Document Open Access Logo

Consistency for Counting Quantifiers

Authors Florent R. Madelaine, Barnaby Martin



PDF
Thumbnail PDF

File

LIPIcs.MFCS.2018.11.pdf
  • Filesize: 454 kB
  • 13 pages

Document Identifiers

Author Details

Florent R. Madelaine
  • LIMOS, Université d'Auvergne, Clermont-Ferrand, France
Barnaby Martin
  • Department of Computer Science, Durham University, U.K.

Cite AsGet BibTex

Florent R. Madelaine and Barnaby Martin. Consistency for Counting Quantifiers. In 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 117, pp. 11:1-11:13, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)
https://doi.org/10.4230/LIPIcs.MFCS.2018.11

Abstract

We apply the algebraic approach for Constraint Satisfaction Problems (CSPs) with counting quantifiers, developed by Bulatov and Hedayaty, for the first time to obtain classifications for computational complexity. We develop the consistency approach for expanding polymorphisms to deduce that, if H has an expanding majority polymorphism, then the corresponding CSP with counting quantifiers is tractable. We elaborate some applications of our result, in particular deriving a complexity classification for partially reflexive graphs endowed with all unary relations. For each such structure, either the corresponding CSP with counting quantifiers is in P, or it is NP-hard.

Subject Classification

ACM Subject Classification
  • Theory of computation → Constraint and logic programming
Keywords
  • Quantified Constraints
  • Constraint Satisfaction
  • Logic in Computer Science
  • Universal Algebra
  • Computational Complexity

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads

References

  1. Manuel Bodirsky, Víctor Dalmau, Barnaby Martin, and Michael Pinsker. Distance constraint satisfaction problems. In Mathematical Foundations of Computer Science 2010, 35th International Symposium, MFCS 2010, Brno, Czech Republic, August 23-27, 2010. Proceedings, pages 162-173, 2010. URL: http://dx.doi.org/10.1007/978-3-642-15155-2_16.
  2. F. Börner, A. Krokhin, A. Bulatov, and P. Jeavons. Quantified constraints and surjective polymorphisms. Technical Report PRG-RR-02-11, Oxford University, 2002. Google Scholar
  3. Richard C. Brewster, Tomás Feder, Pavol Hell, Jing Huang, and Gary MacGillivray. Near-unanimity functions and varieties of reflexive graphs. SIAM J. Discrete Math., 22(3):938-960, 2008. URL: http://dx.doi.org/10.1137/S0895480103436748.
  4. A. Bulatov. Tractable conservative constraint satisfaction problems. In Proceedings of LICS'03, pages 321-330, 2003. Google Scholar
  5. A. Bulatov, A. Krokhin, and P. G. Jeavons. Classifying the complexity of constraints using finite algebras. SIAM Journal on Computing, 34:720-742, 2005. Google Scholar
  6. Andrei A. Bulatov. A dichotomy theorem for nonuniform CSPs. Proc. FOCS 2017, pages 319-330, 2017. Google Scholar
  7. Andrei A. Bulatov and Amir Hedayaty. Counting predicates, subset surjective functions, and counting csps. In 42nd IEEE International Symposium on Multiple-Valued Logic, ISMVL 2012, pages 331-336, 2012. Google Scholar
  8. Andrei A. Bulatov and Amir Hedayaty. Galois correspondence for counting quantifiers. Multiple-Valued Logic and Soft Computing, 24:405-424, 2015. Google Scholar
  9. Jin-Yi Cai, Martin Fürer, and Neil Immerman. An optimal lower bound on the number of variables for graph identification. Combinatorica, 12(4):389-410, Dec 1992. URL: http://dx.doi.org/10.1007/BF01305232.
  10. Hubie Chen. The complexity of quantified constraint satisfaction: Collapsibility, sink algebras, and the three-element case. SIAM J. Comput., 37(5):1674-1701, 2008. URL: http://dx.doi.org/10.1137/060668572.
  11. Robin Clark and Murray Grossman. Number sense and quantifier interpretation. Topoi, 26(1):51-62, 2007. Google Scholar
  12. H.-D. Ebbinghaus and J. Flum. Finite Model Theory. Springer, 1999. 2nd edition. Google Scholar
  13. T. Feder and M. Vardi. The computational structure of monotone monadic SNP and constraint satisfaction: A study through Datalog and group theory. SIAM Journal on Computing, 28:57-104, 1999. Google Scholar
  14. Tomás Feder, Pavol Hell, and Jing Huang. List homomorphisms and circular arc graphs. Combinatorica, 19(4):487-505, 1999. URL: http://link.springer.de/link/service/journals/00493/bibs/9019004/90190487.htm.
  15. Tomás Feder, Pavol Hell, and Jing Huang. Bi-arc graphs and the complexity of list homomorphisms. J. Graph Theory, 42:61-80, 2003. Google Scholar
  16. E.C. Freuder. A sufficient condition for backtrack-free search. Journal of the ACM, 29(1):24-32, 1982. Google Scholar
  17. P. Hell and J. Nešetřil. Graphs and Homomorphisms. OUP, 2004. Google Scholar
  18. Peter Jeavons, David Cohen, and Martin Cooper. Constraints, consistency and closure. AI, 101(1-2):251-265, 1998. Google Scholar
  19. P. G. Kolaitis and M. Y. Vardi. Finite Model Theory and Its Applications (Texts in Theoretical Computer Science. An EATCS Series), chapter A logical Approach to Constraint Satisfaction. Springer-Verlag New York, Inc., 2005. Google Scholar
  20. Benoit Larose, Barnaby Martin, and Daniël Paulusma. Surjective h-colouring over reflexive digraphs. In Rolf Niedermeier and Brigitte Vallée, editors, 35th Symposium on Theoretical Aspects of Computer Science, STACS 2018, February 28 to March 3, 2018, Caen, France, volume 96 of LIPIcs, pages 49:1-49:14. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2018. URL: http://dx.doi.org/10.4230/LIPIcs.STACS.2018.49.
  21. Barnaby Martin, Florent R. Madelaine, and Juraj Stacho. Constraint satisfaction with counting quantifiers. SIAM J. Discrete Math., 29(2):1065-1113, 2015. URL: http://dx.doi.org/10.1137/140981332.
  22. M. Otto. Bounded variable logics and counting - A study in finite models, volume 9. Springer-Verlag, 1997. IX+183 pages. Google Scholar
  23. Dmitriy Zhuk. A proof of CSP dichotomy conjecture. Proc. FOCS 2017, pages 331-342, 2017. Google Scholar
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail