![]() This point of view will be adopted quite often in the sequel. ![]() The key idea of this chapter is to give a purely algebraic definition of recognizable sets. In the second chapter, we shift to a more algebraic point of view. Other proofs of McNaughton's theorem are given later on. Although this construction is rather involved, we have chosen to place it at the very beginning of the book because it is straightforward. A proof of McNaughton theorem is given, using Safra's determinization algorithm. It covers the necessary elements of the theory of automata on finite words such as Kleene's theorem. The first chapter contains the definitions of rational expressions, Büchi and Muller automata and recognizable sets. It is unlikely to cover the whole content in one single course, but a selection with emphasis either on topology, or on logic, or on automata and semigroups, is possible. ![]() The book can be used to lecture and the authors have used the manuscript for several years for graduate courses in computer science. The dependence between chapters is not too strong, making it possible to read some chapters independently from other ones. No particular background is required to read it, except for a standard mathematical culture. ![]() The book is intended for researchers or advanced students in mathematics or computer science. All proofs are given in detail, with a few, duly mentionned, exceptions. Actually, if several surveys have appeared on infinite words, this book is the first manuel devoted to this topic. It gathers for the first time the basic results with the advanced ones. This book presents a comprehensive treatment of all aspects of this theory. The notion of an infinite sequence is of interest to model the behavior of systems which are supposed to work endless, as operating systems for example. Many other results have appeared since then and the theory has known an important increase of interest motivated by applications to problems in computer science. This difficult result had been conjectured by David Muller while working at questions related to oscillating circuits. Later on, Robert McNaughton proved the equivalence of deterministic and non-deterministic automata, a natural extension of the corresponding result for finite words. He actually showed that all properties of the integers expressible in this logic can also be defined in terms of finite automata. He was able to prove the decidability of this theory. Working on weak logical theories of the integers, he was lead to consider the monadic second order theory of the successor function on the integers. The motivation for this generalization originates in the early work of Richard Büchi in the sixties. The other possible extension is the subject of this book: Finite sequences of symbols are replaced by infinite sequences. It can also be a real number corresponding to some probability of the word. This value can be an integer counting the number of paths labeled by this word in an automaton or the integer represented by this word in some basis. Words are replaced by functions associating to each word some numerical value. The theory of formal series is one of them. There are at least two possible extensions of this theory. The possible practical applications include lexical analysis, text processing and sofware verification. The elementary theory of automata allows both the specification and the verification of simple properties of finite sequences of symbols. This diversity appears in the material presented in this book which covers topics related to computer science, algebra, logic, topology and game theory. $\Sigma^0_1$-complete (for Turing reductions) but not $\Pi^0_1$.Īutomata theory arose as an interdisciplinary field, with roots in several scientific domains such as pure mathematics, electronics and computer science. We also give an example of a minimal subshift whose language is Problem of context-free languages is not decidable even for minimal $\Pi^0_1$ Minimal subsystems, refuting a conjecture in in another direction, and show that the model-checking We consider partitioned graphs, by which we mean finite directed graphs with a partitioned edge set $)$ in terms of the zeta functions of the topological Markov shifts and the generating functions of the Markov codes.We show that there exists a universal subshift having only a finite number of
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