Analysis of behaviour of automata

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. The automata theory finds multiple applications in programming, technical cybernetics, theory of dynamic systems, etc. Classical problems of this theory are the direct problems: analysis of processes of transformation of information 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 an experiment 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 body of known fragments of particular forms possessing their characteristic properties. We introduce the fragment?automaton relation, consider properties of classes of fragments of a given automaton. We introduce the notion of a cofragment (forbidden fragment) of an automaton, consider properties of classes of automata which have a given fragment?cofragment pair. 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 values of these components. It is demonstrated that identifiers provide us with a powerful tool for solving the problems we consider in this paper.
Next, we introduce the general notion of representation of the reference automaton to within a given precision (similarity) relative to a priori class of automata as a pair (fragment, cofragment) of the reference automaton which can be a pair (fragment, cofragment) of an automaton in a priori class if and only if it is similar to the reference one. It is shown that this concept covers and generalises a series of particular notions (control, recognising experiments, questionnaire languages, $k$-sets, etc.) known in the automata theory. We suggest a natural classification of representations. We carry out a systematic study of conditions for existence and structure of representations.
We obtain precise conditions for existence of representations of general form and their particular classes in terms of relations between properties of a priori class, class of automata similar to the reference one, and the class of automata indistinguishable from the reference automaton for the corresponding indistinguishability relation. We obtain conditions for existence of nontrivial representations for various classes, including questionnaire languages and control experiments.
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.