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The paper is concerned with propositional calculi having arbitrary modus inference operations that are analogous to the modus ponens operation. For these calculi the question on the existence and cardinality of sets of generators is examined. A criterion for propositional calculi with arbitrary modus inference operations to be finitely generated is put forward. For some calculi, the existence of a finite complete system of tautologies is shown to imply the existence of a basis of arbitrarily large finite cardinality. We show the existence of propositional calculi with countable basis, without any basis, and calculi having a complete subsystem without any basis.

A new fast approximation to the Catalan constant is constructed, which makes it possible to calculate this constant with a complexity bound that is close to optimal. The work was supported partially by the RFBR (grant No 13-01-00657).

The notion of cycle indices of an automaton is introduced as a pair of positive parameters (b, q) effectively evaluated by the automaton. It is shown that the totality of periods of constant automata which may be represented as superpositions of a given automaton, Boolean functions, and the delay element, coincides with the variety of divisors of the members of the geometric progression b, bq, bq 2 ,... This suggests an algorithm for checking the expressibility of constant automata, as well as a theorem on the verification of the expressibility of the totality of all automata with a limited number of states.

The paper is concerned with definability of functions and relations in the FE language of functional equations of a countable-valued logic. The class of relations definable by functional equations over the set of functions {0, x + 1} is shown to coincide with the class ∑ 1 1 of the analytical Kleene hierarchy. Two extensions of the FE language are proposed that are equivalent in their expressibility to the FE language with functional constants for all homogeneous functions. This research was carried out with the financial support of the Russian Foundation for Basic Research (grant no. 13-01-00958).

We consider formal languages defined by bigram matrices and study connections between various properties of such languages, directed graphs and Eulerian circuits in these graphs. We also formulate criteria of non-emptiness, finiteness and infiniteness of languages and conditions of language regularity.

We consider the problem of identification of states of the contacts by means of experiments consisting in arranging these contacts into two-pole circuits with subsequent ”testing” of these circuits, i. e. in finding Boolean functions realized by the circuits constructed.

In the 1980s, E. S. Golod construction of infinite-dimensional nilalgebras was adapted by the author for the construction of nonnilpotent subalgebras generated by two elements. In appropriate 2-generated subgroups of adjoint p-groups some infinite subgroups generated by a pair of conjugate elements of order p, p is an odd prime, were found. In this paper, this construction is generalized. We give a condition that guarantees that finitely generated subalgebra of nilalgebra is not nilpotent. The infinite subgroups of the Golod group generated by involutions are constructed. The work was supported by the Ministry of Education grant, the theme No. 1.34.11 and by the Krasnoyarsk State Pedagogical University V. P.Astafieva, grant NSH No. 10.

Let S n be the semigroup of mappings of the set of n elements into itself, A be a fixed subset of the set of natural numbers ℕ, and V n (A) be the set of mappings from S n with cycle lengths belonging to A. Mappings from V n (A) are called A-mappings. Consider a random mapping σ n uniformly distributed on V n (A). Let λ n be the number of cyclic points of σ n . It is supposed that the set A has an asymptotic nonnegative density ς. We describe the asymptotic behaviour of the cardinality of the set V n (A) and prove a limit theorem for the sequence of random variables λ n as n → ∞.