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.STACS.2022.37
URN: urn:nbn:de:0030-drops-158474
URL: https://drops.dagstuhl.de/opus/volltexte/2022/15847/
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Idziak, Paweł M. ; Kawałek, Piotr ; Krzaczkowski, Jacek

Satisfiability of Circuits and Equations over Finite Malcev Algebras

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Abstract

We show that the satisfiability of circuits over finite Malcev algebra A is NP-complete or A is nilpotent. This strengthens the result from our earlier paper [Idziak and Krzaczkowski, 2018] where nilpotency has been enforced, however with the use of a stronger assumption that no homomorphic image of A has NP-complete circuits satisfiability. Our methods are moreover strong enough to extend our result of [Idziak et al., 2021] from groups to Malcev algebras. Namely we show that tractability of checking if an equation over such an algebra A has a solution enforces its nice structure: A must have a nilpotent congruence ν such that also the quotient algebra A/ν is nilpotent. Otherwise, if A has no such congruence ν then the Exponential Time Hypothesis yields a quasipolynomial lower bound. Both our results contain important steps towards a full characterization of finite algebras with tractable circuit satisfiability as well as equation satisfiability.

BibTeX - Entry

@InProceedings{idziak_et_al:LIPIcs.STACS.2022.37,
  author =	{Idziak, Pawe{\l} M. and Kawa{\l}ek, Piotr and Krzaczkowski, Jacek},
  title =	{{Satisfiability of Circuits and Equations over Finite Malcev Algebras}},
  booktitle =	{39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)},
  pages =	{37:1--37:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-222-8},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{219},
  editor =	{Berenbrink, Petra and Monmege, Benjamin},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2022/15847},
  URN =		{urn:nbn:de:0030-drops-158474},
  doi =		{10.4230/LIPIcs.STACS.2022.37},
  annote =	{Keywords: Circuit satisfiability, solving equations, Exponential Time Hypothesis}
}

Keywords: Circuit satisfiability, solving equations, Exponential Time Hypothesis
Collection: 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)
Issue Date: 2022
Date of publication: 09.03.2022


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