ) 4 .1 co m p le xit y o f 2 =8 s p lit -r a d ix a lg o r it h m fo r q < 2 0 in c a s e o f q = 1 5 , we h a ve c+15 = 1 6 8 ; c £ 15 = 3 0 a n d fr o m [7 ] c + t = 1 8 0 ; c £ t = 6 0 a n d t h e r e fo r e c+n = 11 4 pq 2 p ¡ 15 16 q 2 p ¡ 1 8 2 p ³ c+t ¡ 8 c+q ´ + 1 4 c+t + +( ¡ 1 ) p 2 p=2 [7 ( 1 5 q ¡ 2 c+t ) c o s ( p a r c t a n ( p 7 ) ) + p 7 ( 2 c+t ¡ 2 7 q ) s in ( p a r c t a n ( p 7 ) ) ] c£n ( 1 5 £ 2 p ) = 54 p 1 5 £ 2 p ¡ 17 16 1 5 £ 2 p ¡ 15 16 ( ¡1 ) p 2 p=2 ( ) + 4 5 +( ¡ 1 ) p 2 p=2 [7 ( 2 5 q ¡ 2 c£t ) c o s ( p a r c t a n ( p 7 ) ) + p 7 ( 2 c£t ¡ 4 5 q ) s in ( p a r c t a n ( p 7 ) ) ]: ( 2 8 ) 5 . co m p a r is o n th e n u m b e r o f a d d it io n s a n d m u lt ip lic a t io n s r e qu ir e d fo r c o m p u t in g d ft fo r va r io u s le n g t h s a r e p r e s e n t e d in ta b le 1 ( 2 =4 s p lit -r a d ix a lg o r it h m ) . a s a n e xa m p le a r a n g e fr o m 2 5 6 t o 2 0 4 8 is c h o s e n . u s in g o n ly c o n ve n t io n a l a lg o r it h m fo r 2 p we h a ve o n ly c o m p u t e d ft o f 2 5 6 ; 5 1 2 ; 1 0 2 4 ; 2 0 4 8 s iz e s . if s iz e is n o t e qu a l t o t h e s e va lu e s we n e e d t o p a d t h e in p u t d a t a u p t o n e xt 2 p. u s in g q £ 2 p a lg o r it h m a llo ws t o c o ve r t h e r a n g e 2 5 6 ¡ 1 0 2 4 wit h 2 7 n e w p o in t s . th is a p p r o a c h a llo ws t o s ig n ī c a n t ly r e d u c e t h e n u m b e r o f a r it h m e t ic o p e r a t io n s . to ¯ n d o u t t h e q fo r wh ic h t h e a lg o r it h m b e c o m e s t h e m o s t e ± c ie n t in t e r m s o f t h e n u m b e r o f a r it h m e t ic o p e r a t io n s , ¯ r s t o f a ll we c u t t h e va lu e s o f q fo r wh ic h cn1 > cn2, b u t n1 < n2, wh e r e cn = c + n +c £ n . it is e a s y t o s e e t h a t o n ly fo r 1 ; 3 ; 5 ; 9 ; 1 3 ; 1 5 c o n d it io n p r e s e n t e d a b o ve is t r u e . fo r g e t t in g m o r e a c c u r a t e r e s u lt s we c o m p a r e t h e va lu e o f en = cn n . fin a lly we g e t en ( 9 £ 2 p ) < en ( 9 £ 3 p ) < en ( 9 £ 1 5 p ) < en ( 1 £ 2 p ) < < en ( 5 £ 2 p ) < en ( 1 5 £ 2 p ) : 3 0 about complexity of fft algorithms for length of q £ 2p table 1: number of arithmetic operations required by 2=4 split-radix algorithm for the dft length 256 ¡ 1024 n q p add. mul. count 256 1 8 5380 1284 6664 272 17 4 6832 1720 8552 288 9 5 6036 1196 7232 304 19 4 9200 1672 10872 320 5 6 6736 1880 8616 352 11 5 9468 2204 11672 360 45 3 7812 1140 8952 384 3 7 8028 2192 10220 416 13 5 10852 2372 13224 448 7 6 10992 2760 13752 480 15 5 10956 2112 13068 512 1 9 12292 3076 15368 544 17 5 15092 4052 19144 576 9 6 13584 3136 16720 608 19 5 19996 4028 24024 640 5 7 15172 4580 19752 704 11 6 20784 5288 26072 720 45 4 17424 3000 20424 768 3 8 18096 5396 23492 832 13 6 23888 5784 29672 896 7 7 24364 6668 31032 960 15 6 24432 5460 29892 1024 1 10 27652 7172 34824 1088 17 6 33040 9464 42504 1152 9 7 30228 7724 37952 1216 19 6 43184 9576 52760 1280 5 8 33744 10840 44584 1408 11 7 45308 12380 57688 1440 45 5 38628 7620 46248 1536 3 9 40284 12816 53100 1664 13 7 52196 13700 65896 1792 7 8 53488 15688 69176 1920 15 7 53964 13344 67308 2048 1 11 61444 16388 77832 in ta b le 2 a r e p r e s e n t e d t h e c o m p a r is o n r e s u lt s fo r t h e va r io u s le n g t h d fts ( 2 =8 s p lit -r a d ix a lg o r it h m ) . r. barseghyan 3 1 table 2: number of arithmetic operations required by 2=4 split-radix algorithm for the dft length 256 ¡ 2048 n q p add. mul. count 256 1 8 5380 1284 6664 360 45 3 8048 1360 9408 480 15 5 11256 2520 13776 512 1 9 12292 3076 15368 720 45 4 18076 3620 21696 960 15 6 25212 6180 31392 1024 1 10 27652 7172 34824 1440 45 5 39696 8640 48336 1920 15 7 55824 14880 70704 2048 1 11 61444 16388 77832 6 . co n c lu s io n in c a s e o f lo o kin g fo r c o m p u t a t io n a lly e ± c ie n t a lg o r it h m in t e r m s o f n u m b e r o f m u lt ip lic a t io n s in g e n e r a l c a s e we n e e d t o c h o o s e 2 =8 s p lit -r a d ix fft a lg o r it h m , b e c a u s e c o e ± c ie n t o f n lo g 2 is s m a lle r . e ± c ie n c y o f a lg o r it h m s in t e r m s o f t o t a l n u m b e r o f a r it h m e t ic o p e r a t io n s is d is c u s s e d b e lo w. th e t o t a l n u m b e r o f a r it h m e t ic o p e r a t io n s r e qu ir e d b y 2 = 4 s p lit -r a d ix a lg o r it h m c a n b e c o m p u t e d u s in g ( 1 2 ) a n d is p r e s e n t e d b e lo w cn ( 2 =4 ) = 4 pq 2 p ¡ 2 p ( 6 q ¡ c+q ) + 8 q: ( 2 9 ) th e t o t a l n u m b e r o f a r it h m e t ic o p e r a t io n s r e qu ir e d fo r c o m p u t a t io n 2 =8 s p lit -r a d ix a lg o r it h m c a n b e r e t r ie ve d fr o m ( 2 8 ) cn ( 2 =8 ) = 4 pq 2 p ¡ 2 p ( 5 2 q ¡ ( 1 8 ct ¡ cq ) ) + 8 q+ +2 ( ¡ 1 ) p 2 p=2 £ [7 ( 2 0 q ¡ ct ) c o s ( ® ) + + p 7 ( ct ¡ 3 6 q ) s in ( ® ) :] ( 3 0 ) fo r g e t t in g a c o m p u t a t io n a lly e ± c ie n t a lg o r it h m , we n e e d t o c a lc u la t e cn ( 2 =4 ) ¡ cn ( 2 = 8 ) u s in g ( 2 9 ) a n d ( 3 0 ) . fo r s im p lic it y, o n ly t h e c o e ± c ie n t s fo r 2 p a n d q £ 2 p a r e in c lu d e d cn ( 2 =4 ) ¡ cn ( 2 =8 ) = ( 72 q ¡ 1 8 ct ) 2 p < 0 ct > 2 8 q: in o t h e r wo r d s we c a n s a y t h a t if ct is g r e a t e r t h a n 2 8 q, 2 =4 s p lit -r a d ix a lg o r it h m is m o r e e ± c ie n t c o m p a r e d t o cn ( 2 =8 ) . refer ences [1 ] e . o. b r ig h a m , the f ast f ourier applications, e n g le wo o d cli®s , n j, p r e n t ic e -h a ll, 1 9 8 8 . [2 ] j. k . e r s o y, f ourier-r elated transforms. f ast algorithms and applications, e n g le wo o d cli®s , n j, p r e n t ic e -h a ll,1 9 9 7 . 3 2 about complexity of fft algorithms for length of q £ 2p [3 ] j. w . co o le y a n d j. w . tu ke y, \ a n a lg o r it h m fo r t h e m a c h in e c o m p u t a t io n o f t h e c o m p le x fo u r ie r s e r ie s ," m ath. computation, p p . 2 9 7 { 3 0 1 , 1 9 6 5 . [4 ] r . y a vn e , \ a n e c o n o m ic a l m e t h o d fo r c a lc u la t in g t h e d is c r e t e fo u r ie r t r a n s fo r m ," in p roc. af ip s, vo l. 3 3 , p p . 1 1 5 { 1 2 5 , 1 9 6 8 . [5 ] m. fr ig o a n d s . g. jo h n s o n , \ a m o d i¯ e d s p lit -r a d ix fft wit h fe we r a r it h m e t ic o p e r a t io n s " , ieee trans. signal p rocessing, vo l. 5 5 , p p . 1 1 1 -1 1 9 , 2 2 0 7 . [6 ] g. b i a n d y . q. ch e n , \ fa s t d ft a lg o r it h m fo r le n g t h n = q £ 2 p," ie e e trans. circuits and systems ii, vo l. 4 5 , n o . 6 , p p . 6 8 5 6 9 0 , 1 9 9 8 . [7 ] g. b i, g. l i a n d x . l i, \ a u n i¯ e d e xp r e s s io n fo r s p lit -r a d ix d ft a lg o r it h m s ," p p . 3 2 3 3 2 6 , 2 0 1 0 . [8 ] p . d u h a m e l, \ im p le m e n t a t io n o f s p lit -r a d ix fft a lg o r it h m s fo r c o m p le x, r e a l, a n d r e a ls ym m e t r ic d a t a " , ie e e trans. acoust., speech, signal p rocess., vo l. 3 4 , p p . 2 8 5 2 9 5 , a p r il, 1 9 8 6 . [9 ] h . v . s o r e n s e n , m. t. h e id e m a n a n d c. s . b u r r u s , " on c o m p u t in g t h e s p lit -r a d ix fft," ie e e tr a n s . a c o u s t ., s p e e c h , s ig n a l p r o c e s s ., vo l. 3 4 ,p p . 1 5 2 1 5 6 , fe b . 1 9 8 6 . [1 0 ] s . b o u g u e z e l, m. om a ir a n d m. n . s . s wa m y, \ a n e w r a d ix-2 / 8 fft a lg o r it h m fo r le n g t h -q £ 2 m d fts ," ie e e trans. circuits and systems i, vo l. 5 1 , n o . 1 , p p . 1 7 2 3 1 7 3 2 , 2 0 0 4 . [1 1 ] s . b o u g u e z e l, m. om a ir a n d m. n . s . s wa m y, \ a g e n e r a l c la s s o f s p lit -r a d ix fft a lg o r it h m s fo r t h e c o m p u t a t io n o f t h e d ft o f le n g t h -2 m," ie e e trans. signal p rocessing, vo l. 5 5 , n o . 8 , p p . 4 1 2 7 4 1 3 8 , 2 0 0 7 . [1 2 ] on lin e : [a va ila b le ] h t t p :/ / m a xim a .s o u r c e fo r g e .n e t [1 3 ] i. s e le s n ic k a n d s . b u r r u s , \ p r o g r a m s fo r p r im e l e n g t h ffts ," h t t p :/ / c n x.o r g / c o n t e n t / m 1 8 1 3 7 / 1 .5 / . [1 4 ] r . b a r s e g h ya n , \ co m p le xit y o f t h e c o m p o s it e le n g t h fft a lg o r it h m s " , p roceedings of international conference csit 2015, y e r e va n , a r m e n ia , 2 0 1 5 . submitted 22.08.2017, accepted 06.12.2017. q £ 2 p »ñï³ñáõãû³ý ü²¼-µ³ñ¹áõãû³ý ù³ëçý è. ´³ñë»õû³ý ²ù÷á÷áõù ²ßë³ï³ýùáõù ëï³óí³í ¿ éá·³ñçãù³ï³ý µ³ý³ó¨á, áñá ï³ù³û³ï³ý q £ 2 p-»ñï³ñáõãû³ý í»ïïáñý»ñç ñ³ù³ñ ãáõûé ¿ ï³éçë ñ³ßí³ñï»é üáõñû»ç ¹çëïñ»ï ó¨³÷áëáõãû³ý (ü²ò) ñ³ù³ñ ³ýññ³å»ßï ·áñíáõáõãûáõýý»ñç ×ß·ñçï ù³ý³ïá, áñï»õ q-ý ï»ýï ãçí ¿: î ñëîæíîñòè àëãîðèòìîâ áïô äëÿ äëèíû q £ 2 p ð. áàðñåãÿí àííîòàöèÿ â ýòîé ñòàòüå âûâåäåíà ëîãàðèôìè÷åñêàÿ ôîðìóëà, êîòîðàÿ ïîçâîëÿåò âû÷èñëèòü òî÷íîå êîëè÷åñòâî íåîáõîäèìûõ îïåðàöèé äëÿ âû÷èñëåíèÿ äèñêðåòíîãî ïðåîáðàçîâàíèÿ ôóðüå (dft) äëÿ âåêòîðîâ äëèíû q £ 2 p, ãäå q íå÷åòíîå öåëîå ÷èñëî. microsoft word robert.doc математические вопросы кибернетики и вычислительной техники 30, 92--104, 2008. 92 построение ассоциативных правил путем цепного раздробления n-мерного единичного куба цепями левон асланян, роберт хачатрян институт проблем информатики и автоматизации нан ра lasl@sci.am, robert@simartek.am аннотация в работе решена задача поиска ассотиативных правил и приведен альтернативный алгоритму apriori метод решения этой задачи, путем цепного раздробления n-мерного куба, по технике анселя. описаны инстументы для работы над цепями, выделенные из результатов тонояна. приведено краткое описание программной реализации альтернативного подхода. литература [1] коробков б. к., “о монотонных функциях алгебры логики”, сб. ‘проблемы кибернетики’, вып. 13, м., ‘наука’, стр. 5-28, 1965. [2] ансель ж., “о числе монотонных булевых функций n переменных”, ‘кибернетический сборник’, новая серия, вып. 5, м., ‘мир’, стр. 53-5, 1968. [3] тоноян г. п., “разбиение вершин n-мерного единичного куба на цепи и расшифровка монотонных булевых функций”, журнал вычислительной математики и математической физики, том. 19, n% 6, стр. 1532-1542, 1976. [4] kotsiantis s. and kanellopoulos d., “association rules mining: a recent overview”, gests international transactions on computer science and engineering, vol. 32 (1), pp. 71-82, 2006. л. асланян, р. хачатрян 93 ²ëáóç³ïçí ï³ýáýý»ñç ï³éáõóáõù` n-ã³÷³ýç ùç³íáñ ëáñ³ý³ñ¹á ßõã³ý»ñç ïñáñ»éáõ »õ³ý³ïáí è. ²ëé³ýû³ý, è.ê³ã³ïñû³ý ²ù÷á÷áõù ²ßë³ï³ýùáõù éáõíí³í ¿ ³ëáóç³ïçí ï³ýáýý»ñç ÷ýïñù³ý ëý¹çñá ¨ µ»ñí³í ¿ ñ³ûïýç apriori ³é·áñçãùç ³ûéáýïñ³ýù³ûçý ï³ñµ»ñ³ïá ³û¹ ëý¹ñç éáõíù³ý ñ³ù³ñ: ²ûý ñçùýí³í ¿ n-ã³÷³ýç ùç³íáñ ëáñ³ý³ñ¹á ßõã³ý»ñç ïñáñ»éáõ íñ³, ²ýë»éç ïáõùçó ³é³ç³ñïí³í »õ³ý³ïáí: üï³ñ³·ñí³í »ý ·áñíçùý»ñ ßõã³ý»ñç ñ»ï ³ßë³ï»éáõ ñ³ù³ñ, áñáýù ¹áõñë »ý µ»ñí»é îáýáû³ýç ïáõùçó ëï³óí³í ³ñ¹ûáõýùý»ñçó: ´»ñí³í ¿ íñ³·ñ³ûçý ñ³ù³ï³ñ·ç ï³ñ× ýï³ñ³·çñ, áñï»õ çñ³ï³ý³óí³í ¿ ëý¹ñç ³ûéáýïñ³ýù³ûçý éáõíù³ý ï³ñµ»ñ³ïá: microsoft word crom.doc ìàòåìàòè÷åñêèå âîïðîñû êèáåðíåòèêè è âû÷èñëèòåëüíîé òåõíèêè 30, 47--53, 2008. 47 создание контрольных точек и восстановление mpi программ мгер ю. мовсисян ереванский физический институт им. а.и.алиханяна mher.movsisyan@gmail.com àííîòàöèÿ выполнение программ на вычислительных кластерах обычно занимает довольно большое время. в процессе выполнения может возникнуть потребность изменения физического местоположения отдельных процессов параллельной программы или временная остановка всей программы. в этой статье описана разработанная система crom (checkpointing and recovery of mpi), которая предоставляет возможность создания контрольных точек для остановки и последующего возобновления выполнения mpi программы. функциональность создания контрольных точек и восстановления реализована в виде дополнительных компонент mpich2 и не требует изменений в коде mpi программы. ëèòåðàòóðà [1] m. movsisyan, v. sahakyan, “transparent checkpointing protocol for mpi programs with decentralized initiator”, csit 2007, pp. 227-229. [2] message passing interface forum, “mpi: a message-passing interface standard”, version 1.1, june 1995. http://www.mpi-forum.org/docs/docs.html [3] message passing interface forum, “mpi-2: extensions to the message-passing interface”, july 1997, http://www.mpi-forum.org/docs/docs.html [4] mpich2, http://www-unix.mcs.anl.gov/mpi/mpich2/ [5] open mpi, http://www.open-mpi.org/ [6] m. chandy and l. lamport, “distributed snapshots: determining global states of distributed systems”, in acm transactions on computing systems, 3(1): pp. 63-75, 1985. [7] myrinet, http://www.myri.com/myrinet/overview/ [8] the mpich team argonne national laboratory, “process management in mpich2” draft 2.1. march 30, 2007. [9] berkeley lab checkpoint/restart (blcr), http://ftg.lbl.gov/checkpointrestart/checkpointrestart.shtml [10] h. hargrove and c. duell, “berkeley lab checkpoint/restart (blcr) for linux clusters”, in proceedings of scidac 2006: june 2006. создание контрольных точек и восстановление mpi программ 48 [11] j. duell, p. hargrove, and e. roman, “the design and implementation of berkeley lab’s linux checkpoint/restart”, technical report lbnl-54941, lawrence berkeley national laboratory, 2003. [12] m. elnozahy, l. alvisi, y. m. wang, and d. b. johnson, “a survey of rollback-recovery protocols in message passing systems”, technical report cmu-cs-96-181, school of computer science, carnegie mellon university, pittsburgh, pa, usa, 1996. êïáõ·ù³ý ï»ï»ñç ëï»õíáõùá ¨ í»ñ³ï³ý·ýáõùá mpi íñ³·ñ»ñáõù ø. øáíëçëû³ý ²ù÷á÷áõù ð³ßíáõ³ï³ý ïé³ëï»ñý»ñáõù íñ³·ñ»ñç ï³ï³ñáõùá ëáíáñ³µ³ñ ï¨áõù ¿ µ³í³ï³ýçý »ñï³ñ£ ìñ³·ñ»ñç ï³ï³ñù³ý å³ù³ý³ï ï³ñáõ »ý í³·»é ½áõ·³ñ»é íñ³·ñç áýã³óùý»ñç ýç½çï³ï³ý ¹çñùç ÷á÷áëáõãû³ý ï³ù ³ùµáõç íñ³·ñç ï³ï³ñù³ý å³ù³ý³ï³íáñ ¹³¹³ñ»óù³ý ³ýññ³å»ßïáõãûáõýý»ñ£ ²ûë ñá¹í³íáõù ýï³ñ³·ñíáõù ¿ ñ»õçý³ïç ïáõùçó ùß³ïí³í crom (checkpointing and recovery of mpi) ñ³ù³ï³ñ·á, áñá ñý³ñ³íáñáõãûáõý ¿ áýó»éáõù å³ñå³ý»é ï³ï³ñíáõ mpi íñ³·çñá ¨ í»ñ³ãáõ³ñï»é£ ä³ñå³ýù³ý ¨ í»ñ³ãáõ³ïù³ý ýáõýïóçáý³éáõãûáõýá çñ³·áñíí³í ¿ mpich2 µ³õ³¹ñçãý»ñç ï»ëùáí ¨ ãç å³ñ³ýçáõù mpi íñ³·ñç ïá¹ç ÷á÷áëáõãûáõý£ mathematical problems of computer science 45, 67--76, 2016. on transitive closures of two-dimensional strongly positive arithmetical sets1 seda n. manukian institute for informatics and automation problems of nas ra e-mail: zaslav@ipia.sci.am abstract the notions of positive and strongly positive arithmetical set are considered in ([1]-[3]). it is noted in [3] that the transitive closure of any 2-dimensional strongly positive set is primitive recursive. in this article a more strong statement is proved: the transitive closure of any 2-dimensional strongly positive set is defined by an arithmetical formula in the signature (0, =, <, 𝑆𝑆), where 𝑆𝑆(𝑥𝑥) = 𝑥𝑥 + 1. besides, it is proved that the class of two-dimensional strongly positive sets and the class of transitive closures of such sets do not coincide with the class of two-dimensional arithmetical sets expressible by the formulas in the signature (0, = , <, 𝑆𝑆). keywords: positive, strongly positive, arithmetical set, dimension, signature. 1. introduction the notion of strongly positive arithmetical set is defined and investigated in [3]. it is proved in [3] that for any 𝑛𝑛 ≥ 3 there exists a 2𝑛𝑛-dimensional strongly positive set such that its transitive closure is not recursive. it is noted in [3] (without a proof) that the transitive closure of any 2dimensional strongly positive set is primitive recursive. below a stronger statement is proved: the transitive closure of any 2-dimensional strongly positive set can be defined by an arithmetical formula in the signature (0, =, <, 𝑆𝑆), where 𝑆𝑆(𝑥𝑥) = 𝑥𝑥 + 1(see below, theorem 1). it is proved also that the class of the mentioned transitive closures does not coincide with the class of sets expressible by arithmetical formulas in the signature (0, =, <, 𝑆𝑆). for example, it is proved that 1 this work was supported by state committee of science, mess ra in frame of the research project scs 15t1b238. 67 mailto:zaslav@ipia.sci.am 68 on transitive closures of two-dimensional strongly positive arithmetical sets the set {(𝑥𝑥, 𝑦𝑦)/𝑦𝑦 = 𝑥𝑥 + 2} is not strongly positive and cannot be represented as the transitive closure of some strongly positive set (see below, theorem 2). 2. main definitions and results by 𝑁𝑁 we denote the set of all non-negative integers, 𝑁𝑁 = {0,1,2, … }. by 𝑁𝑁𝑛𝑛, where 𝑛𝑛 ≥ 1, we denote the set of 𝑛𝑛-tuples (𝑥𝑥1, 𝑥𝑥2, … 𝑥𝑥𝑛𝑛), where 𝑥𝑥𝑖𝑖 ∈ 𝑁𝑁 for 1 ≤ 𝑖𝑖 ≤ 𝑛𝑛. an 𝑛𝑛-dimensional arithmetical set, where 𝑛𝑛 ≥ 1, is defined as any subset of 𝑁𝑁𝑛𝑛. an 𝑛𝑛-dimensional arithmetical predicate is defined as a predicate 𝑃𝑃, which is true on some set 𝐴𝐴 ⊆ 𝑁𝑁𝑛𝑛 and false on the set 𝑁𝑁𝑛𝑛\𝐴𝐴. if the mentioned relation between 𝐴𝐴 and 𝑃𝑃 takes place, then we say that 𝑃𝑃 is the representing predicate for 𝐴𝐴, and 𝐴𝐴 is the set of truth for 𝑃𝑃. the notions of primitive recursive set and recursive set are defined in a usual way (see [4][6]). the notion of arithmetical formula on a given signature (on the base of logical operations &,∨, ⊃, ¬, ∀, ∃) is defined in a usual way, ([2]-[6]). we will consider arithmetical formulas in the signatures (0, =, 𝑆𝑆) and (0, =, <, 𝑆𝑆), where 𝑆𝑆(𝑥𝑥) = 𝑥𝑥 + 1 for 𝑥𝑥 ∈ 𝑁𝑁. the deductive systems in the signatures (0, =, 𝑆𝑆) and (0, =, <, 𝑆𝑆) are defined as in [6]; we will denote these deductive systems correspondingly by deds and dedl. as it is proved in [6], these deductive systems are complete. we say that the formulas 𝐹𝐹 and 𝐺𝐺 in the corresponding signatures are equivalent if the formula (𝐹𝐹 ⊃ 𝐺𝐺)&(𝐺𝐺 ⊃ 𝐹𝐹) is deducible in the corresponding deductive system. we will consider the formulas in the mentioned signatures up to their equivalence. the relation “𝑛𝑛-dimesional arithmetical set 𝐴𝐴 is defined by an arithmetical formula 𝐹𝐹” is given in a usual way (see [2]-[6]) (in [2] this relation is called as follows: “𝑘𝑘-dimensional arithmetical set 𝐴𝐴 is represented (or representable) by a formula 𝐹𝐹"). the notion of transitive closure 𝐴𝐴∗ for an arithmetical set 𝐴𝐴 having an even dimension 2𝑘𝑘 (where 𝑘𝑘 ≥ 1) is defined in a usual way (see, for example, [3], [8]). let us recall that the following statement holds (see [3], lemma 3.4, and [8], p.72): if 𝐴𝐴 is a 2𝑘𝑘-dimensional set, 𝐴𝐴 ⊆ 𝑁𝑁2𝑘𝑘, where 𝑘𝑘 ≥ 1, then (𝑥𝑥1, 𝑥𝑥2, … , 𝑥𝑥𝑘𝑘, 𝑦𝑦1, 𝑦𝑦2, … , 𝑦𝑦𝑘𝑘) ∈ 𝐴𝐴∗ if and only if there exists a sequence (𝑄𝑄1, 𝑄𝑄2, … , 𝑄𝑄𝑚𝑚) of 𝑘𝑘-tuples such that 𝑚𝑚 ≥ 2, 𝑄𝑄1 = (𝑥𝑥1, 𝑥𝑥2, … , 𝑥𝑥𝑘𝑘), 𝑄𝑄𝑚𝑚 = (𝑦𝑦1, 𝑦𝑦2, … , 𝑦𝑦𝑘𝑘), (𝑄𝑄𝑖𝑖, 𝑄𝑄𝑖𝑖+1) ∈ 𝐴𝐴 for 1 ≤ 𝑖𝑖 ≤ 𝑚𝑚 − 1. below we will say that the sequence (𝑄𝑄1, 𝑄𝑄2, … , 𝑄𝑄𝑚𝑚), having the mentioned properties is a sequence establishing the value (𝑄𝑄1, 𝑄𝑄𝑚𝑚) ∈ 𝐴𝐴∗ of the transitive closure 𝐴𝐴∗ (or, shortly, etc-sequence). in what follows we will consider etc-secuences only for the case 𝑘𝑘 = 1. the notion of strongly positive arithmetical set is defined as in [3]. let us recall that an 𝑚𝑚-dimensional arithmetical set, where 𝑚𝑚 ≥ 1, is said to be strongly positive if it is defined by an arithmetical formula 𝐹𝐹 which is constructed by logical operators & and ∨ from subformulas having the forms (𝑥𝑥 = 𝑎𝑎) (where 𝑎𝑎 is a constant, 𝑎𝑎 ∈ 𝑁𝑁), 𝑥𝑥 = 𝑦𝑦, 𝑦𝑦 = 𝑆𝑆(𝑥𝑥), ¬(𝑥𝑥 = 0), where 𝑥𝑥 and 𝑦𝑦 are variables. s. manukian 69 theorem 1: the transitive closure of any 2-dimensional strongly positive set can be defined by an arithmetical formula in the signature (0, =, <, 𝑆𝑆). theorem 2: the set {(𝑥𝑥, 𝑦𝑦)/ 𝑦𝑦 = 𝑆𝑆(𝑆𝑆(𝑥𝑥))} is not strongly positive and cannot be represented as a transitive closure of some strongly positive set. 3. proofs of theorems we consider the properties of 2-dimensional strongly positive sets. let 𝜋𝜋 be any set of such kind. by 𝜂𝜂 we denote the representing predicate for 𝜋𝜋. using the definition of strongly positive set we conclude that the predicate 𝜂𝜂 can be expressed by an arithmetical formula 𝐹𝐹 having the form 𝐹𝐹1 ∨ 𝐹𝐹2 ∨ … ∨ 𝐹𝐹𝑚𝑚, where any 𝐹𝐹𝑖𝑖 is the conjuction of subformulas having the following forms: 𝑥𝑥 = 𝑎𝑎, 𝑦𝑦 = 𝑏𝑏, (where 𝑎𝑎 and 𝑏𝑏 are constants, 𝑎𝑎 ∈ 𝑁𝑁, 𝑏𝑏 ∈ 𝑁𝑁), 𝑥𝑥 = 𝑦𝑦, 𝑦𝑦 = 𝑆𝑆(𝑥𝑥), 𝑥𝑥 = 𝑆𝑆(𝑦𝑦), ¬(𝑥𝑥 = 0), ¬(𝑦𝑦 = 0). the predicate expressed by the formula 𝐹𝐹𝑖𝑖 we denote by 𝜂𝜂𝑖𝑖; the set of truth for 𝜂𝜂𝑖𝑖 we denote by 𝜋𝜋𝑖𝑖. the following equalities hold: 𝜂𝜂(𝑥𝑥, 𝑦𝑦) ≡ 𝜂𝜂1(𝑥𝑥, 𝑦𝑦) ∨ 𝜂𝜂2(𝑥𝑥, 𝑦𝑦) ∨ … ∨ 𝜂𝜂𝑚𝑚(𝑥𝑥, 𝑦𝑦); 𝜋𝜋 = 𝜋𝜋1 ∪ 𝜋𝜋2 ∪ … ∪ 𝜋𝜋𝑚𝑚. clearly, all the predicates 𝜂𝜂, 𝜂𝜂1, 𝜂𝜂2, … , 𝜂𝜂𝑚𝑚, and all the sets 𝜋𝜋, 𝜋𝜋1, 𝜋𝜋2, … , 𝜋𝜋𝑚𝑚 are expressible by arithmetical formulas in the signature (0, =, 𝑆𝑆). let us note that if some 𝐹𝐹𝑖𝑖 includes simultaneously some two subformulas of the forms 𝑥𝑥 = 𝑦𝑦, 𝑦𝑦 = 𝑆𝑆(𝑥𝑥), 𝑥𝑥 = 𝑆𝑆(𝑦𝑦), then the corresponding predicate 𝜂𝜂𝑖𝑖 is identically false, hence, 𝐹𝐹𝑖𝑖 can be deleted from the structure of 𝐹𝐹, similarly, 𝐹𝐹𝑖𝑖 can be deleted from the structure of 𝐹𝐹 if it includes subformulas of the forms 𝑥𝑥 = 𝑎𝑎1 and 𝑥𝑥 = 𝑎𝑎2 where 𝑎𝑎1 ≠ 𝑎𝑎2 or subformulas of the forms 𝑦𝑦 = 𝑏𝑏1 and 𝑦𝑦 = 𝑏𝑏2 where 𝑏𝑏1 ≠ 𝑏𝑏2. let us consider possible forms of the formula 𝐹𝐹𝑖𝑖. we do not consider the cases mentioned above when 𝐹𝐹𝑖𝑖 can be deleted from the structure of 𝐹𝐹. (case 1). 𝐹𝐹𝑖𝑖 contains the subformulas 𝑥𝑥 = 𝑎𝑎, 𝑦𝑦 = 𝑆𝑆(𝑥𝑥), and, possibly, ¬(𝑥𝑥 = 0), ¬(𝑦𝑦 = 0). in this case 𝜋𝜋𝑖𝑖 is either empty, or contains the single pair (𝑎𝑎, 𝑎𝑎 + 1) (for example, 𝜋𝜋𝑖𝑖 is empty if 𝐹𝐹𝑖𝑖 has the form (𝑥𝑥 = 0 & 𝑦𝑦 = 𝑆𝑆(𝑥𝑥)& ¬(𝑥𝑥 = 0))). (case 2). 𝐹𝐹𝑖𝑖 contains the subformulas 𝑦𝑦 = 𝑏𝑏, 𝑦𝑦 = 𝑆𝑆(𝑥𝑥), and, possibly, ¬(𝑥𝑥 = 0), ¬(𝑦𝑦 = 0). in this case 𝜋𝜋𝑖𝑖 is either empty or contains the single pair (𝑏𝑏 − 1, 𝑏𝑏), where 𝑏𝑏 > 0. (case 3). 𝐹𝐹𝑖𝑖 contains the subformulas 𝑦𝑦 = 𝑆𝑆(𝑥𝑥), and, possibly, ¬(𝑥𝑥 = 0), ¬(𝑦𝑦 = 0) (we suppose that 𝐹𝐹𝑖𝑖 contains no subformula having one of the forms 𝑥𝑥 = 𝑎𝑎, 𝑦𝑦 = 𝑏𝑏, 𝑥𝑥 = 𝑦𝑦, 𝑥𝑥 = 𝑆𝑆(𝑦𝑦)). in this case all pairs of numbers having the form (𝑥𝑥, 𝑥𝑥 + 1), where 𝑥𝑥 > 0, belong to 𝜋𝜋𝑖𝑖. the statement (0,1) ∈ 𝜋𝜋𝑖𝑖 is true if and only if the subformula ¬(𝑥𝑥 = 0) is not included in 𝐹𝐹𝑖𝑖. (case 4). 𝐹𝐹𝑖𝑖 contains the subformulas 𝑦𝑦 = 𝑏𝑏, 𝑥𝑥 = 𝑆𝑆(𝑦𝑦), and, possibly, ¬(𝑥𝑥 = 0), ¬(𝑦𝑦 = 0). in this case 𝜋𝜋𝑖𝑖 is either empty, or contains the single pair (𝑏𝑏 + 1, 𝑏𝑏). (case 5). 𝐹𝐹𝑖𝑖 contains the subformulas 𝑥𝑥 = 𝑎𝑎, 𝑥𝑥 = 𝑆𝑆(𝑦𝑦), and, possibly, ¬(𝑥𝑥 = 0), ¬(𝑦𝑦 = 0). in this case 𝜋𝜋𝑖𝑖 is either empty, or contains the single pair (𝑎𝑎, 𝑎𝑎 − 1), where 𝑎𝑎 > 0. (case 6). 𝐹𝐹𝑖𝑖 contains the subformulas 𝑥𝑥 = 𝑆𝑆(𝑦𝑦), and, possibly, ¬(𝑥𝑥 = 0), ¬(𝑦𝑦 = 0) (we suppose that 𝐹𝐹𝑖𝑖 contains no subformulas having one of the forms 𝑥𝑥 = 𝑎𝑎, 𝑦𝑦 = 𝑏𝑏, 𝑥𝑥 = 𝑦𝑦, 𝑦𝑦 = 𝑆𝑆(𝑥𝑥)). 70 on transitive closures of two-dimensional strongly positive arithmetical sets in this case all pairs of numbers having the form (𝑥𝑥 + 1, 𝑥𝑥), where 𝑥𝑥 > 0, belong to 𝜋𝜋𝑖𝑖. the statement (1,0) ∈ 𝜋𝜋𝑖𝑖 is true if and only if the subformula ¬(𝑦𝑦 = 0) is not included in 𝐹𝐹𝑖𝑖. (case 7). 𝐹𝐹𝑖𝑖 contains the subformulas 𝑥𝑥 = 𝑎𝑎, 𝑦𝑦 = 𝑏𝑏, and, possibly, 𝑥𝑥 = 𝑦𝑦 ¬(𝑥𝑥 = 0), ¬(𝑦𝑦 = 0). in this case 𝜋𝜋𝑖𝑖 is either empty, or contains the single pair (𝑎𝑎, 𝑏𝑏). (case 8). 𝐹𝐹𝑖𝑖 contains the subformulas 𝑥𝑥 = 𝑎𝑎, 𝑥𝑥 = 𝑦𝑦, and, possibly, ¬(𝑥𝑥 = 0), ¬(𝑦𝑦 = 0). in this case 𝜋𝜋𝑖𝑖 is either empty, or contains the single pair (𝑎𝑎, 𝑎𝑎). (case 9). 𝐹𝐹𝑖𝑖 contains the subformulas 𝑦𝑦 = 𝑏𝑏, 𝑥𝑥 = 𝑦𝑦, and, possibly, ¬(𝑥𝑥 = 0), ¬(𝑦𝑦 = 0). in this case 𝜋𝜋𝑖𝑖 is either empty, or contains the single pair (𝑏𝑏, 𝑏𝑏). (case 10). 𝐹𝐹𝑖𝑖 contains the subformulas 𝑥𝑥 = 𝑎𝑎, and, possibly, ¬(𝑥𝑥 = 0), ¬(𝑦𝑦 = 0) (we suppose that 𝐹𝐹𝑖𝑖 contains no subformulas having one of the forms, 𝑦𝑦 = 𝑏𝑏, 𝑥𝑥 = 𝑦𝑦, 𝑦𝑦 = 𝑆𝑆(𝑥𝑥), 𝑥𝑥 = 𝑆𝑆(𝑦𝑦)). in this case 𝜋𝜋𝑖𝑖 is empty when 𝑎𝑎 = 0 and the subformula ¬(𝑥𝑥 = 0) is included in 𝐹𝐹𝑖𝑖. in the opposite case 𝜋𝜋𝑖𝑖 contains all pairs (𝑎𝑎, 𝑦𝑦), where y> 0. the statement (𝑎𝑎, 0) ∈ 𝜋𝜋𝑖𝑖 is true (for 𝑎𝑎 > 0) if and only if the subformula ¬(𝑦𝑦 = 0) is not included in 𝐹𝐹𝑖𝑖. (case 11). 𝐹𝐹𝑖𝑖 contains the subformulas 𝑦𝑦 = 𝑏𝑏, and, possibly, ¬(𝑥𝑥 = 0), ¬(𝑦𝑦 = 0) (we suppose that 𝐹𝐹𝑖𝑖 contains no subformulas having one of the forms, x= 𝑎𝑎, 𝑥𝑥 = 𝑦𝑦, 𝑦𝑦 = 𝑆𝑆(𝑥𝑥), 𝑥𝑥 = 𝑆𝑆(𝑦𝑦)). in this case 𝜋𝜋𝑖𝑖 is empty when 𝑏𝑏 = 0, and the subformula ¬(𝑦𝑦 = 0) is included in 𝐹𝐹𝑖𝑖. in the opposite case 𝜋𝜋𝑖𝑖 contains all pairs (𝑥𝑥, 𝑏𝑏), where x> 0. the statement (0, 𝑏𝑏) ∈ 𝜋𝜋𝑖𝑖 is true (for 𝑏𝑏 ≠ 0) if and only if the subformula ¬(𝑥𝑥 = 0) is not included in 𝐹𝐹𝑖𝑖. (case 12). 𝐹𝐹𝑖𝑖 contains the subformulas 𝑥𝑥 = 𝑦𝑦, and, possibly, ¬(𝑥𝑥 = 0), ¬(𝑦𝑦 = 0) (we suppose that 𝐹𝐹𝑖𝑖 contains no subformulas having one of the forms, x= 𝑎𝑎, y = 𝑏𝑏, 𝑦𝑦 = 𝑆𝑆(𝑥𝑥), 𝑥𝑥 = 𝑆𝑆(𝑦𝑦)). in this case 𝜋𝜋𝑖𝑖 contains all the pairs having the form (𝑥𝑥, 𝑥𝑥), where x > 0. the statement (0,0) ∈ 𝜋𝜋𝑖𝑖 is true if and only if the subformulas ¬(𝑥𝑥 = 0) and ¬(𝑦𝑦 = 0) are not included in 𝐹𝐹𝑖𝑖. it is easily seen that all the variants of the structure of 𝜋𝜋𝑖𝑖 are exhausted in the cases 1-12. now we will consider the variants of the structure of 𝜋𝜋∗. as it is proved in [3] (see [3], lemma 3.4) the statement (𝑥𝑥, 𝑦𝑦) ∈ 𝜋𝜋∗is true if and only if there exists an etc-sequence (𝑞𝑞1, 𝑞𝑞2, … , 𝑞𝑞𝑟𝑟) such that 𝑞𝑞1 = 𝑥𝑥, 𝑞𝑞𝑟𝑟 = 𝑦𝑦, (𝑞𝑞𝑖𝑖, 𝑞𝑞𝑖𝑖+1) ∈ 𝜋𝜋 for 1 ≤ 𝑖𝑖 < 𝑟𝑟. without loss of generality we may suppose that any considered etc-sequence (𝑞𝑞1, 𝑞𝑞2, … , 𝑞𝑞𝑟𝑟) where 𝑟𝑟 ≥ 3, satisfies the condition 𝑞𝑞𝑖𝑖 ≠ 𝑞𝑞𝑗𝑗 when 𝑖𝑖 ≠ 𝑗𝑗 (otherwise the given etc-sequence may be replaced by a shorter sequence having the same properties). let us consider the number 𝑑𝑑 such that 𝑑𝑑 = 𝑑𝑑1 + 1, where 𝑑𝑑1 is the maximum of the numbers 𝑎𝑎 and 𝑏𝑏 in the formulas 𝑥𝑥 = 𝑎𝑎 and 𝑦𝑦 = 𝑏𝑏 included in 𝐹𝐹. if no formula of such forms is included in 𝐹𝐹, then we admit 𝑑𝑑 = 3. we will use below some classification of pairs (𝑥𝑥, 𝑦𝑦) such that 𝑥𝑥 ∈ 𝑁𝑁, 𝑦𝑦 ∈ 𝑁𝑁. we say that (𝑥𝑥, 𝑦𝑦) belongs to the subset 𝑆𝑆1 if 𝑥𝑥 ≤ 𝑑𝑑, 𝑦𝑦 ≤ 𝑑𝑑. in a similar way we define the subsets 𝑆𝑆2, 𝑆𝑆3, 𝑆𝑆4 as sets of pairs (𝑥𝑥, 𝑦𝑦) such that (𝑥𝑥, 𝑦𝑦) ∈ 𝑆𝑆2 if 𝑥𝑥 > 𝑑𝑑, 𝑦𝑦 ≤ 𝑑𝑑; (𝑥𝑥, 𝑦𝑦) ∈ 𝑆𝑆3 if 𝑥𝑥 ≤ 𝑑𝑑, 𝑦𝑦 > 𝑑𝑑; (𝑥𝑥, 𝑦𝑦) ∈ 𝑆𝑆4 if 𝑥𝑥 > 𝑑𝑑, 𝑦𝑦 > 𝑑𝑑. the sets 𝑆𝑆1∗, 𝑆𝑆2∗, 𝑆𝑆3∗, 𝑆𝑆4∗ are defined coorrespondingly as 𝑆𝑆1 ∩ 𝜋𝜋∗, 𝑆𝑆2 ∩ 𝜋𝜋∗, 𝑆𝑆3 ∩ 𝜋𝜋∗, 𝑆𝑆4 ∩ 𝜋𝜋∗. a pair (𝑥𝑥, 𝑦𝑦) is said to be increasing if 𝑥𝑥 < 𝑦𝑦 and decreasing if 𝑥𝑥 > 𝑦𝑦. lemma 3.1: if the number ℎ satisfies the condition ℎ ≥ 𝑑𝑑, and the pair (𝑥𝑥, 𝑦𝑦) ∈ 𝜋𝜋∗ satisfies the conditions 𝑥𝑥 ≤ ℎ, 𝑦𝑦 ≤ ℎ, then there exists an etc-sequence (𝑞𝑞1, 𝑞𝑞2, … , 𝑞𝑞𝑟𝑟) for the value (𝑥𝑥, 𝑦𝑦) ∈ 𝜋𝜋∗ such that 𝑞𝑞1 = 𝑥𝑥, 𝑞𝑞𝑟𝑟 = 𝑦𝑦, (𝑞𝑞𝑖𝑖, 𝑞𝑞𝑖𝑖+1) ∈ 𝜋𝜋 for 1 ≤ 𝑖𝑖 < 𝑟𝑟, 𝑞𝑞𝑖𝑖 ≤ ℎ for 1 ≤ 𝑖𝑖 ≤ 𝑟𝑟. s. manukian 71 proof: as it follows from the condition (𝑥𝑥, 𝑦𝑦) ∈ 𝜋𝜋∗, there exists an etcsequence (𝑞𝑞1, 𝑞𝑞2, … , 𝑞𝑞𝑟𝑟) such that 𝑞𝑞1 = 𝑥𝑥, 𝑞𝑞𝑟𝑟 = 𝑦𝑦, (𝑞𝑞𝑖𝑖, 𝑞𝑞𝑖𝑖+1) ∈ 𝜋𝜋 for 1 ≤ 𝑖𝑖 < 𝑟𝑟. if 𝑞𝑞𝑖𝑖 ≤ ℎ for 1 ≤ 𝑖𝑖 ≤ 𝑟𝑟, then the statement of lemma is satisfied. in the opposite case let 𝑘𝑘 be the minimal index in the sequence (𝑞𝑞1, 𝑞𝑞2, … , 𝑞𝑞𝑟𝑟) such that 𝑘𝑘 > 1, 𝑞𝑞𝑘𝑘 > ℎ. let 𝑙𝑙 be the minimal index such that 𝑙𝑙 ≥ 𝑘𝑘, 𝑞𝑞𝑙𝑙+1 ≤ ℎ. clearly, any number 𝑞𝑞𝑗𝑗, where 𝑘𝑘 ≤ 𝑗𝑗 ≤ 𝑙𝑙 satisfies the condition 𝑞𝑞𝑗𝑗 > ℎ(note that the case 𝑘𝑘 = 𝑙𝑙 is not excluded). the following statements hold: 𝑞𝑞𝑘𝑘−1 ≤ ℎ, 𝑞𝑞𝑙𝑙+1 ≤ ℎ, the pair (𝑞𝑞𝑘𝑘−1, 𝑞𝑞𝑘𝑘) is increasing, the pair (𝑞𝑞𝑙𝑙, 𝑞𝑞𝑙𝑙+1) is decreasing. but any increasing pair (𝑥𝑥, 𝑦𝑦) ∈ 𝜋𝜋 such that 𝑥𝑥 ≥ ℎ, 𝑦𝑦 ≥ ℎ, 𝑥𝑥 < 𝑦𝑦 should satisfy the conditions of (case 3) or (case 10) mentioned above. therefore, either 𝑞𝑞𝑘𝑘−1 = ℎ, 𝑞𝑞𝑘𝑘 = ℎ + 1 (case 3) or 𝑞𝑞𝑘𝑘−1 = 𝑎𝑎, where 𝑎𝑎 is the number contained in a formula 𝑥𝑥 = 𝑎𝑎 included in 𝐹𝐹. similarly, any decreasing pair (𝑥𝑥, 𝑦𝑦) ∈ 𝜋𝜋 such that 𝑥𝑥 ≥ ℎ, 𝑦𝑦 ≥ ℎ, 𝑥𝑥 > 𝑦𝑦, satisfies the conditions of (case 6) or (case 11) mentioned above. therefore either 𝑞𝑞𝑙𝑙 = ℎ + 1, 𝑞𝑞𝑙𝑙+1 = ℎ (case 6) or 𝑞𝑞𝑙𝑙+1 = 𝑏𝑏, where 𝑏𝑏 is the number contained in a formula 𝑦𝑦 = 𝑏𝑏 included in 𝐹𝐹 (case 11). now if 𝑞𝑞𝑘𝑘−1 = 𝑞𝑞𝑙𝑙+1 = ℎ, 𝑞𝑞𝑘𝑘 = 𝑞𝑞𝑙𝑙 = ℎ + 1, (case 3, case 6) then the segment (𝑞𝑞𝑘𝑘−1, 𝑞𝑞𝑘𝑘, 𝑞𝑞𝑘𝑘+1, … , 𝑞𝑞𝑙𝑙, 𝑞𝑞𝑙𝑙+1) of the etc-sequence (𝑞𝑞1, 𝑞𝑞2, … , 𝑞𝑞𝑟𝑟) can be replaced by the single number 𝑞𝑞𝑘𝑘−1 = 𝑞𝑞𝑙𝑙+1 = ℎ. if 𝑞𝑞𝑘𝑘−1 = 𝑎𝑎, then the mentioned segment can be replaced by the segment (𝑎𝑎, 𝑞𝑞𝑙𝑙+1) (case 10). if 𝑞𝑞𝑙𝑙+1 = 𝑏𝑏, then the mentioned segment can be replaced by the segment (𝑞𝑞𝑘𝑘−1, 𝑏𝑏) (case 11). clearly the sequence obtained by these replacements is an etcsequence for the value (𝑥𝑥, 𝑦𝑦) ∈ 𝜋𝜋∗. transforming in this way any segment of the sequence (𝑞𝑞1, 𝑞𝑞2, … , 𝑞𝑞𝑟𝑟) containing members greater than ℎ, we obtain the etc-sequence satisfying the conditions of lemma. this completes the proof. corollary 1: there is only finite number of etc-sequences obtained by the transformations described in lemma 3.1. indeed, the length of such sequence (without repetitions of members) is ≤ ℎ + 1, and any member of such sequence is ≤ ℎ. corollary 2: the set 𝑆𝑆1∗ can be defined by arithmetical formula in the signature (0, =, <, 𝑆𝑆) (even in (0, =, 𝑆𝑆)). indeed, applying corollary 1 to the case when ℎ = 𝑑𝑑, we conclude that the set of pairs (𝑥𝑥, 𝑦𝑦) such that 𝑥𝑥 ≤ 𝑑𝑑, 𝑦𝑦 ≤ 𝑑𝑑, (𝑥𝑥, 𝑦𝑦) ∈ 𝜋𝜋∗ is finite, hence, it can be defined by a formula having the form �(𝑥𝑥 = 𝑥𝑥1)&(𝑦𝑦 = 𝑦𝑦1)� ∨ �(𝑥𝑥 = 𝑥𝑥2)&(𝑦𝑦 = 𝑦𝑦2)� ∨ … ∨ �(𝑥𝑥 = 𝑥𝑥𝑚𝑚)&(𝑦𝑦 = 𝑦𝑦𝑚𝑚)�, where all 𝑥𝑥𝑖𝑖 and 𝑦𝑦𝑖𝑖 are constants. if this set is empty, then it can be defined by the formula (𝑥𝑥 = 𝑦𝑦)&(𝑥𝑥 = 𝑆𝑆(𝑦𝑦)). note. if 𝑥𝑥 ≤ 𝑑𝑑, 𝑦𝑦 ≤ 𝑑𝑑, then the statement (𝑥𝑥, 𝑦𝑦) ∈ 𝜋𝜋∗ may be tested constructively. the method of testing is actually given in corollary 1. lemma 3.2: if some pair (𝑥𝑥0, 𝑦𝑦0), where 𝑥𝑥0 > 𝑑𝑑, 𝑦𝑦0 ≤ 𝑑𝑑 belongs to 𝜋𝜋∗, then any pair (𝑥𝑥, 𝑦𝑦0), where 𝑥𝑥 > 𝑑𝑑, belongs to 𝜋𝜋∗. proof: if some pair (𝑥𝑥0, 𝑦𝑦0) satisfies the conditions of lemma, then there exists an etc-sequence (𝑞𝑞1, 𝑞𝑞2, … , 𝑞𝑞𝑟𝑟) such that 𝑞𝑞1 = 𝑥𝑥0, 𝑞𝑞𝑟𝑟 = 𝑦𝑦0, (𝑞𝑞𝑖𝑖, 𝑞𝑞𝑖𝑖+1) ∈ 𝜋𝜋 for 1 ≤ 𝑖𝑖 < 𝑟𝑟. let 𝑘𝑘 be the minimal index in the sequence (𝑞𝑞1, 𝑞𝑞2, … , 𝑞𝑞𝑟𝑟) such that 𝑘𝑘 < 𝑟𝑟, 𝑞𝑞𝑘𝑘+1 ≤ 𝑑𝑑 (the case 𝑘𝑘 = 1 is not excluded). without loss of generality we may suprose that 𝑞𝑞𝑖𝑖 ≤ 𝑑𝑑 for 𝑘𝑘 + 1 ≤ 𝑖𝑖 ≤ 𝑟𝑟 (indeed, in the opposite case the segment (𝑞𝑞𝑘𝑘+1, 𝑞𝑞𝑘𝑘+2, … , 𝑞𝑞𝑟𝑟) can be transformed by the method described in the proof of lemma 3.1). 