Two definitions of the equivalence of automata

1. In the comprehensive literature on the theory of automata the concept of equivalence of automata is based on the action of an automaton. Two automata are equivalent if? roughly speaking ? they make the sane. Obviously, there arises the problem of minimizing the automaton. S. Ginsburg x and W. M. Gluskov2 ? si? milarly to other authors ? assune that the set of inputs is invariant and they mini? mize only the set of states of automata. This is practically justified since the reduction of the set of states leads to the reduction of the number of technical elements when we build an automaton. In my paper On equivalence of automata (Zeszyty Naukowe WSP w Opolu, Ma tematyka IV, 1964) I was concerned with the minimization of the set of inputs which, having no (or very insignificant) influence on the simplicity of the frame of an auto? maton, leads to the simplification of the input alphabet and, therefore, simplifies the work of the operator. I introduced ?ere the concept of an original as an automaton with the smallest sets of states acd inputs and, moreover, suitably modified concept of the equivalence of automata. However, the argunents of the above-mentioned prper concerned only the finite automata. In the present paper I transfer the concept of equivalence in the said extended sense on arbitrary automata. I was inspirited to this work by Professor Czesiaw Ryll-Nardzewski. Two ways lead to the concept of equivalence of automata: the investigation of the relationship between the input words and the corresponding output words or the homomorphic mapping of an automaton into itself. The first way corresponds to the following shortly expressed equivalence: to make the same. The second way is more general, for in this case the time is not assumed to be discrete. The second definition of equivalence may be made equivalent to the first one if we assume that the time is discrete. 2. We shall now consider the second way leading to the concept of equivalence of automata.