On some problems in the theory of partial automata

The paper shows how some semigroup-theory methods can be used in automata problems. The theory of automata is closely connected with the theory of semigroups (or, to be more exact, with the theory of representations of semigroups by transformations). The concept of automaton without outputs is equivalent to the concept of representa­ tion of free semigroup by transformations. If we consider everywhere or not every­ where defined, one-valued or many-valued transformations, we obtain full or partial, deterministic or nondeterministic automata. All this was discussed by the author in his lecture at the International symposium on relay circuits and finite automata in Moscow, September 1962 [1]. Many results in the theory of transformation semigroups may be interpreted as results on automata (and vice versa). Unfortunately, the main concepts and results of the theory of transformation semigroups are almost unknown among the specia­ lists in the automata theory. Quite a few recent results on automata turn out to be well known after being translated into semigroup language. The aim of this paper is to present some results on transformation semigroups as results on automata. This semigroup-theoretic results have been partly published in [4, 5]. The main ideas of these results, the underlying point of view (the so-called "relation algebras") were exposed in [2] and, in a much shorter form, in [3]. The output function of an automaton does not play any r61e in this paper. We consider automata without outputs. A (finite) automaton is an ordered triple A = (X, S, 5) where X is a (finite) set of input symbols, S is a (finite) set of inner states and <5 is a transition function, i.e. a partial mapping of the set X x S into S. If <5 is everywhere defined, the automaton A is called full. If 8(x, s) is not defined, it means that the automaton is destroyed when the input is x while the inner state is s. One can consider multi-valued 8. In this case the automaton A, which is in the state s, goes to some state from the set <5 where