Block Products and Nesting Negations in FO²

Abstract

The alternation hierarchy in two-variable first-order logic FO²[ < ] over words was recently shown to be decidable by Kufleitner and Weil, and independently by Krebs and Straubing. In this paper we consider a similar hierarchy, reminiscent of the half levels of the dot-depth hierarchy or the Straubing-Thérien hierarchy. The fragment $\Sigma^2_m$ of FO² is defined by disallowing universal quantifiers and having at most m − 1 nested negations. One can view $\Sigma^2_m$ as the formulas in FO² which have at most m blocks of quantifiers on every path of their parse tree, and the first block is existential. Thus, the m ^th level of the FO²-alternation hierarchy is the Boolean closure of $\Sigma^2_m$. We give an effective characterization of $\Sigma^2_m$, i.e., for every integer m one can decide whether a given regular language is definable by a two-variable first-order formula with negation nesting depth at most m. More precisely, for every m we give ω-terms U _m and V _m such that an FO²-definable language is in $\Sigma^2_m$ if and only if its ordered syntactic monoid satisfies the identity U _m  ≤ V _m . Among other techniques, the proof relies on an extension of block products to ordered monoids.