” 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 49, 41–48, 2018. dynamic task scheduling based on abelian sandpile and rotor-router models hayk e. nahapetyan and suren s. poghosyan institute for informatics and automation problems of nas ra e-mail: hayknahapetyan@yahoo.com, psuren55@yandex.ru abstract this study is dedicated to the possible usage of self-organized criticality models in large-scale computing systems for load balancing and energy-awareness. methods and software tools aimed at modeling and visualization of dynamic tasks scheduling in virtual distributed systems constructed over sandpile and rotor-router models, are also presented. keywords: asm, rotor-router decentralized systems, dynamic task scheduling. 1. introduction the concept of self-organized criticality was first introduced by bak, tang and wiesenfeld in 1987 [3], and gave rise to growing interest in the study of self-organizing systems. bak et al. argued that in many natural phenomena, the dissipative dynamics of the system is such that it drives the system to a critical state, thereby leading to ubiquitous power law behaviors. the sandpile models, being a class of cellular automata, are among the simplest theoretical models, which exhibit self-organized criticality. a special subclass of interest consists of so called abelian sandpile models (asm). the abelian sandpile and rotor-router models were discovered several times by researchers in different communities operating independently. the abelian sandpile model was invented by dhar [1], where the rotor-router model, a deterministic analogue of random walk, was first defined by priezzhev et al. under the name of eulerian walkers [2]. the abelian property means that the final stable state of the ca is independent of the order in which the updates of cells are carried out. this property plays a key role during the numerical, as well as analytical studies of the asm [4] – [7]. there are a number of solutions for tasks scheduling and load balancing based on the sandpile model [8]–[10]. anyway, for large-scale real-time computing systems, critical performance constraints are imposed by the environment, and the correctness depends not only on the logical result of the computation, but also on the time at which the results are produced with keeping energy-awareness. newer solutions based on rotor-router model may provide a better solution in the area. besides, the paper presents appropriate software packages for simulating and verifying large-scale cluster systems relied on sandpile and rotor-router-based tasks scheduling. the scheduler and load balancer possessing the above characteristics are constructed on an agent system in which the agents render the cells of a cellular automaton. thus, agents 41 42 dynamic task scheduling based on abelian sandpile and rotor-router models are essential components of the architecture, where the topology for interconnecting and where collaborating the agents is another issue for consideration. as a model for simulation, a 2-dimensional lattice is investigated, where every agent denotes itself as a computer node with a private computing resource. depended on the assigned workload, the node itself makes a decision whether to migrate tasks to adjacent/neighboring nodes or not. detailed description of the proposed model is given in the third section. 2. sandpile model consider an undirected graph g = (v, e) described with the set of vertices v = {v1, v2, . . . , vn} and the set of edges e. each vertex vi ∈ v is assigned a variable hi which takes integer values and represents the height of the sand at that vertex. hmaxi denotes the maximal allowed height for the vertex vi in the graph g. for a d-dimensional lattice, we take hmaxi = 2d + 1. ct denotes the set of heights hi, which determines the configuration of the system at a given discrete time t . a configuration is called stable, if all heights satisfy hi < h max i . the vertex vi is called closed, if h max i = deg(vi), where deg(vi) indicates the degree of vi. the dynamics of the system is defined by the following rules. consider a stable configuration ct at a given time t . we add a grain of sand to a random vertex vi ∈ v by setting hi to hi + 1 (we assume that the vertex is chosen randomly with a uniform distribution on the set v ). this new configuration, if stable, defines ct+1. if hi ≥ h max i , then the vi becomes unstable and topples losing h max i grains of sand, while all neighbors of vi receive one grain. note that if the vertex is open, then the system loses grains. during the toppling of the closed vertices, the number of grains is conserved. note also that toppling of a vertex may cause some of its neighboring vertices to become unstable. in this case, those vertices also topple according to the same toppling rule. once all unstable vertices are toppled, a new stable configuration ct+1 is obtained. if the finite connected graph g has at least one open vertex, then all vertices become stable after a finite number of topplings. moreover, the new stable configuration is independent of the toppling order. let âi be an operator, which acts on sandpile configurations and adds a grain to vertex i. it can be easily shown that âiâj = âjâi. this is the reason why the sandpile model is called abelian. 2.1 rotor-router model to define the rotor-router model on a directed graph g, for each vertex of g, fix a cyclic ordering of the outgoing edges. to each vertex v we associate a rotor (v ) chosen from among the outgoing edges from v . a chip performs a walk on g according to the rotor-router rule: if the chip is at v , we first increment the rotor (v ) to its successor e = (v, w) in the cyclic ordering of outgoing edges from v, and then route the chip along e to w. if the chip ever reaches a sink, i.e., a vertex of g with no outgoing edges, the chip will stop there; otherwise, the chip continues walking forever. h. nahapetyan and s. poghosyan 43 fig 1. rotor-router example. 3. sand-scheduler in this section, we are going to describe a software tool that has a purpose of modeling and visualizing dynamic tasks scheduling in distributed systems. as already mentioned, the scheduling algorithm used by this tool is based on two well-known models: abelian sandpile model (asm) and rotor-router model. in original formulation of asm, each site on a finite grid has an associated value that corresponds to the slope of the pile. this slope builds up as ”grains of sand” (or ”chips”) are randomly placed onto the pile, until the slope exceeds a specific threshold value at which time that site collapses and transfers its sand grains to its adjacent sites, increasing their slope. bak, tang, and wiesenfeld considered the process of successive random placement of sand grains on the grid; each such placement of sand at a particular site may have no effect,or it may cause avalanches, which may have a cascading effect on many sites. the original interest behind the model stemmed from the fact that in simulations on lattices, it is attracted to its critical state, at which point the correlation length of the system and the correlation time of the system go to infinity, without any fine tuning of a system parameter. in the sandpile model dropping another grain of sand onto the pile may cause nothing to happen, or it may cause the entire pile to collapse in a massive slide. we use the above mentioned property of sandpile in order to dynamically schedule tasks, based on the background process of avalanches that is visible in debug enabled state. in our workload, tasks may arrive in a group of up to 7 tasks and be assigned to some node in the system. these tasks in a group are typically a set of multiple instances of the same sequential program. that is why the tasks in a group are independent of each other and can be executed in parallel. all the tasks in a group have only one important property, assigned by ti, which shows the number of rounds needed for the task to be executed in a system. all the tasks of a given group have the same ti. preliminaries for the sandpile-based dynamic scheduling problem are the following: 44 dynamic task scheduling based on abelian sandpile and rotor-router models fig 2. sand-scheduler “debug” enabled. • each task has its own required execution time ti. • there is a set p of homogeneous processors, where |p | = nxm (10x10 in our example). • each processor can execute at most k task simultaneously at any given time. (k = 3 in our example) • the nodes are connected between them and only interacting with the small subset of the neighbours (at most 4). • each node has a working queue q, |q| ≤ 4 that gets filled up when all the resources are taken. • the total execution time for any task is said to be ti = ti + si, where si is the time required for the task to be scheduled(find empty slot in nodes) and start its execution. 3.1 sandpile-based mode in this mode, we study the system in a critical state, and the state of the system is being reconfigured periodically. it is a cellular automaton, which models the process of dropping on grains of sand on a surface and the collapsing of grains due to the increase of the height of the slope. this process is going on regardless of the number of assigned tasks to the system. the tasks are pretty similar to the grains of sand but they do not participate in avalanches themselves. when the grains of the current node reach the maximum value, they start to topple. in case there are tasks assigned to the node at that moment and these tasks are not yet ready to be executed (all the computing resources of the node are reserved), they will be toppled with the sands and move to another node along with the corresponding grain of sand. the transition rule in this model is triggered when the height of the current grain is bigger than the configured value for the system (4 in our study). this process of avalanches is going on indefinitely, meanwhile, this system is dynamically balancing the distribution of tasks among all buckets. h. nahapetyan and s. poghosyan 45 fig 3. sand-scheduler debug mode. green executing tasks, red waiting tasks, blue fictive sands for simulating avalanches, black fictive sands that are in critical state. another option that this scheduler supports, is the fault-tolerance. during the execution, we can disable an arbitrary node or nodes, meanwhile, the scheduler will keep working without this kind of failures that are common in large-scale computational systems. the “energy aware mode is a modification of this scheduler in order to reduce the number of nodes that are powered on. depending on the number of non-scheduled tasks in the system at any given moment of time, only the required (min count of nodes needed for executing tasks at the same time) count of nodes are powered on. fig 4. sand-scheduler. 46 dynamic task scheduling based on abelian sandpile and rotor-router models 3.2 rotor-router mode the software package developed implements one more scheduling mode based on the rotorrouter model. relying on priezzev’s [12, 13] dhar’s studies [14], we can make sure that grains in rotor-router configuration are equally distributed. so, if we change grains with tasks we can be sure that load balancing will be provided in cluster systems. moreover, we are pushing forward a hypothesis that even for tasks with execution timing(the task will be removed after execution and free up space) in rotor-router configuration tasks will be equally distributed in the system, which is visible via software package described in this paper. fig 5.sand-scheduler. non of the tasks is executed yet. fig 6. sand-scheduler. first tasks have been executed. h. nahapetyan and s. poghosyan 47 4. conclusion in this paper, possible usage of asm and rotor-router model in cluster systems has been discussed. also, appropriate software tools have been developed for cluster simulation and for visualization of tasks dissemination. perspectives of this work are to deploy asm and rotorrouter-based algorithms for task distribution on real systems and obtain a comparative analysis between real-world solutions. 5. acknowledgement the authors are grateful to prof. yu. h. shoukourian and dr. y. alaverdyan for important discussions and critical remarks at all stages of the work. this work was supported by the state committee of science mes ra, in the frames of the research project no. 16yr-1b008. references [1] d. dhar, “self-organized critical state of sandpile automaton models”, phys. rev. lett., vol. 64, no. 14, pp. 1613–1616, 1990. [2] b. priezzhev, d. dhar, a. dhar and s. krishnamurthy, “eulerian walkers as a model of self-organized criticality”, phys. rev. lett., vol. 77, pp. 50795082, 1996. [3] p. bak, c. tang and k. wiesenfeld,“self-organized criticality: an explanation of the 1/f noise”,phys. rev. lett., vol.59, no. 4, pp. 381384, 1987. [4] v. s. poghosyan, s. y. grigorev, v. b. priezzhev and p. ruelle, “pair correlations in the sandpile model: a check of logarithmic conformal field theory”, phys. lett. b, vol. 659, pp. 768772, 2008. [5] su. s. poghosyan, v. s. poghosyan, v. b. priezzhev and p. ruelle, “numerical study of correspondence between the dissipative and fixed-energy abelian sandpile models”, phys.rev. e, 84, 066119, 2011. [6] v. s. poghosyan, s. s. poghosyan and h. e. nahapetyan, “the investigation of models of self-organized systems by parallel programming methods based on the example of an abelian sandpile model”, proc. csit conference 2013, yerevan armenia, sept. 23-27, pp. 260-262, 2013. [7] h. nahapetyan, j.-pierre jessel, s. poghosyan and y. shoukourian,“a multi user and multi purpose ca simulator”,phys. rev. lett., vol.59, no. 4, pp. 381384, 1987. proc. csit conference 2017, yerevan armenia, sept. 23-27, pp. 260-262. [8] y. rabani, a. sinclair, and r. wanka, “local divergence of markov chains and the analysis of iterative load-balancing schemes”, in ieee symp. on foundations of computer science, pp. 694705, 1998. [9] j. l. j. laredo, p. bouvry, f. guinand, b. dorronsoro and c. fernandes, “the sandpile scheduler”, cluster computing vol.17, pp 191204, 2014. [10] j. gsior and f. seredyski, “a sandpile cellular automata-based scheduler and load balancer”, journal of computational science, vol.21, pp. 460-468, 2017. [11] l. levine and y. peres, “asymptotics for rotor-router aggregation and the divisible sandpile”, potential analysis, 30: 1. https://doi.org/10.1007/s11118-008-9104-6 4 8 dynamic task scheduling based on abelian sandpile and rotor-router models [1 2 ] a . m. p o vo lo t s ky, v . b . p r ie z z h e v a n d r . r . s h c h e r b a ko v, \ d yn a m ic s o f e u le r ia n wa lke r s " , p hysical review e , vo l.5 8 , d oi:h t t p s :/ / d o i.o r g / 1 0 .1 1 0 3 / p h ys r e ve .5 8 .5 4 4 9 [1 3 ] v . b . p r ie z z h e v, \ s e lf-o r g a n iz e d c r it ic a lit y in s e lf-d ir e c t in g wa lks " , a r x iv:c o n d m a t / 9 6 0 5 0 9 4 [1 4 ] d . d h a r , \ th e o r e t ic a l s t u d ie s o f s e lf-o r g a n iz e d c r it ic a lit y" , p hysica a: statistical m echanics and its applications, vo l. 3 6 9 , n o . 1 , p p . 2 9 -7 0 , 2 0 0 6 submitted 04.09.2017, accepted 15.01.2018. ¸çý³ùçï ³é³ç³¹ñ³ýùý»ñç åé³ý³íáñáõù` ñçùýí³í ²µ»éû³ý ³í³½³ïáõûïç ¨ rotor-router ùá¹»éý»ñç íñ³ ð. ü³ñ³å»ïû³ý ¨ ê. äáõáëû³ý ²ù÷á÷áõù ²ûë áõëáõùý³ëçñáõãûáõýá ýíçñí³í ¿ ëáßáñ³í³í³é ñ³ßíáõ³ï³ý ñ³ù³ï³ñ·»ñáõù çýùý³ï³½ù³ï»ñå ïñçïçï³ï³ý ùá¹»éý»ñç ñý³ñ³íáñ û·ï³·áñíù³ýá í³ýñ³µ»éýí³íáõãû³ý µ³ßëù³ý ¨ ¿ý»ñ·³ëý³ûáõáõãû³ý ýå³ï³ïý»ñáí: ü»ñï³û³óí³í »ý ù»ãá¹ý»ñ ¨ íñ³·ñ³ûçý ·áñíçùý»ñ, áñáýù ùçïí³í »ý ³í³½³ïáõûïç ¨ éáïáñ-éááõï»ñ ùá¹»éý»ñç ñçù³ý íñ³ ï³éáõóí³í íçñïáõ³é µ³ßëí³í ñ³ù³ï³ñ·»ñáõù ¹çý³ùçï ëý¹çñý»ñç åé³ý³íáñù³ýá ¨ ï»ë³µ»ñù³ýá: äèíàìè÷åñêîå ïëàíèðîâàíèå çàäà÷, îñíîâàííûõ íà ìîäåëÿõ àáåëåâîé ïåñ÷àíîé êó÷è è ðîòîð-ðîóòåðà ã. íàãàïåòÿí è ñ. ïîãîñÿàí àííîòàöèÿ èññëåäîâàíèå ïîñâÿùåíî âîçìîæíîìó èñïîëüçîâàíèþ ñàìîîðãàíèçóþùèõñÿ êðèòè÷íûõ ìîäåëåé â øèðîêîìàñøòàáíûõ ñèñòåìàõ äëÿ áàëàíñèðîâêè íàãðóçêè è ýíåðãîñáåðåæåíèÿ. òàêæå ïðåäñòàâëåíû ìåòîäû è ïðîãðàììíûå ñðåäñòâà, ïðåäíàçíà÷åííûå äëÿ ìîäåëèðîâàíèÿ è âèçóàëèçàöèè ïëàíèðîâàíèÿ äèíàìè÷åñêèõ çàäà÷ â âèðòóàëüíûõ ðàñïðåäåëåííûõ ñèñòåìàõ, ïîñòðîåííûõ íà ìîäåëÿõ ïåñ÷àíîé êó÷è è ðîòîð-ðîóòåðà. 051 hayk'sabstract_1 petros's_article.dvi mathematical problems of computer science 49, 7{17, 2018. on locally-b alanced 2-p ar titions of complete m ultipar tite gr aphs a r a m h . gh a r ib ya n y a n d p e t r o s a . p e t r o s ya n yz ydepartment of informatics and applied mathematics, ysu, zinstitute for informatics and automation problems of nas ra e-mail: aramgharibyan@gmail.com, petros petrosyan@ysu.am, pet petros@ipia.sci.am abstract a 2-partition of a graph g is a function f : v (g) ! fwhite; blackg. a 2-partition f of a graph g is locally-balanced with an open neighborhood if for every v 2 v (g), jjfu 2 ng(v): f(u) = whitegj ¡ jfu 2 ng(v): f(u) = blackgjj · 1; where ng(v) = fu 2 v (g): uv 2 e(g)g. a 2-partition f0 of a graph g is locallybalanced with a closed neighborhood if for every v 2 v (g), ¯̄ jfu 2 ng[v]: f 0(u) = whitegj ¡ jfu 2 ng[v]: f0(u) = blackgj ¯̄ · 1; where ng[v] = ng(v) [ fvg. in this paper we give necessary and su±cient conditions for the existence of locally-balanced 2-partitions of complete multipartite graphs. keywords: 2-partition, locally-balanced 2-partition, equitable coloring, complete multipartite graph. 1 . in t r o d u c t io n a ll g r a p h s c o n s id e r e d in t h is wo r k a r e ¯ n it e , u n d ir e c t e d , a n d h a ve n o lo o p s o r m u lt ip le e d g e s . l e t v ( g) a n d e ( g) d e n o t e t h e s e t s o f ve r t ic e s a n d e d g e s o f a g r a p h g, r e s p e c t ive ly. th e s e t o f n e ig h b o r s o f a ve r t e x v in g is d e n o t e d b y ng ( v ) . l e t ng[v] = ng ( v ) [fvg. a g r a p h g is c a lle d a c o m p le t e n-p a r t it e ( n ¸ 2 ) g r a p h if it s ve r t ic e s c a n b e p a r t it io n e d in t o n n o n e m p t y in d e p e n d e n t s e t s x1; : : : ; xn s u c h t h a t e a c h ve r t e x in xi is a d ja c e n t t o a ll t h e o t h e r ve r t ic e s in xj fo r 1 · i < j · n. l e t kr1;r2;:::;rn d e n o t e a c o m p le t e n-p a r t it e g r a p h wit h in d e p e n d e n t s e t s x1; x2; : : : ; xn o f s iz e s r1; r2; : : : ; rn. th e t e r m s a n d c o n c e p t s t h a t we d o n o t d e ¯ n e c a n b e fo u n d in [8 , 1 5 ]. a 2 -partition o f a g r a p h g is a fu n c t io n f : v ( g) ! fwhite; b lackg. a 2 -p a r t it io n f o f a g r a p h g is locally-balanced with an open neighborhood if fo r e ve r y v 2 v ( g ) , jjfu 2 ng ( v ) : f ( u ) = whitegj ¡ jfu 2 ng ( v ) : f ( u) = b lackgjj · 1 : a 2 -p a r t it io n f 0 o f a g r a p h g is locally-balanced with a closed neighborhood if fo r e ve r y v 2 v ( g) , jjfu 2 ng[v]: f 0 ( u ) = whitegj ¡ jfu 2 ng[v]: f 0 ( u) = b lackgjj · 1 : 7 8 on locally-balanced 2-partitions of complete multipartite graphs th e c o n c e p t o f lo c a lly-b a la n c e d 2 -p a r t it io n o f g r a p h s wa s in t r o d u c e d b y b a likya n a n d k a m a lia n [1 2 ] in 2 0 0 5 , a n d it c a n b e c o n s id e r e d a s a s p e c ia l c a s e o f e qu it a b le c o lo r in g s o f h yp e r g r a p h s [1 ]. in [1 ], b e r g e o b t a in e d s o m e s u ± c ie n t c o n d it io n s fo r t h e e xis t e n c e o f e qu it a b le c o lo r in g s o f h yp e r g r a p h s . in [7 , 9 , 1 0 , 1 4 ], t h e a u t h o r s c o n s id e r e d t h e p r o b le m s o f t h e e xis t e n c e a n d c o n s t r u c t io n o f p r o p e r ve r t e x-c o lo r in g o f a g r a p h fo r wh ic h t h e n u m b e r o f ve r t ic e s in a n y t wo c o lo r c la s s e s d i®e r b y a t m o s t o n e . in [1 1 ], 2 -ve r t e x-c o lo r in g s o f g r a p h s we r e c o n s id e r e d fo r wh ic h e a c h ve r t e x is a d ja c e n t t o t h e s a m e n u m b e r o f ve r t ic e s o f e ve r y c o lo r . in p a r t ic u la r , in [1 1 ], it wa s p r o ve d t h a t t h e p r o b le m o f t h e e xis t e n c e o f s u c h a c o lo r in g is np -c o m p le t e e ve n fo r t h e ( 2 p; 2 q ) -b ir e g u la r ( p; q ¸ 2 ) b ip a r t it e g r a p h s . in [1 2 ], b a likya n a n d k a m a lia n p r o ve d t h a t t h e p r o b le m o f e xis t e n c e o f lo c a lly-b a la n c e d 2 -p a r t it io n wit h a n o p e n n e ig h b o r h o o d o f b ip a r t it e g r a p h s wit h m a xim u m d e g r e e 3 is np -c o m p le t e . l a t e r , t h e y a ls o p r o ve d [1 3 ] t h e s im ila r r e s u lt fo r lo c a lly-b a la n c e d 2 -p a r t it io n s wit h a c lo s e d n e ig h b o r h o o d . in [2 , 3 , 4 ], t h e n e c e s s a r y a n d s u ± c ie n t c o n d it io n s fo r t h e e xis t e n c e o f lo c a llyb a la n c e d 2 -p a r t it io n s o f t r e e s we r e o b t a in e d . in [6 ], t h e a u t h o r s o b t a in e d s o m e n e c e s s a r y c o n d it io n s fo r t h e e xis t e n c e o f lo c a lly-b a la n c e d 2 -p a r t it io n s o f e u le r ia n g r a p h s . in p a r t ic u la r , t h e y p r o ve d s o m e r e s u lt s o n t h e e xis t e n c e o f lo c a lly-b a la n c e d 2 -p a r t it io n s o f r o o k's g r a p h s a n d c yc le p o we r s . in t h e p r e s e n t p a p e r we g ive n e c e s s a r y a n d s u ± c ie n t c o n d it io n s fo r t h e e xis t e n c e o f lo c a lly-b a la n c e d 2 -p a r t it io n s o f c o m p le t e m u lt ip a r t it e g r a p h s . a p r e lim in a r y ve r s io n o f t h is p a p e r wa s p r e s e n t e d a t t h e 6 t h p o lis h co m b in a t o r ia l co n fe r e n c e , b e d le wo , p o la n d , 2 0 1 6 [5 ]. 2 . ma in r e s u lt s b e fo r e we fo r m u la t e a n d p r o ve o u r r e s u lt s , we in t r o d u c e s o m e t e r m in o lo g y a n d n o t a t io n . fo r a n y 2 -p a r t it io n ' o f a g r a p h g, we d e ¯ n e ' a s fo llo ws : fo r a n y v 2 v ( g) , le t '( v ) = ( b lack, if '( v ) = white, white, if '( v ) = b lack. if ' is a 2 -p a r t it io n o f a g r a p h g a n d v 2 v ( g) , t h e n d e ¯ n e # ( v ) a n d # [v] a s fo llo ws : # ( v ) = jfu 2 ng ( v ) : '( u ) = whitegj ¡ jfu 2 ng ( v ) : '( u ) = b lackgj, # [v] = jfu 2 ng[v]: '( u) = whitegj ¡ jfu 2 ng[v]: '( u ) = b lackgj. cle a r ly, ' is a lo c a lly-b a la n c e d 2 -p a r t it io n wit h a n o p e n n e ig h b o r h o o d ( wit h a c lo s e d n e ig h b o r h o o d ) if fo r e ve r y v 2 v ( g ) , j # ( v ) j · 1 ( j # [v]j · 1 ) . if g is a c o m p le t e n-p a r t it e g r a p h a n d x is a p a r t o f g, t h e n x is c a lle d a n o d d ( e ve n ) p a r t if jxj is o d d ( jxj is e ve n ) . if g is a c o m p le t e n-p a r t it e g r a p h , t h e n b y m1, m2, a n d m¸3 we d e n o t e t h e n u m b e r o f p a r t s o f g wit h o n e ve r t e x, t wo ve r t ic e s a n d a t le a s t t h r e e ve r t ic e s , r e s p e c t ive ly. if ' is a 2 -p a r t it io n o f a c o m p le t e n-p a r t it e g r a p h g a n d x is a p a r t o f g, t h e n b y wx ( bx ) we d e n o t e t h e n u m b e r o f white ( b lack ) ve r t ic e s o f t h e p a r t x. if ' is a 2 -p a r t it io n o f a c o m p le t e n-p a r t it e g r a p h g, t h e n b y w ( b ) we d e n o t e t h e n u m b e r o f white ( b lack ) ve r t ic e s in g. w e b e g in wit h t h e fo llo win g s im p le le m m a . lemma 1. if ' is a locally-balanced 2-partition of g, then ' is also a locally-balanced 2partition of g. lemma 2. if kr1;r2:::;rn has a locally-balanced 2 -partition with an open neighborhood, then for any part x, jwx ¡ bxj · 1 . a. gharibyan and p. petrosyan 9 p r oof. l e t ' b e a a lo c a lly-b a la n c e d 2 -p a r t it io n wit h a n o p e n n e ig h b o r h o o d o f kr1;r2:::;rn . s u p p o s e , t o t h e c o n t r a r y, t h a t t h e r e e xis t s a p a r t x0 s u c h t h a t e it h e r wx0 ¡ bx0 ¸ 2 o r bx0 ¡ wx0 ¸ 2 . b y l e m m a 1 ., we c a n a s s u m e t h a t t h e r e e xis t s a p a r t x0 s u c h t h a t wx0 ¡ bx0 ¸ 2 ( 1 ) it is e a s y t o s e e t h a t fo r a n y v 2 x, # ( v ) = w ¡ b ¡ wx + bx . fr o m t h is a n d t a kin g in t o a c c o u n t t h a t fo r a n y v 2 v ( kr1;r2:::;rn ) ; ¡ 1 · # ( v ) · 1 , we o b t a in t h a t fo r a n y p a r t x, ¡ 1 · w ¡ b ¡ wx + bx · 1 : ( 2 ) b y ( 1 ) a n d ( 2 ) , we h a ve w ¡ b ¸ 1 : ( 3 ) fo r a n y p a r t y ( y 6= x0 ) , b y ( 2 ) , we h a ve ¡ 1 · w ¡ b ¡ wy + by · 1 . fr o m t h is a n d ( 3 ) , we o b t a in t h a t fo r a n y p a r t y ( y 6= x0 ) , wy ¡ by ¸ 0 : ( 4 ) l e t u s c o n s id e r a n y v 2 x ( x 6= x0 ) . cle a r ly, # ( v ) = x y;y 6=x ( wy ¡ by ) = x y;y 6=x;x0 ( wy ¡ by ) + wx0 ¡ bx0 : b y ( 1 ) a n d ( 4 ) , we g e t # ( v ) ¸ 2 , wh ic h is a c o n t r a d ic t io n . t heor em 1. the graph kr1;r2:::;rn has a locally-balanced 2-partition with an open neighborhood if and only if the number of odd parts is even or one. p r oof. a s s u m e t h a t kr1;r2:::;rn h a s k e ve n p a r t s a n d s o d d p a r t s . s u p p o s e t h a t kr1;r2:::;rn h a s a lo c a lly-b a la n c e d 2 -p a r t it io n wit h a n o p e n n e ig h b o r h o o d , b u t c o n t a in s o d d n u m b e r s ( s ¸ 3 ) o d d p a r t s . l e m m a 2 im p lie s t h a t fo r a n y p a r t x o f kr1;r2:::;rn , jwx ¡ bxj · 1 . w e d e c o m p o s e a ll t h e ve r t ic e s o f t h e g r a p h in t o t h r e e g r o u p s a s fo llo ws : 1 . a ll e ve n p a r t s xi ( h e r e , we h a ve wxi ¡ bxi = 0 , b y l e m m a 2 ) ; 2 . a ll o d d p a r t s x0i wit h wx0i > bx 0 i ( h e r e , we h a ve wx0 i ¡ bx0 i = 1 , b y l e m m a 2 ) ; 3 . a ll o d d p a r t s x00i wit h wx00i < bx00i ( h e r e , we h a ve wx00i ¡ bx00i = ¡ 1 , b y l e m m a 2 .) . l e t x1; x2; : : : ; xk b e p a r t s o f t h e ¯ r s t g r o u p ; x 0 1; x 0 2; : : : ; x 0 m b e p a r t s o f t h e s e c o n d g r o u p ; x001 ; x 00 2 ; : : : ; x 00 t b e p a r t s o f t h e t h ir d g r o u p . w it h o u t lo s s o f g e n e r a lit y, we m a y a s s u m e t h a t m > t. co n s id e r t wo c a s e s . 1 0 on locally-balanced 2-partitions of complete multipartite graphs ca s e 1 : t = 0 . cle a r ly, m ¸ 3 . l e t u s c o n s id e r a ve r t e x v 2 x01. b y 1 a n d 2 , we h a ve # ( v ) = x y;y 6=x01 ( wy ¡ by ) = kx i=1 ( wxi ¡ bxi ) + mx i=2 ( wx0 i ¡ bx0 i ) = m ¡ 1 ¸ 2 ; wh ic h is a c o n t r a d ic t io n . ca s e 2 : t > 0 . l e t u s c o n s id e r a ve r t e x v 2 x001 . b y 1 , 2 a n d 3 , we h a ve # ( v ) = x y;y 6=x00 1 ( wy ¡ by ) = kx i=1 ( wxi ¡ bxi ) + mx i=1 ( wx0i ¡ bx0i ) + tx i=2 ( wx00i ¡ bx00i ) = m ¡ ( t ¡ 1 ) = ( m ¡ t) + 1 ¸ 2 ; wh ic h is a c o n t r a d ic t io n . n o w we s h o w t h a t if t h e n u m b e r o f o d d p a r t s is e ve n o r o n e , t h e n kr1;r2:::;rn h a s a lo c a lly-b a la n c e d 2 -p a r t it io n wit h a n o p e n n e ig h b o r h o o d . fir s t we c o lo r e ve n p a r t s u n ifo r m ly. n e xt we c o n s id e r t wo c a s e s . ca s e a ) th e n u m b e r o f o d d p a r t s is 2 l. fir s t l o d d p a r t s we c o lo r a s fo llo ws : if x is s u c h a p a r t wit h 2 p + 1 ( p ¸ 0 ) ve r t ic e s , t h e n a n y p ve r t ic e s g e t t h e c o lo r b lack a n d t h e r e s t ve r t ic e s g e t t h e c o lo r white. n e xt l o d d p a r t s we c o lo r s im ila r ly b y t a kin g t h e c o lo r white in s t e a d o f b lack, a n d vic e ve r s a . ca s e b ) x is t h e o n ly o d d p a r t wit h 2 p + 1 ( p ¸ 0 ) ve r t ic e s . in t h is c a s e we c o lo r a n y p ve r t ic e s o f x wit h t h e c o lo r b lack a n d t h e r e s t ve r t ic e s wit h t h e c o lo r white. it is e a s y t o s e e t h a t t h e a b o ve -m e n t io n e d 2 -p a r t it io n is a lo c a lly-b a la n c e d 2 -p a r t it io n o f kr1;r2:::;rn wit h a n o p e n n e ig h b o r h o o d . t heor em 2. if m1 = 0 , then the graph kr1;r2:::;rn has a locally-balanced 2-partition with a closed neighborhood if and only if it has no odd part. p r oof. l e t ' b e a lo c a lly-b a la n c e d 2 -p a r t it io n wit h a c lo s e d n e ig h b o r h o o d o f kr1;r2:::;rn . s u p p o s e , t o t h e c o n t r a r y, t h a t t h e r e e xis t s a n o d d p a r t x. s in c e m1 = 0 , we h a ve jxj ¸ 3 . w it h o u t lo s s o f g e n e r a lit y, we m a y a s s u m e t h a t wx > bx . w e c o n s id e r t wo c a s e s . ca s e 1 : bx = 0 . co n s id e r a ve r t e x v 2 x. cle a r ly, '( v ) = white. th is im p lie s t h a t # [v] = w ¡ b ¡ wx + bx + 1 : s in c e ' is a lo c a lly-b a la n c e d 2 -p a r t it io n wit h a c lo s e d n e ig h b o r h o o d o f kr1;r2:::;rn , we h a ve ¡ 1 · w ¡ b ¡ wx + bx + 1 · 1 : th is im p lie s t h a t a. gharibyan and p. petrosyan 1 1 w ¡ b ¸ wx ¡ 2 > 0 . ca s e 2 : bx > 0 . co n s id e r a ve r t e x v 2 x wit h '( v ) = b lack. th is im p lie s t h a t # [v] = w ¡ b ¡ wx + bx ¡ 1 . s in c e ' is a lo c a lly-b a la n c e d 2 -p a r t it io n wit h a c lo s e d n e ig h b o r h o o d o f kr1;r2:::;rn , we h a ve ¡1 · w ¡ b ¡ wx + bx ¡ 1 · 1 . th is im p lie s t h a t w ¡ b ¸ wx ¡ bx > 0 . in a n y c a s e we o b t a in w ¡ b > 0 : ( 5 ) fir s t we c o n s id e r t h e c a s e wh e n fo r a n y p a r t y , wy ¸ by . l e t u s c o n s id e r a ve r t e x v 2 x0 ( x0 6= x ) wit h '( v ) = white. th is im p lie s t h a t # [v] = x y;y 6=x0 ( wy ¡ by ) + 1 = wx ¡ bx + x y;y 6=x;x0 ( wy ¡ by ) + 1 ¸ 2 ; wh ic h is a c o n t r a d ic t io n . s o , we m a y a s s u m e t h a t t h e r e e xis t s a p a r t x0 s u c h t h a t wx0 < bx0. if wx0 = 0 , t h e n we c o n s id e r a ve r t e x v 2 x0. cle a r ly, '( v ) = b lack. fr o m t h is a n d t a kin g in t o a c c o u n t t h a t ' is a lo c a lly-b a la n c e d 2 -p a r t it io n wit h a c lo s e d n e ig h b o r h o o d o f kr1;r2:::;rn , we h a ve ¡ 1 · w ¡ b ¡ wx0 + bx0 ¡ 1 · 1 : h e n c e , w ¡ b · 2 ¡ bx0 · 0 , wh ic h c o n t r a d ic t s ( 5 ) . if wx0 > 0 , t h e n we c o n s id e r a ve r t e x v 2 x0 wit h '( v ) = white. fr o m t h is a n d t a kin g in t o a c c o u n t t h a t ' is a lo c a lly-b a la n c e d 2 -p a r t it io n wit h a c lo s e d n e ig h b o r h o o d o f kr1;r2:::;rn , we h a ve ¡1 · w ¡ b ¡ wx0 + bx0 + 1 · 1 : h e n c e , w ¡ b · wx0 ¡ bx0 < 0 , wh ic h c o n t r a d ic t s ( 5 ) . l e t kr1;r2:::;rn b e a c o m p le t e n-p a r t it e g r a p h wit h o u t a n o d d p a r t . in t h is c a s e we c o lo r e a c h p a r t u n ifo r m ly. lemma 3. if m1 > 0 and kr1;r2:::;rn has a locally-balanced 2-partition with a closed neighborhood, then jw ¡ bj · 1 . 1 2 on locally-balanced 2-partitions of complete multipartite graphs p r oof. l e t ' b e a lo c a lly-b a la n c e d 2 -p a r t it io n wit h a c lo s e d n e ig h b o r h o o d o f kr1;r2:::;rn . co n s id e r a ve r t e x v o f t h e p a r t x wit h jxj = 1 . s in c e ' is a lo c a lly-b a la n c e d 2 -p a r t it io n wit h a c lo s e d n e ig h b o r h o o d o f kr1;r2:::;rn , we h a ve # [v] = w ¡ b, a n d h e n c e ¡1 · w ¡ b · 1 : remar k 1. clearly, l emma 3. implies that if jv ( kr1;r2:::;rn ) j is even, then w = b, and if jv ( kr1;r2:::;rn ) j is odd, then jw ¡ bj = § 1 (in fact, by l emma 1., we may assume that w ¡ b = 1 ). t heor em 3. if m1 > 0 and jv ( kr1;r2:::;rn ) j is even, then kr1;r2:::;rn has a locally-balanced 2-partition with a closed neighborhood if and only if it has no odd part x with at least three vertices. p r oof. l e t ' b e a lo c a lly-b a la n c e d 2 -p a r t it io n wit h a c lo s e d n e ig h b o r h o o d o f kr1;r2:::;rn . s u p p o s e t h a t t h e r e e xis t s a n o d d p a r t x wit h a t le a s t t h r e e ve r t ic e s . w e m a y a s s u m e t h a t wx > bx . w e c o n s id e r t wo c a s e s . ca s e 1 : bx = 0 . co n s id e r a ve r t e x v 2 x. cle a r ly, '( v ) = white. th is im p lie s t h a t # [v] = w ¡ b ¡ wx + bx + 1 : b y r e m a r k 1 , we g e t # [v] = 1 ¡ wx · ¡ 2 , wh ic h is a c o n t r a d ic t io n . ca s e 2 : bx > 0 . co n s id e r a ve r t e x v 2 x wit h '( v ) = b lack. th is im p lie s t h a t # [v] = w ¡ b ¡ wx + bx ¡ 1 : b y r e m a r k 1 a n d t a kin g in t o a c c o u n t t h a t wx > bx , we o b t a in # [v] = bx ¡ wx ¡ 1 · 2 ; wh ic h is a c o n t r a d ic t io n . n o w le t kr1;r2:::;rn b e a c o m p le t e n-p a r t it e g r a p h wit h m1 > 0 o d d p a r t s . cle a r ly, m1 is e ve n . e a c h e ve n p a r t we c o lo r u n ifo r m ly. th e n we c o lo r m1 2 o d d p a r t s wit h t h e c o lo r b lack a n d t h e o t h e r m1 2 o d d p a r t s wit h t h e c o lo r white. it is e a s y t o s e e t h a t t h is 2 -p a r t it io n is a lo c a lly-b a la n c e d 2 -p a r t it io n wit h a c lo s e d n e ig h b o r h o o d o f kr1;r2:::;rn . t heor em 4. if the graph kr1;r2:::;rn has an odd number of vertices, m1 > 0 and there exists a part x such that jxj = 2 + 2 k (k 2 n ) , then kr1;r2:::;rn has no locally-balanced 2 -partition with a closed neighborhood. p r oof. s u p p o s e t h a t ' is a lo c a lly-b a la n c e d 2 -p a r t it io n wit h a c lo s e d n e ig h b o r h o o d o f kr1;r2:::;rn . b y r e m a r k 1 a n d l e m m a 1 , we g e t w ¡ b = 1 : a. gharibyan and p. petrosyan 1 3 l e t u s c o n s id e r fo u r c a s e s . ca s e 1 : bx = 0 : l e t u s c o n s id e r a ve r t e x v 2 x. s in c e ' is a lo c a lly-b a la n c e d 2 -p a r t it io n wit h a c lo s e d n e ig h b o r h o o d o f kr1;r2:::;rn , we h a ve # [v] = w ¡ b ¡ wx + bx + 1 = 2 ¡ wx · ¡2 ; wh ic h is a c o n t r a d ic t io n . ca s e 2 : wx = 0 : l e t u s c o n s id e r a ve r t e x v 2 x. s in c e ' is a lo c a lly-b a la n c e d 2 -p a r t it io n wit h a c lo s e d n e ig h b o r h o o d o f kr1;r2:::;rn , we h a ve # [v] = w ¡ b ¡ wx + bx ¡ 1 = bx ¸ 4 ; wh ic h is a c o n t r a d ic t io n . ca s e 3 : wx > bx > 0 : l e t u s c o n s id e r a ve r t e x v 2 x wit h '( v ) = b lack. s in c e ' is a lo c a lly-b a la n c e d 2 -p a r t it io n wit h a c lo s e d n e ig h b o r h o o d o f kr1;r2:::;rn , we h a ve # [v] = w ¡ b ¡ wx + bx ¡ 1 = bx ¡ wx · ¡ 2 ; wh ic h is a c o n t r a d ic t io n . ca s e 4 : 0 < wx · bx . l e t u s c o n s id e r a ve r t e x v 2 x wit h '( v ) = white. s in c e ' is a lo c a lly-b a la n c e d 2 -p a r t it io n wit h a c lo s e d n e ig h b o r h o o d o f kr1;r2:::;rn , we h a ve # [v] = w ¡ b ¡ wx + bx + 1 = 2 + bx ¡ wx ¸ 2 ; wh ic h is a c o n t r a d ic t io n . t heor em 5. if the graph kr1;r2:::;rn has an odd number of vertices, m1 > 0 and each part x of the graph has either two vertices or an odd number of vertices, then kr1;r2:::;rn has a locally-balanced 2 -partition with a closed neighborhood if and only if m1 ¸ 2 m2 + m¸3 ¡ 1 . p r oof. s u p p o s e t h a t ' is a lo c a lly-b a la n c e d 2 -p a r t it io n wit h a c lo s e d n e ig h b o r h o o d o f kr1;r2:::;rn . b y r e m a r k 1 a n d l e m m a 1 , we h a ve w ¡ b = 1 : ( 6 ) l e t u s c o n s id e r a p a r t x wit h o n ly t wo ve r t ic e s u a n d v. if '( v ) = white a n d '( u ) = b lack, t h e n s in c e ' is a lo c a lly-b a la n c e d 2 -p a r t it io n wit h a c lo s e d n e ig h b o r h o o d o f kr1;r2:::;rn , we h a ve # [v] = w ¡ b ¡ wx + bx + 1 = 2 ; a c o n t r a d ic t io n . s im ila r ly, if '( v ) = b lack a n d '( u ) = white, t h e n we c o n s id e r t h e ve r t e x u. if '( v ) = '( u ) = b lack, t h e n s in c e ' is a lo c a lly-b a la n c e d 2 -p a r t it io n wit h a c lo s e d n e ig h b o r h o o d o f kr1;r2:::;rn , we h a ve # [v] = w ¡ b ¡ wx + bx ¡ 1 = 2 ; 1 4 on locally-balanced 2-partitions of complete multipartite graphs a c o n t r a d ic t io n . th is im p lie s t h a t '( v ) = '( u ) = white. l e t x b e a n o d d p a r t wit h a t le a s t t h r e e ve r t ic e s . if wx = 0 , t h e n b y c o n s id e r in g a ve r t e x v 2 x a n d t a kin g in t o a c c o u n t t h a t ' is a lo c a lly-b a la n c e d 2 -p a r t it io n wit h a c lo s e d n e ig h b o r h o o d o f kr1;r2:::;rn , we o b t a in # [v] = w ¡ b ¡ wx + bx ¡ 1 = bx ¸ 3 ; a c o n t r a d ic t io n . if 0 < wx < bx , t h e n b y c o n s id e r in g a ve r t e x v 2 x wit h '( v ) = white a n d t a kin g in t o a c c o u n t t h a t ' is a lo c a lly-b a la n c e d 2 -p a r t it io n wit h a c lo s e d n e ig h b o r h o o d o f kr1;r2:::;rn , we o b t a in # [v] = w ¡ b ¡ wx + bx + 1 = 2 + bx ¡ wx ¸ 3 ; a c o n t r a d ic t io n . th is s h o ws t h a t wx > bx : ( 7 ) l e t u s c o n s id e r t wo c a s e s . ca s e 1 : jxj = 3 . b y ( 7 ) , we h a ve t h a t t h e r e a r e t wo p o s s ib le c a s e s : a ll ve r t ic e s o f x a r e white o r t wo o f t h e m a r e white a n d t h e la s t o n e is b lack. if wx = 3 a n d bx = 0 , t h e n fo r a n y v 2 x, we h a ve # [v] = w ¡ b ¡ wx + bx + 1 = ¡1 : th is im p lie s t h a t fo r a n y v 2 x, ¡1 · # [v] · 1 : if wx = 2 a n d bx = 1 , t h e n fo r a n y v 2 x, we h a ve e it h e r # [v] = w ¡b¡wx +bx +1 = 1 ( if '( v ) = white ) o r # [v] = w ¡ b ¡ wx + bx ¡ 1 = ¡ 1 ( if '( v ) = b lack ) . th is im p lie s t h a t fo r a n y v 2 x, ¡1 · # [v] · 1 : ca s e 2 : jxj ¸ 5 . b y ( 7 ) , we h a ve wx ¡ bx ¸ 1 . l e t u s s h o w t h a t wx ¡ bx = 1 . s u p p o s e , t o t h e c o n t r a r y, t h a t wx ¡ bx ¸ 3 . if bx = 0 , t h e n fo r a n y v 2 x, we h a ve # [v] = w ¡ b ¡ wx + bx + 1 · ¡ 3 ; wh ic h is a c o n t r a d ic t io n . if wx > bx > 0 . fo r a n y v 2 x wit h '( v ) = b lack, we h a ve # [v] = w ¡ b ¡ wx + bx ¡ 1 · ¡3 ; wh ic h is a c o n t r a d ic t io n . th is s h o ws t h a t wx ¡ bx = 1 . s o , we o b t a in t h a t if x is a p a r t wit h a t le a s t t wo ve r t ic e s , t h e n we h a ve t h e fo llo win g t h r e e p o s s ib le c a s e s : a ) if x is a n o d d p a r t wit h t h r e e ve r t ic e s , t h e n e it h e r wx ¡ bx = 1 o r wx ¡ bx = 3 ; ( 8 ) a. gharibyan and p. petrosyan 1 5 b ) if x is a n o d d p a r t wit h a t le a s t ¯ ve ve r t ic e s , t h e n wx ¡ bx = 1 ; ( 9 ) c ) if x h a s t wo ve r t ic e s , t h e n wx ¡ bx = 2 : ( 1 0 ) b y ( 6 ) , ( 8 ) ,( 9 ) ,( 1 0 ) , we g e t m1 = x x;jxj=1 ( bx + wx ) ¸ x x;jxj=1 ( bx ¡ wx ) = b ¡ w + x x;jxj=2 ( wx ¡ bx ) + x x;wx ¡bx =1 ( wx ¡bx ) + x x;wx ¡bx =3 ( wx ¡bx ) = b¡w +2 jfx: x is a p a r t wit h jxj = 2 gj+ jfx: x is a p a r t wit h wx ¡ bx = 1 gj + 3 jfx: x is a p a r t wit h wx ¡ bx = 3 gj ¸ b ¡ w + 2 jfx: x is a p a r t wit h jxj = 2 gj + jfx: x is a p a r t wit h wx ¡ bx = 1 gj+ jfx: x is a p a r t wit h wx ¡ bx = 3 gj = b ¡ w + 2 jfx: x is a p a r t wit h jxj = 2 gj+ jfx: x is a n o d d p a r t wit h a t le a s t t h r e e ve r t ic e s gj = ¡ 1 + 2 m2 + m¸3: n o w le t m1 ¸ 2 m2 + m¸3 ¡ 1 . w e c o n s t r u c t a 2 -p a r t it io n o f kr1;r2:::;rn a s fo llo ws : 1 ) e a c h ve r t e x v 2 x ( jxj = 2 ) , we c o lo r wit h t h e c o lo r white; 2 ) fo r a n y o d d p a r t x wit h jxj = 2 p + 1 ( p 2 n ) ve r t ic e s , we c o lo r p + 1 ve r t ic e s o f x wit h t h e c o lo r white, a n d t h e r e s t p ve r t ic e s we c o lo r wit h t h e c o lo r b lack; 3 ) w e t a ke a n y ( 2 m2 + m¸3 ¡ 1 ) p a r t s wit h o n ly o n e ve r t e x o f t h e g r a p h a n d we c o lo r e a c h ve r t e x in t h e s e p a r t s wit h t h e c o lo r b lack. s in c e jv ( kr1;r2:::;rn ) j is o d d a n d m1 ¸ 2 m2 + m¸3 ¡ 1 , we h a ve t h a t m1 ¡ ( 2 m2 + m¸3 ¡ 1 ) is e ve n a n d n o n -n e g a t ive . th e ve r t ic e s o f t h e r e m a in in g m1 ¡ ( 2 m2 + m¸3 ¡ 1 ) p a r t s we c o lo r u n ifo r m ly. it is n o t d i± c u lt t o s e e t h a t t h is 2 -p a r t it io n is a lo c a lly-b a la n c e d 2 -p a r t it io n wit h a c lo s e d n e ig h b o r h o o d o f kr1;r2:::;rn . 