” 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×³ïý»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. 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[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 é»½áý³ýëáí ß»ù³ûçý ñ³ù³ï³ñ·áõù ². ð³ëáõùû³ý, ì. âáíù³ëû³ý, ð. ð³ñáû³ý ²ù÷á÷áõù ¸çï³ñïí³í ¿ çýýáñù³óç³ûç ñ³õáñ¹ù³ý ³ñ³·áõãû³ý ï³ëí³íáõãûáõýá ³õùáõïý»ñç ù³ï³ñ¹³ïçó ëïáë³ëïçï é»½áý³ïáñáí 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 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[30] j. chung, j. huh, a. nagrani, t. afouras and a. zisserman, “spot the conversation: speaker diarization in the wild”, arxiv preprint arxiv:2007.01216, 2020. [31] d. park, w. chan, y. zhang, c. chiu, b. zoph, e. cubuk and q. le, “specaugment: a simple data augmentation method for automatic speech recognition”, arxiv preprint arxiv:1904.08779, 2019. [32] t. ko, v. peddinti, d. povey, m. seltzer and s. khudanpur, “a study on data augmentation of reverberant speech for robust speech recognition”, proceedings of the ieee international conference on acoustics, speech and signal processing (icassp), pp. 5220-5224, 2017. davit karamyan, grigor kirakosyan and saten harutyunyan 6 7 [3 3 ] d . s n yd e r , g. ch e n a n d d . p o ve y, \ mu s a n : a m u s ic , s p e e c h , a n d n o is e c o r p u s " , arxiv p reprint arxiv:1510.08484, 2 0 1 5 . [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 . [2 ] 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 . [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 , " 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 , ma y p p . 7 3 7 -7 5 6 , 2 0 0 4 . [4 ] s . h a m d io u i, z. a l-a r s a n d a .j. va n d e go o r , " te s t in g s t a t ic a n d 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 vl si test symposium, p p . 3 9 5 -4 0 0 , 2 0 0 2 . [5 ] r . d . a d a m s , e . s . co o le y, " a n a lys is o f d e c e p t ive r e a d d e s t r u c t ive m e m o r y fa u lt m o d e l a n d r e c o m m e n d e d t e s t in g " , p roc. of ie e e north atlantic test w orkshop, p p . 2 7 -3 2 , 1 9 9 6 . [6 ] s . h a m d io u i, r . w a d s wo r t h , j. d . r e ye s , a .j. va n d e go o r , " im p o r t a n c e o f d yn a m ic fa u lt s fo r n e w s r a m t e c h n o lo g ie s " in p roc. of ie e e e uropean test w orkshop, p p . 2 9 -3 4 , 2 0 0 3 . 1 8 h. avetisyan, g. harutunyan, v. vardanian 1 9 [7 ] s . h a m d io u i, g.n . ga yd a d jie v, a n d a .j. va n d e go o r , " a fa u lt p r im it ive b a s e d a n a lys is o f d yn a m ic m e m o r y fa u lt s " , ie e e 14th annual w orkshop on circuits, systems and signal p rocessing, v e ld h o ve n , t h e n e t h e r la n d s , p p . 8 4 -8 9 , 2 0 0 3 . [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 ëçýã»½ç ïáýëïñáõïïçí ï»ëáõãû³ýá, ù³ëý³íáñ³å»ë fp ¹³ßïç íñ³ pt ³ëïç×³ýç 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 åë¨¹á-å³ï³ñ³ï³ý ãí»ñç ·»ý»ñ³ïáñ mixmax-ç ñ³ù³ñ ü. ø³ñïçñáëû³ý, ¶. î³ñû³ý ¨ ü. ²ïáåáí ²ù÷á÷áõù äë¨¹á-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 â»µçß¨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×³ïý»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×³ïí³í 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×³ïí³í 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×³ýá 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 ±-é»¹áõïóç³ûç ·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 ±-é»¹áõïóç³ûç ·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 ±-é»¹áõïóç³ûç ·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