Volume 25, Issue 6

We consider a method that modifies regular expressions in order to solve the “exponential explosion” problem on the number of states of the finite automaton that recognizes a set of regular languages defined by the union of regular expressions of the form .∗ R 1 .∗ R 2 .∗, where R 1 and R 2 are arbitrary regular expressions. We estimate the growth functions of regular languages from a subclass of the described class of regular languages and propose a method for estimation of relative growth of the number of words for the modification of a language defined by a pair of regular expressions.We also analyse practical eficiency of the proposed modification method and estimation method for the case of regular expressions from the system Snort.

Let (Z* n ) be a branching process in independent identically distributed random environments with (conditioned on the environments) geometric distribution of the number of offsprings and immigration of independent identically distributed numbers of new particles at moments of extinction. Supposing that the increments of the accompanying random walk (S n ) and numbers of immigrants satisfy right-hand Cramér condition we obtain the asymptotics of large deviation probabilities P(ln Z* n ≥ θn).

We are concerned with problems of the existence of periodic structures in words from formal languages. We consider both squares (that is, fragments of the form xx, where x is an arbitrary word) and squares with one mismatch (that is, fragments of the form xy, where a word x differs from a word y by exactly one letter). Given natural numbers l 0 and l 1 , we study conditions for the existence of arbitrarily long words not containing squares with length larger than l 0 and squares with one mismatch and length larger than l 1 . For all possible pairs l 1 ≥ l 0 a minimal alphabet cardinality is found which permits to construct such a word. This research was carried out with the financial support of the Russian Foundation for Basic Research (grant no. 14-01-00598) and of the Branch of Mathematics of the Russian Academy of Sciences Program “Algebraic and combinatorial methods of mathematical cybernetics and new generation information systems” (the project “Problem of optimal synthesis of control systems”).

Suficient conditions for the joint distribution of frequences of elements on the cycle of the random multicyclic sequence modulo 4 to converge to the limit distribution are given. This limit distribution and some of its properties are described.

The paper is concerned with checking problems for failure and state diagnostics of N gates implementing in a working condition a given Boolean function f(x 1 , . . . , x n ). This problem is solved by composing single-output circuits from these gates and analyzing the output values of these circuits on all input tuples of variables. An arbitrary constant malfunction at the output of any single gate is allowed. It is required to minimize the number of circuits required for a check for failure and determination of all states of all gates. Exact values for the minimal possible number of such circuits are obtained.

From the set of all permutations of the degree n with a given number N ≤ n of cycles two permutations are choosed randomly, uniformly and independently. The cycles of each permutation are numbered in some of N! possible ways. We study the coincidence probability of the cycle lengths of permutations for a given numbering. This probability up to a suitably selected renumbering of cycles of the first permutation equals to the probability of similarity of these permutations. The asymptotic estimates of the coincidence probability of the cycle lengths are obtained for five types of relations between N, n → ∞.