1 6 on locally-balanced 2-partitions of complete multipartite graphs refer ences [1 ] c. b e r g e , gr a p h s a n d h yp e r g r a p h s , e ls e vie r s c ie n c e l t d , 1 9 8 5 . 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[1 3 ] s .v . b a likya n a n d r .r . k a m a lia n , \ on np -c o m p le t e n e s s o f t h e p r o b le m o f e xis t e n c e o f lo c a lly-b a la n c e d 2 -p a r t it io n fo r b ip a r t it e g r a p h s g wit h ¢ ( g) = 4 u n d e r t h e e xt e n d e d d e ¯ n it io n o f t h e n e ig h b o u r h o o d o f a ve r t e x" , d oklady nan r a, vo l. 1 0 6 , n o . 3 , p p . 2 1 8 -2 2 6 , 2 0 0 6 . [1 4 ] d . d e w e r r a , \ on g o o d a n d e qu it a b le c o lo r in g s " , in cahiers du c.e .r .o., vo l. 1 7 , p p . 4 1 7 -4 2 6 , 1 9 7 5 . [1 5 ] d .b . w e s t , in t r o d u c t io n t o gr a p h th e o r y, p r e n t ic e -h a ll, n e w je r s e y, 2 0 0 1 . submitted 02.10.2017, accepted 26.01.2018. a. gharibyan and p. petrosyan 1 7 èñçí µ³½ù³ïáõù³ýç ·ñ³ýý»ñç éáï³é-ñ³í³ë³ñ³ïßéí³í 2-ïñáñáõùý»ñç ù³ëçý ². ô³ñçµû³ý ¨ ä. ä»ïñáëû³ý ²ù÷á÷áõù f : v ( g) ! fwhite; b lackg ýáõýïóç³ý ïáãíáõù ¿ g ·ñ³ýç 2-ïñáñáõù: g ·ñ³ýç f 2-ïñáñáõùá ïáãíáõù ¿ éáï³é-ñ³í³ë³ñ³ïßéí³í 2-ïñáñáõù µ³ó ßñç³ï³ûùáí, »ã» ï³ù³û³ï³ý v 2 v ( g ) ·³·³ãç ñ³ù³ñ ï»õç áõýç jjfu 2 ng ( v ) : f ( u ) = whitegj ¡ jfu 2 ng ( v ) : f ( u) = b lackgjj · 1 ; áñï»õ ng ( v ) = fu 2 v ( g ) : uv 2 e ( g ) g: g ·ñ³ýç f 0 2-ïñáñáõùá ïáãíáõù ¿ éáï³éñ³í³ë³ñ³ïßéí³í ïñáñáõù ÷³ï ßñç³ï³ûùáí, »ã» ï³ù³û³ï³ý v 2 v ( g) ·³·³ãç ñ³ù³ñ ï»õç áõýç jjfu 2 ng[v]: f 0 ( u ) = whitegj ¡ jfu 2 ng[v]: f 0 ( u) = b lackgjj · 1 ; áñï»õ ng[v] = ng ( v ) [ fvg: ²ûë ³ßë³ï³ýùáõù ïñíáõù »ý éñçí µ³½ù³ïáõù³ýç ·ñ³ýý»ñç éáï³é-ñ³í³ë³ñ³ïßéí³í ïñáñáõùý»ñç ·áûáõãû³ý ³ýññ³å»ßï ¨ µ³í³ñ³ñ å³ûù³ýý»ñ: î ëîêàëüíî-ñáàëàíñèðîâàííûx 2-ðàçáèåíèÿx ïîëíûx ìíîãîäîëüíûx ãðàôîâ à. ãàðèáÿí è ï. ïåòðîñÿí àííîòàöèÿ 2-ðàçáèåíèåì ãðàôà g íàçûâàåòñÿ ôóíêöèÿ f : v ( g) ! fwhite; b lackg. 2-ðàçáèåíèå f ãðàôà g íàçûâàåòñÿ ëîêàëüíî-ñáàëàíñèðîâàííûì ñ îòêðûòîé îêðåñòíîñòüþ, åñëè äëÿ ëþáîé âåðøèíû v 2 v ( g) jjfu 2 ng ( v ) : f ( u ) = whitegj ¡ jfu 2 ng ( v ) : f ( u) = b lackgjj · 1 ; ãäå ng ( v ) = fu 2 v ( g ) : uv 2 e ( g) g. 2-ðàçáèåíèå f 0 ãðàôà g íàçûâàåòñÿ ëîêàëüíîñáàëàíñèðîâàííûì ñ çàêðûòîé îêðåñòíîñòüþ, åñëè äëÿ ëþáîé âåðøèíû v 2 v ( g ) jjfu 2 ng[v]: f 0 ( u ) = whitegj ¡ jfu 2 ng[v]: f 0 ( u) = b lackgjj · 1 ; ãäå ng[v] = ng ( v ) [ fvg. â íàñòîÿùåé ðàáîòå äàþòñÿ íåîáõîäèìûå è äîñòàòî÷íûå óñëîâèÿ ñóùåñòâîâàíèÿ ëîêàëüíî-ñáàëàíñèðîâàííûõ 2-ðàçáèåíèé ïîëíûõ ìíîãîäîëüíûõ ãðàôîâ. 84 mathematical problems of computer science 58, 84–90, 2022. doi: 10.51408/1963-0095 udc 004.62:004.946 data processing and persistence in virtual reality systems arman a. hovhannisyan national polytechnic university of armenia e-mail: aahovhannisyan1@gmail.com abstract data processing and persistence are key aspects of developing a virtual reality system. in this paper, an improvement is offered to the distance calculation algorithm of the unity engine. additionally, data persistence mechanisms provided by the unity engine are reviewed, and file system is selected as an appropriate option. storage of object coordinates to the file system is implemented. the results provide a baseline for developing a system for creating virtual stands for professional research. keywords: virtual reality, data management, file system, serialization. article info: received 29 march 2022; received in revised form 9 october 2022; accepted 17 november 2022. 1. introduction in virtual reality, data is represented both in primitive types (int, float, string) and complex types provided by the engine. object positions in space are determined in the cartesian coordinate system [1] (see fig. 1). a common task is to compare the distance of 2 points from a given point 𝐴(𝑥1, 𝑦1, 𝑧1). the unity engine scripting api [2] provides a complex data type called vector3 to store object coordinates, along with its vector3.distance() method to calculate distance between 2 points. given the points (𝑥2, 𝑦2 , 𝑧2), 𝐶(𝑥3, 𝑦3 , 𝑧3), this method may be used to accomplish the task, comparing the following values: vector3.distance(b, a), vector3.distance(c, a). a more efficient solution may be applied using the formula of the distance between 2 points [1]: 𝑑𝐴𝐵 = √(𝑥2 − 𝑥1) 2 + (𝑦2 − 𝑦1) 2 + (𝑧2 − 𝑧1) 2 , (1) a. hovhannisyan 85 𝑑𝐴𝐶 = √(𝑥3 − 𝑥1) 2 + (𝑦3 − 𝑦1) 2 + (𝑧3 − 𝑧1) 2 (2) instead of comparing values for 𝑑𝐴𝐵 and 𝑑𝐴𝐶 , the radicands may be compared, saving cpu time on unnecessary calculations. fig. 1. object positions in space. to have persistent data between sessions, the user progress has to be stored on the disk. there are several methods of managing data storage, including sql database, playerprefs and os file system. on specific events during the runtime, which are to be defined, data containing all the current values have to be stored. these events may include user interaction, object state mutation, or events may be set to trigger on specific timestamps, e.g., every 10 seconds. then, on the next program run, these stored values have to be fetched and transmitted to the engine to render the objects in the same state and position, as they were when the last event was triggered. 2. persistent data to have persistent data between sessions, the user progress has to be stored on the disk. below are listed several methods of managing data storage. 1. sql database sql is useful when there is relational data. it supports queries to fetch related data sets. in our case, we have just objects that need to be memorized and then retrieved on the next run. such simple operations are easier to implement and faster in work on file system. sql is a dedicated software and isn’t an integrated part of operating systems, as file system is. also, a connection to sql service should be kept active during the runtime. 2. playerprefs playerprefs is a class provided by unity engine that stores player preferences between game sessions. it stores values in the os registry. though it is possible to store data using this data processing and persistence in virtual reality systems 86 method, it is not recommended to do so. this method should be used for data, that can be afforded to lose, such as user settings and preferences. sensitive and relatively big data should not be kept in registries. 3. file system to store data in files, it needs to be formatted in some way. it may be serialized [3] to binary format and written to a file. that data will then be successfully deserialized and used in the application. but since binary is not human-readable, it makes this format insufficient. moreover, it is not possible to edit the saved data manually. using json data type allows bypassing these problems. taking into account the points mentioned above, it was decided to handle data storage using file system and serialization, so every time data needs to be stored, it is serialized to json format and written to a file (see fig. 2). then, to restore the state in the application, the file is read and data is deserialized to object (see fig. 3). fig. 2. storing data. fig. 3. using stored data. 3. saving object position a specific and common example is persisting the object position. in this example, we have a cube placed on a table (see fig. 4). the origin (0, 0, 0) can be located on the ground. a. hovhannisyan 87 fig. 4. cube. to save the cube position after it is replaced, a class called savemanager is created, which contains 2 methods: save and load. these methods use a file called "position.dat" to write/read data. savemanager.cs public class savemanager { public static void save(vector3 pos) { string path = path.combine(application.persistentdatapath, "position.dat"); file.writealltext(path, jsonutility.tojson(pos)); } public static vector3 load() { string path = path.combine(application.persistentdatapath, "position.dat"); string result = file.readalltext(path); vector3 pos = jsonutility.fromjson(result); return pos; } } the save and load methods would then be invoked from a script, which is bound to the object. the save method would be bound to the xr grab interactable component [4] “select exited” event to save data every time the object is released. the load method would be invoked from the start method to set object positions from the saved data on a fresh program run. cubescript.cs public class cubescript : monobehaviour { // start is called before the first frame update void start() { vector3 position = savemanager.load(); transform.position = position; } public void save() { savemanager.save(transform.position); } } data processing and persistence in virtual reality systems 88 fig. 5. cube position changed via left controller. the saved file position.dat: {"x":-0.0030583017505705358,"y":0.838373601436615,"z":-2.0934112071990969} after restarting the program, we still have the cube in its new place (see fig. 5). now we can modify this file content, and set the coordinates to (0, 0, 0). {"x":0,"y":0,"z":0} after modifying and saving the file, and running the program again, we can see that the cube appears on the origin as expected (see fig. 6). fig. 6. cube position manually set to origin. 4. conclusion in this article, an improvement to the unity engine distance calculation algorithm was suggested. additionally, data types provided by the unity engine were reviewed. data storage options were compared and decided to use the os file system and data serialization. as an example, a cube position storage and loading were implemented. this method will be used also for custom complex data types to store, marking the class representing the data type as serializable. a. hovhannisyan 89 references [1] wikipedia, (2012) cartesian coordinate system. [online]. available: https://en.wikipedia.org/wiki/cartesian_coordinate_system [2] unity engine scripting api reference. [online]. available: https://docs.unity3d.com/scriptreference/ [3] microsoft docs, (2021) serialization (c#). [online]. available: https://docs.microsoft.com/en-us/dotnet/csharp/programming-guide/concepts/serialization/ [4] the khronos group inc., “the openxr specification”. [5] unity learning, (2020) create with vr. [online]. available: https://learn.unity.com/course/create-with-vr?uv=2020.3 տվյալների մշակումը և պահպանումը վիրտուալ իրականության համակարգերում արման ա․ հովհաննիսյան հայաստանի ազգային պոլիտեխնիկական համալսարան e-mail: aahovhannisyan1@gmail.com ամփոփում տվյալների մշակումը և պահպանումը վիրտուալ իրականության համակարգի զարգացման հիմնական ասպեկտներ են։ այս հոդվածում առաջարկվում է unity շարժիչի հեռավորության հաշվարկման ալգորիթմի լավարկում։ բացի այդ, դիտարկվում են unity շարժիչի կողմից տրամադրվող տվյալների պահպանման մեխանիզմները, և որպես նպատակահարմար տարբերակ, ընտրվում է ֆայլային համակարգը։ իրականացվում է օբյեկտի կոորդինատների պահպանումը ֆայլային համակարգում։ ստացված արդյունքները հիմք են հանդիսանում մասնագիտական հետազոտությունների վիրտուալ ստենդների ստեղծման համակարգի մշակման համար։ բանալի բառեր՝ վիրտուալ իրականություն, տվյալների կառավարում, ֆայլային համակարգ, սերիալիզացիա https://docs.unity3d.com/scriptreference/ https://docs.microsoft.com/en-us/dotnet/csharp/programming-guide/concepts/serialization/ data processing and persistence in virtual reality systems 90 обработка и сохранение данных в системах виртуальной реальности арман а. оганесян национальный политехнический университет армении e-mail: aahovhannisyan1@gmail.com аннотация обработка и сохранение данных являются ключевыми аспектами разработки системы виртуальной реальности. в данной статье предлагается улучшение алгоритма расчета расстояний unity engine. кроме того, рассматриваются механизмы сохранения данных, предоставляемые unity engine, и в качестве подходящего варианта выбирается файловая система. реализуется хранение координат объекта в файловой системе. результаты обеспечивают основу для разработки системы создания виртуальных стендов для профессиональных исследований. ключевые слова: виртуальная реальность, управление данными, файловая система, сериализация mailto:aahovhannisyan1@gmail.com mathematical problems of computer science 59, 45–56, 2023. doi: 10.51408/1963-0101 udc 004.891.3 expert knowledge-based rgt solvers for software testing mane p. buniatyan1, sedrak v. grigoryan2 and emma h. danielyan3 1synopsys armenia, yerevan, armenia 2institute for informatics and automation problems of nas ra,yerevan, armenia 3epam systems inc., yerevan, armenia e-mail: buniatyanmane@gmail.com, addressforsd@gmail.com, emma danielyan@yahoo.com abstract program testing is a way of assessing the quality of software and reducing the risk of software failure in operation [1]. quality issues can cause as financial loss as well as harm to human lives (e.g., when the bug is in medical instruments, cars, etc.). so, it is very hard to underestimate the importance of testing. there are multiple testing techniques, which are split into 3 major categories. one of them includes experience-based techniques. test cases and scenarios used in experience-based testing are derived from the tester’s knowledge and intuition, as well as their experience with similar applications and technologies. these techniques can be helpful in identifying tests that are not identified easily by other more systematic techniques. depending on the tester’s approach and experience, experience-based techniques may achieve widely varying degrees of coverage and effectiveness [1]. we propose a method for automation of experience-based testing via a class of combinatorial problems (rgt class). a solver is developed for the class. it acquires expert knowledge and elaborates effective strategies for rgt problems [2]. the proposed method generates test cases dynamically based on the response of the program. the adequacy of the method is being experimented for ”blender” open-source application, which has python api allowing to experiment with testing and analyze test results. keywords: rgt class, rgt solver, software testing, expert systems. article info: received 25 september 2022; sent for review 11 october 2022; accepted 07 february 2023. acknowledgement: the authors express their deep gratitude to dr. edward pogossian for his contribution and constructive comments to the work. 1. introduction software testing is an approach to assess the quality of software and to reduce the risk of its failure in operation [1]. 45 46 expert knowledge-based rgt solvers for software testing in [1], testing techniques are divided into 3 groups: black-box, white-box and experiencebased techniques. in the case of the last one, test cases are based on the testers’ knowledge and intuition, on experience with similar applications and technologies. these techniques are efficient in identifying tests that are not identified easily by other more systematic techniques as well as when there is a limited testing time or incomplete specifications [1]. according to the world quality report 2021-2022 [3], one of the current trends in quality assurance and software testing is test automation. test automation has the following benefits [1]: • saves time by reducing repetitive manual work • provides greater consistency and repeatability • allows to evaluate the situation more objectively based on static measures, coverage reports, etc. • provides more accurate information about the current state of testing based on gathered statistics, test progress, defect rates and performance. there is a way to automate test case generation, known as the model-based testing (mbt). mbt is a technique for generating a test suite from requirements [4]. instead of individual tests creation, testers create models that allow generating test cases automatically. these methods can be used in regression testing and are especially useful when the system changes frequently. in this case, the test suite can be regenerated easily by adjusting the model instead of readjusting each test case separately. mbt has three important components [4]: • a model (requirement, information, workflow, architectural, behavioral, configuration, deployment, performance, risk, environment, and usage models [5]) • a test-generation algorithm • tools generating a supporting infrastructure (including the expected output). mbt tools are meant to generate test suites by manipulating either with input data or behavior without handling both simultaneously. generated test cases do not provide ways to test the system dynamically (the choice of modules to testing depends on the previous test results). software testing can be considered as a combinatorial problem between a tester and states of a program. hence, testing can be also considered as a representative of reproducible game tree (rgt) class problems. rgt is a class of combinatorial problems, for which the space of solutions is a reproducible game tree. these problems meet the following requirements [6]: • there are interacting actors (players, competitors, etc.) performing identified types of actions in specified moments of time and specified types of situations • there are identified benefits for each actor • there are descriptions of situations in which actors act in and are transformed after actions. m. buniatyan, s. grigoryan and e. danielyan 47 for such problems with a given arbitrary situation x and an actor a, who is going to act in x, we can generate a corresponding game tree gt(x, a) comprising all the games started from x. games represent all possible sequences of legal actions for players and situations that they can create from the given initial, or the root situation x. in our consideration, the games are finite and end with one of the goal situations of the problem [6]. assuming that a plays according to a deterministic program, a strategy, the gt(x, a) represents, in fact, all possible performance trees of the strategies from x. in that sense, the gt(x, a) determines the space of all possible solutions from the situation x. with the given criterion k to evaluate the quality of strategies, we can define the best strategy s*(x, a) and the corresponding best action of a from x [6]. rgt class includes important problems like computer networks intrusion protection, optimal management and marketing strategy elaboration in competitive environments, testing of programs, defense of military units from various types of attacks, communication problems, certain types of teaching, chess and chess-like games [2]. one of the advantages of rgt class is that these problems are reducible to the standard kernel problems k. kmethodology multiplies the achievements for particular problems of this class. distributed development of this methodology is possible. k-methodology enhances the effectiveness of rgt solvers providing answers to urgent rgt questions including the following ones [2]: • measurement of the effectiveness of solvers • analysis and typification of combating knowledge • construction of knowledge-based solvers • regular acquisition of rgt expert knowledge and enhancing the effectiveness of solvers. the validity of k-methodology was proved for certain rgt problems including chess, network intrusion protection, navy defense from attacks, management, marketing etc. [2]. rgt solver is a software that acquires expert knowledge and elaborates effective strategies for rgt problems [2]. it is a universal tool for solving rgt-class problems. strategy searching and game tree. as already mentioned, the space of solutions for rgt problems is a reproducible game tree, and with the given criteria, we can evaluate and choose the best possible actions in given situations for the given actor. as the combinatorial complexity of the mentioned problems is huge, we need to reduce the game tree. otherwise, the computer’s computational resources (memory and storage) will not be enough to solve them. c. shannon suggested reducing the tree by building it until the resources are expired. it is not an effective way because we waste our resources to compute steps that will not improve the current situation. another approach, suggested by m. botvinnik, is to consider only those steps that have potential benefit in the current case, i.e., we should not examine the steps that have no meaning. we can evaluate the possible usefulness of an action with the knowledge (without reviewing the opponent’s answers) and choose the most profitable one. then, by checking the opponent’s potential actions, we can build the game tree and choose the best move in a given situation [7]. the solver builds the game tree, evaluates situations with the knowledge, then chooses the best action using the minimax algorithm. 48 expert knowledge-based rgt solvers for software testing the purpose of this paper: testing of programs can be considered as an rgt problem, and rgt solver can be used for experience-based testing as an expert system when the corresponding knowledge is available. in this work, we aim to provide a definition of testing problems as rgt problems, a way of formulating knowledge, and an approach for proper assessment of tested programs, which also covers the drawbacks of model-based testing approaches (in particular, combining different behaviors and input data, running both functional and non-functional tests at the same time, and generating tests dynamically). thus, the following open questions are addressed: 1. what kind of knowledge are we going to use, who are the actors as well as what are their possible actions? 2. how to evaluate each situation, what kind of goals each actor has, etc.? overall, this leads to proposing a model for representing an experience-based testing as an rgt problem. 2. reduction of program testing to rgt class in rgt problems, it is essential to define the situations, the actors, the actions, and benefits for each of them. let’s define these terms for program testing. the actors in software testing are the system under test (i.e., the program) and the tester. note, that unlike some other problems in the rgt class (e.g., like chess), where the opponent tries to make counteraction, in testing the program just responds to the tester’s actions. the actions are any valid elementary operations that can be performed with the program. while building the ”game” tree, the solver dynamically combines these actions, creates test cases and executes them depending on the response of the program. note, that not all combinations of the elementary operations are meaningful from the perspective of the user (e.g., actions that have no effect or are not connected with each other). that is why we need to find a way to control these combinations. the actions of the program are actually only responses to the tester’s actions. the situations are the current states of the program. we can estimate the current situations with [0;1] numbers, where 0 means that no bugs are found, 1that the program is in a critical state and is not usable. the numbers in-between 0 and 1 are intermediate values, and situations with values closer to 1 are worse than situations with values closer to 0. we suggest the following criteria for evaluating the current state of programs (these criteria can be expanded later): • existence of bugs (difference between expected and observed results): different bugs have different importance; when the main functionalities of the program do not work as expected, the program becomes useless (e.g., if the user is not able to log into a social network, save the result of the accomplished job, do a transfer in the banking system, etc.). • performance degradation: we all would like to have fast, high performing programs, but unfortunately it is not always possible. performance degradation in a part of the program that is used frequently will cause to slowdown the work, but if it is in a part m. buniatyan, s. grigoryan and e. danielyan 49 that can be done without human interaction and/or is performing rarely, then it can be acceptable. • security: this is essential for some programs (e.g., banking system, strategic information storing, transfers, etc.). • crashes and hangovers: this is always bad, and in some cases, they can even cause to a fatal problem, like losing the whole work performed. in most situations, this is not acceptable. we need to take into account the number of problems, as well as their severity and importance, the sequence of actions causing the problem (i.e., how frequently the problem occurs in ”real life”). a bug in a very important functionality is worse than a crash that users might not even encounter, but, on the other hand, having lots of ”minor” issues in the program is also not acceptable. when one of the main functionalities does not meet the requirements mentioned above, the program is in a critical state, and it cannot be delivered to customers. the importance of each functionality is considered as a multiplier for the appropriate criterion. the current state of the program can be measured with the following evaluation function: st = mc ∗ c + mb ∗ b + mp ∗ p + ms ∗ s {1}, where mc, mb, mp, ms ∈ [0; 1], c, b, p, s = {0 | 1}. c, b, p and s are boolean variables, that show the existence of crashes/hangovers, bugs, performance degradations or security problems respectively (1 if the mentioned problems occurred, otherwise 0). mc, mb, mp and ms are multipliers for the occurred problems (they show the importance of the broken functionality). any occurred problem is counted only once, so if, for example, a crash occurs, even if it relates to a security problem or it is a bug (obviously, it is not an expected result) we will consider c = 1, b = 0, s = 0 and p = 0. if the current state of the program is bigger than 1, we consider it as 1. 3. rgt expert knowledge formatting for testing error guessing, exploratory testing and checklist-based testing are representatives of experience-based techniques [1]. considering the characteristics of each of these techniques, we propose the following usage of the solver: by reviewing issues occurred before, the usage of the program and its main functionalities, we create checklists. in the solver, checklists are represented as plans, and the checklists’ actions as goals. based on the coverage reports, the source files responsible for each action can be defined. these connections help to prioritize the created checklists. the user can also define priorities depending on the module he/she is most interested in. checklists lead to the creation of a game tree. each branch in the tree is a test case. it is important to mention that actions in the checklists are general, i.e., many elementary actions can correspond to one action in the checklist. it allows you to combine multiple actions and build a tree. checklists define if it still needs to proceed to the next steps or not in case of a defect occurrences in the current step. multipliers in formula {1} are also given as knowledge for the solver. they show the importance of user action. note, that multipliers should be defined for both elementary 50 expert knowledge-based rgt solvers for software testing and checklist actions. the same elementary action in different situations can have different importance, e.g., if the user tries to save a text file it is more important to save the text than the style. we multiply both multipliers to get one for the action. imagine that in the example below, mb = 0.8 for the elementary action “save” and for the following checklists of actions ”open the program, add text, save”, “open an existing text file, change the style, save”. let’s say we have mb = 1 for the “save” in the checklist1 and mb = 0.6 for the “save” in the checklist2. in this case, if the program is not able to save the text, we will have mb=1*0.8=0.8 and for the second case: mb=0.6*0.8 = 0.48. thus, the first case will be considered worse than the second one. in the case of performance degradation, we need to multiply mp with the coefficient showing how many times the performance was slowed down or how much longer it takes to perform the same action. e.g., if the performance is 2x slower than expected, we need to multiply mp with 2. the testing continues until a. the given time is expired, b. all/chosen checklists are checked or c. if the program gets into a critical state. 4. rgt solver experiments in program testing we have chosen the blender program as a system under test. it is an open-source 3d modeling program with a python interface that can be used for testing. in order to understand how the program testing solver works and how the knowledge and checklists can be represented, let’s study an example. to understand how the knowledge and checklists can be represented, let us review an example. the checklist below checks some of the main functionalities of the program: fig. 1. checklist example to keep it simple, we just added a few basic operations, but this list can be enlarged if needed. the operations in this checklist can be independent, like lines 6 and 7. but if this was a checklist based on the previous failures or a user story, then all steps would depend on each other. this checklist could be used if we had limited testing time and could only check the main operations to make sure that there were no critical issues (like a smoke test). the first line of the checklist (i.e., the comment) represents the name of the checklist and the source file which is associated with the checklist (here, as we don’t know the corresponding m. buniatyan, s. grigoryan and e. danielyan 51 source file, we put x.cpp just to show the structure of the checklist. the source file is not mandatory). if some multipliers are absent in the checklist, we assign 0 to them (e.g., ms=0 for all actions in checklist below, because they could not lead to security problems). the variable nextstep is used to determine whether the next step should be performed or not in case of bug in the current step (e.g., if the user is not able to move the 3d cursor it is still somewhere in the scene and the user can add objects). in line 3 we open the program. if it crashes it is a critical state for the program, thus mc=1. next to mp there is the expected time the operation should take (mp/5s/). if it takes 25 seconds, we multiply mp by 5. as this operation is not repeatable and happens only once, when the work starts, its performance is not very important, but yet the user cannot wait for about 10 minutes to start working. as the performance depends on the users’ computer, the performance parameters are defined for minimum system requirements of the program. in the example above, we just used values based on local resources. in line 4, we need to move the 3d cursor. 