Structure Theorem and Strict Alternation Hierarchy for FO² on Words

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 FO$^2[<]$ 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 structure theorems show exactly what is expressible using a given quantifier depth, $n$, and using $m$ blocks of alternating quantifiers, for any $mleq n$. 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 since it was asked in [Etessami, Vardi, and Wilke 1997].