Block products and nesting negations in FO2

The alternation hierarchy in two-variable first-order logic FO 2 [∈ < ∈] 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 of FO 2 is defined by disallowing universal quantifiers and having at most m∈-∈1 nested negations. One can view as the formulas in FO 2 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 FO 2 -alternation hierarchy is the Boolean closure of. We give an effective characterization of, 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 FO 2 -definable language is in 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. © 2014 Springer International Publishing Switzerland