3d cursor position defines where the object is being added. it can also be used as a 3d view orientation to define where to move objects, to move the pivot point to the 3d cursor, as the rotation point in the spin tool, etc. so, it is a quite important feature, but in case it does not work users can still find workarounds. note that there is no expected time next to mp for this action. it is because this action should work simultaneously with the click (i.e., should not take noticeable time). like other actions in the checklist, this is one of the basic operations, so crash is unacceptable here, thus mc = 0.9. note that all the multipliers here are conditional and this is just an example. in real world example, probably, multipliers should be chosen more thoroughly. nextstep is 1 here, because even if the 3d cursor cannot be moved, we are still able to add an object. to perform this step using the python api we do the following: fig. 2. elementary operation: move 3d cursor 52 expert knowledge-based rgt solvers for software testing this is an elementary operation for moving the 3d cursor. the first line comment shows the corresponding general operation (in the checklist) and the multiplier. as in this case only 1 elementary operation corresponds to the checklist operation, its multiplier is 1. note that the case is not always the same (the coordinates are randomly generated) and the test also checks if the operation was performed successfully or not. in the 5th line of the checklist, we have the ”add object” operation. many elementary operations correspond to this operation (see fig. 3): there are lots of groups of objects, and each group itself contains various objects. fig. 3. add object the python code below is an example of the “add object” operation. it adds a cube in the current location of the cursor. as all objects can be used for creating different 3d models, and their importance is dependent on what exactly the user tries to create m=1 for all objects. note that if the object is not added then we cannot perform the next action, i.e., we cannot change its geometry. the last command in the checklist is “change geometry”. first of all, the user should switch to the edit mode in order to change the object’s geometry, i.e., move the object’s vertices, edges and faces, and then perform the corresponding operations. for this general action, there are 3 possible elementary actions (move vertices, edges, faces). all of them are important while creating a 3d model, but considering the fact that if a user is not able to move the edge, he/she can choose vertices of the edge and move them together (so that the edge will be moved), or choose all edges/vertices of a face and move it. the most important one in those operations is moving vertices, and then edges, then surfaces. thus, in this case, m. buniatyan, s. grigoryan and e. danielyan 53 fig. 4. elementary operation: add object the multiplier for each operation will be different: fig. 5. elementary operation: change geometry for the given example, the solver moves the 3d cursor to different positions, adds different objects, changes their geometry, and makes sure that these operations work as expected for different objects (i.e., checks that the python tests are passing). to check how the solver behaves if the operation does not work, we can simply use assertnotequal function instead of assertequal (e.g., instead of “assertequal(bpy.context.scene.cursor.location.x, x)” we can write “assertnotequal( bpy.context.scene.cursor.location.x, x)”). the solver will combine different elementary tests together, create test cases and run them. to run tests, we use the following command: in order to use the solver for different programs, we use a configuration file, which defines how to run tests (e.g., paths to test cases, checklists and elementary operations). 54 expert knowledge-based rgt solvers for software testing fig. 6. command for running a test 5. conclusion we propose a new approach for test automation and test results evaluation considering the testing of programs as a rgt-class problem. in this work: 1. tools defining the types of knowledge for testing the target application are described. the described knowledge is being integrated into rgt solver and being used to run test cases, test scenarios with later evaluation of test results. 2. an approach for evaluating the state of the program during the testing is proposed. 3. the adequacy of the proposed approach is being experimented with the open-source blender application. 4. the proposed approach solves drawbacks of the model-based testing approach, namely, allows to generate test cases dynamically. the described solution is generic for the rgt solver and can be used for testing various applications. references [1] k. olsen and m. posthuma and s. ulrich, “ certified tester foundation level syllalbus”, international software testing qualifications board, pp. 56–62, 2019. [2] e. pogossian, constructing models of being by cognizing. yerevan, pp. 150–159, 2020. [3] world quality report, capgemini, sogeti, micro focus, pp 16–37, 2021 [4] d. rakhi, j. ashish, n. karunanithi, j. leaton, c. lott, g. patton and b. horowitz, “model-based testing in practice”, proceedings of the 1999 international conference on software engineering (ieee cat. no.99cb37002), los angeles, ca, usa, 1999, pp. 285-294, doi: 10.1145/302405.302640. [5] i. schieferdecker and a. hoffmann, model-based testing, ieee software 29.1, pp. 14–18, 2012. [6] e. pogossian, v. vahradyan a. grigoryan, on competing agents consistent with expert knowledge, proceedings of second international workshop, ais-adm 2007,autonomous intelligent systems: multi-agents and data mining, st. petersburg, russia, pp. 229–241, 2007. [7] m. botvinnik, computers in chess: solving inexact search problems, springer-verlag, new york, 1983. m. buniatyan, s. grigoryan and e. danielyan 5 5 öáñó³·çï³ï³ý ·çï»éçùý»ñç íñ³ ñçùýí³í rgt solver-ç ïçñ³éáõùá íñ³·ñ³ûçý ³å³ñáíù³ý ã»ëï³íáñù³ý ëý¹ñáõù ø³ý» ä. ´áõýç³ãû³ý1, ê»¹ñ³ï ì. ¶ñç·áñû³ý 2 ¨ ¾ùù³ ð. ¸³ýç»éû³ý3 1synopsys ð³û³ëï³ý, ºñ¨³ý 2ðð ¶²² æýýáñù³ïçï³ûç ¨ ³íïáù³ï³óù³ý åñáµé»ùý»ñç çýëïçïáõï, ºñ¨³ý, ð³û³ëï³ý 3 epam ð³û³ëï³ý, ºñ¨³ý e-mail: buniatyanmane@gmail.com, addressforsd@gmail.com, emma danielyan@yahoo.com ²ù÷á÷áõù â»ëï³íáñáõùá íñ³·ñç áñ³ïá ·ý³ñ³ï»éáõ ¨ ß³ñ³·áñíù³ý ù»ç íñ³·ñ³ûçý ³å³ñáíù³ý ó³ëáõù³ý éçëï»ñá ýí³½»óý»éáõ ùççáó ¿: ìñ³·ñáõù ëë³éý»ñç ³éï³ûáõãûáõýá ï³ñáõ ¿ µ»ñ»é çýãå»ë ýçý³ýë³ï³ý ïáñáõëïý»ñç, ³ûýå»ë ¿é ù³ñ¹ï³ûçý ½áñ»ñç (ûñçý³ï, µåßï³ï³ý ë³ñù³íáñáõùý»ñç ï³ù ù»ù»ý³ý»ñáõù ³éï³ ëë³éý»ñá): ²ûëåçëáí, µ³ñ¹ ¿ ã»ñ³·ý³ñ³ï»é ã»ëï³íáñù³ý ï³ñ¨áñáõãûáõýá: â»ëï³íáñù³ý ùáï»óáõùý»ñá ï³ñ»éç ¿ µ³å³ý»é 3 ñçùý³ï³ý ëùµ»ñç, áñáýóçó ù»ïá ÷áñóç íñ³ ñçùýí³í (experience-based) ã»ëï³íáñáõùý ¿: ²ûë å³ñ³·³ûáõù ã»ëï»ñá ëï»õííáõù »ý‘ ñçùýí»éáí ã»ëï³íáñáõç ·ç»éçùý»ñç ¨ çýïáõçóç³ç, çýãå»ë ý³¨ ý³ëïçýáõù ýù³ý³ïçå íñ³·ñ»ñç ñ»ï áõý»ó³í ÷áñóç íñ³: öáñóç íñ³ ñçùýí³í ùáï»óáõùý»ñý û·ýáõù »ý µ³ó³ñ³ûï»é ³ûýåçëç ëë³éý»ñ, áñáýù ß³ï µ³ñ¹ ¿ ñ³ûïý³µ»ñ»é ³í»éç ñ³ù³ï³ñ·í³í ùáï»óáõùý»ñáí: ²ûë ³ßë³ï³ýùáõù ù»ýù ³é³ç³ñïáõù »ýù ÷áñóç íñ³ ñçùýí³í ã»ëï³íáñù³ý ³íïáù³ï³óáõù` û·ï³·áñí»éáí ïáùµçý³ïáñ ëý¹çñý»ñç rgt ¹³ëá: rgt ¹³ëç ëý¹çñý»ñç éáõíù³ý ñ³ù³ñ ùß³ïíáõù ¿rgt solver-áª íñ³·ñ³ûçý ÷³ã»ã, áñá ïáõï³ïáõù ¿ ÷áñó³·çï³ï³ý ·çï»éçùý»ñ ¨ ëï»õíáõù ¿ ³ñ¹ûáõý³í»ï é³½ù³í³ñáõãûáõýý»ñ rgt ¹³ëç ëý¹çñý»ñç éáõíù³ý ñ³ù³ñ: ²é³ç³ñïáõù »ýù rgt solver-ý û·ï³·áñí»é íñ³·ñ»ñç ã»ëï³íáñù³ý ëý¹ñáõù: solver-á ëï»õíáõù ¿ ã»ëï³ûçý çñí³ç×³ïý»ñ` ï³ëí³í íñ³·ñç ³ñó³·³ýùçó/å³ï³ëë³ýçó ¨ ·ý³ñ³ïáõù ¿ ¹ñ³ýù áëï ý³ë³å»ë ë³ñù³ýí³í ã³÷³ýçßý»ñç: ²ûë ùáï»óù³ý ³¹»ïí³ïáõãûáõýá ÷áñó³ñïíáõù ¿ »é³ã³÷ ùá¹»é³íáñù³ý »blender» íñ³·ñç ùççáóáí: ´³ý³éç µ³é»ñ` rgt ¹³ë, rgt solver, íñ³·ñ³ûçý ³å³ñáíù³ý ã»ëï³íáñáõù, ÷áñó³·çï³ï³ý ñ³ù³ï³ñ·»ñ: 5 6 expert knowledge-based rgt solvers for software testing rgt solver íà îñíîâå ýêñïåðòíûõ çíàíèé äëÿ òåñòèðîâàíèÿ ïðîãðàììíîãî îáåñïå÷åíèÿ ìàíå ï. áóíèàòÿí1, ñåäðàê â. ãðèãîðÿí2 è åììà ã. äàíèåëÿí3 1synopsys àðìåíèÿ, åðåâàí 2èíñòèòóò ïðîáëåì èíôîðìàòèêè è àâòîìàòèçàöèè íàí ðà, åðåâàí, àðìåíèÿ 3epam àðìåíèÿ, åðåâàí e-mail: buniatyanmane@gmail.com, addressforsd@gmail.com, emma danielyan@yahoo.com àííîòàöèÿ òåñòèðîâàíèå ïðîãðàìì-ýòî ñïîñîá îöåíêè êà÷åñòâà ïðîãðàììíîãî îáåñïå÷åíèÿ è ñíèæåíèÿ ðèñêà îòêàçà ïðîãðàììíîãî îáåñïå÷åíèÿ â ðàáîòå. î÷åíü òðóäíî íåäîîöåíèòü âàæíîñòü òåñòèðîâàíèÿ: ïðîáëåìû ñ êà÷åñòâîì ïðîãðàìì ìîãóò ïðèâåñòè êàê ê ôèíàíñîâûì ïîòåðÿì, òàê è íàíåñòè óùåðá çäîðîâüþ ëþäåé (íàïðèìåð, êîãäà îøèáêà íàõîäèòñÿ â ìåäèöèíñêèõ ïðèáîðàõ, àâòîìîáèëÿõ è ò. ä.). ìåòîäû òåñòèðîâàíèÿ ìîæíî ïîäðàçäåëèòü íà 3 îñíîâíûå ãðóïïû. îäíà èç íèõ ýòî ìåòîäû, îñíîâàííûå íà îïûòå. çäåñü òåñòîâûå ïðèìåðû ñîçäàþòñÿ íà îñíîâå çíàíèé è èíòóèöèè òåñòèðîâùèêà, à òàêæå íà åãî îïûòå ðàáîòû ñ àíàëîãè÷íûìè ïðèëîæåíèÿìè è òåõíîëîãèÿìè. ýòè ìåòîäû ìîãóò áûòü ïîëåçíû ïðè îïðåäåëåíèè òåñòîâ, êîòîðûå íå ëåãêî èäåíòèôèöèðîâàòü äðóãèìè áîëåå ñèñòåìàòè÷åñêèìè ïîäõîäàìè ê òåñòèðîâàíèþ. â çàâèñèìîñòè îò ïîäõîäà è îïûòà òåñòèðîâùèêà, ýòè ìåòîäû ìîãóò îáåñïå÷èâàòü øèðîêóþ ñòåïåíü ïîêðûòèÿ è ýôôåêòèâíîñòü òåñòèðîâàíèÿ. â äàííîé ñòàòüå ìû ïðåäëàãàåì ìåòîä òåñòèðîâàíèÿ íà îñíîâå îïûòà (àâòîìàòèçàöèÿ òåñòèðîâàíèÿ) ÷åðåç êëàññ êîìáèíàòîðíûõ çàäà÷ (rgt êëàññ). rgt êëàññ âêëþ÷àåò òàêèå âàæíûå çàäà÷è, êàê çàùèòà îò âòîðæåíèé â êîìïüþòåðíûå ñåòè, ðàçðàáîòêà îïòèìàëüíîé ñòðàòåãèè óïðàâëåíèÿ è ìàðêåòèíãà â êîíêóðåíòíîé ñðåäå, òåñòèðîâàíèå ïðîãðàìì, çàùèòà âîèíñêèõ ÷àñòåé îò ðàçëè÷íûõ òèïîâ àòàê, ïðîáëåìû ñî ñâÿçüþ, îòäåëüíûå âèäû îáó÷åíèÿ, øàõìàòû è øàõìàòîïîäîáíûå èãðû. rgt solver ýòî ïðîãðàììà, êîòîðàÿ íàêàïëèâàåò ýêñïåðòíûå çíàíèÿ è ðàçðàáàòûâàåò ýôôåêòèâíûå ñòðàòåãèè äëÿ ðåøåíèÿ çàäà÷ êëàññà rgt. â êà÷åñòâå ýêñïåðòíîé ñèñòåìû äëÿ òåñòèðîâàíèÿ, îñíîâàííîãî íà îïûòå, ïðåäëàãàåòñÿ èñïîëüçîâàòü rgt solver. solver ãåíåðèðóåò òåñòîâûå ñèòóàöèè íà îñíîâå îòâåòà/ðåàêöèè ïðîãðàììû è îöåíèâàåò èõ ïî ðÿäó çàðàíåå îïðåäåëåííûõ êðèòåðèåâ. àäåêâàòíîñòü ìåòîäà ïîêàçàíà íà ïðèìåðå ïðèëîæåíèÿ ñ îòêðûòûì èñõîäíûì êîäîì ”áëåíäåð”. êëþ÷åâûå ñëîâà: rgt êëàññ, rgt solver, òåñòèðîâàíèå ïðîãðàììíîãî îáåñïå÷åíèÿ, çíàíèÿ, ýêñïåðòíûå ñèñòåìû. 05_սեդրակ_59 05 mathematical problems of computer science 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 начиная с начала 2000 года осуществляется внедрение ghis в здравоохранении, в рамках принятого проекта о реформирование информ mathematical problems of computer science 49, 92--96, 2018. info-communication systems based on e-mail and sms technologies andranik e. mkhitaryan institute for informatics and automation problems of nas ra e-mail: and.mkhitaryan@gmail.com abstract the features of the widespread systems for the exchange of messages in computer and cellular networks, the prerequisites for the creation of combined messaging systems using ip and gsm technologies are considered. the peculiarities of the construction of hybrid e-mail / sms info-communication systems are considered. keywords: mail2sms, e-mail, sms, notification. 1. introduction electronic mail is a system to send/receive text messages between computers by the internet. despite the fast delivery of the letter to the addressee, e-mail is a typical "on demand" system. the reading of the delivered message by the addressee depends on the time when he/she requests it from the e-mail provider. another worldwide system for text message exchange is sms. unlike e-mail, sms is not a postal system: the size of the transmitted message is limited, message delivery is possible only in the coverage area of the cellular network or according to the roaming agreements of the local cellular operator. the undeniable advantage of sms technology is the delivery of the message directly into “the pocket” of a mobile subscriber. the limitations typical of e-mail and sms can be overcome by the use of integrated, "in one bottle," mutually complementary network and sms technologies. at the same time, the problem of "long-range" of a particular sms-service is solved using the internet as a transport for sms messages (websms). 2. solution the problem of "on demand" can be solved by the use of e-mail related sms-systems to promptly notify the recipient of incoming e-mails in his mailbox [1], [2]. these technologies 92 a. mkhitaryan 93 usually use special program selectors, built into the mail server that allocate messages for which a notification is required to be sent to the recipient (according to the permissive list with the corresponding e-mail addresses on the server) and provides necessary data to form an sms notification for the particular sms server, which has a gateway to the cellular network of the region. sms server in the internet environment is equipped with a gsm modem a gateway to a given cellular network, which provides the formation and transmission of sms notifications. in the considered case, the initiative of selecting from among the incoming messages those for which it is desirable to receive an sms notification, is assigned to the recipient. another approach is possible when creating a notification system: in this case, an application for sending an sms notification to the addressee is sent directly by the sender when preparing the letter [3]. in such cases, the message should contain certain signs indicating the need to send the letter to the addressee together with sms notification. bearing in mind that email notification systems are based on the use of sms technologies, it is natural to combine the info-communication resources of e-mail and sms technologies in the form of one service for the user of an e-mail. in such systems, the ability of e-mail account users to send sms notifications to the specified cellular networks is not the only possible way, because in these systems it is easy to integrate "in one package" for an e-mail user: a wide range of services related to the use of sms technologies (transmission of sms notifications via e-mail, group mailings of sms, etc.). using such a hybrid "e-mail + sms" technology allows creating a unified system for the user of webmail to access all the info-communication resources used in this mail system, both e-mail and sms. 3. multiple approaches there are two possible approaches to creating such hybrid systems: a) systems with the selection of incoming mail from the mail server according to certain characteristics and transfer of the necessary data to sms server for configuration and sending sms (figure 1). the criterion for selecting an incoming message containing a request for operations with sms can be marked by the sender in the "subject" section. accordingly, when composing a letter in the subject section, a command line is formed containing a request for the gsm controller to perform a specified specific operation. the selector of mail server selects e-mails containing a command in the subject section from the incoming e-mail stream. from the letter, in turn, the attributes of the letter are highlighted and together with the content of the subject are transferred from the mail server to the gsm controller for further processing in accordance with the command contained in the command line. fig. 1. structure of the first approach. info-communicational systems based on email and sms technologies 94 b) systems that operate with copies of letters with the necessary target designations sent directly to the sms server to create and send an sms to the recipient (figure 2). in this case, the system is presented in the form of an independent e-mail / sms infocommunication resource, not directly connected to webmail services that uses the e-mail as a transport of access to the resource for the user. user access to the system is done directly from the webmail page of the user, without requiring additional access to other resources. during the preparation of the letter, the user needs to send a copy of the letter to the mail address of the system (to request a notification) or send an email directly to the system. such a solution does not require a binding to a specific e-mail provider. these provisions are used in the unimail-asnet project [4]. the server acts as an ordinary e-mail user. for the convenience of webmail users, the mechanisms for accessing and managing sms options are combined with e-mail preparation operations with a minimum number of request commands to the unimail server. all operations with the system are made directly from the webmail user's page when composing the letter. in the highlighted markers of the "subject" field, the command of the message contains the information (phone numbers of the addressees, management of the "black list") for the unimail server to perform the corresponding operations. a copy of the letter should be sent to the e-mail address of the server for further processing. the choice of the mode requested by the user (notification or sending of sms) is made automatically by the system, analyzing the address part of the message copy being processed. commands with descriptions:  *notice to…(phone numbers)* parallel to the letter send sms notification to the addressee/addressees  *sms to… (phone numbers)* send an sms to the addressee/addressees  *add to black list* include in the "black list" of the blocked recipient  *del from black list* exclude the blocked addressee from the "black list" fig. 2. structure of the second approach. a. mkhitaryan 95 4. conclusion besides many undisputable advantages, e-mail is a typical “on demand” info-communication resource. however, another technology – sms, does not have such a problem and allows you to send text messages directly to the addressee’s mobile phone. the use of e-mail/sms hybrid system will help users overcome that disadvantage of e-mail. in the article two different approaches of such systems are presented. both advantages and disadvantages of both models are discussed. the possible design for them and the use cases are described. references [1] д. геворкян, а. нанасян и к. хачатрян, «новые web ресурсы asnet.am», proceedings of international conference of computer science and information technologies, csit-2011, ереван, pp. 311-312, 2011. [2] d. gevorkyan, k. khachatryan, a. nanassian, a. petrosyan, g.petrosyan, v. sahakyan and e. vardanyan, “mail informerselective incoming instant phone notification system”, proceedings of international conference computer science and information technologies, csit, yerevan, armenia, pp. 466-467, 2009. [3] а. нанасян и к. хачатрян «mail2sms.asnet.am – система оповещения о входящих письмах», proceedings of international conference of computer science and information technologies csit-2013. ереван, 2013. [4] а. мхитарян, э. матвеев, а. нанасян, в. саакян и a. petrosyan, “гибридная инфокоммуникационная email/sms система unimail”, proceedings of international conference of computer science and information technologies, csit, yerevan, armenia, pp. 389-391, 2017. submitted 22.11.2017, accepted 12.02.2018. տեղեկատվական հաղորդակցման համակարգեր հիմնված էլ. փոստի և sms-ի տեխնոլոգիաների վրա ա. մխիթարյան ամփոփում աշխատանքում դիտարկվել են համակարգչային և բջջային կապերով տեքստային հաղորդագրություններ ուղարկելու համար նախատեսված մեխանիզմների առանձնահատկությունները, ip և gsm տեխնոլոգիաների համատեղ կիրառմամբ հաղորդագրությունների փոխանցման համակարգի ստեղծման նախադրյալները։ info-communicational systems based on email and sms technologies 96 դիտարկվել են e-mail/sms հիբրիդային տեղեկատվական հաղորդակցման համակարգի ստեղծման առանձնահատկությունները։ инфо-коммуникационные системы на базе технологий e-mail и sms а․ мхитарян аннотация рассмотрены особенности распространенных систем обмена письменными сообщениями в компьютерных и сотовых сетях, предпосылки создания совмещенных систем передачи сообщений, использующих технологии ip и gsm сетей. рассмотрены особенности построения гибридных e-mail/sms инфокоммуникационных систем. mathematical problems of computer science 58, 42–51, 2022. doi:10.51408/1963-0091 udc 519.6, 004.9 a brief comparison between white box, targeted adversarial attacks in deep neural networks grigor v. bezirganyan and henrik t. sergoyan department of mathematics, technical university of munich, muinch, german e-mail: grigor.bezirganyan@tum.de, henrik.sergoyan@tum.de abstract today, neural networks are used in various domains, in most of which it is critical to have reliable and correct output. this is why adversarial attacks make deep neural networks less reliable to be used in safety-critical areas. hence, it is important to study the potential attack methods to be able to develop much more robust networks. in this paper, we review four white box, targeted adversarial attacks, and compare them in terms of their misclassification rate, targeted misclassification rate, attack duration, and imperceptibility. our goal is to find the attack(s), which would be efficient, generate adversarial samples with small perturbations, and be undetectable to the human eye. keywords: adversarial attacks, robustness, machine learning, deep learning. article info: received 26 aprile 2022; received in revised form 4 july 2022; accepted 29 july 2022. 1. introduction nowadays, deep neural networks are becoming more and more popular to solve problems in various domains, including safety-critical areas such as medicine, self-driving cars, etc. unfortunately, techniques to fool deep learning models have recently come out to provide incorrect outputs [1]. particularly, in the image classification domain, an attacker can create an altered image, which will be misclassified by a model but will be classified correctly by a human. this altered image is often referred to as an adversarial example, and this process as an adversarial attack. to be protected against such attacks, researchers try to create methods to make the models more robust against such perturbations. studying adversarial attacks and their potential helps us develop better countermeasures against them. in this paper, we will discuss some of the adversarial algorithms and test them against an image classification model. we then compare the results of the experiments in terms of their misclassification rate, targeted misclassification rate, attack duration, and imperceptibility. 42 g. bezirganyan and h. sergoyan 43 in poisoning attacks, the attacker tries to insert fake samples (i.e., data samples with wrong labels) into the training dataset, which will make the model learn on those fake samples and output wrong results. this kind of attack is possible when the attacker has the means to import those fake samples into the training set. in contrast, in evasion attacks, the attacker does not need access to the dataset. in this case, the attacker creates adversarial samples, which are similar and hard to distinguish by a human from the original samples but are misclassified by the trained model. based on how much information the attacker has about the model, attacks can be classified into white-box, black-box, and gray-box attacks. in the white box scenario, the attacker has full knowledge about the model architecture and uses this knowledge to generate adversarial examples. in contrast, in the black-box setup, the attacker does not know the architecture. instead, the attacker observes the output of the model from the given input. some of the attacks assume access to the soft labels (i.e., probability or likelihood score of belonging to a class), while others try to generate examples based on only hard labels (i.e., class labels without the score). in the gray-box setting, the attacker has an access to the original model and trains a generative model on it. when the generative model is ready, the attacker uses that model to generate adversarial samples. hence, the original model is no more needed. recently, in [2] another category was introduced, called no-box attacks. in contrast to black-box attacks, the attacker cannot query the model, instead, he has a small number of samples from the same domain as the victim. the authors train an auto-encoder on those samples and then generate the adversarial examples using the features learned from the auto-encoder. in the targeted attack, the attacker tries to misclassify the given sample into a specific target label. in contrast, in non-targeted attacks, the attacker tries to classify the sample into any other class. in this paper, we try to overview some of the adversarial attack techniques and, running experiments in the same setting, compare them based on: • misclassification: what percentage of the adversarial samples were misclassified • targeted misclassification: what percentage of the adversarial samples were successfully misclassified to the target class • imperceptibility: how much the adversarial example looks like the original image • duration of the attack: how long it takes to generate an adversarial example 1.1 definitions and notations 1.1.1 poisoning attacks vs evasion attacks 1.1.2 attacker’s knowledge of the model 1.1.3 targeted vs non-targeted attacks 1.2 our goal and contribution 44 a brief comparison between white box, targeted adversarial attacks in deep neural networks in this paper, we will concentrate only on white-box and target attacks. in particular, we will discuss and experiment with the fast gradient sign method [1], projected gradient descent [3], autopgd [4], and fw + dual lmo [5]. we chose these attack methods as fgsm is one of the first and simplest methods, which is still popular today. pgd is the most popular, as even many new state-of-the-art methods are modified versions of the pgd attack. autopgd, being one of those variations, achieves state-of-the-art results according to the authors. and while these attacks use ℓp norms, we also chose fw + dual lmo as an example of an attack that uses another norm (wasserstain norm in this case). 2. attack mechanisms in this section, we will briefly overview the attacks, which will be used for experimentation further in the paper. in our attacks, we are given a set of input images x ∈ rn×n, and our goal is to craft an adversarial example x′ ∈ rn×n that will be misclassified by the deep learning model f : rn×n → n. since we are discussing targeted attacks, we want to misclassify the adversarial sample into our desired target class t ∈ n instead of the original class y ∈ n. furthermore, the perturbation we add to the image should be as small as possible, not to be detected by a human. so, we can formulate the problem in the following way: given a neural network f : rn×n → n, input image x ∈ rn×n with a label y ∈ n, a distance function || · || and a perturbation budget ϵ ∈ r find an x′ ∈ r such that f(x′) = t ̸= y s.t. ||x′ − x|| ≤ ϵ. (1) in our case, the distance functions will be l1, l2, l∞ distances or the wasserstein distance. since we can access the gradients of the network in the white-box setting, what most of the gradient-based attacks do, is to fix the network weight and maximize the loss by updating the image. for that, they add a small perturbation η ∈ rn×n to the original image: x′ = x + η the most efficient way to maximize the loss would be to add noise in the same direction as the gradients. [1] introduced an attack method, where they do exactly that: add a perturbation in a direction that will increase the loss function l between the adversarial example and the original label x′ = x + ϵ · sign(∇xl(θ, x, y)). (2) we can see that in this way the maximum allowed perturbation is added, while still being in the ϵ ball. for a targeted setting, the update step will become: x′ = x − ϵ · sign(∇xl(θ, x, t)) in other words, a perturbation is added to minimize the loss between the adversarial sample and the target class t. 2.1 fast gradient sign method (fgsm) g. bezirganyan and h. sergoyan 45 the projected gradient descent attack (pgd) or basic iterative method (bim) was introduced in [3], where they transformed the fgsm [1] one-step attack into an iterative one by performing the update step (2) multiple times with a small step size α ∈ rn×n. this will work better, as the fgsm adds the maximum allowed perturbation, but does not guarantee to maximize the loss within the allowed ϵ−ball. in contrast, in an iterative approach, the algorithm is more likely to find the maxima. to ensure that the adversarial sample remains in the ϵ neighborhood, pgd projects the sample back to the ϵ ball after each update step. in other words, it performs projected gradient descent (or ascent) on the input sample. the update steps for targeted and untargeted attacks will be as follows: x(i+1) = πϵ(x (i) + α · sign(∇x(i)l(θ, x (i), y))) (3) x(i+1) = πϵ(x (i) − α · sign(∇x(i)l(θ, x (i), t))) (4) so, the attacker tries to find a perturbation that either finds the maximum loss between x′ and y (3) (untargeted attack), or the minimum loss between x′ and t (4) (targeted attack). it has recently been suggested [4] that the cross-entropy loss and the fixed step size of the pgd attack [3] may be two reasons for its potential failure. they propose an alternative loss function and a new gradient-based method, auto-pgd, which does not require a fixed step size. they divide their method into two phases: an exploration phase and an exploitation phase. during the exploration phase, they search for good initial points, while in the exploitation phase, they try to maximize the accumulated knowledge. the step size value depends on the trend of optimization. if the objective function decreases rapidly, then the step size does not need to be changed, otherwise, if it decreases slowly, the step size is reduced. the wasserstein adversarial attack was introduced in [6]. here they proposed to use the wasserstein distance instead of the commonly used ℓp distances. for images, the wasserstein distance can be seen as the cost of redistributing pixel mass. for example, while rotations change ℓp norms dramatically, they only slightly change the wasserstein distance. so, what their algorithm does, is to do a pgd attack [3], but instead of projecting on an ℓp norm, they project on the wasserstein ball. however, since the projection onto the wasserstein ball is computationally expensive, they make an approximation by performing modified sinkhorn iterations [7]. [5] improved the algorithm by introducing an exact but still efficient projection operator. they also introduce an adversary generating method based on the frank-wolfe [8] method equipped with a suitable linear minimization oracle and show that it works very fast for wasserstein constraints. in this paper, we will use that frank-wolfe method (fw + dual lmo) for the experiments. 2.2 projected gradient descent 2.3 auto-projected gradient descent 2.4 wasserstein attack 46 a brief comparison between white box, targeted adversarial attacks in deep neural networks in this experiment, our goal is to run fgsm [1], pdg [3], autopgd [4], and fw + dual lmo [5] attacks on the same environment and compare them in terms of misclassification, targeted misclassification, attack duration, and imperceptibility. we are performing our experiments on a pre-trained resnet-18 [9] classifier on the cifar10 dataset [10], with initial 92.4% accuracy on the test set. we generate the adversarial examples on a server with an nvidia geforce gtx 1080-ti gpu. we use the adversarial robustness toolkit (art) [11] for fgsm [1] and pgd [3] and autopgd [4] attacks, and the original implementation by the authors for fw + dual lmo [5]. we run each of the adversarial attacks with a set of epsilon values in ϵ ∈ (0, 0.5] and for all target classes. we use ℓp norms for fgsm, pgd, and autopgd, and we use the wasserstein distance for the fw + dual lmo. all the other hyper-parameters are left to their default values. for the fw + dual lmo, in the original implementation, there was no option for targeted attacks. hence, we modified their implementation and added the option for target attacks. for that we converted the problem: maximize l(f(x′), y) subject to ||x′ − x|| ≤ ϵ to minimize l(f(x′), t) subject to ||x′ − x|| ≤ ϵ we log the duration of the attack, the misclassification rate, and the targeted misclassification rate for later comparison. the source code for the experiment can be found https : //github.com/bezirganyan/adversarialarenahere. we first look at the average misclassification and targeted misclassification scores that each of our models was able to achieve for some ϵ ∈ (0, 0.5]. in table 1, we can see average misclassification and targeted misclassification rates for the best epsilon of each attack. as we can see from the ℓp attacks, the ℓ∞ norm yields the highest scores in our setup. hence, from now on we will use the ℓ∞ norm for further comparisons. note that this does not mean that the ℓ∞ norm is better since we could get similar scores and similar perturbations for higher ϵ values under other norms, as the ℓ∞ attack will add a higher amount of perturbation under the same epsilon. furthermore, we can see that from the ℓp attacks in terms of targeted misclassification rate, the pgd, and autopgd attacks yield very high scores leaving the fgsm attack behind with a huge margin. in general, pgd and autopgd attacks behave almost identically in 3. experiments 3.1 goal 3.2 setup 4. results 4.1 targeted misclassification and misclassification rate g. bezirganyan and h. sergoyan 47 our experiments. we hypothesize that this is because we are testing on an undefended model, on which they both reach their maximum potential limit. the developers of the art framework confirmed that on their tests on defended models in an untargeted setting, autopgd behaved slightly better. we, hence, plan to test and compare the models on a defended model in our future work. in fig. 1, we can see the misclassification and targeted misclassification rates of the attacks for different epsilons and under the ℓ∞ norm. we can see that in terms of misclassification and targeted misclassification rates the pgd and autopgd attack perform best within the ℓp attacks by having around 90% misclassification rate even for very small epsilon. fig. 1. average misclassification and targeted misclassification rates for different ϵ values under ℓ∞ and wasserstein (fw) norms. furthermore, we can see that for the fgsm attack, the targeted misclassification does not increase monotonically. the reason for this can be that since the fgsm is not an iterative algorithm and performs just one step, it overshoots when the epsilon is too big and misses the target class. the fw+dual lmo attack performs best in terms of both misclassification and targeted misclassification rates. nevertheless, we cannot compare the amount of perturbation under ℓ∞ and wasserstein norms, since they imply different amounts of changes to the image. hence, we will need to combine these results with the visual ones to be able to make a fair comparison. in table 2, we can see the time duration needed to generate an adversarial example. being a simple one-step attack, fgsm leads the competition followed by the pgd and autopgd 4.2 duration 48 a brief comparison between white box, targeted adversarial attacks in deep neural networks table 1: average misclassification and average targeted misclassification rates for different norms attack norm miscl targ. miscl. autopgd ℓ1 0.0832 0.1100 pgd ℓ1 0.0810 0.1086 fgsm ℓ1 0.0879 0.1116 autopgd ℓ2 0.8968 0.9977 fgsm ℓ2 0.6157 0.3914 pgd ℓ2 0.8927 0.9925 autopgd ℓ∞ 0.9000 1.0000 fgsm ℓ∞ 0.9149 0.5515 pgd ℓ∞ 0.9022 1.0000 fw was 0.9000 1.0000 attacks. pgd, which performs much better than fgsm in terms of targeted misclassification rate, is around 71 times slower. the slowest is the fw + dual lmo attack, which performs around 400 times slower than the fgsm attack. in table 2, we can see the time duration needed to generate an adversarial example. being a simple one-step attack, fgsm leads the competition followed by the pgd and autopgd attacks. pgd, which performs much better than fgsm in terms of targeted misclassification rate, is around 71 times slower. the slowest is the fw + dual lmo attack, which performs around 400 times slower than the fgsm attack. table 2: duration of generating an adversarial example in seconds. fgsm pgd autopgd fw+dual lmo 0.7 50 87 338 one of the most important aspects of adversarial attacks is that they should be undetected by the human eye. hence, in this section, we study how detectable are the adversarial samples generated by the attacks. to visualize the results, we chose the smallest ϵ for each of our attacks, under which our model showed at least 80% misclassification. you can see the visualizations in the figures 2 and 3. we can see that in the examples generated by the fgsm attack, although the original image is still well visible, the perturbation is easily detectable to us. for pgd, autopgd, and fw + dual lmo attacks, however, the perturbations are 4.3 duration 4.4 imperceptibility g. bezirganyan and h. sergoyan 49 hardly visible. in fact, from fig. 3 it is noticeable that pgd and autopgd attacks apply small perturbations uniformly over the image. while the fw + dual lmo attack perturbs only small portions of the image, the perturbations are much more visible. fig. 2. adversarial samples on an image with original label 4 (deer). fig. 3. perturbations added to the image with original label 4 (deer). 