Structure Theorem and Strict Alternation Hierarchy for FO2 on Words

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Abstract

It is well-known that every first-order property on words is expressible using at most three variables. The subclass of properties expressible with only two variables is also quite interesting and wellstudied. We prove precise structure theorems that characterize the exact expressive power of first-order logic with two variables on words. Our results apply to FO2[<] and FO 2[<; Suc], the latter of which includes the binary successor relation in addition to the linear ordering on string positions.

For both languages, our structuretheorems showexactly whatis expressible using a given quantifier depth, n, and using m blocks of alternating quantifiers, for any m ≤ n. Using these characterizations, we prove, among other results, that there is a strict hierarchy of alternating quantifiers for both languages. The question whether there was such a hierarchy had been completely open. As another consequence of our structural results, we show that satisfiability for FO2[<], which is NEXP-complete in general, becomes NP-complete once we only consider alphabets of a bounded size.