Volume 13, Issue 3

We analyse the state of the art of a rather new field of automata theory—the study of behaviour of automata in labyrinths; more than a hundred publications devoted to this topic have been published. We consider key notions, problems, achievements, methods of problem solutions, and open problems in an important direction of this study, the behaviour of independent systems of automata in labyrinths. In a series of cases, we give base assertions in a more strong form and give a more general presentation than the authors of the corresponding papers do. New results are also contained in this survey.

In this study, we construct the mathematical model of a covert channel of agent interaction in the wide area network and in a closed segment of a local area network. The channel transmits information through an IPsec-based protective device with the use of encapsulation and enciphering of packets. We analyse the asymptotic behaviour of the time needed to learn the agent the language of information transmission. We prove that if the number n of nodes in an arbitrary segment and the number m of segments grow without limits, then the learning time is O ( m 2 n ln n ) under some conditions.

We give examples where we realise the idea to use a non-commutative operation in the schemes of public distribution of keys suggested by V. M. Sidelnikov.

For the matrix A T A , where A is the matrix of restrictions of a multi-index transportation problem, we give the characteristic polynomial and the base of eigenvectors. We discuss eventual applications of the results obtained.

In this paper we study the maximally non-Hamiltonian graphs (MNH graphs), that is, the graphs whose property to be non-Hamiltonian ceases to exist as soon as an edge is added. We give rules how to construct MNH graphs which make it possible, starting from MNH graphs with some properties, to get MNH graphs of greater order.

In each finite simple sporadic group, excepting the Baby Monster group B , the Monster group M , the McLaughlin group McL and Mathieu groups M 11 , M 22 , M 23 , three generating involutions, two of which commute, are found. If G is one of the groups M 12 , M 24 , HS , J 1 , J 2 , J 3 , then we give pairs of numbers p, q, p ≤ q , such that p = | ik |, q = |jk | for some involutions i , j , k with condition | ij | = 2 generating the group G . The triples of involutions mentioned above are found with the use of the system of computer algebra GAP. Recall that any two involutions of the triple of involutions generating either McL , or M 11 , or M 22 , or M 23 do not commute.

We estimate the number r ( n,m ) of functions of n -valued logic in m + 1 variables which are local transition functions of reversible homogeneous structures with arbitrary fixed neighbourhood pattern consisting of m vectors. It follows from the results obtained in the paper that if n → ∞, then ln r ( n,m ) ∼ n m +1 ln n uniformly in m .

We consider the class of all partitions of a set of n elements into N blocks. Provided that the uniform distribution is given on this class and n, N → ∞, we describe the asymptotic behaviour of the mathematical expectation and variance and prove Poisson and local normal limit theorems for the random variable equal to the number of blocks of a given size in a partition chosen at random. We find asymptotic expansions of the number of partitions of a set of n elements into N blocks with exactly k = k ( n, N ) blocks of a given size.

We consider a tuple of states of an ( s – 1)-order Markov chain whose transition probabilities depend on a small part of s – 1 preceding states. We obtain limit distributions of certain χ 2 -statistics X and Y based on frequencies of tuples of states of the Markov chain. For the statistic X , frequencies of tuples of only those states are used on which the transition probabilities depend, and for the statistic Y , frequencies of s -tuples without gaps. The statistical test with statistic X which distinguishes the hypotheses H 1 (a high-order Markov chain) and H 0 (an independent equiprobable sequence) appears to be more powerful than the test with statistic Y . The statistic Z of the Neyman–Pearson test, as well as X , depends only on frequencies of tuples with gaps. The statistics X and Y are calculated without use of distribution parameters under the hypothesis H 1 , and their probabilities of errors of the first and second kinds depend only on the non-centrality parameter, which is a function of transition probabilities. Thus, for these statistics the hypothesis H 1 can be considered as composite.