72 on transitive closures of two-dimensional strongly positive arithmetical sets the pair (𝑞𝑞𝑘𝑘, 𝑞𝑞𝑘𝑘+1) is decreasing, hence, either 𝑞𝑞𝑘𝑘+1 = 𝑏𝑏, where 𝑏𝑏 is the number in a formula 𝑦𝑦 = 𝑏𝑏 included in 𝐹𝐹, (case 11 considered above), or 𝑞𝑞𝑘𝑘+1 = 𝑞𝑞𝑘𝑘 − 1 (case 6 considered above). now the etc-sequence for the pair (𝑥𝑥, 𝑦𝑦0) where 𝑥𝑥 > 𝑑𝑑 is obtained from the sequence (𝑞𝑞1, 𝑞𝑞2, … , 𝑞𝑞𝑟𝑟) as follows: if 𝑞𝑞𝑘𝑘+1 = 𝑏𝑏, then the segment (𝑞𝑞1, 𝑞𝑞2, … , 𝑞𝑞𝑘𝑘+1) is replaced by the segment (𝑥𝑥, 𝑞𝑞𝑘𝑘+1) (see case 11); if 𝑞𝑞𝑘𝑘 = 𝑞𝑞𝑘𝑘+1 + 1, then any pair (𝑥𝑥 + 1, 𝑥𝑥), where 𝑥𝑥 > 0, belongs to 𝜋𝜋 (see case 6) hence, we can obtain the required etc-sequence replacing in the sequence (𝑞𝑞1, 𝑞𝑞2, … , 𝑞𝑞𝑟𝑟) the segment (𝑞𝑞1, 𝑞𝑞2, … , 𝑞𝑞𝑘𝑘+1) by the segment (x, x – 1, … , 𝑞𝑞𝑘𝑘+1 + 1, 𝑞𝑞𝑘𝑘+1). it is easily seen that the sequence obtained by the mentioned replacements is an etc-sequence for the value (𝑥𝑥, 𝑦𝑦0) ∈ 𝜋𝜋∗. this complets the proof. corollary: the set 𝑆𝑆2∗ can be defined by an arithmetical formula in the signature (0, =, <, 𝑆𝑆). indeed, applying lemma 3.1 and its corollaries to the case when ℎ = 𝑑𝑑 + 1, we obtain the complete list of pairs (𝑥𝑥, 𝑦𝑦) ∈ 𝜋𝜋∗ such that 𝑥𝑥 ≤ 𝑑𝑑 + 1, 𝑦𝑦 ≤ 𝑑𝑑 + 1. in particular we obtain the complete list of pairs having the property (𝑑𝑑 + 1, 𝑦𝑦0) ∈ 𝑆𝑆2∗, where 𝑦𝑦0 ≤ 𝑑𝑑. using lemma 3.2 we conclude that any pair (𝑑𝑑 + 1, 𝑦𝑦0) ∈ 𝑆𝑆2∗ where 𝑦𝑦0 ≤ 𝑑𝑑 generates the set {(𝑥𝑥, 𝑦𝑦)/(𝑥𝑥 > 𝑑𝑑)&(𝑦𝑦 = 𝑦𝑦0)}, contained in 𝑆𝑆2∗, so the set 𝑆𝑆2∗ is the union of sets having this form for all 𝑦𝑦0 ≤ 𝑑𝑑 such that (𝑑𝑑 + 1, 𝑦𝑦0) ∈ 𝑆𝑆2∗. but any set {(𝑥𝑥, 𝑦𝑦)/(𝑥𝑥 > 𝑑𝑑)&(𝑦𝑦 = 𝑦𝑦0)} is defined by the formula (𝑥𝑥 > 𝑑𝑑)&(𝑦𝑦 = 𝑦𝑦0), so the set 𝑆𝑆2∗ is defined by the disjunction of these formulas. this completes the proof. lemma 3.3: if some pair (𝑥𝑥0, 𝑦𝑦0), where 𝑥𝑥0 ≤ 𝑑𝑑, 𝑦𝑦0 > 𝑑𝑑, belongs to 𝜋𝜋∗, then any pair (𝑥𝑥0, 𝑦𝑦), where 𝑦𝑦 > 𝑑𝑑 belongs to 𝜋𝜋∗. the proof is similar to that of lemma 3.2. we use the etc-sequence (𝑞𝑞1, 𝑞𝑞2, … , 𝑞𝑞𝑟𝑟) such that 𝑞𝑞1 = 𝑥𝑥0, 𝑞𝑞𝑟𝑟 = 𝑦𝑦0, (𝑞𝑞𝑖𝑖, 𝑞𝑞𝑖𝑖+1) ∈ 𝜋𝜋 for 1 ≤ 𝑖𝑖 < 𝑟𝑟. let 𝑙𝑙 be the maximal index in the sequence (𝑞𝑞1, 𝑞𝑞2, … , 𝑞𝑞𝑟𝑟) such that 𝑞𝑞𝑙𝑙 ≤ 𝑑𝑑. without loss of generality we may suppose that 𝑞𝑞𝑖𝑖 ≤ 𝑑𝑑 when 1 ≤ 𝑖𝑖 ≤ 𝑙𝑙 (otherwise the segment (𝑞𝑞1, 𝑞𝑞2, … , 𝑞𝑞𝑙𝑙) may be transformed by the method used in the proof of lemma 3.1). the pair (𝑞𝑞𝑙𝑙, 𝑞𝑞𝑙𝑙+1) is increasing, therefore either 𝑞𝑞𝑙𝑙 = 𝑎𝑎, where 𝑎𝑎 is the number in a formula 𝑥𝑥 = 𝑎𝑎 included in 𝐹𝐹 (case 10) or 𝑞𝑞𝑙𝑙+1 = 𝑞𝑞𝑙𝑙 + 1 (case 3). the etc-sequence for establishing the statement (𝑥𝑥0, 𝑦𝑦) ∈ 𝜋𝜋∗ (where 𝑦𝑦 > 𝑑𝑑) is obtained from the sequence (𝑞𝑞1, 𝑞𝑞2, … , 𝑞𝑞𝑟𝑟) by the following replasements: either the segment (𝑞𝑞𝑙𝑙, 𝑞𝑞𝑙𝑙+1, … , 𝑞𝑞𝑟𝑟) is replaced by the segment (𝑎𝑎, 𝑦𝑦) (case 10), or this segment is replased by the segment (𝑞𝑞𝑙𝑙, 𝑞𝑞𝑙𝑙 + 1, … , 𝑦𝑦 − 1, 𝑦𝑦) (case 3). this completes the proof. corollary: the set 𝑆𝑆3∗ can be defined by an arithmetical formula in the signature (0, =, <, 𝑆𝑆). the proof is similar to the proof of corollary of lemma 3.2. in what follows we will say that a formula having the form 𝑦𝑦 = 𝑆𝑆(𝑥𝑥), 𝑥𝑥 = 𝑆𝑆(𝑦𝑦), or 𝑥𝑥 = 𝑦𝑦 is contained in a special way in some 𝐹𝐹𝑖𝑖 if this formula is contained in 𝐹𝐹𝑖𝑖 and the conditions described correspondingly in (case 3), (case 6) or (case 12) mentioned above are satisfied. lemma 3.4: if the formula 𝑦𝑦 = 𝑆𝑆(𝑥𝑥) is contained in a special way in some 𝐹𝐹𝑖𝑖 then any pair (𝑥𝑥, 𝑦𝑦) such that 𝑥𝑥 < 𝑦𝑦, 𝑥𝑥 > 0, belongs to 𝜋𝜋∗. indeed, for establishing this statement it is sufficient to consider the etc-sequence (𝑥𝑥, 𝑥𝑥 + 1, … , 𝑦𝑦 − 1, 𝑦𝑦). lemma 3.5: if the formula 𝑥𝑥 = 𝑆𝑆(𝑦𝑦) is contained in a special way in some 𝐹𝐹𝑖𝑖, then any pair (𝑥𝑥, 𝑦𝑦) such that 𝑥𝑥 > 𝑦𝑦, 𝑦𝑦 > 0 belongs to 𝜋𝜋∗. s. manukian 73 for establishing this statement it is sufficient to consider the etc-sequence (𝑥𝑥, 𝑥𝑥 − 1, … , 𝑦𝑦 − 1, 𝑦𝑦). lemma 3.6: if the formula 𝑦𝑦 = 𝑆𝑆(𝑥𝑥) is contained in a special way in some 𝐹𝐹𝑖𝑖, and the formula 𝑥𝑥 = 𝑆𝑆(𝑦𝑦) is contained in a special way in some 𝐹𝐹𝑗𝑗, where 𝑖𝑖 ≠ 𝑗𝑗, then any pair (𝑥𝑥, 𝑦𝑦), where 𝑥𝑥 > 0, 𝑦𝑦 > 0 belongs to 𝜋𝜋∗. for establishing this statement it is sufficient to consider the etc-sequence (𝑥𝑥, 𝑥𝑥 + 1, … , 𝑧𝑧, … , 𝑦𝑦 − 1, 𝑦𝑦), where 𝑧𝑧 = max (𝑥𝑥, 𝑦𝑦) + 1. lemma 3.7: the set 𝑆𝑆4∗ can be defined by an arithmetical formula in the signature (0, =, <, 𝑆𝑆). proof: if the set 𝑆𝑆4∗is empty, then it is defined, for example, by the formula (𝑥𝑥 = 𝑦𝑦)&(𝑦𝑦 = 𝑆𝑆(𝑥𝑥)). otherwise, there exists a pair (𝑥𝑥0, 𝑦𝑦0) ∈ 𝑆𝑆4∗, that is 𝑥𝑥0 > 𝑑𝑑, 𝑦𝑦0 > 𝑑𝑑, (𝑥𝑥0, 𝑦𝑦0) ∈ 𝜋𝜋∗. hence, there exists an etc-sequence (𝑞𝑞1, 𝑞𝑞2, … , 𝑞𝑞𝑟𝑟) such that 𝑞𝑞1 = 𝑥𝑥0, 𝑞𝑞𝑟𝑟 = 𝑦𝑦0, (𝑞𝑞𝑖𝑖, 𝑞𝑞𝑖𝑖+1) ∈ 𝜋𝜋 for 1 ≤ 𝑖𝑖 < 𝑟𝑟. we will distinguish two cases: (𝛼𝛼) there exists such 𝑖𝑖 that 1 ≤ 𝑖𝑖 ≤ 𝑟𝑟, 𝑞𝑞𝑖𝑖 ≤ 𝑑𝑑. (𝛽𝛽) 𝑞𝑞𝑖𝑖 > 𝑑𝑑 for any 𝑖𝑖, where 1 ≤ 𝑖𝑖 ≤ 𝑟𝑟. let us consider the case (𝛼𝛼).we denote the number 𝑞𝑞𝑖𝑖 such that 𝑞𝑞𝑖𝑖 ≤ 𝑑𝑑 by 𝑧𝑧. the pair (𝑥𝑥0, 𝑧𝑧), where 𝑥𝑥0 > 𝑑𝑑 belongs to 𝜋𝜋∗, therefore, using lemma 3.2 we conclude that any pair (𝑥𝑥, 𝑧𝑧), where 𝑥𝑥 > 𝑑𝑑, belongs to 𝜋𝜋∗. the pair (𝑧𝑧, 𝑦𝑦0), where 𝑦𝑦0 > 𝑑𝑑, belongs to 𝜋𝜋∗, therefore, using lemma 3.3 we conclude that any pair (𝑧𝑧, 𝑦𝑦), where 𝑦𝑦 > 𝑑𝑑 belongs to 𝜋𝜋∗. hence, any pair (𝑥𝑥, 𝑦𝑦), where 𝑥𝑥 > 𝑑𝑑, 𝑦𝑦 > 𝑑𝑑, belongs to 𝜋𝜋∗. so in the case (𝛼𝛼) the set 𝑆𝑆4∗ is defined by the formula (𝑥𝑥 > 𝑑𝑑)&(𝑦𝑦 > 𝑑𝑑). let us note that similar conclusion concerning the set 𝑆𝑆4∗ can be made if there exists any etc-sequence (𝑞𝑞1, 𝑞𝑞2, … , 𝑞𝑞𝑟𝑟) such that 𝑞𝑞1 > 𝑑𝑑, 𝑞𝑞𝑟𝑟 > 𝑑𝑑 and 𝑞𝑞𝑖𝑖 ≤ 𝑑𝑑 for some 𝑖𝑖, 1 < 𝑖𝑖 < 𝑟𝑟. now let us consider the case (𝛽𝛽). we will investigate the properties of all etc-sequences (𝑞𝑞1, 𝑞𝑞2, … , 𝑞𝑞𝑟𝑟) such that 𝑞𝑞𝑖𝑖 > 𝑑𝑑 for 1 ≤ 𝑖𝑖 ≤ 𝑟𝑟, and (𝑞𝑞𝑖𝑖, 𝑞𝑞𝑖𝑖+1) ∈ 𝜋𝜋 for 1 ≤ 𝑖𝑖 < 𝑟𝑟. we distingmish the following subcases: (𝛽𝛽1), (𝛽𝛽2), (𝛽𝛽3), (𝛽𝛽4). (𝛽𝛽1) in some etc-sequence (𝑞𝑞1, 𝑞𝑞2, … , 𝑞𝑞𝑟𝑟) of the mentioned kind there exists an index 𝑖𝑖 such that 1 ≤ 𝑖𝑖 < 𝑟𝑟, 𝑞𝑞𝑖𝑖+1 = 𝑞𝑞𝑖𝑖 + 1, but there is no etc-sequence of the mentioned kind containing an index 𝑗𝑗 such that 𝑞𝑞𝑗𝑗+1 = 𝑞𝑞𝑗𝑗 − 1. (𝛽𝛽2) in some etc-sequence (𝑞𝑞1, 𝑞𝑞2, … , 𝑞𝑞𝑟𝑟) of the mentioned kind there exists an index 𝑖𝑖 such that 1 ≤ 𝑖𝑖 < 𝑟𝑟, 𝑞𝑞𝑖𝑖+1 = 𝑞𝑞𝑖𝑖 − 1, but there is no etc-sequence of the mentioned kind containing an index 𝑗𝑗 such that 𝑞𝑞𝑗𝑗+1 = 𝑞𝑞𝑗𝑗 + 1. (𝛽𝛽3) in some etc-sequence (𝑞𝑞1, 𝑞𝑞2, … , 𝑞𝑞𝑟𝑟) of the mentioned kind there exists an index 𝑖𝑖 such that 1 ≤ 𝑖𝑖 < 𝑟𝑟, 𝑞𝑞𝑖𝑖+1 = 𝑞𝑞𝑖𝑖 + 1; besides, in some etc-sequence (𝑞𝑞1, 𝑞𝑞2, … , 𝑞𝑞𝑡𝑡) of the mentioned kind there exists an index 𝑗𝑗 such that 1 ≤ 𝑗𝑗 < 𝑡𝑡, 𝑞𝑞𝑗𝑗+1 = 𝑞𝑞𝑗𝑗 − 1. (𝛽𝛽4) there is no etc-sequence of the mentioned kind satisfying the conditions described in the subcases (𝛽𝛽1)-( 𝛽𝛽3). clearly, the subcase (𝛽𝛽1) takes place if some 𝐹𝐹𝑖𝑖 in the structure of 𝐹𝐹 has the form (𝑦𝑦 = 𝑆𝑆(𝑥𝑥)), but there is no 𝐹𝐹𝑗𝑗 having the form (𝑥𝑥 = 𝑆𝑆(𝑦𝑦)). similarly, the subcase (𝛽𝛽2) takes place if some 𝐹𝐹𝑖𝑖 in the structure of 𝐹𝐹 has the form (𝑥𝑥 = 𝑆𝑆(𝑦𝑦)), but there is no 𝐹𝐹𝑗𝑗 having the form (𝑦𝑦 = 𝑆𝑆(𝑥𝑥)). the subcase (𝛽𝛽3) takes place if some 𝐹𝐹𝑖𝑖 and 𝐹𝐹𝑗𝑗 in the structure of 𝐹𝐹 have the forms, correspondingly 74 on transitive closures of two-dimensional strongly positive arithmetical sets (𝑦𝑦 = 𝑆𝑆(𝑥𝑥)) and (𝑥𝑥 = 𝑆𝑆(𝑦𝑦)). the subcase (𝛽𝛽4) takes place if the formula 𝐹𝐹 contains no 𝐹𝐹𝑖𝑖 having one of the mentioned forms. it is easily seen that in the subcase (𝛽𝛽1) the set 𝑆𝑆4∗ is defined by the formula (𝑥𝑥 > 𝑑𝑑)&(𝑦𝑦 > 𝑑𝑑)&(𝑥𝑥 < 𝑦𝑦) or by the formula (𝑥𝑥 > 𝑑𝑑)&(𝑦𝑦 > 𝑑𝑑)&(𝑥𝑥 ≤ 𝑦𝑦) (see lemma 3.4). in the subcase (𝛽𝛽2) the set 𝑆𝑆4∗ is defined by the formula (𝑥𝑥 > 𝑑𝑑)&(𝑦𝑦 > 𝑑𝑑)&(𝑥𝑥 > 𝑦𝑦) or by the formula (𝑥𝑥 > 𝑑𝑑)&(𝑦𝑦 > 𝑑𝑑)&(𝑥𝑥 ≥ 𝑦𝑦) (see lemma 3.5). let us note that the inequalities 𝑥𝑥 ≤ 𝑦𝑦 and 𝑥𝑥 ≥ 𝑦𝑦 are obtained in the subcases (𝛽𝛽1) and (𝛽𝛽2) if some 𝐹𝐹𝑖𝑖 in the structure of 𝐹𝐹 has the form 𝑥𝑥 = 𝑦𝑦 (see case 12 mentioned above). in the subcase (𝛽𝛽3) the set 𝑆𝑆4∗ is defined by the formula (𝑥𝑥 > 𝑑𝑑)&(𝑦𝑦 > 𝑑𝑑) (see lemma 3.6). in the subcase (𝛽𝛽4) the set 𝑆𝑆4∗ is either empty or is defined by the formula (𝑥𝑥 > 𝑑𝑑)&(𝑦𝑦 > 𝑑𝑑)&(𝑥𝑥 = 𝑦𝑦). this completes the proof. proof of theorem 1. as it is established in lemmas 3.1-3.7, the sets 𝑆𝑆1∗, 𝑆𝑆2∗, 𝑆𝑆3∗, 𝑆𝑆4∗, are defined by formulas in the signature {0, =, <, 𝑆𝑆}. hence, the set 𝜋𝜋∗ = 𝑆𝑆1∗ ∪ 𝑆𝑆2∗ ∪ 𝑆𝑆3∗ ∪ 𝑆𝑆4∗ is defined by the disjunction of the mentioned formulas. this completes the proof. proof of theorem 2. let 𝐴𝐴 be the set {(𝑥𝑥, 𝑦𝑦)/𝑦𝑦 = 𝑆𝑆(𝑆𝑆(𝑥𝑥))}, let 𝐵𝐵 be any 2-dimensional strongly positive set, let 𝐵𝐵∗ be the transitive closure of 𝐵𝐵. we define the number 𝑑𝑑 for the set 𝐵𝐵 by the method given above. by 𝐷𝐷 we denote the set {(𝑥𝑥, 𝑦𝑦)/(𝑥𝑥 > 𝑑𝑑)&(𝑦𝑦 > 𝑑𝑑)}. using lemma 3.7 we conclude that the set 𝐵𝐵∗ ∩ 𝐷𝐷 either is empty or is defined by one of the following formulas: (𝑥𝑥 > 𝑑𝑑)&(𝑦𝑦 > 𝑑𝑑)&(𝑥𝑥 > 𝑦𝑦), (𝑥𝑥 > 𝑑𝑑)&(𝑦𝑦 > 𝑑𝑑)&(𝑥𝑥 < 𝑦𝑦), (𝑥𝑥 > 𝑑𝑑)&(𝑦𝑦 > 𝑑𝑑)&(𝑥𝑥 = 𝑦𝑦), or by the disjunction of some of these formulas. similarly, using lemmas 3.4-3.6 we conclude that the set 𝐵𝐵 ∩ 𝐷𝐷 either is empty or is defined by one of the following formulas: (𝑥𝑥 > 𝑑𝑑)&(𝑦𝑦 > 𝑑𝑑)&(𝑦𝑦 = 𝑆𝑆(𝑥𝑥)), (𝑥𝑥 > 𝑑𝑑)&(𝑦𝑦 > 𝑑𝑑)&(𝑥𝑥 = 𝑆𝑆(𝑦𝑦)), (𝑥𝑥 > 𝑑𝑑)&(𝑦𝑦 > 𝑑𝑑)&(𝑥𝑥 = 𝑦𝑦), or by the disjuction of some of these formulas. therefore, in all the cases the set 𝐴𝐴 ∩ 𝐷𝐷 is different form 𝐵𝐵 ∩ 𝐷𝐷 and 𝐵𝐵∗ ∩ 𝐷𝐷. hence, 𝐴𝐴 ≠ 𝐵𝐵 and 𝐴𝐴 ≠ 𝐵𝐵∗. this completes the proof. note 1. the statement of theorem 2 is true also for any set defined by the formula 𝑦𝑦 = 𝑆𝑆(𝑆𝑆 … 𝑆𝑆(𝑥𝑥) … ), where the symbol 𝑆𝑆 is repeated 𝑛𝑛 ≥ 2 times. the proof is similar to that of theorem 2. note 2. obviously, any set defined by a formula in the signature (0, =, <, 𝑆𝑆) is primitive recursive, however, the reverse is not true (for example, the set of even numbers is primitive recurcive, but it cannot be defined by arithmetical formula in the signature (0, =, <, 𝑆𝑆) (see [6])). so the statement of theorem 1 is stronger than the statement of theorem 2 in [3]. references [1] s. n. manukian, “on the representation of recursively enumerable sets in weak arithmetics”, transactions of the iiap of nas of ra, mathematical problems of computer science, vol. 27, pp. 90-110, 2006. [2] s. n. manukian, “on an algebraic classification of multidimensional recursively enumerable sets expressible in formal arithmetical systems”, transactions of the iiap of nas of ra, mathematical problems of computer science, vol. 41, pp. 103-113, 20014. s. manukian 75 [3] s. n.manukian, “on strongly positive multidimensional arithmetical sets”, transactions of the iiap of nas of ra, mathematical problems of computer science, vol. 43, pp. 3241, 2015. [4] s. c. kleene, introduction to metamathematics, d.van nostrand comp., inc., new york – toronto, 1952. [5] e. mendelson, introduction to mathematical logic, d.van nostrand comp., inc., princeton – toronto – new york – london, 1964. [6] h. b. enderton, a mathematical introduction to logic, 2nd edition, san diego, harcourt, academic press, 2001. [7] g. s. tseytin, “one method of representation for the theory of algorithms and enumerable sets”, transactions of steklov institute of the acad. sci. ussr (in russian), vol. 72, pp. 6998, 1964. [8] a. i. maltsev, algorithms and recursive functions, 2nd edition (in russian), m.,”nauka”, 1986. submitted 04.10.2015, accepted 15.01.2016 խիստ պոզիտիվ երկչափ թվաբանական բազմությունների տրանզիտիվ փակումների մասին ս. մանուկյան ամփոփում պոզիտիվ և խիստ պոզիտիվ թվաբանական բազմությունների գաղափարները սահմանված են [1]-[3] հոդվածներում: [3] հոդվածում նշված է, որ ցանկացած երկչափ խիստ պոզիտիվ բազմության տրանզիտիվ փակումը պարզագույն անդրադարձ է: այս հոդվածում ապացուցվում է ավելի ուժեղ պնդում, այսինքն՝ ցանկացած երկչափ խիստ պոզիտիվ բազմության տրանզիտիվ փակումը նկարագրվում է թվաբանական բանաձևի միջոցով (0, =, <, 𝑆𝑆) սիգնատուրայում (որտեղ 𝑆𝑆(𝑥𝑥) = 𝑥𝑥 + 1): բացի դրանից ապացուցվում է, որ երկչափ խիստ պոզիտիվ բազմությունների դասը և այդ բազմությունների տրանզիտիվ փակումների դասը չեն համընկնում (0, =, <, 𝑆𝑆) սիգնատուրայում արտահայտվող թվաբանական բազմությունների դասի հետ: 76 on transitive closures of two-dimensional strongly positive arithmetical sets о транзитивных замыканиях строго позитивных арифметических множеств размерности 2 с. манукян аннотация понятия позитивного и строго позитивного множества рассматриваются в [1]-[3]. в [3] указано, что транзитивное замыкание всякого строго позитивного множества размерности 2 примитивно рекурсивно. в этой статье доказывается более сильное утверждение: транзитивное замыкание всякого строго позитивного множества размерности 2 задается арифметической формулой в сигнатуре (0, =, <, 𝑆𝑆), где 𝑆𝑆(𝑥𝑥) = 𝑥𝑥 + 1. доказывается также, что класс строго позитивных множеств размерности 2 и класс транзитивных замыканий таких множеств не совпадают с классом арифметических множеств размерности 2, задаваемых посредством арифметических формул в сигнатуре (0, =, <, 𝑆𝑆). начиная с начала 2000 года осуществляется внедрение ghis в здравоохранении, в рамках принятого проекта о реформирование информ mathematical problems of computer science 46, 87--91, 2016. linear orderings of tridimensional grids david h. muradian institute for informatics and automation problems of nas ra e-mail: david.h.muradian@gmail.com abstract the minimal linear arrangement problem (minla) is defined as follows: given a graph g, find a linear ordering (layout) 𝜑𝜑 for the vertices of g on a line such that the sum of the edge lengths is minimized over all orderings. edge length for an edge (x, y) is defined as | 𝜑𝜑(𝑥𝑥) − 𝜑𝜑(𝑦𝑦)|. in this paper we describe the class of minimal orderings of the special case of tridimensional grids – cartesian product of three simple paths, when one of them consists of two vertices. keywords: linear ordering, minimal linear arrangement problem, grids, wirelength. 1. introduction given a graph g=(x,u), a layout 𝜑𝜑 is a one-to-one mapping 𝜑𝜑 : x →{1,…,|x|}. for a given graph g=( x,u) and a layout 𝜑𝜑, we define 𝐸𝐸𝜑𝜑(𝐺𝐺) = � |𝜑𝜑(𝑥𝑥) − 𝜑𝜑(𝑦𝑦)| (𝑥𝑥,𝑦𝑦)∈𝑈𝑈 , as a wirelength of 𝜑𝜑. we define also wirelength of g as 𝐸𝐸(𝐺𝐺) = min 𝜑𝜑 𝐸𝐸𝜑𝜑(𝐺𝐺), where 𝜑𝜑 ranges over all layouts of g, and a layout 𝜑𝜑0 is called minimal if 𝐸𝐸𝜑𝜑0(𝐺𝐺)= 𝐸𝐸(𝐺𝐺). let’s denote by 𝜙𝜙𝐺𝐺 𝐸𝐸 the class of minimal layouts of g. let 𝑋𝑋′, 𝑋𝑋′′ ⊂ 𝑋𝑋 be nonempty disjoint sets, 𝑘𝑘 ∈ 1, 𝑁𝑁����� and 𝜑𝜑 be some layout of g. let’s denote: 𝑋𝑋𝜑𝜑𝑘𝑘 = {𝜑𝜑−1(1), 𝜑𝜑−1(2), … , 𝜑𝜑−1(𝑘𝑘), } 𝜔𝜔(𝑋𝑋′, 𝑋𝑋′′) = |{(𝑥𝑥, 𝑦𝑦) ∈ 𝑈𝑈 ⁄ 𝑥𝑥 ∈ 𝑋𝑋′; 𝑦𝑦 ∈ 𝑋𝑋′′ }| δ𝜑𝜑(𝑋𝑋′) = 1 |𝑋𝑋′| (|{(𝑥𝑥, 𝑦𝑦) ∈ 𝑈𝑈 ⁄ 𝑥𝑥 ∈ 𝑋𝑋′; 𝑦𝑦 ∉ 𝑋𝑋′; 𝜑𝜑(𝑥𝑥) < 𝜑𝜑(𝑦𝑦) }| − |{(𝑥𝑥, 𝑦𝑦) ∈ 𝑈𝑈 ⁄ 𝑥𝑥 ∈ 𝑋𝑋′; 𝑦𝑦 ∉ 𝑋𝑋′; 𝜑𝜑(𝑥𝑥) > 𝜑𝜑(𝑦𝑦) }|) 87 linear orderings of tridimensional grids 88 definition: we say that a set 𝑋𝑋′ (𝑋𝑋′ ⊂ 𝑋𝑋) is compact with respect to layout 𝜑𝜑, if max 𝑥𝑥∈𝑋𝑋′ 𝜑𝜑(𝑥𝑥) − min 𝑥𝑥∈𝑋𝑋′ 𝜑𝜑(𝑥𝑥) = |𝑋𝑋′| − 1. definition: we say that a set 𝑋𝑋′ (𝑋𝑋′ ⊂ 𝑋𝑋) directly goes behind the set 𝑋𝑋′′ (𝑋𝑋′′ ⊂ 𝑋𝑋) (this is denoted by 𝑋𝑋′ 𝜑𝜑 ← 𝑋𝑋′′), if 𝑋𝑋′, 𝑋𝑋′′ are compact and 𝑚𝑚𝑚𝑚𝑥𝑥 𝑥𝑥∈𝑋𝑋′ 𝜑𝜑(𝑥𝑥) = 𝑚𝑚𝑚𝑚𝑚𝑚 𝑥𝑥∈𝑋𝑋′′ 𝜑𝜑(𝑥𝑥) + 1. definition: we say that the sets 𝑋𝑋′, 𝑋𝑋′′ are independent of one another, if 𝜔𝜔(𝑋𝑋′, 𝑋𝑋′′) = 0. in the present paper the following lemma from [1] will play an essential role. lemma: if 𝜔𝜔(𝑋𝑋′, 𝑋𝑋′′) = 0, 𝑋𝑋′ 𝜑𝜑 ← 𝑋𝑋′′ and 𝜑𝜑 is a minimal layout, then . δ𝜑𝜑(𝑋𝑋′) ≤ δ𝜑𝜑(𝑋𝑋′′) let 𝜑𝜑 be some layout of g=(x,u) and 𝐺𝐺′ be an induced subgraph with vertex set 𝑋𝑋′ ⊂ 𝑋𝑋. let the vertices of 𝑋𝑋′ have the following numbers at the layout 𝜑𝜑: 𝑚𝑚1 < 𝑚𝑚2 < ⋯ < 𝑚𝑚�𝑋𝑋′�. consider the following layout 𝜑𝜑′: 𝜑𝜑′�𝜑𝜑−1(𝑚𝑚𝑖𝑖)� = 𝑚𝑚 �𝑚𝑚 = 1, |𝑋𝑋′|���������. definition: we say that a subgraph 𝐺𝐺′ is ordered minimally at 𝜑𝜑, if 𝜑𝜑′ is a minimal layout for 𝐺𝐺′. consider the graph p2,m,n with the vertex set 𝛱𝛱2,𝑚𝑚,𝑛𝑛 = �𝑥𝑥𝑖𝑖,𝑗𝑗,𝑘𝑘/ 𝑚𝑚 = 1,2����; 𝑗𝑗 = 1, 𝑚𝑚������; 𝑘𝑘 = 1, 𝑚𝑚������ and the edge set u, where �𝑥𝑥𝑖𝑖,𝑗𝑗,𝑘𝑘, 𝑥𝑥𝑖𝑖′,𝑗𝑗′,𝑘𝑘′� ∈ 𝑈𝑈 if and only if |𝑚𝑚 − 𝑚𝑚 ′| + |𝑗𝑗 − 𝑗𝑗′| + |𝑘𝑘 − 𝑘𝑘′| = 1. let’s denote 𝛱𝛱𝑖𝑖1,𝑗𝑗1,𝑘𝑘1 𝑖𝑖2,𝑗𝑗2,𝑘𝑘2=�𝑥𝑥𝑖𝑖,𝑗𝑗,𝑘𝑘 / 𝑚𝑚1 ≤ 𝑚𝑚 ≤ 𝑚𝑚2; 𝑗𝑗1 ≤ 𝑗𝑗 ≤ 𝑗𝑗2; 𝑘𝑘1 ≤ 𝑘𝑘 ≤ 𝑘𝑘2�, ω0 = �𝑥𝑥1,1,1, 𝑥𝑥1,𝑚𝑚,1, 𝑥𝑥1,1,𝑛𝑛, 𝑥𝑥1,𝑚𝑚,𝑛𝑛, 𝑥𝑥2,1,1, 𝑥𝑥2,𝑚𝑚,1, 𝑥𝑥2,1,𝑛𝑛, 𝑥𝑥2,𝑚𝑚,𝑛𝑛� where 1 ≤ 𝑚𝑚1 ≤ 𝑚𝑚2 ≤ 2; 1 ≤ 𝑗𝑗1 ≤ 𝑗𝑗2 ≤ 𝑚𝑚; 1 ≤ 𝑘𝑘1 ≤ 𝑘𝑘2 ≤ 𝑚𝑚. definition: we say that the set 𝑋𝑋′ ⊂ 𝛱𝛱2,𝑚𝑚,𝑛𝑛 is concise with respect to 𝑥𝑥1,1,1, if for every 𝑥𝑥𝑖𝑖,𝑗𝑗,𝑘𝑘 ∈ 𝑋𝑋′ we have 𝛱𝛱1,1,1 𝑖𝑖,𝑗𝑗,𝑘𝑘 ⊆ 𝑋𝑋′. definition: we say that a layout φ is concise with respect to 𝑥𝑥1,1,1, if for every 𝑘𝑘 ∈ 1,2𝑚𝑚𝑚𝑚�������� the set 𝑋𝑋𝜑𝜑𝑘𝑘 is concise with respect to 𝑥𝑥1,1,1. similarly one can define conciseness of sets and layouts with respect to other vertices from ω0. the following statements are valid. 1. if 𝜑𝜑 ∈ 𝜙𝜙p2,m,n 𝐸𝐸 , then for every 𝑘𝑘 ∈ 1,2𝑚𝑚𝑚𝑚�������� the set 𝑋𝑋𝜑𝜑𝑘𝑘 is concise with respect to at least one vertex from ω0. 2. for each vertex from ω0, there is a minimal, concise with respect to its layout. we will leave out the proofs of the above statements as they are very similar to analogous statements from [1] d. muradian 89 the following theorem is a main result of this paper. theorem: let 𝜑𝜑 be concise with respect to 𝑥𝑥1,1,1. then 𝜑𝜑 is minimal if and only if for each i,j (𝑚𝑚 ∈ 1, 𝑚𝑚�����; 𝑗𝑗 ∈ 1, 𝑚𝑚������) 𝑥𝑥1,𝑖𝑖,𝑗𝑗 𝜑𝜑 ← 𝑥𝑥2,𝑖𝑖,𝑗𝑗 and the subgraphs induced by the sets 𝛱𝛱1,1,1 1,𝑚𝑚,𝑛𝑛, 𝛱𝛱2,1,1 2,𝑚𝑚,𝑛𝑛 are ordered minimally at 𝜑𝜑. proof: only taking into consideration conciseness of 𝜑𝜑 with respect to 𝑥𝑥1,1,1, the set 𝛱𝛱2,𝑚𝑚,𝑛𝑛 is divided into subsets 𝛱𝛱𝑖𝑖 (regarding δ𝜑𝜑(𝑥𝑥)): 𝛱𝛱0 = �𝑥𝑥1,1,1� 𝛱𝛱1 = 𝛱𝛱1,2,1 1,𝑚𝑚−1,1 ∪ 𝛱𝛱1,1,2 1,1,𝑛𝑛−1; 𝛱𝛱2 = 𝛱𝛱1,2,2 1,𝑚𝑚−1,𝑛𝑛−1 ∪ �𝑥𝑥1,𝑚𝑚,1, 𝑥𝑥1,1,𝑛𝑛, 𝑥𝑥2,1,1�; 𝛱𝛱3 = 𝛱𝛱1,𝑚𝑚,2 1,𝑚𝑚,𝑛𝑛−1 ∪ 𝛱𝛱1,2,𝑛𝑛 1,𝑚𝑚−1,𝑛𝑛 ∪ 𝛱𝛱2,2,1 2,𝑚𝑚−1,1 ∪ 𝛱𝛱2,1,2 2,1,𝑛𝑛−1; 𝛱𝛱4 = 𝛱𝛱2,2,2 2,𝑚𝑚−1,𝑛𝑛−1 ∪ �𝑥𝑥1,𝑚𝑚,𝑛𝑛, 𝑥𝑥2,𝑚𝑚,1, 𝑥𝑥2,1,𝑛𝑛�; 𝛱𝛱5 = 𝛱𝛱2,𝑚𝑚,2 2,𝑚𝑚,𝑛𝑛−1 ∪ 𝛱𝛱2,2,𝑛𝑛 2,𝑚𝑚−1,𝑛𝑛; 𝛱𝛱6 = �𝑥𝑥2,𝑚𝑚,𝑛𝑛�. and δ𝜑𝜑(𝑥𝑥) = 3 − 𝑚𝑚 at 𝑥𝑥 ∈ 𝛱𝛱𝑖𝑖. at first we will prove that 𝑥𝑥1,1,1 𝜑𝜑 ← 𝑥𝑥2,1,1, i.e., φ(𝑥𝑥2,1,1)=2. let’s assume the reverse: 𝑥𝑥1,1,1 𝜑𝜑 ← 𝑆𝑆 𝜑𝜑 ← 𝑥𝑥2,1,1, and 𝑆𝑆 ≠ ∅. consider a case 𝑥𝑥1,𝑚𝑚,𝑛𝑛 ∉ 𝑆𝑆. we have δ𝜑𝜑(𝑆𝑆) ≤ δ𝜑𝜑�𝑥𝑥2,1,1� = 1 by the lemma. it is easy to see that for every set 𝑋𝑋′: δ𝜑𝜑(𝑋𝑋′) = 1 |𝑋𝑋′| � δ𝜑𝜑(𝑥𝑥). 𝑥𝑥∈𝑋𝑋′ as 𝜑𝜑 is concise with respect to 𝑥𝑥1,1,1, then from 𝑥𝑥1,𝑚𝑚,𝑖𝑖 ∈ 𝑆𝑆 follows 𝑥𝑥1,1,𝑖𝑖 ∈ 𝑆𝑆, where 𝑚𝑚 ∈ 2, 𝑚𝑚 − 1���������� (and from 𝑥𝑥1,𝑗𝑗,𝑛𝑛 ∈ 𝑆𝑆 follows 𝑥𝑥1,𝑗𝑗,1 ∈ 𝑆𝑆, where 𝑗𝑗 ∈ 2, 𝑚𝑚 − 1�����������). therefore, δ𝜑𝜑(𝑆𝑆) ≥ 1 and δ𝜑𝜑(𝑆𝑆) = 1 if and only if 𝑆𝑆 = 𝛱𝛱1,1,1 1,𝑚𝑚,𝑛𝑛\�𝑥𝑥1,1,1, 𝑥𝑥1,𝑚𝑚,𝑛𝑛�. let 𝑆𝑆 𝜑𝜑 ← 𝑅𝑅 𝜑𝜑 ← 𝑥𝑥1,𝑚𝑚,𝑛𝑛 (obviously 𝑥𝑥2,1,1 ∈ 𝑅𝑅). easy to see that 𝜔𝜔�𝑅𝑅, 𝑥𝑥1,𝑚𝑚,𝑛𝑛� = 0, δ𝜑𝜑�𝑥𝑥1,𝑚𝑚,𝑛𝑛� = −1, and from the conciseness of 𝜑𝜑 we have δ𝜑𝜑(𝑅𝑅) > −1, which contradicts the lemma. let’s now consider the case 𝑥𝑥1,𝑚𝑚,𝑛𝑛 ∈ 𝑆𝑆. then 𝛱𝛱1,1,1 1,𝑚𝑚,𝑛𝑛 𝜑𝜑 ← 𝛱𝛱2,1,1 2,𝑚𝑚,𝑛𝑛 and the subgraphs 𝐺𝐺1, 𝐺𝐺2 induced with them are ordered minimally at 𝜑𝜑. really, it is easy to see that for every ordering ψ, for which 𝛱𝛱1,1,1 1,𝑚𝑚,𝑛𝑛 𝜓𝜓 ← 𝛱𝛱2,1,1 2,𝑚𝑚,𝑛𝑛, we will have 𝐸𝐸𝜓𝜓(𝛱𝛱2,𝑚𝑚,𝑛𝑛) = 𝑚𝑚2𝑚𝑚2 + 𝐸𝐸𝜓𝜓1(𝐺𝐺1) + 𝐸𝐸𝜓𝜓2(𝐺𝐺2), where 𝜓𝜓1(𝑥𝑥) = 𝜓𝜓(𝑥𝑥) when 𝑥𝑥 ∈ 𝛱𝛱1,1,1 1,𝑚𝑚,𝑛𝑛 and 𝜓𝜓2(𝑥𝑥) = 𝜓𝜓(𝑥𝑥) − 𝑚𝑚𝑚𝑚 when 𝑥𝑥 ∈ 𝛱𝛱2,1,1 2,𝑚𝑚,𝑛𝑛. therefore, 𝜑𝜑 ∈ 𝜙𝜙p2,m,n 𝐸𝐸 if and only if 𝜓𝜓1 ∈ 𝜙𝜙𝐺𝐺1 𝐸𝐸 , 𝜓𝜓2 ∈ 𝜙𝜙𝐺𝐺2 𝐸𝐸 . so 𝐺𝐺1, 𝐺𝐺2 at 𝜑𝜑 are ordered minimally. then from [1] we will have the following. if m ≤ n, then a) at m > 4: 𝛱𝛱1,1,1 1,𝜆𝜆0,𝜆𝜆0 𝜑𝜑 ← 𝛱𝛱1,1,1 1,𝑚𝑚,𝑛𝑛\𝛱𝛱1,1,1 1,𝜆𝜆0,𝜆𝜆0 𝜑𝜑 ← 𝛱𝛱2,1,1 2,𝜆𝜆0,𝜆𝜆0, where 2 ≤ 𝜆𝜆0 < 1 2 m; b) at m < 4: 𝛱𝛱1,1,1 1,𝑚𝑚,1 𝜑𝜑 ← 𝛱𝛱1,1,1 1,𝑚𝑚,𝑛𝑛\𝛱𝛱1,1,1 1,𝑚𝑚,1 𝜑𝜑 ← 𝛱𝛱2,1,1 2,𝑚𝑚,1; c) at m = 4: the case а) or b) is happened. it is not difficult to compute: δ𝜑𝜑 �𝛱𝛱1,1,1 1,𝑚𝑚,𝑛𝑛\𝛱𝛱1,1,1 1,𝜆𝜆0,𝜆𝜆0� = 𝑚𝑚𝑚𝑚 − 𝜆𝜆02 − 2𝜆𝜆0 𝑚𝑚𝑚𝑚 − 𝜆𝜆0 2 = 1 − 2𝜆𝜆0 𝑚𝑚𝑚𝑚 − 𝜆𝜆0 2 > 0; linear orderings of tridimensional grids 90 δ𝜑𝜑 �𝛱𝛱2,1,1 2,𝜆𝜆0,𝜆𝜆0� = 2𝜆𝜆0 − 𝜆𝜆02 𝜆𝜆0 2 = 2 𝜆𝜆0 − 1 ≤ 0; δ𝜑𝜑�𝛱𝛱1,1,1 1,𝑚𝑚,𝑛𝑛\𝛱𝛱1,1,1 1,𝑚𝑚,1� = 𝑚𝑚(𝑚𝑚 − 1) − 𝑚𝑚 𝑚𝑚(𝑚𝑚 − 1) = 𝑚𝑚 − 2 𝑚𝑚 − 1 > 0; δ𝜑𝜑�𝛱𝛱2,1,1 2,𝑚𝑚,1� = 0. the last relations obviously contradict the lemma. therefore, 𝑥𝑥1,1,1 𝜑𝜑 ← 𝑥𝑥2,1,1. now let’s show, that 𝑥𝑥1,𝑖𝑖,𝑗𝑗 𝜑𝜑 ← 𝑥𝑥2,𝑖𝑖,𝑗𝑗 for each i,j (𝑚𝑚 ∈ 1, 𝑚𝑚�����; 𝑗𝑗 ∈ 1, 𝑚𝑚������). we will say that the vertices 𝑥𝑥1,𝑖𝑖,𝑗𝑗, 𝑥𝑥2,𝑖𝑖,𝑗𝑗 are neighbors. let’s assume the reverse. let z be a vertex with the smallest number, which does not directly goes behind its neighbor (denote the latter by y). so we have 𝑦𝑦 𝜑𝜑 ← 𝑆𝑆 𝜑𝜑 ← 𝑧𝑧; s ≠ ∅; δ𝜑𝜑(𝑦𝑦) = δ𝜑𝜑(𝑧𝑧) + 2. by the definition of z every vertex from 𝛱𝛱2,1,1 2,𝑚𝑚,𝑛𝑛⋂𝑆𝑆 directly goes behind its neighbor. let �𝛱𝛱2,1,1 2,𝑚𝑚,𝑛𝑛⋂𝑆𝑆 � = 𝑘𝑘 and 𝑦𝑦 𝜑𝜑 ← 𝑀𝑀1 𝜑𝜑 ← 𝑁𝑁1 𝜑𝜑 ← … 𝜑𝜑 ← 𝑀𝑀𝑘𝑘 𝜑𝜑 ← 𝑁𝑁𝑘𝑘 𝜑𝜑 ← 𝑀𝑀𝑘𝑘+1 𝜑𝜑 ← 𝑧𝑧, where 𝑁𝑁𝑖𝑖 − one pair of neighbors, and 𝑀𝑀𝑗𝑗 ⊂ 𝛱𝛱1,1,1 1,𝑚𝑚,𝑛𝑛. then as 𝜑𝜑 is concise, we have 𝜔𝜔(𝑆𝑆, 𝑧𝑧) = 0; 𝜔𝜔(𝑦𝑦, 𝑁𝑁𝑖𝑖) = 0; 𝜔𝜔( 𝑀𝑀𝑖𝑖, 𝑁𝑁𝑖𝑖) = 0, (1) at 1 ≤i ≤ j ≤ k. notice, that y and s cannot be independent of one another. otherwise, by the lemma we would have δ𝜑𝜑(𝑦𝑦) ≤ δ𝜑𝜑(𝑆𝑆) ≤ δ𝜑𝜑(𝑧𝑧) which would contradict the relation δ𝜑𝜑(𝑦𝑦) = δ𝜑𝜑(𝑧𝑧) + 2. therefore: ⋃𝑀𝑀𝑖𝑖≠ ∅. let’s show, that δ𝜑𝜑(𝑆𝑆) > −1. let’s assume the reverse: δ𝜑𝜑(𝑆𝑆) ≤ −1 . then, as δ𝜑𝜑(𝑁𝑁𝑖𝑖) ≥ −1 for every 𝑚𝑚 ∈ 1, 𝑘𝑘�����, then ⋃𝑀𝑀𝑖𝑖 consists of a unique vertex 𝑥𝑥1,𝑚𝑚,𝑛𝑛. since 𝜔𝜔(𝑦𝑦, 𝑆𝑆) ≠ 0 , then by (1) we will have 𝑦𝑦 ∈ 𝛱𝛱3. therefore, δ𝜑𝜑(𝑧𝑧) = −2. but from the conciseness of 𝜑𝜑 we can conclude, that 𝑥𝑥1,𝑚𝑚,𝑛𝑛 𝜑𝜑 ←𝑧𝑧 , which contradicts the lemma. from δ𝜑𝜑(𝑆𝑆) > −1 we have δ𝜑𝜑(𝑧𝑧) = 0 (δ𝜑𝜑(𝑦𝑦) = 2). notice, that δ𝜑𝜑(𝑁𝑁𝑖𝑖) takes values from {-1;0;1}. let’s assume, that δ𝜑𝜑(𝑁𝑁𝑖𝑖) ≥ 0 for each 𝑚𝑚 ∈ 1, 𝑘𝑘�����. then it is easy to see, that δ𝜑𝜑(𝑆𝑆) > 0, which is not possible by the lemma. therefore, there would be 𝑁𝑁𝑖𝑖, for which δ𝜑𝜑(𝑁𝑁𝑖𝑖) = −1. let 𝑁𝑁𝑝𝑝 be a pair with the smallest index from {𝑁𝑁𝑖𝑖}𝑖𝑖=1,𝑘𝑘���� , for which δ𝜑𝜑(𝑁𝑁𝑖𝑖) = −1. we have p ≥ 1 . we will prove by induction that 𝑀𝑀𝑖𝑖 = ∅ for all 𝑚𝑚 ∈ 2, 𝑝𝑝�����. really, 𝑀𝑀𝑝𝑝 = ∅ by the lemma and (1). let the sets 𝑀𝑀𝑖𝑖+1, 𝑀𝑀𝑖𝑖+2, … , 𝑀𝑀𝑝𝑝 be empty. we have 𝑀𝑀𝑖𝑖 𝜑𝜑 ← ⋃ 𝑁𝑁𝑗𝑗 𝑝𝑝 𝑗𝑗=𝑖𝑖+1 ; δ𝜑𝜑(𝑀𝑀𝑖𝑖)≥0; δ𝜑𝜑�⋃ 𝑁𝑁𝑗𝑗 𝑝𝑝 𝑗𝑗=𝑖𝑖+1 � < 0, and by the lemma we will have 𝑀𝑀𝑖𝑖 = ∅ . then 𝑦𝑦 𝜑𝜑 ← 𝑀𝑀1 𝜑𝜑 ←⋃ 𝑁𝑁𝑗𝑗 𝑝𝑝 𝑗𝑗=1 . but δ𝜑𝜑(𝑦𝑦⋃𝑀𝑀1) > 0 , δ𝜑𝜑�⋃ 𝑁𝑁𝑗𝑗 𝑝𝑝 𝑗𝑗=1 � < 0, which contradicts the lemma. thus, we obtained that 𝑥𝑥1,𝑖𝑖,𝑗𝑗 𝜑𝜑 ← 𝑥𝑥2,𝑖𝑖,𝑗𝑗 for each i,j (𝑚𝑚 ∈ 1, 𝑚𝑚�����; 𝑗𝑗 ∈ 1, 𝑚𝑚������), i.e., vertices of 𝐺𝐺1 got odd numbers, while the vertices of 𝐺𝐺2 – even numbers. let’s define layouts 𝜑𝜑1 and 𝜑𝜑2 for the graphs 𝐺𝐺1, 𝐺𝐺2: 𝜑𝜑1�𝜑𝜑−1(2𝑘𝑘 − 1)� = 𝑘𝑘; 𝜑𝜑2�𝜑𝜑−1(2𝑘𝑘)� = 𝑘𝑘; for each ∈ 1, 𝑚𝑚𝑚𝑚������� . then it is easy to see, that 𝐸𝐸𝜑𝜑(𝑃𝑃2,𝑚𝑚,𝑛𝑛) = 𝑚𝑚𝑚𝑚 + 2𝐸𝐸𝜑𝜑1(𝐺𝐺1) + 2𝐸𝐸𝜑𝜑2(𝐺𝐺2). (2) therefore, 𝜑𝜑 is minimal if and only if 𝜑𝜑 ∈ 𝜙𝜙𝐺𝐺𝑖𝑖 𝐸𝐸 (𝑚𝑚 ∈ 1,2����). this proves the theorem. substituting the formula of 𝐸𝐸(𝐺𝐺𝑖𝑖) from [1] into (2), at m ≤ n we will have: d. muradian 91 𝐸𝐸𝜑𝜑(𝑃𝑃2,𝑚𝑚,𝑛𝑛) = 𝑚𝑚𝑚𝑚 + 4 �− 2 3 𝜆𝜆0 3 + 2𝑚𝑚𝜆𝜆02 − �𝑚𝑚2 + 𝑚𝑚 − 2 3 �𝜆𝜆0 + 𝑚𝑚(𝑚𝑚2 + 𝑚𝑚 − 1) − 𝑚𝑚�, where �𝑚𝑚 − �𝑚𝑚 2 2 − 𝑚𝑚 2 + 1 4 � ≤ 𝜆𝜆0 ≤ �𝑚𝑚 + 1 2 − �𝑚𝑚 2 2 − 𝑚𝑚 2 + 1 4 �. references 1. d. o. muradian and t. e. piliposyan, “minimal numberings of vertices of a rectangular lattice”, akad. nauk. armjan.ssr 1, in russian, vol.70, pp. 21-27, 1980. submitted 07.07.2016, accepted 26.10.2016. գծային համարակալումներ եռաչափ ցանցերի համար դ. մուրադյան ամփոփում գրաֆի մինիմալ համարակալում գտնելու խնդիրը սահմանվում է հետևյալ կերպ: պահանջվում է գտնել տրված գրաֆի գագաթների այնպիսի տեղաբաշխում թվային առանցքի վրա, որ էջերի երկարությունների գումարը լինի նվազագույն, որտեղ էջի երկարությունը նրան կից գագաթների համարների տարբերության բացարձակ արժեքն է: այս աշխատանքում նկարագրվում է մինիմալ համարակալումների դասը եռաչափ ցանցերի մի մասնավոր դեպքի՝ երեք պարզ շղթաների դեկարտյան արտադրյալի համար, որոնցից մեկն ունի երկու գագաթ: линейные нумерации трехмерных решеток д. мурадян аннотация в работе описывается класс минимальных по длине нумераций частного случая трехмерных решеток – декартого произведения трех простых цепей, когда один из них состоит из двух вершин. минимальная длина нумерациия графа определяется следующим образом: для данного графа g требуется найти такую линейную нумерацию его вершин, чтобы сумма длин ребер (абсолютное число разности номеров инцидентных ей вершин) была минимальна относительно всевозможных нумераций графа. 1. introduction references начиная с начала 2000 года осуществляется внедрение ghis в здравоохранении, в рамках принятого проекта о реформирование информ mathematical problems of computer science 48, 105--111, 2017. further results for encoding and decoding procedures of asymmetric low magnitude error correcting codes hamlet k. khachatrian institute for informatics and automation problems of nas ra e-mail: hamletkh@ipia.sci.am abstract in this paper an implementation of encoding and decoding procedures for double ±1 error correcting optimal linear codes over rings 𝑍𝑍7 and 𝑍𝑍9 is presented. keywords: error correcting codes, asymmetric low magnitude error correcting codes, encoding and decoding procedures. 