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Track B: Automata, Logic, Semantics, and Theory of Programming

DOI: 10.4230/LIPIcs.ICALP.2022.127

Satisfiability Problems for Finite Groups

Authors: Paweł M. Idziak, Piotr Kawałek, Jacek Krzaczkowski, and Armin Weiß

Published in: LIPIcs, Volume 229, 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)

Abstract

Over twenty years ago, Goldmann and Russell initiated the study of the complexity of the equation satisfiability problem (PolSat and the NUDFA program satisfiability problem (ProgramSat) in finite groups. They showed that these problems are in 𝖯 for nilpotent groups while they are NP-complete for non-solvable groups. In this work we completely characterize finite groups for which the problem ProgramSat can be solved in randomized polynomial time under the assumptions of the Randomized Exponential Time Hypothesis and the Constant Degree Hypothesis. We also determine the complexity of PolSat for a wide class of finite groups. As a by-product, we obtain a classification for ListPolSat, a version of PolSat where each variable can be restricted to an arbitrary subset. Finally, we also prove unconditional algorithms for these problems in certain cases.

Cite as

Paweł M. Idziak, Piotr Kawałek, Jacek Krzaczkowski, and Armin Weiß. Satisfiability Problems for Finite Groups. In 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 229, pp. 127:1-127:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{idziak_et_al:LIPIcs.ICALP.2022.127,
  author =	{Idziak, Pawe{\l} M. and Kawa{\l}ek, Piotr and Krzaczkowski, Jacek and Wei{\ss}, Armin},
  title =	{{Satisfiability Problems for Finite Groups}},
  booktitle =	{49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)},
  pages =	{127:1--127:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-235-8},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{229},
  editor =	{Boja\'{n}czyk, Miko{\l}aj and Merelli, Emanuela and Woodruff, David P.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2022.127},
  URN =		{urn:nbn:de:0030-drops-164685},
  doi =		{10.4230/LIPIcs.ICALP.2022.127},
  annote =	{Keywords: Satisifiability, Solvable groups, ProgramSat, PolSat, Exponential Time Hypothesis}
}

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DOI: 10.4230/LIPIcs.STACS.2022.37

Satisfiability of Circuits and Equations over Finite Malcev Algebras

Authors: Paweł M. Idziak, Piotr Kawałek, and Jacek Krzaczkowski

Published in: LIPIcs, Volume 219, 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)

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.

Cite as

Paweł M. Idziak, Piotr Kawałek, and Jacek Krzaczkowski. Satisfiability of Circuits and Equations over Finite Malcev Algebras. In 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 219, pp. 37:1-37:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2022.37},
  URN =		{urn:nbn:de:0030-drops-158474},
  doi =		{10.4230/LIPIcs.STACS.2022.37},
  annote =	{Keywords: Circuit satisfiability, solving equations, Exponential Time Hypothesis}
}

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Documents authored by Idziak, Paweł M.

Satisfiability Problems for Finite Groups

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Satisfiability of Circuits and Equations over Finite Malcev Algebras

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