5. conclusion and future work we compared different attack methods with different metrics. the champion of the comparison is the pgd attack. although being a very simple attack, it performs very well in terms of misclassification and targeted misclassification rates, is fast, and is almost nondetectable by the human eye in our experiments. autopgd, while yielding similar results, is much slower, and hence, comes in second place in our comparison. fw + dual lmo attack performed very well in terms of duration, misclassification, and targeted misclassification 50 a brief comparison between white box, targeted adversarial attacks in deep neural networks rates, but the perturbations were much more noticeable. the fgsm attack was the fastest with a high misclassification rate but came last in terms of imperceptibility. since we’ve covered only a small portion of attacks, we plan to extend the attack list by adding more well-known or state-of-the-art methods and extend the experiment domain to black-box attacks as well. furthermore, we plan to test these attacks on a defended model and compare their performances. particularly, we are interested to see the difference between autopgd and pgd attacks on a defended model. references [1] i. j. goodfellow, j. shlens, and c. szegedy, “explaining and harnessing adversarial examples,” in 3rd international conference on learning representations, iclr 2015 conference track proceedings, 2015. [2] q. li, y. guo, and h. chen, “practical no-box adversarial attacks against dnns,” preproceedings advances in neural information processing systems, vol. 33, 2020. [3] a. kurakin, i. j. goodfellow, and s. bengio, “adversarial machine learning at scale,” in 5th international conference on learning representations, iclr 2017 conference track proceedings, 2017. [4] f. croce and m. hein, “reliable evaluation of adversarial robustness with an ensemble of diverse parameter-free attacks,” arxiv preprint arxiv:2003.01690, 2020. [5] k. wu, a. wang, and y. yu, “stronger and faster wasserstein adversarial attacks,” in international conference on machine learning, pp. 10377–10387, pmlr, 2020. [6] e. wong, f. r. schmidt, and j. zico kolter, “wasserstein adversarial examples via projected sinkhorn iterations,” in 36th international conference on machine learning, icml 2019, 2019. [7] m. cuturi, “sinkhorn distances: lightspeed computation of optimal transport,” advances in neural information processing systems, vol. 26, pp. 2292–2300, 2013. [8] m. frank, p. wolfe, et al., “an algorithm for quadratic programming,” naval research logistics quarterly, vol. 3, no. 1-2, pp. 95–110, 1956. [9] k. he, x. zhang, s. ren, and j. sun, “deep residual learning for image recognition,” in proceedings of the ieee computer society conference on computer vision and pattern recognition, 2016. [10] a. krizhevsky, g. hinton, et al., “learning multiple layers of features from tiny images,” 2009. [11] m.-i. nicolae, m. sinn, m. n. tran, b. buesser, a. rawat, m. wistuba, v. zantedeschi, n. baracaldo, b. chen, h. ludwig, i. molloy, and b. edwards, “adversarial robustness toolbox v1.2.0,” corr, vol. 1807.01069, 2018. g. bezirganyan and h. sergoyan 5 1 êåçï³ï ïáõ÷áí, ãçñ³ë³íáñí³í ùñó³ïó³ûçý ñ³ñó³ïáõùý»ñç ñ³ù³éáï ñ³ù»ù³ïáõãûáõýá ëáñá ý»ûñáý³ûçý ó³ýó»ñáõù ¶ñç·áñ ´»½çñ·³ýû³ý ¨ ð»ýñçï ê»ñ·áû³ý øûáõýë»ýç î»ëýçï³ï³ý ð³ù³éë³ñ³ý e-mail: grigor.bezirganyan@tum.de, henrik.sergoyan@tum.de ²ù÷á÷áõù êðàòêîå ñðàâíåíèå ìåæäó ”áåëûì ÿùèêîì”, öåëåâûìè ñîñòÿçàòåëüíûìè àòàêàìè ïðîòèâíèêà â ãëóáîêèõ íåéðîííûõ ñåòÿõ ãðèãîð áåçèðãàíÿí è ãåíðèê ñåðãîÿí òåõíè÷åñêèé óíèâåðñèòåò ìþíõåíà e-mail: grigor.bezirganyan@tum.de, henrik.sergoyan@tum.de àííîòàöèÿ ñåãîäíÿ íåéðîííûå ñåòè èñïîëüçóþòñÿ â ðàçëè÷íûõ îáëàñòÿõ, â áîëüøèíñòâå èç êîòîðûõ âàæíî èìåòü íàäåæíûé è ïðàâèëüíûé âûâîä. âîò ïîýòîìó ñîñòÿçàòåëüíûå àòàêè äåëàþò ãëóáîêèå íåéðîííûå ñåòè ìåíåå íàäåæíûìè äëÿ èñïîëüçîâàíèÿ â îáëàñòÿõ, ãäå áåçîïàñíîñòü èìååò ðåøàþùåå çíà÷åíèå. ñëåäîâàòåëüíî, âàæíî èçó÷èòü ïîòåíöèàëüíûå ìåòîäû àòàêè, ÷òîáû èìåòü âîçìîæíîñòü ðàçðàáàòûâàòü ãîðàçäî áîëåå íàäåæíûå ñåòè. â ýòîé ñòàòüå ìû ðàññìàòðèâàåì ÷åòûðå ”áåëûõ ÿùèêà” öåëåíàïðàâëåííûå ñîñòÿçàòåëüíûå àòàêè è ñðàâíèâàåì èõ ñ òî÷êè çðåíèÿ ÷àñòîòû îøèáî÷íûõ êëàññèôèêàöèé, ÷àñòîòû öåëåâûõ îøèáî÷íûõ êëàññèôèêàöèé, äëèòåëüíîñòè àòàêè è íåçàìåòíîñòè. íàøà öåëü íàéòè àòàêè, êîòîðûå áûëè áû ýôôåêòèâíû è ãåíåðèðîâàëè áû ñîñòÿçàòåëüíûå âûáîðêè ñ íåáîëüøèìè âîçìóùåíèÿìè è íå îáíàðóæèâàëèñü áû ÷åëîâå÷åñêèì ãëàçîì. êëþ÷åâûå ñëîâà: ñîñòÿçàòåëüíûå àòàêè, íàäåæíîñòü, ìàøèííîå îáó÷åíèå, ãëóáîêîå îáó÷åíèå. ²ûëûñ ý»ûñáý³ûçý ó³ýó»ñý û·ï³·áñííáõù »ý ï³ñµ»ñ ³ëå³ñ»½ý»ñáõù, áñáýóçó ß³ï»ñáõù ï³ñ¨áñ ¿ áõý»ý³é ñáõë³éç ¨ ×ß·ñçï ³ñ¹ûáõýù: ²ñ³ ã» çýãáõ ùñó³ïó³ûçý ñ³ñó³ïáõùý»ñá ý»ûñáý³ûçý ó³ýó»ñá ¹³ñóýáõù »ý ³í»éç ùçã ñáõë³éç‘ µ³ñóñ ³ýíï³ý·áõãû³ý ù³ï³ñ¹³ï å³ñ³ýçáõ ïçñáõûãý»ñáõù: ð»ï¨³µ³ñ, ï³ñ¨áñ ¿ áõëáõùý³ëçñ»é ñ³ñó³ïù³ý ñý³ñ³íáñ ù»ãá¹ý»ñá` ³í»éç ï³ûáõý ¨ ³ýíï³ý· ó³ýó»ñ ùß³ï»éáõ ñ³ù³ñ: ²ûë ñá¹í³íáõù ù»ýù ùýý³ñïáõù »ýù ãáñë ëåçï³ï?ïáõ÷áí, ãçñ³ë³íáñí³í ùñó³ïó³ûçý ñ³ñó³ïáõùý»ñ ¨ ñ³ù»ù³ïáõù ¹ñ³ýù çñ»ýó ëë³é ¹³ë³ï³ñ·ù³ý ³ëïç×³ýç, ýå³ï³ï³ûçý ëë³é ¹³ë³ï³ñ·ù³ý ³ñ³·áõãû³ý ³ëïç×³ýç, ñ³ñó³ïù³ý ï¨áõáõãû³ý ¨ ³ýýï³ï»éçáõãû³ý ³éáõùáí: ø»ñ ýå³ï³ïý ¿ ·ïý»é ñ³ñó³ïáõù(ý»ñ), áñáýù ³ñ¹ûáõý³í»ï ïéçý»ý ¨ ïëï»õí»ý ùñó³ïó³ûçý ûñçý³ïý»ñ‘ ÷áùñ ß»õáõùý»ñáí ¨ ³ýýï³ï»éç ù³ñ¹áõ ³ãùç ñ³ù³ñ: ´³ý³éç µ³é»ñ` ùñó³ïó³ûçý ñ³ñó³ïáõùý»ñ, ï³ûáõýáõãûáõý, ù»ù»ý³û³ï³ý áõëáõóáõù, ëáñá áõëáõóáõù: 04_sergoyan_33__52__copy_ 04 mathematical problems of computer science 53, 29–38, 2020. udc 519.6 estimating time of driver arrival with gradient boosting algorithms and deep neural networks henrik t. sergoyan american university of armenia e-mail: henriksergoyan@gmail.com abstract customer experience and resource management determine the degree to which transportation service providers can compete in today’s heavily saturated markets. the paper investigates and suggests a new methodology to optimize calculations for estimated time of arrival (from now on eta, meaning the time it will take for the driver to reach the designated location) based on the data provided by gg collected from rides made in 2018. gg is a transportation service providing company, and it currently uses the open source routing machine (osrm) which exhibits significant errors in the prediction phase. this paper shows that implementing algorithms such as xgboost, catboost, and neural networks for the said task will improve the accuracy of estimation. paper discusses the benefits and drawbacks of each model and then considers the performance of the stacking algorithm that combines several models into one. thus, using those techniques, final results showed that mean squared error (mse) was decreased by 54% compared to the current gg model. keywords: gg, estimated time of driver arrival, osrm, xgboost, catboost, neural networks. 1 introduction eta calculation is a relatively significant product differentiator for firms competing in the automated ride service hailing industry. eta computation is particularly difficult for cities with underdeveloped traffic and road management services such as yerevan, thus requiring novel methodologies and datasets to design models that can be deployed for use by relatively fickle customers. gg is one of the major transportation service providers in yerevan and this paper uses data provided by gg to predict eta. while gg currently estimates the time of arrival, it does so with a significant error. this paper discusses optimization techniques to calculate estimated arrival time with more accurate and flexible algorithms. 29 30 estimating time of driver arrival with gradient boosting algorithms and deep neural networks 1.1 problem setting and description gg has an application that enables drivers to receive orders to transport customers and goods. having chosen a pick-up location through the app, the customer gets an eta for the driver. the task is to improve eta estimation so that it is more accurate than the baseline osrm model, which gg currently employs. to do so, the paper uses several joint machine learning algorithms such as xgboost, catboost, and neural networks. the dataset provided by gg consists of more than 2.4m orders occured (received) in 2018. the number of unique drivers exceeds 6000. in addition to spatio-temporal dataset, the hourly data about weather conditions were used to better capture possible traffic jams. 1.2 structure of the paper the paper consists of five central sections. the first section provides the problem statement and description, the overall summary of the paper structure representing the current method gg uses for estimating the arrival time of the driver while describing its drawbacks. after giving full understanding about the reason for further investigations, the next 3 sections, give detailed information and clarification of alternative models and outline advantages and disadvantages of each of them showing the results run on the 1-year information containing data. based on the preceding sections the fifth part of the paper shows comparisons of all 3 suggested models and interprets the most optimal solution of the problem based on 3 methods. 1.3 description of osrm the open source routing machine is a routing engine that works to find the shortest paths in the road networks, and it supports linux, freebsd, windows and mac os x platforms. the primary function of the environment is to compute and output the shortest path between the origin and the destination points, on the basis of which the system can compute the estimated time within a couple of milliseconds for the transport to reach a particular destination. by implementing contraction hierarchies, pure route computation takes the minimal time of those calculations. the most significant part of the calculation is finding the route and transmitting the geometry over the network. the decision-making capacity of the system is based on routing algorithms with road network data from the osm (openstreetmap) project (see [1]). the osrm optimization algorithm prioritizes the speed with which calculations are made. data obtained from osm consist of 3 main components; namely nodes, ways and relations between them. nodes determine the geometric structure of the path. ways are 2d polylines consisting of line segments. several lines can share the same node if they intersect with each other. a relation relates nodes or ways to turn restrictions, routes, and other features [2]. in these graphs, nodes represent directions of an osm segment and graph edges connect graph nodes by describing the transition from one specific point to another. here is how it works: h. sergoyan 31 fig. 1. osrm visual implementation. even though this method seems to expedite the process of calculating eta, the major drawback is that it does not capture the impact of weather conditions, traffic conditions and other real-life situations that depend on human factors. thus, this paper concentrates on improving eta calculation using information concerning weather and concrete data generated by drivers on actual routes. 2 machine learning approach to the problem 2.1 related work managing traffic among the urban population is becoming an increasingly significant issue for major municipalities. thus, one of the fundamental problems to be solved is the efficient management of road traffic by minimizing congestion and assisting travelers with real-time information. some urban areas currently use datasets that include information concerning gps coordinates of origins and destinations, travel time, travel distance, pickup date, trip start-end times and the total fare to organize the road management. by analyzing data, route optimizing environments provide information for travelers related to the optimal routes, road conditions, and locations of incidents [3], [4]. ishan jindal, tony (zhiwei) qin, xuewen che, matthew nokleby, jieping ye studied this topic and used the unified neural network approach to estimate travel time and distance for a taxi trip. the data they used were structurally similar to those provided by gg, so this paper takes its results as an additional benchmark and aims to improve upon it [5]. their research considers the waiting time at intersections for travel time estimations. the method used is a particular case of the path-based approach, where they add predictions of waiting time at intersections of sub-paths including the neighbor-based method by averaging the travel time for all the samples in the training data that have the same origin, destination, and time of day. thus, the paper focuses on predicting travel time and distance from a source to a target as a function of the time of day based on nyc travel trip historical data. having a large amount of training data, the authors build a unified neural network learning model, which jointly learns the travel time and distance between the origin and destination [5]. to understand what the travel time is, one needs to think of the time taken by the driver to move from the initial location to the final location including velocity changes or stops 32 estimating time of driver arrival with gradient boosting algorithms and deep neural networks made by him/her depending on some conditions. travel distance is the path taken by the driver to reach from the primary point to the final location. thus, the travel time depends on the origin and destination at a particular time of the day [4]. 2.2 xgboost: extreme gradient boosting xgboost is a decision-tree-based ensemble machine learning algorithm that uses a gradient boosting framework. in prediction problems involving unstructured data (images, text, etc.), artificial neural networks tend to outperform all other algorithms or structures. however, when it comes to small-to-medium structured/tabular data, decision-tree-based algorithms are considered best-in-class (see [4]). xgboost algorithm was developed as a research project at the university of washington. tianqi chen and carlos guestrin presented their paper at sigkdd conference in 2016 and caught the machine learning world by fire. since its introduction, this algorithm has not only been credited with winning numerous kaggle competitions but also for being the driving force under the hood for several cutting-edge industry applications. xgboost is an ensemble learning method, and the main principle behind it is that a group of weak learners come together to form a keen learner. bagging and boosting are two widely used ensemble learners [4], [6]. 2.2.1 bagging while decision trees are one of the most easily interpretable models, they exhibit highly variable behavior. therefore, several decision trees are being generated in parallel and form the base learners of bagging technique. data sampled with replacement is fed to these learners for training. the final prediction is the averaged output from all the learners. the main principle behind the ensemble model is that a group of weak learners come together to form a strong learner [6]. 2.2.2 boosting in boosting, the trees are built sequentially so that each subsequent tree aims to reduce the errors of the previous tree. each tree learns from its predecessors and updates the residual errors. hence, the tree that grows next in the sequence will learn from an updated version of the residuals [6]. using xgboost, in this case, has several advantages. first of all, xgboost has a feature of out-of-core computing that helps to handle massive datasets that do not fit into memory. moreover, because of a block structure in its system design, xgboost can make use of multiple cores on the cpu and gpu. data are sorted and stored in in-memory units called blocks. that enables the data layout to be reused by subsequent iterations, instead of computing it again. finally, xgboost has an option to penalize complex models through both l1 and l2 regularizations. regularization helps in preventing overfitting [6]. 2.3 catboost: categorical boosting catboost (categorical boosting) is a state-of-the-art open-source gradient boosting on decision trees library developed by yandex. the algorithm is spawned from xgboost, but it has its advantages compared to all other gradient boosting algorithms. although xgboost h. sergoyan 33 became a game-changing algorithm in machine learning, it indeed, has its drawbacks. for example, xgboost has a problem of dealing with categorical features, and therefore, one-hot encoding is used, which adds a new binary feature for each category. however, in the case of high cardinality features, such as user id, partner id, which were present in the data, such a technique leads to an infeasibly large number of new features. catboost deals with this issue and is now considered the most innovative algorithm for processing categorical features. at the end of the paper, it is shown how well catboost deals with categorical features in the data by looking at the final accuracies of the models [7]. 2.4 nn: neural networks neural networks were used as the last method to predict the time of arrival of a driver. the main advantage of neural nets is that they can detect patterns that no other algorithms can discover. the logic behind it differs significantly from the aforementioned boosting algorithms [8]. • the network architecture includes an input layer, a hidden layer(s) and an output layer. it is also called mlp (multi-layer perceptron) because of the multiple layers. • the hidden layer can be seen as a distillation layer that concentrates some of the essential patterns from the inputs and passes them onto the next layer to view. it renders the network faster and more efficient by identifying only the critical information from the inputs and leaving out redundant information. • the activation function serves for two notable purposes [8]: ◦ captures non-linear relationships between the inputs; ◦ helps to convert the input into a more useful output. 2.5 feature engineering and final results to train the above-mentioned models, annual data generated by gg taxi orders were used. the data consists of more than 2m unique orders received in 2018. the features of the data were the driver id, the coordinates where he takes the order, the coordinates of the order denoted by the user, the time-stamp when the order was created, the actual duration of the trip, and the estimated duration of the trip by osrm. however, several extra features were added to the data, such as dummy variables of weekdays or weather information. it is important to mention that weather data contains hourly historic information about weather conditions, such as temperature, humidity, pressure, wind speed and facts about raining or snowing. the data contained outliers and missing values, and each of them happened because of failure of a mobile program. however, all this misleading information was identified and removed. 34 estimating time of driver arrival with gradient boosting algorithms and deep neural networks 2.5.1 training process and evaluation the evaluation of the proposed model and all the considered baselines are on the mean absolute error (mae). the task of each model was to minimize mae. mae = n∑ i=1 |yi −zi| n cross validation was used to prevent overfitting while training the xgboost and catboost models. moreover, as these algorithms require specifications of a lot of parameters, grid search was used to identify the best parameters for each model. below are bar graphs depicting the importance of features used to train xgboost and catboost algorithms. both models indicate that the most important feature is the distance. however, the remaining features are represented as having varying levels of importance for both respective models. for example, catboost illustrates that the second most important feature is partner id, while the second most important feature for xgboost was the hour of the day in which the trip occurred. it is worthwhile to mention that both models indicate that osrm estimates (”from start to take”) are important features. this makes considerable sense, since osrm estimates include information concerning the waiting time at traffic lights, the number of turns, the angle of turns, and the mean velocity between turns. fig. 2. importance of features for xgboost algorithm. h. sergoyan 35 fig. 3. importance of features for catboost algorithm. neural networks built with keras library require some intuitive approach to identify the number of neurons, hidden layers, and types of activation functions for each layer. based on experimentation, the optimal number of neurons and the most appropriate activation function to minimize mae were determined. as the last stroke of the brush, the predictions of those trained models and actual time were combined in one dataset and a linear regression model was trained on the said dataset. in machine learning, this process is usually called stacking, where it is expected that the ensemble of several models provides a better result than each of those models independently. however, this example clearly illustrates that this is not always the case. although our stacked model outperforms xgboost and neural networks, it still has a slightly higher mae than catboost regressor. this fact can be interpreted in the following way: our three models mostly pick up and model the same information quite similarly and therefore, we did not get much out of an ensemble. below the performance of each model for train and test sets is presented. table 1: mae results for each method. methods train test osrm 205.796 205.787 xgboost 98.77 98.96 catboost 92.34 94.97 nn 98.09 98.34 stack 93.03 95.01 3 conclusion in this paper, we proposed a new approach that formulates the eta calculation as a pure regression problem. as such, we train catboost, xgboost, and neural networks and successfully improve upon benchmarks set by the osrm architecture used by gg and the unified neural network approach. the catboost model outperformed xgboost and neural network. thus, while osrm exhibits errors of 205.796 seconds and 205.787 seconds for 36 estimating time of driver arrival with gradient boosting algorithms and deep neural networks test and train data respectively, catboost model exhibits errors of 92.34 and 94.97 seconds on train and test data respectively. we believe that by increasing the number and quality of features and using more powerful models we can register significant improvements in predictive performance. references [1] m. dodge and r. kitchin, “crowdsourced cartography: mapping experience and knowledge”, environment and planning a: economy and space, vol. 45, no. 1, pp. 19–36, 2013. [2] s. huber and ch. rust, “calculate travel time and distance with openstreetmap data using the opensource routing machine (osrm)”, the state journal, vol. 16, no. 2, pp. 416–423, 2016. [3] y. li and k. fu and zh. wang and c. shahabi and j. ye and y. liu, “multi-task representation learning for travel time estimation”, proceedings of the 24th acm sigkdd international conference on knowledge discovery and data mining, pp. 1695–1704, 2018. [4] zh. wang and k. fu and j. ye, “learning to estimate the travel time”, proceedings of the 24th acm sigkdd international conference on knowledge discovery and data mining, pp. 858–866, 2018. [5] i. jindal and t. qin and x. chen and m. nokleby and j. ye, “a unified neural network approach for estimating travel time and distance for a taxi trip”, arxiv:1710.04350v1, [stat.ml], 2017. [6] t. chen and c. guestrin, “xgboost: a scalable tree boosting system”, proceedings of the 22nd acm sigkdd international conference on knowledge discovery and data mining, pp. 785–794, 2016. [7] l. prokhorenkova and g. gusev and a. vorobev and a. v. dorogush and a. gulin, “catboost: unbiased boosting with categorical features”, proceedings of the 32nd international conference on neural information processing systems, pp. 6639–6649, 2018. [8] d. wang and j. zhang and w. cao and j. li and y. zheng, “when will you arrive? estimating travel time based on deep neural networks”, aaai, pp 2500–2507, 2018. submitted 10.12.2019, accepted 18.04.2020. h. sergoyan 3 7 ì³ñáñ¹ç å³ù³ýù³ý å³ù³ý³ïç ùáï³ñïáõùá áõå»õ³óí³í ·ñ³¹ç»ýïç ³é·áñçãùý»ñç ¨ ëáñá ý»ûñáý³ûçý ó³ýó»ñç ùççáóáí ð»ýñçï î. ê»ñ·áû³ý ºñ¨³ýç å»ï³ï³ý ñ³ù³éë³ñ³ý e-mail: henriksergoyan@gmail.com ²ù÷á÷áõù îöåíêà âðåìåíè ïðèáûòèÿ âîäèòåëÿ ñ ïîìîùüþ àëãîðèòìîâ ãðàäèåíòíîãî óñèëåíèÿ è ãëóáîêèõ íåéðîííûõ ñåòåé ãåíðèõ ò. ñåðãîÿí àìåðèêàíñêèé óíèâåðñèòåò àðìåíèè e-mail:henriksergoyan@gmail.com àííîòàöèÿ îáñëóæèâàíèå êëèåíòîâ è óïðàâëåíèå ðåñóðñàìè îïðåäåëÿþò ñòåïåíü, â êîòîðîé ïîñòàâùèêè òðàíñïîðòíûõ óñëóã ìîãóò êîíêóðèðîâàòü íà ñåãîäíÿøíèõ ñèëüíî íàñûùåííûõ ðûíêàõ. â ýòîé ðàáîòå èññëåäóåòñÿ è ïðåäëàãàåòñÿ íîâàÿ ìåòîäîëîãèÿ îïòèìèçàöèè äëÿ ðàñ÷åòîâ ïðåäïîëàãàåìîãî âðåìåíè ïðèáûòèÿ âîäèòåëÿ íà îñíîâå ïðîøëîãîäíèõ äàííûõ, ïðåäîñòàâëåííûõ gg, ñîáðàííûõ èç ïîåçäîê âîäèòåëåé êîìïàíèè. gg ÿâëÿåòñÿ êîìïàíèåé, ïðåäîñòàâëÿþùåé òðàíñïîðòíûå óñëóãè, è â íàñòîÿùåå âðåìÿ îíà èñïîëüçóåò the open source routing machine (osrm), êîòîðûé íà ýòàïå ïðîãíîçèðîâàíèÿ äåìîíñòðèðóåò ð³×³ëáñ¹ý»ñç ëå³ë³ñïáõùá ¨ é»ëáõñëý»ñç ï³é³í³ñáõùá áñáßáõù »ý ïñ³ýëåáñï³ûçý í³é³ûáõãûáõýý»ñ ù³ïáõóáõ áýï»ñáõãûáõýý»ñç ùñó³ïó»éáõ ï³ñáõáõãû³ý ³ëïç×³ýá ³ûëûñí³ ù»í ¨ ñ³·»ó³í ßáõï³ý»ñáõù: ²ûë ³ßë³ï³ýùý áõëáõùý³ëçñáõù ¨ ³é³ç³ñïáõù ¿ ýáñ ù»ãá¹³µ³ýáõãûáõý í³ñáñ¹ç å³ù³ýù³ý ï¨áõáõãûáõýá ñ³ßí³ñï»éáõ ñ³ù³ñ` ñçùýí»éáí gg-ç 2018 ãí³ï³ýç áõõ¨áñáõãûáõýý»ñçó ñ³í³ùí³í ïíû³éý»ñç íñ³: gg-ý ïñ³ýëåáñï³ûçý í³é³ûáõãûáõýý»ñ ù³ïáõóáõ áýï»ñáõãûáõý ¿ ¨ ý»ñï³ûáõùë û·ï³·áñíáõù ¿ the open source routing machine (osrm) ñ³ù³ï³ñ·á, áñá ï³ýë³ï»ëù³ý å³ù³ý³ï ³ßë³ïáõù ¿ ¿³ï³ý ëë³éý»ñáí: ²ûë ³ßë³ï³ýùáõù óáõûó ¿ ïñí³í, áñ ýßí³í ³é³ç³¹ñ³ýùç ñ³ù³ñ ³ûýåçëç ³é·áñçãùý»ñç çñ³ï³ý³óáõùá, çýãåçëçù »ý xgboost, catboost ¨ ëáñá ý»ûñáý³ûçý ó³ýó»ñá, ïµ³ñ»é³íç å³ù³ý³ïç ×ß·ñïáõãûáõýá: ²ßë³ï³ýùáõù ùýý³ñïíáõù »ý ûáõñ³ù³ýãûáõñ ùá¹»éç ³é³í»éáõãûáõýý»ñá ¨ ã»ñáõãûáõýý»ñá ¨ ³ûýáõñ»ï»õ ¹çï³ñïíáõù ¿ ïáõï³ï³ûçý ³é·áñçãùç ï³ñµ»ñ³ïá, áñá ùç³íáñáõù ¿ ùç ù³ýç ùá¹»éý»ñá ù»ïç ù»ç: ²ûëåçëáí, ³û¹ ï»ëýçï³ûç ïçñ³éù³ùµ ëï³óí³í í»ñçý³ï³ý ³ñ¹ûáõýùý»ñá óáõûó »ý ïí»é, áñ ùçççý ù³é³ïáõë³ûçý ß»õáõùá gg-áõù ïçñ³éíáõ ý³ëáñ¹ ùá¹»éç ýï³ïù³ùµ ýí³½»é ¿ 54%-áí: ´³ý³éç µ³é»ñ` gg, í³ñáñ¹ç å³ù³ýù³ý ï¨áõáõãûáõý, osrm, xgboost, catboost, ý»ûñáý³ûçý ó³ýó»ñ: 3 8 estimating time of driver arrival with gradient boosting algorithms and deep neural networks äîâîëüíî ñóùåñòâåííûå îøèáêè. ýòà ðàáîòà ïîêàçûâàåò, ÷òî ðåàëèçàöèÿ àëãîðèòìîâ, òàêèõ êàê xgboost,catboost è neural networks äëÿ óêàçàííîé çàäà÷è, ïîâûñèò òî÷íîñòü ðàñ÷åòà. â ñòàòüå îáñóæäàþòñÿ ïðåèìóùåñòâà è íåäîñòàòêè êàæäîé ìîäåëè, à çàòåì ðàññìàòðèâàåòñÿ ïðîèçâîäèòåëüíîñòü àëãîðèòìà ñóììèðîâàíèÿ, êîòîðûé îáúåäèíÿåò íåñêîëüêî ìîäåëåé â îäíó. òàêèì îáðàçîì, îêîí÷àòåëüíûå ðåçóëüòàòû, ïîëó÷åííûå ñ èñïîëüçîâàíèåì ýòèõ ìåòîäîâ, ïîêàçàëè, ÷òî ñðåäíÿÿ êâàäðàòè÷åñêàÿ îøèáêà óìåíüøèëàñü íà 54% ïî ñðàâíåíèþ ñ òåêóùåé ìîäåëüþ gg. êëþ÷åâûå ñëîâà: gg, ðàñ÷åòíîå âðåìÿ ïðèáûòèÿ âîäèòåëÿ, osrm, xgboost, catboost, neural networks 04_sergoyan_53 (1) 04 d:\user\...\main.dvi mathematical problems of computer science 49, 110{114, 2018. on m ultiple h ypotheses lao t esting with rejection of decision for t wo dependent objects e vg u e n i h a r o u t u n ia n , a r a m y e s a ya n a n d n a r in e h a r u t yu n ya n institute for informatics and automation problems of nas ra e-mail: eghishe@sci.am, armfrance@yahoo.fr, narineharutyunyan57@gmail.com abstract multiple statistical hypotheses testing with possibility of rejecting of decision is considered for model consisting of two dependent objects characterized by joint discrete probability distribution. the matrix of error probabilities exponents (reliabilities) of asymptotically optimal tests is studied. keywords: multiple hypotheses testing, optimal tests, rejection option, two object. 1 . in t r o d u c t io n th is p a p e r is d e vo t e d t o t h e s t u d y o f c h a r a c t e r is t ic s o f lo g a r it h m ic a lly a s ym p t o t ic a lly o p t im a l ( l a o) h yp o t h e s e s t e s t in g wit h p o s s ib ilit y o f r e je c t io n o f d e c is io n fo r t h e m o d e l o f t wo d e p e n d e n t o b je c t s wit h jo in t p t o b a b ilit y d is t r ib u t io n s ( p d s ) . th e c o r r e s p o n d e n c e p r e s e n t s a c o m p le m e n t t o p r o b le m s s t u d ie d in [1 , 2 ], wh e r e t h e c o m p o n e n t s o f r a n d o m ve c t o r c h a r a c t e r iz in g t wo o b je c t s we r e in d e p e n d e n t , s o it wa s p o s s ib le t o c o n s id e r t h e t e s t p r o c e d u r e wit h s e p a r a t e t e s t s fo r t wo o b je c t s . in [2 ] t wo d i®e r e n t m o d e ls a r e s t u d ie d : ¯ r s t , wh e n r e je c t io n wa s a llo we d o n ly fo r o n e o f t h e o b je c t s a n d s e c o n d , wh e n r e je c t io n wa s a llo we d fo r b o t h o b je c t s . th e p r o b le m wit h a n a lo g o u s s t a t e m e n t fo r o n e a r b it r a r ily va r yin g o b je c t wit h s id e in fo r m a t io n wa s e xa m in e d in [3 ]. it is wo r t h t o r e c a ll t h e p r e vio u s r e s u lt s o n l a o t e s t in g o f m a n y h yp o t h e s e s p u b lis h e d in [4 -6 ]. me t h o d s a n d b a s ic r e s u lt s o f l a o t e s t in g a r e a ls o p r e s e n t e d in b o o ks [7 -9 ] 2 . p r o b le m fo r m u la t io n a n d r e s u lt l e t p ( x ) b e a s p a c e o f a ll p r o b a b ilit y d is t r ib u t io n s g( x ) o n ¯ n it e s e t x . l e t ( x1; x2 ) b e r a n d o m ve c t o r t a kin g va lu e s in t h e s e t x £ x wit h o n e o f m 2; m ¸ 2 jo in t p d s gm1;m2 2 p ( x £ x ) , m1; m2 = 1 ; m. l e t ( x1; x2 ) 4 = ( ( x11; x 2 1 ) ; :::; ( x 1 n; x 2 n ) ; :::; ( x 1 n ; x 2 n ) ) , x 1 n; x 2 n 2 x ; n = 1 ; n, b e a ve c t o r o f r e s u lt s o f n in d e p e n d e n t o b s e r va t io n s o f t h e ve c t o r ( x1; x2 ) , it is c a lle d a s a m p le . th e s t a t is t ic ia n h a s t o d e t e r m in e u n kn o wn p d s fr o m t h e s e t o f h yp o t h e s e s : hm1;m2 : g = gm1;m2, m1; m2 = 1 ; m o r wit h d r a w t o d o a n y ju d g e m e n t u s in g o b t a in e d s a m p le . 1 1 0 e. haroutunian, a. yesayan and n. harutyunyan 1 1 1 w e c a ll t h is p r o c e d u r e a c o m p o u n d t e s t a n d d e n o t e it b y ©n . th e t e s t ©n c a n b e d e ¯ n e d b y t h e d ivis io n o f t h e s p a c e x n £ x n in t o m 2 + 1 d is jo in t s u b s e t s , wh e r e am1;m2, m1; m2 = 1 ; m , c o n t a in s a ll ve c t o r s ( x1; x2 ) fo r wh ic h t h e h yp o t h e s is hm1;m2 is a d o p t e d , a n d am +1 c o n t a in s a ll ve c t o r s fo r wh ic h we r e fu s e t o t a ke a c e r t a in a n s we r . l e t ®l1;l2jm1;m2 ( ©n ) b e t h e p r o b a b ilit y o f t h e e r r o n e o u s a c c e p t a n c e o f t h e h yp o t h e s is hl1;l2 b y t h e t e s t ©n p r o vid e d t h a t t h e h yp o t h e s is hm1;m2 is t r u e , wh e r e ( m1; m2 ) 6= ( l1; l2 ) , m1; m2; l1; l2 = 1 ; m, ®l1;l2jm1;m2 ( ©n ) = g n m1;m2 ( al1;l2 ) : w h e n t h e h yp o t h e s is hm1;m2 is t r u e , b u t we d e c lin e t h e d e c is io n c o n c e r n in g t o t h e h yp o t h e s e s , t h e c o r r e s p o n d in g p r o b a b ilit y o f e r r o r is : ®m +1;m+1jm1;m2 ( ©n ) = g n m1;m2 ( am +1 ) : th e p r o b a b ilit y n o t t o a c c e p t a t r u e h yp o t h e s e s hm1;m2, m1; m2 = 1 ; m is t h e fo llo win g : ®m1;m2jm1;m2 ( ©n ) = x (l1;l2) 6=(m1;m2); l1;l2=1;m ; (l1;l2)=(m+1;m+1) ®l1;l2jm1;m2 ( ©n ) : ( 1 ) w e s t u d y t h e c o r r e s p o n d in g r e lia b ilit ie s el1;l2jm1;m2 ( ©) o f t h e s e qu e n c e o f t e s t s ©, el1;l2jm1;m2 ( ©) 4 = lim n!1 ¡ 1 n lo g ®l1;l2jm1;m2 ( ©n ) ; m1; m2; l1; l2 = 1 ; m; ( l1; l2 ) = ( m + 1 ; m + 1 ) : ( 2 ) d e ¯ n it io n s ( 1 ) a n d ( 2 ) im p ly t h a t em1;m2jm1;m2 ( ©) = m in (l1;l2) 6=(m1;m2) el1;l2jm1;m2 ( ©) ; m1; m2; l1; l2 = 1 ; m; ( l1; l2 ) = ( m + 1 ; m + 1 ) : ( 3 ) w e c a ll t h e t e s t s e qu e n c e ©¤ l a o fo r t h e m o d e l wit h t wo o b je c t s if fo r t h e g ive n p o s it ive va lu e s o f c e r t a in p a r t o f e le m e n t s o f t h e r e lia b ilit y m a t r ix e ( ©¤ ) t h e p r o c e d u r e ©¤ p r o vid e s m a xim a l va lu e s fo r a ll o t h e r e le m e n t s o f it . fo r m = 2 t h e m a t r ix will b e a s fo llo ws : e ( ©) = 0 bbb@ e1;1j1;1 e1;2j1;1e2;1j1;1 e2;2j1;1 e3;3j1;1 e1;1j1;2 e1;2j1;2e2;1j1;2 e2;2j1;2 e3;3j1;2 e1;1j2;1 e1;2j2;1e2;1j2;1 e2;2j2;1 e3;3j2;1 e1;1j2;2 e1;2j2;2e2;1j2;2 e2;2j2;2 e3;3j2;2 1 ccca : w it h t h e g ive n e le m e n t s e1;1j1;1; e1;2j1;2; e2;1j2;1; e2;2j2;2 we d e ¯ n e t h e r e g io n s o f a c c e p t a n c e o f t h e t e s t . in t h e g e n e r a l c a s e o f m h yp o t h e s e s fo r g ive n r e lia b ilit ie s e1;1j1;1; e1;2j1;2; e2;1j2;1; :::; em;mjm;m we d e ¯ n e t h e fo llo win g r e g io n s : rm1;m2 4 = fq : d ( qjjgm1;m2 ) · em1;m2jm1;m2g; m1; m2 = 1 ; m; ( 4 ) 1 1 2 on multiple hypotheses lao testing with rejection of decision for two dependent objects rm+1;m +1 4 = fq : d ( qjjgm1;m2 ) > em1;m2jm1;m2 ; m1; m2 = 1 ; mg; ( 5 ) e¤m1;m2jm1;m2 = e ¤ m1;m2jm1;m2 ( em1;m2jm1;m2 ) 4 = 4 = em1;m2jm1;m2; m1; m2 = 1 ; m; ( 6 ) e¤l1;l2jm1;m2 = e ¤ l1;l2jm1;m2 ( el1;l2jl1;l2 ) 4 = in f q2r l1;l2 d ( qjjgm1;m2 ) ; l1; l2; m1; m2 = 1 ; m; ( m1; m2 ) 6= ( l1; l2 ) ( 7 ) e¤m +1;m+1jm1;m2 = e ¤ m+1;m+1jm1;m2 ( e1;1j1;1; e1;2j1;2; :::; em;mjm;m ) 4 = in f q2r m +1;m+1 d ( qjjgm1;m2 ) ; m1; m2 = 1 ; m: ( 8 ) l e t u s d e n o t e b y ( m1; m2 ) ¡ t h e s e t o f a ll p a ir in d ic e s in r o w o f ( m1; m2 ) va r yin g fr o m ( 1 ; 1 ) t ill p r e vio u s o f ( m1; m2 ) a n d b y ( m1; m2 ) + t h e s e t o f a ll p a ir in d ic e s in r o w o f ( m1; m2 ) va r yin g fr o m n e xt o f ( m1; m2 ) t ill ( m; m ) . t heor em: if all distributions gm1;m2 = fgm1;m2 ( x1; x2 ) ; x1; x2 2 x g, m1; m2 = 1 ; m , are di®erent in the sense that d ( gl1;l2jjgm1;m2 ) > 0 , and the positive numbers e1;1j1;1; e2;2j2;2; :::; em;mjm;m are such that the following inequalities hold e1;1j1;1 < m in l1;l2=1;m ; (l1;l2) 6=(1;1) d ( gl1;l2jjg1;1 ) ; ( 9 ) em1;m2jm1;m2 < m in [ m in (l1;l2)2(m1;m2)¡ e¤l1;l2jm1;m2 ( el1;l2jl1;l2 ) ; m in (l1;l2)2(m1;m2)+ d ( gl1;l2jjgm1;m2 ) ]; m1; m2 = 1 ; m; ( m1; m2 ) 6= ( 1 ; 1 ) ; ( m1; m2 ) 6= ( m; m ) ; ( 1 0 ) em;mjm;m < m in l1;l2=1;m ; (l1;l2) 6=(m;m ) e¤l1;l2jm;m ( el1;l2jl1;l2 ) ; ( 1 1 ) then there exists a lao sequence of tests, all elements of the reliability matrix of which e ¤ = fe¤l1;l2jm1;m2g are positive and are de¯ned in ( 6 ) ¡ ( 8 ) . w hen one of the inequalities ( 9 ) ¡ ( 1 1 ) is violated, then at least one element of the matrix ( 6 ) ¡ ( 8 ) is equal to 0 . th e p r o o f o f t h e t h e o r e m c o n s is t s in p r e s e n t a t io n o f t h e p r o b le m fo r t wo o b je c t s a s a p r o b le m fo r o n e c a p a c io u s o b je c t . if we r e n u m e r a t e a s fo llo ws ( 1 ; 1 ) = 1 ; ( 1 ; 2 ) = 2 ; :::; ( 1 ; m ) = m; ( 2 ; 1 ) = m + 1 ; :::; ( 2 ; m ) = 2 m; :::; ( m; m ) = m 2 a n d d e n o t e ( x1; x2 ) = y , x £ x = y we will h a ve p r o b le m o f m 2 h yp o t h e s e s t e s t in g fo r o n e o b je c t wit h p o s s ib ilit y o f d e c is io n r e je c t io n . s o u s in g t h is n u m e r a t io n we will h a ve t h e c o r r e s p o n d in g e r r o r p r o b a b ilit ie s a n d r e lia b ilit ie s fo r l = 1 ; m 2 + 1 ; ; m = 1 ; m 2, wh e n we a p p ly th e o r e m 2 o f [3 ]. ge n e r a liz a t io n o f t h e r e s u lt is p o s s ib le in m a n y d ir e c t io n s . e. haroutunian, a. yesayan and n. harutyunyan 1 1 3 refer ences [1 ] e . a . h a r o u t u n ia n , p . m. h a ko b ya n a n d a . o. y e s s a ya n , \ on m u lt ip le h yp o t h e s e s l a o t e s t in g wit h r e je c t io n o f d e c is io n fo r m a n y in d e p e n d e n t o b je c t s " , p roceedings of international conference csit 2011, p p . 