The expressive power of existential first order sentences of Büchi's sequential calculus

Jean-Eric Pin (1)

1 LIAFA - Laboratoire d'informatique Algorithmique : Fondements et Applications

The aim of this paper is to study the first order theory of the successor, interpreted on finite words. More specifically, we complete the study of the hierarchy based on quantifier 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 effectively. We give a semigroup theoretic description of the expressive power of 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 field, is not in the scope of Eilenberg's variety theorem, since B?1-definable 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.

Mots clés: existential formula; Büchi's sequential calculus; finite automaton; regular language

Domaines: Logique en informatique [cs.LO]

Jean-Eric Pin. The expressive power of existential first order sentences of Büchi's sequential calculus. Discrete Mathematics, 2005, 291, pp.155-174. ⟨hal-00112833⟩