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On cofinal and definite automata

Cofinal or directable automata were introduced in [1] and further investigated in [2, 7, 8, 9]. Cofinal automata are automata whose states can be directed to a single state by a suitable input word. We will call a cofinal automaton definite if there is an integer n such that all input words of length greater than or equal n direct the state set to a single state. Perles et al. [10] investigated definite events and definite automata. In particular they used shift registers, a special type of definite automata, in their discussion of the synthesis problem. Moreover, Stoklosa [12, 13] investigated these automata from an algebraic point of view. In section 2 of this paper we will prove a graph theoretic property of shift registers, namely that the transition diagram of a shift register contains a hamiltonian circle. In section 3 we apply this result in order to investigate the determination whether an arbitrary automaton is cofinal or not. In section 4 we determine the structure of all strongly definite automata with the aid of shift registers. Finally, in section 5, we characterize the general structure of definite automata. Let us give precise definitions first.