1 1 7 { 1 2 0 , y e r e va n 2 0 1 1 . [2 ] e . h a r o u t u n ia n , p . h a ko b ya n , a . y e s s a ya n a n d n . h a r u t yu n ya n \ on mu lt ip le h yp o t h e s e s l a o te s t in g w it h l ib e r t y o f r e je c t io n o f d e c is io n fo r two in d e p e n d e n t ob je c t s " ,... [3 ] e . h a r o u t u n ia n , p . h a ko b ya n a n d a . y e s s a ya n , \ ma n y h yp o t h e s e s l a o t e s t in g wit h r e je c t io n o f d e c is io n fo r a r b it r a r ily va r yin g o b je c t " , transactions of iiap of nas of r a and of ysu, m athematical p roblems of computer science, vo l. 3 5 , p p . 7 7 -8 5 , 2 0 1 1 . [4 ] e . a . h a r o u t u n ia n , \ ma n y s t a t is t ic a l h yp o t h e s e s : in t e r d e p e n d e n c e o f o p t im a l t e s t 's e r r o r p r o b a b ilit ie s e xp o n e n t s " , ( in r u s s ia n ) , a b s t r a c t o f t h e r e p o r t o n t h e 3 r d a llu n io n s c h o o l-s e m in a r , \p rogram-algorithmical software for applied multi-variate statistical analysis", ts a kh ka d z o r , p a r t 2 , p p . 1 7 7 { 1 7 8 , 1 9 8 8 . [5 ] e . a . h a r o u t u n ia n , \ l o g a r it h m ic a lly a s ym p t o t ic a lly o p t im a l t e s t in g o f m u lt ip le s t a t is t ic a l h yp o t h e s e s " , p roblems of control and information theory, vo l. 1 9 ( 5 -6 ) , p p . 4 1 3 { 4 2 1 , 1 9 9 0 . [6 ] e . h a r o u t u n ia n a n d p . h a ko b ya n , \ mu lt ip le h yp o t h e s e s l a o t e s t in g fo r m a n y in d e p e n d e n t o b je c t " , in t e r n a t io n a l jo u r n a l \ s c h o la r ly r e s e a r c h e xc h a n g e " , vo l. 2 0 0 9 , p p . 1 -6 , 2 0 0 9 . [7 ] i. cs is z ¶a r a n d j. k äo r n e r , information theory: coding theorems for discrete memoryless systems, s e c o n d e d it io n , ca m b r id g e u n ive r s it y p r e s s , 2 0 1 1 . [8 ] r . e . b la h u t , p rinciples and p ractice of information theory, a d d is o n -w e s le y, r e a d in g , ma , 1 9 8 7 . [9 ] t. m. co ve r a n d j. a . th o m a s , e lements of information theory. s e c o n d e d it io n , n e w y o r k, w ile y, 2 0 0 6 . submitted 10.10.2017, accepted 15.02.2018. ´³½ù³ïç í³ñï³íý»ñç áñáßáõùçó ññ³å³ñù³ùµ è²ú ëïáõ·ù³ý ù³ëçý »ñïáõ ï³ëû³é ûµû»ïïý»ñç ¹»åùáõù º. ð³ñáõãûáõýû³ý, ². ºë³û³ý ¨ ü. ð³ñáõãûáõýû³ý ²ù÷á÷áõù ð³ù³ï»õ áý¹ñ³ï ñ³í³ý³ï³ý³ûçý µ³ßëù³ùµ µýáõã³·ñíáõ »ñïáõ ï³ëû³é ûµû»ïïý»ñç ýï³ïù³ùµ ¹çï³ñïíáõù ¿ µ³½ù³ïç í³ñï³íý»ñç áñáßáõùçó ññ³å³ñù³ý ñý³ñ³íáñáõãû³ùµ ëïáõ·áõùá: àõëáõùý³ëçñí»é ¿ ëë³éç ñ³í³ý³ï³ýáõãûáõýý»ñç ³ëçùåïáïáñ»ý ûåïçù³é óáõóçãý»ñç (ñáõë³éçáõãûáõýý»ñç) ù³ïñçóá: 1 1 4 on multiple hypotheses lao testing with rejection of decision for two dependent objects î lao òåñòèðîâàíèè ìíîãèx ãèïîòåç ñ îòêàçîì îò ðåøåíèÿ äëÿ äâóõ çàâèñèìûõ îáúåêòîâ å. àðóòþíÿí, à. åñàÿí è í. àðóòþíÿí àííîòàöèÿ äëÿ ìîäåëè ñîñòîÿùeé èç äâóõ çàâèñèìûõ îáúåêòîâ, xàðàêòåðèçóåìûx ñîâìåñòíûì äèñêðåòíûì ðàñïðåäåëåíèåì âåðîÿòíîñòåé ðàññìàòðèâàåòñÿ òåñòèðîâàíèå ìíîãèx ñòàòèñòè÷åñêèõ ãèïîòåç ñ âîçìîæíîñòüþ îòêàçà îò ðåøåíèÿ. èçó÷åíà ìàòðèöà àñèìïòîòè÷åñêè îïòèìàëüíûõ ýêñïîíåíò âåðîÿòíîñòåé îøèáîê (íàäåæíîñòåé). начиная с начала 2000 года осуществляется внедрение ghis в здравоохранении, в рамках принятого проекта о реформирование информ mathematical problems of computer science 49, 66--73, 2018. construction of linear codes over rings 𝑍𝑍𝑚𝑚 correcting double ±1 or ±2 errors gurgen h. khachatrian* and hamlet k. khachatrian ** * american university of armenia **institute for informatics and automation problems of nas ra e-mail: gurgenkh@aua.am, hamletkh@ipia.sci.am abstract in this paper a construction of double ±1 and ±2 errors correcting linear optimal and quasi-optimal codes over rings 𝑍𝑍5, 𝑍𝑍7 and 𝑍𝑍9 is presented with the limitation that both errors have the same amplitude in absolute value. keywords: error correcting codes, codes over the rings 𝑍𝑍𝐴𝐴 , errors with magnitude ±1 and ±2. 1. introduction from a practical point of view the codes over rings 𝑍𝑍2𝑚𝑚 or 𝑍𝑍2𝑚𝑚+1 are interesting, because they can be used in 22𝑚𝑚 – qam (quadrature amplitude modulation) schemes. codes over finite rings, particularly over integer residue rings and their applications in coding theory, have been studied for a long time. errors occuring in the channel usually have a limited magnitude, and this effect is particularly applicable to flash memories. there have been a couple of papers regarding the optimal ±1 single error correcting codes over alphabet 𝑍𝑍𝑚𝑚 [1,2]. in this paper we will consider on errors with the magnitude ±1 or ±2. there have been a couple of papers regarding the ±1 and ±2 single error correcting codes [1, 3]. in this paper we will construct codes correcting double ±1 or ±2 errors with the limitation that both errors have the same amplitude in absolute value over the rings 𝑍𝑍𝑚𝑚. they are based on optimal codes with 4 parity check symbols correcting double errors over rings 𝑍𝑍𝑚𝑚 of value ±1 , adding only one parity check symbol. in this case, we cannot correct double errors with different magnitudes, for example, one error with a magnitude of +1 and the other one with a magnitude of +2. the optimality criteria for linear codes over the fixed ring 𝑍𝑍𝑚𝑚 can be considered in two ways (see [4]). first of all, recall that the code of the length n is optimal-1 if it has a minimum possible number of parity check symbols. secondly, the optimality-2 criteria for the code is that for a given number of parity check symbols, it has a maximum possible length. our constructed 66 mailto:gurgenkh@aua.am g. khachatrian and h. khachatrian 67 code will satisfy the first optimality criteria. in this case, we do not know codes, which satisfy the optimality criteria – 2. from [4] we know the optimal code (12, 8) correcting double ±1 errors over the ring 𝑍𝑍5. the linear code (12, 8) correcting double errors over the ring 𝑍𝑍5 of value ±1, presented in [4] satisfies the optimality criteria -1: 𝐻𝐻 = � 1 1 1 1 1 0 1 2 3 4 1 1 0 1 2 3 4 2 2 2 2 2 1 1 3 2 4 4 2 3 2 4 4 2 1 1 1 1 1 1 1 3 2 4 4 2 0 4 �. this code was given by the parity check matrix 𝐻𝐻, which has 8 information and 4 parity check symbols. in this case the number of combinations for each code word that can be corrected is (1 + 12 ∗ 2 + (12 𝑐𝑐ℎ𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜 2) ∗ 4) = 289. thus, we have that 289 ∗ 58 ≤ 512 and the cardinality of the best possible code is 512/289 < 59 . in this paper a construction of the optimal code (13, 8) over the ring 𝑍𝑍5 correcting either double ±1 or ±2 errors is presented. 2. construction of optimal 𝑪𝑪 (𝟏𝟏𝟏𝟏, 𝟖𝟖) linear code over ring 𝑍𝑍5 our purpose is to construct an optimal linear code over the ring 𝑍𝑍5 correcting double errors of the type ±1 or ±2 . our constructed code will correct only double errors with the same magnitude (if both errors have magnitude ±1 , or ±2). it is well known, that a linear code given by the parity check matrix 𝐻𝐻, can correct up to two errors of the type ±1, only when 𝐻𝐻 has a property according to which all the syndromes resulting from adding and subtracting operations between any two columns of the matrix 𝐻𝐻 are different: ± ℎ𝑖𝑖𝑖𝑖 ± ℎ𝑖𝑖𝑚𝑚 ≠ ± ℎ𝑖𝑖𝑖𝑖 ± ℎ𝑖𝑖𝑖𝑖 (𝑗𝑗, 𝑚𝑚) ≠ (𝑙𝑙, 𝑘𝑘). in this case, for correcting up to two errors of the type ±1 or ±2, the syndromes of our parity check matrix 𝐻𝐻 must satisfy the following condition too: ±2 ∗ ℎ𝑖𝑖𝑖𝑖 ± 2 ∗ ℎ𝑖𝑖𝑚𝑚 ≠ ±ℎ𝑖𝑖𝑖𝑖 ± ℎ𝑖𝑖𝑖𝑖 (𝑗𝑗, 𝑚𝑚) ≠ (𝑙𝑙, 𝑘𝑘). to construct this kind of matrix 𝐻𝐻, at first we will use the first 10 columns of the optimal linear code (12, 8) correcting double ±1 errors: � 1 1 1 1 1 0 1 2 3 4 0 1 2 3 4 2 2 2 2 2 3 2 4 4 2 3 2 4 4 2 1 1 1 1 1 3 2 4 4 2 �. construction of linear codes over rings 𝑍𝑍𝑚𝑚 correcting double ±1 or ±2 errors 68 for this matrix we know, that all syndromes ± ℎ𝑖𝑖𝑖𝑖 ± ℎ𝑖𝑖𝑚𝑚 ≠ ±ℎ𝑖𝑖𝑖𝑖 ± ℎ𝑖𝑖𝑖𝑖 and ±2 ∗ ℎ𝑖𝑖𝑖𝑖 ± 2 ∗ ℎ𝑖𝑖𝑚𝑚 ≠ ±2(ℎ𝑖𝑖𝑖𝑖 ± ℎ𝑖𝑖𝑖𝑖) are different. however, in this case there can be matches between the syndromes: ±2 ∗ ℎ𝑖𝑖𝑖𝑖 ± 2 ∗ ℎ𝑖𝑖𝑚𝑚 𝑎𝑎𝑎𝑎𝑎𝑎 ± ℎ𝑖𝑖𝑖𝑖 ± ℎ𝑖𝑖𝑖𝑖 . to solve this problem, we need to add one parity check symbol to the code. consequently, we need to add one additional row to the matrix, which will be the following row: [1 2 3 0 4 1 2 3 0 4]. after adding this row to the matrix, all the corresponding ±2 ∗ ℎ𝑖𝑖𝑖𝑖 ± 2 ∗ ℎ𝑖𝑖𝑚𝑚 ≠ ±ℎ𝑖𝑖𝑖𝑖 ± ℎ𝑖𝑖𝑖𝑖 syndromes will be different. now the code given by the parity check matrix below, can correct up to two errors of the type ±1 or ±2 : ⎣ ⎢ ⎢ ⎢ ⎡ 1 1 1 1 1 0 1 2 3 4 0 1 2 3 4 2 2 2 2 2 3 2 4 4 2 3 2 4 4 2 1 1 1 1 1 3 2 4 4 2 1 2 3 0 4 1 2 3 0 4⎦ ⎥ ⎥ ⎥ ⎤ . our constructed code has a length of 10, from which 5 are parity check symbols, and the other 5 information symbols. this constructed code (10, 5) over the ring 𝑍𝑍5 can correct up to two errors of the type ±1 and ±2, but it is not optimal in the sense that it has a minimal possible number of parity check symbols. to make this code optimal, we at least need to have a length of 13, therefore, we need to add 3 additional columns to the parity check matrix 𝐻𝐻. we have found the corresponding 3 columns by computer search: ⎣ ⎢ ⎢ ⎢ ⎡ 1 4 4 1 0 4 1 2 3 4 3 0 2 3 2⎦ ⎥ ⎥ ⎥ ⎤ . so, we obtain the parity check matrix 𝐻𝐻5 with a length of 13. the code given by this parity check matrix 𝐻𝐻5 can correct up to two errors of the type ±1 or ±2: 𝐻𝐻5 = ⎣ ⎢ ⎢ ⎢ ⎡ 1 1 1 1 1 0 1 2 3 4 1 4 4 0 1 2 3 4 2 2 2 2 2 1 0 4 3 2 4 4 2 3 2 4 4 2 1 2 3 1 1 1 1 1 3 2 4 4 2 4 3 0 1 2 3 0 4 1 2 3 0 4 2 3 2⎦ ⎥ ⎥ ⎥ ⎤ . now we can prove the following lemma: lemma 2.1. a linear code (13, 8) over the ring 𝑍𝑍5, given by the parity check matrix 𝐻𝐻5 with 5 parity check symbols, correcting up to two errors of the type ±1 or ±2, is optimal in the sense that it has a minimal possible number of parity check symbols. g. khachatrian and h. khachatrian 69 proof. it can be checked that a linear code over the ring 𝑍𝑍5, given by the parity check matrix h5, can correct up to two errors of the type ±1 or ±2. this means that the syndromes of parity check matrix h5 satisfy the condition: ±2 ∗ ℎ𝑖𝑖𝑖𝑖 ± 2 ∗ ℎ𝑖𝑖𝑚𝑚 ≠ ±ℎ𝑖𝑖𝑖𝑖 ± ℎ𝑖𝑖𝑖𝑖 (𝑗𝑗, 𝑚𝑚) ≠ (𝑙𝑙, 𝑘𝑘). in this case, we have 8 possible types of double errors:  +1 𝑎𝑎𝑎𝑎𝑎𝑎 + 1  +1 𝑎𝑎𝑎𝑎𝑎𝑎 − 1  −1 𝑎𝑎𝑎𝑎𝑎𝑎 + 1  −1 𝑎𝑎𝑎𝑎𝑎𝑎 − 1  +2 𝑎𝑎𝑎𝑎𝑎𝑎 + 2  +2 𝑎𝑎𝑎𝑎𝑎𝑎 − 2  −2 𝑎𝑎𝑎𝑎𝑎𝑎 + 2  −2 𝑎𝑎𝑎𝑎𝑎𝑎 − 2 additionally, we have 4 more cases for single errors of the type ±1 and ±2. in this case, the number of combinations for each code word that can be corrected is (1 + 13 ∗ 4 + (13 𝑐𝑐ℎ𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜 2) ∗ 8) = 677. thus, we have that 677 ∗ 58 ≤ 513 and the cardinality of the best possible code is 513/677 < 59 . 3. construction of 𝑪𝑪 (𝟏𝟏𝟏𝟏, 𝟏𝟏𝟏𝟏) and 𝑪𝑪 (𝟏𝟏𝟏𝟏, 𝟏𝟏𝟏𝟏) codes over rings 𝑍𝑍7 and 𝑍𝑍9 in this paragraph we will construct asymmetric low magnitude error correcting codes over different rings 𝑍𝑍𝑚𝑚. as it was mentioned above, in this paper we intend to construct codes correcting double ±1 and ±2 errors with the same magnitude. in the previous paragraph we constructed the optimal code c (13, 8) over the ring 𝑍𝑍5 based on the optimal code c (12, 8) correcting only double ±1 errors by adding only one parity check symbol. codes constructed during this paragraph will not be optimal in the sense that they have a minimal possible number of parity check symbols. 3.1 construction of code 𝑪𝑪 (𝟏𝟏𝟏𝟏, 𝟏𝟏𝟏𝟏) over ring 𝑍𝑍7 it was shown in [5] that the optimal code c (16, 12) over the ring 𝑍𝑍7, correcting double ±1 errors, has the following parity check matrix: � 1 1 1 1 1 1 1 0 1 2 3 4 5 6 1 1 6 5 4 3 2 1 0 2 2 2 2 2 2 2 1 1 4 3 6 6 3 4 2 4 3 6 6 3 4 2 1 6 1 1 1 1 1 1 1 4 3 6 6 3 4 2 0 0 �. to construct the code c (17, 12), correcting double ±1 and ±2 errors with the same magnitude, we need to take the first 14 columns of the parity check matrix: construction of linear codes over rings 𝑍𝑍𝑚𝑚 correcting double ±1 or ±2 errors 70 � 1 1 1 1 1 1 1 0 1 2 3 4 5 6 6 5 4 3 2 1 0 2 2 2 2 2 2 2 4 3 6 6 3 4 2 4 3 6 6 3 4 2 1 1 1 1 1 1 1 4 3 6 6 3 4 2 � and add one parity check symbol to this code. consequently, we need to add one additional row to the parity check matrix, which will be the following row: [2 3 4 0 1 6 5 2 3 4 0 1 6 5]. now the code given by the following parity check matrix, can correct up to two errors of the type ±1 and ±2: ⎣ ⎢ ⎢ ⎢ ⎡ 1 1 1 1 1 1 1 0 1 2 3 4 5 6 6 5 4 3 2 1 0 2 2 2 2 2 2 2 4 3 6 6 3 4 2 4 3 6 6 3 4 2 1 1 1 1 1 1 1 4 3 6 6 3 4 2 2 3 4 0 1 6 5 2 3 4 0 1 6 5⎦ ⎥ ⎥ ⎥ ⎤ . our newly constructed code given by this parity check matrix has a length of 14, from which 5 are parity check symbols and 9 information symbols. we want to have the same number of information symbols as in code c (16, 12) with 4 parity check symbols and 12 information symbols. so, we need to add 3 additional columns to the matrix to have 12 information symbols. we found the corresponding 3 columns by computer search: ⎣ ⎢ ⎢ ⎢ ⎡ 6 6 6 1 2 3 5 4 1 3 1 4 5 5 4⎦ ⎥ ⎥ ⎥ ⎤ . so, we obtain the parity check matrix 𝐻𝐻7 , with a length of 17: 𝐻𝐻7 = ⎣ ⎢ ⎢ ⎢ ⎡ 1 1 1 1 1 1 1 0 1 2 3 4 5 6 6 6 6 6 5 4 3 2 1 0 2 2 2 2 2 2 2 1 2 3 4 3 6 6 3 4 2 4 3 6 6 3 4 2 5 4 1 1 1 1 1 1 1 1 4 3 6 6 3 4 2 3 1 4 2 3 4 0 1 6 5 2 3 4 0 1 6 5 5 5 4⎦ ⎥ ⎥ ⎥ ⎤ . the code given by this parity check matrix 𝐻𝐻7 with 5 parity check symbols and 12 information symbols, can correct up to two errors of the type ±1 and ±2 with the same magnitude. in this case the number of combinations for each code word that can be corrected is (1 + 17 ∗ 4 + (17 𝑐𝑐ℎ𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜 2) ∗ 8) = 1157. as such, using this code 𝐶𝐶(17, 12) over the ring 𝑍𝑍7, correcting double ±1 and ±2 errors with the same magnitude, we can correct all 1157 errors, which can occur in the code word. this code does not satisfy the optimality criteria-1, since an equation 717/1157 < 713 is not satisfied in g. khachatrian and h. khachatrian 71 this case which means that the minimum number of parity check symbols should be 4 not 5, i.e., we have one extra parity check symbol. we will refer to this kind of codes as quasi-optimal. 3.2 construction of code 𝑪𝑪 (𝟏𝟏𝟏𝟏, 𝟏𝟏𝟏𝟏) over ring 𝑍𝑍9 again, from [5] we know the optimal code c (20, 16) over the ring 𝑍𝑍9, correcting double ±1 errors with 4 parity check symbols and 16 information symbols, which was given below presented by the following parity check matrix : � 1 1 1 1 1 1 1 1 8 7 6 5 4 3 2 1 1 1 2 4 7 6 5 4 3 2 1 0 2 2 2 2 2 2 2 2 1 1 2 4 7 3 2 4 4 2 3 7 7 3 2 4 4 2 3 7 1 1 2 4 1 1 1 1 1 1 1 1 7 3 2 4 4 2 3 7 6 3 7 2 �. to construct the code c (21, 16), correcting double ±1 and ±2 errors with the same magnitude, we need to take the first 16 columns of the matrix above in the same way as we did in the previous construction for the code c (17, 12) over the ring 𝑍𝑍7: � 1 1 1 1 1 1 1 1 8 7 6 5 4 3 2 1 7 6 5 4 3 2 1 0 2 2 2 2 2 2 2 2 7 3 2 4 4 2 3 7 7 3 2 4 4 2 3 7 1 1 1 1 1 1 1 1 7 3 2 4 4 2 3 7 � and add one parity check symbol to this code. consequently, we need to add one additional row to the matrix, which will be the following row: [2 3 4 0 1 8 7 3 2 3 4 0 1 8 7 3]. now the code given by the following parity check matrix, can correct up to two errors of the type ±1 and ±2 with the same magnitude: ⎣ ⎢ ⎢ ⎢ ⎡ 1 1 1 1 1 1 1 1 8 7 6 5 4 3 2 1 7 6 5 4 3 2 1 0 2 2 2 2 2 2 2 2 7 3 2 4 4 2 3 7 7 3 2 4 4 2 3 7 1 1 1 1 1 1 1 1 7 3 2 4 4 2 3 7 2 3 4 0 1 8 7 3 2 3 4 0 1 8 7 3⎦ ⎥ ⎥ ⎥ ⎤ . our newly constructed code given by this parity check matrix has a length of 16, from which 5 are parity check symbols, and 11 information symbols. again, we want to have the same number of information symbols as in code c (20, 16) with 4 parity check symbols and 16 information symbols correcting double ±1 errors. so, we need to add 5 additional columns to the matrix to have 16 information symbols. we found the corresponding 5 columns by computer search: construction of linear codes over rings 𝑍𝑍𝑚𝑚 correcting double ±1 or ±2 errors 72 ⎣ ⎢ ⎢ ⎢ ⎡ 6 6 8 8 8 2 4 0 2 8 2 3 2 2 6 3 7 1 7 1 3 7 1 7 1⎦ ⎥ ⎥ ⎥ ⎤ . the code given by this parity check matrix 𝐻𝐻9 with 5 parity check symbols and 16 information symbols, can correct up to two errors of the type ±1 and ±2 with the same magnitude: 𝐻𝐻9 = ⎣ ⎢ ⎢ ⎢ ⎡ 1 1 1 1 1 1 1 1 8 7 6 5 4 3 2 1 6 6 8 8 8 7 6 5 4 3 2 1 0 2 2 2 2 2 2 2 2 2 4 0 2 8 7 3 2 4 4 2 3 7 7 3 2 4 4 2 3 7 2 3 2 2 6 1 1 1 1 1 1 1 1 7 3 2 4 4 2 3 7 3 7 1 7 1 2 3 4 0 1 8 7 3 2 3 4 0 1 8 7 3 3 7 1 7 1⎦ ⎥ ⎥ ⎥ ⎤ . in this case the number of error combinations for each code word that can be corrected is (1 + 21 ∗ 4 + (21𝑐𝑐ℎ𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜 2) ∗ 8) = 1765. as such, using this code 𝐶𝐶(21, 16) over the ring 𝑍𝑍9 correcting double ±1 and ±2 errors with the same magnitude, we can correct all 1765 errors which can occur in the code word. again, in this case, similar to the 𝐶𝐶(17, 12) code over the ring 𝑍𝑍7 we have a quasioptimal code. 4. conclusion in this paper a construction of double ±1 and ±2 error correcting linear optimal code over the ring 𝑍𝑍5 and quasi-optimal codes over the rings 𝑍𝑍7 and 𝑍𝑍9 are constructed. we plan to investigate if the approach presented in this paper can be extended to construct codes for larger alphabets as well as to construct quasi-optimal codes with higher code rates. references [1] s. martirossian, “single error correcting close packed and perfect codes”, proc.1st intas int. seminar coding theory and combinatorics, armenia, pp. 90-115, 1996. [2] h. kostadinov, n. manev and h. morita, “on ±1 error correctable codes”, ieice trans.fundamentals, vol. e93-a, pp. 2578-2761, 2010. [3] a. j. han vinck and h. morita, “codes over the ring of integers modulo m”, ieice trans.fundamentals , vol. e81-a, pp. 2013-2018, 1998. [4] g. khachatrian and h. morita, “construction of optimal ±1 double error correcting linear codes over ring z5”, 3th international workshop on advances in communications, boppard, germany, pp. 10-12, may 2014. g. khachatrian and h. khachatrian 73 [5] g. khachatrian and h. khachatrian ”construction of double ±1 error correcting linear optimal codes over rings 𝑍𝑍7 and 𝑍𝑍9”, mathematical problems of computer science, vol. 45, pp. 106--110, 2016. submitted 26.10.2017, accepted 07.02.2018. ±𝟏𝟏 և ±𝟏𝟏 մեծությամբ կրկնակի սխալներ ուղղող գծային կոդերի կառուցումներ տարբեր մեծության 𝑍𝑍𝑚𝑚 օղակներում գ. խաչատրյան և հ. խաչատրյան ամփոփում այս հոդվածի շրջանակներում ներկայացված են 𝑍𝑍5, 𝑍𝑍7 և 𝑍𝑍9 օղակներում ±1 կամ ±2 միևնույն մեծությամբ կրկնակի սխալներ ուղղող օպտիմալ և քվազի-օպտիմալ կոդերի կառուցումներ: построение кодов исправляющих двойные ошибки размера ±𝟏𝟏 и ±𝟏𝟏 в 𝑍𝑍𝑚𝑚 кольцах с разными величинами г. хачатрян и г. хачатрян аннотация в данной статье представлено построение оптимальных и квази-оптимальных линейных кодов в кольцах 𝑍𝑍5, 𝑍𝑍7 и 𝑍𝑍9, исправляющих двойные ошибки размера ±1 и ±2 с одинаковыми величинами. construction of linear codes over rings ,𝑍-𝑚. correcting double ±1 or ±2 errors gurgen h. khachatrian* and hamlet k. khachatrian ** in this paper a construction of double ±1 and ±2 errors correcting linear optimal and quasi-optimal codes over rings ,𝑍-5., ,𝑍-7. and ,𝑍-9. is presented with the limitation that both errors have the same amplitude in absolute value. ,,1-4-4-1-0-4-1-2-3-4-3-0-2-3-2... 69 mathematical problems of computer science 59, 69–81, 2023. doi: 10.51408/1963-0103 udc 004.725, 004.852 research of model increasing reliability intrusion detection systems timur v. jamgharyan national polytechnic university of armenia, yrevan, armenia e-mail: t.jamgharyan@yandex.ru abstract the paper presents the results of the using, a recurrent neural network to detect malicious software as part of the snort intrusion detection system.the research was conducted on datasets generated on the basis of athena, dyre, engrat, grum, mimikatz, surtr malware exploiting vulnerability cve-2022-20685 in the snort intrusion detection system. processing of input traffic data was carried out before the frag-3 and modbus preprocessors. the method of k nearest neighbors was used as a mathematical apparatus. the simulation of the developed software at different iterations. all research results are presented in https://github.com/t-jn keywords: machine learning, dataset, malware, preprocessor, metasploit, k nearest neighbors method, intrusion detection system. article info: received 8 january 2023; send to review 7 february 2023; accepted 7 march 2023. 1. introduction the intrusion detection systems (ids) include many different software components designed to detect various types of traffic with an embedded malicious component. detection is carried out according to a set of rules that are configured based on the threat model and security policies. the security architecture of the network infrastructure (ni) is built taking into account possible attacks according to various models։ triad cia (confindentiality, integrity, availability, cia), parker's hexad [1]. network ids, unlike host ids, detect attacks directed at the network segment and contain a set of complementary rules and security scripts that can neutralize an attack on the network. unlike host-based ids, network-based ids require more computing resources due to the fact that a larger set of rules and detectors is activated during their operation [2]. when using host ids in the infrastructure for a fleet of computing systems running linux os, can disable https://github.com/t-jn research of model increasing reliability intrusion detection systems 70 the rules for windows (or another os), but hardly possible for a network ids, since different operating systems are used in the infrastructure. modern ids are able to detect various types of attacks at different levels of the osi (open system interconnection, osi) model: bad traffic, system scanning, the use of known exploits to attack over various protocols, various backdoors, various known malware [3]. a significant limitation of systems for analyzing network traffic and the state of ni is the algorithmic and functional determinism inherent in them. an important issue of infrastructure security is the reliability of the processed data of the ids itself (data reliability – is, the property of the processed data not to have hidden errors [4]). the processing of data streams in the ids itself is determined by the functioning algorithms, data presentation formats, and the formalization of signature classifiers. protecting the ids signature database (both remote and local) is also one of the most important tasks. if the signatures database has been attacked for availability, then when a new vulnerability appears, the ids will not receive the necessary signature and the infrastructure perimeter will become vulnerable [5]. the development of m2m (machine-to-machine, m2m) and ml (machine learning, ml) technologies has increased the capabilities of both attack and defense tools. various researchers are conducting research on increasing (improving) various parameters of ids with ml [6, 7, 8]. one of the parameters that improves when using ml modules as part of a standard ids is its variability. unlike deterministic ids, ids with ml are capable of forming a multi-criteria sample on the basis of which the detector operation scheme is formed within the given constraints. but ids with ml have certain limitations when integrating them into the ni architecture. in particular, ml ids are very sensitive to various implementations of «noise attacks» («noise attack» is a variant of an availability attack in which a large number of random and meaningless fragmented packets are sent to the attacked system, some of which contain malware [9]). a dangerous consequence of a «noise attack» on a ml network ids is that attackers «attack» it for a long time with streams of datasets that cause false positives, «teach» the ml ids discriminator to be immune to this type of traffic (creating a cyclic chain of operations: false positive--true negative--false negative--true positive, which overload both the ids itself and the siem system (security information and event management, siem). various manufacturers combine ids modules into different classes, which allows you to quickly reconfigure the ids itself for specific tasks. in particular, for snort open source ids, there are many different types of preprocessors (frag-3, stream, performance monitor, smtp, pop, imap, ssh, dns, dce/rpc, sip preprocessors, reputation preprocessor, modbus preprocessor) each of which is functionally is responsible for handling the given protocol and/or data type.  ids preprocessor is a software module that receives data from the network traffic decoding module and outputs them to the input of intrusion detection modules. as stated in the article «attacks on machine learning systems» [10], the most vulnerable part of the ml ids is the traditional ids component (the deterministic part of the ids). ml systems, like any other, will be hacked using vulnerabilities in these traditional components. the use of ml at the preprocessor level is due to the fact that when developing an ids with ml, it is not enough to create a functioning model that can detect a threat not described in a set of rules (signatures) or generate new ones based on «known» signatures, but it is also necessary to protect the ids itself from probable infection with malware that can compromise the reliability of the results issued by ids․the choice of using a neural network at the preprocessor level is also due to the fact that the ids, which has a neural network in its component composition after the preprocessor, is able to protect the ni, since malware not detected by standard datasets (described in the signature/rule database) will be detected with varying probability neural network. but with a «noise attack», the target is the ids itself, which, when taken out of the reliable functioning mode, will no longer detect malware. undescribed at the preprocessor level, t. jamgharyan 71 malicious data embedded in ids can be detected using performance preprocessors that evaluate various kinds of statistics. but the problem is that, having determined the type of network ids, attackers can design an attack taking into account the work of preprocessors, and malware embedded in the ids itself will not go beyond the allowable statistical deviations. a lot of research has been devoted to the task of applying machine learning as part of ids, but only a small part of them explores the use of machine learning at the preprocessor level. this limitation, in particular, is due to the fact that the «response» of the neural network is probabilistic in nature and it is necessary to introduce clear boundaries for the neural network itself. otherwise, the neural network will be an event generator, which will be classified as an attack by the ids detection modules. thus, there is a recursion to the problem of stability and integrity of both the ids and the ni as a whole [11]. this research explores the potential of a recurrent neural network (rnn) to detect malware at the preprocessor level. the choice in the research of rnn from the entire set of neural networks is determined by the fact that rnn form a directed sequence between elements, which allows processing a series of events in time (this characteristic allows granular processing of fragmented datasets). the relevance of the work lies in the ever-increasing role of ids with ml in the ni security architecture and the increasing security requirements of the ids itself. the use of a neural network at the preprocessor level will increase the reliability of malware detection results without affecting the main ids signature database, which will reduce the attack surface for the ids itself. the novelty of the research lies in the application of the k nearest neighbors (k nearest neighbors, knn) method to detect malware in ids before preprocessors.  the k nearest neighbors method is a metric algorithm for classifying objects. malicious software athena, dyre, engrat, grum, mimikatz, surtr obtained from publicly available sources was used as calibration data [12--15]. the choice of the knn method is determined by the fact that it is necessary to minimize the value of the preprocessor error, and for this it is necessary to carry out a preliminary grouping and classification of unknown input datasets in normalized traffic.  traffic normalization modification of packets of protocols of the transport, and network levels for their subsequent processing by ids detection modules. 2. formulation and description the problem it is necessary to detect a malicious dataset in normalized traffic. the mathematical model construction was carried out on the basis of the formulas obtained in the sources [16,17]. there are network traffic 𝑋 inputs that contain malware fragments (1). 𝑋𝑚 = {(𝑥1, 𝑦1 ), … , (𝑥𝑚 , 𝑦𝑚 )}, (1) where, 𝑥𝑚network traffic datasets that do not contain malicious components, 𝑦𝑚network traffic datasets containing malicious components, 𝑚number of the analyzed packet of the input dataset. on the set of input traffic data sets, the distance function 𝑥𝜌(𝑦, 𝑦′) is given. the greater the value of the distance function, the less similar the entities are 𝑦, 𝑦′, where 𝑦′the minimum size of a malware dataset that can be uniquely identified and classified with respect to 𝑦. for any entity 𝜐 in the data package, arrange the objects 𝑥𝑖 in ascending order (2). 𝜌(𝜐, 𝑥1;𝜐 ) ≤ 𝜌(𝜐, 𝑥2;𝜐 ) ≤ ⋯ ≤ 𝜌(𝜐, 𝑥𝑚;𝜐 ), (2) research of model increasing reliability intrusion detection systems 72 where 𝑥𝑖;𝜐 the set of network traffic data that is the 𝑖-th neighbor of the entity 𝜐. similarly for the 𝑖 -th neighbor of the entity 𝜐 in the dataset 𝑦𝑖;𝜐. using the formula (3 from the source [17], we determine the malicious knn components for the traffic arriving in the ni. 𝛼(𝜐) = arg max 𝑦∈𝑌 ∑[𝑦(𝑥𝑖;𝜐 ) = 𝑦] 𝑚 𝑖=1 𝜔(𝑖, 𝜐), (3) where, 𝜔(𝑖, 𝜐)a given weight function that evaluates the degree of importance of the 𝑖-th neighbor for the classification of the entity 𝜐. by changing the 𝜔(𝑖, 𝜐) value, you can get different versions of the k nearest neighbors method (4). 𝜔(𝑖, 𝜐) = [𝑖 ≤ 𝑘]. (4) when 𝜔(𝑖, 𝜐) = [𝑖 = 1] malware is detected only in the given single value 𝜔. that is, the rnn is only able to detect the malware datasets it was trained on. a graphical representation of a rnn is shown in fig. 1. fig. 1. recurrent neural network. attackers can load malware into the ids itself not in a single package, but in fragments (using the built-in frag-3 preprocessor as an internal attack tool), then the research task of grouping and classifying malware fragments arises. standard ids do not cope with this task very effectively, but ml ids, in the presence of a training set, are able to solve this problem. the disadvantage of ml ids is that they can produce unreliable results if the preprocessor responsible for a particular type of traffic/protocol is «damaged» as a result of a «noise attack». a particular danger lies in the fact that any traffic entering the ids preprocessors (both ml and deterministic) is not checked for malicious components, since the task of the preprocessor is to «reformat» traffic for processing by detectors. 3. task statement it is necessary to develop and programmatically implement an algorithm and, based on it, software that integrates a rnn capable of solving the problem of grouping and classification with the ids preprocessor. t. jamgharyan 73 4. boundary conditions 1. the smallest fragment of the malware file (𝜉) that can be classified 𝜉 = 20𝑏𝑦𝑡𝑒 (detection was carried out using context-piecewise hashing (context triggered piecewise hashing, ctph), which is discussed in detail in [18]. 2. the delay in the processed module should not cause a «signal race». traffic from the output of the preprocessor module to the input of the detection modules must be sent synchronously. as part of this condition, an additional restriction has been introduced only udp (user datagram protocol, udp) traffic is processed. 