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The behaviour of automata in labyrinths is a rather new field of automata theory, but more than one hundred papers devoted to this topic have been published. In this paper, we consider the key notions, problems, achievements, methods to solve problems, and open problems related to an important direction of this field, the behaviour of collectives 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.

We consider the problem of generating strings that belong to certain languages and satisfy some additional restrictions. Languages are defined by formal grammars and automata. The following formulation of this problem as a decision one is proposed: for a language represented by a formal grammar or an automaton and a pair of strings, determine whether there exists a string in this language that lies lexicographically between these strings. It is proved that this problem is NLOGSPACE-complete for deterministic and nondeterministic finite automata and for linear context-free grammars; P-complete for context-free grammars of the general form; NP-complete for alternating finite automata, for conjunctive grammars and for linear conjunctive grammars; PSPACE-complete for context-sensitive grammars and linear bounded automata.

We show that the minimal circuit of functional elements over the basis { x → y , } which realises a linear function of n variables consists of 4 n - 4 elements.

We suggest algorithms for constructing models of a finite automaton which are new automata such that the equations describing the functioning of these models are consequences of the equations describing the functioning of the initial automaton. These new automata are constructed with the use of powers of the initial automaton and some additional functions on the output words. Applications of the results are related to the problem on determining the state of an automaton on the base of the input sequences and the corresponding output sequences.

We give a complete description of the limit behaviour of the number of cycles of length r in a random permutation of degree n with N cycles in all domains of variations of the parameters n , N , r , thus essentially extending the known results concerning the limit behaviour of the number of such cycles.

We give conditions for emergence of a giant cycle in a random permutation with a given number of cycles and obtain limit distributions of the maximum cycle lengths in all domains of variation of the parameters.