1. introduction codes over finite rings, particularly over integer residue rings and their applications in coding theory, have been studied for a long time. errors happening in the channel are basically asymmetrical; they also have a limited magnitude, and this effect is particularly applicable to flash memories. there have been a couple of papers regarding the optimal ±1 single error correcting codes over the alphabet 𝑍𝑍𝑚𝑚 [1, 2]. also there are some papers regarding the construction of optimal double ±1 error correcting codes [3, 4]. here, we propose to construct encoding and decoding algorithms for the optimal codes correcting double ±1 errors. in [5] you can see the construction of encoding and decoding procedures for the optimal linear code (12, 8) over ring 𝑍𝑍5, which was given by parity check matrix 𝐻𝐻5: 𝐻𝐻5 = � 1 1 1 1 1 0 1 2 3 4 1 1 0 1 2 3 4 2 2 2 2 2 1 1 3 2 4 4 2 3 2 4 4 2 1 1 1 1 1 1 1 3 2 4 4 2 0 4 �. in this case the number of combinations for each code word that can be corrected is: (1 + 12 ∗ 2 + (12 𝑐𝑐ℎ𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜 2) ∗ 4) = 289. 105 further results for encoding and decoding procedures of asymmetric low error correcting codes 106 implementation of codes over large alphabets is much more difficult compared with small alphabets. in this paper we construct encoding and decoding procedures for the codes (16, 12) and (20, 16) over rings 𝑍𝑍7 and 𝑍𝑍9, which are developed in [4]. using this codes we can correct consequently 512 and 800 errors of type ±1 in any vectors from 𝑍𝑍7 and 𝑍𝑍9 with lengths 12 and 16 by adding only 4 parity check symbols. 2. presentation of the codes (16, 12) and (20, 16) over rings 𝒁𝒁𝟕𝟕 and 𝒁𝒁𝟗𝟗 in [4] you can see the construction of optimal linear codes over rings z7 and z9 correcting double ±1 errors. 2.1 code (16, 12) over ring 𝒁𝒁𝟕𝟕 let a linear (16, 12) code over ring z7 be given by the following parity check matrix 𝐻𝐻7: 𝐻𝐻7 = � 1 1 1 1 1 1 1 0 1 2 3 4 5 6 1 1 6 5 4 3 2 1 0 2 2 2 2 2 2 2 1 1 4 3 6 6 3 4 2 4 3 6 6 3 4 2 1 6 1 1 1 1 1 1 1 4 3 6 6 3 4 2 0 0 � . a linear code over ring z7 , with 12 information and 4 parity check symbols, given by the parity check matrix 𝐻𝐻7 can correct up to two errors of the type ±1, because 𝐻𝐻7 has a property according to which all the syndromes resulting from adding and subtracting operations between any two columns of the matrix 𝐻𝐻7 are different (±hi ± hj and hi ≠ hj). this code is optimal in the sense that it has a minimal possible number of parity check symbols. in this case the number of combinations for each code word that can be corrected is: 16 ∗ 2 + (16 𝑐𝑐ℎ𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜 2) ∗ 4 = 512. 2.2 code (20, 16) over ring 𝒁𝒁𝟗𝟗 the parity check matrix 𝐻𝐻9 for an optimal linear code (20, 16) correcting double errors of the type ±1 over ring z9 has the following form: 𝐻𝐻9 = � 1 1 1 1 1 1 1 1 8 7 6 5 4 3 2 1 1 1 2 4 7 6 5 4 3 2 1 0 2 2 2 2 2 2 2 2 1 1 2 4 7 3 2 4 4 2 3 7 7 3 2 4 4 2 3 7 1 1 2 4 1 1 1 1 1 1 1 1 7 3 2 4 4 2 3 7 6 3 7 2 �. a linear code over ring z9 , with 16 information and 4 parity check symbols, given by the parity check matrix 𝐻𝐻9 can correct up to two errors of the type ±1. this code is optimal too in the sense that it has a minimal possible number of parity check symbols. in this case the number of combinations for each code word that can be corrected is: ( 20 ∗ 2 + (20 𝑐𝑐ℎ𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜 2) ∗ 4) = 800 . in the next chapter we will construct encoding and decoding procedures for these two optimal linear codes. h. khachatrian 107 3. encoding and decoding procedures 3.1 code (16, 12) for encoding every message in z7 we must have the generator matrix 𝐺𝐺7. for this we should construct a combinatorial equivalent matrix 𝐻𝐻′7 from parity check matrix 𝐻𝐻7 of the code (16, 12): 𝐻𝐻′7 = � 1 0 0 0 5 2 5 1 5 2 5 0 1 1 6 1 0 1 0 0 2 1 5 5 0 6 4 1 4 6 0 4 0 0 1 0 0 0 0 0 4 1 6 5 5 6 5 5 0 0 0 1 1 5 5 2 3 1 1 3 0 6 2 3 �. here all syndromes will be different, too. we know the theorem, which says, that if h′ = [−pt|in−k], then g = [ik|p] (the reverse statement is also true), where ik is a 𝑘𝑘 ∗ 𝑘𝑘 identity matrix and p is a 𝑘𝑘 ∗ (𝑛𝑛 − 𝑘𝑘) matrix, 𝐺𝐺𝐻𝐻′𝑇𝑇 = 0. (1) thus, we can construct the generator matrix 𝐺𝐺7: 𝐺𝐺7 = ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎡ 2 5 0 6 1 0 0 0 0 0 0 0 0 0 0 0 5 6 0 2 0 1 0 0 0 0 0 0 0 0 0 0 2 2 0 2 0 0 1 0 0 0 0 0 0 0 0 0 6 2 0 5 0 0 0 1 0 0 0 0 0 0 0 0 2 0 3 4 0 0 0 0 1 0 0 0 0 0 0 0 5 1 6 6 0 0 0 0 0 1 0 0 0 0 0 0 2 3 1 6 0 0 0 0 0 0 1 0 0 0 0 0 0 6 2 4 0 0 0 0 0 0 0 1 0 0 0 0 6 3 2 0 0 0 0 0 0 0 0 0 1 0 0 0 6 1 1 1 0 0 0 0 0 0 0 0 0 1 0 0 1 0 2 5 0 0 0 0 0 0 0 0 0 0 1 0 6 3 2 4 0 0 0 0 0 0 0 0 0 0 0 1⎦ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎤ . encoding procedure: in our scheme the message was presented by 12-tuples in z7. 𝑣𝑣 = (𝑎𝑎1, 𝑎𝑎2, 𝑎𝑎3, … , 𝑎𝑎12) is an arbitrary 12-tuple, and consider the vector 𝑢𝑢 that is the linear combination of columns 𝐺𝐺7 with 𝑎𝑎𝑖𝑖 is the 𝑖𝑖𝑡𝑡ℎ coefficient. 𝑢𝑢 = 𝑣𝑣𝐺𝐺 = (𝑐𝑐1, 𝑐𝑐2, 𝑐𝑐3, 𝑐𝑐4, 𝑎𝑎1, 𝑎𝑎2, 𝑎𝑎3, … , 𝑎𝑎12), where the first 4 components of the code vector are the parity check symbols and the next 12 components are information symbols, where 𝑐𝑐𝑗𝑗 = �∑ 𝑎𝑎𝑖𝑖𝑝𝑝𝑖𝑖𝑗𝑗 𝑘𝑘 𝑖𝑖=1 �𝑚𝑚𝑜𝑜𝑚𝑚7. (2) let us show the example to describe how we do these procedures. further results for encoding and decoding procedures of asymmetric low error correcting codes 108 example. let (0 1 2 6 4 0 6 5 4 1 2 2) be the message vector in z7. from (2) we can obtain parity check symbols by multiplying this message vector with the columns of the matrix 𝐺𝐺7. for example, the first parity check symbol is c1: 𝑐𝑐1 = (0 ∗ 2) + (1 ∗ 5) + (2 ∗ 2) + (6 ∗ 6) + (4 ∗ 2) + (0 ∗ 5) + (6 ∗ 2) + (5 ∗ 0) + (4 ∗ 6) + (1 ∗ 6) + (2 ∗ 1) + (2 ∗ 6) = 0 + 5 + 4 + 1 + 1 + 0 + 5 + 0 + 3 + 6 + 2 + 5 = 4(𝑚𝑚𝑜𝑜𝑚𝑚7). (all operations are in z7.) similarly, we can find other 3 parity check symbols: 𝑐𝑐2 = 5, 𝑐𝑐3 = 3, 𝑐𝑐4 = 1. after performing other multiple operations with matrix 𝐺𝐺7 we obtain this encoded vector: (4 5 3 1 0 1 2 6 4 0 6 5 4 1 2 2). as we can see in this code, the encoded message (codeword) has the length 16, from which the first 4 are parity check symbols, and the last 12 are information symbols. decoding procedure: now we will show how a decoding procedure will be implemented using the parity check matrix 𝐻𝐻′7, if during the message sending process the errors occured in the codewords. we will describe the decoding procedure by 3 steps: 1. receiver multiplies the vector with every row of matrix h′7 and gets the syndrome s = 𝐯𝐯h′. if s = (0,0,0,0) then there were not any errors in the received vector. 2. if the calculated syndrome s is a nonzero vector, then there are some occurred errors. these codes can correct only up to two errors with magnitude ±1. we know that all possible syndromes of matrix h′7 are different (±hi ± hj and hi ≠ hj). after calculating the syndrome the receiver knows from which two columns of the matrix h′7 the syndrome was resulted, consequently, it can find the two corresponding components of the vector, where the error was occurred and the direction of the error (if +hi, then upward direction or if −hi downward direction). on the other hand, if in the table of syndromes we do not have the resulted syndrome, then we cannot correct this kind of errors. 3. after finding the error components the receiver adds or subtracts 1 from them and obtains the sent code vector (c1, c2, c3, c4, a1, a2, a3, … , a12). so (a1, a2, a3, … , a12) is our message vector. an example. (4 5 3 1 0 1 2 6 4 0 6 5 4 1 2 2) is an encoded vector from the previous example. let 2 errors occur in the channel, and the receiver gets the vector (4 5 2 1 0 1 2 6 4 0 6 5 4 1 2 1). after performing multiple operations with rows of matrix 𝐻𝐻′7 the receiver obtains the syndrome(6 3 1 4). next from the table of syndromes it finds the corresponding columns, now they are 3 and 16. hence, the syndrome (6 3 1 4) was resulted from adding a negated column 3 of matrix 𝐻𝐻′7 to the negated column 16: 0 −1 0 −4 −1 −5 0 −3 = −1 −4 −6 −3 (𝑚𝑚𝑜𝑜𝑚𝑚7) = (6 3 1 4) (because in 𝑍𝑍7 0 = 7, −1 = 6, −2 = 5 , −3 = 4, −5 = 2, −6 = 1). hence, the error positions of encoded vector are 3 and 16 (both have a downward direction). h. khachatrian 109 so, it adds 1 to 𝑡𝑡ℎ𝑜𝑜 3rd component and 1 to 16𝑡𝑡ℎ of vector (4 5 𝟐𝟐 1 0 1 2 6 4 0 6 5 4 1 2 𝟏𝟏) and obtains the sent encoded vector (4 5 3 1 0 1 2 6 4 0 6 5 4 1 2 2). consequently, the message vector (code word) is (0 1 2 6 4 0 6 5 4 1 2 2) as we have in the example of the encoding procedure. using this code we can find and correct all possible 512 errors of the type ±1 in every vector over ring 𝑍𝑍7. 3.2 encoding and decoding for the code (20, 16) for the code (20, 16) over the ring z9 correcting double errors of the type ±1 we can do the same encoding and decoding processes as we did for the code (16, 12). in this case, the parity check matrix 𝐻𝐻′9 and the generator matrix 𝐺𝐺9 will have the following form: 𝐻𝐻′9 = � 1 0 0 0 6 7 8 5 0 6 6 0 7 3 5 4 7 4 7 4 0 1 0 0 0 6 0 2 4 7 1 4 1 1 1 1 7 4 3 3 0 0 1 0 6 4 5 6 6 3 0 6 3 5 6 3 7 1 4 5 0 0 0 1 1 4 1 5 2 1 6 8 6 3 0 6 4 1 7 0 �, 𝐺𝐺9= ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎡ 3 0 3 8 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 3 5 5 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 4 8 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 4 7 3 4 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 5 3 7 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 3 2 6 8 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 3 8 0 3 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 5 3 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 2 8 6 3 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 6 8 4 6 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 4 8 3 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 5 8 6 3 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 2 2 2 5 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 5 5 8 8 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 2 6 5 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 5 6 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1⎦ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎤ unlike the previous case for the code (16, 12) over ring 𝑍𝑍7, in this case the message was presented by 16-tuples in 𝑍𝑍9. the encoded vector 𝑢𝑢 (codeword) has a length 20: 𝑢𝑢 = 𝑣𝑣𝐺𝐺 = (𝑐𝑐1, 𝑐𝑐2, 𝑐𝑐3, 𝑐𝑐4, 𝑎𝑎1, 𝑎𝑎2, 𝑎𝑎3, … , 𝑎𝑎16), where the first four are the parity check symbols: 𝑐𝑐𝑗𝑗 = �∑ 𝑎𝑎𝑖𝑖𝑝𝑝𝑖𝑖𝑗𝑗 𝑘𝑘 𝑖𝑖=1 �𝑚𝑚𝑜𝑜𝑚𝑚9 and the next 16 are information symbols. using this code we can find and correct all possible 800 errors of the type ±1 in every vector over ring 𝑍𝑍9. further results for encoding and decoding procedures of asymmetric low error correcting codes 110 4. conclusion in this paper an implementation of encoding and decoding procedures of optimal (16, 12) and (20, 16) linear codes over ring 𝑍𝑍7 and 𝑍𝑍9 correcting double ±1 errors is presented. we propose that this approach can be extended for implementation of similar procedures for the optimal codes over other rings 𝑍𝑍𝑛𝑛 and the research in this direction will follow. references [1] s. martirossian, “single error correcting close packed and perfect codes”, proc.1st intas int. seminar coding theory and combinatorics, armenia, pp. 90-115, 1996. [2] h. kostadinov, n.manev and h.morita, “on ±1 error correctable codes”, ieice trans.fundamentals, vol. e93-a, pp. 2578-2761, 2010. [3] g. khachatrian and h. morita, “construction of optimal 1 double error correcting linear codes over ring z5 ”, 3rd international workshop on advances in communications, boppard, germany, pp. 10-12, may 2014. [4] g. khachatryan and h. khachatryan, “construction of double ±1 error correcting linear optimal codes over rings 𝑍𝑍7 and 𝑍𝑍9” , mathematical problems of computer science, vol. 45, pp. 106—110, 2016. [5] h. khachatryan, “encoding and decoding procedures for double ±1 error correcting linear code over ring 𝑍𝑍5”, mathematical problems of computer science, vol. 43, pp. 57—61, 2015. submitted 17.09.2017, accepted 04.12.2017. այլ արդյունքներ ասիմետրիկ փոքր մեծությամբ սխալներ ուղղող կոդերով կոդավորման և ապակոդավորման ալգորիթմների համար հ. խաչատրյան ամփոփում այս հոդվածի շրջանակներում ներկայացված են կոդավորման և ապակոդավորման ալգորիթմները 𝑍𝑍7 և 𝑍𝑍9 օղակներում կառուցված ասիմետրիկ փոքր ամպլիտուդայով սխալներ ուղղող կոդերի համար: h. khachatrian 111 алгоритмы кодирования и декодирования для кодов исправляющих асимметричные двойные ошибки г. хачатрян аннотация в данной статье представлены алгоритмы кодирования и декодирования для кодов в кольцах 𝑍𝑍7 и 𝑍𝑍9 исправляющих двойные асимметричные ошибки. hamlet k. khachatrian начиная с начала 2000 года осуществляется внедрение ghis в здравоохранении, в рамках принятого проекта о реформирование информ mathematical problems of computer science 48, 64--73, 2017. effective and accurate binary clone detection hayk k. aslanyan ivannikov institute for system programming of the ras e-mail: hayk@ispras.ru abstract software developers usually copy and paste a particular piece of code as they prefer to use a pre-written or a partial solution as a basis for solving their problem. however, it can lead to various errors, as well as increase the size of the source and binary code. finding similar parts of code (clones) in binary code becomes more applicable when the source code is not available. additionally, a compiler can copy some parts of code during various transformations and create code clones, which do not exist in the source code. detection of binary code clones is used for malware analysis, finding semantic errors, detecting copyright violation, etc. this article discusses the existing methods of binary code clones detection and introduces a new method for binary clone detection. it consists of three main stages. the first stage is based on binnavi platform [1] and generates program dependence graphs for each binary function. graphs are generated based on reil [2] (reverse engineering intermediate language) platform-independent language. reil representation is supported for several architectures (x86, x86-64, arm, mips, ppc), thus ensuring the independence of the tool from the target architecture. the second stage detects clones based on previously created graphs. a polynomial heuristic algorithm is suggested for finding the maximum common subgraph of two program dependence graphs. at the third stage, the obtained clones are visualized for manual analysis. keywords: binary code clone, program static analysis, program dependence graph. 1. introduction the reuse of code fragments (a continuous sequence of code lines) is often encountered in software development process. there are several approaches for finding code clones, which are based on text [3], lexical [4], syntactic [5]—[7] and semantic [8]—[14] analysis of the program. however, all these methods analyze source code and the task of finding clones in binary code is less studied. these approaches are not applicable for binary code as binary code is platform dependent and registers and direct memory access are used, while in source code only variables are considered. detection of binary code clones has many practical applications such as finding functionally similar parts of the program, malware, semantic errors and copyright violations. binary code clones are divided into three main types. the first type of binary code clones are code fragments that completely match. the second type of binary code clones are code fragments, 64 h. aslanyan 65 which can differ in types, values, names of data and registers. the third type of binary code clones are code fragments that can differ in types, names of data and registers, and may also differ in some instructions (in a particular fragment some instructions may be added or removed). examples of assembly code clones (for x86 architecture) are shown in fig.1. the clone of the first type is exactly matched fragments. the clone of the second type uses ecx register instead of eax. the clone of the third type uses ecx register instead of eax and has one deleted instruction (imul eax, ebp+var_4]). fig.1. examples of code clones (x86 assembler). 2. binary code clone detection approaches text-based approach. jang et al. [15] proposed a fingerprinting algorithm called bitshred based on bloom filters to cluster malware samples. bitshred consists of three phases: shredding a file, creating a fingerprint, and comparing fingerprints. in the shredding phase, bitshred divides all executable code sections into fragments. then for each fragment fingerprint is calculated based on bloom filters [16]. at the last stage it compares fingerprints by the following ratio: j(a,b) = s(bfa∧bfb) / s(bfa∨bfb), where s(bf) is the count of set bits in the bf. finally, fragments, having a higher similarity score, are clustered together. the algorithm finds clones of only the first type. token-based approach. a. schulman [17] proposed a system that creates a hash for each function in a binary file. matched hashes that occur in more than one file indicate a clone of the code. hashes are based on opcodes and location labels of the opcodes. karim et al. [18] algorithm also considers hashes on instructions opcodes, but they are calculated using n-grams. the algorithms of the approach allow finding clones of the first and second types. metrics-based approach. the system, created by d. bruschi et al. [19], finds clones in binary files to detect malicious programs. first, the binary file is disassembled and normalized, then the dead code is deleted and the code is split into fragments. for each fragment, a metric is constructed based on the control flow graph. at the last stage, clones are detected by comparing the obtained metrics. effective and accurate binary clone detection 66 sæbjørnsen et al. [20] after obtaining the assembler from the binary file, create intermediate representations of the assembly code. then, a binary code is partitioned into overlapping code segments that consist of a block of contiguous assembly from a function. then they create a normalized instruction sequence, abstracting the information of memory location and registers. next, it performs clone detection on the normalized sequence. they define two methods for creating clone clusters. the first method is an exact match that uses a hash for each code region, and a clone exists if there are any repeated hash values. the second method is an inexact match, which extracts a set of features from a code region and looks for other code regions with the same feature set. they count the number of occurrences of each feature to create a feature vector for each region. next, they use locality-sensitive hashing (lsh) (andoni and indyk, [21]) on each region and perform a distance calculation for clustering based on features for inexact matching. based on this work, m. farhadi et al. [22] created a system for detecting clones of malicious code in programs. algorithms of the approach detect all three types of clones. structural-based approach. t. dullien et al. [23] proposed a system for comparing binary files based on structural analysis, to search for malicious code. the algorithm consists of two stages: the generation of several hashes of malicious code and the recognition of similarity between different sections of code based on the control flow graph. y. david and e.yahav [24] detect similarity among functions based on decomposition of functions into tracelets, which are continuous, partial traces of execution and are obtained from control flow graph. to measure similarity between two tracelets, they define a set of simple rewrite rules and measure how many rewrites are required to reach from one tracelet to another. they do this step by encoding the problem as a constraint-solving problem and measure distance using the number of constraints that have to be violated to reach a match. algorithms of the approach detect all three types of clones. behavior-based approach. in [25] d.e. krutz and e.shahab propose a new approach for code clone detecting, which also detects semantic clones. they use concolic analysis, which combines concrete and symbolic values in order to traverse all possible paths. 3. tool architecture for binary code clone detection the proposed model takes into account the following requirements:  finding all types of clones;  independence from the target architecture;  scalability: the size of the analyzed programs can reach hundreds of mb;  a large percentage of true positives (> 90%). it is allowed to specify values of two variables: the minimum number of instructions for clones (mn) and the minimum percentage of similarity (mp) of clones. the work of the tool is divided into three main stages: 1. the first stage is based on ida pro disassembler [26] and binnavi static analysis platform [1]. the ida pro disassembler is used as a tool for restoring structures and the control flow of the program. at this stage, machine code is translated to reil representation, then, pdg (program dependence graph) is generated for each function. 2. at the second stage, the assembler code clones are detected taking into account the parameters of the mn and mp. 3. at the third stage code clones and their corresponding pdgs are visualized. h. aslanyan 67 the main advantage of the proposed tool is that it is based on a semantic approach, which is more correct, than other approaches. 3.1 generation of pdgs binnavi platform is used to generate program dependence graphs for each function. binnavi provides an interface for generating and using various intermediate representations of the program based on reil, including the generation of the control flow graph, the generation of the call graph and data dependency graph. as a part of the tool, a new functionality has been added to the binnavi platform, which allows each function to automatically generate a control flow graph and a data flow graph and merge them into program dependence graph (fig. 2). the vertices of pdg correspond to reil instructions, and the edges are data and control dependencies between instructions. each vertex has id, which is the opcode of its reil instruction. fig. 2. example of pdg. 3.2 heuristic algorithm for maximum common subgraph detection of two pdgs at this stage a maximum common subgraph is calculated for each pair of pdgs with heuristic algorithm. this algorithm is named tracebasedslice. it uses several procedures, which are defined below. let 𝐺𝐺(𝑉𝑉, 𝐸𝐸) be a pdg and 𝑋𝑋 and 𝑌𝑌 be any sets of pdg vertices, such that 𝑋𝑋 ⊆ 𝑉𝑉. two vertices can be matching candidates, if the first vertex’s id, predecessors’ and successors’ amounts are equal to the second vertex’s id, predecessors’ and successors’ amount. definition 1: getpredecessors(x,y) procedure returns empty set if y is empty, otherwise returns vertices from x, which are not in y and are predecessors for y’s vertices. effective and accurate binary clone detection 68 definition 2: getsuccessors(x,y) procedure returns empty set if y is empty, otherwise returns vertices from x, which are not in y and are successors for y’s vertices. definition 3: sortvertices(x) procedure sorts vertices of x by their ids, predecessors count, successors count and binary address. definition 4: makecorrespondence (x, y) procedure returns pairs of vertices from sorted x and y sets, which are matching candidates. it considers vertices’ ids, predecessors, successors count and based on merging algorithm. definition 5: makeonecorrespondence(x, y) procedure returns a pair of vertices from sorted x and y sets, which are matching candidates. definition 6: for any m∈x and n∈y checkpredecessors(x,y,m,n) condition is satisfied, if predecessors of m from x and predecessors of n from y have the same set of ids. definition 7: for any m∈x and n∈y checksuccessors(x,y,m,n) condition is satisfied, if successors of m from x and successors of n from y have the same set of ids. definition 8: inducedsubgraph(x, g) procedure returns induces subgraph of x in g. procedure tracebasedslice input: pair of pdgs g1 (v1, e1), g2 (v2, e2) output: maximum common subgraph of g1 and g2 1. matchednodes1 ⊆ 𝑉𝑉1, matchednodes2 ⊆ 𝑉𝑉2 2. matchednodes1 ← ∅, matchednodes2 ← ∅ 3. nopredecessor1←{n∈v1:n hasn't predecessor} 4. nopredecessor2←{n∈v2:n hasn't predecessor} 5. continuematching ← true 6. while (continuematching) 7. continuematching←false 8. tempmatching⊆ 𝑉𝑉1 × 𝑉𝑉2 9. tempmatching← ∅ 10. sortedneighbours1← sortvertices (getpredecessors (v1, matchednodes1)) 11. sortedneighbours2← sortvertices (getpredecessors (v2, matchednodes2)) 12. tempmatching←makecorrespondence(sortedneighbours1, sortedneighbours2) 13. sortedneighbours1← sortvertices (getsuccessors (v1, matchednodes1)) 14. sortedneighbours2← sortvertices (getsuccessors (v2, matchednodes2)) 15. tempmatching←tempmatching ∪ makecorrespondence(sortedneighbours1, sortedneighbours2) 16. if tempmatching is empty 17. tempmatching←makeonecorrespondence (nopredecessor1, nopredecessor2) 18. if tempmatching is not empty 19. continuematching←true 20. for all (v1, v2) ∈ tempmatching 21. if checkpredecessors(matchednodes1, matchednodes2, v1, v2) and checksuccessors(matchednodes1, matchednodes2, v1, v2) 22. matchednodes1 ←matchednodes1 ∪ {v1} 23. matchednodes2←matchednodes2 ∪ {v2} 24. return inducedsubgraph(matchednodes1, g1), inducedsubgraph(matchednodes2, g2) h. aslanyan 69 after detecting the maximum common subgraph, the tracebasedslice procedure returns the common part of two functions. if it satisfies mn and mp, then the obtained results are saved and visualized. 