Ensembles reconnaissables de mots biinfinis

Abstract

The purpose of automata theory is to study and classify those properties of words that may be defined by a finite structure, say a finite automaton or a finite monoid. It seems natural to consider the same problem for infinite words. This amounts to studying the asymptotic behaviour of finite automata. As is well-known, this breaks the equivalence between determinism and non-determinism of finite automata. The study of the infinite behaviour of finite automata is based on a deep theorem due to B-&-uuml;chi and Mc Naughton: the recognizable sets of infinite words are the finite boolean combinations of deterministic ones (i.e. recognized by deterministic automata). The aim of this paper is to build an analogous theorem for two-sided infinite sequences. We define a biinfinite word as the equivalence class under the shift of a two-sided infinite sequence. The recognizable sets of biinfinite words are defined in a natural way and one is led to a two-sided notion of determinism. This notion seems to be new and justifies the consideration of biinfinite words. The main result of this paper is the extension to biinfinite words of the theorem of B-&-uuml;chi and Mc Naughton: the recognizable sets of biinfinite words are the finite boolean combinations of deterministic ones (Theorem 3.1). There exist three available proofs of B-&-uuml;chi-Mc Naughton's theorem. The original one by Mc Naughton [4] is hard to read. The proof given by Eilenberg in his book [2] has been constructed by Sch-&-uuml;tzenberger and Eilenberg from Mc Naughton's proof; it is similar to that of Rabin [5]. Finally, Sch-&-uuml;tzenberger gave a further proof in [6], which makes the argument more direct by using the methods of the theory of finite monoids. The proof of our main result follows closely Sch-&-uuml;tzenberger's method. This method allows to reduce the two-sided case to the one-sided case, although this seems very difficult to obtain directly. In the first section, we briefly recall the theory of the one-sided infinite behaviour of finite automata. In particular, we give Sch-&-uuml;tzenberger's proof of B-&-uuml;chi-Mc Naughton's theorem. The elements of this proof are used in the proof of our main result. In the second section we define the notions of biinfinite word, biautomaton and deterministic biautomaton. The last section contains the proof of our main result.