Structure Theorem and Strict Alternation Hierarchy for FO2 on Words

It is well-known that every first-order property on words is expressible using atmost three variables. The subclass of properties expressible with only two variables is alsoquite interesting and well-studied. We prove precise structure theorems that characterizethe exact expressive power of first-order logic with two variables on words. Our resultsapply to both the case with and without a successor relation.For both languages, our structure theorems show exactly what is expressible using agiven 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 hierarchyof alternating quantifiers for both languages. The question whether there was such ahierarchy had been completely open. As another consequence of our structural results,we show that satisfiability for first-order logic with two variables without successor, whichis NEXP-complete in general, becomes NP-complete once we only consider alphabets of abounded size.