3. the hardware must support the parallel computing mode. the developed software connects the rnn to frag-3 and modbus preprocessors (frag3 preprocessor for defragmenting an ip packet, modbus preprocessor for processing data from a variety of devices operating in scada networks (supervisory control and data acquisition, scada).since the frag-3 preprocessor is designed to build packages, using a trained rnn can neutralize the process of «assembling» malicious packages inside the ids, increasing the level of reliability of its functioning. on fig.2 shows a diagram of the snort ids with the proposed data processing software implemented on rnn. fig. 2. snort ids with developed data processing software. research of model increasing reliability intrusion detection systems 74 5. description of the module the network traffic coming from the decoders is directed to the preprocessor processing module (standard operation of the snort ids). the traffic that should processed by the frag 3 and modbus preprocessors is sent to the developed module based on the rnn. after processing according to the developed algorithm, this traffic is again sent to the standard detection modules. the task of the module is to carry out the primary «cut-off» of possible malware and protect the ids itself from being modified by malware. the developed algorithm is shown in fig. 3. fig. 3. developed algorithm. algorithm operation the software that searches for fragmented malware receives network traffic datasets from a decoder (snort ids a low-level interceptor) as input. only traffic that must be processed by the frag-3 and modbus preprocessors is subject to processing. step 1. converting received datasets to «data frame». this conversion is necessary to speed up the work of the rnn, since the traffic not processed by the developed module goes directly to t. jamgharyan 75 the preprocessor module and the processing delay should not exceed the boundary conditions (boundary condition 2). step 2 phase 1. calculation of the distance from the target object, which must be classified to each of the sample objects (traffic). computing a distance metric between likely malware datasets. all calculations are performed in parallel mode (boundary condition 3),  2.1 k=0 calculation of the distance metric and detection of malicious datasets is not performed, since the classification of malicious and non-malicious datasets is impossible,  2.2 k=1 the distance between malicious and non-malicious datasets is constant (k=const). only those malicious datasets that fall within the specified distance metric are detected,  2.3 k=m continuous detection mode.upper limit: the value of m that the hardware can handle,  2.4 k>m malicious datasets are not detected,  2.5 kfeature:install odl-restconf odl-l2switch-all odl-mdsal-apidocs odl-dlux-all odl-openflowplugin-all a. manasyan 95 they provide a graphical user interface of opendaylight software, as well as the necessary tools and devices. after activating them, immediately after setting the appropriate settings in our physical openvswitch network, we see a virtualized version of our network. to establish a "controller" connection in our physical network, we have previously configured the openvswitch openflow device by giving it the ip address of the controller by typing the following command: ovs-vsctl set-controller br0 tcp: 192.168.18.129:6633, where 192.168.18.129 is the ip address of the controller and it can be different for different devices, 6633 is the connection port and the protocol that controls data transfer over tcp. thanks to this, it was able to communicate with other devices. fig․ 2 ․ example of a virtual network in opendaylight. fig. 2 shows a virtualized version of the physical network in opendaylight. the picture clearly shows all the devices in our network that are connected to the openflow protocol support device, openvswitch. it is thanks to the openflow protocol that our sdn controller sees our entire physical network. openflow is a protocol for managing data processing, which is transmitted over the network through routers and switches using sdn technology. fast packet forwarding (data forwarding) on a classic router or switch and high-level routing decisions (control operations) are made on the same device. the openflow switch separates these two functions. data redirection is performed by the switch itself, while routing decisions are entrusted to a separate controller, usually a standard server. after clicking on the network topology, yang automatically shows us the confrest api url it uses to get this information: fig․ 3․ confrest api url network management automation through virtualization 96 by clicking the send button(fig.3), we can see the topology of our operational network. fig. 4․ operating network topology. in fig. 4, we can see information about our current topology, including the mac (media access control) and ip addresses of our hosts. so, you do not need to enter the device to see them every time, but you can see them from one control panel of sdn. when we send network traffic, all the information about it is mentioned in the flow tables of the sdn: how many packages were sent to us, how many arrived, how many dropped on the way, and what errors we encountered. and all that information we can see in the nodes of fig.5. fig.5. node connector statistics. a. manasyan 97 5. conclusion this paper proposes a solution for network optimization. as a result of the research, we concluded that automation improves the speed of it operations in response to analytical change. the ability to monitor operations, just as needed, provides greater visual control of the network, and transparency of processes within it. network automation improves work efficiency, reduces human error, increases access to network services, and provides better customer service. research has shown that the sdn solution includes the automatic integration of private networks, and the transmission of information over all available channels, without loss of application speed and quality. as a result of the study, it became clear that network automation can be implemented regardless of its type, which facilitates its transition. network virtualization is a more all-encompassing version of virtualization that makes it possible to convert physical network hardware into software that can easily be transitioned to different domains as needed, increasing flexibility and scalability for the network. i came to the conclusion that its use on the network will be of great benefit to network administrators. references [1] how can network automation be improved with sdn. [online]. available: https://www.acadiatech.com/blog/how-can-network-automation-improve-with-sdn/ [1] w. braun and m. menth, "software-defined networking using openflow: protocols applications and architectural design choices", future internet, vol. 6, no. 2, pp. 302-336, 2014, [online]. available: http://www.mdpi.com/1999-5903/6/2/302 [2] s. jena. difference between software defined network and traditional network. [online]. available: https://www.geeksforgeeks.org/difference-between-softwaredefined-network-and-traditional-network/ [3] j. doherty, sdn and nfv simplified: a visual guide to understanding software defined networks and network function virtualization, addison-wesley professional, march 2016. [4] software-defined sd-wan technologies. [online]. available: https://tadviser.com/index.php/article:sd-wan_%28software_defined%29_softwaredefined_wan https://www.tadviser.ru/images/thumb/9/96/idc_sd-wan_market_share_snapshot_blog_ii-1024x876.jpg/840px-idc_sd-wan_market_share_snapshot_blog_ii-1024x876.jpg https://www.tadviser.ru/images/thumb/9/96/idc_sd-wan_market_share_snapshot_blog_ii-1024x876.jpg/840px-idc_sd-wan_market_share_snapshot_blog_ii-1024x876.jpg https://www.acadiatech.com/blog/how-can-network-automation-improve-with-sdn/ http://www.mdpi.com/1999-5903/6/2/302 https://www.geeksforgeeks.org/difference-between-software-defined-network-and-traditional-network/ https://www.geeksforgeeks.org/difference-between-software-defined-network-and-traditional-network/ https://learning.oreilly.com/search/?query=author%3a%22jim%20doherty%22&sort=relevance&highlight=true https://learning.oreilly.com/library/view/sdn-and-nfv/9780134307398/ https://learning.oreilly.com/library/view/sdn-and-nfv/9780134307398/ https://learning.oreilly.com/library/publisher/uuid/1495afeb-554e-4ba9-84fb-f260848d173e https://tadviser.com/index.php/article:sd-wan_%28software_defined%29_software-defined_wan https://tadviser.com/index.php/article:sd-wan_%28software_defined%29_software-defined_wan https://habr.com/ru/company/hpe/blog/255363/ https://habr.com/ru/company/hpe/blog/255363/ network management automation through virtualization 98 ցանցի ղեկավարման ավտոմատացում վիրտուալացման միջոցով արուսյակ դ․ մանասյան հհ գաա ինֆորմատիկայի և ավտոմատացման պրոբլեմների ինստիտուտ e-mail:armanasyan@iiap.sci.am ամփոփում հետազոտության նպատակն էր մշակել ցանցի ղեկավարման ավտոմատացման մեթոդներ՝ վերլուծելով դրանց վիրտուալ անալոգը: աշխատանքում հիմնավորվում է այս մոտեցման արդիականությունը, վեր են հանվում առավելություններն ու թերությունները, ընդգծվում են առկա խնդիրները և առաջարկվում են դրանց լուծման ուղիներ։ արդյունքում ներկայացվել է ցանցերի վիրտուալացման արդյունավետությունը՝ փորձնական ցանցի օրինակով: բանալի բառեր՝ ցանց, ավտոմատացում, վիրտուալացում, sdn (ծրագրակողմնորոշված ցանց), opendaylight (ծրագրային ապահովում), openflow (արձանագրություն)։ автоматизация управления сетью за счет виртуализации арусяк д. манасян институт проблем информатики и автоматизации нан ра e-mail:armanasyan@iiap.sci.am аннотация цель исследования заключалась в разработке методов автоматизации управления сетью путем анализа ее виртуального аналога. в работе обосновывается актуальность такого подхода, выявляются преимущества и недостатки, подчеркиваются существующие проблемы и предлагаются пути их решения. в результате была показана эффективность виртуализации сети на примере пилотной сети. ключевые слова: сеть, автоматизация, виртуализация, sdn (программноопределяемая сеть), opendaylight (программное обеспечение), openflow (протокол). 1. introduction 5. conclusion ամփոփում mathematical problems of computer science 49, 97–102, 2018. existence of maximum entropy problem solution in a general n-dimensional case ruben a. gevorgyan and narek d. margaryan yerevan state university ruben−gevorgyan@yahoo.com, narek−margaryan@outlook.com abstract in the following paper, we will define conditions, which need to be satisfied in order for the maximum entropy problem applied in european call options to have a solution in a general n-dimensional case. we will also find a minimum right boundary for the price range in order to have at least one risk neutral measure satisfying the option pricing formula. the results significantly reduce the computational time of optimization algorithms used in maximum entropy problem. keywords: entropy, boundary, distribution, options. 1. introduction the maximum entropy methodology has recently started to become a quite popular tool with a huge potential of application in different fields [1, 2, 3, 4]. the core of the theory is based on shannon’s classical definition of information entropy [5], which is a crucial foundation in information theory. the maximum entropy approach has been broadly studied for its application in finance and financial extrapolation [6], and there have been significant contributions to its development since then, including the application of legendre transforms [7], partially finite convex programming [8], the employment of risk neutral moments [9], as well as the application of the problem as a non-parametric approach in american options pricing [10]. by theory, in the discrete case, the price of a european call option should be equal to the mathematical expectation of future pay-offs’ discounted value, thus lying in their convex hull. in reality, actual market prices may be biased from the theoretical ones [11] and lie out of the convex hull. we will concentrate on the derivation of conditions for the existence of solution which will not only reduce the computational time but will also result in an automated distribution recovery process [12, 13, 14] and will later allow us to develop algorithmic trading strategies that train on huge data sets. 2. outline of the problem consider having european call options for n different strike prices. let us denote the vector of strike prices with k and the vector of future states with x. x and k needn’t be of the 97 98 existence of maximum entropy problem solution in a general n-dimensional case same dimension. maximum entropy methodology seeks a risk neutral probability measure p, such that ap = b, (1) n∑ i=1 pi = 1, pi ≥ 0, (2) s(p) = n∑ i=1 piln(pi) is maximal, (3) where b is the vector of current option prices’ future values for each strike and a is (x1 − k1) + (x2 − k1)+ . . . (xn − k1)+ ... ... ... ... (x1 − kn)+ (x2 − kn)+ . . . (xn − kn)+   , (4) where (x)+ = max(x, 0). the probability vector p and the vector of future states x have the same dimension, in fact pi is the probability mass assigned to the future state xi. the distribution of future states will change as we change the state vector x. the question that interests us is what kind of state vector should be considered in order for a probability measure satisfying (1), (2) to exist in the first place. it is obvious that the greater the number of a’s linearly independent columns is, the bigger will their convex hull be, and so the more p vectors may exist satisfying (1), (2). so first of all we will consider the state vector (k1, . . . , kn, kn + t) for some arbitrary t. matrix a will now have the form below.  0 k2 − k1 . . . kn − k1 kn − k1 + t 0 0 . . . kn − k2 kn − k2 + t ... ... ... ... ... 0 0 . . . kn − kn−1 kn − kn−1 + t 0 0 . . . 0 t   . (5) we will denote a’s columns by a0, a1, . . . , an. let α(t) denote the angle between an and i, where i is the unit vector (1, . . . , 1). it is easy to show that limt→∞ cos α(t) = 1, so in order to see if any t exists, s.t. (1), (2) are satisfied, we will consider i instead of an, assuming that the angle between b and i isn’t 0 (this assumption holds throughout the text). let’s consider the following n + 1 hyperplane vector pairs (we denote hyperplanes by hp(·)).  hp(a1, a2, . . . , an−1, i), a0 hp(a0, a2, . . . , an−1, i), a1 ... hp(a0, a1, . . . , an−2, i), an−1 hp(a0, a1, . . . , an−2, an−1), i . (6) for each hyperplane above, we will denote by ni its normal “pointing” in the direction of the associated vector ai (note that a0 = (0, . . . , 0))  ⟨n0 − a1, a0 − a1⟩ ≥ 0 ⟨n1, a1⟩ ≥ 0 ... ⟨nn, an⟩ ≥ 0 , (7) r. gevorgyan and n. margaryan 99 where ⟨·, ·⟩ denotes the scalar (dot) product. 3. existence of solution the following proposition is obvious. proposition 1. there exists a finite t, s.t. (1), (2) are satisfied if and only if the following inequalities take place.   ⟨n0 − a1, b − a1⟩ ≥ 0 ⟨n1, b⟩ ≥ 0 ... ⟨nn, b⟩ ≥ 0 . (8) we now proceed to finding a minimal value for t, s.t. conditions (1) and (2) are satisfied. b represents the vector of prices and, thus its components are non-negative. assume that (8) takes place. if the last component of b, bn is 0, then the minimal value of t for which b ∈ conv(a0, a1, . . . , an−1, an) is 0 (conv(·) denotes the convex hull). in case bn is greater than 0, we will use the following lemmas (note that an = an−1 + ti). lemma 1. ∃µ > 0, s.t. ∀t for which b ∈ conv(a0, . . . , an−1, an), t ≥ µ > 0. proof. assume the opposite, then ∀ϵ > 0 ∃t0 < ϵ, s.t. ∃γ0, . . . , γn, γi ≥ 0, ∑n i=0 γi = 1, for which γ0a0 + . . . + γnan = b. let r = inf q∈conv(a0,...,an−1) ρ(b, q), ϵ = r ρ(b, i) , where ρ is the euclidean distance. for the ϵ above there exists 0 < t0 < ϵ, s.t. γ0a0+. . .+ γnan = b ⇔ γ0a0 + . . . + (γn−1 + γn)an−1 + t0γni = b ⇒ ρ(γ0a0 + . . . + (γn−1 + γn)an−1, b) = ρ(b − t0γni, b) ≤ t0ρ(i, b) < r, resulting in a contradiction. lemma 2. if for some t0 b ∈ conv(a0, . . . , an), then this also holds for any t > t0. proof. let t > t0, γ ′ n−1 = γn−1 + γn t−t0 t , γ ′ n = γn t0 t , then γ ′ n−1 + γ ′ n = γn−1 + γn and γ0a0 + . . . + γ ′ n−1an−1 + γ ′ nan = γ0a0 + . . . + γnan = b we now know that the set t of all possible t’s for which b ∈ conv(a0, . . . , an) is bounded from below by a positive number and unbounded from above. the next lemma proves that for t = inf t b is again in the convex hull conv(a0, . . . , an). lemma 3. let t be the set of all t’s, s.t. b ∈ conv(a0, . . . , an), then t = inf t ∈ t. proof. let’s assume the opposite. as t is the infimum of t, then for ∀ϵ > 0 ∃t0 ∈ t, s.t. 0 < t0 − t < ϵ. let r = inf q∈conv(a0,...,an−1,an−1+ti) ρ(b, q), 100 existence of maximum entropy problem solution in a general n-dimensional case ϵ = r ρ(b, i) . by the assumption there exists a t0 < t+ϵ, s.t. b = γ0a0 +. . .+γnan. let a ′ n = an−1 +ti, then ρ(γ0a0 + . . . + γn−1an−1 + γna ′ n, b) = ρ(b + (t − t0)i, b) ≤ (t0 − t)ρ(i, b) < r, resulting in a contradiction. based on the lemmas we may now formulate the main theorem of the article. theorem 1. if condition (8) is satisfied, the angle between b and i isn’t 0, then b ∈ conv(a0, . . . , an), where an = an−1 + ti and γn−1 = 0 in the linear representation of b by vectors a0, . . . an. the minimal value of t, t is given by t = bn(kn − kn−1) bn−1 − bn . (9) proof. we only need to show that γn−1 = 0. assume it’s not, then b = γ0a0 + . . . + γnan = γ0a0 + . . . + γn−1an−1 + γn(an−1 + ti) = γ0a0 + . . . + (γn + γn−1)an−1 + tγn γn + γn−1 (γn + γn−1)i = γ0a0 + . . . + (γn + γn−1)(an−1 + tγn γn + γn−1 i). as γn > 0, then tγn γn + γn−1 < t, which is a contradiction. having known that an−1 doesn’t ”participate” in the linear representation of b, we only need to find the value of t, s.t. b ∈ hp(a0, . . . , an−2, an). for that we will find the normal n of the hyperplane and solve ⟨n, b⟩ = 0 for t. we find n by observing the determinant of the following matrix based on the vectors from the hyperplane.  k2 − k1 0 . . . . . . 0 k3 − k1 k3 − k2 0 . . . 0 ... kn−1 − k1 kn−1 − k2 . . . 0 0 kn − k1 + t kn − k2 + t . . . kn − kn−1 + t t e1 e2 . . . en−1 en   . (10) the determinant is (−1)2n−1t(k2 − k1) . . . (kn−1 − kn−2)en−1+ (−1)2n(kn − kn−1 + t)(k2 − k1) . . . (kn−1 − kn−2)en. so n = (0, . . . , 0, −t, kn − kn−1 + t), and therefore ⟨n, b⟩ = 0 ⇔ t = bn(kn − kn−1) bn−1 − bn . r. gevorgyan and n. margaryan 101 4. conclusion as a result we obtained a way of checking whether a solution to the maximum entropy problem applied in european call options exists, before starting the optimization. if (8) takes place, then in order for the solution to exist, the right bound of future states vector must be greater than or equal to the value of t described in the theorem above. checking the existence of solution prevents the user from unknowingly proceeding to the stage of entropy maximization over an empty set of discrete probability distributions, which would yield unpredictable results. references [1] y. alhassid, n. agmon and r. d. levine, “an upper bound for the entropy and its applications to the maximal entropy problem”, chem. phys. lett., vol. 53, pp. 22, 1978. [2] y.alhassid, n. agmon and r. d. levine, “an algorithm for finding the distribution of maximal entropy”, journal of computational physics, vol. 30, pp. 250-258, 1979. [3] r. d. levine and m. tribus, “the maximum entropy formalism”, cambridge ma: mit press, pp. 207-209, 1978. [4] y. alhassid and r. d. levine, “experimental and inherent uncertainties in the information theoretic approach”, chem. phys. lett., vol. 73, pp. 16-20, 1980. [5] c. e. shannon, “a mathematical theory of communication”, bell systems technical journal, vol. 27, pp. 379-423, 1948. [6] p. w. buchen and m. kelly, “the maximum entropy distribution of an asset inferred from option prices”, journal of financial and quantitative analysis, vol. 31, pp. 143159, 1996. [7] c. neri and l. schneider, “a family of maximum entropy densities matching call options prices”, arxiv:1102.0224v1 [q-fin.pr], 2011. [8] j. borwein, r. choksi and p. marechal, “probability distributions of assets inferred from option prices via the principle of maximum entropy”, siam j. optim., vol. 14, no 2, pp. 464-478, 2003. [9] l. s. rompolis, “a new method of employing the principle of maximum entropy to retrieve the risk neutral density”, http://web.xrh.unipi.gr/attachments/seminars/2008 [10] yu. xishen and li yang, “pricing american options using a nonparametric entropy approach”, hindawi publishing corporation, pp. 16, article id 369795, 2014. [11] m.rubinstein, “implied binomial trees.” finance working paper, vol. 49, no 3, pp. 771-818, 1994. [12] n. d. margaryan “assessment of asset price distributions using maximum entropy method”, proc. of engineering academy of armenia, vol. 14, no 1, pp. 57-61, 2017. [13] n. d. margaryan “an algorithmic approach to solving the maximum entropy problem”, proc. of engineering academy of armenia, vol. 14, no 3, pp. 371-374, 2017. [14] n. d. margaryan “a boundary for the existence of solution to the maximum entropy problem applied in european call options”, proc. of the yerevan state university, vol. 52, no 1, pp. 3-7, 2018. submitted 18.10.2017, accepted 12.02.2018. 1 0 2 existence of maximum entropy problem solution in a general n-dimensional case àý¹ñ³ýáõñ n-ã³÷³ýç ¹»åùáõù ³é³í»é³·áõûý ¿ýïñáåç³ûç ëý¹ñç éáõíù³ý ·áûáõãûáõýá è. ¶¨áñ·û³ý ¨ ü. ø³ñ·³ñû³ý ²ù÷á÷áõù ð»ï¨û³é ³ßë³ï³ýùáõù ïë³ñù³ý»ýù å³ûù³ýý»ñ, áñáýó µ³í³ñ³ñí³íáõãûáõýý ³ýññ³å»ßï ¿ áý¹ñ³ýáõñ n-ã³÷³ýç ¹»åùáõù ºíñáå³ï³ý ûåóçáýý»ñáõù ïçñ³éíáõ ³é³í»é³·áõûý ¿ýïñáåç³ûç ëý¹ñç éáõíù³ý ·áûáõãû³ý ñ³ù³ñ: ü³¨ ï·ïý»ýù ·ý³ûçý ùçç³ï³ûùç ýí³½³·áõûý ³ç³ïáõùû³ý ë³ñù³ýá` ûåóçáýý»ñç ·ý³·áû³óù³ý µ³ý³ó¨çý µ³í³ñ³ñáõ ³éýí³½ý ù»ï éçëïçó ã»½áù ã³÷ç ·áûáõãû³ý ñ³ù³ñ: êï³óí³í ³ñ¹ûáõýùý»ñá ½·³éçáñ»ý ýí³½»óýáõù »ý ³é³í»é³·áõûý ¿ýïñáåç³ûç ëý¹ñç ù»ç û·ï³·áñííáõ ûåïçùç½³óçáý ³é·áñçãùý»ñç ñ³ßí³ñï³ûçý å³ù³ý³ïá: ñóùåñòâîâàíèå ðåøåíèÿ ïðîáëåìû ìàêñèìàëüíîé ýíòðîïèè â îáùåì n-ìåðíîì ñëó÷àå ð. ãåâîðãÿí è í. ìàðãàðÿí àííîòàöèÿ â ñëåäóþùåé ñòàòüå ìû îïðåäåëèì óñëîâèÿ, âûïîëíåíèå êîòîðûõ íåîáõîäèìî äëÿ ñóùåñòâîâàíèÿ ðåøåíèÿ ïðîáëåìû ìàêñèìàëüíîé ýíòðîïèè, ïðèìåíÿåìîé â åâðîïåéñêèõ îïöèîíàõ, â îáùåì n-ìåðíîì ñëó÷àå. ìû òàêæå íàéäåì ìèíèìàëüíóþ ïðàâóþ ãðàíèöó äëÿ öåíîâîãî äèàïàçîíà, êîòîðàÿ íåîáõîäèìà äëÿ ñóùåñòâîâàíèÿ õîòÿ áû îäíîé ðèñê-íåéòðàëüíîé ìåðû óäîâëåòâîðÿþùåé ôîðìóëå öåíîîáðàçîâàíèÿ îïöèîíîâ. ïîëó÷åííûå ðåçóëüòàòû çíà÷èòåëüíî óìåíüøàþò âû÷èñëèòåëüíîå âðåìÿ îïòèìèçàöèîííûõ àëãîðèòìîâ, èñïîëüçóåìûõ â çàäà÷å ìàêñèìàëüíîé ýíòðîïèè. article abstract_n mathematical problems of computer science 59, 27–34, 2023. doi: 10.51408/1963-0099 udc 510.64 the relationship between the proof complexities of linear proofs in quantified sequent calculus and substitution frege systems hakob a. tamazyan yerevan state university, yerevan, armenia e-mail: hakob.tamazyan@ysu.am abstract it has formerly been proved that there is an exponential speed-up in the number of lines of the quantified propositional sequent calculus over substitution frege systems when considering proofs as trees. this paper shows that a linear proof of any quantifierfree tautology in quantified propositional sequent calculus can be transformed into a linear proof of the same tautology in a substitution frege systems with no more than polynomially increasing proof lines and size. keywords: sequent systems, frege systems, proof size, number of proof lines, exponential speed-up. article info: received 23 march 2023; sent for review 2 april 2023; accepted 19 may 2023. 1. introduction the existence of a propositional proof system that has proofs of polynomial size for all tautologies is equivalent to the equation np = co-np [1]. this observation has gained attention in recent years, leading to the examination of new proof systems. through the discovery of new systems, the computational power of existing ones is gaining a greater understanding. a hierarchy of proof systems has been established based on two complexity measures (size and lines), and the relationships between these systems are being explored. alessandra carbone in [2] compared the number of derivation lines in the form of a tree in some propositional calculus systems and revealed a distinctive property of the quantified propositional sequent calculus (qpk system). namely, for some sequences of formulas, the qpk system has an exponential speed-up by lines with respect to the substitution sequent calculus (spk system) and substitution frege systems (sf systems) when proofs are considered as trees. it was shown in [3] that the lines of linear proofs of the same formulae families in all three systems are the same by order. later, in [4], the same result was achieved if one considers the sizes of linear proofs of the same formulae families for comparison. 27 28 the relationship between the proof complexities of linear proofs in qpk and substitution sf in this paper, the relationship between the proof complexities of linear proofs in qpk and sf has been investigated for all quantifier-free tautologies: it turns out that qpk system has no significant advantage over sf when only linear proofs are considered. specifically, after the transformation of linear qpk-proof of a quantifier-free tautology into a linear sfproof of the same tautology by some algorithm, both complexities (the number of lines and sizes) of linear proofs in sf can increase polynomially at most. 2. preliminaries first and foremost, lets define several proof systems according to [1, 5, 6]. the frege system f uses a denumerable set of propositional variables, a finite, complete set of propositional connectives. it has a finite set of inference rules defined by a figure of the form a1a2...am b (the rules of inference with zero hypotheses are the schemes of axioms). f must be sound and complete, i.e., for each rule of inference a1a2...am b every truth-value assignment, satisfying a1a2...am, also satisfies b, and f must prove every tautology. the substitution frege system sf is defined by adding to f the substitution rule a(p) a(b) where simultaneous substitution of the formula b is allowed for the variable p. the lk sequent calculus was introduced by gentzen [7] for first-order logic. each line in lk-proof is a sequent: a sequent is written in the form: a1, . . . , an → b1, . . . , bm where a1, . . . , an and b1, . . . , bm are formulas. we denote these sequences of formulas by capital greek letters γ, ∆, etc. as a quantifier symbol in lk, we will include only the universal quantification ∀. the existential quantification symbol ∃ will be added by the following definition: (∃x)a(x) ≡ ¬(∀x)¬a(x). the inference rules of the sequent calculus lk are as follows: • initial sequents are sequents of the following form: a → a where a is any formula. • structural rules: weakening : left γ → ∆ a, γ → ∆ weakening : right γ → ∆ γ → ∆, a exchange : left γ1, a, b, γ2 → ∆ γ1, b, a, γ2 → ∆ exchange : right γ → ∆1, a, b, ∆2 γ → ∆1, b, a, ∆2 contraction : left γ1, a, a, γ2 → ∆ γ1, a, γ2 → ∆ contraction : right γ → ∆1, a, a, ∆2 γ → ∆1, a, ∆2 h. tamazyan 29 • logical rules: ¬ : left γ → ∆, a ¬a, γ → ∆ ¬ : right a, γ → ∆ γ → ∆, ¬a ∧ : left a, b, γ → ∆ a ∧ b, γ → ∆ ∧ : right γ → ∆, a γ → ∆, b γ → ∆, a ∧ b ∨ : left a, γ → ∆ b, γ → ∆ a ∨ b, γ → ∆ ∨ : right γ → ∆, a, b γ → ∆, a ∨ b ⊃: left γ → ∆, a b, γ → ∆ a ⊃ b, γ → ∆ ⊃: right a, γ → ∆, b γ → ∆, a ⊃ b • the cut rule: γ → ∆, a a, γ → ∆ γ → ∆ let us denote by pk the sequent calculus lk, where the rules are restricted to propositional logic. the substitution system spk is defined as the propositional sequent calculus pk with an additional substitution rule: sbp γ → ∆, a(p) γ → ∆, a(b) , where simultaneous substitution of the formula b is allowed for the variable p, and p does not appear in γ, ∆. the quantifier system qpk is defined as the propositional sequent calculus pk, where new quantification rules are added: ∀ : left a(b), γ → ∆ (∀q)a(q), γ → ∆ ∀ : right γ → ∆, a(p) γ → ∆, (∀q)a(q) where b is any formula such that no free variable occurrence in b becomes bounded in a(b), and with the restriction that the atom p does not occur freely in the lower sequents of ∀ : right. notice that the the following two inferences can be derived in qpk system using the definition of the quantifier ∃: ∃ : left a(p), γ → ∆ (∃q)a(q), γ → ∆ a(p), γ → ∆ γ → ∆, ¬a(p) γ → ∆, (∀q)(q) ¬(∀q)¬a(q), γ → ∆ ∃ : right γ → ∆, a(b) γ → ∆, (∃q)a(q) γ → ∆, a(b) ¬a(b), γ → ∆ (∀q)(q), γ → ∆ γ → ∆, ¬(∀q)¬a(q) 30 the relationship between the proof complexities of linear proofs in qpk and substitution sf 3. main results for a given linear proof in qpk with n number of lines and proof size s, one can always find a linear proof in spk of the same tautology having o(n2) lines and o(s5) proof size. first of all, notice that for any linear proof in spk, there exists a linear proof in qpk of the same tautology with the same number of lines. the sequent (∀p)a(p), γ → ∆, a(b) is provable for all a, b, and the sequent γ → ∆, (∀p)a(p) is derivable from ∆ → ∆, a(p). hence, after combining them through a cut rule, one derives γ → ∆, a(b). here we examine the relationship between these systems in the opposite scenario. lemma. for n, m ≥ 0 and p not appeared in γ, ∆, the following inference γ, a1(p), . . . , an(p) → ∆, an+1(p), . . . , an+m(p) γ, a1(b), . . . , an(b) → ∆, an+1(b), . . . , an+m(b) can be achieved in spk system with o(n + m) lines using the substitution rule only once. proof. first, let’s prove these additional inferences: 1. γ → ∆, ¬a a, γ → ∆ γ → ∆, ¬a a → a γ → ∆, ¬a ¬a, a → a, γ → ∆ 2. γ → ∆, a ∨ b γ → ∆, a, b γ → ∆, a ∨ b a → a b → b γ → ∆, a ∨ b a → a, b b → b γ → ∆, a ∨ b a → a, b b → a, b γ → ∆, a ∨ b a ∨ b → a, b γ → ∆, a, b 3. γ, a ∧ b → ∆ γ, a, b → ∆ γ, a ∧ b → ∆ a → a b → b γ, a ∧ b → ∆ a, b → a b → b γ, a ∧ b → ∆ a, b → a a, b → b γ, a ∧ b → ∆ a, b → a ∧ b γ, a, b → ∆ h. tamazyan 31 the final proof will look like this: γ, a1(p), . . . , an(p) → ∆, an+1(p), . . . , an+m(p) γ, a1(p) ∧ a2(p), . . . , an(p) → ∆, an+1(p), . . . , an+m(p).... γ, a1(p) ∧ . . . ∧ an(p) → ∆, an+1(p), . . . , an+m(p).... γ, a1(p) ∧ . . . ∧ an(p) → ∆, an+1(p) ∨ . . . ∨ an+m(p) γ → ∆, an+1(p) ∨ . . . ∨ an+m(p), ¬(a1(p) ∧ . . . ∧ an(p)) γ → ∆, an+1(p) ∨ . . . ∨ an+m(p) ∨ ¬(a1(p) ∧ . . . ∧ an(p)) γ → ∆, an+1(b) ∨ . . . ∨ an+m(b) ∨ ¬(a1(b) ∧ . . . ∧ an(b)) γ → ∆, an+1(b) ∨ . . . ∨ an+m(b), ¬(a1(b) ∧ . . . ∧ anb) γ, a1(b) ∧ . . . ∧ an(b) → ∆, an+1(b) ∨ . . . ∨ an+m(b).... γ, a1(b), . . . , an(b) → ∆, an+1(b), . . . , an+m(b) note that in this proof the substitution rule is applied only once. theorem 1. for a given linear proof in qpk of some quantifier-free tautology with n number of lines, there exists a linear proof in spk of the same tautology having o(n2) number of lines. proof. suppose p is a given linear proof in qpk. since p is the proof of a quantifierfree tautology, if a formula with a quantifier appears in the proof, then it must disappear at some point in the next lines. these formulas can appear either by quantification rules or by weakening rules, and the cut rule is the only inference rule capable of removing a formula from the sequent. notice that if we apply the cut rule to two sequents and some formula a with a quantifier is removed, then it is impossible that both of these sequents got this quantifier by the ∀ : left rule. first of all, we will remove all applications of the ∀ : left rule in the proof of p . let (∀q)a(q) be some formula or subformula in the proof. suppose it appeared by ∀ : right rule that infers γ → ∆, (∀q)a(q) from γ → ∆, a(p). since p does not occur free in sequent γ → ∆, (∀q)a(q), instead of the ∀ : right rule, we can apply the substitution rule to γ → ∆, a(p) and substitute p with some new variable k that did not appear throughout the proof. if (∀q)a(q) appeared by weakening rules, we will replace it with the formula a(k), where k is again some new variable that did not appear throughout the proof. according to the previously mentioned claim, the formula (∀q)a(q) should have been removed at some point via the cut rule. therefore, just before the application of cut rule, we will substitute the variable k with the corresponding matching formula to be able to apply the cut rule successfully. this substitution is allowed since k does not appear in the remaining formulas of the sequent. this removal of formulas with quantifiers from the proof can have the following effects. firstly, since these formulas have been replaced with different ones, the contraction rule can not be applied to these replacements anymore, as they can differ from each other. therefore, instead of applying the contraction rule to them, in the next lines we will apply the same inference rules to both of them. as these formulas should disappear in one of the next lines by the cut rule, we will apply the cut-elimination rule twice so that both of them 32 the relationship between the proof complexities of linear proofs in qpk and substitution sf will be removed. there are o(n) applications of the contraction rule, then after this change, the number of lines will become o(n2). however, according to the lemma, the number of applications of the substitution rule will not change and will remain o(n). secondly, the ∀ : left rule that transformed some sequent a(b), γ → ∆ into (∀q)a(q), γ → ∆, will not be applied to the proof, and the formula b will appear in the next lines. hence, there might be an application of the substitution rule in these next lines that substitutes some variable x into some formula c so that x also appears in the formula b. this means that besides the formula c, there can also be other formulas with the variable x in the sequent. therefore, to fix this, we will apply the substitution to these formulas too. considering that the number of applications of the ∀ : left rule was o(n) and removing each application of the contraction rule adds just one formula to the sequent, the number of such formulas in the sequent will be o(n). therefore, according to the lemma, each such substitution will require o(n) additional lines. since there are o(n) applications of the substitution rule, this change will add o(n2) number of lines to our proof. this will conclude the transformation process, and the transformed spk proof will have o(n2) lines. theorem 2. for a given linear proof in qpk of some quantifier-free tautology with a proof size s, there exists a linear proof in spk of the same tautology having o(s5) proof size. proof. suppose p is a given linear proof in qpk with n number of lines and proof size s. let p ′ be the transformed spk proof according to the process described above. to calculate its size, let’s dive into the transformation process step by step. we replaced each application of the ∀ : right rule with a substitution rule to substitute one variable with another. the formulas with quantifiers that appeared by weakening rules have been replaced by formulas with the same size. afterwards, we added a substitution before the application of the cut rule to match the corresponding formula. all these steps change the number of proof lines and the proof size linearly. let’s denote them by n′, s′, respectively. moreover, we removed all applications of the ∀ : left rule. therefore, if some application of the ∀ : left rule transformed the sequent a(b), γ → ∆ into (∀q)a(q), γ → ∆, then after the removal, the formula b will appear in the next lines. this will increase the proof size by at most n′ · |a(b)|, where |a(b)| is the size of the formula a(b). removing the ith application of the ∀ : left rule increases the proof size by at most n′ ·|ai(bi)|, then removing all of them will add no more than ∑ i n′ · |ai(bi)| = n′ · ∑ i |ai(bi)| ≤ n′ · s′ ≤ s′2 to the proof size. as s′ is o(s), after this step, the proof size will be o(s2) and the number of lines will remain o(n). removing applications of the contraction rule has the following two effects on the proof size. first of all, it will keep the eliminated formula in a sequent, so it will appear in the next lines. the added proof size can be calculated completely like the previous method. since the number of applications of the contraction rule is o(n) and the proof size is o(s2), this change will make the proof size o(s3). the number of lines will remain o(n). the second effect of removing applications of the contraction rule will be applying the same inference rules to both formulas. since the proof size is o(s3), then applying the same h. tamazyan 33 inference rule to the previously eliminated formula can increase the proof size by o(s3). the number of applications of the contraction rule is o(n), and since n ≤ s, the overall proof size will become o(s4). finally, the removal of the ∀ : left rule causes some substitution steps to also substitute the same variable in several other formulas of the same sequent. notice that all these substitution steps were ∀ : right rule replacements that substitute one variable with another, as otherwise we won’t face such a problem. each such substitution that simultaneously substitutes the same variable in these sequent formulas required o(n) lines. if the ith such substitution is applied to the sequent si, then this change will overall add no more than∑ i c · n · |si| = c · n · ∑ i |si| ≤ c · s · ∑ i |si| to the proof size, where |si| is the size of the sequent si and c is some constant. ∑ i |si| is smaller than the current proof size, therefore the transformed spk proof will have o(s5) size. corollary. since the system spk is polynomially equivalent to the system sf, there is a transformation of a linear proof of any quantifier-free tautology in qpk into a linear proof in the system sf that increases the proof lines and size at most polynomially. 4. conclusion this work described an algorithm according to which any qpk linear proof can be transformed into a sf linear proof by increasing its lines and size to at most a polynomial extent. the obtained results show that the qpk system does not have a substantial advantage over the system sf in terms of linear proofs. references [1] s. a. cook and a. r. reckhow, “the relative efficiency of propositional proof systems”, symbolic logic, vol. 44, pp. 36–50, 1979. [2] a. carbone, “quantified propositional logic and the number of lines of tree-like proofs”, studia logica, vol. 64, pp. 315–321, 2000. [3] h. a. tamazyan and a. a. chubaryan, “on proof complexities relations in some systems of propositional calculus, mathematical problems of computer science, vol. 54, pp. 138–146, 2020. [4] l. a. apinyan and a. a chubaryan, “on sizes of linear and tree-like proofs for any formulae families in some systems of propositional calculus”, mathematical problems of computer science, vol. 57, pp. 47–55, 2022. [5] p. pudlák, the lengths of proofs, in s. buss (ed.), handbook of proof theory, elsevier, vol. 137, pp. 547-637, 1998. [6] j. kraj́ıc̆ek, proof complexity, encyclopedia of mathematics and its applications, cambridge university press, vol. 170, 2019. [7] g. gentzen, “die widerspruchsfreiheit der reinen zahlentheorie”, mathematische annalen, vol. 112, pp. 493–565, 1936. 