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DOI: 10.4230/LIPIcs.CSL.2018.28

An Algebraic Decision Procedure for Two-Variable Logic with a Between Relation

Authors: Andreas Krebs, Kamal Lodaya, Paritosh K. Pandya, and Howard Straubing

Published in: LIPIcs, Volume 119, 27th EACSL Annual Conference on Computer Science Logic (CSL 2018)

Abstract

In earlier work (LICS 2016), the authors introduced two-variable first-order logic supplemented by a binary relation that allows one to say that a letter appears between two positions. We found an effective algebraic criterion that is a necessary condition for definability in this logic, and conjectured that the criterion is also sufficient, although we proved this only in the case of two-letter alphabets. Here we prove the general conjecture. The proof is quite different from the arguments in the earlier work, and required the development of novel techniques concerning factorizations of words. We extend the results to binary relations specifying that a factor appears between two positions.

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Andreas Krebs, Kamal Lodaya, Paritosh K. Pandya, and Howard Straubing. An Algebraic Decision Procedure for Two-Variable Logic with a Between Relation. In 27th EACSL Annual Conference on Computer Science Logic (CSL 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 119, pp. 28:1-28:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{krebs_et_al:LIPIcs.CSL.2018.28,
  author =	{Krebs, Andreas and Lodaya, Kamal and Pandya, Paritosh K. and Straubing, Howard},
  title =	{{An Algebraic Decision Procedure for Two-Variable Logic with a Between Relation}},
  booktitle =	{27th EACSL Annual Conference on Computer Science Logic (CSL 2018)},
  pages =	{28:1--28:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-088-0},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{119},
  editor =	{Ghica, Dan R. and Jung, Achim},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2018.28},
  URN =		{urn:nbn:de:0030-drops-96953},
  doi =		{10.4230/LIPIcs.CSL.2018.28},
  annote =	{Keywords: two-variable logic, finite model theory, algebraic automata theory}
}

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DOI: 10.4230/LIPIcs.FSTTCS.2012.86

An effective characterization of the alternation hierarchy in two-variable logic

Authors: Andreas Krebs and Howard Straubing

Published in: LIPIcs, Volume 18, IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2012)

Abstract

We characterize the languages in the individual levels of the quantifier alternation hierarchy of first-order logic with two variables by identities. This implies decidability of the individual levels. More generally we show that two-sided semidirect products with J as the right-hand factor preserve decidability.

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Andreas Krebs and Howard Straubing. An effective characterization of the alternation hierarchy in two-variable logic. In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2012). Leibniz International Proceedings in Informatics (LIPIcs), Volume 18, pp. 86-98, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2012)

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@InProceedings{krebs_et_al:LIPIcs.FSTTCS.2012.86,
  author =	{Krebs, Andreas and Straubing, Howard},
  title =	{{An effective characterization of the alternation hierarchy in two-variable logic}},
  booktitle =	{IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2012)},
  pages =	{86--98},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-47-7},
  ISSN =	{1868-8969},
  year =	{2012},
  volume =	{18},
  editor =	{D'Souza, Deepak and Radhakrishnan, Jaikumar and Telikepalli, Kavitha},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2012.86},
  URN =		{urn:nbn:de:0030-drops-38501},
  doi =		{10.4230/LIPIcs.FSTTCS.2012.86},
  annote =	{Keywords: FO\underline2, Quantifier Alternation, J, Pseudovarities, Identities}
}

Document

DOI: 10.4230/LIPIcs.CSL.2011.525

Algebraic Characterization of the Alternation Hierarchy in FO^2[<] on Finite Words

Authors: Howard Straubing

Published in: LIPIcs, Volume 12, Computer Science Logic (CSL'11) - 25th International Workshop/20th Annual Conference of the EACSL (2011)

Abstract

We give an algebraic characterization of the quantifier alternation hierarchy in first-order two-variable logic on finite words. As a result, we obtain a new proof that this hierarchy is strict. We also show that the first two levels of the hierarchy have decidable membership problems, and conjecture an algebraic decision procedure for the other levels.

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Howard Straubing. Algebraic Characterization of the Alternation Hierarchy in FO^2[<] on Finite Words. In Computer Science Logic (CSL'11) - 25th International Workshop/20th Annual Conference of the EACSL. Leibniz International Proceedings in Informatics (LIPIcs), Volume 12, pp. 525-537, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2011)

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@InProceedings{straubing:LIPIcs.CSL.2011.525,
  author =	{Straubing, Howard},
  title =	{{Algebraic Characterization of the Alternation Hierarchy in FO^2\lbrack\langle\rbrack on Finite Words}},
  booktitle =	{Computer Science Logic (CSL'11) - 25th International Workshop/20th Annual Conference of the EACSL},
  pages =	{525--537},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-32-3},
  ISSN =	{1868-8969},
  year =	{2011},
  volume =	{12},
  editor =	{Bezem, Marc},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2011.525},
  URN =		{urn:nbn:de:0030-drops-32549},
  doi =		{10.4230/LIPIcs.CSL.2011.525},
  annote =	{Keywords: automata, finite model theory}
}

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Documents authored by Straubing, Howard

An Algebraic Decision Procedure for Two-Variable Logic with a Between Relation

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An effective characterization of the alternation hierarchy in two-variable logic

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Algebraic Characterization of the Alternation Hierarchy in FO^2[<] on Finite Words

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