On logical hierarchies within FO2-definable languages

Manfred Kufleitner (1) , Pascal Weil (2)

1 Institut für Formale Methoden der Informatik
2 LaBRI - Laboratoire Bordelais de Recherche en Informatique

We consider the class of languages defined in the 2-variable fragment of the first-order logic of the linear order. Many interesting characterizations of this class are known, as well as the fact that restricting the number of alternations yields an infinite hierarchy. We show that each level of this hierarchy forms a variety of languages and hence, it admits an algebraic characterization. With this tool, we show that the quantifier alternation hierarchy over FO2[<] is decidable within one unit. For this purpose, we relate each level of the hierarchy with decidable varieties of languages, which can be defined in terms of iterated deterministic and co-deterministic products. A crucial notion in this process is that of condensed rankers, a refinement of the rankers of Weis and Immerman and the turtle languages of Schwentick, Therien and Vollmer.

Domaines: Logique en informatique [cs.LO]; Théorie et langage formel [cs.FL]

Manfred Kufleitner, Pascal Weil. On logical hierarchies within FO2-definable languages. 2011. ⟨hal-00567171v1⟩