3 4 the relationship between the proof complexities of linear proofs in qpk and substitution sf ¶í³ûçý ³ñï³íáõùý»ñç µ³ñ¹áõãûáõýý»ñç ï³åá í³í³éçãý»ñáí ë»ïí»ýóç³é ñ³ù³ï³ñ·áõù ¨ ï»õ³¹ñù³ý ï³ýáýáí üñ»·»ç ñ³ù³ï³ñ·»ñáõù ð³ïáµ ². â³ù³½û³ý ºñ¨³ýç å»ï³ï³ý ñ³ù³éë³ñ³ý, ºñ¨³ý, ð³û³ëï³ý e-mail: hakob.tamazyan@ysu.am ²ù÷á÷áõù ñâÿçü ìåæäó ñëîæíîñòÿìè äîêàçàòåëüñòâ ëèíåéíûõ âûâîäîâ â ñèñòåìå ñåêâåíöèàëüíîãî èñ÷èñëåíèÿ ñ êâàíòîðàìè è ñèñòåìàõ ôðåãå ñ ïðàâèëîì ïîäñòàíîâêè àêîá à. òàìàçÿí åðåâàíñêèé ãîñóäàðñòâåííûé óíèâåðñèòåò, åðåâàí, àðìåíèÿ e-mail: hakob.tamazyan@ysu.am àííîòàöèÿ ðàíåå áûëî äîêàçàíî, ÷òî ñóùåñòâóåò ýêñïîíåíöèàëüíîå óñêîðåíèå êîëè÷åñòâà øàãîâ â ñèñòåìå ñåêâåíöèàëüíîãî èñ÷èñëåíèÿ âûñêàçûâàíèé ñ êâàíòîðàìè ïî ñðàâíåíèþ ñ ñèñòåìàìè ôðåãå ñ ïðàâèëîì ïîäñòàíîâêè, êîãäà ìû ðàññìàòðèâàåì âûâîäû â âèäå äåðåâüåâ. ýòà ñòàòüÿ ïîêàçûâàåò, ÷òî ëèíåéíûé âûâîä ëþáîé áåñêâàíòîðíîé òàâòîëîãèè â ñèñòåìå ñåêâåíöèàëüíîãî èñ÷èñëåíèÿ âûñêàçûâàíèé ñ êâàíòîðàìè ìîæíî ïðåâðàòèòü â ëèíåéíûé âûâîä òîé æå òàâòîëîãèè â ñèñòåìàõ ôðåãå ñ ïðàâèëîì ïîäñòàíîâêè ñ íå áîëåå ÷åì ïîëèíîìèàëüíî âîçðàñòàþùèì êîëè÷åñòâîì øàãîâ è äëèíîé âûâîäà. êëþ÷åâûå ñëîâà: ñåêâåíöèàëüíûå ñèñòåìû, ñèñòåìû ôðåãå, äëèíà âûâîäà, êîëè÷åñòâî øàãîâ âûâîäà, ýêñïîíåíöèàëüíîå óñêîðåíèå. ü³ëïçýáõù ³å³óáõóí»é ¿, áñ í³í³éçãý»ñáí ë»ïí»ýóç³é ñ³ù³ï³ñ·áõù ³éï³ ¿ ù³ûé»ñç ù³ý³ïç ¿ùëåáý»ýóç³é ³ñ³·³óáõù ï»õ³¹ñù³ý ï³ýáýáí üñ»·»ç ñ³ù³ï³ñ·»ñç ýï³ïù³ùµ, »ñµ ¹çï³ñïáõù »ýù í³é³ûçý ³ñï³íáõùý»ñá: ²ûë ñá¹í³íá óáõûó ¿ ï³éçë, áñ ³é³ýó í³í³éçãý»ñç, ó³ýï³ó³í ýáõûý³µ³ýáõãû³ý ·í³ûçý ³ñï³íáõùá í³í³éçãý»ñáí ë»ïí»ýóç³é ñ³ù³ï³ñ·áõù ñý³ñ³íáñ ¿ í»ñ³í»é ýáõûý ýáõûý³µ³ýáõãû³ý ·í³ûçý ³ñï³íù³ý ï»õ³¹ñù³ý ï³ýáýáí üñ»·»ç ñ³ù³ï³ñ·»ñáõù` áõý»ý³éáí ³ñï³íù³ý ù³ûé»ñç ù³ý³ïç ¨ »ñï³ñáõãû³ý ³é³í»é³·áõûý µ³½ù³ý¹³ù³ûçý ³×: ´³ý³éç µ³é»ñ` ë»ïí»ýóç³é ñ³ù³ï³ñ·»ñ, üñ»·»ç ñ³ù³ï³ñ·»ñ, ³ñï³íù³ý »ñï³ñáõãûáõý, ³ñï³íù³ý ù³ûé»ñç ù³ý³ï, ¿ùëåáý»ýóç³é ³ñ³·³óáõù: 03_chub_59_27_34 03 mathematical problems of computer science 59, 57–68, 2023. doi: 10.51408/1963-0102 udc 004.934 making speaker diarization system noise tolerant davit s. karamyan1,2, grigor a. kirakosyan2,3 and saten a. harutyunyan2 1russian-armenian university, yerevan, armenia 2krisp.ai, yerevan 3institute of mathematics of nas ra, yerevan, armenia e-mail: {dkaramyan, gkirakosyan, sharutyunyan }@krisp.ai abstract the goal of speaker diarization is to identify and separate different speakers in a multi-speaker audio recording. however, noise in the recording can interfere with the accuracy of these systems. in this paper, we explore methods such as multi-condition training, consistency regularization, and teacher-student techniques to improve the resilience of speaker embedding extractors to noise. we test the effectiveness of these methods on speaker verification and speaker diarization tasks and demonstrate that they lead to improved performance in the presence of noise and reverberation. to test the speaker verification and diarization system under noisy and reverberant conditions, we created augmented versions of the voxceleb1 cleaned test and voxconverse dev datasets by adding noise and echo with different snr values. our results show that, on average, we can achieve a 19.1% relative improvement in speaker recognition using the teacher-student method and a 17% relative improvement in speaker diarization using consistency regularization compared to a multi-condition trained baseline. keywords: speaker recognition, speaker diarization, noise robustness, teacherstudent, consistency regularization. article info: received 9 january 2023; send to review 30 january 2023, received in revised form 11 april 2023; accepted 17 april 2023. acknowledgement: this research was supported by krisp.ai. 1. introduction and related work speaker recognition (sr) is a broad field of study that addresses two major tasks: speaker identification and speaker verification. speaker identification is the task of identifying a person, whereas speaker verification is the task of determining whether the speaker is who they claim to be. in this study, we focus on far-field, text-independent speaker recognition, where the speaker’s identity is determined by the speaking style rather than the content of the speech. typically, such speaker recognition systems operate on unconstrained speech utterances that are converted into a fixed-length vector known as speaker embedding. many speech0-processing tasks use speaker embedding such as speaker diarization (sd) [1, 2], automatic speech recognition (asr) [3], and speech synthesis [4, 5]. 57 58 making speaker diarization system noise tolerant in recent years, deep neural networks have actively been employed for speaker embedding extractors since d-vector [6] was proposed. subsequently, the x-vector [7] was widely used because of the superior performance achieved by employing statistical pooling and time delay neural network (tdnn). other architectures such as resnet-based convolutional neural networks and cnns with cross-convolutional layers [8, 9] were employed for capturing the traits of speech. in addition, to deal with variable-length inputs, transformer [10], cnn-lstm [11] and a slew of variants of tdnn [12] were applied for dnn-based speaker embedding extractors. finally, to reduce the computational complexity and make the models smaller, [13, 14] employed 1d depth-wise separable convolutions for the speaker recognition task. metric learning techniques have been successful in speaker recognition tasks. these methods aim to create speaker embeddings with small distances between embeddings of the same speaker and large distances between embeddings of different speakers since unsupervised clustering will be applied to embeddings later in the speaker diarization pipeline. the triplet loss was proposed in [15] which required a careful selection of a triplet because the effectiveness of the performance depended on the contrast between negative and query samples. the prototypical loss was proposed in [16], where many negative samples were used and the euclidean distance between the centroid of all negative samples and the query embedding was maximized. in the generalized end-to-end loss [17], every utterance in the mini-batch functions as a query as opposed to just one in the prototypical loss. the angular prototypical (ap) loss [18] used only one utterance from each class as the query like the prototypical loss, but with a cosine similarity-based metric. the primary use case for speaker embeddings is speaker diarization. speaker diarization is the process of dividing an input audio stream into homogeneous segments according to the speaker’s identity. a typical speaker diarization system usually consists of several steps: (1) speech segmentation, where the input audio is segmented into short sections that are assumed to have a single speaker, and the non-speech sections are filtered out by voice activity detection (vad), (2) speaker embedding extractor, where speaker embeddings are extracted from segmented sections, (3) clustering, where the extracted audio embeddings are grouped [1] into clusters based on the number of speakers present in the audio recording, and optionally, (4) resegmentation step is performed to further refine clustering results. in real-world environment, noise causes significant degradations to the performance of speaker diarization systems, and is, hence, a major problem requiring special attention. the goal of noise-tolerant speaker diarization is to achieve improved performance in noisy environments. a recent work [19] tackles this problem using the auto-encoder architecture as a dimensionality reduction module. they extract two low-dimensional codes from speaker embeddings, representing the speaker identity and irrelevant noise information, then remove the noise factors. to our knowledge, there hasn’t been a lot of research done in this particular area. asr systems also suffer deterioration due to audio noise, and this has been the subject of extensive research [20, 21, 22], some of which inspired us. in this paper, we explore several approaches, borrowed from unsupervised domain adaptation, to make the speaker recognition models noise tolerant. in particular, we apply teacherstudent and consistency regularization techniques on speaker recognition and diarization tasks and compare them with multi-condition training when various noise augmentations are used. we were inspired by the significant results of this work for teacher-student [22], where clean and noisy audios are fed to the teacher and the student, respectively, to enforce similarity between the output distributions. consistency regularization is a commonly-used d. karamyan, g. kirakosyan and s. harutyunyan 59 technique amongst a variety of tasks in machine learning. this work [20] applies it in a manner similar to that mentioned previously, only here clean and noisy inputs are both fed to the student model. in the paragraphs that follow, we’ll discuss in detail how we apply these concepts to obtain noise-robust speaker recognition and diarization. 2. improving noise robustness of speaker diarization system there are several ways to improve the performance of speaker diarization systems in noisy and reverberant environments. for instance, work in [1] proposed the sequence of refinement operations to smooth and denoise data in the similarity space. in this work, we will focus only on the speaker embedding extraction part, and we are going to use unsupervised domain adaptation techniques to make the model noise tolerant. given a training dataset consisting of pairs (xi, yi) where xi represents an audio signal and yi represents the speaker id. our goal is to learn a parametrized function fθ, which should be able to compress any given audio into a d-dimensional vector, also known as a speaker embedding. moreover, if two audio signals are spoken by the same speaker, then the cosine similarity between their corresponding embeddings should be higher. conversely, if the two audios are spoken by different speakers, the cosine similarity between their embeddings should be lower. the additive angular margin (aam) loss, as proposed in [23], is a prevalent method for training speaker embedding extractors. the aim of the aam loss is to minimize the angle between speaker embeddings belonging to the same speaker while simultaneously maximizing the angle between speaker embeddings belonging to different speakers. 2.1 consistency regularization the core idea behind consistency regularization (cr) is to make sure that the network produces similar embeddings for the augmented versions of the same unlabeled utterance [20, 24, 25]. it is enforced by an additional regularization term in the loss function: lcr = 1 n n∑ i=1 |fθ(a(xi)) − fθ(a(xi))|22, where fθ is an embedding extractor with parameters θ, n represents the total number of training examples within the dataset. by a(x) we denote a stochastic operation that augments the audio in such a way that the speaker identity remains the same. so the difference is most likely non-zero. the final form of loss is a weighted combination of laam and lcr as shown below: l = (1 − α)laam + αlcr, where α is a hyperparameter taking values between 0 and 1. 2.2 teacher-student one critical problem with lcr loss is that it is not stable because of unstable target. to mitigate unstable target problem, the teacher-student model was proposed in [26], where two separate models were used: a student network with θ parameters and a teacher with 60 making speaker diarization system noise tolerant θ′ parameters. on unlabeled examples, the teacher network provides the learning target for the student network: lts = 1 n n∑ i=1 |fstudentθ (a(xi)) − f teacher θ′ (a(xi))| 2 2. student is trained as usual. teacher model is not trained via back-propagation. instead, its weights are updated at each iteration using the weights from the student network. again, the final loss is a weighted combination of laam and lts as shown below: l = (1 − α)laam + αlts. 2.3 knowledge distillation if the teacher model is already trained, it is desirable that its weights remain constant. this training setup is known as ”knowledge distillation”, where the student model is trained to mimic a pre-trained, larger model [27]. 3. experiments 3.1 model architecture in all experiments, we will use the speakernet [13] architecture as the backbone model. speakernet models are made up of 1d depth-wise separable convolutional layers. on top of the model, a statistical pooling layer is used to obtain a fixed-length vector. the proposed variation of speakernet (speakernet-m) has fewer parameters (5m) when compared to sota and shows very similar performance on voxceleb1 [28] trial files when compared to sota systems. the model provides embeddings of size 256 for a given audio sample. in teacher-student experiments, both the teacher and the student have the same architecture. 3.2 datasets the voxceleb1 [28] and voxceleb2 [29] datasets are widely recognized benchmarks in the field of speaker recognition. these datasets have pre-defined development and test sets, which allow for an objective and consistent evaluation of speaker recognition models. we trained our speaker recognition models using only the development part, which consisted of 7205 distinct speakers. for evaluation of speaker embeddings quality, we use voxceleb1 cleaned test trial file. the test trial file contains a list of audio pairs, and the model’s performance is evaluated based on its ability to correctly determine whether the two recordings belong to the same speaker or not. to evaluate speaker diarization, we use the voxconverse [30] development set. the dataset statistics are shown in table 1. 3.3 metrics the equal error rate (eer) metric is used to evaluate the speaker verification. this is the rate used to determine the threshold value for a system when its false acceptance rate and d. karamyan, g. kirakosyan and s. harutyunyan 61 table 1: statistics of datasets used for training speakernet. dataset # speakers duration (h) # utterances voxceleb1 1211 340.4 148642 voxceleb2 5994 2359.77 1,092,009 false rejection rate are equal. we calculate eer on voxceleb1 cleaned test trial file under original, noisy and echo conditions. for diarization evaluation purposes, we used diarization error rate (der). this is the sum of three error terms: false alarm (fa), missed detection (ms) and speaker confusion error rate (cer). similar to the previous works [12, 14], we use collar 0.25 sec and ignore overlap speech regions for confusion error rate calculation. we test the diarization system in original, noisy, and echo scenarios, just like we do for speaker verification. both eer and der are calculated using the cosine similarity back-end. 3.4 experiment setup 3.4.1 input features our audio pre-processing procedure is identical to the one described in the speakernet paper [13]. for each frame window of 20 ms, shifted by 10 ms, 64-dimensional acoustic features were calculated from the speech recordings. each utterance fed to the encoder has a size t × 64, where t is the number of frames in a given audio sample. we crop speech segments into random chunks from 3 to 8 seconds. with larger chunks, the model converges faster. 3.4.2 clean teacher our first baseline is a clean teacher trained on voxceleb1 and voxceleb2 datasets with additive angular margin loss. we set the aam loss hyperparameters to s = 30 and m = 0.2, as it was shown in [13, 14], these values give the best results. to avoid overfitting, we added specaugment [31] to the training pipeline, which randomly masks blocks of frequency and time channels. 3.4.3 noisy teacher our second baseline is a noisy teacher trained with the same objective as a clean baseline, and with the additional augmentation steps described below: • no augment: leave the utterance unchanged • rir augment: reverberate an input audio using an impulse response from rirs dataset [32] • noise augment: add noise from musan [33] dataset with signal-to-noise (snr) values randomly chosen from 0-50db • rir-noise augment: apply noise and echo perturbations to the same audio at the same time 62 making speaker diarization system noise tolerant • speed augment: speed perturbation with 0.95x and 1.05x speeds rir, noise, and rir-noise augmentations all have a probability of 0.25 and are mutually exclusive. speed augmentation is applied independently with a probability of 0.1. 3.4.4 consistency regularization we add an extra mean squared loss between embeddings for the augmented and nonaugmented versions of the same utterance to the aam loss during training. we set the α hyperparameter in the final loss to 0.1. 3.4.5 teacher-student in order to supervise the student model, we choose our clean-teacher baseline as the teacher. we did not update teacher weights during the training and no perturbations were applied to the input of the teacher model. the flow chart of teacher-student training is presented in fig. 1. during the training procedure, in addition to the aam loss, the mean squared loss between the student and teacher-produced embeddings is minimized. we set the α hyperparameter in the final loss to 0.1. fig.1. flow chart of teacher-student learning for improving noise robustness of sr. 3.4.6 optimization all models are trained for 200 epochs with an sgd optimizer, with an initial learning rate (lr) of 0.08 using a cosine annealing lr scheduler on 4 a100 gpus. 3.5 evaluations 3.5.1 speaker verification all the experiment findings are displayed in table 2. the results of the original speakernet and the pre-trained checkpoint1 publicly released by nvidia are also provided for comparison. 1https: //catalog.ngc.nvidia.com/orgs/nvidia/teams/nemo/models/speakerverification_speakernet d. karamyan, g. kirakosyan and s. harutyunyan 63 the pre-trained checkpoint was trained solely with noise augmentation using the abovementioned datasets. in order to examine the speaker verification system under noisy and reverberant conditions, we created augmented versions of voxceleb1 clean test trials by injecting noise and echo with different snr values. table 2: comparison of different speaker verification models under noise and reverb conditions. the results are reported in equal error rates. the more aggressively noise has been applied, the lower the snr values were. a noise level of 0 db indicates that the sound and the noise have the same energy. model orig 0db 5db 10db rir speakernet [13] 2.14 speakernet (nvidia) 1.92 9.75 5.43 3.61 16.5 clean teacher 1.87 12.9 6.94 4.21 16.5 noisy teacher 2.6 9.35 5.84 4.23 12.74 consistency reg. 1.76 8.05 4.40 3.13 12.26 teacher-student 1.73 9.16 4.79 3.26 9.18 table 2 showcases the effectiveness of the methods applied. we can see that training the speakernet model with data augmentation (noisy teacher) improves the results in the noisy/reverberant environment with a small deterioration of eer on the original (not perturbed) audios. the teacher-student method achieves the lowest eer scores in original and reverberant cases (rir), whereas the consistency regularization method shows the best results for noisy audios. using the teacher-student method, we were able to improve the eer by an average of 19.1% compared to the multi-condition trained model. with consistency regularization, we were able to improve the eer by an average of 14.8% compared to the multi-condition trained model. 3.5.2 speaker diarization we employ our trained speakernet models for speaker diarization task to see which model has the smallest performance degradation in noisy conditions. we found that the optimal sliding window size and shift for speech segmentation are 1.5 and 0.5 seconds, respectively. in addition, diarization experiments are based on oracle vad to evaluate the vad-independent performance. the affinity matrix a is constructed using the cosine similarity between segment embeddings. we further apply the following sequence of refinement operations to the affinity matrix a: • row-wise thresholding: for each row, keep top-12 largest elements and set the rest to 0 • symmetrization: y = 1 2 (a + at ) • diffusion: y = aat we use the spectral clustering method [34] to obtain speaker labels. to get a full picture, we present the diarization results for both known (oracle) and unknown numbers of speakers. in the latter case, we utilize the maximal eigen-gap approach to determine the number of speakers [1]. 64 making speaker diarization system noise tolerant table 3: comparison of speaker diarization systems with various speaker embedding extractors under noise and reverberant conditions. the results are reported in diarization error rate (der). model known #speakers unknown #speakers 0db 5db 10db rir orig avg 0db 5db 10db rir orig avg clean teacher 12.13 4.48 1.96 2.44 1.26 4.45 15.44 7.59 2.74 4.48 1.78 6.40 noisy teacher 9.20 4.49 3.13 3.12 1.57 4.30 13.09 7.94 4.18 4.14 1.95 6.26 consistency reg. 9.50 3.46 2.0 2.50 1.45 3.78 13.40 4.90 2.57 3.45 1.67 5.20 teacher-student 9.84 3.41 2.11 2.43 1.36 3.83 13.99 6.17 3.09 3.52 1.61 5.67 in order to assess the performance of the speaker diarization system under noisy and reverberant conditions, we modified the voxconverse dev dataset by adding noise and echo at various signal-to-noise ratios. the results, shown in table 3, indicate that the teacher-student and consistency regularization methods generally outperform the multi-condition baseline model for both scenarios involving known and unknown numbers of speakers. in particular, when the number of speakers is unknown, we observed approximately 17% and 9.5% relative performance improvements for the consistency regularization and teacher-student methods, respectively, compared to the multi-condition baseline. however, it is worth noting that in certain specific scenarios, the baseline models may outperform the models with the overall best average performance. 4. conclusions in this research, we explore ways to increase the accuracy of speaker recognition and speaker diarization in noisy and reverberant environments, such as multi-condition, teacher-student, and consistency regularization. the key component of the methods used is the additional regularization term between embeddings for augmented and non-augmented versions of the same utterance. through the use of teacher-student and consistency regularization, we were able to improve the performance of speakernet on speaker recognition and diarization tasks in noisy and reverberant situations. references [1] q. wang, c. downey, l. wan, p. mansfield and i. moreno, “speaker diarization with lstm”, 2018 ieee international conference on acoustics, speech and signal processing (icassp). pp. 5239-5243, 2018. [2] x. anguera, s. bozonnet, n. evans, c. fredouille, g. friedland, and o. vinyals, “speaker diarization: a review of recent research”, ieee transactions on audio, speech, and language processing, vol. 20, pp. 356-370, 2012. [3] q. wang, h. muckenhirn, k. wilson, p. sridhar, z. wu, j. hershey, r. saurous, r. weiss, y. jia, and i. moreno, “voicefilter: targeted voice separation by speakerconditioned spectrogram masking”, arxiv preprint arxiv:1810.04826, 2018. [4] y. jia, y. zhang, r. weiss, q. wang, j. shen, f. ren, p. nguyen, r. pang, i. lopez moreno, y. wu, and others, “transfer learning from speaker verification to multid. karamyan, g. kirakosyan and s. harutyunyan 65 speaker text-to-speech synthesis”, advances in neural information processing systems, vol. 31, 2018. 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[24] a. vanyan and h. khachatrian, “deep semi-supervised image classification algorithms: a survey”, j. univers. comput. sci., vol. 27, pp. 1390-1407, 2021. [25] s. laine and t. aila, “temporal ensembling for semi-supervised learning”, arxiv preprint arxiv:1610.02242, 2016. [26] a. tarvainen and h. valpola, “mean teachers are better role models: weight-averaged consistency targets improve semi-supervised deep learning results”, advances in neural information processing systems, vol.30, 2017. [27] g. hinton, o. vinyals and j. dean, “distilling the knowledge in a neural network. arxiv preprint arxiv:1503.02531, 2015. [28] a. nagrani, j. chung and a. zisserman, “voxceleb: a large-scale speaker identification dataset”, arxiv preprint arxiv:1706.08612, 2017. [29] j. chung, a. nagrani and a. zisserman, “voxceleb2: deep speaker recognition”, arxiv preprint arxiv:1806.05622, 2018. [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 начиная с начала 2000 года осуществляется внедрение ghis в здравоохранении, в рамках принятого проекта о реформирование информ mathematical problems of computer science 49, 115--122, 2018. object-oriented modeling of matching to systemic classifiers sedrak v. grigoryan, nairi p. hakobyan and hovhannes s. vrtanesyan institute for informatics and automation problems of nas ra e-mail: addressforsd@gmail.com, hakobyannairi@gmail.com, hovhannesvrtanesyan@gmail.com abstract in this paper we present a version of object-oriented implementation of models of constructive regularized mental systems, mentals, and systemic classifiers introduced in [1] as well as algorithms for matching them to situations. we experiment the adequacy of the models and algorithms for the chess representing kernels of the class of combinatorial problems, where space of solutions can be represented by reproducible game trees (rgt). keywords: modelling, systemic classifiers, mental, matching, chess. 1. introduction 1.1. a number of researches ([1, 3]) search for systemic solutions of combinatorial problems (such as chess), as it is stated that those problems cannot be adequately solved by parametric methods [2]. we follow the line of approach given in [1], trying to provide a programmatic implementation of the model, suggest matching algorithms for these implementations, as well as provide evidence of their adequacy. 1.2. as it is described in [1], mental systems represent realities, in particular utilities, but have varying effectiveness with respect to the goals and are processed to support utilization and gaining benefits from utilities. it is stated that a mighty way of enhancement of effectiveness of mss, and thus, cognizers, is the regularization of classifiers induced by mental doers and mental systems, while classifiers cl of members x of communities c are regularized in c if accompanied by ontological in c methods, instructions allowing x regularly provide positive samples of inputs of cl, as well as let the members of c do the same by communicating with x. doers, we assume, are realities having inoutput parts and for available inrealities, i.e, realities at the input parts, either elaborating certain output realities or staying passive. classifiers of roots and utilities are identified as root and induced goals. 115 mailto:addressforsd@gmail.com mailto:hakobyannairi@gmail.com mailto:hovhannesvrtanesyan@gmail.com object-oriented modeling of matching to systemic classifiers 116 1.3. to model mentals and systemic classifiers, we consider a class of regularized competition problems, where the space of solutions can be represented by reproducible game trees (rgt). rgt is a class of problems, which includes the following requirements: a. there are (a) interacting actors (players, competitors, etc.) performing (b) identified types of actions in the (c) specified types of situations; b. there are identified utilities, goals for each actor; c. actions for each actor are defined. [4] 1.4. we consider the implementation of the given in [1] models of mentals and systemic classifiers for rgt problems, particularly for the kernel rgt problem, chess, intensively studied since shannon’s pioneer work in 1949 [9]. in what follows, we present an implementation for the models of systemic classifiers induced by mentals, provide algorithms of matching for the given models, as well as the ways to experiment their adequacy for the presentation of chess mental systems. 2. implementation of systemic classifiers 2.1. natural languages are systemic and comprehensive by their coverage of msystems, but classifiers are not constructive and model only fuzzy ones, because they determine not the positives of msystems, but only ids of positives and ids of rels between them [1]. oop languages are covering mdoers as well but are more systemic with respect to algorithms since they involve attribute/parent/doing relationships corresponding to have/be/do ones in natural languages [10]. we follow the line of object-oriented implementation of systemic classifiers based on have/be/do relationships that in contrast with the previous versions of implementations [5, 6] aim to be comprehensive with respect to specification of mentals [1]. fig. 1. oop (left) and have/be/do (right) presentations. fig. 2. nuclears for chess. s. grigoryan, n. hakobyan and h. vrtanesyan 117 2.1.1 we start the implementation of systemic classifiers from nuclears, which are presented as oop classes containing only one rule. it is similar to primitive types in oop. for instance, in chess, we define the following as nuclears: coordinates, color, and figure type. nuclears are similar to nuclear abstracts defined in [5]. 2.1.2 to make the matching algorithms easier, we introduce basic classifiers, which we define as the minimal units to appear in the situation. in chess, fields of the chess boards, which contain or do not contain a figure are basic classifiers. these classifiers can only have nuclears as attributes. we create -have relations between minimal classifiers and their attributes. in case of parent existence (for simplicity, the current implementation allows having only one parent for each type of classifier) we create a -be relation between minimal classifiers and their parent. 2.1.3 more complex systemic classifiers, which include as attributes any other types of classifiers, we name compositeclassifiers. similar to minimal classifiers, they can have only one parent with a -be relation and -have relations with their attributes. they are similar to oop classes. as in oop classes, this type of classifiers can also be virtual. in oop this is achieved by having one undefined method, we achieved it by having undefined attributes. an attribute is called undefined if in its description there is at least one rule that is not specified (not specified rules have the value ‘?’). only for compositeclassifiers we implement negation. there are 2 types of negations: negation of general concept and negation for a specific instance. in the first case, compositeclassifier is generally negated if there is no such concept at all. for example, if we want to check whether there is a ‘check’ concept on the chess board, this is a general negation. in the second case, negation refers only to the exact instance, say, to the concept ‘there is no figure on e4 field’. compositeclassifiers are similar to complex abstracts defined in [5] but provide more flexible presentations. 2.1.4 we develop sets similar to arrays in oop and similar to sets described in [6] with restriction that they can be only continuous and represent sets of classifiers. for simplicity, we assume that sets consist only of compositeclassifiers (since they can only represent any other type of classifier), that are connected to the sets with -have relations. sets have 3 rules on their instances: 1. impose restrictions on lower and upper bounds; 2. define continuity with respect to attributes instances; 3. indicate the direction. 2.1.5 actions are similar to oop functions that describe some algorithm. they are connected with -do relation with an actor that makes an action (precondition). in contrast to the actions described in [5], we represent actions as a composition of the main precondition classifier, which is the actor, other precondition classifiers and rules describing the algorithm, how to change the precondition and provide an instance of classifier as an output. fig. 3. presentation of "empty line" set, which is virtual, where orange colored nodes are sets. fig. 4. presentation of actions, which perform similar to methods in oop. object-oriented modeling of matching to systemic classifiers 118 2.1.6 dynamic classifiers describe the transition from one state to another similar to [7]. for example, in chess it is impossible to describe the concept ‘king cannot escape’ with classifiers that describe static chess concepts. these types have 2 attributes – precondition and postcondition, both of them are compositeclassifiers and are connected with -have relations. dynamicclassifiers, the same as previously described classifiers, can have only one parent with be relation. 2.1.7 goals also have 2 attributes – precondition and postcondition with -have relations. they can also have one parent with -be relation. in addition, there is also an evaluating function, which is defined by rules and maximal depth of tree to evaluate. 2.1.8 plans include goals as attributes connected by -have relations, and the goals are ordered by priorities. 2.1.9 corresponders have a compositeclassifier as an attribute with a -have relation and actions with a -do relation. some real life concepts, like factories, cars implementation require both functionality and attributes existence. for instance, the concept ‘car’ should have 4 wheels (attribute) and must be able to be driven. 2.2 experimenting models for chess we will demonstrate the algorithm on the example of the chess concept ‘check’ (shown in figure 3). ‘check’ is a compositeclassifier. attributes of ‘check’ are ‘king’ and ‘field is under attack’. ‘king’ is a minclassifier and asfollows from its definition, there are nuclears as attributes – 2 ‘coordinates’, ‘figure type’ and ‘figure color’. ‘king’ has a parent ‘figure’. ‘field is under attack’ is a virtual compositeclassifier, where attributes are ‘field’, ‘attacker’, and it has children ‘field is under attack of pawn’, ‘field is under attack of bishop’ etc. ‘field’ is a minclassifier with all the described nuclears with attributes. ‘field’ is a parent for ‘figure’. ‘attacker’ is a ‘figure’. we’ll define one of ‘field is under attack’’s children, the others are defined equally. fig. 5 a goal from plan of "mate by rook". fig. 6. presentation of a machine tool, which has both actions to perform and attributes. fig. 7. systemic presentation of "check" chess concept in solver18. blue nodes present composites (including virtual "field is under attack" concept), red nodes present mindoers, green nodes present nuclears and blue nodes present actions. s. grigoryan, n. hakobyan and h. vrtanesyan 119 ‘field is under attack of pawn’ is a compositeclassifier with attributes ‘field’ and ‘attacker’, where ‘attacker’ is a ‘pawn’. 3. matching mental system to situations fig. 8. presentation of situations. we represent situations as instances of nuclears (for example, for chess it has a figure color: “white, black, no color”, figure type: “pawn, bishop, knight, rook, queen, king, no figure”, coordinates nuclear has two children – x and y both having 1-8 coordinates). nuclears instances are specified in groups with groupids. a set of nuclear instances with group ids describes the situation as described in [8], but, on the contrary, here we describe the side that will act in the given situation. 3.1 general matching 3.1.1 matching nuclears matching nuclears is performed by checking its rule, if the rule is satisfied it’s being matched, and an instance of active nuclear with the given value is sent forward through the system. 3.1.2 matching minimal classifiers matching of minclassifiers is done by matching its attributes. 3.1.3 matching composite classifiers matching composites is done by matching its attributes and local rules checking. if all the attributes are matched and rules are satisfied, then the composite is matched. virtual composites are created and matched as described in [8], usages, which are the uses of virtually specified composite classifiers, are matched by their virtual specification matching and additional rules checking, while specifications matched as regular composites trigger matching of their parent pure virtual composite classifiers. 