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DOI: 10.4230/LIPIcs.STACS.2013.305

Quantifier Alternation in Two-Variable First-Order Logic with Successor Is Decidable

Authors: Manfred Kufleitner and Alexander Lauser

Published in: LIPIcs, Volume 20, 30th International Symposium on Theoretical Aspects of Computer Science (STACS 2013)

Abstract

We consider the quantifier alternation hierarchy within two-variable first-order logic FO^2[<,suc] over finite words with linear order and binary successor predicate. We give a single identity of omega-terms for each level of this hierarchy. This shows that for a given regular language and a non-negative integer~$m$ it is decidable whether the language is definable by a formula in FO^2[<,suc] which has at most m quantifier alternations. We also consider the alternation hierarchy of unary temporal logic TL[X,F,Y,P] defined by the maximal number of nested negations. This hierarchy coincides with the FO^2[<,suc] quantifier alternation hierarchy.

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Manfred Kufleitner and Alexander Lauser. Quantifier Alternation in Two-Variable First-Order Logic with Successor Is Decidable. In 30th International Symposium on Theoretical Aspects of Computer Science (STACS 2013). Leibniz International Proceedings in Informatics (LIPIcs), Volume 20, pp. 305-316, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2013)

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@InProceedings{kufleitner_et_al:LIPIcs.STACS.2013.305,
  author =	{Kufleitner, Manfred and Lauser, Alexander},
  title =	{{Quantifier Alternation in Two-Variable First-Order Logic with Successor Is Decidable}},
  booktitle =	{30th International Symposium on Theoretical Aspects of Computer Science (STACS 2013)},
  pages =	{305--316},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-50-7},
  ISSN =	{1868-8969},
  year =	{2013},
  volume =	{20},
  editor =	{Portier, Natacha and Wilke, Thomas},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2013.305},
  URN =		{urn:nbn:de:0030-drops-39438},
  doi =		{10.4230/LIPIcs.STACS.2013.305},
  annote =	{Keywords: automata theory, semigroups, regular languages, first-order logic}
}

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DOI: 10.4230/LIPIcs.STACS.2011.356

First-order Fragments with Successor over Infinite Words

Authors: Jakub Kallas, Manfred Kufleitner, and Alexander Lauser

Published in: LIPIcs, Volume 9, 28th International Symposium on Theoretical Aspects of Computer Science (STACS 2011)

Abstract

We consider fragments of first-order logic and as models we allow finite and infinite words simultaneously. The only binary relations apart from equality are order comparison < and the successor predicate +1. We give characterizations of the fragments Sigma_2 = Sigma_2[<,+1] and FO^2 = FO^2[<,+1] in terms of algebraic and topological properties. To this end we introduce the factor topology over infinite words. It turns out that a language $L$ is in FO^2 cap Sigma_2 if and only if $L$ is the interior of an FO^2 language. Symmetrically, a language is in FO^2 cap Pi_2 if and only if it is the topological closure of an FO^2 language. The fragment Delta_2 = Sigma_2 cap Pi_2 contains exactly the clopen languages in FO^2. In particular, over infinite words Delta_2 is a strict subclass of FO^2. Our characterizations yield decidability of the membership problem for all these fragments over finite and infinite words; and as a corollary we also obtain decidability for infinite words. Moreover, we give a new decidable algebraic characterization of dot-depth 3/2 over finite words. Decidability of dot-depth 3/2 over finite words was first shown by Glasser and Schmitz in STACS 2000, and decidability of the membership problem for FO^2 over infinite words was shown 1998 by Wilke in his habilitation thesis whereas decidability of Sigma_2 over infinite words is new.

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Jakub Kallas, Manfred Kufleitner, and Alexander Lauser. First-order Fragments with Successor over Infinite Words. In 28th International Symposium on Theoretical Aspects of Computer Science (STACS 2011). Leibniz International Proceedings in Informatics (LIPIcs), Volume 9, pp. 356-367, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2011)

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@InProceedings{kallas_et_al:LIPIcs.STACS.2011.356,
  author =	{Kallas, Jakub and Kufleitner, Manfred and Lauser, Alexander},
  title =	{{First-order Fragments with Successor over Infinite Words}},
  booktitle =	{28th International Symposium on Theoretical Aspects of Computer Science (STACS 2011)},
  pages =	{356--367},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-25-5},
  ISSN =	{1868-8969},
  year =	{2011},
  volume =	{9},
  editor =	{Schwentick, Thomas and D\"{u}rr, Christoph},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2011.356},
  URN =		{urn:nbn:de:0030-drops-30267},
  doi =		{10.4230/LIPIcs.STACS.2011.356},
  annote =	{Keywords: infinite words, regular languages, first-order logic, automata theory, semi-groups, topology}
}

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Quantifier Alternation in Two-Variable First-Order Logic with Successor Is Decidable

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First-order Fragments with Successor over Infinite Words

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