Short Definitions in Constraint Languages

Authors Jakub Bulín , Michael Kompatscher

Thumbnail PDF


  • Filesize: 0.72 MB
  • 15 pages

Document Identifiers

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.

Cite AsGet BibTex

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


  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    PDF Downloads


  1. Erhard Aichinger, Peter Mayr, and Ralph McKenzie. On the number of finite algebraic structures. Journal of the European Mathematical Society, 16(8):1673-1686, September 2014. URL:
  2. Kirby A. Baker and Alden F. Pixley. Polynomial interpolation and the Chinese Remainder Theorem for algebraic systems. Mathematische Zeitschrift, 143(2):165-174, June 1975. URL:
  3. Libor Barto, Andrei Krokhin, and Ross Willard. Polymorphisms, and How to Use Them. In Andrei Krokhin and Stanislav Zivny, editors, The Constraint Satisfaction Problem: Complexity and Approximability, volume 7 of Dagstuhl Follow-Ups, pages 1-44. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, Dagstuhl, Germany, 2017. URL:
  4. Libor Barto, Jakub Opršal, and Michael Pinsker. The wonderland of reflections. Israel Journal of Mathematics, 223(1):363-398, 2018. URL:
  5. Clifford Bergman. Universal Algebra: Fundamentals and Selected Topics. Chapman and Hall/CRC, New York, November 2011. URL:
  6. Joel Berman, Paweł Idziak, Petar Marković, Ralph McKenzie, Matthew Valeriote, and Ross Willard. Varieties with few subalgebras of powers. Transactions of the American Mathematical Society, 362(3):1445-1473, March 2010. URL:
  7. V. G. Bodnarčuk, L. A. Kalužnin, V. N. Kotov, and B. A. Romov. Galois theory for Post algebras, part I and II. Cybernetics, 5:243-539, 1969. Google Scholar
  8. Zarathustra Brady. Notes on CSPs and Polymorphisms, October 2022. arXiv:2210.07383 [cs, math]. URL:
  9. Andrei Bulatov, Peter Mayr, and Ágnes Szendrei. The Subpower Membership Problem for Finite Algebras with Cube Terms. Logical Methods in Computer Science, Volume 15, Issue 1, February 2019. URL:
  10. Andrei A. Bulatov. A Dichotomy Theorem for Nonuniform CSPs. In 2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS), pages 319-330, October 2017. URL:
  11. S. Burris and H. P. Sankappanavar. A Course in Universal Algebra. Springer New York, November 1981. Google Scholar
  12. Hubie Chen. The expressive rate of constraints. Annals of Mathematics and Artificial Intelligence, 44(4):341-352, August 2005. URL:
  13. Tomás Feder and Moshe Y. Vardi. The Computational Structure of Monotone Monadic SNP and Constraint Satisfaction: A Study through Datalog and Group Theory. SIAM Journal on Computing, 28(1):57-104, January 1998. URL:
  14. Ralph Freese and Ralph McKenzie. Commutator Theory for Congruence Modular Varieties. CUP Archive, August 1987. Google Scholar
  15. David Geiger. Closed systems of functions and predicates. Pacific Journal of Mathematics, 27:95-100, 1968. Google Scholar
  16. Paweł Idziak, Petar Marković, Ralph McKenzie, Matthew Valeriote, and Ross Willard. Tractability and Learnability Arising from Algebras with Few Subpowers. SIAM Journal on Computing, 39(7):3023-3037, January 2010. URL:
  17. Peter Jeavons. On the algebraic structure of combinatorial problems. Theoretical Computer Science, 200(1):185-204, June 1998. URL:
  18. Peter Jeavons, David Cohen, and Martin C. Cooper. Constraints, consistency and closure. Artificial Intelligence, 101(1-2):251-265, May 1998. Google Scholar
  19. Peter Jeavons, David Cohen, and Marc Gyssens. Closure properties of constraints. Journal of the ACM, 44(4):527-548, July 1997. URL:
  20. Keith A. Kearnes and Ágnes Szendrei. Clones of algebras with parallelogram terms. International Journal of Algebra and Computation, 22(01):1250005, February 2012. URL:
  21. Dexter Kozen. Complexity of finitely presented algebras. In Proceedings of the ninth annual ACM symposium on Theory of computing, STOC '77, pages 164-177, New York, NY, USA, May 1977. Association for Computing Machinery. URL:
  22. Marcin Kozik. A finite set of functions with an EXPTIME-complete composition problem. Theoretical Computer Science, 407(1):330-341, November 2008. URL:
  23. Victor Lagerkvist and Magnus Wahlström. Polynomially Closed Co-clones. In 2014 IEEE 44th International Symposium on Multiple-Valued Logic, pages 85-90, May 2014. URL:
  24. Peter Mayr. The subpower membership problem for Mal'cev algebras. International Journal of Algebra and Computation, 22(07):1250075, November 2012. URL:
  25. Thomas J. Schaefer. The complexity of satisfiability problems. In Proceedings of the tenth annual ACM symposium on Theory of computing, STOC '78, pages 216-226, New York, NY, USA, May 1978. Association for Computing Machinery. URL:
  26. Charles C. Sims. Computational methods in the study of permutation groups. In John Leech, editor, Computational Problems in Abstract Algebra, pages 169-183. Pergamon, January 1970. URL:
  27. Dmitriy Zhuk. A Proof of CSP Dichotomy Conjecture. In 2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS), pages 331-342, October 2017. URL:
  28. Dmitriy Zhuk. A Proof of the CSP Dichotomy Conjecture. Journal of the ACM, 67(5):30:1-30:78, August 2020. URL:
  29. Dmitriy N. Zhuk. Key (critical) relations preserved by a weak near-unanimity function. Algebra universalis, 77(2):191-235, April 2017. URL:
Questions / Remarks / Feedback

Feedback for Dagstuhl Publishing

Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail