Symbolic Dynamics and Finite Automata

In this survey, we present some connections between notions and results in automata theory and other ones in symbolic dynamics. 1 I n t r o d u c t i o n Symbolic dynamics is a field born with the work in topology of Marston Morse at the beginning of the years 20 [25]. It is, according to Morse, an "algebra and geometry of recurrence". The idea is to study the sequences of symbols obtained by scanning the successive regions of a surface met while following a trajectory starting from a given point. A further paper by Morse and Itedlund ([24]) gives the basic results of this theory. Later, the theory was developed by many authors as a branch of ergodic theory (see for example the collected works in [32] or [7]). One of the main directions of research has been the problem of the isomorphism of subshifts of finite type (see below the definition of these objects). It is not yet completely solved although the last results of Kim and Roush [21] indicate a counterexample to a long-standing conjecture formulated by F. Williams [34]. There are many links between symbolic dynamics and the theory of automata, as pointed out by R. Adler and B. Weiss ([33]). The interplay is interesting to explore since it brings new problems and methods. In this paper, I will present some of these interconnections and report some of the new results that have been obtained in this direction together with some interesting open problems. The material presented here does not cover all existing connections of this kind. There are in particular interesting links between symbolic dynamics and representation of numbers that are not presented here (see [17]). There are also important connections with cellular automata (see e.g. [11]). 2 W h a t is s y m b o l i c d y n a m i c s a b o u t ? We present in this Section a short introduction to the concepts of symbolic dynamics. For a more detailed exposition, we refer to [32]. We consider the set A 77 of two-sided infinite words as a topological space with respect to the usual product topology. The shift transformation ~ acts on A 7] bijectively. It associates to x E A 77 the element y = g(z) E A 77 defined for n C T ] b y Yn = ~n.t -1