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

Author Peter Mayr

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Peter Mayr
  • Department of Mathematics, University of Colorado Boulder, CO, USA
  • Institute for Algebra, Johannes Kepler Universitรคt Linz, Austria


I want to thank E. Aichinger for discussions on this problem and the referees for their diligent reading and comments.

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Peter Mayr. On the Complexity Dichotomy for the Satisfiability of Systems of Term Equations over Finite Algebras. In 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 272, pp. 66:1-66:12, Schloss Dagstuhl โ€“ Leibniz-Zentrum fรผr Informatik (2023)


For a fixed finite algebra ๐€, we consider the decision problem SysTerm(๐€): does a given system of term equations have a solution in ๐€? This is equivalent to a constraint satisfaction problem (CSP) for a relational structure whose relations are the graphs of the basic operations of ๐€. From the complexity dichotomy for CSP over fixed finite templates due to Bulatov [Bulatov, 2017] and Zhuk [Zhuk, 2017], it follows that SysTerm(๐€) for a finite algebra ๐€ is in P if ๐€ has a not necessarily idempotent Taylor polymorphism and is NP-complete otherwise. More explicitly, we show that for a finite algebra ๐€ in a congruence modular variety (e.g. for a quasigroup), SysTerm(๐€) is in P if the core of ๐€ is abelian and is NP-complete otherwise. Given ๐€ by the graphs of its basic operations, we show that this condition for tractability can be decided in quasi-polynomial time.

Subject Classification

ACM Subject Classification
  • Theory of computation โ†’ Problems, reductions and completeness
  • systems of equations
  • general algebras
  • constraint satisfaction


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