Properties of systems of defining relations for automata

Abstract

We suggest a canonical system of defining relations for finite everywhere defined outputless automata. We construct a procedure to pass from an arbitrary finite system of defining relations to a canonical one and, as a corollary, a procedure to check whether a finite system of pairs of words is a defining system for a given automaton or not. We also suggest a procedure to pass from a traversal of all arcs of the automaton graph to a system of defining relations and vice versa. 1.Defining relations for outputless automata as sets of equalities of words have been studied in [1], where it is shown that problems similar to the Thue problems for semigroups are solvable for automata. In [2], both equalities and inequalities of words are considered, and their connection with test experiments with finite Mealy automata is established. A procedure is also suggested to construct, beginning with a finite everywhere defined initial automaton, a finite system of defining relations.In this paper, for initially connected everywhere defined finite outputless automata we introduce a canonical system of defining relations and give a procedure to construct it and to go from an arbitrary finite system of defining relations to the canonical system. On this base we suggest a procedure to check whether an arbitrary finite set of pairs of words is a system of defining relations for a given automata or not.We find sharp bounds for the number of canonical systems, for the number of pairs of words in them, and for the total length of all words included into these systems.With the use of a canonical system, we study interrelations between a traversal of all arcs of the graph of an automaton and its systems of defining relations. We suggest procedures to construct on the basis of a traversal a system of defining relations and vice versa. 2.By an automaton, unless otherwise stated, is meant a finite determinate initial everywhere defined (semi)automaton [3]. All automata under consideration are over the same input alphabet X. Let X * denote the set of all words of finite length over this alphabet. Let p = x 1 . . . x k , and let d( p), d( p) = k, be the length of the word p. Let e be the empty word, d(e) = 0. A word p is said to be a prefix of a word p if p = p z for some word z ∈ X * . Let P ∪ Q and P Q denote, respectively, the union and the concatenation of sets P, Q ⊆ X * .Let A = (A, X, δ, a 0 ) be an automaton, where A is a finite set of states, a 0 is the initial state, δ : A × X → A is the transition function, δ(a, x) stands for ax. We extend the

The presentation of deterministic and strongly deterministic graphs

“…In the research of the presentation of finite automata without output in [6], a special presentation of the automaton (the canonical system of defining relations) was proposed, and the problem of characterization (comparing this automaton with the reference automaton only by its presentation) was solved. This solution used a special procedure for transforming the presentation to the canonical system of defining relations of the reference automaton.…”

“…We also found a criterion for sets of words, according to which, if this algorithm builds a graph, this graph will necessarily be strongly deterministic. Similarly to the canonical system of defining relations [6], we propose a special defining pair, which is called canonical. The numerical characteristics of the power of the components of the canonical defining pair and the lower bounds on the volume of these components are found.…”

“…. , 6, the receptacle is the vertex with label 1, and the inflorescence lacks edges (1,5) and (1,6).…”