3.3 visualization of binary code clones the last stage of the tool is visualization of clones. the purpose of the created graphical interface is to demonstrate the assembler code of the obtained clones, the percentage of their similarity, the corresponding graphs and maximum common subgraph (fig. 3). fig. 3. visualization of detected clones 4. results to assess the effectiveness of the tool, it is tested on various real projects. two successive versions are analyzed for each project. the tool analyzes the pairs of functions, which have the same name in old and new versions of projects. as the same function most likely will change at a few instructions in new version, then they should be clones. if the function is not changed, then they should be clones of the second type. the target machine for testing is intel i5, 4 cores, ram 16 gb. the results on several programs are shown in table 1. effective and accurate binary clone detection 70 table 1: results project name version binaries sizes (mb) functions count with the same name analyze time detected clones count (mp = 50%) detected clones count (mp = 90%) old new old new grep 3.0 3.1 0.74 0.74 308 10s 307 299 gdb 8.0 8.1 49 66 12707 6m 14s 11894 10683 findutils 4.4.1 4.4.2 1.1 1.1 484 12s 484 482 gcc 4.9.0 5.4.0 3.2 3.4 1041 50s 956 662 git 2.6.0 2.9.5 9.4 9.8 3257 1m 15s 3098 2627 bison 2.3 2.4 1.3 1.5 498 19s 498 497 5. conclusion in this paper, the main 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[25] d. e. krutz and e. shihab, "cccd: concolic code clone detection," wcre, 2013. d. e. krutz and e. shihab, "cccd: concolic code clone detection," 2013 20th working conference on reverse engineering, koblenz, 2013, pp. 489-490. [26] https://www.hex-rays.com/products/ida. effective and accurate binary clone detection 72 [27] h. k. aslanyan, s. f. kurmangaleev, v. g. vardanyan, m. s. arutunian and s. s. sargsyan, "platform-independent and scalable tool for binary code clone detection," trudy ispran/proc. isp ras, vol. 1, no. 2, pp. 215-226, 2016. submitted 02.08.2017, accepted 23.11.2017. երկուական կոդի կլոնների արդյունավետ և ճշգրիտ որոնում հ․ ասլանյան ամփոփում ծրագրային ապահովման մշակողները հաճախ պատճենում և տեղադրում են որոշակի կոդ, քանի որ նրանք նախընտրում են օգտագործել նախապես եղած լուծումը կամ մասնակի լուծումը որպես այլ խնդրի լուծման հիմք: այնուամենայնիվ, դա կարող է հանգեցնել տարբեր սխալների, ինչպես նաև նախնական և երկուական կոդերի մեծացմանը: կոդի նմանատիպ մասերը (կլոններ) երկուական կոդում գտնելու խնդիրը դառնում է ավելի կիրառելի, երբ նախնական կոդը հասանելի չէ: բացի այդ, տարբեր ձևափոխությունների ընթացքում կոմպիլյատորը կարող է պատճենել կոդի որոշ մասեր և ստեղծել կոդի կլոններ, որոնք գոյություն չունեն նախնական կոդում: երկուական կոդի կլոնների հայտնաբերումը կարող է օգտագործվել վնասակար կոդի, սիմվոլիկ սխալների, հեղինակային իրավունքի խախտումների հայտնաբերման համար: հոդվածում քննարկվում է երկուական կոդերի հայտնաբերման գոյություն ունեցող մեթոդները և ներկայացվում է երկուական կոդի կլոնների հայտնաբերման նոր մեթոդ: այն բաղկացած է երեք հիմնական փուլերից: առաջին փուլը հիմնված է binnavi համակարգի վրա և կառուցում է ծրագրի կախվածության գրաֆներ յուրաքանչյուր երկուական ֆունկցիայի համար: գրաֆները ստեղծվում են` հիմնվելով reil (reverse engineering intermediate language) ճարտարապետությունից անկախ լեզվի վրա: reil ներկայացումը կարելի է ստանալ մի քանի ճարտարապետությունների ասեմբլերների համար (x86, x86-64, arm, mips, ppc), դրանով ապահովելով գործիքի անկախությունը կոնկրետ ճարտարապետությունից: երկրորդ փուլը հայտնաբերում է կլոններ՝ նախկինում ստեղծված գրաֆների հիման վրա: երկու ծրագրի կախվածությունների գրաֆների առավելագույն ընդհանուր ենթագրաֆի հայտնաբերման համար առաջարկվում է բազմանդամային մոտարկող ալգորիթմ: երրորդ փուլում ստացված կլոնները ցուցադրվում են հետագա վերլուծության համար: h. aslanyan 73 эффективное и точное обнаружение клонов бинарного кода а. асланян аннотация разработчики программного обеспечения обычно копируют и вставляют определенный фрагмент кода, поскольку предпочитают использовать предварительно написанное решение или частичное решение в качестве основы для решения своей проблемы. однако это может привести к различным ошибкам, а также увеличить размер исходного и бинарного кода. поиск похожих частей кода (клонов) в двоичном коде становится более применимым, если исходный код недоступен. кроме того, компилятор во время различных преобразований может скопировать некоторые части кода и создать клоны кода, которых нет в исходном коде. обнаружение клонов бинарного кода используется для нахождения вредоносного кода, поиска семантических ошибок, обнаружения нарушения авторских прав и т. д. в этой статье обсуждаются существующие методы обнаружения клонов бинарных кодов и вводится новый метод их обнаружения. он состоит из трех основных этапов. первый этап основан на платформе binnavi и генерирует графы зависимостей программы для каждой двоичной функции. графы создаются на основе независимого от платформы языка reil (reverse engineering intermediate language). представление reil поддерживается для нескольких архитектур (x86, x86-64, arm, mips, ppc), что обеспечивает независимость инструмента от целевой архитектуры. второй этап обнаруживает клоны на основе ранее созданных графов. предлагается полиномиальный эвристический алгоритм для нахождения максимального общего подграфа двух графов зависимостей программы. на третьем этапе полученные клоны визуализируются для ручного анализа. references microsoft word m. kyrexian_10.doc mathematical problems of computer science 34, 2010. 31 construction of irreducible, normal and primitive polynomials over finite fields melsik kyureghyan institute for informatics and automation problems cryptography has been experiencing a drastic development in contemporary times. this is highly conditioned by fast growth of network systems and progress of communication technologies for which information security has become a problem of special importance. secure, trustworthy and reliable delivery of information is a central problem in information theory and coding theory with applications in computer science and telecommunication. as a rule, error correcting/detecting codes used to provide reliable transmission of data over an unreliable communication channel are unstable and do not ensure required data protection. this has necessitated employment of joint coding: cryptographic coding for data encryption and error correcting/detecting coding to improve reliability of communication on a channel. both types of coding are applied sequentially, guaranteeing security of information during communications. modern cryptography intersects the disciplines of mathematics, computer science and telecommunication, and is a cornerstone of computer and communications security. it addresses a wide range of problems, such as: a) electronic signatures are commonly used for signing digital documents and ensuring that downloaded applications are provided by a trusted source, and are central to the operation of public key infrastructures and many network security schemes. development of easy-realizable fast algorithms for electronic signature is one of hard problems. b) modern block ciphers are widely used to provide encryption of quantities of information, and/or a cryptographic checksum to ensure the contents have not been altered. design of easy-realizable fast algorithms for block ciphers remains a problem of high importance. the problem of explicitly constructing irreducible, normal and primitive polynomials over galois fields is one of the challenging problems in computer algebra, coding theory, cryptography and theory of finite fields and plays a major role in modern engineering, primarily due to wide use of such polynomials in variety of coding, cryptographic and computational applications. moreover, recent advances in these areas have awakened an even more interest to the subject of such polynomials. researchers of data coding laboratory have conducted research on the theoretical foundations of cryptography, the application of cryptography to network and system security. some theoretic and practical problems in this area have been attacked, particularly: 32 1. several recurrent methods of constructing irreducible polynomials over finite fields have been proposed recently [1]. 2. some effective algorithms, employing recurrent methods to construct primitive polynomials over finite fields have been considered [1-2]. 3. security characteristics of block ciphers safer+ and safer++ of safer family have been investigated from theoretical and practical point of view. the coordinate permutation chosen for use in both ciphers safer+ and safer++ is the “armenian shuffle”, which is used in place of ‘hadamard shuffle’ employed in the previous ciphers of the safer family. ‘armenian shuffle’ not only provides even better diffusion of safer+ and safer++ and enhances strength of these ciphers against both differential and linear cryptanalysis, but also runs significantly faster as it allows fewer number of encryption rounds in the cipher. this property enables building new cryptosystems, equivalent to safer+ and safer++ by their crypto characteristics, which would exceed previous ciphers in speed and would be similarly strongly secure against cryptanalysis [4-5]. 4. new public-key encryption and public-key digital signature schemes have been proposed based on discrete logarithm problem. for the public-key encryption scheme it has been shown that the given algorithm has an advantage over well known el-gamal public-key encryption scheme in terms of complexity of implementation and bandwidth efficiency. the specific of presented digital signature scheme is that the signature is addressed from a given user with a given public key to another user with a different public key so that only the recipient will be able to verify the signature from a specified user. the complexity of implementation is similar to digital signature standard algorithm (dsa) [3]. references [1]m. k. kyuregyan and g. m kyureghyan. irreducible compositions of polynomials over finite fields, arxiv: 1008.18637v1 [math.nt] 11 august 2010. accepted for publication in designs, codes and cryptography, an international journal, editors-in-chief: d. jungnickel; j.d. key; p. wild. [2] m. k. kyuregyan. “iterated constructions of irreducible polynomials over finite fields with linearly independent roots”, j. finite fields and their applications, vol. 10, issue 3 (2004), pp. 323-341. [3] khachatrian g., kyuregyan m., new public key encryption and signature scheme. proceedings of russian-germanarmenian workshop applications of information theory, coding and security. yerevan, armenia april 14-16, 2010 pp. 31-34. http://ipia.sci.am/web/waitsc2010.htm [4] j. l. massey, g. h. khachatrian and m. k. kuregian, nomination of safer+ as candidate algorithm for the advanced encryption standard (aes), submission document from cylink corporation to nist, june 1998. [5] j. l. massey, g. h. khachatrian and m. k. kuregian, nomination of safer++ as candidate algorithm for the new european schemes for signatures, integrity, and encryption (nessie), submission document from cylink corporation, 2000. начиная с начала 2000 года осуществляется внедрение ghis в здравоохранении, в рамках принятого проекта о реформирование информ mathematical problems of computer science 50, 67--75, 2018. feedback driven grammar-based fuzzing seryozha a. asryan institute for informatics and automation problems of nas ra e-mail: asryan@ispras.ru abstract in this paper, we present a method for grammar-based fuzzing, which improves its penetration power. it is based on input data generation using a fuzzer feedback. several other methods are prone to create an initial set of acceptable test cases before the actual fuzzing process, and hence are unable to use the runtime information to increase the generated input’s quality. the proposed method uses the coverage information gathered for each input sample and guides grammar-based input generation. this method uses more than 120 bnf (backus-naur form) grammar rules described in antlr (another tool for language recognition) platform. experimental results show that our method feedback driven random test generation, has higher code coverage capabilities compared with the existing methods. keywords: fuzzing, bnf grammars, structured data, automated test generation 1. introduction correctness of compilers is crucial to most software projects. random testing, or fuzzing, has emerged as an important tool for finding bugs in compilers and runtimes. the main idea behind fuzzing is to feed target application (program under test) with a large amount of mutated test inputs to trigger unintended program behavior like hangs and crashes. the existing fuzzing approaches can be classified in three basic categories – blackbox fuzzing, whitebox fuzzing and graybox fuzzing. blackbox fuzzers had no knowledge about the program’s internal structure and, hence, are less effective. sometimes they can use grammars to generate inputs with specific characteristics. the second type of fuzzers usually combine fuzzing with heavy-weighted symbolic execution to improve the effectiveness by applying a symbolic engine in cases where fuzzer is unable to explore a new execution path (i.e., increase the code coverage). graybox fuzzing is something in between. it uses a light-weighted program instrumentation to extract partial information of the program to generate guided input samples without sacrificing execution speed. unfortunately, when it comes to testing applications with complex structured-inputs, fuzzers have several limitations. examples of such applications can be compilers, translators and interpreters. these applications have a multi-pass design and process input in multiple stages 67 feedback driven grammar-based fuzzing 68 (lexer, parser etc.). because of complicated checks and a huge amount of possible execution paths at the first stage, fuzzers are mostly unable to generate inputs that could exercise code beyond the first stage. currently there are several methods and instruments [1-6, 8-11] that use whitebox fuzzing or grammar-based data generation approaches to address the challenge. csmith [1] has been successfully used to identify hundreds of bugs in c/c++ compilers, however this and other similar approaches have significant drawbacks. csmith couples input generation logic with target programming language specifications. producing inputs based on this strategy requires expert knowledge and a significant engineering effort, which needs to be repeated from scratch for each new language. for example, to support a new programming language, this method requires the definition of a corresponding grammar and manual implementation of language features. grammar-based whitebox fuzzing [2] enhances the whitebox fuzzing technique by using the input grammar specification to construct valid test cases. it presents a dynamic test generation algorithm that uses symbolic execution to directly generate grammar-based constraints, the satisfiability of which is checked using a custom grammar-based constraint solver. the two main disadvantages of this method are the long runtime and the small set of available grammars. another instrument syntesk (syntax testing kit) [3] implements unitesk [4] technology. using bnf grammars, syntesk generates two sets of programs. the first set contains test cases, which will be accepted by compiler. in the second set instruments collect invalid programs, which will be rejected by the compiler. gramfuzz [5] can automatically detect grammar rules based on the provided input samples. after that, those grammars are used for further data generation. this instrument is mainly used for web browsers fuzzing. in its first step gramfuzz considers a set of inputs of html, css and javascript. then it tries to extract bnf rules and construct corresponding ast (abstract syntax tree). during fuzzing, inputs are generated by replacing the ast nodes with available elements (duplications are also accepted). the main limitations of this instrument are: not all bnf rules can be extracted from the provided samples, not all generated inputs are valid programs for parser. another instrument ifuzzer [6] finds bugs in javascript interpreters. it uses evolutionary computing techniques, such as genetic algorithms [7], to guide the fuzzer in generating uncommon input code fragments that may trigger exceptional behavior in the interpreter. ifuzzer gets as an input the context-free grammar of a particular language and a test suite with valid inputs. based on that grammar, it generates parse trees and extracts code fragments (fragment pool) from a given testsuite. the initial population for a genetic algorithm consists of random selection of programs, from the input test samples. for each new generation, ifuzzer uses the fitness function on inputs from the previous generation to determine a set of inputs upon which the new generation should be constructed. elements of the next generation are created by selecting random code fragments of the appropriate input code for replacement. replacement was performed by choosing a random member from the fragment pool. fitness function consists of a fuzzer feedback (whether the program crashed or not) and input complexity. this method uses only the final state of program execution as feedback and doesn’t consider information about the program execution paths (code coverage information), as well as the correlation between program paths. instrument superion [8] proposes a grammar-aware coverage-based grey-box fuzzing. this instrument uses the grammar of its test inputs to parse each input into an ast. based on the constructed ast, superion performs a trimming operation to reduce the size of the inputs, iteratively removing each subtree in the ast of a test input and observing coverage differences. it uses also two types of mutation strategies. the first one is afl’s [9] standart dictionary mutation. the second one replaces one subtree in the ast of a test input with the subtree from itself or another test input in the queue. one of the limitations of superion is that it needs well-documented grammars, as well as an initial set of valid test cases. the paper [10] proposes the instrument blendfuzz that uses grammars to create syntactically valid test inputs and guide test generation. this approach s. asryan 69 consists of two stages. the first stage requires an initial set of valid inputs and a corresponding grammar. according to the provided grammar, blendfuzz breaks test cases into grammatical fragments, which will be used as basic building blocks in the next stage. in the second one, generated code fragments are used to mutate the existing test cases. more specifically, blendfuzz selects a grammatical fragment of one input and replaces it with another one in the pool. this procedure is repeated systematically to generate a large set of input samples. although blendfuzz is quite successful in practice, it is based on random testing technique and doesn’t incorporate results of program execution. the paper [11] describes a learn&fuzz algorithm that uses sample inputs and neural-network-based statistical machine-learning techniques to automatically generate input grammars for grammar-based fuzzing. this algorithm can also generate new inputs based on the probability distribution of the learnt model. learn&fuzz algorithm is trained over a corpus of pdf files to generate test inputs for the microsoft edge pdf parser. the results can vary depending on different input formats and training sets. after studying the variety of methods mentioned above, we came to the conclusion that these methods have several limitations:  input generation strategy requires expert knowledge and a significant engineering effort, which needs to be repeated from scratch for each new language [1]  input generation is based only on the usage of the final state of the program execution (whether the program crashed or not) and doesn’t consider information about the program execution paths (code coverage information), as well as correlation between program paths [6]  method needs well-documented grammars and an initial set of valid test cases [8]  test generation is based on random testing technique and doesn’t use feedback of program execution [10]. in this paper, we propose a method for generating input data based on target bnf grammars. we develop this method on top of our previous paper [12], which increases the overall effectiveness and percentage of correctly generated input samples that will successfully pass the parsing stage. our work makes the following contributions:  we use the push down automata representation of bnf rules and fuzzer edge coverage information to direct the input generation towards increasing coverage.  using the coverage information, we add weighted values to edges (transitions) of push down automata of each input sample. then we use that information at the input generation stage to produce inputs with higher chances to explore new paths of program under test. the rest of this paper is organized as follows: section 2 describes our approach to grammarbased data generation. section 3 provides a detailed description of the implemented model of interaction between a fuzzer and an input generation component (sd-gen – structured data generation). section 4 presents the results of the performed experiments, and comparison with other methods. finally, section 5 presents our conclusion. 2. feedback-driven data generation 2.1 antlr’s grammar structure antlr platform provides bnf grammars for more than 120 different languages. as we discussed in our previous paper [12], each of these grammars has its own set of pushdown automata representations. feedback driven grammar-based fuzzing 70 figure 1 shows an example of a rule from bnf grammar and its corresponding pushdown automata. the rule is described with its name (rule name), followed by a single alternative, terminated with a semicolon, or it can also have alternatives. alternatives are either a list of rule elements or an empty list. in the picture above, “object” is a rule, which has “pair”, “ ‘{’ ‘}’ ” as its alternatives. in antlr platform, each rule should have a pushdown automata representation. for “object” rule (1) antlr would generate pushdown automata (2). we use this representation to distinguish interesting rules, which were used to build inputs to exercise new paths in target program during its fuzzing. 2.2 guided test case generation it is difficult to generate test cases for compilers, because their inputs are highly structured. vast majority of existing compiler fuzzing systems generate a set of inputs before actually starting the testing process. hence, they are not able to change their data generation models based on runtime information. despite the fact that some instruments use bnf grammars, without instrumentation feedback from the testing system, the generated samples will cover random parts of the target program. our test generation system tries to overcome these problems by creating dependencies between the generated input structure and information based on the target code coverage. the method used in our previous version of sd-gen (paper [12]) to generate data was based on the following algorithm. it takes two main parameters as input – depth (d) for each rule (that is, the maximum number of recursive selection); length (l) of generated input. additionally, we use two counters cd=0 and cl=0 to store the current depth and length values. 1. select a random rule from the bnf grammar and mark it as a start rule. add that rule to rlist. set the value of cl to the number of nodes in automata of start rule. go to step 2. 2. walk through rlist and select the first non-terminal symbol (nt node). if there is no nt node go to step 6, otherwise continue to step 3. object: ‘{‘ pair ( ‘,‘ pair ) * ‘}‘ | ‘{‘ ‘}‘ ; pair: string ‘:‘ value ; fig. 1. grammar rule representation in antlr. s. asryan 71 3. on the corresponding automata of nt, the rule algorithm selects a random path using the depth first search algorithm (dfs). then it replaces the nt rule in rlist with the constructed path and updates cl and cd values. 4. if cl and cd values reached their limits (cl < l and cd < d values), go to step 5, if not, go to step 2. 5. for each non-terminal node in rlist, the algorithm constructs the shortest terminal path and replaces it in rlist. 6. store the content of rlist in output file. to implement guided input generation, we added weight values to each transition of existing automata (figure 2). the above figure shows the automata for “object” rule of some bnf grammar, with weights on its edges. due to those weights we are able to make modification in our algorithm:  instead of choosing a random path on the corresponding automata of nt rule in step 3, now we select a path with the highest weight value. this can be achieved by choosing each transition using its weight as measurement for probability, i.e., select “2512” rather than “2523” because the weight of the second edge is less than the weight of the first one. to prevent the algorithm from selecting the same transition on every iteration, we decrease its weight by the appropriate value. 3. implementing sd-gen as plugin for fuzzing tool the current implementation of sd-gen as a fuzzer mutation plugin supports the configuration file that can be used to manually specify the number of testcases to be generated and its iteration counts. the bnf data generation plugin is designed to manage the execution of sd-gen instrument based on the assigned parameters. this plugin has the following main responsibilities:  run sd-gen instrument, which generates a set of bnf-based inputs using information about the transition weights received from the fuzzer. if such information does not yet exist, a set of completely random inputs is generated  for every constructed input, the plugin sends its rule transition information (including weights, if they exist) to the fuzzer engine. on the other hand, after detecting an interesting input, the fuzzer retrieves its rule transition information and updates it based on the target execution coverage results. each time when sdgen is called, it checks whether there is information about transitions weights or not. if the 4 4 6 6 3 2 3 1 4 6 3 3 2 1 5 3 4 4 fig. 2. automata of bnf rule with weight values. feedback driven grammar-based fuzzing 72 required information is available, it will choose the rules and transitions of that rule based on the exited data. doing this iteratively, improves the generated data accuracy and makes it possible to change the test case generation strategies in runtime, hence significantly increasing the impact of the constructed inputs on the target code coverage. the bnf data generation plugin is called in two different phases. first, it acts as a mutation plugin to continuously generate new inputs, and second, it is activated whenever fuzzing detects an interesting input (increases coverage or finds some crashes/hangs) sample to update the corresponding weight values. 4. results this section provides comparison results of the proposed method with our previous method and also with the existing methods. for analysis purpose we use several well-known compilers and interpreters. one of the main parameters of measurement is the application code coverage, as well as the number of generated inputs to gain that coverage. as shown in table 1, almost in all cases, our method was able to achieve more code coverage with less inputs. in case of gcc/g++, csmith gets more coverage results due to its way of generating inputs sample, which, starting from the initial sample, are all valid program instances. application name our previous method feedback driven grammarbased fuzzing (our current method) execution count coverage info. (%) gcc-7.1 24959 26107 ~9800 +4.6 g++-7.1 26154 30103 ~8400 +15.1 python-2.7 7561 7962 ~41000 +5.3 php-v7.1.7 2036 2107 ~325000 +3.5 luac-5.3.4 13395 16274 ~9700 +21.5 gfortran-7.1 24060 24950 ~5300 +3.7 5. conclusion fuzzing is a powerful technique for testing an application by randomly mutating its input values. however, for the certain type of application (compiler, interpreters) it is hard to generate test cases that will not fail at first levels of execution. our proposed method implements guided data generation using bnf grammars. we are able to generate new input testcases based on target application feedback on each input sample. it allows us to make modifications in our testcase generation strategies while continuing the fuzzing process. this method results in improvement of target code coverage and increases the analyses effectiveness. table. 