Semigroups and Automata on Infinite Words

Abstract

This paper is an introduction to the algebraic theory of infinite words. Infinite words are widely used in computer science, in particular to model the behaviour of programs or circuits. From a mathematical point of view, they have a rich structure, at the cross-roads of logic, topology and algebra. This paper emphasizes the combinatorial and algebraic aspects of this theory but the interested reader is referred to the survey articles [34, 44] or to the report [30] for more information on the other aspects. In particular, the important topic of the complexity of the algorithms on infinite words is not treated in this paper. The paper is written with the perspective of generalizing the results on recognizable sets of finite words to infinite words. This does not exactly follow the historical development of the theory, but it gives a good idea of the type of problems that occur in this field. Some of these problems are still open, or have been solved quite recently so that the definitions and results presented below may not be as yet finalized. The first result to be generalized is the equivalence between finite automata, finite deterministic automata and rational expressions. If one adds infinite iteration (“omega” operation) to the standard rational operations, union, product and star, one gets a natural definition of the ω-rational sets of infinite words that extends the definition of rational sets of finite words. Buchi [5] was the first to propose a definition of finite automata acting on infinite words. This definition suffices to extend Kleene’s theorem to infinite words: the sets of infinite words recognized by finite Buchi automata are exactly the ω-rational sets. This result is now known as Buchi’s theorem. However, Buchi’s definition is not totally satisfying since deterministic Buchi automata are not equivalent to non deterministic ones. The con-