License: Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.MFCS.2023.66
URN: urn:nbn:de:0030-drops-186007
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2023/18600/
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Mayr, Peter

On the Complexity Dichotomy for the Satisfiability of Systems of Term Equations over Finite Algebras

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LIPIcs-MFCS-2023-66.pdf (0.6 MB)

Abstract

For a fixed finite algebra ?, we consider the decision problem SysTerm(?): does a given system of term equations have a solution in ?? This is equivalent to a constraint satisfaction problem (CSP) for a relational structure whose relations are the graphs of the basic operations of ?. From the complexity dichotomy for CSP over fixed finite templates due to Bulatov [Bulatov, 2017] and Zhuk [Zhuk, 2017], it follows that SysTerm(?) for a finite algebra ? is in P if ? has a not necessarily idempotent Taylor polymorphism and is NP-complete otherwise. More explicitly, we show that for a finite algebra ? in a congruence modular variety (e.g. for a quasigroup), SysTerm(?) is in P if the core of ? is abelian and is NP-complete otherwise. Given ? by the graphs of its basic operations, we show that this condition for tractability can be decided in quasi-polynomial time.

BibTeX - Entry

@InProceedings{mayr:LIPIcs.MFCS.2023.66,
  author =	{Mayr, Peter},
  title =	{{On the Complexity Dichotomy for the Satisfiability of Systems of Term Equations over Finite Algebras}},
  booktitle =	{48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023)},
  pages =	{66:1--66:12},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-292-1},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{272},
  editor =	{Leroux, J\'{e}r\^{o}me and Lombardy, Sylvain and Peleg, David},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2023/18600},
  URN =		{urn:nbn:de:0030-drops-186007},
  doi =		{10.4230/LIPIcs.MFCS.2023.66},
  annote =	{Keywords: systems of equations, general algebras, constraint satisfaction}
}

Keywords: systems of equations, general algebras, constraint satisfaction
Collection: 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023)
Issue Date: 2023
Date of publication: 21.08.2023

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