1. experimental results for testing of the proposed method s. asryan 73 references [1] xuejun yang, yang chen, eric eide and john regehr, “finding and understanding bugs in c compilers”, pldi '11 proceedings of the 32nd acm sigplan conference on programming language design and implementation, san jose, california, usa, pp. 283-294, 2011. [2] p. godefroid, m. y. levin and d. molnar, "grammar-based whitebox fuzzing", pldi '08 proceedings of the 29th acm sigplan conference on programming language design and implementation, tucson, az, usa, pp. 206-215, 2008. [3] с. в. зеленов и н. в. пакулин, “верификация компиляторов – систематический подход”, proceedings of the institute for system programming, russia, vol. 13, no. 1. стр. 47-64, 2007. [4] i. b. bourdonov, a. s. kossatchev, v. v. kuliamin and a. k. petrenko, "unitesk test suite", fme '02 proceedings of the international symposium of formal methods europe on formal methods getting it right, london, uk, pp. 77-88, 2002. [5] t. guo, p. zhang, x. wang and q. wei, "gramfuzz: fuzzing testing of web browsers based on grammar analysis and structural mutation", second international conference on informatics & applications (icia), pp. 212-215, 2013. [6] s. veggalam, s. rawat, i. haller and h. bos, “ifuzzer: an evolutionary interpreter fuzzer using genetic programming”, proceedings of computer security 21st european symposium on research in computer security, pp. 581-601, sep. 2016 [7] r. i. mckay, n. x. hoai, p. a. whigham, y. shan and m. o’neill, “grammar-based genetic programming: a survey,” genetic programming and evolvable machines, vol. 11, pp. 365–396, may 2010. [8] j. wang, b. chen, l. wei and y. liu, “superion: grammar-aware greybox fuzzing”, 2018. [9] m. zalewski, web-page american fuzzy lop [10] d. yang, y. zhang and q. liu, “blendfuzz: a model-based framework for fuzz testing programs with grammatical inputs”, ieee 11th international conference on trust, security and privacy in computing and communications, pp. 1070-1076, london, june 2012. [11] p. godefroid, h. peleg and r. singh, “learn&fuzz: machine learning for input fuzzing”, proceedings of the 32nd ieee/acm international conference on automated software engineering, pp. 50-59, usa, nov. 2017 [12] s. sargsyan, sh. kurmangaleev, m. mehrabyan, m. mishechkin, t. ghukasyan and s. asryan, “grammar-based fuzzing", proceedings of ivannikov memorial workshop, pp. 32-36, armenia, may 2018. submitted 05.08.2018, accepted 04. 12.2018. feedback driven grammar-based fuzzing 74 ֆազինգի մեթոդ` կոնտեքստից ազատ քերականությունների օգտագործմամբ. տվյալների գեներացիա ֆազերի հետ հետադարձ կապի միջոցով ս. ասրյան ամփոփում մեր օրերում ֆազինգը համարվում է ավտոմատ թեստավորման ամենաարդյունավետ և ամենահայտնի մեթոդներից մեկը։ սակայն գոյություն ունեցող մեթոդներն ունեն սահմանափակ հնարավորություններ հետազոտելու այնպիսի ծրագրային համակարգեր (օր.՝ կոմպիլյատոորներ), որոնք մշակում են բարդ կառուցվածք ունեցող տվյալներ։ հոդվածում ներկայացված է ֆազինգի մեթոդ, որն օգտագօրծում է կոնտեքստից ազատ քերականությունները մուտքային տվյալների կառուցման համար՝ հիմնվելով ֆազինգի ընթացքում ստացված տվյալների վրա։ գոյություն ունեցող մոթոդներից շատերը հակված են կառուցել տվյալների ամբողջական հավաքածուներ, մինչ ֆազինգի սկիզբը, որի հետևանքով անհնար է դառնում օգտագործել կատարման ընթացքում ստացված ինֆորմացիան կառուցվող տվյալների որակը բարձրացնելու համար։ առաջարկվող մեթոդի նպատակն է ուղղորդել քերականության վրա հիմնված մուտքային տվյալների կառուցումը՝ օգտագործելով ինֆորմացիա ուսումնասիրվող ծրագրային համակարգի կոդի ծածկույթի մասին։ այս մեթոդն օգտագործում է ավելի քան 120 քերականությունների bnf (bakus-naur form) ներկայացումներ, նկարագրված antlr ծրագրային համակարգում։ փորձերի արդյունքները ցույց են տալիս, որ ներկայացված մեթոդի միջոցով կառուցված տվյալների շնորհիվ հնարավոր է ստանալ հետազոտվող ծրագրային ապահովման կոդի ավելի մեծ ծածկույթ, քան մյուս հայտնի մեթոդները։ фаззинг с использованием грамматических правил: генерация данных на основе обратной связи с фаззером с. асрян аннотация в наши дни фаззинг является одним из наиболее эффективных и широко используемых методов автоматического динамического анализа. несмотря на это, существующие методы имеют ограничения при тестировании приложений (компиляторы и интерпретаторы) обрабатывающие входные данные, имеющие сложную структуру. s. asryan 75 в статье представлен метод фаззинга на основе грамматических правил. метод основан на генерации входных данных программы, используя обратную связь с фаззером (информацию в результате выполнения фаззинга). множество других методов склонны создавать начальный набор тестовых примеров до начала процесса фаззинга, и, следовательно, не могут использовать информацию, доступную во время выполнения фаззинга, для повышения качества генерируемых тестовых примеров. предлагаемый метод использует покрытие кода программы, собранный для каждого тестового примера и направляет процесс построения новых входных данных (на основе грамматик). данный метод использует бнф (форма бэкуса — наура) представления более 120 грамматик, описанных в платформе antlr (another tool for language recognition). проведенные тестирования показывают, что метод генерации случайных тестов учитывая обратную связь с фаззером, позволяет достичь большего покрытия кода, чем существующие методы. microsoft word e. haroutunian_6.doc mathematical problems of computer science 34, 2010. 20 on close interaction of information theory and mathematical statistics e. a. haroutunian iiap nas ra dedicated to the 70-th anniversary of academician yuri haik shoukourian the first publications of the author (being scientific secretary of the computing centre of the armenian academy of science) was [1], where the notion of the e -capacity of the discrete memoryless channel (dmc) was introduced. the paper [2], the most frequently cited by information theory specialists, presents method of derivation of sphere packing bound for error probability of the dmc. in 1987 the author came back to computing centre renamed as institute for informatics and automation problems and directed in course of long epoch of twenty years (from 1986) by yury shoukourian. in 1992 the author defended the thesis of the doctor of science in moscow. in recent years e. haroutunian prepared more than ten candidates and one doctor of science specialized in information theory and mathematical statistics. international publications of last 20 years of the author and of his disciples are presented below. they can be classified to three following clusters. publications in information theory [1, 2, 4, 6-14, 19, 20, 22, 29-35] are devoted to study of estimates of error probability of various complex information transmission and protection systems. the second group includes investigations of different statistical problems and applications [3, 5, 15-18, 21, 23-28, 37-39]. the third section puts together papers devoted to interaction of methods of information theory and mathematical statistics [36, 40-43]. except cited below international publications more than forty scientific works were published in transactions of iiap and other republican editions. the research in the noted fields will be continued by staff of the laboratory of information theory and applied statistics of iiap in near future. references 1. e. a. haroutunian, “upper estimate of transmission rate for memoryless channel with countable number of output signals under given error probability exponent,” (in russian), 3rd all union conf. on theory of information transmission and coding, uzhgorod, publishing hous of the uzbek ac.sc., tashkent, pp. 83--86, 1967. 2. e. a. haroutunian, “estimates of the error probability exponent for a semi-continuous memoryless channel,” (in russian), probl. inform. transm., vol. 4, no. 4, pp. 37—48, 1968. 3. e. a. haroutunian, “on asymptotically optimal criteria for markov chains,” (in russian), first world congress of bernoulli society, section 2, vol. 2, no. 3, pp. 153-156, 1990. 4. m. e. haroutunian, “e-capacity of arbitrarily varying channel with informed encoder,” probl. inform. transm., (in russian), vol. 26, no. 4, pp. 16-23, 1990. 5. e. a. haroutunian, “logarithmically asymptotically optimal testing of multiple statistical hypotheses,” probl. contr. and inform. theory, vol. 19, no. 5--6, pp. 413-421, 1990. 21 6. e. a. haroutunian, r. sh. maroutian, “  ,e  -achievable rates for multiple descriptions of random varying source,” probl. contr. and inform. theory, vol. 20, no. 2, pp. 165--178, 1991. 7. m. e. haroutunian, “bounds of e-capacity for the channel with random parameter,” probl. inform. transm., (in russian), vol. 27, no. 1, pp. 14--23, 1991. 8. e. a. haroutunian, m. e. haroutunian, “channel with random parameter," proc. of 12-th prague conf. on inform. theory, statis. decision func. random proc., p. 20, 1994. 9. a. n. harutyunyan, e. a. haroutunian, “on properties of rate-reliability-distortion function,” ieee trans. inform. theory, vol. 50, no. 11, pp. 2768—2769, 1996. 10. e. a. haroutunian, m. e. haroutunian, “bounds of e-capacity region for restricted two-way channel,” (in russian), probl. inform. transm., vol. 34, no. 3, pp. 7--16, 1998. 11. e. a. haroutunian, a. n. haroutunian, a. r. kazarian, (ghazaryan) “on rate-reliabilities-distortions function of source with many receivers,” proc. joint session 6-th prague symp. asymptotic statist. and 13-th prague conf. inform. theory, statist. decision func. random proc., prague, vol. 1, pp. 217-220, 1998. 12. e. a. haroutunian, a. n. harutyunyan, “successive refinement of information with reliability criterion," in proc. ieee intern. symp. inform. theory, sorrento, italy, p. 205, 2000. 13. e. a. haroutunian, a. n. harutyunyan, a. r. ghazaryan, “on rate-reliability-distortion function for robust descriptions system,” ieee trans. inform. theory, vol. 46, no. 7, pp. 2690--2697, 2000. 14. e. haroutunian, a. ghazaryan, “on the shannon cipher system with wiretapper guessing subject to distortion and reliability requirements”, ieee international symposium on information theory, losanna, switzerland, p.324, 2002. 15. h. shahumyan, e. haroutunian, “on the most informative indices selection method”, proceedings of the intern. congress of mathematicians, beijing, p. 189, 2002. 16. n. ajabyan, f. topsoe, e. haroutunian, “on application of entropy analysis to spatiotemporal evolution of ecological models”, proceedings of the international conference on csit, yerevan. pp. 190-195, 2003. 17. h. shahumyan, e. haroutunian, “armenian web site on applied statistics”, proceedings of the international conference csit, yerevan, pp. 449-451, 2003. 18. i. safaryan, e. haroutunian, “common approach to the distributions mixture identification and dependence models analysis”, proceedings of international conference csit, yerevan, pp. 184-186, 2003. 19. m. e. haroutunian, “on multiple-access channel with random parameter,” proceedings of international conference csit, yerevan, armenia, pp. 174--178, 2003. 20. a. n. harutyunyan, e. a. haroutunian, “on properties of rate-reliability-distortion function”, ieee trans. on inform. theory, vol. 11, pp. 2768–2773, 2004. 21. e. a. haroutunian, p. s. avetisyan, i. a. safaryan, a. v. manasyan, “analysis of the multidimensional distributions non-homogeneity in social-economic models of teaching”, proceedings of vi (jubilee) international school-seminar “multivariate statistical analysis and econometrics”, pp. 5354, tsahkadzor, june 2004. 22. m. e. haroutunian, s. a. tonoyan, “random coding bound of information hiding e-capacity,” trans. of ieee intern. symp. infrom. theory, chicago, usa, p. 536, 2004. 23. r. ahlswede, e. haroutunian, “testing of hypothesis and identification”, electronic notes in discrete mathematics, 21, pp. 185 -189, 2005. 24. r. ahlswede, e. aloyan, e. haroutunian, “on logarithmically asymptotically optimal hypothesis testing for arbitrarily varying source with side information”, electronic notes in discrete mathematics 21, pp. 91-95, 2005. 25. e. haroutunian, “reliability in multiple hypotheses testing and identification problem”, nato science series: computer and system sciences, vol. 198, ios press, pp. 189 – 201, 2005. 26. e. haroutunian, h. shahumyan, a. manasyan, “armenian statistical web lab”, proceedings international conference csit, pp. 591-594, yerevan, armenia, 2005. 27. r. f. ahlswede, e. aloyan, e. a. haroutunian, “on logarithmically asymptotically optimal hypothesis testing for arbitrarily varying source with side information,” lecture notes in computer 22 science, vol. 4123, “general theory of information transfer and combinatorics”, springer, pp. 457-461, 2006. 28. r. f. ahlswede, e. a. haroutunian, “on logarithmically asymptotically optimal testing of hypotheses and identification”, lecture notes in computer science, vol. 4123, “general theory of information transfer and combinatorics”, springer, pp. 462-478, 2006. 29. m. haroutunian, “bounds of e-capacity for multiple-access channel with random parameter”, lecture notes in computer science, vol. 4123, ``general theory of information transfer and combinatorics", springer, pp. 166--183, 2006. 30. a. n. harutyunyan , “notes on conditions for successive refinement of information", lecture notes in computer science, vol. 4123, “general theory of information transfer and combinatorics”, springer, pp. 130--138, 2006. 31. a. n. harutyunyan, a.j. han vinck, “error exponent in avs coding”, proc. of ieee int. symp. inform. theory, seattle, wa, july 9--14, pp. 2166--2170, 2006. 32. e. haroutunian, “on bounds for ecapacity of dmc”, ieee trans. on inform. theory, v. 53. no. 11, pp. 4210—4220, 2007. 33. e. haroutunian, “reliability approach in wiretapper guessing theory”, “aspects of network and information security”, nato science for peace and security series, information and communication security, vol. 17, pp. 248-260, ios press, 2008. 34. m. e. haroutunian, “e-capacity of information hiding systems”, “aspects of network and information security”, nato science for peace and security series, information and communication security, vol. 17, pp. 261-273, ios press, 2008. 35. a. n. harutunyan, “remarks on e -optimal rate function in dms coding”, “aspects of network and information security”, nato science for peace and security series, information and communication security, vol. 17, pp. 308-314, ios press, 2008. 36. e. haroutunian, m. haroutunian, a. harutyunyan, “reliability criteria in information theory and in statistical hypothesis testing”, foundations and trends in communications and information theory, vol. 4, no. 2-3, pp. 1-171, 2008. 37. e. haroutunian, n. grigoryan, “on arbitrarily varying markov source coding and hypothesis lao testing by non-informed statistician”, proceedings of ieee int. symp. inform. theory, seoul, south korea, june 28--july 3, pp. 981-985, 2009. 38. e. haroutunian, p. hakobyan, “multiple hypotheses lao testing for many independent object”, international journal “scholarly research exchange”, vol. 2009, pp. 1--6, 2009. 39. e. haroutunian, p. hakobyan, "on hypotheses optimal testing for many independent objects", proceedings of international conference csit, pp. 141-144, yerevan, 2009. 40. m. grigoryan, a. n. harutyunyan, “multiple hypothesis testing for arbitrarily varying sources", proceedings of international workshop on applications of information theory, coding and security, yerevan, armenia, pp. 67-70, 2010. 41. a. n. harutyunyan, n. m. grigoryan, “chernoff bounds of hypothesis testing for arbitrarily varying sources", proceedings of international workshop on applications of information theory, coding and security, yerevan, armenia, pp. 71-74, 2010. 42. n. m. grigoryan, a. n. harutyunyan, “error exponents in multiple hypothesis testing for arbitrarily varying sources”, ieee international workshop on information theory, dublin, ireland, pp. 1-5, 2010. 43. e. a. haroutunian, “information theory and statistics”, international encyclopedia of statistical science, springer-verlag, pp. 643-645, 2010. microsoft word article1.doc ìàòåìàòè÷åñêèå âîïðîñû êèáåðíåòèêè è âû÷èñëèòåëüíîé òåõíèêè 31, 163—166, 2008. 163 о методике построения корпоративных сетей передачи данных на базе технологии mpls vpn нана д. григорян государственный педагогический университет им. х.абовяна zarasak@dolphin.am аннотация статья посвящена попытке анализа методики построения корпоративных сетей передачи данных на базе технологии mpls vpn (multiprotocol label switching virtual private network) организации вкс на базе многопротокольной коммутации ip-пакетов по меткам. технология mpls позволяет обеспечить управляемость, легкую наращиваемость, надежность и высокую готовность решений, необходимых для предоставления мультисервисных услуг организациям. данный метод целесообразно использовать при создании кспд единой информационной системы армении, целью которой является повышение эффективности контроля и надзора за безопасностью и качеством продукции и услуг при их создании, поступлении и обращении на потребительском рынке армении. литература [1] м. шестаков, принципы построения корпоративных сетей передачи данных, «компьютерра», № 256, 1997 г. [2] м. захватов, построение виртуальных частных сетей (vpn) на базе технологии mpls. руководство cisco., 2001 г. [3] edited by jeff doyle and matt kolon. mcgraw-hill/osborne // juniper networks routers: the complete reference, 2002. [4] ivan pepelnjak, jim guichard elnjak, jeff apcar. cisco systems // mpls and vpn architectures, vol. 2, 2004. [5] r. aggarwal et al., multicast in 2547 vpns and vpls, internet draft, work in progress, http://www.ietf.org/internet-drafts/draft-raggarwa-l3vpn-mvpn-vpls-mcast-01.txt november, 2004. [6] r. aggarwal, d. papadimitriou, s. yasukawa (editors) et al., extensions to rsvp-te for point to multipoint te lsps // internet draft, work in progress, http://www.ietf.org/internetdrafts/draft-ietf-mpls-rsvp-te-p2mp-01.txt november, 2004. [7] в. олвейк, структура и реализация современной технологии mpls., 2004 г. [8] y. serbest, ray qiu, rob nath. supporting ip multicast over vpls, february, 2005, http://www.ietf.org/ internet-drafts/draft-serbest-l2vpn-vpls-mcast-02.txt о методике построения корпоративных сетей передачи данных на базе mpls vpn 164 pls ï»ëýáéá·ç³ûç ñçù³ý íñ³ ïíû³éý»ñç ñ³õáñ¹ù³ý ïáñåáñ³ïçí ó³ýó»ñç ï³éáõóù³ý ù»ãá¹çï³ûç í»ñéáõíáõãûáõýá ü. ¶ñç·áñû³ý ²ù÷á÷áõù ðá¹í³íá mpls ï»ëýáéá·ç³ûç ñçù³ý íñ³ ïíû³éý»ñç ñ³õáñ¹ù³ý ïáñåáñ³ïçí ó³ýó»ñç ï³éáõóù³ý ù»ãá¹çï³ûç í»ñéáõíáõãû³ý ÷áñó ¿: mpls ï»ëýáéá·ç³ý ãáõûé ¿ ï³éçë ³å³ñáí»é ³ûý áñáßáõùý»ñç ï³ñ·³íáñ»éçáõãûáõýá, ³ñ³· ³×ý áõ ñáõë³éçáõãûáõýá, áñáýù ³ýññ³å»ßï »ý ï³½ù³ï»ñåáõãûáõýý»ñçý µ³½ù³½³ý í³é³ûáõãûáõýý»ñ ù³ïáõó»éáõ ñ³ù³ñ: îíû³é ù»ãá¹á ýå³ï³ï³ñ³ñù³ñ ¿ û·ï³·áñí»é ð³û³ëï³ýç ùç³ëý³ï³ý ï»õ»ï³ïí³ï³ý ñ³ù³ï³ñ·ç ïíû³éý»ñç ñ³õáñ¹ù³ý ïáñåáñ³ïçí ó³ýó ëï»õí»éáõ å³ù³ý³ï: начиная с начала 2000 года осуществляется внедрение ghis в здравоохранении, в рамках принятого проекта о реформирование информ mathematical problems of computer science 45, 35--43, 2016. analysis of experiments of a new approach for test quality evaluation mariam e. haroutunian, varazdat k. avetisyan institute for informatics and automation problems of nas ra e-mail: armar@ipia.sci.am, avetvarazdat@gmail.com abstract in the previous paper [1] we suggested a new model of test quality evaluation based on information measures such as shannon entropy and average mutual information. to establish the practical bounds of these measures and the required number of examinees, some experiments were conducted. in this paper the analysis of these experiments are provided. keywords: test quality, shannon entropy, average mutual information, classical test theory, item response theory. 1. introduction test developers are basically concerned about the quality of test items and how examinees respond to it when constructing tests. test theories and related models provide a frame of reference for doing test design work or solving other practical problems. a good test model might specify the precise relationships among test items and ability scores, so that careful design work can be done to produce desired test score distribution and errors of the size that can be allowed. a good test theory or model can also handle errors of measurements by helping understand the role that measurement errors play in estimating examinee’s ability and correlations between variables and true scores or ability scores. there are two currently popular statistical frameworks to address test data analysis and test quality evaluation: classical test theory (ctt) [2] and item response theory (irt) [3]. ctt is a theory about test scores that introduces three concepts test score, true score and error score. in the ctt, the notion of ability is expressed by the true score, which is defined as "the expected value of observed performance on the test of interest." an examinee's ability is defined only in terms of a particular test. when the test is "hard," the examinee will appear to have low ability; when the test is "easy," the examinee will appear to have higher ability. ctt was the dominant statistical approach for 35 mailto:armar@ipia.sci.am mailto:avetvarazdat@gmail.com analysis of the experiments of a new approach for test quality evaluation 36 testing data until lord and novick (1968) placed it in the context with several other statistical theories of mental test scores, notably irt. irt is a model-based measurement statistical theory in which the performance of an examinee on a test item can be predicted (or explained) by a set of factors called traits, latent traits, or abilities; and the relationship between the examinees' item performance and the set of traits underlying item performance can be described by a monotonically increasing function called an item characteristic function or item characteristic curve (icc). each of these approaches has its advantages and disadvantages [4]. for example, in ctt item parameters are dependent on the examinee sample from which they are obtained, but in irt these parameters are examinee group independent. but on the other hand, in case of ctt smaller examinee sample sizes are required for analysis and the methods are simpler compared to irt. besides the existing ctt and irt models, we have developed a new approach [1] based on information measures such as shannon entropy and average mutual information. the main idea of the new approach is the following. suppose that the test consists of n items, each item can be considered as a binary random variable (rv) x1, x2, .., xn with probabilities p for correct answers and 1 − p, for incorrect answers: 𝑋𝑋𝑖𝑖 = � 1 𝑤𝑤𝑤𝑤𝑤𝑤ℎ 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑤𝑤𝑤𝑤𝑝𝑝 𝑝𝑝𝑖𝑖, 0 𝑤𝑤𝑤𝑤𝑤𝑤ℎ 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑤𝑤𝑤𝑤𝑝𝑝 1 − 𝑝𝑝𝑖𝑖, 𝑤𝑤 = 1, 𝑁𝑁.������ we consider shannon entropy of rv 𝑋𝑋𝑖𝑖 𝐻𝐻(𝑋𝑋𝑖𝑖) = −�𝑝𝑝(𝑥𝑥𝑖𝑖 ) log 𝑝𝑝(𝑥𝑥𝑖𝑖 𝑥𝑥𝑖𝑖 ) and the average mutual information of two items: 𝐼𝐼�𝑋𝑋𝑖𝑖 ∧ 𝑋𝑋𝑗𝑗 � = � 𝑝𝑝(𝑥𝑥𝑖𝑖 , 𝑥𝑥𝑗𝑗 ) log 𝑝𝑝(𝑥𝑥𝑖𝑖, 𝑥𝑥𝑗𝑗 ) 𝑝𝑝(𝑥𝑥𝑖𝑖) ∗ 𝑝𝑝(𝑥𝑥𝑗𝑗 )𝑥𝑥𝑖𝑖,𝑥𝑥𝑗𝑗 = 𝐻𝐻(𝑋𝑋𝑖𝑖) − 𝐻𝐻�𝑋𝑋𝑖𝑖 | 𝑋𝑋𝑗𝑗� = 𝐻𝐻�𝑋𝑋𝑗𝑗� − 𝐻𝐻�𝑋𝑋𝑗𝑗 | 𝑋𝑋𝑖𝑖�. our test quality evaluation model consists of the following methods: method 1. if the value of 𝐻𝐻(𝑋𝑋𝑖𝑖) is close to 0, it means that we have a bad test item, which can be very easy or very difficult. if the value of 𝐻𝐻(𝑋𝑋𝑖𝑖) is close to 1 we have a good test item. method 2. if the value of 𝐼𝐼�𝑋𝑋𝑖𝑖 ∧ 𝑋𝑋𝑗𝑗 � is close to 0, it means that there is independency of test items xi and xj . in case of values close to min�𝐻𝐻(𝑋𝑋𝑖𝑖), 𝐻𝐻�𝑋𝑋𝑗𝑗�� xi and xj items repeat each other. method 3. if t h e value of conditional entropy �𝐻𝐻�𝑋𝑋𝑗𝑗 | 𝑋𝑋𝑖𝑖�� is close to 𝐻𝐻(𝑋𝑋𝑖𝑖 ), then xi and xj are independent. however, several questions remain open. 1. how precisely our model evaluates the quality of test items and how comparable is it to ctt and irt estimation methods? 2. which are the permissible limits of 𝐻𝐻(𝑋𝑋𝑖𝑖) and 𝐼𝐼�𝑋𝑋𝑖𝑖 ∧ 𝑋𝑋𝑗𝑗 � ? m. haroutunian, v. avetisyan 37 3. which is the sufficient number of the examinee samples for precise evaluation? the answers to these questions can be found experimentally. 2. description of experiments the results of school final exams of armenian language and literature held in 2008 were selected for testing. the results were provided in encrypted form by the center for assessment and testing. four test-results are chosen to be analyzed. each test consists of 80 items, and the number of schoolchildren who participated in the examination process is 2000. the names of the first 50 𝑋𝑋𝑖𝑖 items are 𝐴𝐴1, 𝐴𝐴2, … 𝐴𝐴50 and the names of the last 30 𝑋𝑋𝑖𝑖 items are 𝐵𝐵1, 𝐵𝐵2, … 𝐵𝐵30. for analysis test quality evaluation system developed by us was used in [5]. for each item of four tests the 𝐻𝐻(𝑋𝑋𝑤𝑤), ctt difficulty index [2] and irt b parameter [3] values have been calculated, the comparability of the mentioned parameters observed and the permissible limits of 𝐻𝐻(𝑋𝑋𝑖𝑖)defined. difficulty is defined in both ctt and irt. in ctt the difficulty index p is the proportion of examinees who answer the item correctly. for multiple-choice, true/false, and other items that are scored as right (1 point) or wrong (0 points), item difficulty is the proportion of examinees who answered the item correctly. it ranges from 0 to 1. item difficulty for a polytomous item (an item scored in more than two ordinal categories) is simply the item mean or average item score. it ranges between the minimum and the maximum possible item scores. in irt the difficulty index b (irt b parameter) is on the same metric as the proficiencies or traits. this metric is arbitrary, but often it is anchored so that the proficiency distribution in a designated group has a mean of 0 and standard deviation of 1. the item difficulty identifies the proficiency at which about 50% of the examinees are expected to answer the item correctly. to observe the dependency of 𝐻𝐻(𝑋𝑋𝑖𝑖) and 𝐼𝐼�𝑋𝑋𝑖𝑖 ∧ 𝑋𝑋𝑗𝑗� values on the examinee sample size and define the enough number of examinee samples five experiments are carried out for each test. the analysis was conducted by choosing the same test at random based on 500, 300, 200, 100, 50 participants’ results. for each test 𝐼𝐼�𝑋𝑋𝑖𝑖 ∧ 𝑋𝑋𝑗𝑗� and ctt correlation coefficient 𝑅𝑅�𝑋𝑋𝑖𝑖 , 𝑋𝑋𝑗𝑗� between 𝑋𝑋𝑖𝑖 and 𝑋𝑋𝑗𝑗 items [2] was calculated, their compatibility was observed and the permissible limits of 𝐼𝐼�𝑋𝑋𝑖𝑖 ∧ 𝑋𝑋𝑗𝑗� were defined. correlation coefficient ranges from -1 +1. coefficient value should be small or equal to 0.3. if coefficient value is close to +1, it means that test items repeat each other and one of that items should be removed from the test. the negative correlation means there is an independency of test items. 