The Half-Levels of the FO2 Alternation Hierarchy

Abstract

The alternation hierarchy in two-variable first-order logic FO2[<] over words was shown to be decidable by Kufleitner and Weil, and independently by Krebs and Straubing. 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. The Boolean closure of \({{\Sigma }^{2}_{m}}\) yields the m th level of the FO2-alternation hierarchy. 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 in \({{\Sigma }^{2}_{m}}\). Among other techniques, the proof relies on an extension of block products to ordered monoids.

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Acknowledgments

This work was supported by the German Research Foundation (DFG) under grant DI 435/5-2.

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Correspondence to Manfred Kufleitner.

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This is the revised full version of a conference paper at CSR 2014 [8].

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Fleischer, L., Kufleitner, M. & Lauser, A. The Half-Levels of the FO2 Alternation Hierarchy. Theory Comput Syst 61, 352–370 (2017). https://doi.org/10.1007/s00224-016-9712-2

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Keywords

  • Regular language
  • Finite monoid
  • Positive variety
  • First-order logic