Structure Theorem and Strict Alternation Hierarchy for FO^2 on Words

TL;DR

This paper provides a detailed structural characterization of two-variable first-order logic on words, establishing a strict alternation hierarchy and analyzing complexity reductions for satisfiability problems.

Contribution

It introduces precise structure theorems for FO^2 on words, revealing the exact expressive power and hierarchy of quantifier alternations, and simplifies satisfiability complexity on bounded alphabets.

Findings

Established a strict hierarchy of quantifier alternations for FO^2 on words.

Proved the expressive power of FO^2 with quantifier depth n and m blocks of alternation.

Showed satisfiability complexity drops from NEXP to NP on bounded alphabets.

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 well-studied. We prove precise structure theorems that characterize the exact expressive power of first-order logic with two variables on words. Our results apply to both the case with and without a successor relation. For both languages, our structure theorems show exactly what is expressible using a given quantifier depth, n, and using m blocks of alternating quantifiers, for any m \leq 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 first-order logic with two variables without successor, which is NEXP-complete in general, becomes NP-complete once we only consider alphabets of a bounded size.