License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.MFCS.2019.36
URN: urn:nbn:de:0030-drops-109808
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2019/10980/
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Lagerkvist, Victor ; Nordh, Gustav

On the Strength of Uniqueness Quantification in Primitive Positive Formulas

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LIPIcs-MFCS-2019-36.pdf (0.6 MB)

Abstract

Uniqueness quantification (Exists!) is a quantifier in first-order logic where one requires that exactly one element exists satisfying a given property. In this paper we investigate the strength of uniqueness quantification when it is used in place of existential quantification in conjunctive formulas over a given set of relations Gamma, so-called primitive positive definitions (pp-definitions). We fully classify the Boolean sets of relations where uniqueness quantification has the same strength as existential quantification in pp-definitions and give several results valid for arbitrary finite domains. We also consider applications of Exists!-quantified pp-definitions in computer science, which can be used to study the computational complexity of problems where the number of solutions is important. Using our classification we give a new and simplified proof of the trichotomy theorem for the unique satisfiability problem, and prove a general result for the unique constraint satisfaction problem. Studying these problems in a more rigorous framework also turns out to be advantageous in the context of lower bounds, and we relate the complexity of these problems to the exponential-time hypothesis.

BibTeX - Entry

@InProceedings{lagerkvist_et_al:LIPIcs:2019:10980,
  author =	{Victor Lagerkvist and Gustav Nordh},
  title =	{{On the Strength of Uniqueness Quantification in Primitive Positive Formulas}},
  booktitle =	{44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)},
  pages =	{36:1--36:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-117-7},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{138},
  editor =	{Peter Rossmanith and Pinar Heggernes and Joost-Pieter Katoen},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2019/10980},
  URN =		{urn:nbn:de:0030-drops-109808},
  doi =		{10.4230/LIPIcs.MFCS.2019.36},
  annote =	{Keywords: Primitive positive definitions, clone theory, constraint satisfaction problems}
}

Keywords: Primitive positive definitions, clone theory, constraint satisfaction problems
Collection: 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)
Issue Date: 2019
Date of publication: 20.08.2019

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