3. the analysis of results. based on the first experiment results comparison of 𝐻𝐻(𝑋𝑋𝑖𝑖), ctt difficulty index p and irt b parameter values of test1 are shown in figure 1. according to ctt test items for which difficulty values are between 0.3 and 0.74 interval are good items (not easy and not very difficult 34 items), and based on the analyzed data we can see that 𝐻𝐻(𝑋𝑋𝑖𝑖) values of test 1 for these items are between 0.82 and 1.0. for easy test items, difficulty values are between 0.75 and 0.9 (29 items), 𝐻𝐻(𝑋𝑋𝑖𝑖) values are between 0.48 and 0.81. for very easy test items difficulty values are between 0.9 and 1.0 (14 items), 𝐻𝐻(𝑋𝑋𝑖𝑖) values are between 0.12 and 0.47. approximately the same results were obtained for the tests 2, 3 and 4. as we can see in case of 𝐻𝐻(𝑋𝑋𝑖𝑖)’s large values close to 1 irt b parameter gets large values. analysis of the experiments of a new approach for test quality evaluation 38 fig. 1. 𝐻𝐻(𝑋𝑋𝑖𝑖), ctt difficulty parameters p and irt b parameters of test1. h(xi) irt b difficulty p -5 -4 -3 -2 -1 0 1 2 3 a11a21b19 a3 a48 b6 b9 a50 a4 b12a14a13a47b24a10 a8 a26a24b21a40a25b11b14a46b16b28 m. haroutunian, v. avetisyan 39 to analyze the dependency of 𝐻𝐻(𝑋𝑋𝑖𝑖) values on the number of examinee sample we draw 𝐻𝐻(𝑋𝑋𝑖𝑖) graphics of each test based on the results of five experiments. the graphics are shown in figure 2. fig. 2. 𝐻𝐻(𝑋𝑋𝑖𝑖) graphics for five experiments of test1. the maximum differences of 𝐻𝐻(𝑋𝑋𝑖𝑖) values are presented in table1. table 1. maximum difference of 𝐻𝐻(𝑋𝑋𝑖𝑖) values 500 300 200 100 50 500 0.089 0.07 0.135 0.2 300 0.089 0.13 0.15 0.19 200 0.07 0.13 0.16 100 0.135 0.15 0.16 0.3 50 0.2 0.19 0.23 0.3 while decreasing the examinee sample size until 100, it is obvious that the differences of 𝐻𝐻(𝑋𝑋𝑖𝑖) values are small and the maximum difference is 0.15. but when examinee sample size is decreased more than 100, the difference is close to 0.3, and in case of values equal to 50 the difference is close to 0.3. with the same principle for each test the mutual information 𝐼𝐼�𝑋𝑋𝑖𝑖 ∧ 𝑋𝑋𝑗𝑗� was calculated and the dependency of 𝐼𝐼�𝑋𝑋𝑖𝑖 ∧ 𝑋𝑋𝑗𝑗� values on the number of examinee sample was observed. the graphics are shown in figure 3. analysis of the experiments of a new approach for test quality evaluation 40 fig. 3. 𝐼𝐼�𝑋𝑋𝑖𝑖 ∧ 𝑋𝑋𝑗𝑗� for five experiments of test1 a1 item. the average mutual information and correlation between test items also have been analyzed. the graphics based on some items’ data are presented in figure 4 and figure 5. fig. 4. correlation �𝑅𝑅�𝑋𝑋𝑖𝑖 , 𝑋𝑋𝑗𝑗�� and average mutual information �𝐼𝐼�𝑋𝑋𝑖𝑖 ∧ 𝑋𝑋𝑗𝑗�� between a1 item and other 80 items. a1 (500) a1 (300 a1 (200) a1 (100) a1 (50) 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 a27b15a12a31 b8 b21a33b25a32 b1 b23a10a16a46 b4 a17a24b13a47b29a14b17 b9 b12 r(x,y) i(x^y)x10 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 a4 a7 a1 0 a1 3 a1 6 a1 9 a2 2 a2 5 a2 8 a3 1 a3 4 a3 7 a4 0 a4 3 a4 6 a4 9 b2 b5 b8 b1 1 b1 4 b1 7 b2 0 b2 3 b2 6 b2 9 m. haroutunian, v. avetisyan 41 fig. 5. correlation �𝑅𝑅�𝑋𝑋𝑖𝑖 , 𝑋𝑋𝑗𝑗�� and average mutual information �𝐼𝐼�𝑋𝑋𝑖𝑖 ∧ 𝑋𝑋𝑗𝑗�� between a11 item and other 80 items. table 2. a11 a5 a8 a15 a26 a31 b18 b19 b20 b23 r(x,y) -0.0083 -0.0186 -0.0581 -0.0195 -0.0385 -0.0005 -0.0151 0.0219 -0.0177 i(x∧y) 0.0011 0.0013 0.0049 0.0001 0.00001 0.0001 0.000004 0.0004 0.00006 for item a11 the negative correlation with other items and average mutual information values are presented in table 2. when we compare items’ correlations and average mutual information graphics, it is easy to see that the results are comparable. for example, from the correlation matrix and graphics shown in figure 5 we can see that a11 test item has negative correlation with other test items, for these items the values of mutual information are presented in table2. if correlation values are negative, average mutual information values are small enough, but from test we should remove those items, so the smallest permissible limit value of average mutual information should be 0.005. 4. conclusion in this research while analyzing the data, the following has been determined. 1. the methods suggested by us are correctly defining the quality of test items and the results are comparable with ctt and irt estimation methods. 2. simpler mathematical analysis is needed compared to irt. 3. 𝐼𝐼�𝑋𝑋𝑖𝑖 ∧ 𝑋𝑋𝑗𝑗� describes the dependence of test items which does not have an equivalent in irt. 4. for test good items 𝐻𝐻(𝑋𝑋𝑖𝑖) values should be between 0.8 and 1.0, for fairly good items (easy) values are between 0.45 and 0.8, and for bad test items 𝐻𝐻(𝑋𝑋𝑖𝑖) values are between 0 and 0.45. r(x,y) i(x^y)x10 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25 a1 a4 a7 a1 0 a1 3 a1 6 a1 9 a2 2 a2 5 a2 8 a3 1 a3 4 a3 7 a4 0 a4 3 a4 6 a4 9 b2 b5 b8 b1 1 b1 4 b1 7 b2 0 b2 3 b2 6 b2 9 analysis of the experiments of a new approach for test quality evaluation 42 5. for 𝐼𝐼�𝑋𝑋𝑖𝑖 ∧ 𝑋𝑋𝑗𝑗 � preferable are the values smaller than 0.05 and greater than 0.005 0.005≤ 𝐼𝐼�𝑋𝑋𝑖𝑖 ∧ 𝑋𝑋𝑗𝑗 � ≤ 0.05 6. smaller sample sizes are required in comparison with ctt and irt. the sample size should be more than 100. in ctt the sample size is between 200 and 500, and in irt it depends on the irt model, but samples over 500 are needed. references [1] m. haroutunian and v. avetisyan, “new approach for test quality evaluation based on shannon information measures”, transactions of ipia of nas ra, mathematical problems of computer science, vol. 44, pp. 7-21, 2015. [2] m. b. chelishkova, theory and practice of pedagogical tests constructing, moscow: logos, 2002. [3] c. demars, item response theory. oxford university press; 1 edition, 2010. [4] k. hambleton and w. jones, “comparison of classical test theory and item response theory and their applications to test development”, educational measurement: issues and practice, vol. 12, no. 3, pp. 38-47, 1993. [5] m. haroutunian and v. avetisyan, “development of the test quality evaluation system”, proceedings of the international conference on computer science and information technologies (csit 2015), yerevan, armenia, september 28-october 2, pp. 372--375, 2015. submitted 10.10.2015, accepted 20.01.2016 թեստի որակի գնահատման նոր մոտեցման փորձարկումների վերլուծություն մ․ հարությունյան,վ․ ավետիսյան ամփոփում նախորդ հոդվածում [1] հեղինակների կողմից առաջարկվել է թեստերի որակի գնահատման նոր մոդել՝ հիմնված շենոնի էնտրոպիայի և միջին փոխադարձ ինֆորմացիայի վրա։ այս մեծությունների սահմանային արժեքները և թեստավորման մասնակիցների բավարար քանակը որոշելու համար կատարվել են փորձարկումներ։ հոդվածում ներկայացված է այդ փորձարկումների վերլուծությունը, որից հետևում է, որ հաշվարկները ավելի պարզ են irt-ի համեմատությամբ, թեստավորման արդյունքների վերլուծության համար պահանջվում է մասնակիցների ավելի քիչ քանակ, քան` ctt-ում և irt-ում։ m. haroutunian, v. avetisyan 43 анализ экспериментов нового подхода для оценки качества теста м. арутюнян, в. аветисян аннотация в предыдущей статье [1] авторами была предложена новая модель оценки качества теста на основе энтропии шэннона и средней взаимной информации. для того, чтобы установить практические пределы этих величин и необходимое количество экзаменуемых, были проведены эксперименты. в данной статье представлен анализ этих экспериментов, из которого следует, что расчеты более просты по сравнению с irt, для анализа результатов тестирований требуется меньшее количество экзаменуемых, чем в стт и irt. analysis of experiments of a new approach for test quality evaluation mariam e. haroutunian, varazdat k. avetisyan 2. description of experiments d:\sbornik\...\rev.dvi mathematical problems of computer science 30, 5{17, 2008. an e ±cient m ethod for gener ation of m ar ch t ests b ased on for mulas gurgen harutunyany, davit melkumyany, hasmik elchyanz, valery vardaniany y virage logic e-mail: fgurgen, davit.melkumyan, valery.vardaniang@viragelogic.com zyerevan state university e-mail: hasmik.elchyan@yahoo.com abstract a general method for generation of minimal march tests to detect or diagnose any subclass of simple static or dynamic faults in static rams is described. the proposed method is shown to generate all possible march tests satisfying certain necessary conditions for detection of faults. a correspondence between march tests and natural numbers is established that allows construct a formula that enables generation of all march tests detecting certain faults. as an example, the method is applied for construction of new minimal march tests for detection of several subclasses of three-operation dynamic faults. the method can be generalized for detection/diagnosis of any subset of static or dynamic faults. refer ences [1 ] a .j. va n d e go o r , te s t in g s e m ic o n d u c t o r m e m o r ie s : th e o r y a n d p r a c t ic e , j ohn w iley & sons, 1 9 9 1 . [2 ] a . j. va n d e go o r , b . s m it , " th e a u t o m a t ic g e n e r a t io n o f m a r c h t e s t s " , ie e e international w orkshop m emory technology d esign and testing, p p . 1 3 1 -1 3 6 , 1 9 9 3 . [3 ] k . za r r in e h , s . j. u p a d h ya ya , a n d s . ch a kr a va r t y, " a n e w fr a m e wo r k fo r g e n e r a t in g o p t im a l m a r c h t e s t s fo r m e m o r y a r r a ys " , p roc. int. conf. (itc), p p . 7 3 -8 2 , 1 9 9 8 . [4 ] a . b e n s o , s . d i ca r lo , g. d i n a t a le , p . p r in e t o , " a n o p t im a l a lg o r it h m fo r t h e a u t o m a t ic g e n e r a t io n o f m a r c h t e s t s " , d ate 2002, ie e e d esign, automation and test in e urope conference and e xhibition, p p . 9 3 8 9 3 9 , 2 0 0 2 . [5 ] s . m. a l-h a r b i, s . k . gu p t a , " a n e ± c ie n t m e t h o d o lo g y fo r g e n e r a t in g o p t im a l a n d u n ifo r m m a r c h t e s t s " , ie e e vl si test symposium, p p . 2 3 1 -2 3 7 , 2 0 0 1 . [6 ] c.-f. w u , c.-t. h u a n g , a n d c.-w . w u , " r a ms e s : a fa s t m e m o r y fa u lt s im u la t o r " , p roc. int. symp. d efect and f ault tolerance in vl si systems (d f t), albuquerque, p p . 1 6 5 -1 7 3 , n o v. 1 9 9 9 . [7 ] j.-f. l i, k .-l . ch e n g , c.-t. h u a n g , a n d c.-w . w u , " ma r c h -b a s e d r a m d ia g n o s t ic a lg o r it h m s fo r s t u c k-a t a n d c o u p lin g fa u lt s " , p roc. ie e e itc, p p . 7 5 8 -7 6 7 , 2 0 0 1 . 5 6 an e±cient method for generation of march tests based on formulas [8 ] t. gyo n jya n , v . a . v a r d a n ia n , " a n e ± c ie n t a lg o r it h m fo r g e n e r a t in g m in im a l m a r c h t e s t s fo r fa u lt d e t e c t io n a n d d ia g n o s is in s t a t ic r a n d o m a c c e s s m e m o r ie s " , international d esign and test w orkshop, d ubai, p p . 1 9 -2 0 , 2 0 0 6 . [9 ] a . b e n s o , a . b o s io , s . d i ca r lo , g. d i n a t a le , p . p r in e t t o , " a u t o m a t ic m a r c h t e s t s g e n e r a t io n fo r s t a t ic a n d d yn a m ic fa u lt s in s r a ms " , [1 0 ] g. h a r u t u n ya n , v . a . v a r d a n ia n , y . zo r ia n , " min im a l m a r c h t e s t s fo r d yn a m ic fa u lt s in r a n d o m a c c e s s m e m o r ie s " , j ournal of e lectronic testing: theory and applications, v o l. 2 3 , n u m b e r 1 , p p . 5 5 -7 4 , 2 0 0 7 . [1 1 ] l . d ilillo , p . gir a r d , s . p r a vo s s o u d o vit c h , a . v ir a z e l, m. b a s t ia n , " r e s is t ive -o p e n d e fe c t in je c t io n in s r a m c o r e -c e ll: a n a lys is a n d c o m p a r is o n b e t we e n 0 .1 3 ¹m a n d 9 0 n m t e c h n o lo g ie s " , d esign automation conference, p p . 8 5 7 -8 6 2 , 2 0 0 5 . [1 2 ] l . d ilillo , p . gir a r d , s . p r a vo s s o u d o vit c h , a . v ir a z e l, s . b o r r i, m. h a g e -h a s s a n , " d yn a m ic r e a d d e s t r u c t ive a u lt in e m b e d d e d -s r a ms : a n a lys is a n d m a r c h t e s t s o lu t io n s " , p roc. ie e e e uropean test symposium, 2 0 0 4 . [1 3 ] s . h a m d io u i, a .j. va n d e go o r , m. r o d g e r s , " ma r c h s s : a t e s t fo r a ll s t a t ic s im p le fa u lt s " , r ecords of ie e e int. w orkshop m td t, p p . 9 5 -1 0 0 , 2 0 0 2 . [1 4 ] s . h a m d io u i, a .j. va n d e go o r , m. r o d g e r s , " l in ke d fa u lt s in r a n d o m a c c e s s m e m o r ie s : c o n c e p t , fa u lt m o d e ls , t e s t a lg o r it h m s , a n d in d u s t r ia l r e s u lt s " , ie e e trans. cad , vo l. 2 3 , n o . 5 , p p . 7 3 7 -7 5 6 , 2 0 0 4 . [1 5 ] g. h a r u t u n ya n , v . a . v a r d a n ia n , y . zo r ia n , " min im a l m a r c h t e s t s fo r u n lin ke d s t a t ic fa u lt s in r a n d o m a c c e s s m e m o r ie s " , p roc. 23rd ie e e vl si test symposium, p alm springs, ca, usa, p p . 5 3 -5 9 , 2 0 0 5 . [1 6 ] a . j. va n d e go o r , z. a l-a r s , " fu n c t io n a l m e m o r y fa u lt s a fo r m a l n o t a t io n a n d a t a xo n o m y" , p roc. ie e e vl si test symposium, m ontreal, canada, p p . 2 8 1 -2 9 0 , 2 0 0 0 . [1 7 ] a h o , s e t h i, u llm a n , co m p ile r s : p r in c ip le s , te c h n iqu e s , a n d to o ls , addison-w esley, is b n 0 -2 0 1 -1 0 0 8 8 -6 , 1 9 8 6 . ´³ý³ó¨»ñç ùççáóáí ù³ñß ï»ëï»ñç ï³éáõóù³ý ³ñ¹ûáõý³í»ï ù»ãá¹ ¶. ð³ñáõãûáõýû³ý, ¸. ø»éùáõùû³ý, ð. ¾éãû³ý, ì. ì³ñ¹³ýû³ý ²ù÷á÷áõù üï³ñ³·ñí³í ¿ ñçßíáõ ë³ñù»ñáõù ëï³ïçï ï³ù ¹çý³ùçï ³ýë³ñùáõãûáõýý»ñç ï³ù³û³ï³ý »ýã³µ³½ùáõãûáõý ñ³ûïý³µ»ñáõ ï³ù ³ëïáñáßáõ ùçýçù³é ù³ñß ï»ëï»ñç ï³éáõóù³ý ù»ãá¹: ²é³ç³¹ñí³í ù»ãá¹áõù ï³éáõóíáõù »ý áñáß³ïç ³ýññ³å»ßï å³ûù³ýý»ñçý µ³í³ñ³ñáõ µáéáñ ù³ñß ï»ëï»ñá: ø³ñß ï»ëï»ñç ¨ µý³ï³ý ãí»ñç ùçç¨ ïñí»é ¿ ³ñï³å³ïï»ñáõù, áñá ñý³ñ³íáñáõãûáõý ¿ ï³éçë ï³éáõó»é µ³ý³ó¨: ´³ý³ó¨ç ùççáóáí ëï³óíáõù »ý µáéáñ ³ûý ù³ñß ï»ëï»ñá, áñáýù ñ³ûïý³µ»ñáõù ï³ù ³ëïáñáßáõù »ý ïñí³í ³ýë³ñùáõãûáõýý»ñá: ø»ãá¹á ïçñ³éí»é ¿ »ñ»ù ·áñíáõáõãû³ùµ ¹çý³ùçï ³ýë³ñùáõãûáõýý»ñç áñáß³ïç »ýã³¹³ëá ñ³ûï³µ»ñáõ ùçýçù³é ù³ñß ï»ëï»ñ ëï³ý³éáõ ñ³ù³ñ: ø»ãá¹á ñý³ñ³íáñáõãûáõý ¿ ï³éçë ï³éáõó»é ï³ù³û³ï³ý ëï³ïçï ¨ ¹çý³ùçï ³ýë³ñùáõãûáõýý»ñ ñ³ûï³µ»ñáõ ï³ù ³ëïáñáßáõ ùçýçù³é ù³ñß ï»ëï»ñ: mathematical problems of computer science 59, 16–26, 2023. doi: 10.51408/1963-0098 udc 519.714 rdnf oriented analytics to random boolean functions levon h. aslanyan, irinaa. arsenyan, vilik m. karakhanyan and hasmik a. sahakyan institute for informatics and automation problems of nas ra,yerevan, armenia e-mail: kavilik@gmail.com abstract dominant areas of computer science and computation systems are intensively linked to the hypercube-related studies and interpretations. this article presents some transformations and analytics for some example algorithms and boolean domain problems. our focus is on the methodology of complexity evaluation and integration of several types of postulations concerning special hypercube structures. our primary goal is to demonstrate the usual formulas and analytics in this area, giving the necessary set of common formulas often used for complexity estimations and approximations. the basic example under considered is the boolean minimization problem, in terms of the average complexity of the so-called reduced disjunctive normal form (also referred to as complete, prime irredundant, or blake canonical form). in fact, combinatorial counterparts of the disjunctive normal form complexities are investigated in terms of sets of their maximal intervals. the results obtained compose the basis of logical separation classification algorithmic technology of pattern recognition. in fact, these considerations are not only general tools of minimization investigations of boolean functions, but they also prove useful structures, models, and analytics for constraint logic programming, machine learning, decision policy optimization and other domains of computer science. keywords: boolean function, hypercube, complexity, asymptotic, reduced disjunctive normal form. article info: received 14 february 2023; sent for review 10 march 2023; accepted 11 april 2023. 1. hypercube and related structures the metric theory of boolean functions (bf) [1], [2] arose in the 70’s, in parallel with the emergence of broader design and implementation ideas for mechanical and electronic computation devices. it was then that it turned out that the system of binary representation of numbers is the most optimal, both from the point of view of the algorithmic implementation of arithmetic calculations and also from the point of view of developing physical carriers of performing these calculations [3]. bf – functions with only binary variables, and also 16 l. aslanyan, i. arsenyan, v. karakhanyan and h. sahakyan 17 with values in the domain {0, 1}, although simple among the other similar mathematical concepts, they are quite complex in solving problems associated with their transformations and optimization. the metric theory of boolean functions provides the necessary knowledge for coding, transforming and implementing binary functions. although the way to minimal bf representations are and remains difficult, a rather complete picture of the main forms of function representation of functions has been obtained, and the basic role here takes the concept of disjunctive normal forms. successive steps of several transformations of functions are found to achieve minimal forms as a chain from the table or formula representation to the reduced d.n.f., then to the deadlock forms and finally – the minimal structures. the accompanying structures and bottlenecks of achieving acceptable optimization are investigated intensively [1], [4]–[7]. here we will not cover the whole theory but will pay attention to one fundamental construction, – to the concept of reduced disjunctive normal forms (r.d.n.f.) of boolean functions. r.d.n.f. is the collection of all minimal conjunctions and geometrically the system of all maximum intervals/sub-cubes of functions. these forms are a universal concept, and they also arise in problems such as circuit design from set of functional elements (logical part of chip design), in the theory of pattern recognition (logic separation algorithm, and generation of logical regularities) [8]–[11], in biological models of heredity and mutations (phylogeny, parsimony) [12, 13], etc. turning to the complexity characterization of structures associated with the reduced disjunctive normal form, where two types are usually considered: the largest and most typical characteristics, we will focus on the second component. in a concise survey of the domain, the initial studies of [5], [14], and [15], should be mentioned, that give the formulas of average numbers of maximal intervals in boolean functions. [16], [17] extended these results to the case of partially defined boolean functions. an alternative track of papers in these topics includes the articles [18], [19], [20]. current research on the topics of bf and complexities might be demonstrated through the papers [21]–[26]. methodologically, in studies in the area of bf, it should be taken into account that the function determination domain, as well as the number of functions itself, are finite, depending on the number of the variables – the dimensionality. so, considering the parameter π(f) over the functions, we get the split of these functions into finite classes by the values of this parameter. these are the rates and intensity of the accepted values of the parameter π(f). in some cases, it is convenient to refer to these valuations as probabilistic distributions, which is not obligatorily but is convenient in some contexts. in this concern, there appears a link to the model of random boolean functions and the combinatorial theories initiated by a. renyi and p. erdos [27], [28]. 1.1 concepts and definitions in the binary domain elementary conjunction, direction. let α̃ and β̃ – be arbitrary vertices of the ndimensional unite cube. and let ji, i = 1, 2, · · · , r be all coordinates, those where αji = βji. consider the formula k(x1, x2, · · · , xn) = r∧ i=1 x σji ji , with σji = αji, i = 1, 2, · · · , r. we say that k is an elementary conjunction stretched on the pair of vertices α̃ and β̃ of the n-dimensional unit cube en. the number of literals in k is the rank of k. the geometrical counterpart of k is a sub-cube defined as follows. assign 0 values to all but j1, j2, · · · , jr coordinates and denote this vertex by v0. similarly, assign 18 rdnf oriented analytics to random boolean functions these coordinates by the value 1, obtaining the vertex v1. these are the minimal and maximal vertices that belong to k, and they determine a unique sub-cube of all truth vertices of k. n − r, the number of variable coordinates of k is the size of its sub-cube. let λ = {j1, j2, · · · , jr} be a collection of r indices drawn up of variables x1, x2, · · · , xn, and let λ̄ be the complementary to the λ set of indices. conjunctions of the form ∧r i=1 x σji ji and the corresponding intervals will be called conjunctions and intervals of the direction λ. for a fixed r there are crn different directions, and each of them is determined by the appropriate selection of an r subset {j1, j2, · · · , jr} of the set {1, 2, ..., n}. the individual interval in the direction {j1, j2, · · · , jr} appears in result of assigning the values σ1, σ2, · · · , σr to the variables xj1, xj2, · · · , xjr. fig. 1. geometry of hypercube. this also means that there are 2n−r conjunctions and intervals in one of the r-directions. the collection λ̄ of indices defines another set of directions. let f be an arbitrary logical formula and m ⊆ bn. we say that f absorbs or covers m if on each tuple α̃ ∈ m the formula f accepts the unite (true) value. let α̃ ∈ en be an arbitrary vertex. call the value | α̃ |= ∑n i=1 αi the module or the weight of α̃. the set of all vertices β̃ ∈ en, with ρ(α̃, β̃) =| α̃ ⊕ β̃ |= k, call the k–the layer of en in relation to the vertex α̃ (⊕ – mentions mod2 summation). intervals nk1 and nk2, k1(x1, x2, · · · , xn) = r∧ i=1 x σ1 ji ji and k2(x1, x2, · · · , xn) = r∧ i=1 x σ2 ji ji of the same size and the same direction we call neighbors if ρ(σ̃1, σ̃2) = 1, where ρ – be the hamming distance, ρ(σ̃1, σ̃2) = ∑r i=1 | σ1ji − σ 2 ji | . let then ji0 is the number of that unique coordinate for which σ1ji0 ̸= σ2ji0 . then we say that the conjunctions k 1 and k2 (or the pair of neighbor intervals corresponding to them) joined by the coordinate xji0 , and, as a result, l. aslanyan, i. arsenyan, v. karakhanyan and h. sahakyan 19 a new conjunction (interval) appears: r∧ i ̸=i0,i=1 x σji ji . partition the variable set x1, x2, · · · , xn in an arbitrary manner into two nonempty groups: xi1, xi2, · · · , xk as the first group, and xik+1, xik+2, · · · , xin as the second. then, the n-dimensional unit cube en may be represented as the cartesian multiplication bk ×bn−k of two sub-cubes: bk and bn−k generated correspondingly by the sets of variables xi1, xi2, · · · , xik and xik+1, xik+2, · · · , xin. let us enumerate the vertices of b n−k by the layers relative to the vertex 0̃ of bn−k. enumeration among the vertices of a particular layer is arbitrary, but the first group that is enumerated by low numbers is layer zero, then the first layer, and so on. additional ordering among layer vertices may use lexicographic order, binary value based order, etc. consider an arbitrary k-dimensional sub-cube bk of en, the first k-dimensional interval bk1 in the direction of b k. list the neighbor intervals to the considered one, bk1, bk2, b k 3, · · · , bkn−k+1. let f be an arbitrary (partially defined) boolean function that satisfies the following conditions: α) bk1 doesn’t contain zero value vertices of f : (∀α̃ ∈ bk1, f(α̃) ̸= 0), β) each of the neighbor with bk1 interval contains at least one ‘unit’ value vertex f : (∀j, j = 2, 3, · · · , n − k + 1 ∃α̃ ∈ bkj , f(α̃) = 1), γ) bk1 contains at least one ‘unit’ vertex of f : (∃α̃ ∈ bk1, f(α̃) = 1). in conditions α), β), γ), we say that bk1 is a maximal interval of the function f. d.n.f., composed of all elementary conjunctions, corresponding to maximal intervals of function f is named the reduced disjunctive normal form of f. the number of disjunctive members of this formula is considered as its complexity. denoting by rk(f) the number of all maximal k–intervals of the function f we get the formula of complexity of the reduced disjunctive normal form of f: n∑ k=0 rk(f). 