The FO^2 alternation hierarchy is decidable

Abstract

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.

References in corpus (1)

• Logic Meets Algebra: the Case of Regular Languages

Cited by in corpus (6)

• Quantifier Alternation in Two-Variable First-Order Logic with Successor Is Decidable
• The Word Problem for Omega-Terms over the Trotter-Weil Hierarchy
• The Join Levels of the Trotter-Weil Hierarchy are Decidable
• Nesting negations in FO2 over infinite words
• From algebra to logic: there and back again -- the story of a hierarchy
• Non-locality of the meet levels of the Trotter-Weil Hierarchy