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This survey contains results concerning behavioural and abstract theory of finite automata. Finite automata and related constructions are cornerstone concepts of discrete mathematics and mathematical cybernetics. Automata theory finds multiple applications in programming, technical cybernetics, the theory of dynamic systems, etc. Classical problems of this theory are the direct problems: analysis of processes of information transformation carried out by automata and properties of automata, and the inverse problems: synthesis of automata of prescribed properties and identification (restoring, recognition, deciphering, control and diagnosis) of an automaton by means of experiments with it. The fundamental notion which underlies the study of the above problems is the notion of a fragment which is a natural extension of a great number of known fragments of particular forms possessing their characteristic properties. Another key notion is the notion of an identifier of unobservable components of functioning of an automaton, that is, of a fragment which allows for unambiguous determination of these components. The subject matter reflected in this survey is being developed intensively, both in the direction of solving mathematical problems and in applied studies related to synthesis, testing and verification of complex software and hardware systems. This survey contains a series of final results, but they can be treated as a regular step in studying problems of analysis and synthesis of automata by their behaviour, which are continuously filled with new content and require additional resources to solve them.

In this paper we present relatively simple proofs of the following new upper bounds: c N , where c < 2 is a constant and N is the number of variables, for MAX-SAT for formulas with constant clause density; 2 K /6 , where K is the number of clauses, for MAX-2-SAT; 2 N /6.7 for ( n , 3)-MAX-2-SAT. All bounds are proved by the splitting method. The main two techniques are combined complexity measures and clause learning.

We consider a class of identities with permutations of three variables in a quasigroup ( Q ,·), each of which leads to an isotopy of the quasigroup to a group (abelian group). With the use of such identities, a criterion of isotopy of a quasigroup to a group (abelian group) is formulated, and a set of identities with permutations is given which lead to a special type of linearity (alinearity) of a quasigroup over a group (abelian group). It follows from these results that in the Belousov identity, which characterises quasigroups isotopic to a group (abelian group), two out of five variables (one out of four variables) can be fixed in arbitrary way. The obtained results give a possibility to describe an infinite number of identities in a primitive quasigroup ( Q ,·, \, /) leading to an isotopy of a quasigroup ( Q ,·) to a group or to its linearity of a given type.

We consider branching random walks with continuous time on integer lattices such that the particles born and die at a unique point. Under the assumption that the walk is symmetric and homogeneous, we derive integral and differential equations for the dynamics of local probabilities of continuation of the process in arbitrary nodes of the lattice, as well as probabilities of survival of the population of particles, for lattices of any dimension. In the critical case, we study the asymptotic behaviour, as t → ∞, of local probabilities, probabilities of survival of the population of particles, and conditional distributions of the population size on Z and Z 2 .

We investigate the joint distribution of the number of dense series in a random sequence over a finite alphabet. With the use of the Chen–Stein method, we find estimates of the distance in variation between the distribution of the vector of the numbers of dense series of ones of given lengths and the accompanying multidimensional Poisson distribution. These estimates give a possibility to prove limit theorems of Poisson type for the numbers of dense series of ones of given lengths and of lengths no less than a given length, for the number of intervals densely filled by ones, a limit theorem for the maximal length of dense series of ones, and a limit theorem for the number of dense series of ones of a given weight.