Representations of finite automata

Grunskii, I. S.

doi:10.1007/bf01072098

Cited by 2 publications

(15 citation statements)

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“…Among these problems are the problem to find the internal state of an automaton by experiments with it [12,16], the problem to synthesise an automaton by a given fragment (experiment [17,18], questionnaire languages [20,21]), the problems to compare behaviours of automata (equivalence, indistinguishability by a single or multiple experiment), the problems of control and recognition of automata. These connections are considered in more detail in studies [6,11,22], where it is also shown that the identifiers are a powerful and adequate tool in investigating properties of fragments of an automaton and especially of its representations.…”

Section: Automaton State Identifiersmentioning

confidence: 96%

“…Then it is said that R describes the class A.R/ or that R defines A to within A.R/. In the framework of the classes F .A/ and A.R/, in studies [6,22] properties of fragments of various forms are systematically investigated.…”

Section: Automaton Fragmentsmentioning

confidence: 99%

“…In this connection, a quite wide spectrum of problems arises related to classification of the experiments, to the questions of solvability of problems of recognising one or another property of the automaton, to estimating the complexity of minimal experiments sufficient to solve recognition problems, and to estimating the complexity of construction of such experiments. The theory of experiments is being studied intensively, a number of fundamental results have been obtained [2][3][4][5][6][7][8]. It has found a widespread application and has a pronounced effect on development of a series of mathematical and technical disciplines (such as technical diagnosis, theory of dynamical systems, theory of formal languages and grammatics, theoretical programming, etc.…”

Section: Introductionmentioning

confidence: 99%

“…Methods and results of this stage are quite completely presented in [7]. One of the authors (I. S. Grunskii) was probably the first who referred to these sequences and responses to them as state identifiers, began their systematic study [4] and paid attention to the existence of identifiers of other unobservable components of behaviour (for example, input sequences [6]). At the second stage, many particular ways to construct control and recognition experiments were obtained in the cases where they were known to exist.…”

Section: Introductionmentioning

confidence: 99%

“…While analysing the precision of description of an automaton by fragments, an important part is played by the so-called identifiers of automaton properties (states, inputs, outputs) [6], fragments which allow for unambiguous identification of these properties. Identifiers are again examples of descriptors.…”

Section: Introductionmentioning

confidence: 99%

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Analysis of behaviour of automata

Kudryavtsev¹,

Grunskii²,

Kozlovskii³

2009

Discrete Mathematics and Applications

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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. 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 fragmentcofragment 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.

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Section: Automaton State Identifiersmentioning

confidence: 96%

Section: Automaton Fragmentsmentioning

confidence: 99%