An effective characterization of the alternation hierarchy in two-variable logic

We characterize the languages in the individual levels of the quantifier alternation hierarchy of first-order logic with two variables by identities. This implies decidability of the individual levels. More generally we show that the two-sided semidirect product of a decidable variety with the variety J is decidable

Linguistics

Contains reports on two research projects.National Science Foundation (Grant G-7364 and Grant G-13903)National Institutes of Health (Grant MH-04737-02

Algebraic Characterization of the Alternation Hierarchy in FO^2[<] on Finite Words

We give an algebraic characterization of the quantifier alternation hierarchy in first-order two-variable logic on finite words. As a result, we obtain a new proof that this hierarchy is strict. We also show that the first two levels of the hierarchy have decidable membership problems, and conjecture an algebraic decision procedure for the other levels