2. on the maximum number of k-dimensional maximal intervals of rbf consider the class p2(n) of all boolean functions of n variables x1, x2, · · · , xn. let p, 0 < p < 1 be a fixed number, and fp – the probability distribution on p2(n), generated in the following way. the function f ∈ p2(n) is induced as a result of a randomized experiment, where the values of the function on vertices of en are derived randomly. the value 1 appears with a probability p and the 0 value – with a complementary probability 1−p. the vertices of en take part in this experiment independently of each other, and the probabilistic distribution fp over the set of boolean functions is generated in this way. the probability of an individual boolean function f under the distribution fp depends on the balance between the 0 and 1 values of the function f (the volumes of the sets n{ and en−n{). for f ∈ p2(n), this probability is equal to p|n{|(1 − p)2 n−|n{|. when p = 1/2 this probability is simply 1/22 n and the corresponding distribution becomes the uniform distribution over 20 rdnf oriented analytics to random boolean functions the p2(n). we introduce the notation rk(f) for the number of k-dimensional maximal intervals of the function f ∈ p2(n). and let rk(n, p) be the average value of the number of k-dimensional maximal intervals of functions f ∈ p2(n) under the distribution fp. it is easy to make sure, that rk(n, p) = ∑ f∈p2(n) fp(f) ∗ rk(f) (1) the number rk(n, p) in the expression (1) is given by its definition as a sum over all functions of f ∈ p2(n), counting all their k-dimensional maximal intervals and taking into account the probabilities of f in the distribution fp. further evidence of these constructions is provided by the following scheme: fig. 2. this figure presents the bipartite graph of functions and k-dimensional maximal intervals. upper line functions are placed in order of the number of their ”true” values, from 0 to 2k. different functions include different numbers of k-dimensional maximal intervals and have different probabilities under the distribution fp. instead, each interval presented in the bottom line is connected to the same number of functions. this is because the sizes of intervals is the same. the order of intervals is by groups of intervals, that belong to the same direction. numeration inside the functions with the same number of ”ones” and inside the groups of intervals of the same direction is arbitrary. following [5], we change the order of counting in 1, first considering all k-dimensional intervals in en. we relay two events to these intervals: the one, about their maximality, and then the second, about the set of functions that accept the first event about maximality. in this regard, it is also convenient to split the en in parts: the current k-dimensional interval k and its all n−k neighboring k-dimensional intervals k1, k2, · · · , kn−k. this part, the current interval and its neighbors, covers an area e1 of 2k(n − k + 1) vertices of en. and the second part that we consider, consists of the complementary area e2 to e1 up to en. the probability of maximality of k for the function f becomes the product of probability of maximality of k together with the conditional probability of f when k is given to be maximal. the first probability equals p2 k (1 − p2k)n−k. the first and second parts consist of events, and their sums of probabilities are equal to 1 as a probabilistic distribution. now, when we sum up the mentioned conditional probabilities with all f, we get the probability 1, and the final probability of maximality of k, under the conditions of fp, becomes p2 k (1 − p2k)n−k. it reminds us to take this probability for all k-dimensional intervals, obtaining the following equivalent form for (1), l. aslanyan, i. arsenyan, v. karakhanyan and h. sahakyan 21 rk(n, p) = c k n2 n−kp2 k (1 − p2 k ) n−k . (2) theorem 1. rk(n, p) is a concave function of the parameter k in the interval [0, n]. it is important to know the behavior of the function rk(n, p defined on the interval [1, n]. initially, it is useful to calculate the values of the function at the boundary points of the domain of definition: k = 0, 1, ..., n − 1, n. we give these values both for the arbitrary p and the value 1/2. table 1: values of rk(n, p) on boundary points, such as k = 0, 1, ..., n − 1, n. boundary point values of rk(n, p) dimension k of maximal interval rk(n, p) rk(n, 1/2) k = 0 2np(1 − p)n 1/2 k = 1 n2n−1p2(1 − p2)n−1 (n/4)(3/2)n−1 ... ... ... k = n − 1 n2n−1p2n−1(1 − p2n−1) n2n−1(1 − 1/22n−1)/22n−1) k = n p2 n 1/22 n as we can see, both the left and right boundary point values of the interval (0, n) are small, but there is a noticeable increase from left to right at the left end, and a decrease from left to right at the right end. to get a complete picture of the behavior, consider a number of special intermediate point values of the function at: k1 = log 1 −logp , k0 = log logn −logp , and k1 = log n −logp . the technical element of choosing of these values is in simple evaluation of sub-formula ek = 2 2k, which is an important functional part of the 1. substituting k1, k0, and k2 into ek we get: ek1 = 1/2, ek0 = 1/n, ek2 = 1/2 n. (3) let us start the proof of postulations 1-3. for this, conduct a preliminary analysis of the expression (2) for rk(n, p). consider an arbitrary integer value function k(n) that obeys the restriction 0 ≤ k(n) ≤ n, and substitute it into the expression 2. we are interested in the behaviour of the received function rk(n)(n, p) depending on the parameter k(n) as n → ∞. first let’s make sure that with the increase of k the expression rk(n, p) increases monotonically by the k ≤ [k0], and then it decreases, when ]k0[≤ k. by doing this we compose the relation rk = rk+1(n, p) rk(n, p) = (n − k)p2k(1 + p2k)n−k 2(k + 1)(1 − p2k+1) . (4) this expression can be considered for an arbitrary (not only for the integer) assignment to the parameter k. we will follow by checking if this function is concave in the interval 0 < k < n for large n. the direct way of this is to derive the expression of the fraction 22 rdnf oriented analytics to random boolean functions rk and treat it for a possible constant/zero value of it. in such consideration, the most important role takes the part ak = (n − k)p2 k of the base expression 4. substituting k0 into ak we obtain that (n − k0)p2 k0 = (n − k0)p ( logn −logp ) = (n − k0)2 logp( −logn logp ) = n−k0 n , which is converging to 1 as n → ∞. with the help of formulas in section 3. we see that the part bk = (1 + p 2k)n−k of (4) is limited at the point k0: (6) gives (1 + p 2k0 )n → e as n → ∞, so that (1 + p2 k0 )n−k0 also tends to e. compose the fraction bk+1/bk in the following form: bk+1/bk = ( 1 + p2 k p2 k )n−k−1 ( 1 + p2 k )n−k = ( 1+p2 k p2 k 1+p2 k )n−k 1 + p2 k p2 k (5) fig. 3. differential of growing rk(n, p). note that the fraction 1+p 2k p2 k 1+p2 k is less than 1, so its n − k degree is also less than 1. and the denominator of (5) is greater than 1 so that, finally, the expression (5) is less than 1 for all k, which means a monotonic decrease of the expression rk in (5). in general, as k increases, all the factors of (4), other than bk, decrease monotonically and, besides this, as n → ∞ , this expression tends to zero at the point k0 and grows infinitely when k = k0 − 1. finally, we receive that with increasing k, for the beginning, ik(n, p) increases, achieving its maximal value at the point [k0] or ]k0[, and, then, it decreases. 3. on the dependency of number of k-dimensional maximal intervals on k consider the parameter k2 = log n −logp. since 0 < p < 1, we have k2 = logn + c, where c represents an absolute constant determined by the fixed value of p. we intend to obtain an asymptotic formula for ik(n, p) by the n → ∞ for the values of k of the form k2 + const. we make use of the following expressions ckn ∼ nk k! , (1 − p2k) ∼ 1, and n! ∼ nne−n √ 2πn as n → ∞, which are based on the formulas 1. if 0 ≤ x ≤ 1 and 0 ≤ y, then exp(x(1 − x 2 )y) ≤ (1 + x)y ≤ exp(xy). (6) l. aslanyan, i. arsenyan, v. karakhanyan and h. sahakyan 23 2. if 0 ≤ x ≤ 1 and 0 ≤ y, then (1 − x)y ≤ exp(−xy); and (7) exp(−x(1 − x)y) ≤ (1 − x)y, when additionally 0 ≤ x ≤ 1/2. 3. if x and y be natural numbers, and x ≤ y, then (1 − x y ) x−1 2 ≤ x−1∏ i=1 (1 − i y ) ≤ (1 − x 2y )x−1. (8) and are valid for the mentioned values of the parameter k, and for this reason ik(n, p) ∼ nkek2n−kp2 k kk √ 2πk = ĩk(n, p). (9) theorem 2. the probability, that functions of the class p2(n) under the distribution fp have maximal intervals of sizes k, k < [k1] or k > [k2], where k1 = log 1 −logp and k2 = log n −logp tends to zero with n → ∞. on the right side of (9) we have expression, that depends on the continuous argument k, and which is equivalent to the expression ik(n, p) for the integer values of the parameter k, of the form k2 + const. in the mentioned area, ĩk(n, p) decreases monotonically with the increase of k, ĩk2(n, p) tends to infinity, and ĩk2+1(n, p) tends to zero, when n → ∞, so that ik(n, p) → 0, for values k >]k2[ and ik(n, p) → ∞ for values k0 ≤ k ≤ [k2], by n → ∞. let us also denote, that we do not insist that i]k2[(n, p) as n → ∞ converges to any appropriate value. in what follows, we will use the first chebyshev inequality (1). the first inequality lets formulate an extension of a postulation from [29] for the case of the probability distribution fp. actually, if to consider the expression ik(f), as a parameter of π(f) then for the values k >]k2[ ik(n, p) → 0 by n → ∞, and taking into the force the first inequality for the arbitrary ϵ(n) ≥ 0 p(ik(f) ≥ ϵ(n)) → 0 when n → ∞. a similar situation takes place in the region of small values of the parameter k. for the value k = k1 and p = 1/2 by the (3) p 2k1 = 1/2 and rk1(n, p) → ∞ as n → ∞. for p > 1/2, already for the value k1 − 1, we observe that rk1−1(n, p) → 0 as n → ∞. this is just because 2n−k1+1 1−p2k1−1 is a decreasing exponent, which together with ckn tends to 0. 4. conclusion this article has two goals: first, it considers the set of formulas needed to analyze the complexity of structures associated with a multidimensional unit cube, providing the necessary transformations and approximations for these formulas. further, the paper considers a typical study for this field using these formulas. the problem under consideration estimates the complexity of the reduced disjunctive normal form of boolean functions on average, or, what is the same, for almost the entire class of functions. 24 rdnf oriented analytics to random boolean functions references [1] yu. i. zhuravlev, “set-theoretical methods of algebra of logic, problemi kibernetiki, vol. 8, pp. 544, 1962. 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[29] l. h. aslanyan, “on implementation of reduced disjunctive normal form in the problem of extension of partial boolean functions”, junior researcher, natural sciences, yerevan state university, vol. 20, no. 2, pp. 6575, 1974. 2 6 rdnf oriented analytics to random boolean functions ä³ï³ñ³ï³ý µáõéû³ý ýáõýïóç³ý»ñç î¸üò ïáõùýáñáßí³í í»ñéáõíáõãûáõý è¨áý ð. ²ëé³ýû³ý, æñçý³ ². ²ñë»ýû³ý, ìçéçï ø. î³ñ³ë³ýû³ý, ð³ëùçï ². ê³ñ³ïû³ý ðð ¶²² æýýáñù³ïçï³ûç ¨ ³íïáù³ï³óù³ý åñáµé»ùý»ñç çýëïçïáõï, ºñ¨³ý, ð³û³ëï³ý e-mail: kavilik@gmail.com ²ù÷á÷áõù àíàëèòèêà îðèåíòèðîâàííàÿ íà ñäíô ñëó÷àéíûõ áóëåâûõ ôóíêöèé ëåâîí à. àñëàíÿí, èðèíà à. àðñåíÿí, âèëèê ì. êàðàõàíÿí, àñìèê à. ñààêÿí èíñòèòóò ïðîáëåì èíôîðìàòèêè è àâòîìàòèçàöèè íàí ðà, åðåâàí, àðìåíèÿ e-mail: kavilik@gmail.com àííîòàöèÿ äàííàÿ ñòàòüÿ ïðåñëåäóåò äâå öåëè: âî-ïåðâûõ, â íåé ðàññìàòðèâàåòñÿ íàáîð ôîðìóë, íåîáõîäèìûõ äëÿ àíàëèçà ñëîæíîñòè ñòðóêòóð, ñâÿçàííûõ ñ ìíîãîìåðíûì åäèíè÷íûì êóáîì, ïðåäîñòàâëÿÿ íåîáõîäèìûå ïðåîáðàçîâàíèÿ è àïïðîêñèìàöèè äëÿ ýòèõ ôîðìóë. äàëåå, â ñòàòüå ðàññìàòðèâàåòñÿ òèïè÷íîå èññëåäîâàíèå äëÿ äàííîé îáëàñòè ñ èñïîëüçîâàíèåì ýòèõ ôîðìóë. ðàññìàòðèâàåìàÿ ïðîáëåìà îöåíèâàåò ñëîæíîñòü ñîêðàùåííîé äèçúþíêòèâíîé íîðìàëüíîé ôîðìû áóëåâûõ ôóíêöèé â ñðåäíåì, èëè, ÷òî òî æå ñàìîå, ïî÷òè äëÿ âñåãî êëàññà ôóíêöèé. êëþ÷åâûå ñëîâà: áóëåâà ôóíêöèÿ, ìíîãîìåðíûé åäèíè÷íûé êóá, ñëîæíîñòü, àñèìïòîòèêà, ñîêðàùåííàÿ äèçúþíêòèâíàÿ íîðìàëüíàÿ ôîðìàå. ²ûë ñá¹í³íý áõýç »ñïáõ ýå³ï³ï, ý³ë ³ûý ùýý³ñïáõù ¿ µ³ý³ó¨»ñç ùç ß³ñù, áñáýù ³ýññ³å»ßï »ý µ³½ù³ã³÷ ùç³íáñ ëáñ³ý³ñ¹ç ñ»ï ï³åí³í ï³éáõóí³íùý»ñç µ³ñ¹áõãûáõýá í»ñéáõí»éáõ ñ³ù³ñ` ³å³ñáí»éáí ³ýññ³å»ßï ÷áë³ï»ñåáõùý»ñ ¨ ùáï³ñïáõùý»ñ ³ûë µ³ý³ó¨»ñç ñ³ù³ñ: ²í»éçý, ñá¹í³íá ùýý³ñïáõù ¿ ³ûë áéáñïç ñ³ù³ñ áñáß µýáñáß áõëáõùý³ëçñáõãûáõý` û·ï³·áñí»éáí ³ûë µ³ý³ó¨»ñá: øýý³ñïíáõ áýã³ó³ï³ñ·á ·ý³ñ³ïáõù ¿ µáõéû³ý ýáõýïóç³ý»ñç ïñ׳ïí³í ¹ç½ûáõýïïçí ýáñù³é ó¨ç µ³ñ¹áõãûáõýá ùçççýáõù ï³ù, áñ ýáõûýý ¿, ¹³ëç ·ñ»ã» µáéáñ ýáõýïóç³ý»ñç ñ³ù³ñ: ´³ý³éç µ³é»ñ ´áõéû³ý ýáõýïóç³, µ³½ù³ã³÷ ùç³íáñ ëáñ³ý³ñ¹, µ³ñ¹áõãûáõý, ³ëçùåïáïçï³, ïñ׳ïí³í ¹ç½ûáõýïïçí ýáñù³é ó¨: 02_vilik_59_16_26 02 d:\sbornik\...\gen_haar_tr2.dvi mathematical problems of computer science 31, 79{89, 2008. fast gener alized h aar t r ansfor ms h a ko b g. s a r u kh a n ya n institue for informatics and automation problems of nas of ra e-mail: hakop@ipia.sci.am abstract the fast generalized haar transform algorithms of orders 4n; 3n; and 5n are presented. refer ences [1 ] a h m e d n ., r a o k .r . or t h o g o n a l tr a n s fo r m s fo r d ig it a l s ig n a l p r o c e s s in g . s p r in g e r v e r la g , n e w y o r k, 1 9 7 5 . [2 ] h a a r z. zu r th e o r ie d e r o r t h o g o n a le n fu n kt io n e n s ys t e m e , ma t h . a n n ., vo l 6 9 , p .3 3 1 3 7 1 , 1 9 1 4 . [3 ] r a o k .r ., n a r a s im h a m , r e vu lu r i k . a fa m ily o f d is c r e t e h a a r t r a n s fo r m s , co m p u t . e le c t r . e n g r g ., vo l. 2 , p .3 6 7 -3 6 8 , 1 9 7 5 . [4 ] s e b e r r y j., zh a n g x .m. s o m e o r t h o g o n a l d e s ig n s a n d c o m p le x h a d a m a r d m a t r ic e s b y u s in g t wo h a d a m a r d m a t r ic e s . a u s t r a l. j. o f co m b in . th e o r y., n o .4 , p .9 3 -1 0 2 , 1 9 9 1 . [5 ] co r in t h io s m.j. a n e w c la s s o f g e n e r a l-b a s e m a t r ic e s a n d a fo r m a lis m fo r o p t im a l p a r a lle l/ p ip e lin e d c o m p u t e r a r c h it e c t u r e . e le c t r ic a l a n d co m p u t e r e n g in e e r in g , 1 9 9 3 . ca n a d ia n co n fe r e n c e o n 1 4 -1 7 s e p t ., vo l.2 , p .8 5 1 -8 5 6 , 1 9 9 3 . [6 ] min g yo n g zh o u , zh o n g ka n l iu , h a m a h . a r e s o lu t io n -c o n t r o lla b le h a r m o n ic a l r e t r ie va l a p p r o a c h o n t h e ch r e s t e n s o n d is c r e t e s p a c e . ie e e tr a n s a c t io n s , s ig n a l p r o c e s s in g , vo l.4 2 , is s u e 5 , p .1 2 8 1 -1 2 8 4 , 1 9 9 4 . [7 ] mo r a g a , c., p o s wig j. p r o p e r t ie s o f t h e zh a n g -h a r t le y s p e c t r u m o f p a t t e r n s . p r o c e e d in g s o f t h e twe n t ie t h in t e r n a t io n a l s ym p o s iu m , mu lt ip le -v a lu e d l o g ic , p .6 2 -6 8 , 1 9 9 0 . [8 ] a g a ia n s ., b a ja d ia n h . ge n e r a liz e d o r t h o g o n a l h a a r s ys t e m s : s yn t h e s is , me t r ic a n d co m p u t in g p r o p e r t ie s . h a a r me m o r ia l co n fe r e n c e , vo l.1 , p .9 7 -1 1 3 , co llo g . ma t h . s o c . ja n o s b o lya i, 4 9 , n o r t h -h o lla n d , a m s t e r d a m , n e w y o r k, 1 9 8 7 . [9 ] ma lla t s . a w a ve le t to u r o f s ig n a l p r o c e s s in g . a c a d e m ic p r e s s , 2 0 0 1 . ð³³ñç áý¹ñ³ýñ³óí³í ³ñ³· ó¨³÷áëáõãûáõýý»ñ ð. ê³ñáõë³ýû³ý ²ù÷á÷áõù ðá¹í³íáõù ý»ñï³û³óí³í »ý 4 n; 3 n; ¨ 5 n ï³ñ·ç ð³³ñç áý¹ñ³ýñ³óí³í ³ñ³· ó¨³÷áëáõãûáõýý»ñç ³é·áñçãùý»ñá: 7 9 d:\sbornik\...\article.dvi mathematical problems of computer science 31, 122{129, 2008. i nter val t otal color ings of cer tain gr aphs p e t r o s a . p e t r o s ya n institute for informatics and automation problems of nas of ra, e-mail: pet petros@fipia.sci.am, yahoo.comg abstract an interval total t¡coloring of a graph g is a total coloring of g with colors 1; 2; : : : ; t such that at least one vertex or edge of g is colored by i; i = 1; 2; : : : ; t; and the edges incident with each vertex v together with v are colored by (dg(v) + 1) consecutive colors, where dg(v) is the degree of the vertex v in g. it is proved that complete graphs, complete bipartite graphs and n¡dimensional cubes have interval total colorings and bounds are found for the possible number of colors in such colorings. refer ences [1 ] a .s . a s r a t ia n , t.m.j. d e n le y, r . h a g g kvis t , b ip a r t it e gr a p h s a n d th e ir a p p lic a t io n s , ca m b r id g e tr a c t s in ma t h e m a t ic s , 1 3 1 , ca m b r id g e u n ive r s it y p r e s s , 1 9 9 8 . [2 ] a .s . a s r a t ia n , r .r . k a m a lia n , \ in t e r va l c o lo r in g s o f e d g e s o f a m u lt ig r a p h " , appl. m ath. 5, p p . 2 5 -3 4 , 1 9 8 7 . [3 ] a .s . a s r a t ia n , r .r . k a m a lia n , \ in ve s t ig a t io n o n in t e r va l e d g e -c o lo r in g s o f g r a p h s " , j ournal of combin. theory, ser. b , vo l. 6 2 , p p . 3 4 -4 3 , 1 9 9 4 . [4 ] m. b e h z a d , gr a p h s a n d th e ir ch r o m a t ic n u m b e r s , p h .d . t h e s is , mic h ig a n s t a t e u n ive r s it y, 1 9 6 5 . [5 ] m. b e h z a d , g. ch a r t r a n d , j.k . co o p e r , \ th e c o lo u r n u m b e r s o f c o m p le t e g r a p h s " , j ournal of l ondon m ath. soc. 42, p p . 2 2 6 -2 2 8 , 1 9 6 7 . [6 ] o.v . b o r o d in , \ on t h e t o t a l c o lo u r in g o f p la n a r g r a p h s " , j . r eine angew. m ath. 394, p p . 1 8 0 -1 8 5 , 1 9 8 9 . [7 ] o.v . b o r o d in , a .v . k o s t o c h ka , d .r . w o o d a ll, \ to t a l c o lo r in g s o f p la n a r g r a p h s wit h la r g e m a xim u m d e g r e e " , j ournal of graph theory 26, p p . 5 3 -5 9 , 1 9 9 7 . [8 ] k .h . ch e w, h .p . y a p , \ to t a l c h r o m a t ic n u m b e r o f c o m p le t e r¡ p a r t it e g r a p h s " , j ournal of graph theory 16, p p . 6 2 9 -6 3 4 , 1 9 9 2 . [9 ] t.r . je n s e n , b .to ft , gr a p h co lo r in g p r o b le m s , w ile y in t e r s c ie n c e s e r ie s in d is c r e t e ma t h e m a t ic s a n d op t im iz a t io n , 1 9 9 5 . [1 0 ] a .v . k o s t o c h ka , \ th e t o t a l c o lo r in g o f a m u lt ig r a p h s wit h m a xim a l d e g r e e 4 " , d iscrete m athematics 17, p p . 1 6 1 -1 6 3 , 1 9 7 7 . [1 1 ] a .v . k o s t o c h ka , \ th e t o t a l c o lo r in g o f a m u lt ig r a p h s wit h m a xim a l d e g r e e 5 is a t m o s t s e ve n " , d iscrete m athematics 162, p p . 1 9 9 -2 1 4 , 1 9 9 6 . [1 2 ] p .a . p e t r o s ya n , \ in t e r va l e d g e c o lo u r in g s o f c o m p le t e g r a p h s a n d n¡ c u b e s " , m athematical p roblems of computer science, vo l. 2 5 , p p . 5 -8 , 2 0 0 6 . [1 3 ] m. r o s e n fe ld , \ on t h e t o t a l c o lo r in g o f c e r t a in g r a p h s " , israel j . m ath. 9, p p . 3 9 6 -4 0 2 , 1 9 7 1 . 1 2 2 p. petrosyan 1 2 3 [1 4 ] d .p . s a n d e r s , y . zh a o , \ on t o t a l 9 -c o lo r in g p la n a r g r a p h s o f m a xim u m d e g r e e s e ve n " , j ournal of graph theory 31, p p . 6 7 -7 3 ,1 9 9 9 . [1 5 ] n . v ija ya d it ya , \ on t h e t o t a l c h r o m a t ic n u m b e r o f a g r a p h " , j ournal of l ondon m ath. soc. (2) 3, p p . 4 0 5 -4 0 8 , 1 9 7 1 . [1 6 ] v .g. v iz in g , \ on a n e s t im a t e o f t h e c h r o m a t ic c la s s o f a p¡ g r a p h " , d iskret. analiz 3, p p . 2 5 -3 0 , 1 9 6 4 . [1 7 ] d .b . w e s t , in t r o d u c t io n t o gr a p h th e o r y, p r e n t ic e -h a ll, n e w je r s e y, 1 9 9 6 . [1 8 ] h .p . y a p , to t a l co lo r in g s o f gr a p h s , l e c t u r e n o t e s in ma t h e m a t ic s 1 6 2 3 , s p r in g e r v e r la g , b e r lin , 1 9 9 6 . [1 9 ] z. zh a n g , j. zh a n d , j. w a n g , \ th e t o t a l c h r o m a t ic n u m b e r s o f s o m e g r a p h s " , scientia sinica a 31, p p . 1 4 3 4 -1 4 4 1 , 1 9 8 8 . àñáß ·ñ³ýý»ñç ùçç³ï³ûù³ûçý éç³ï³ï³ñ ý»ñïáõùý»ñ ä. ä»ïñáëû³ý ²ù÷á÷áõù g ·ñ³ýç éç³ï³ï³ñ ý»ñïáõùá 1 ; 2 ; :::t ·áõûý»ñáí ï³ýí³ý»ýù ùçç³ï³ûù³ûçý éç³ï³ï³ñ t-ý»ñïáõù, »ã» ³ù»ý ùç i ·áõûýáí, i = 1 ; 2 ; :::t ý»ñïí³í ¿ ³éýí³½ý ù»ï ·³·³ã ï³ù ïáõ ¨ ûáõñ³ù³ýãûáõñ ·³·³ãçý ïçó ïáõ»ñá ¨ ·³·³ãá ý»ñïí³í »ý ( dg ( v ) +1 ) ñ³çáñ¹³ï³ý ·áõûý»ñáí, áñï»õ dg ( v ) áí ýß³ý³ïí³í ¿ ·³·³ãç ³ëïç׳ýá g ·ñ³ýáõù: ²å³óáõóí³í ¿, áñ éñçí ·ñ³ýý»ñá, éñçí »ñïïáõù³ýç ·ñ³ýý»ñá ¨ ã³÷³ýç ëáñ³ý³ñ¹á áõý»ý ùçç³ï³ûù³ûçý éç³ï³ï³ñ ý»ñïáõùý»ñ ¨ ·ïýí³í »ý ·ý³ñ³ï³ï³ýý»ñ ³û¹ ý»ñïáõùý»ñç ù»ç ù³ëý³ïóáõ ·áõûý»ñç ñý³ñ³íáñ ãíç ñ³ù³ñ: d:\user\...\article_final.dvi mathematical problems of computer science 49, 103{109, 2018. on dependence of i nter pr etation algor ithms of t yped functional p r ogr ams on canonical n otion of ±-reduction d a vit a . gr ig o r ya n department of informatics and applied mathematics, ysu e-mail: david.grigoryan.a@gmail.com abstract in this paper the interpretation algorithms of typed functional programs are considered. the interpretation algorithm is based on substitutions, ¯-reduction and canonical ±-reduction. the basic semantics of typed functional program is a function with indeterminate values of arguments, which is the main component of its least solution. if the value of the basic semantics for some values of arguments is indeterminate, then the interpretation algorithm either stops with the value ?, or works endlessly. it is shown that seven known interpretation algorithms are ?-depend on canonical notion of ±-reduction. here are these algorithms: fs (of full substitution), pes (of parallel external substitution), les (of left external substitution), pis (of parallel inner substitution), lis (of left inner substitution), act (active algorithm), pas (passive algorithm). keywords: typed functional program, canonical ±-reduction, interpretation algorithm, ?-dependency. 1 . typ e d ¸-te r m s , ca n o n ic a l n o t io n o f ±-r e d u c t io n , typ e d fu n c t io n a l p r o g r a m s th e d e ¯ n it io n s o f t h is s e c t io n c a n b e fo u n d in [1 ; 2 ; 3 ]. l e t m b e a p a r t ia lly o r d e r e d s e t , wh ic h h a s a le a s t e le m e n t ?, wh ic h c o r r e s p o n d s t o t h e in d e t e r m in a t e va lu e . e a c h e le m e n t o f m is c o m p a r a b le o n ly wit h it s e lf a n d wit h ?, wh ic h is t h e le a s t e le m e n t o f m. l e t u s d e ¯ n e t h e s e t o f t yp e s ( d e n o t e d b y t ypes) . 1 . m 2 t ypes, 2 . if ¯,®1,...,®k 2 t ypes ( k > 0 ) , t h e n t h e s e t o f a ll m o n o t o n ic m a p p in g s fr o m ®1 £:::£®k in t o ¯ ( d e n o t e d b y [®1 £ ::: £ ®k ! ¯]) b e lo n g s t o t ypes. l e t ® 2 t ypes, t h e n t h e o r d e r o f t yp e ® ( d e n o t e d b y ord ( ®) ) will b e a n a t u r a l n u m b e r , wh ic h is d e ¯ n e d in t h e fo llo win g wa y: if ® = m t h e n ord ( ®) = 0 , if ® = [®1 £ ::: £ ®k ! ¯], wh e r e ¯; ®1; :::; ®k 2 t ypes; k > 0 , t h e n ord( ®) = max ( ord ( ®1 ) ; :::; ord( ®k ) ; ord( ¯ ) ) + 1 . if x is a va r ia b le o f t yp e ® a n d c o n s t a n t c 2 ®, t h e n ord( x ) = ord ( c) = ord( ® ) : l e t ® 2 typ e s a n d v® b e a c o u n t a b le s e t o f va r ia b le s o f t yp e ®, t h e n v = s ®2t ypes v® is t h e s e t o f a ll va r ia b le s . th e s e t o f a ll t e r m s , d e n o t e d b y ¤ = s ®2t ypes ¤ ®, wh e r e ¤ ® is t h e s e t o f t e r m s o f t yp e ®, is d e ¯ n e d in t h e fo llo win g wa y: 1 0 3 1 0 4 on dependence of interpretation algorithms of typed functional programs on canonical notion 1 . if c 2 ®; ® 2 typ e s , t h e n c 2 ¤ ®, 2 . if x 2 v®, ® 2 typ e s , t h e n x 2 ¤ ®, 3 . if ¿ 2 ¤ [®1£:::£®k!¯], ti 2 ¤ ®i , wh e r e ¯; ®i 2 t ypes; i = 1 ; :::; k; k ¸ 1 , t h e n ¿ ( t1; :::; tk ) 2 ¤ ¯ ( t h e o p e r a t io n o f a p p lic a t io n , ( t1; :::; tk ) is t h e s c o p e o f t h e a p p lic a t o r ¿ ) , 4 . if ¿ 2 ¤ ¯; xi 2 v®i wh e r e ¯; ®i 2 t ypes, i 6= j ) xi 6= xj; i; j = 1 ; :::; k; k ¸ 1 t h e n ¸x1:::xk[¿ ] 2 ¤ [®1£:::£®k!¯] ( t h e o p e r a t io n o f a b s t r a c t io n , ¿ is t h e s c o p e o f t h e a b s t r a c t o r ¸x1:::xk ) . th e n o t io n o f fr e e a n d b o u n d o c c u r r e n c e s o f va r ia b le s a s we ll a s fr e e a n d b o u n d va r ia b le a r e in t r o d u c e d in t h e c o n ve n t io n a l wa y. th e s e t o f a ll fr e e va r ia b le s in t h e t e r m t is d e n o t e d b y f v(t). te r m s t1 a n d t2 a r e s a id t o b e c o n g r u e n t ( wh ic h is d e n o t e d b y t1 ´ t2 ) if o n e t e r m c a n b e o b t a in e d fr o m t h e o t h e r b y r e n a m in g b o u n d va r ia b le s . th e o c c u r r e n c e o f fr e e va r ia b le in t h e t e r m is c a lle d in t e r n a l if it d o e s n o t e n t e r t h e a p p lic a t o r , t h e s c o p e o f wh ic h c o n t a in s a fr e e o c c u r r e n c e o f s o m e va r ia b le . th e o c c u r r e n c e o f fr e e va r ia b le in t h e t e r m is c a lle d e xt e r n a l if it d o e s n o t e n t e r t h e s c o p e o f t h e a p p lic a t o r t h a t c o n t a in s a fr e e o c c u r r e n c e o f s o m e va r ia b le . l e t t 2 ¤ ®; ® 2 t ypes a n d f v ( t ) ½ fy1; :::; yng; y0 = ( y01 ; :::; y0n ) , wh e r e yi 2 v¯i ; y0i 2 ¯i; ¯i 2 t ypes; i = 1 ; :::; n; n ¸ 0 : th e va lu e o f t h e t e r m t fo r t h e va lu e s o f t h e va r ia b le s y1; :::yn e qu a l t o y0 = ( y 0 1; :::; y 0 n ) , is d e n o t e d b y v aly0 ( t) a n d is d e ¯ n e d in t h e c o n ve n t io n a l wa y. l e t t e r m s t1; t2 2 ¤ ®; ® 2 t ypes, f v ( t1 ) [f v ( t2 ) = fy1; :::; yng; yi 2 v¯i ; ¯i 2 t ypes; i = 1 ; :::; n; n ¸ 0 , t h e n t e r m s t1 a n d t2 a r e c a lle d e qu iva le n t ( d e n o t e d b y t1 » t2 ) if fo r a n y y0 = ( y 0 1; :::; y 0 n ) , wh e r e y 0 i 2 v¯i ; i = 1 ; :::; n we h a ve t h e fo llo win g : v aly0 ( t1 ) = v aly0 ( t2 ) . a t e r m t 2 ¤ ®; ® 2 t ypes, is c a lle d a c o n s t a n t t e r m wit h va lu e a 2 ® if t » a. fu r t h e r , we a s s u m e t h a t m is a r e c u r s ive s e t a n d t h e c o n s id e r e d t e r m s u s e va r ia b le s o f a n y o r d e r a n d c o n s t a n t s o f o r d e r · 1 , wh e r e c o n s t a n t s o f o r d e r 1 a r e s t r o n g c o m p u t a b le , m o n o t o n ic fu n c t io n s wit h in d e t e r m in a t e va lu e s o f a r g u m e n t s . a fu n c t io n f : m k ! m; k ¸ 1 , wit h in d e t e r m in a t e va lu e s o f a r g u m e n t s , is s a id t o b e s t r o n g c o m p u t a b le if t h e r e e xis t s a n a lg o r it h m , wh ic h s t o p s wit h t h e va lu e f ( m1; :::; mk ) 2 m fo r a ll m1; :::; mk 2 m, s e e [2 ]. a t e r m t 2 ¤ wit h a ¯ xe d o c c u r r e n c e o f a s u b t e r m ¿1 2 ¤ ®, wh e r e ® 2 t ypes, is d e n o t e d b y t¿1 , a n d a t e r m wit h t h is o c c u r r e n c e o f ¿1 r e p la c e d b y ¿2, wh e r e ¿22 ¤ ®, is d e n o t e d b y t¿2. to s h o w m u t u a lly d i®e r e n t va r ia b le s o f in t e r e s t x1; :::; xk; k ¸ 1 , o f a t e r m t, t h e n o t a t io n t[x1; :::; xk] is u s e d . th e n o t a t io n t[t1; :::; tk] d e n o t e s t h e t e r m o b t a in e d b y t h e s im u lt a n e o u s s u b s t it u t io n o f t h e t e r m s t1; :::; tk fo r a ll fr e e o c c u r r e n c e s o f t h e va r ia b le s x1; :::; xk, r e s p e c t ive ly, wh e r e xi 2 v®i ; i 6= j ) xi 6´ xj, ti 2 ¤ ®i ; ®i 2 t ypes; i; j = 1 ; ::; k; k ¸ 1 . a t e r m o f t h e fo r m ¸x1:::xk[¿ [x1; :::; xk]]( t1; :::; tk ) , wh e r e xi 2 v®; i 6= j ) xi 6´ xj ; ¿ 2 ¤ ; ti 2 ¤ ®i ; ®i 2 t ypes; i; j = 1 ; :::; k; k ¸ 1 , is c a lle d a ¯-r e d e x, it s c o n vo lu t io n is t h e t e r m ¿ [t1; :::; tk]. th e s e t o f a ll p a ir s ( ¿0; ¿1 ) , wh e r e ¿0 is a ¯-r e d e x a n d ¿1 is it s c o n vo lu t io n , is c a lle d a n o t io n o f ¯-r e d u c t io n a n d is d e n o t e d b y ¯. a o n e -s t e p ¯-r e d u c t io n ( !¯ ) a n d ¯r e d u c t io n ( !!¯ ) a r e d e ¯ n e d in t h e c o n ve n t io n a l wa y. a t e r m c o n t a in in g n o ¯-r e d e xe s is c a lle d a ¯-n o r m a l fo r m . th e s e t o f a ll ¯-n o r m a l fo r m s is d e n o t e d b y ¯-n f . ±-r e d e x h a s a fo r m f ( t1; :::; tk ) , wh e r e f 2 [m k ! m ], ti 2 ¤ m ; i = 1 ; :::; k; k ¸ 1 , it s c o n vo lu t io n is e it h e r m 2 m a n d in t h is c a s e f ( t1; :::; tk ) » m o r a s u b t e r m ti a n d in t h is c a s e f ( t1; :::; tk ) » ti; i = 1 ; :::; k. a ¯ xe d s e t o f t e r m p a ir s ( ¿0; ¿1 ) , wh e r e ¿0 is a ±-r e d e x a n d ¿1 is it s c o n vo lu t io n , is c a lle d a n o t io n o f ±-r e d u c t io n a n d is d e n o t e d b y ±. a o n e -s t e p ±-r e d u c t io n ( !± ) a n d ±-r e d u c t io n ( !!± ) a r e d e ¯ n e d in t h e c o n ve n t io n a l wa y. d. grigoryna 1 0 5 a o n e -s t e p ¯±-r e d u c t io n ( !¯± ) a n d ¯±-r e d u c t io n ( !!¯± ) a r e d e ¯ n e d in t h e c o n ve n t io n a l wa y. a t e r m c o n t a in in g n o ¯±-r e d e xe s is c a lle d a n o r m a l fo r m . th e s e t o f a ll n o r m a l fo r m s is d e n o t e d b y nf . a n o t io n o f ±-r e d u c t io n is c a lle d a s in g le -va lu e d n o t io n o f ±-r e d u c t io n , if ± is a s in g le va lu e d r e la t io n , i.e ., if ( ¿0; ¿1 ) 2 ± a n d ( ¿0; ¿2 ) 2 ±, t h e n ¿1 ´ ¿2, wh e r e ¿0; ¿1; ¿2 2 ¤ m . a n o t io n o f ±-r e d u c t io n is c a lle d a n e ®e c t ive n o t io n o f ±-r e d u c t io n if t h e r e e xis t s a n a lg o r it h m , wh ic h fo r a n y t e r m f ( t1; :::; tk ) , wh e r e f 2 [m k ! m ], ti 2 ¤ m ; i = 1 ; :::; k; k ¸ 1 , g ive s it s c o n vo lu t io n if f ( t1; :::; tk ) is a ±-r e d e x a n d s t o p s wit h a n e g a t ive a n s we r o t h e r wis e . de¯nition 1. [3 ] an e®ective, single-valued notion of ±-reduction is called a canonical notion of ±-reduction if: 1. t 2 ¯-nf ,t » m; m 2 m n f?g ) t !! ±m, 2. t 2 ¯-nf ; f v ( t ) = ;; t » ? ) t !!± ?. t heor em 1. (on canonical notion of ±-reduction). f or every recursive set of strong computable, monotonic functions with indeterminate values of arguments there exists a canonical notion of ±-reduction. p r o ve d in [3 ]. typ e d fu n c t io n a l p r o g r a m p is t h e fo llo win g s ys t e m o f e qu a t io n s : p 8 >< >: f1 = t1[f1; :::; fn] ::: fn = tn[f1; :::; fn] ( 1 ) wh e r e fi 2 v®i ; i 6= j ) fi 6´ fj; ti[f1; :::; fn] 2 ¤ ®i , fv ( ti[f1; :::; fn]) ½ ff1; :::; fng, ®i 2 t ypes; i; j = 1 ; :::; n; n ¸ 1 ; ®1 = [m k ! m ]; k ¸ 1 . e ve r y t yp e d fu n c t io n a l p r o g r a m p h a s t h e le a s t s o lu t io n ( s e e [1 ]) . l e t ( f1; :::; fn ) 2 ®1 £ ::: £ ®n b e t h e le a s t s o lu t io n o f p , t h e n t h e ¯ r s t c o m p o n e n t f1 2 [m k ! m] o f t h e le a s t s o lu t io n is s a id t o b e t h e b a s ic s e m a n t ic s o f t h e p r o g r a m p a n d is d e n o t e d b y fp. f ix ( p ) = f ( m1; :::; mk; m ) j fp ( m1; :::; mk ) = m; wh e r e m; m1; :::; mk 2 m; k ¸ 1 g. 2 . in t e r p r e t a t io n a lg o r it h m s , ?