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The Complexity of Checking Quasi-Identities over Finite Algebras with a Mal'cev Term

Authors Erhard Aichinger , Simon Grünbacher

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Erhard Aichinger
  • Institute for Algebra, Johannes Kepler Universität Linz, Austria
Simon Grünbacher
  • Institute for Algebra, Johannes Kepler Universität Linz, Austria


The authors thank M. Behrisch for discussions on this problem, the referees of a previous version for their criticism that helped to improve the result, and the referees of the present version for several suggestions improving the presentation.

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Erhard Aichinger and Simon Grünbacher. The Complexity of Checking Quasi-Identities over Finite Algebras with a Mal'cev Term. In 40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 254, pp. 4:1-4:12, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)


We consider finite algebraic structures and ask whether every solution of a given system of equations satisfies some other equation. This can be formulated as checking the validity of certain first order formulae called quasi-identities. Checking the validity of quasi-identities is closely linked to solving systems of equations. For systems of equations over finite algebras with finitely many fundamental operations, a complete P/NPC dichotomy is known, while the situation appears to be more complicated for single equations. The complexity of checking the validity of a quasi-identity lies between the complexity of term equivalence (checking whether two terms induce the same function) and the complexity of solving systems of polynomial equations. We prove that for each finite algebra with a Mal'cev term and finitely many fundamental operations, checking the validity of quasi-identities is coNP-complete if the algebra is not abelian, and in P when the algebra is abelian.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Combinatorial algorithms
  • Theory of computation → Complexity classes
  • quasi-identities
  • conditional identities
  • systems of equations


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