The Subpower Membership Problem of 2-Nilpotent Algebras

Author Michael Kompatscher

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Michael Kompatscher
  • Department of Algebra, Faculty of Mathematics and Physics, Charles University, Prague, Czech Republic


I would like to thank Peter Mayr for introducing me to difference clonoids and giving several helpful comments on earlier versions of this paper. Furthermore I would like to thank the anonymous referees for their many helpful remarks.

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Michael Kompatscher. The Subpower Membership Problem of 2-Nilpotent Algebras. In 41st International Symposium on Theoretical Aspects of Computer Science (STACS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 289, pp. 46:1-46:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


The subpower membership problem SMP(𝐀) of a finite algebraic structure 𝐀 asks whether a given partial function from Aⁿ to A can be interpolated by a term operation of 𝐀, or not. While this problem can be EXPTIME-complete in general, Willard asked whether it is always solvable in polynomial time if 𝐀 is a Mal'tsev algebra. In particular, this includes many important structures studied in abstract algebra, such as groups, quasigroups, rings, Boolean algebras. In this paper we give an affirmative answer to Willard’s question for a big class of 2-nilpotent Mal'tsev algebras. We furthermore develop tools that might be essential in answering the question for general nilpotent Mal'tsev algebras in the future.

Subject Classification

ACM Subject Classification
  • Theory of computation → Algebraic complexity theory
  • subpower membership problem
  • Mal'tsev algebra
  • compact representation
  • nilpotence
  • clonoids


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