The expressive power of existential rst order sentences of B uchi ' s sequential calculus

The aim of this paper is to study the rst order theory of the successor, interpreted on nite words. More speci cally, we complete the study of the hierarchy based on quanti er alternations (or n-hierarchy). It was known (Thomas, 1982) that this hierarchy collapses at level 2, but the expressive power of the lower levels was not characterized e ectively. We give a semigroup theoretic description of the expressive power of 1, the existential formulas, and B 1, the boolean combinations of existential formulas. Our characterization is algebraic and makes use of the syntactic semigroup, but contrary to a number of results in this eld, is not in the scope of Eilenberg's variety theorem, since B 1-de nable languages are not closed under residuals. An important consequence is the following: given one of the levels of the hierarchy, there is polynomial time algorithm to decide whether the language accepted by a deterministic n-state automaton is expressible by a sentence of this level.