-d e p e n d e n c e th e in p u t o f t h e in t e r p r e t a t io n a lg o r it h m a is a p r o g r a m p o f t h e fo r m ( 1 ) , a t e r m f1 ( m1; :::; mk ) , wh e r e mi 2 m , i = 1 ; :::; k; k ¸ 1 a n d a c a n o n ic a l n o t io n o f ±-r e d u c t io n . a lg o r it h m a s t o p s wit h t h e r e s u lt m 2 m o r wo r ks in ¯ n it e ly. a lg o r it h m a u s e s t h r e e kin d s o f o p e r a t io n s : s u b s t it u t io n o f t h e t e r m s t1; :::; tn in s t e a d o f s o m e fr e e o c c u r r e n c e s o f va r ia b le s f1; :::; fn, o n e -s t e p ¯-r e d u c t io n a n d o n e -s t e p ±-r e d u c t io n . p roca ( p ) = f ( m1; :::; mk; m ) j a lg o r it h m a s t o p s fo r t h e p r o g r a m p a n d t h e t e r m f1 ( m1; :::; mk ) wit h t h e r e s u lt m, wh e r e m; m1; :::; mk 2 m; k ¸ 1 g in t e r p r e t a t io n a lg o r it h m a is c o n s is t e n t if fo r a n y p r o g r a m p a n d fo r a n y c a n o n ic a l n o t io n o f ±-r e d u c t io n we h a ve : p roca ( p ) ½ f ix ( p ) . t heor em 2. e very interpretation algorithm a is consistent. 1 0 6 on dependence of interpretation algorithms of typed functional programs on canonical notion p r oof. fo llo ws fr o m t h e r e s u lt s o f [4 ]. de¯nition 2. an interpretation algorithm a ?-depends on canonical notion of ±-reduction if there exist ±1 and ±2 canonical notions of ±-reduction, program p and m1; :::; mk 2 m; k ¸ 1 such that: ( m1; :::; mk; ? ) 2 p roca ( p ) for ±1 and ( m1; :::; mk; ? ) 62 p roca ( p ) for ±2. to s h o w a s e qu e n c e fi1 ; :::; fis; s ¸ 1 o f s o m e fr e e o c c u r e n c e s o f va r ia b le s o f t h e s e t ff1; :::; fng in t h e t e r m t, t h e n o t io n t < fi1; :::; fis > is u s e d . th e n o t io n t < ti1; :::; tis > d e n o t e s t h e t e r m o b t a in e d b y t h e s im u lt a n e o u s s u b s t it u t io n o f t h e t e r m s ti1; :::; tis fo r fr e e o c c u r e n c e s fi1 ; :::; fis, r e s p e c t ive ly. de¯nition 3. (interpretation algorithms) f s, p e s, l e s, p is, l is, p as, act s te p 1 : f s, p e s, l e s, p is, l is, p as, act : if t 2 n f a n d f v ( t ) \ ff1; :::; fng = ; t h e n t. e ls e g o t o s t e p 2 . s te p 2 : f s, p e s, l e s, p is, l is : if t ´ t¿ wh e r e ¿ is a le ft m o s t r e d e x ( ±-r e d e x o r ¯-r e d e x) , t h e n a( p; t¿0 ) , wh e r e ¿ 0 is t h e c o n vo lu t io n o f t h e ¿ , a 2 ff s; p es; les; p is; lisg, e ls e g o t o s t e p 3 . act, p as : if t ´ tfi ; ( 0 · i · n) , wh e r e tfi is t h e t e r m t wit h a ¯ xe d le ft m o s t fr e e o c c u r r e n c e o f a va r ia b le o f t h e s e t ff1; :::; fng, wh ic h is lo c a t e d t o t h e le ft o f t h e le ft m o s t r e d e x, t h e n a ( p; tti ) , wh e r e a 2 fact; p asg, e ls e g o t o s t e p 3 . s te p 3 : f s : if t ´ t[f1; :::; fn], t h e n f s ( p; t[t1; :::; tn]) . p e s : if t ´ t < fi1 ; :::; fik >, wh e r e fi1 ; :::; fik ; k > 0 is t h e s e qu e n c e o f a ll e xt e r n a l fr e e o c c u r r e n c e s o f va r ia b le s o f t h e s e t ff1; :::; fng, t h e n p es ( p; t < ti1; :::; tik >) . l e s : if t ´ tfi , wh e r e fi is t h e le ft m o s t e xt e r n a l fr e e o c c u r r e n c e o f a va r ia b le o f t h e s e t ff1; :::; fng, t h e n les ( p; tti ) . p is : if t ´ t < fi1; :::; fik >, wh e r e fi1; :::; fik ; k > 0 is t h e s e qu e n c e o f a ll in t e r n a l fr e e o c c u r r e n c e s o f va r ia b le s o f t h e s e t ff1; :::; fng, t h e n p is ( p; t < ti1 ; :::; tik >) . l is : if t ´ tfi , wh e r e fi is t h e le ft m o s t in t e r n a l fr e e o c c u r r e n c e o f a va r ia b le o f t h e s e t ff1; :::; fng, t h e n lis ( p; tti ) . act : if t ´ t¿ wh e r e ¿ ´ ¸x1:::xr[¿[x1; :::; xr]]( ¿1; :::; ¿r ) a n d ¿ is a le ft m o s t r e d e x, t h e n act ( p; t¿ [act (p;¿1);:::;act (p;¿r)] ) , e ls e g o t o s t e p 4 . p as : if t ´ t¿ wh e r e ¿ ´ ¸x1:::xr[¿[x1; :::; xr]]( ¿1; :::; ¿r ) a n d ¿ is a le ft m o s t r e d e x, t h e n p as ( p; t¿ [¿1;:::;¿r] ) , e ls e g o t o s t e p 4 . s te p 4 : act, p as : if t ´ t¿ wh e r e ¿ is a le ft m o s t r e d e x, wh ic h is a ±-r e d e x, t h e n a ( p; t¿0 ) , wh e r e ¿ 0 is t h e c o n vo lu t io n o f t h e ¿, a 2 fact; p asg. t heor em 3. any interpretation algorithm f s, p e s, l e s, p is, l is, p as, act ?-depends on canonical notion of ±-reduction. p r oof. l e t u s ¯ x m = n [ f?g, wh e r e n = f 0 ; 1 ; 2 ; :::g a n d c = fnot eqg wh e r e not eq 2 [m 2 ! m] is a b u ilt -in fu n c t io n a n d fo r e ve r y m1; m2 2 m we h a ve : not eq ( m1; m2 ) = ( 1 ; if m1; m2 2 n and m1 6= m2 ?; otherwise d. grigoryna 1 0 7 it is e a s y t o s e e t h a t not eq is a s t r o n g c o m p u t a b le , n a t u r a lly e xt e n d e d fu n c t io n wit h in d e t e r m in a t e va lu e s o f a r g u m e n t s ( a fu n c t io n is s a id t o b e n a t u r a lly e xt e n d e d , if it s va lu e is ? wh e n e ve r t h e va lu e o f a t le a s t o n e o f t h e a r g u m e n t s is ? ) . th e r e fo r e , fr o m th e o r e m 1 it fo llo ws t h a t t h e r e e xis t s t h e fo llo win g c a n o n ic a l n o t io n o f ±-r e d u c t io n ± fo r t h e s e t c: ± is : ( not eq ( n1; n2 ) ; 1 ) 2 ±, wh e r e n1; n2 2 n a n d n1 6= n2 ( not eq ( n; n) ; ?) 2 ±, wh e r e n 2 n ( not eq ( n; ?) ; ? ) 2 ±, wh e r e n 2 n ( not eq ( ?; n) ; ? ) 2 ±, wh e r e n 2 n ( not eq ( ?; ? ) ; ? ) 2 ± l e t u s d e ¯ n e t wo c a n o n ic a l n o t io n s o f ±-r e d u c t io n ±1 a n d ±2. ±1 is : ( not eq ( n1; n2 ) ; 1 ) 2 ±1, wh e r e n1; n2 2 n a n d n1 6= n2 ( not eq ( t; t ) ; ?) 2 ±1, wh e r e t 2 ¤ m ( not eq ( t; ? ) ; ?) 2 ±1, wh e r e t 2 ¤ m ( not eq ( ?; t ) ; ?) 2 ±1, wh e r e t 2 ¤ m ±2 is : ( not eq ( n1; n2 ) ; 1 ) 2 ±2, wh e r e n1; n2 2 n a n d n1 6= n2 ( not eq ( m; m ) ; ?) 2 ±2, wh e r e m 2 m ( not eq ( t; ? ) ; ?) 2 ±2, wh e r e t 2 ¤ m ( not eq ( ?; t ) ; ?) 2 ±2, wh e r e t 2 ¤ m it is e a s y t o s e e t h a t ±1 is a n e ®e c t ive , s in g le -va lu e d n o t io n o f ±-r e d u c t io n . th e r e fo r e , t o s h o w t h a t ±1 is a c a n o n ic a l n o t io n o f ±-r e d u c t io n it s u ± c e s t o s h o w t h a t ± ½ ±1. l e t ( ¿1; ¿2 ) 2 ±, wh e r e ¿1; ¿2 2 m , t h e n t h e fo llo win g c a s e s a r e p o s s ib le : ¿1 ´ n1 a n d ¿2 ´ n2, wh e r e n1; n2 2 n a n d n1 6= n2, t h e n it is o b vio u s t h a t ( not eq ( n1; n2 ) ; 1 ) 2 ±1. ¿1 ´ ¿2 ´ n, wh e r e n 2 n , t h e n fr o m ( not eq ( t; t) ; ? ) 2 ±1, wh e r e t 2 ¤ m , fo llo ws t h a t ( not eq ( n; n) ; ?) 2 ±1. ¿1 ´ n a n d ¿2 ´ ?, wh e r e n 2 n , t h e n fr o m ( not eq ( t; ? ) ; ? ) 2 ±1, wh e r e t 2 ¤ m , fo llo ws t h a t ( not eq ( n; ? ) ; ? ) 2 ±1. ¿1 ´ ? a n d ¿2 ´ n, wh e r e n 2 n , t h e n fr o m ( not eq ( ?; t ) ; ? ) 2 ±1, wh e r e t 2 ¤ m , fo llo ws t h a t ( not eq ( ?; n) ; ? ) 2 ±1. ¿1 ´ ? a n d ¿2 ´ ?, t h e n fr o m ( not eq ( t; t) ; ? ) 2 ±1, wh e r e t 2 ¤ m , fo llo ws t h a t ( not eq ( ?; ? ) ; ? ) 2 ±1. it is e a s y t o s e e t h a t ±2 is a n e ®e c t ive , s in g le -va lu e d n o t io n o f ±-r e d u c t io n . th e r e fo r e , t o s h o w t h a t ±2 is a c a n o n ic a l n o t io n o f ±-r e d u c t io n it s u ± c e s t o s h o w t h a t ± ½ ±2. l e t ( ¿1; ¿2 ) 2 ±, wh e r e ¿1; ¿2 2 m , t h e n t h e fo llo win g c a s e s a r e p o s s ib le : ¿1 ´ n1 a n d ¿2 ´ n2, wh e r e n1; n2 2 n a n d n1 6= n2, t h e n it is o b vio u s t h a t ( not eq ( n1; n2 ) ; 1 ) 2 ±2. ¿1 ´ ¿2 ´ n, wh e r e n 2 n, t h e n fr o m ( not eq ( m; m ) ; ? ) 2 ±2, wh e r e m 2 m, fo llo ws t h a t ( not eq ( n; n) ; ? ) 2 ±2. ¿1 ´ n a n d ¿2 ´ ?, wh e r e n 2 n , t h e n fr o m ( not eq ( t; ? ) ; ? ) 2 ±2, wh e r e t 2 ¤ m , fo llo ws t h a t ( not eq ( n; ? ) ; ? ) 2 ±2. 1 0 8 on dependence of interpretation algorithms of typed functional programs on canonical notion ¿1 ´ ? a n d ¿2 ´ n, wh e r e n 2 n , t h e n fr o m ( not eq ( ?; t ) ; ?) 2 ±2, wh e r e t 2 ¤ m , fo llo ws t h a t ( not eq ( ?; n) ; ? ) 2 ±2. ¿1 ´ ? a n d ¿2 ´ ?, t h e n fr o m ( not eq ( m; m) ; ? ) 2 ±2, wh e r e m 2 m, fo llo ws t h a t ( not eq ( ?; ? ) ; ?) 2 ±2. l e t p b e t h e fo llo win g p r o g r a m , wh e r e f1; f2 2 v[m !m]; x 2 vm p ( f1 = ¸x[not eq ( f2 ( x) ; f2 ( x ) ) ] f2 = ¸x[f2 ( x ) ] fo r ±1, p r o g r a m p a n d f s; p es; les; p is; lis; p as; act we h a ve : f1 ( 0 ) ; ¸x[not eq ( f2 ( x) ; f2 ( x) ) ]( 0 ) !¯ not eq ( f2 ( 0 ) ; f2 ( 0 ) ) !±1?; th e r e fo r e ( 0 ; ? ) 2 p rocf s ( p ) , ( 0 ; ? ) 2 p rocp es ( p ) , ( 0 ; ?) 2 p rocles ( p ) , ( 0 ; ? ) 2 p rocp is ( p ) , ( 0 ; ?) 2 p roclis ( p ) , ( 0 ; ?) 2 p rocp as ( p ) , ( 0 ; ? ) 2 p rocact ( p ) . fo r ±2, p r o g r a m p a n d act , lis, p as, les we h a ve : f1 ( 0 ) ; ¸x[not eq ( f2 ( x) ; f2 ( x) ) ]( 0 ) !¯ not eq ( f2 ( 0 ) ; f2 ( 0 ) ) ; not eq ( ¸x[f2 ( x) ]( 0 ) ; f2 ( 0 ) ) !¯ not eq ( f2 ( 0 ) ; f2 ( 0 ) ) ; ... a n d s o o n . th e r e fo r e ( 0 ; ?) 62 p rocles ( p ) , ( 0 ; ? ) 62 p roclis ( p ) , ( 0 ; ? ) 62 p rocp as ( p ) , ( 0 ; ? ) 62 p rocact ( p ) . fo r ±2, p r o g r a m p a n d f s, p es, p is we h a ve : f1 ( 0 ) ; ¸x[not eq ( f2 ( x) ; f2 ( x) ) ]( 0 ) !¯ not eq ( f2 ( 0 ) ; f2 ( 0 ) ) ; not eq ( ¸x[f2 ( x) ]( 0 ) ; ¸x[f2 ( x ) ]( 0 ) ) !¯ not eq ( f2 ( 0 ) ; f2 ( 0 ) ) ) ; ... a n d s o o n . th e r e fo r e ( 0 ; ? ) 62 p rocf s ( p ) , ( 0 ; ? ) 62 p rocp es ( p ) , ( 0 ; ?) 62 p rocp is ( p ) . in c o n c lu s io n , fo r e a c h in t e r p r e t a t io n a lg o r it h m a 2 ff s; p es; les; p is; lis; p as; act g t h e r e e xis t ±1 a n d ±2 c a n o n c ia l n o t io n s o f ±-r e d u c t io n a n d p r o g r a m p s u c h t h a t ( 0 ; ? ) 2 p roca ( p ) fo r ±1 a n d ( 0 ; ? ) 62 p roca ( p ) fo r ±2, t h e r e fo r e a ?-d e p e n d s o n c a n o n ic a l n o t io n o f ±-r e d u c t io n . refer ences [1 ] s . a . n ig iya n \ fu n c t io n a l l a n g u a g e s " , p rogramming and computer software, vo l. 1 7 , n o . 5 , p p . 2 9 0 -2 9 7 , 1 9 9 2 . [2 ] s . a . n ig iya n , \ on n o n -c la s s ic a l t h e o r y o f c o m p u t a b ilit y" , p roceedings of the ysu, p hysical and m athematical sciences, n o .1 , p p .5 2 -6 0 , 2 0 1 5 . [3 ] s . a . n ig iya n a n d t.v .k h o n d ka r ya n , \ on c a n o n ic a l n o t io n o f ±-r e d u c t io n a n d o n t r a n s la t io n o f t yp e d ¸-t e r m s in t o u n t yp e d ¸-t e r m s " , p roceedings of the ysu, p hysical and m athematical sciences, n o . 1 , p p . 4 6 -5 2 , 2 0 1 7 . [4 ] r . y u . h a ko p ia n , \ on p r o c e d u r a l s e m a n t ic s o f s t r o n g t yp e d fu n c t io n a l p r o g r a m s " , p roceedings of ysu, natural sciences, ( in r u s s ia n ) , n o . 3 , p p .5 9 -6 9 , 2 0 0 8 . d. grigoryna 1 0 9 submitted 10.10.2017, accepted 18.01.2018. îçåç½³óí³í ýáõýïóçáý³é íñ³·ñ»ñç çýï»ñåñ»ï³óç³ûç ³é·áñçãùý»ñç ï³ëí³íáõãûáõýá ï³ýáýçï ±-黹áõïóç³ûç ·³õ³÷³ñçó ¸. ¶ñç·áñû³ý ²ù÷á÷áõù ²ßë³ï³ýùáõù ¹çï³ñïí³í »ý ïçåç½³óí³í ýáõýïóçáý³é íñ³·ñ»ñç çýï»ñåñ»ï³óç³ûç ³é·áñçãùý»ñá: æýï»ñåñ»ï³óç³ûç ³é·áñçãùá ñçùýí³í ¿ ï»õ³¹ñù³ý, ¯é»¹áõïóç³ûç ¨ ï³ýáýçï ±-黹áõïóç³ûç ·áñíáõáõãûáõýý»ñç íñ³: îçåç½³óí³í ýáõýïóçáý³é íñ³·ñ»ñç ñçùý³ï³ý ë»ù³ýïçï³ý ³ýáñáß ³ñ·áõù»ýïý»ñáí ýáõýïóç³ ¿, áñá ÷áùñ³·áõûý éáõíù³ý ñçùý³ï³ý µ³õ³¹ñçãý ¿: ºã» ñçùý³ï³ý ë»ù³ýïçï³ûç ³ñå»ùá áñáß ³ñå»ùý»ñç ¹»åùáõù ³ýáñáß ¿, ³å³ çýï»ñåñ»ï³óç³ûç ³é·áñçãùá ï³ù ï³ý· ¿ ³éýáõù ? ³ñå»ùáí, ï³ù ³ßë³ïáõù ¿ ³ýí»ñç: òáõûó ¿ ïñí³í, áñ 7 ñ³ûïýç çýï»ñåñ»ï³óç³ûç ³é·áñçãùý»ñá ?-ï³ëí³í »ý ï³ýáýçï ±-黹áõïóç³ûç ·³õ³÷³ñçó: ²û¹ ³é·áñçãù»ñá ñ»ï¨û³éý »ý` fs (éçñí ï»õ³¹ñù³ý), pes (½áõ·³ñ»é ³ñï³ùçý ï»õ³¹ñù³ý), les (ó³ë ³ñï³ùçý ï»õ³¹ñù³ý), pis (½áõ·³ñ»é ý»ñùçý ï»õ³¹ñù³ý), lis (ó³ë ý»ñùçý ï»õ³¹ñù³ý), act (³ïïçí ³é·áñçãù), pas (å³ëçí ³é·áñçãù). î çàâèñèìîñòè àëãîðèòìîâ èíòåðïðåòàöèè òèïèçèðîâàííûõ ôóíêöèîíàëüíûõ ïðîãðàìì îò êàíîíè÷åñêîãî ïîíÿòèÿ ±-ðåäóêöèè ä. ãðèãîðÿí àííîòàöèÿ â äàííîé ðàáîòå ðàññìàòðèâàþòñÿ èíòåðïðåòàòîðû òèïèçèðîâàííûõ ôóíêöèîíàëüíûõ ïðîãðàìì. àëãîðèòì èíòåðïðåòàöèè îñíîâàí íà ïîäñòàíîâêàõ, ¯ðåäóêöèè è êàíîíè÷åñêîé ±-ðåäóêöèè. îñíîâíàÿ ñåìàíòèêà òèïèçèðîâàííûõ ôóíêöèîíàëüíûõ ïðîãðàìì åñòü ôóíêöèÿ ñ íåîïðåäåëåííûìè çíà÷åíèÿìè àðãóìåíòîâ, êîòîðàÿ ÿâëÿåòñÿ ãëàâíîé êîìïîíåíòîé åå íàèìåíüøåãî ðåøåíèÿ. åñëè çíà÷åíèå îñíîâíîé ñåìàíòèêè, äëÿ íåêîòîðûõ çíà÷åíèé àðãóìåíòîâ, åñòü íåîïðåäåëåííîñòü, òî àëãîðèòì èíòåðïðåòàöèè ëèáî îñòàíàâëèâàåòñÿ ñî çíà÷åíèåì ?, ëèáî ðàáîòàåò áåñêîíå÷íî. ïîêàçàíî, ÷òî ñåìü èçâåñòíûõ àëãîðèòìîâ èíòåðïðåòàöèè ?-çàâèñÿò îò êàíîíè÷åñêîãî ïîíÿòèÿ ±-ðåäóêöèè. âîò ýòè àëãîðèòìû: fs (ïîëíîé ïîäñòàíîâêè), pes (ïàðàëëåëüíîé âíåøíåé ïîäñòàíîâêè), les (ëåâîé âíåøíåé ïîäñòàíîâêè), pis (ïàðàëëåëüíîé âíóòðåííåé ïîäñòàíîâêè), lis (ëåâîé âíóòðåííåé ïîäñòàíîâêè), act (àêòèâíûé àëãîðèòì), pas (ïàññèâíûé àëãîðèòì). mathematical problems of computer science 53, 7–13, 2020. udc 510.6 polynomial bounded proof complexities for some classes of dnf-tautologies garik v. petrosyan yerevan state university e-mail: garik.petrosyan.1@gmail.com abstract in this paper, we present the results on frege proof complexities of some dnftautologies. at first we introduce the notion of complete dnfs and prove that complete dnfs are tautologies, we also show that if the proof complexities for the set of complete dnfs are polynomially bounded, then the set of dnf-tautologies d, for which the number of non-negated variables in every conjunct is o(log(d)), also has polynomially bounded proof complexities. later we show that the set of all balanced dnf-tautologies has polynomial proof complexities. keywords: frege systems, proof complexity, dnf, complete dnf, balanced formulas. 1. introduction one of the most fundamental problems of the proof complexity theory is to find an efficient proof system for classical propositional calculus. there is a widespread understanding that polynomial-time computability is the correct mathematical model of feasible computation. according to the opinion, a truly ”effective” system should have a polynomial-size p(n) proof for every tautology of size n. in [1], cook and reckhow named such a system a super system. they showed that np = conp iff there exists a super system. it is well known that many systems are not super. this question about the frege system, the most natural calculi for propositional logic, is still open. in many papers, some specific sets of tautologies are introduced, and it is shown that the question about polynomially bounded sizes for frege-proofs of all tautologies is reduced to an analogous question for a set of specific tautologies. in particular, lutz strasburger introduced in [2] the notion of balanced formulas and showed that if there are polynomially bounded frege proofs for the set of balanced tautologies, then the frege systems are super. an analogous result for some other class of tautologies is proved in [3]. one of the well-known classes of tautologies is the class of tautologies, given in disjunctive normal form (dnf-tautologies), and it is an open question if the frege-proof complexities for the set of dnf-tautologies have polynomial upper bounds. the frege-proof complexities of some dnf-tautology classes are investigated in this paper. at first the notion of complete 7 8 polynomial bounded proof complexities for some classes of dnf-tautologies dnf is introduced, and it is proved that if the proof complexities for the set of complete dnfs are polynomially bounded, then the set of dnf-tautologies d, for which the number of non-negated variables in every conjunct is o(log(d)), also has polynomially bounded proof complexities. then it is proved that the proof complexities of the set of balanced dnf-tautologies has polynomially bounded proof complexities as well. 2. main notions and notations here we give basic definitions, which are necessary to give main results. definition 1: a frege system f uses a denumerable set of propositional variables, a finite complete set of propositional connectives; f has a finite set of inference rules defined by a figure of the form a1a2...an b (the rules of inference with zero hypotheses are axiom schemes); f must be sound and complete, i.e., for each rule of inference a1a2...an b every truth-value assignment, satisfying a1a2...an, also satisfies b, and f must prove every tautology. the particular choice of a language for the presented propositional formulas is immaterial in this consideration. however, because of some technical reasons, we assume that the language contains the propositional variables pi (i ≥ 1) or pij (i ≥ 1; j ≥ 1) logical connectives ¬,∧,∨,⊃ and parentheses (, ). note that some parentheses and ∧ can be omitted in generally accepted cases. note that our convention for serial disjunction a1∨a2∨...∨an (conjunction a1 ∧a2 ∧ ...∧an) associated from left to right. by |ϕ| we denote the size of the formula ϕ defined as the number of all entries of logical signs in it. it is obvious that the full size of the formula, which is understood to be the number of all symbols, is bounded by some linear function in |ϕ|. in the theory of proof complexity, the four main characteristics of the proof are: tcomplexity (length), defined as the number of proof steps, l-complexity (size), defined as the sum of sizes for all formulas in the proof (size), s-complexity (space), informally defined as the maximum of minimal sum of sizes for formulas on blackboard needed to verify all steps in the proof (formal definitions are, for example, in [2]) and w-complexity (width), defined as the maximum of sizes of the proof formulas. definition 2: let φ be a proof system and ϕ be a tautology. we denote by tφϕ (l φ ϕ, s φ ϕ, w φ ϕ) the minimal possible value of t-complexity (l-complexity, s-complexity, w-complexity) for all -proofs of tautology ϕ. by analogy, we can define the mentioned proof complexity characteristics for the proof of any formula a from premises γ and denote them respectively by t φ γ`a (l φ γ`a, s φ γ`a, w φ γ`a). let m be some set of tautologies. definition 3: we call the φ-proofs of tautologies from the set m t-polynomially (lpolynomially, s-polynomially, w-polynomially) bounded if there is a polynomial p() such that tφϕ ≤ p(|ϕ|) (lφϕ ≤ p(|ϕ|) ,sφϕ ≤ p(|ϕ|) ,wφϕ ≤ p(|ϕ|)) for all tautologies ϕ from m. note that if φ-proofs of tautologies from the set m are l-polynomially bounded they are also t-polynomially, s-polynomially, w-polynomially bounded. following the usual terminology, we call the variables and negated variables literals for classical logic. the conjunct k can be represented simply as a set of literals (no conjunct contains a variable and its negation simultaneously). in [4], the notion of balanced formulas is introduced in the following way. g. petrosyan 9 definition 4: a propositional formula is called balanced if every variable has only two occurrences in it, one positive and one with negative. in [4], it is shown that the problem on l-polynomially bounded sizes of proofs for all tautologies is reduced to the problem on l-polynomially bounded sizes of proofs for all balanced tautologies. 3. main results in this part, frege-proof complexities for some classes of dnf-tautologies are investigated. a. here the notion of complete dnf-tautologies is introduced, and it is proved that if the set of complete dnf-tautologies has l-polynomially therefore also t-polynomially, spolynomially, w-polynomially bounded proofs, then the set of dnf-tautologies d, where the number of non-negated variable in each conjunct is o(log(|d|)), also has l-polynomially therefore also t-polynomially, s-polynomially, w-polynomially bounded proofs. let all variables of a dnf n1 ∨n2 ∨ ...∨nn be negated variables. conjunct k is called representative for d if k contains at least one variable (without negation) from each ni (1 ≤ i ≤ n), and every variable of k is from d. definition 5: dnf d1 = c1 ∨ c2 ∨ ... ∨ cm is called a completion of dnf d2 = n1 ∨ n2 ∨ ...∨nn iff for every representative k of d2 there is a ni (1 ≤ i ≤ n), which is a subset of k, and the expression d = (n1 ∨n2 ∨ ...∨nn)∨(c1 ∨c2 ∨ ...∨cm) is called a complete dnf. theorem 1: complete dnfs are tautologies. proof. let’s assume the opposite: there is a complete dnf d = (n1∨n2∨...∨nn)∨(c1∨ c2∨...∨cm) such that it is not a tautology. if d = 0 in any collection, then n1∨n2∨...∨nn should also be 0 in that collection. n1 ∨n2 ∨ ...∨nn = 0, only if each conjunct is equal to 0, therefore, we have that in each conjunct ni (1 ≤ i ≤ n) there is a literal equal to 0. if we consider the conjunct p constructed by the conjunction of the variables of these literals, as we have a complete dnf, there is a cj (1 ≤ j ≤ m) such that cj is a subset of p , hence cj = 1, but we have assumed that d = 0, which is a contradiction. from this proof it is easy to see that all dnf-tautologies, the conjuncts of which do not contain a variable and a negated variable simultaneously, are complete dnfs. to evaluate the proof complexities of dnf-tautologies by reducing the proof to the proof of complete dnftautologies, we must use some transformations of formulas, therefore we give the following auxiliary statements, which are used to perform those transformations. lemma 1: for all formulas a,b and c, the set of the following formulas 1. a∨b ⊃ c ≡ (a ⊃ c) ∧ (b ⊃ c) 2. a ⊃ a∨b 3. a ⊃ (b ∨c) ≡ (a ⊃ b) ∨ (a ⊃ c) 4. b ⊃ (a ⊃ a∧b) 5. (a ⊃ b) ⊃ ((a∨c) ⊃ (b ∨c) has l-polynomially bounded proofs. 10 polynomial bounded proof complexities for some classes of dnf-tautologies proof. the proof is obvious. most dnf-tautologies , the proof complexities of which are being investigated, have small conjuncts sizes compered to their size. using complete dnfs, we can construct polynomial proofs for such dnf-tautologies. the following theorem gives such a construction. theorem 2: if the set of all complete dnfs has l-polynomially bounded proofs then, the set of dnf-tautologies d, where the number of non-negated variables in each conjunct is o(log(|d|)), also has l-polynomially bounded proofs. proof. let’s consider d dnf-tautology, where the number of non-negated variables in each conjunct is o(log(|d|)). for each conjunct, we separate the sub-conjunct of variables with positive entrance and the set of all such conjuncts and its subsets we denote by p . since the number of non-negated variables in each conjunct is o(log(|d|)), there is such a polynomial p() function that |p| ≤ p(|d|). if dnf is a tautology, it should have at least one conjunct, where all the variables are negated, otherwise it does not cover the 0 point we denote the disjunction of all such conjuncts by n. if d is a tautology, then there is a completion of n such that all conjuncts are from p . if such a completion did not exist, we would take a conjunct constructed by taking one variable from each conjunct of n, and this conjunct would not be covered by p , therefore if we set the values of these variables 1 and the values of all other variables 1, the value of d will be 0. let’s denote that completion by c. n ∨ c is a completion dnf; to prove d, we need to prove n ∨c and n ∨c ⊃ d, the first part follows from our condition. let’s prove the second part n ⊃ d. from lemma 1.1 we get two tautologies to prove n ⊃ d and c ⊃ d. from lemma 1.2 we get the polynomial proof of n ⊃ d. now let’s prove c ⊃ d. suppose c = c1 ∨ c2 ∨ ... ∨ cm, where ci(1 ≤ i ≤ m) is a conjunct. using lemma 1.1, we can convert c ⊃ d into (c1 ⊃ d) ∧ (c2 ⊃ d) ∧ ...∧ (cm ⊃ d). as ci (1 ≤ i ≤ m) is a conjunct, we can prove each one alone and later join them with ∧. note that the number of such ci conjuncts is less than p , therefore it has a polynomial upper bound. using lemma 1.4 and 5, we can reduce the proof of each (ci ⊃ d) (1 ≤ i ≤ m) to a new dnf, which also satisfies the condition that the number of non-negated variables in each conjunct is o(log(|d|)). we can use the same technique to prove these new tautologies and note that their p sets are subsets of our initial p set, therefore we will use only the first p . if we continue the proof in this way, we will have several tautologies, the number of which is polynomial from |d|, and each reduction is performed using polynomial steps and polynomial formulas, therefore the proof is l-polynomially bounded. corollary 3: if the set of complete dnfs has l-polynomially bounded proofs, than the set of dnf-tautologies d, where the number of negated variables in each conjunct is o(log(|d|)), also has l-polynomially bounded proofs. proof. the proof can be obtained from the proof of theorem 2 with slight changes. b. here, the proof complexities of some subset of balanced dnf-tautologies are investigated. definition 6: a dnf-tautology is correct if dnf, obtained from it by removing any conjunct, is no longer a tautology. lemma 2: the number of conjuncts in a balanced correct dnf-tautology with n variables is n+1. g. petrosyan 11 proof. we prove the lemma by induction on number n of variables in a balanced dnftautology d. if n = 1, we have d = ¬p∨p. suppose the statement is valid for a balanced dnf-tautology, which has ≤ n variables. if the number of variables is n + 1, then the correctly balanced dnf should have at least one conjunct with at least two literals pα1i1 p α2 i2 . after assigning α1 to the variable pi1 everywhere in the given dnf, both the number of variables and the number of conjuncts decrease by one, hence the number of conjuncts in the primary dnf should be n + 2. corollary 4: every balanced correct dnf-tautology has at least one conjunct, consisting of one literal. theorem 5: all balanced correct dnf-tautologies have l-polynomially bounded proofs. proof. let us have a balanced correct dnf-tautology d = d1 ∨d2 ∨ ...∨dn+1, which depends on the variables p1,p2, ...,pn. we take ¬d and prove the a contradiction. ¬d = n1 ∧n2 ∧ ...∧nn+1, where ni = ¬di (1 ≤ i ≤ n + 1) and ni = pα1i1 ∨p α2 i2 ∨ ...∨pαrir , where αi ∈ {0, 1} (1 ≤ i ≤ n + 1). based on ni, we construct the formula ei by adding instead of every variable pj from p1,p2, ...,pn which has no occurrence in ni, the formula p1 ∧¬p1 on the j-th place with a disjunction. it is obvious that ni = ei, and this equivalence can be derived with polynomially bounded characteristics of proof complexities. by this notation we have that formula ¬d is equivalent to formula d′ = e1 ∧e2...∧en+1. now we introduce new propositional variables pij (1 ≤ i ≤ n + 1; 1 ≤ j ≤ n), where pij is true if the variable pj occurs in ni and is false for the opposite case, and construct on the base of ei new disjunctions d′i by replacing both the primary literals and the formulas p1 ∧¬p1 with the corresponding variables pij. now we consider the well-known formulas of pigeon hole principle phpn = ∧ni=0 ∨ n−1 k=0 pik ⊃∨ n−1 k=0 ∨0≤i