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Satisfiability of Circuits and Equations over Finite Malcev Algebras

Authors Paweł M. Idziak , Piotr Kawałek , Jacek Krzaczkowski

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Author Details

Paweł M. Idziak
  • Department of Theoretical Computer Science, Faculty of Mathematics and Computer Science, Jagiellonian University, Kraków, Poland
Piotr Kawałek
  • Department of Theoretical Computer Science, Faculty of Mathematics and Computer Science, Jagiellonian University, Kraków, Poland
Jacek Krzaczkowski
  • Department of Computer Science, Faculty of Mathematics, Physics and Computer Science, Maria Curie-Sklodowska University, Lublin, Poland

Funding

The project is partially supported by Polish NCN Grant # 2014/14/A/ST6/00138.
  • Kawałek, Piotr: This research is partially supported by the Priority Research Area SciMat under the program Excellence Initiative – Research University at the Jagiellonian University in Kraków.

Cite As Get BibTex

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) https://doi.org/10.4230/LIPIcs.STACS.2022.37

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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Problems, reductions and completeness
  • Theory of computation → Complexity classes
Keywords
  • Circuit satisfiability
  • solving equations
  • Exponential Time Hypothesis

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