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Short Definitions in Constraint Languages

Authors Jakub Bulín , Michael Kompatscher

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  • 15 pages

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Author Details

Jakub Bulín
  • Department of Theoretical Computer Science and Mathematical Logic, Faculty of Mathematics and Physics, Charles University, Prague, Czech Republic
Michael Kompatscher
  • Department of Algebra, Faculty of Mathematics and Physics, Charles University, Prague, Czech Republic


The authors would like to thank Dmitriy Zhuk for inspiring discussions about critical relations and the anonymous reviewers for their valuable suggestions.

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Jakub Bulín and Michael Kompatscher. Short Definitions in Constraint Languages. In 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 272, pp. 28:1-28:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)


A first-order formula is called primitive positive (pp) if it only admits the use of existential quantifiers and conjunction. Pp-formulas are a central concept in (fixed-template) constraint satisfaction since CSP(Γ) can be viewed as the problem of deciding the primitive positive theory of Γ, and pp-definability captures gadget reductions between CSPs. An important class of tractable constraint languages Γ is characterized by having few subpowers, that is, the number of n-ary relations pp-definable from Γ is bounded by 2^p(n) for some polynomial p(n). In this paper we study a restriction of this property, stating that every pp-definable relation is definable by a pp-formula of polynomial length. We conjecture that the existence of such short definitions is actually equivalent to Γ having few subpowers, and verify this conjecture for a large subclass that, in particular, includes all constraint languages on three-element domains. We furthermore discuss how our conjecture imposes an upper complexity bound of co-NP on the subpower membership problem of algebras with few subpowers.

Subject Classification

ACM Subject Classification
  • Theory of computation → Complexity theory and logic
  • constraint satisfaction
  • primitive positive definability
  • few subpowers
  • polynomially expressive
  • relational clone
  • subpower membership


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