Algebra universalis

, 65:213 | Cite as

Quantified constraint satisfaction and the polynomially generated powers property

  • Hubie ChenEmail author


The quantified constraint satisfaction probem (QCSP) is the problem of deciding, given a relational structure and a sentence consisting of a quantifier prefix followed by a conjunction of atomic formulas, whether or not the sentence is true in the structure. The general computational intractability of the QCSP has led to the study of restricted versions of this problem. In this article, we study restricted versions of the QCSP that arise from prespecifying the relations that may occur via a set of relations called a constraint language. A basic tool used is a correspondence that associates an algebra to each constraint language; this algebra can be used to derive information on the behavior of the constraint language.

We identify a new combinatorial property on algebras, the polynomially generated powers (PGP) property, which we show is tightly connected to QCSP complexity. We also introduce another new property on algebras, switchability, which both implies the PGP property and implies positive complexity results on the QCSP. Our main result is a classification theorem on a class of three-element algebras: each algebra is either switchable and hence has the PGP, or provably lacks the PGP. The description of non-PGP algebras is remarkably simple and robust.

2010 Mathematics Subject Classification

Primary: 03B70 Secondary: 03B10 08A70 68Q17 68T27 

Key words and phrases

quantified constraint satisfaction computational complexity dichotomy theorem 


  1. 1.
    Aspvall B., Plass M.F., Tarjan R.E.: A linear-time algorithm for testing the truth of certain quantified boolean formulas. Information Processing Letters 8(3), 121–123 (1979)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Barto, L., Kozik, M.: Constraint satisfaction problems of bounded width. In: Proceedings of the 50th Annual IEEE Symposium on Foundations of Computer Science, FOCS’09, pp. 595–603 (2009)Google Scholar
  3. 3.
    Berman J., Idziak P., Markovic P., McKenzie R., Valeriote M., Willard R.: Varieties with few subalgebras of powers. Transactions of the American Mathematical Society 362(3), 1445–1473 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Böhler E., Creignou N., Reith S., Vollmer H.: Playing with boolean blocks, part I: Post’s lattice with applications to complexity theory. ACM SIGACT-Newsletter 34(4), 38–52 (2003)CrossRefGoogle Scholar
  5. 5.
    Börner, F., Bulatov, A., Krokhin, A., Jeavons, P.: Quantified constraints: Algorithms and complexity. In: Proceedings of Computer Science Logic 2003, pp. 58–70 (2003)Google Scholar
  6. 6.
    Bulatov, A.: Tractable conservative constraint satisfaction problems. In: Proceedings of 18th IEEE Symposium on Logic in Computer Science (LICS ’03), pp. 321–330 (2003)Google Scholar
  7. 7.
    Bulatov A., Dalmau V.: A simple algorithm for mal’tsev constraints. SIAM Journal of Computing 36(1), 16–27 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Bulatov A., Jeavons P., Krokhin A.: Classifying the complexity of constraints using finite algebras. SIAM Journal of Computing 34(3), 720–742 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Bulatov A.A.: A dichotomy theorem for constraint satisfaction problems on a 3-element set. Journal of the ACM (JACM) 53(1), 66–120 (2006)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Büning H.K., Karpinski M., Flögel A.: Resolution for quantified boolean formulas. Information and Computation 117(1), 12–18 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Chen, H.: The computational complexity of quantiifed constraint satisfaction. Ph.D. thesis, Cornell University, August 2004.Google Scholar
  12. 12.
    Chen, H.: Quantified constraint satisfaction and bounded treewidth. In: Proceedings of European Conference on Artificial Intelligence (ECAI) 2004, pp. 161–165 (2004)Google Scholar
  13. 13.
    Chen H.: The expressive rate of constraints. Annals of Mathematics and Artificial Intelligence 44(4), 341–352 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Chen H.: A rendezvous of logic, complexity, and algebra. SIGACT News 37(4), 85–114 (2006)CrossRefGoogle Scholar
  15. 15.
    Chen H.: The complexity of quantified constraint satisfaction: Collapsibility, sink algebras, and the three-element case. SIAM Journal on Computing 37(5), 1674–1701 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Chen, H., Dalmau, V.: From pebble games to tractability: An ambidextrous consistency algorithm for quantified constraint satisfaction. In: Proceedings of Computer Science Logic (CSL) 2005, pp. 232–247 (2005)Google Scholar
  17. 17.
    Creignou, N., Khanna, S., Sudan, M.: Complexity Classification of Boolean Constraint Satisfaction Problems. SIAM Monographs on Discrete Mathematics and Applications. Society for Industrial and Applied Mathematics (2001)Google Scholar
  18. 18.
    Dalmau, V.: Computational complexity of problems over generalized formulas. Ph.D. Thesis, Universitat Politècnica de CatalunyaGoogle Scholar
  19. 19.
    Dalmau, V.: Generalized majority-minority operations are tractable. In: Proceedings of 20th IEEE Symposium on Logic in Computer Science (LICS ’05), pp. 438–447 (2005)Google Scholar
  20. 20.
    Ebbinghaus, H., Flum, J., Thomas, W.: Mathematical Logic. Springer-Verlag (1984)Google Scholar
  21. 21.
    Gottlob, G., Greco, G., Scarcello, F.: The complexity of quantified constraint satisfaction problems under structural restrictions. In: IJCAI (2005)Google Scholar
  22. 22.
    Grädel E.: Capturing complexity classes by fragments of second order logic. Theoretical Computer Science 101, 35–57 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Idziak, P., Markovic, P., McKenzie, R., Valeriote, M., Willard, R.: Tractability and learnability arising from algebras with few subpowers (extended abstract). In: Proceedings of 22nd IEEE Symposium on Logic in Computer Science (LICS ’07), pp. 213-224 (2007)Google Scholar
  24. 24.
    Jeavons P.: On the algebraic structure of combinatorial problems. Theoretical Computer Science 200, 185–204 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Jeavons P., Cohen D., Gyssens M.: Closure properties of constraints. Journal of the ACM 44, 527–548 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Karpinski, M., Büning, H.K., Schmitt, P.H.: On the computational complexity of quantified horn clauses. In: Proceedings of Computer Science Logic 1987, pp. 129–137 (1987)Google Scholar
  27. 27.
    Kiss, E., Valeriote, M.: On tractability and congruence distributivity. In: Proceedings of 21st IEEE Symposium on Logic in Computer Science (LICS ’06), pp. 221–230 (2006)Google Scholar
  28. 28.
    McKenzie, R., McNulty, G., Taylor, W.: Algebras, Lattices and Varieties, vol. I. Wadsworth and Brooks, California (1987)Google Scholar
  29. 29.
    Pan, G., Vardi, M.Y.: Fixed-parameter hierarchies inside pspace. In: Proceedings of 21st IEEE Symposium on Logic in Computer Science (LICS ’06), pp. 27–36 (2006)Google Scholar
  30. 30.
    Schaefer, T.J.: The complexity of satisfiability problems. In: Proceedings of the ACM Symposium on Theory of Computing (STOC), pp. 216–226 (1978)Google Scholar
  31. 31.
    Szendrei, A.: Clones in Universal Algebra, Seminaires de Mathematiques Superieures, vol. 99. University of Montreal (1986)Google Scholar

Copyright information

© Springer Basel AG 2011

Authors and Affiliations

  1. 1.Departament de Tecnologies de la Informació i les ComunicacionsUniversitat Pompeu FabraBarcelonaSpain

Personalised recommendations