Semigroups and Combinatorial Applications
Semigroups and Combinatorial Applications
Automata, Languages, and Machines
Automata, Languages, and Machines
Automata on Infinite Objects and Church's Problem
Automata on Infinite Objects and Church's Problem
On two-sided infinite fixed points of morphisms
Theoretical Computer Science
MFCS '01 Proceedings of the 26th International Symposium on Mathematical Foundations of Computer Science
Accessibility in Automata on Scattered Linear Orderings
MFCS '02 Proceedings of the 27th International Symposium on Mathematical Foundations of Computer Science
Journal of Computer and System Sciences
Complementation of rational sets on scattered linear orderings of finite rank
Theoretical Computer Science
DLT'02 Proceedings of the 6th international conference on Developments in language theory
A kleene theorem for languages of words indexed by linear orderings
DLT'05 Proceedings of the 9th international conference on Developments in Language Theory
Complementation of rational sets on countable scattered linear orderings
DLT'04 Proceedings of the 8th international conference on Developments in Language Theory
Computational complexity of rule distributions of non-uniform cellular automata
LATA'12 Proceedings of the 6th international conference on Language and Automata Theory and Applications
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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.