Logic, semigroups and automata on words

This is a survey article on the connections between the “sequential calculus” of Büchi, a system which allows to formalize properties of words, and the theory of automata. In the sequential calculus, there is a predicate for each letter and the unique extra non logical predicate is the relation symbol <, which is interpreted as the usual order on the integers. Several famous classes have been classified within this logic. We briefly review the main results concerning second order, which covers classes like PH, NP, P, etc., and then study in more detail the results concerning the monadic second-order and first-order logic. In particular, we survey the results and fascinating open problems dealing with the first-order quantifier hierarchy. We also discuss the first-order logic of one successor and the linear temporal logic. There are in fact three levels of results, since these logics can be interpreted on finite words, infinite words or bi-infinite words. The paper is self-contained. In particular, the necessary background on automata and finite semigroups is presented in a long introductory section, which includes some very recent results on the algebraic theory of infinite words.

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