Block Products and Nesting Negations in FO2

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8476)


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.


regular language finite monoid positive variety first-order logic 


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© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.Formale Methoden der InformatikUniversität StuttgartGermany
  2. 2.Fakultät für InformatikTechnische Universität MünchenGermany

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