The Quantifier Alternation Hierarchy of Synchronous Relations

Diego Figueira (1, 2) , Varun Ramanathan (1) , Pascal Weil (1, 2)

1 LaBRI - Laboratoire Bordelais de Recherche en Informatique
2 CNRS - Centre National de la Recherche Scientifique

The class of synchronous relations, also known as automatic or regular, is one of the most studied subclasses of rational relations. It enjoys many desirable closure properties and is known to be logically characterized: the synchronous relations are exactly those that are defined by a first-order formula on the structure of all finite words, with the prefix, equal-length and last-letter predicates. Here, we study the quantifier alternation hierarchy of this logic. We show that it collapses at level \Sigma_3 and that all levels below admit decidable characterizations. Our results reveal the connections between this hierarchy and the well-known hierarchy of first-order defined languages of finite words.

Mots clés: automatic relations; first-order logic; characterization; quantifier alternation

Domaines: Base de données [cs.DB]; Théorie et langage formel [cs.FL]; Mathématique discrète [cs.DM]; Logique en informatique [cs.LO]

Diego Figueira, Varun Ramanathan, Pascal Weil. The Quantifier Alternation Hierarchy of Synchronous Relations. 2019. ⟨hal-02109188v1⟩