The FO^2 alternation hierarchy is decidable

Manfred Kufleitner (1) , Pascal Weil (2)

1 University of Stuttgart
2 LaBRI - Laboratoire Bordelais de Recherche en Informatique

We consider the two-variable fragment FO^2[<] of first-order logic over finite words. Numerous characterizations of this class are known. Thérien and Wilke have shown that it is decidable whether a given regular language is definable in FO^2[<]. From a practical point of view, as shown by Weis, FO^2[<] is interesting since its satisfiability problem is in NP. Restricting the number of quantifier alternations yields an infinite hierarchy inside the class of FO^2[<]-definable languages. We show that each level of this hierarchy is decidable. For this purpose, we relate each level of the hierarchy with a decidable variety of finite monoids. Our result implies that there are many different ways of climbing up the FO^2[<]-quantifier alternation hierarchy: deterministic and co-deterministic products, Mal'cev products with definite and reverse definite semigroups, iterated block products with J-trivial monoids, and some inductively defined omega-term identities. A combinatorial tool in the process of ascension is that of condensed rankers, a refinement of the rankers of Weis and Immerman and the turtle programs of Schwentick, Thérien, and Vollmer.

Mots clés: quantifier alternation; logical hierarchy; varieties of languages

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

Manfred Kufleitner, Pascal Weil. The FO^2 alternation hierarchy is decidable. Computer Science Logic 2012, 2012, France. pp.426-439, ⟨10.4230/LIPIcs.CSL.2012.426⟩. ⟨hal-00683102⟩