3.1.4 matching sets is done by the checking of each instance of its composite element. if the given numbers of instances that satisfy all the rules defined in set are received, the set instance is considered as matched. basically the set activation is described in [8]. 3.1.5 matching actors and actions actions are activated with the following steps: 1. actor is activated and other precondition attributes are activated, 2: action is activated and can be applied to the situation and modify it. 3.1.6 matching dynamic classifiers matching dynamic classifiers basically described in [6], dynamics are matched when precondition composite is matched, and postcondition for all of the final situations is satisfied. 3.1.7 matching goals goals, as described above, have a precondition, which is composite, this shall be matched in order to consider the goal classifier. when the precondition is matched, the given depth of the tree is generated and final situations are evaluated with the given evaluation function. all the instances of goals that match the postcondition are considered as matched, and their evaluation value is assigned. 3.1.8 matching corresponders matching corresponders is performed by both checking its attributes matching and by checking its action, i.e., comparing the expected postcondition with the output for the given precondition. object-oriented modeling of matching to systemic classifiers 120 in addition, matching of any type of classifiers to situations can be done by matching of their children, i.e., any child matching triggers matching its parent (e.g. “king”, “queen” or any other type of figure matching means matching “figure” as well). general matching process is done through the system for the given situation. at the first step, it finds matching some classifiers, at the next step it continues doing the same until there are no more matched classifiers. after each step matching negated classifiers, dynamics and goals are performed, as described below: 3.2 matching one systemic classifier in some cases, it is just needed to find a certain concept on the situation, say, to see if there’s a check on the chess board or not. check classifier matching is searching exactly for it. in this case, all other classifiers, which are not relevant to the check classifier, are ignored in the system when processing the situation. 3.2.1 negated classifiers matching negated classifiers are composite ones, there are two types of negations in the system, the regular negation shows only a flag in the composite classifier, and when this flag is active and there is no such concept in the situation, then the appropriate classifier instance with all empty values is fired as it is the other case of negation is having a certain negated concept in the situation, then in contrast to the general negation, this needs the main attribute of negated composite to be activated first. then if after finishing the processing of the situation positive presentation of the concept is not matched, the negation instance with the given main attribute is considered as matched. e.g., if “e4 field is not attacked” concept is being searched, then the field with the value e4, which can be defined as a main attribute there is being checked if under attack. if no “field under attack”, where the field is “e4” found, then e4 field is not under attack, which is what we needed. 2.3 matching algorithm by the example of chess let’s discuss an example of matching on an example of ‘check’ described in the previous chapter. on field c3 white king is matched, on field c8 black rook is matched and on field h8 black king is matched. on fields c1-c7, a8, b8, d8-g8 matches ‘field is under attack of rook’, which triggers matching its parent ‘field is under attack’. rook on c8, the free line between rook and white king activates ‘field under attack of rook’, in its turn its triggers matching ‘field under attack’ the ‘field’ attribute of which is the same as ‘king’, consequently, the rule in chess ‘check’ compositeclassifier is also satisfied and ‘check’ is matched. fig. 9. situation where "check" concept is matched. s. grigoryan, n. hakobyan and h. vrtanesyan 121 4. conclusions in the following work models for constructive regularized mental systems were described and their implementations were discussed. 1. object-oriented models for systemic classifiers discussed in [1] were developed, particularly nuclears’, goals’, corresponders’ and other types of classifiers’ models were given, and their adequacy evidence is demonstrated for the example of chess concepts. 2. algorithms for matching situations to each type of systemic classifiers are provided, where there are two types of matching algorithms, a. matching situations to the whole system of classifiers, where all the matched classifiers are activated and b. matching situations to certain classifiers, where only instances of expected classifiers are searched. adequacy of matching algorithms is experimented for the example of chess concepts. acknowledgements authors express their deep gratitude to professor edward pogossian for supervising the work. references [1] e. pogossian, “towards adequate constructive models of mental systems”, international conference in computer sciences and information technologies, yerevan, armenia, pp. 6, 2017. [2] m. botvinnik, “computers in chess: solving in exact search problems”, springer series, in symbolic computation, with appendixes, springer-verlag, new york, 1984. [3] m. botvinnik, about solving approximate problems, (in russian), s. radio, moscow, 1979. [4] e. pogossian, v. vahradyan and a. grigoryan, “on competing agents consistent with expert knowledge”, lecture notes in computer science, ais-adm-07: the international workshop on autonomous intelligent systems agents and data mining, st. petersburg, russia, june 6-7, pp. 229-241, 2007. [5] k. khachatryan and s. grigoryan, “java programs for presentation and acquisition of meanings in ssrgt games”, proceedings of seua annual conference, yerevan, armenia, pp. 127-135, 2013. [6] s. grigoryan, research and development of algorithms and programs of knowledge acquisition and their effective application to resistance problems, pp. 111, yerevan, armenia, 2016. [7] s. grigoryan, “dynamic knowledge integration into hbd knowledge”, international conference in computer sciences and information technologies, yerevan, armenia, pp. 3, 2017. [8] k. khachatryan and s. grigoryan, “java programs for matching situations to the meanings of ssrgt games”, proceedings of seua annual conference, yerevan, armenia, pp. 135-141, 2013. [9] c. shannon, “programming a computer for playing chess”, philosophical magazine, vol. 41, no. 314, 1950. [10] e. pogossian, “on modeling cognition”, international conference in computer sciences and information technologies, yerevan, armenia, pp 194-198, 2011. submitted 11.09.2017, accepted 14.02.2018. object-oriented modeling of matching to systemic classifiers 122 սիստեմիկ դասակարգիչների համապատասխանեցման օբյեկտ-կողմնորոշված մոդելավորում ս. գրիգորյան, ն. հակոբյան և հ. վրթանեսյան ամփոփում աշխատանքում ներկայացնում ենք կառուցողական կանոնակարգված մտավոր համակարգերի, մտավորների (mentals) և սիստեմիկ դասակարգիչների օբյեկտկողմնորոշված մոդելների մի իրականացում, որոնք ներկայացված են [1]-֊ում, մշակված են ալգորիթմներ դրանք իրավիճակներին համապատասխանեցնելու համար։ մոդելների և ալգորիթմների համապատասխանությունը մենք փորձարկում ենք շախմատի համար, որը ներկայացնում է միջուկ կոմբինատոր խնդիրների մի դասի համար, որոնց լուծումների բազմությունը վերարտադրելի ծառ է (rgt): объектно-ориентированное моделирование соответствия системным классификаторам с. григорян, н. акобян и о. вртанесян аннотация в даной работе представлена версия оъектно-ориентированной реализации моделей конструктивно регуляризованных ментальных систем, mentals и системных классификаторов, представленных в [1], а также алгоритмов их сопоставления с ситуациями. адекватность моделей и алгоритмов мы экспериментурием для шахмат, представляющее собой ядро класса комбинаторных задач, где пространство решений представляет собой воспроизводимое дерево. d:\user\...\main.dvi mathematical problems of computer science 49, 26{34, 2018. on a p r oblem of wang concer ning the h amiltonicity of b ipar tite digr aphs s a m ve l k h . d a r b in ya n a n d is ka n d a r a . k a r a p e t ya n institute for informatics and automation problems of nas ra e-mail: amdarbin@ipia.sci.am, isko@ipia.sci.am abstract r. wang (discrete mathematics and theoretical computer science, vol. 19(3), 2017) proposed the following problem. problem. let d be a strongly connected balanced bipartite directed graph of order 2a ¸ 8. suppose that d(x) ¸ 2a ¡ k, d(y) ¸ a + k or d(y) ¸ 2a ¡ k, d(x) ¸ a + k for every pair of vertices fx; yg with a common out-neighbour, where 2 · k · a=2. is d hamiltonian? in this paper, we prove that if a digraph d satis¯es the conditions of this problem, then (i) d contains a cycle factor, (ii) for every vertex x 2 v (d) there exists a vertex y 2 v (d) such that x and y have a common out-neighbour. keywords: digraph, cycle, hamiltonian cycle, bipartite balanced digraph, perfect matching. 1 . in t r o d u c t io n in t h is p a p e r , we c o n s id e r ¯ n it e d ir e c t e d g r a p h s ( d ig r a p h s ) wit h o u t lo o p s a n d m u lt ip le a r c s . a d ig r a p h d is c a lle d h a m ilt o n ia n if it c o n t a in s a h a m ilt o n ia n c yc le , i.e ., a c yc le t h a t in c lu d e s e ve r y ve r t e x o f d. th e ve r t e x s e t a n d t h e a r c s e t o f a d ig r a p h d a r e d e n o t e d b y v ( d ) a n d a( d ) , r e s p e c t ive ly. th e o r d e r o f a d ig r a p h d is t h e n u m b e r o f it s ve r t ic e s . a c yc le fa c t o r in d is a c o lle c t io n o f ve r t e x-d is jo in t c yc le s c1; c2; : : : ; cl s u c h t h a t v ( c1 ) [v ( c2 ) [: : :[v ( cl ) = v ( d ) . a d ig r a p h d is b ip a r t it e if t h e r e e xis t s a p a r t it io n x, y o f v ( d ) in t o t wo p a r t it e s e t s s u c h t h a t e ve r y a r c o f d h a s it s e n d -ve r t ic e s in d i®e r e n t p a r t it e s e t s . it is c a lle d b a la n c e d if jxj = jy j. th e r e a r e a n u m b e r o f c o n d it io n s t h a t g u a r a n t e e t h a t a b ip a r t it e d ig r a p h is h a m ilt o n ia n ( s e e , e .g ., [1 ]-[1 1 ]) . l e t u s r e c a ll t h e fo llo win g d e g r e e c o n d it io n s t h a t g u a r a n t e e t h a t a b a la n c e d b ip a r t it e d ig r a p h is h a m ilt o n ia n . t heor em 1.1. ( a d a m u s , a d a m u s a n d y e o [8 ]) l et d be a balanced bipartite digraph of order 2 a, where a ¸ 2 . then d is hamiltonian provided one of the following holds: (a) d ( u) + d ( v ) ¸ 3 a + 1 for every pair of non-adjacent distinct vertices u and v of d; (b) d is strongly connected and d( u ) + d( v ) ¸ 3 a for every pair of non-adjacent distinct vertices u and v of d; 2 6 s. darbinyan and i. karapetyan 2 7 (c) the minimal degree of d is at least ( 3 a + 1 ) = 2 ; (d) d is strongly connected, and the minimal degree of d is at least 3 a=2 . ob s e r ve t h a t th e o r e m 1 .1 im p o s e s a d e g r e e c o n d it io n o n a ll p a ir s o f n o n -a d ja c e n t ve r t ic e s . in t h e fo llo win g t h e o r e m s a d e g r e e c o n d it io n r e qu ir e s o n ly fo r s o m e p a ir s o f n o n -a d ja c e n t ve r t ic e s . t heor em 1.2. ( j. a d a m u s [9 ]) l et d be a strongly connected balanced bipartite digraph of order 2 a ¸ 6 . if d( x ) +d( y ) ¸ 3 a for every pair of vertices x, y with a common out-neighbour or a common in-neighbour, then d is hamiltonian. n o t ic e t h a t th e o r e m 1 .2 im p r o ve s th e o r e m 1 .1 . s o m e s u ± c ie n t c o n d it io n s fo r t h e e xis t e n c e o f h a m ilt o n ia n c yc le s in a b ip a r t it e t o u r n a m e n t a r e d e s c r ib e d in t h e s u r ve y p a p e r [3 ] b y gu t in . a c h a r a c t e r iz a t io n fo r h a m ilt o n ic it y fo r s e m ic o m p le t e b ip a r t it e d ig r a p h s wa s o b t a in e d in d e p e n d e n t ly b y gu t in [2 ] a n d h äa g g kvis t a n d ma n o u s s a kis [4 ]. t heor em 1.3. ( w a n g [1 0 ]) l et d be a strongly connected balanced bipartite digraph of order 2 a, where a ¸ 1 . suppose that, for every pair of vertices fx; yg with a common outneighbour, either d ( x) ¸ 2 a ¡ 1 and d( y ) ¸ a + 1 or d( y ) ¸ 2 a ¡ 1 and d( x ) ¸ a + 1 . then d is hamiltonian. b e fo r e s t a t in g t h e n e xt t h e o r e m we n e e d t o d e ¯ n e a b a la n c e d b ip a r t it e d ig r a p h o f o r d e r e ig h t . e xample 1. l e t d ( 8 ) b e a b ip a r t it e d ig r a p h wit h p a r t it e s e t s x = fx0; x1; x2; x3g a n d y = fy0; y1; y2; y3g, a n d t h e a r c s e t a ( d ( 8 ) ) c o n t a in s e xa c t ly t h e fo llo win g a r c s : y0x1, y1x0, x2y3, x3y2 a n d a ll t h e a r c s o f t h e fo llo win g 2 -c yc le s : xi $ yi, i 2 [0 ; 3 ], y0 $ x2, y0 $ x3, y1 $ x2 a n d y1 $ x3. it is e a s y t o s e e t h a t d( x2 ) = d( x3 ) = d ( y0 ) = d( y1 ) = 7 a n d d( x0 ) = d( x1 ) = d ( y2 ) = d( y3 ) = 3 ; a n d t h e d o m in a t in g p a ir s in d ( 8 ) a r e : fy0; y1g, fy0; y2g,fy0; y3g,fy1; y2g, fy1; y3g, fx0; x2g, fx0; x3g, fx1; x2g, fx1; x3g a n d fx2; x3g. n o t e t h a t e ve r y d o m in a t in g p a ir s a t is ¯ e s t h e c o n d it io n b1. s in c e x0y0x3y2x2 y1x0 is a c yc le in d ( 8 ) , it is n o t d i± c u lt t o c h e c k t h a t d ( 8 ) is s t r o n g . ob s e r ve t h a t d ( 8 ) is n o t h a m ilt o n ia n . in d e e d , if c is a h a m ilt o n ia n c yc le in d ( 8 ) , t h e n c wo u ld c o n t a in t h e a r c s x1y1 a n d x0y0. th e r e fo r e , c wo u ld c o n t a in t h e p a t h x1y1x0y0 o r t h e p a t h x0y0x1y1, wh ic h is im p o s s ib le s in c e n ¡ ( x0 ) = n ¡ ( x1 ) = fy0; y1g. n o t ic e t h a t t h e d ig r a p h d ( 8 ) d o e s n o t s a t is fy t h e c o n d it io n s o f w a n g 's t h e o r e m . t heor em 1.4. ( d a r b in ya n [1 1 ]) l et d be a strongly connected balanced bipartite digraph of order 2 a ¸ 8 . suppose that maxfd( x ) ; d( y ) g ¸ 2 a ¡ 1 for every pair of vertices x, y with a common out-neighbour. then d is hamiltonian unless d is isomorphic to the digraph d ( 8 ) (for de¯nition of d ( 8 ) , see e xample 1). 2 8 on a problem of wang concerning the hamiltonicity of bipartite digraphs fo r a ¸ 4 th e o r e m 1 .4 im p r o ve s w a n g 's t h e o r e m . a d ig r a p h d is c a lle d p a n c yc lic if it c o n t a in s c yc le s o f e ve r y le n g t h k, 3 · k · jv ( d ) j. a b a la n c e d b ip a r t it e d ig r a p h o f o r d e r 2 a is e ve n p a n c yc lic if it c o n t a in s c yc le s o f e ve r y le n g t h 2 k, 2 · k · a. th e r e a r e va r io u s s u ± c ie n t c o n d it io n s fo r a d ig r a p h ( u n d ir e c t e d g r a p h ) t o b e h a m ilt o n ia n , t h e y a r e a ls o s u ± c ie n t fo r t h e d ig r a p h ( u n d ir e c t e d g r a p h ) t o b e p a n c yc lic . r e c e n t ly, t h e fo llo win g r e s u lt s we r e p r o ve d . t heor em 1.5. ( d a r b in ya n [1 2 ]) l et d be a strongly connected balanced bipartite digraph of order 2 a ¸ 8 other than a directed cycle of length 2 a. if maxfd ( x ) ; d( y ) g ¸ 2 a ¡ 1 for every dominating pair of vertices fx; yg, then either d contains cycles of all even lengths less than or equal to 2 a or d is isomorphic to the digraph d ( 8 ) . t heor em 1.6. ( me s z ka [1 3 ]) l et d be a balanced bipartite digraph of order 2 a ¸ 4 with partite sets x and y . if d ( x) + d ( y ) ¸ 3 a + 1 for every pair of distinct vertices fx; yg either both in x or both in y , then d contains cycles of all even lengths less than or equal to 2 a. t heor em 1.7. ( d a r b in ya n [1 4 ]) l et d be a strongly connected balanced bipartite digraph of order 2 a ¸ 6 with partite sets x and y . if d( x ) + d( y ) ¸ 3 a for every pair of distinct vertices fx; yg either both in x or both in y , then d contains cycles of all even lengths less than or equal to 2 a. t heor em 1.8. ( a d a m u s [1 5 ]) l et d be a strongly connected balanced bipartite digraph of order 2 a ¸ 6 . if d ( x) + d( y ) ¸ 3 a for every pair of distinct vertices fx; yg with a common in-neighbour or a common out-neighbour, then d contains cycles of all even lengths less than or equal to 2 a or a directed cycle of length 2 a. de¯nition 1. l et d be a balanced bipartite digraph of order 2 a, where a ¸ 2 . f or any integer k ¸ 0 , we will say that d satis¯es the condition bk when d( x ) ¸ 2 a ¡ k; d( y ) ¸ a + k or d( x ) ¸ a + k; d( y ) ¸ 2 a ¡ k for any dominating pair of vertices fx; yg in d. in [1 0 ], w a n g p r o p o s e d t h e fo llo win g p r o b le m . p r oblem ( w a n g [1 0 ]) . l et d be a strongly connected balanced bipartite digraph of order 2 a ¸ 8 satisfying the condition bk with 2 · k · a= 2 . is d hamiltonian? b e fo r e s t a t in g t h e n e xt t h e o r e m s we n e e d t o d e ¯ n e a d ig r a p h o f o r d e r t e n . e xample 2. l e t d ( 1 0 ) b e a b ip a r t it e d ig r a p h wit h p a r t it e s e t s x = fx0; x1; x2; x3; x4g a n d y = fy0; y1; y2; y3; y4g s a t is fyin g t h e fo llo win g c o n d it io n s : th e in d u c e d s u b d ig r a p h hfx1; x2; x3; y0; y4gi is a c o m p le t e b ip a r t it e d ig r a p h wit h p a r t it e s e t s fx1; x2; x3g a n d fy0; y4g; fx1; x2; x3g ! fy1; y2; y3g; x4 $ y4; x0 $ y0, x3 $ y1 a n d xi $ yi+1 fo r a ll i 2 [1 ; 3 ]. d ( 1 0 ) c o n t a in s n o o t h e r a r c s . s. darbinyan and i. karapetyan 2 9 it is e a s y t o c h e c k t h a t t h e d ig r a p h d ( 1 0 ) is s t r o n g ly c o n n e c t e d a n d s a t is ¯ e s t h e c o n d it io n b0, b u t t h e u n d e r lyin g u n d ir e c t e d g r a p h o f d ( 1 0 ) is n o t 2 -c o n n e c t e d , a n d d ( 1 0 ) h a s n o c yc le o f le n g t h 8 . ( it fo llo ws fr o m t h e fa c t s t h a t d( x0 ) = d ( x4 ) = 2 , a n d x0 ( x4 ) is o n 2 -c yc le ) . s in c e x1y1x3y3x2y2x1 is a c yc le o f le n g t h 6 , x0 $ y0 a n d x4 $ y4, it is n o t d i± c u lt t o c h e c k t h a t a n y d ig r a p h o b t a in e d fr o m d ( 1 0 ) b y a d d in g a n e w a r c t h e o n e e n d -ve r t e x o f wh ic h is x0 o r x4 c o n t a in s a c yc le o f le n g t h e ig h t . mo r e o ve r , if t o a( d ) we a d d s o m e n e w a r c s o f t h e t yp e yixj , wh e r e i 2 [1 ; 3 ] a n d j 2 [1 ; 3 ], t h e n we a lwa ys o b t a in a d ig r a p h , wh ic h d o e s n o t s a t is fy t h e c o n d it io n b0. t heor em 1.9. ( [1 6 ], [1 7 ]) . l et d be a balanced bipartite digraph of order 2 a ¸ 1 0 other than a directed cycle of length 2 a. suppose that d satis¯es the condition b0, i.e., maxfd( x ) ; d( y ) g ¸ 2 a ¡ 2 for every dominating pair of vertices fx; yg. then d contains cycles of all lengths 2 ; 4 ; : : : ; 2 a ¡ 2 unless d is isomorphic to the digraph d ( 1 0 ) . cle a r ly, t h e e xis t e n c e o f a c yc le fa c t o r is a n e c e s s a r y c o n d it io n fo r a d ig r a p h t o b e h a m ilt o n ia n . in t h is n o t e we p r o ve t h e fo llo win g t h e o r e m . t heor em 1.10. l et d be a strongly connected balanced bipartite digraph of order 2 a ¸ 8 satisfying the condition bk with 2 · k · a= 2 . then d contains a cycle factor. 2 . te r m in o lo g y a n d n o t a t io n te r m in o lo g y a n d n o t a t io n n o t d e s c r ib e d b e lo w fo llo w [1 ]. if xy 2 a ( d ) , t h e n we s a y t h a t x d o m in a t e s y o r y is a n o u t -n e ig h b o u r o f x a n d x is a n in -n e ig h b o u r o f y. l e t x; y b e d is t in c t ve r t ic e s in a d ig r a p h d. th e p a ir fx; yg is c a lle d d o m in a t in g if t h e r e is a ve r t e x z in d s u c h t h a t xz 2 a( d ) a n d yz 2 a ( d ) . in t h is c a s e we s a y t h a t x is a p a r t n e r o f y a n d y is a p a r t n e r o f x. if x 2 v ( d ) a n d a = fxg we s o m e t im e s will wr it e x in s t e a d o f fxg. a ! b m e a n s t h a t e ve r y ve r t e x o f a d o m in a t e s e ve r y ve r t e x o f b. th e n o t a t io n x $ y d e n o t e s t h a t xy 2 a ( d ) a n d yx 2 a ( d ) . l e t n + ( x ) , n¡ ( x) d e n o t e t h e s e t o f o u t -n e ig h b o u r s , r e s p e c t ive ly t h e s e t o f in -n e ig h b o u r s o f a ve r t e x x in a d ig r a p h d. if a µ v ( d ) , t h e n n + ( x; a ) = a \ n + ( x ) , n¡ ( x; a ) = a\n ¡ ( x ) a n d n + ( a) = [x2an + ( x) , n¡ ( a) = [x2an¡ ( x) . th e o u t -d e g r e e o f x is d+ ( x ) = jn + ( x) j a n d d¡ ( x ) = jn¡ ( x) j is t h e in -d e g r e e o f x. s im ila r ly, d+ ( x; a) = jn + ( x; a) j a n d d¡ ( x; a ) = jn¡ ( x; a ) j. th e d e g r e e o f t h e ve r t e x x in d is d e ¯ n e d a s d( x ) = d+ ( x ) + d¡ ( x ) ( s im ila r ly, d( x; a) = d+ ( x; a ) + d¡ ( x; a ) ) . th e p a t h ( r e s p e c t ive ly, t h e c yc le ) c o n s is t in g o f t h e d is t in c t ve r t ic e s x1; x2; : : : ; xm ( m ¸ 2 ) a n d t h e a r c s xixi+1, i 2 [1 ; m ¡ 1 ] ( r e s p e c t ive ly, xixi+1, i 2 [1 ; m ¡ 1 ], a n d xmx1 ) , is d e n o t e d b y x1x2 ¢ ¢ ¢ xm ( r e s p e c t ive ly, x1x2 ¢ ¢ ¢ xmx1 ) . w e s a y t h a t x1x2 ¢ ¢ ¢ xm is a p a t h fr o m x1 t o xm o r is a n ( x1; xm ) -p a t h . give n a ve r t e x x o f a d ir e c t e d p a t h p o r a d ir e c t e d c yc le c, we d e n o t e b y x+ ( r e s p e c t ive ly, b y x¡ ) t h e s u c c e s s o r ( r e s p e c t ive ly, t h e p r e d e c e s s o r ) o f x ( o n p o r c ) , a n d in c a s e o f a m b ig u it y, we p r e c is e p o r c a s a s u b s c r ip t ( t h a t is x+p . . . ) . a d ig r a p h d is s t r o n g ly c o n n e c t e d ( o r , ju s t , s t r o n g ) if t h e r e e xis t s a n ( x; y ) -p a t h in d fo r e ve r y o r d e r e d p a ir o f d is t in c t ve r t ic e s x; y o f d. two d is t in c t ve r t ic e s x a n d y a r e a d ja c e n t if xy 2 a( d ) o r yx 2 a ( d ) ( o r b o t h ) . l e t h b e a n o n -t r ivia l p r o p e r s u b s e t o f ve r t ic e s o f a d ig r a p h d. a n ( x; y ) -p a t h p is a n h-b yp a s s if jv ( p ) j ¸ 3 , x 6= y a n d v ( p ) \ h = fx; yg. 3 0 on a problem of wang concerning the hamiltonicity of bipartite digraphs l e t d b e a b a la n c e d b ip a r t it e d ig r a p h wit h p a r t it e s e t s x a n d y . a m a t c h in g fr o m x t o y is a n in d e p e n d e n t s e t o f a r c s wit h o r ig in in x a n d t e r m in u s in y . ( a s e t o f a r c s wit h n o c o m m o n e n d -ve r t ic e s is c a lle d in d e p e n d e n t ) . if d is b a la n c e d , o n e s a ys t h a t s u c h a m a t c h in g is p e r fe c t if it c o n s is t s o f p r e c is e ly jxj a r c s . th e u n d e r lyin g u n d ir e c t e d g r a p h o f a d ig r a p h d is d e n o t e d b y ug( d ) , it c o n t a in s a n e d g e xy if xy 2 a ( d ) o r yx 2 a( d ) ( o r b o t h ) . 3 . ma in r e s u lt th e o r e m 1 .1 0 is t h e m a in r e s u lt o f t h is p a p e r . p r oof of theor em 1.10. l e t d b e a d ig r a p h s a t is fyin g t h e c o n d it io n s o f t h e t h e o r e m . or e in [1 8 ] ( s e c t io n 8 .6 ) h a s s h o wn t h a t a b a la n c e d b ip a r t it e d ig r a p h d wit h p a r t it e s e t s x a n d y h a s a c yc le fa c t o r if a n d o n ly if d c o n t a in s a p e r fe c t m a t c h in g fr o m x t o y a n d a p e r fe c t m a t c h in g fr o m y t o x. th e r e fo r e , b y t h e we ll-kn o wn k äo n in g -h a ll t h e o r e m ( s e e , e .g ., [1 9 ]) t o s h o w t h a t d c o n t a in s a p e r fe c t m a t c h in g fr o m x t o y , it s u ± c e s t o s h o w t h a t jn + ( s ) j ¸ jsj fo r e ve r y s e t s µ x. l e t s µ x. if jsj = 1 o r jsj = a, t h e n jn + ( s ) j ¸ jsj s in c e d is s t r o n g ly c o n n e c t e d . a s s u m e t h a t 2 · jsj · a ¡ 1 . w e c la im t h a t jn + ( s ) j ¸ jsj. s u p p o s e t h a t t h is is n o t t h e c a s e , i.e ., jn + ( s ) j · jsj ¡ 1 · a ¡ 2 . fr o m t h is a n d s t r o n g ly c o n n e c t e d n e s s o f d it fo llo ws t h a t t h e r e a r e t wo ve r t ic e s x; y 2 s a n d a ve r t e x z 2 n + ( s ) s u c h t h a t fx; yg ! z, i.e ., fx; yg is a d o m in a t in g p a ir . th e r e fo r e , b y c o n d it io n bk, d ( x) ¸ 2 a ¡ k a n d d ( y ) ¸ a + k o r d( x ) ¸ a + k a n d d( y ) ¸ 2 a ¡ k. w it h o u t lo s s o f g e n e r a lit y, we a s s u m e t h a t d( x ) ¸ 2 a ¡ k a n d d( y ) ¸ a + k. th e n 2 a ¡ k · d ( x) · 2 jn + ( s ) j + a ¡ jn + ( s ) j = a + jn + ( s ) j: th e r e fo r e , jn + ( s ) j ¸ a ¡ k a n d jsj ¸ a ¡ k + 1 . p r oposition 1. l e t fu; vg b e a d o m in a t in g p a ir o f ve r t ic e s o f d. th e n fr o m c o n d it io n bk a n d 2 · k · a=2 it fo llo ws t h a t d( u ) ¸ a + k a n d d( vu ) ¸ a + k, i.e ., if a ve r t e x z h a s a p a r t n e r in d , t h e n d ( z ) ¸ a + k. w e c la im t h a t e a c h ve r t e x in y nn + ( s ) h a s n o p a r t n e r in d. in d e e d , le t u b e a n a r b it r a r y ve r t e x in y n n + ( s ) . s in c e jsj ¸ a ¡ k + 1 , we h a ve d( u ) · jsj + 2 ( a ¡ jsj ) = 2 a ¡ jsj · a + k ¡ 1 ; wh ic h c o n t r a d ic t s p r o p o s it io n 1 . th is m e a n s t h a t u h a s n o p a r t n e r in d. w it h o u t lo s s o f g e n e r a lit y, a s s u m e t h a t s = fx1; x2; : : : ; xsg a n d n + ( s ) = fy1; y2; : : : ; ytg: r e c a ll t h a t e ve r y ve r t e x yi wit h t+1 · i · a h a s n o p a r t n e r in d. n o t e t h a t s ¸ t+1 , a¡s · a¡t¡ 1 a n d t h e r e is n o a r c fr o m a ve r t e x o f fx1; x2; : : : ; xsg t o a ve r t e x o f fyt+1; yt+2; : : : ; yag. fr o m t h is a n d s t r o n g ly c o n n e c t e d n e s s o f d it fo llo ws t h a t t h e r e is a ve r t e x xi1 s u c h t h a t yt+1xi1 2 a( d ) . s in c e yt+1 h a s n o p a r t n e r , it fo llo ws t h a t d¡ ( xi1; y nfyt+1g ) = 0 . th e r e fo r e , d( xi1 ) · a + 1 · a + k ¡ 1 s in c e k ¸ 2 . b y p r o p o s it io n 1 , t h is m e a n s t h a t t h e ve r t e x s. darbinyan and i. karapetyan 3 1 xi1 a ls o h a s n o p a r t n e r . s in c e d is s t r o n g ly c o n n e c t e d , t h e r e is a ve r t e x yi2 2 y s u c h t h a t xi1yi2 2 a ( d ) . th e n d¡ ( yi2; x n fxi1g ) = 0 , b e c a u s e o f t h e fa c t t h a t xi1 h a s n o p a r t n e r . th e r e fo r e , d( yi2 ) · a + 1 a n d h e n c e , yi2 a ls o h a s n o p a r t n e r . co n t in u in g t h is p r o c e s s , a s lo n g a s p o s s ib le , a s a r e s u lt we o b t a in a p a t h p = yt+1xi1yi2 xi2 : : : xil yil o r a c yc le c = yt+1xi1yi2xi2 : : : xil yt+1. it is n o t d i± c u lt t o s e e t h a t a ll t h e ve r t ic e s o f t h is p a t h ( c yc le ) h a ve n o p a r t n e r s . if t h e fo r m e r c a s e h o ld s , t h e n x1 is in p , wh ic h is a c o n t r a d ic t io n s in c e x1 h a s a p a r t n e r ( n a m e ly x2 ) . if t h e s e c o n d c a s e h o ld s , t h e n , s in c e e ve r y ve r t e x o f c h a s n o p a r t n e r in d, it fo llo ws t h a t t h e r e is n o a r c fr o m a ve r t e x o f v ( d ) n v ( c ) t o a ve r t e x o f v ( c ) , wh ic h c o n t r a d ic t s t h a t d is s t r o n g ly c o n n e c t e d . th is c o m p le t e s t h e p r o o f o f t h e e xis t e n c e o f a p e r fe c t m a t c h in g fr o m x t o y . th e p r o o f fo r a p e r fe c t m a t c h in g in t h e o p p o s it e d ir e c t io n is a n a lo g o u s . th is c o m p le t e s t h e p r o o f o f t h e t h e o r e m . 4 . r e m a r ks n o w u s in g th e o r e m 1 .1 0 , we p r o ve t h e fo llo win g r e s u lt s ( l e m m a s 3 .1 -3 .3 ) . lemma 3.1. l et d be a strongly connected balanced bipartite digraph of order 2 a ¸ 8 with partite sets x and y satisfying the condition bk, 2 · k · a= 2 . if d is not hamiltonian, then every vertex u 2 v ( d ) has a partner in d. p r oof of lemma 3.1: l e t d b e a d ig r a p h s a t is fyin g t h e c o n d it io n s o f t h e le m m a . fo r a p r o o f b y c o n t r a d ic t io n , s u p p o s e t h a t t h e r e is a ve r t e x x in d, wh ic h h a s n o p a r t n e r . b y th e o r e m 1 .1 0 , d h a s a c yc le fa c t o r , s a y c1, c2, ... , cl. th e n l ¸ 2 s in c e d is n o t h a m ilt o n ia n . w it h o u t lo s s o f g e n e r a lit y, we a s s u m e t h a t x 2 v ( c1 ) . it fo llo ws t h a t d¡ ( x+c1 ) = 1 . th e r e fo r e , d( x+c1 ) · a + 1 . b y p r o p o s it io n 1 , t h is m e a n s t h a t t h e ve r t e x x + c1 a ls o h a s n o p a r t n e r . s im ila r ly, we o b t a in t h a t d( x++c1 ) · a + 1 ( wh e r e x ++ c1 d e n o t e s t h e s u c c e s s o r o f x+c1 o n c1 ) a n d h e n c e , x ++ c1 a ls o h a s n o p a r t n e r in d. co n t in u in g t h is p r o c e s s , we c o n c lu d e t h a t e ve r y ve r t e x o f c1 h a s n o p a r t n e r in d. th is im p lie s t h a t t h e r e is n o a r c fr o m a ve r t e x o f a ( v ( d ) n v ( c1 ) t o a ve r t e x o f v ( c1 ) ) , wh ic h c o n t r a d ic t s t h a t d is s t r o n g ly c o n n e c t e d . th e le m m a is p r o ve d . lemma 3.2. l et d be a strongly connected balanced bipartite digraph of order 2 a ¸ 8 with partite sets x and y satisfying the condition bk, 2 · k · a=2 . if d is not a cycle, then d contains a non-hamiltonian cycle of length at least four. p r oof of lemma 3.2: l e t d b e a d ig r a p h s a t is fyin g t h e c o n d it io n s o f t h e le m m a . fo r a p r o o f b y c o n t r a d ic t io n , s u p p o s e t h a t d c o n t a in s a n o n -h a m ilt o n ia n c yc le o f le n g t h a t le a s t fo u r . if d is h a m ilt o n ia n , t h e n it is n o t d i± c u lt t o s h o w t h a t d c o n t a in s a n o n -h a m ilt o n ia n c yc le o f le n g t h a t le a s t 4 . s o we s u p p o s e , fr o m n o w o n , t h a t d is n o t h a m ilt o n ia n a n d c o n t a in s n o c yc le o f le n g t h a t le a s t 4 . b y th e o r e m 1 .1 0 , d c o n t a in s a c yc le fa c t o r . l e t c1; c2; : : : ; ct b e a m in im a l c yc le fa c t o r o f d ( i.e ., t is a s s m a ll a s p o s s ib le ) . th e n t h e le n g t h o f e ve r y ci is e qu a l t o t wo a n d t = a. l e t ci = xiyixi, wh e r e xi 2 x a n d yi 2 y . b y l e m m a 3 .1 , e ve r y ve r t e x o f d h a s a p a r t n e r . th is m e a n s t h a t fo r e ve r y ve r t e x x 2 v ( d ) , d( x ) ¸ a + k a n d d¡ ( x) ¸ k ¸ 2 , d+ ( x ) ¸ k ¸ 2 . w it h o u t lo s s o f g e n e r a lit y, we a s s u m e t h a t fx1; xjg wit h j 6= 1 is a d o m in a t in g p a ir a n d d( x1 ) ¸ 2 a ¡ k. 3 2 on a problem of wang concerning the hamiltonicity of bipartite digraphs l e t z b e t h e s u b s e t o f y wit h t h e m a xim u m c a r d in a lit y, s u c h t h a t e ve r y ve r t e x o f z t o g e t h e r wit h x1 fo r m s a c yc le o f le n g t h t wo . w it h o u t lo s s o f g e n e r a lit y, we a s s u m e t h a t z = fy1; y2; : : : ; ylg. th e n 2 a ¡ k · d ( x1 ) · 2 l + a ¡ l = a + l. h e n c e , l ¸ a ¡ k. s in c e d c o n t a in s n o c yc le o f le n g t h fo u r , it fo llo ws t h a t t h e ve r t ic e s y1 a n d xi, 2 · i · l, a r e n o t a d ja c e n t . th e r e fo r e , a + k · d( y1 ) · 2 a ¡ 2 l + 2 · 2 k + 2 ; i.e ., k ¸ a ¡ 2 . s in c e a= 2 ¸ k ¸ a ¡ 2 , we h a ve a ¸ 2 k ¸ 2 a ¡ 4 , a · 4 . if a = 4 , t h e n k = a=2 = 2 a n d l = a ¡ k = 2 . it is e a s y t o s e e t h a t d ( x1 ) = d ( y1 ) = 6 , t h e ve r t ic e s y1 a n d xi, 3 · i · 4 , fo r m a c yc le o f le n g t h t wo a n d x1y3 2 a ( d ) o r y3x1 2 a ( d ) . n o w it e a s y t o s e e t h a t d c o n t a in s a c yc le o f le n g t h fo u r . l e m m a 3 .2 is p r o ve d . fo r t h e n e xt le m m a we n e e d t h e fo llo win g le m m a d u e t o b o n d y. b ypass lemma ( l e m m a 3 .1 7 , b o n d y [2 0 ]) . l et d be a strong non-separable (i.e., ug ( d ) is 2-connected) digraph, and let h be a non-trivial proper subdigraph of d. then d contains an h-bypass. remar k: on e c a n p r o ve b yp a s s l e m m a u s in g t h e p r o o f o f th e o r e m 5 .4 .2 [1 ]. n o w we will p r o ve t h e fo llo win g le m m a . lemma 3.3. l et d be a strongly connected balanced bipartite digraph of order 2 a ¸ 8 with partite sets x and y satisfying the condition bk, where 2 · k · a=2 . then the following statements hold: (i) the underlying undirected graph ug ( d ) is 2-connected; (ii) if c is a cycle of length m, 2 · m · 2 a ¡ 2 , then d contains a c-bypass. p r oof of lemma 3.3. ( i) s u p p o s e , o n t h e c o n t r a r y, t h a t d is a s t r o n g ly c o n n e c t e d b a la n c e d b ip a r t it e d ig r a p h o f o r d e r 2 a ¸ 8 wit h p a r t it e s e t s x a n d y s a t is ¯ e s t h e c o n d it io n bk b u t ug ( d ) is n o t 2 -c o n n e c t e d . th e n v ( d ) = e [ f [ fug, wh e r e e a n d f a r e n o n -e m p t y s u b s e t s , e \ f = ;, u =2 e [ f a n d t h e r e is n o a r c b e t we e n e a n d f . s in c e d is s t r o n g , it fo llo ws t h a t t h e r e a r e ve r t ic e s x 2 e a n d y 2 f s u c h t h a t fx; yg ! u, i.e ., fx; yg is a d o m in a t in g p a ir . w it h o u t lo s s o f g e n e r a lit y, we a s s u m e t h a t x; y 2 x. th e n u 2 y . b y c o n d it io n bk, it is e a s y t o s e e t h a t 3 a · d( x ) + d( y ) · 4 + 2 je \ y j + 2 jf \ y j · 2 a + 2 ; wh ic h is a c o n t r a d ic t io n . th is p r o ve s t h a t ug( d ) is 2 -c o n n e c t e d . 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[1 6 ] s . k h . d a r b in ya n , \ on p r e -h a m ilt o n ia n c yc le s in b a la n c e d b ip a r t it e d ig r a p h s " , m athematical problems of computer science, vo l. 4 6 , p p . 7 -1 7 , 2 0 1 4 . [1 7 ] s . k h . d a r b in ya n a n d i. a . k a r a p e t ya n , \ a s u ± c ie n t c o n d it io n fo r p r e -h a m ilt o n ia n cyc le s in b ip a r t it e d ig r a p h s " , csit 2017 r evised selected p apers,ie e e conference proceeding,, p p . 1 0 1 -1 0 9 , d oi:1 0 .1 1 0 9 / cs itte c h n o l.2 0 1 7 .8 3 1 2 1 5 0 . [1 8 ] o. or e , \ th e o r y o f g r a p h s " , a m e r ic a n ma t h e m a t ic a l s o c ie t y, p r o vid e n c e , r .i., a m e r ic a n ma t h e m a t ic a l s o c ie t y co llo qu iu m p u b lic a t io n s , vo l. 3 8 , 1 9 6 2 . [1 9 ] c. b e r g e , graphs and hypergraphs, n o r t h -h o lla n d , a m s t e r d a m , 1 9 7 3 . [2 0 ] j. a . b o n d y, b asic graph theory: p aths and circuits, in h a n d b o o k o f c o m b in a t o r ic s 1-2, e ls e vie r , a m s t e r d a m , 1 9 9 5 . submitted 20.09.2017, accepted 16.01.2018. 3 4 on a problem of wang concerning the hamiltonicity of bipartite digraphs îáùýáñáßí³í »ñïù³ëýû³ ·ñ³ýç ñ³ùçéïáýû³ýáõãû³ý í»ñ³µ»ñû³é ì³ý·ç ëý¹ñç ù³ëçý ê. ¸³ñµçýû³ý ¨ æ. î³ñ³å»ïû³ý ²ù÷á÷áõù ì³ý·á (discrete mathematics and theoretical computer science, vol. 19(3) 2017) ³é³ç³ñï»é ¿ ñ»ï¨û³é ëý¹çññ: êý¹çñ: ¸çóáõù d-ý áõå»õ ï³å³ïóí³í 2 a-·³·³ã³ýç 2 a ¸ 8 ïáõýáñáßí³í »ñïù³ëýû³ ñ³í³ë³ñ³ïßéí³í ·ñ³ý ¿, áñáõù ·³·³ãý»ñç ó³ýï³ó³í fx; yg ñ³õãáõ ½áõû·ç ñ³ù³ñ ï»õç áõý»ý ñ»ï¨û³é ³ýñ³í³ë³ñáõãûáõýý»ñá. d( x ) ¸ 2 a¡k, ¨ d( y ) ¸ a+k ï³ù d( x ) ¸ a + k ¨ d( y ) ¸ 2 a ¡ k, áñï»õ 2 · k · a= 2 : ²ñ¹ûáù d-ý ñ³ùçéïáýû³ý ¿: ü»ñï³ ³ßë³ï³ýùáõù ³å³óáõóí³í ¿, áñ »ã» d ·ñ³ýá µ³í³ñ³ñáõù ¿ ì³ý·ç ëý¹ñç å³ûù³ýý»ñçý, ³å³ (i) d ·ñ³ýá å³ñáõý³ïáõù ¿ óçïé-ý³ïïáñ ¨ ³éýí³½ý ãáñë »ñï³ñáõãû³ùµ áã-ñ³ùçéïáýû³ý óçïé, (ii) d ·ñ³ýç ó³ýï³ó³í x ·³·³ãç ñ³ù³ñ ·áûáõãûáõý áõýç ³ûýåçëç ùç y ·³·³ã, áñ fx; yg á ñ³õãáõ ½áõû· ¿: î çàäà÷å âàíãà î ãàìèëüòîíîâîñòè äâóäîëüíûõ îðãðàôîâ ñ. äàðáèíÿí è è. êàðàïåòÿí àííîòàöèÿ âàíã (discrete mathematics and theoretical computer science, vol. 19(3) 2017) ïðåäëîæèëà ñëåäóþùóþ çàäà÷ó. çàäà÷à: ïóñòü d 2 a -âåðøèííûé 2 a ¸ 8 ñèëüíîñâÿçíûé ñáàëàíñèðîâàííûé äâóäîëüíûé îðãðàô, â êîòîðîì äëÿ ëþáîé ïàðû äîìèíèðóþùèõ âåðøèí fx; yg, d( x ) ¸ 2 a ¡ k, d( y ) ¸ a + k èëè d( y ) ¸ a + k, d ( y ) ¸ 2 a ¡ k, ãäå 2 · k · a=2 . ßâëÿåòñÿ ëè d ãàìèëüòîíîâûì? â íàñòîÿùåé ðàáîòå äîêàçàíî, ÷òî åñëè îðãðàô d óäîâëåòâîðÿåò óñëîâèÿì çàäà÷è âàíãà, òî (i) d ñîäåðæèò öèêë-ôàêòîð è íå-ãàìèëüòîíîâûé öèêë äëèíû ïî êðàéíåé ìåðå 4, (ii) äëÿ êàæäîé âåðøèíû x ñóùåñòâóåò òàêàÿ âåðøèíà y, ÷òî fx; yg ÿâëÿåòñÿ äîìèíèðóþùèì ïàðîì.