skip to main content
research-article

A rendezvous of logic, complexity, and algebra

Published: 14 December 2009 Publication History

Abstract

An emerging area of research studies the complexity of constraint satisfaction problems under restricted constraint languages. This article gives a self-contained, contemporary presentation of Schaefer's theorem on Boolean constraint satisfaction, the inaugural result of this area, as well as analogs of this theorem for quantified formulas. Our exposition makes use of and may serve as an introduction to logical and algebraic tools that have recently come into focus.

References

[1]
Allender, E., Bauland, M., Immerman, N., Schnoor, H., and Vollmer, H. 2005. The complexity of satisfiability problems: Refining Schaefer's theorem. In Proceedings of the 30th International Symposium on Mathematical Foundations of Computer Science (MFCS). Lecture Notes in Computer Science, vol. 3618. Springer, Berlin, Germany, 71--82.
[2]
Aspvall, B., Plass, M. F., and Tarjan, R. E. 1979. A linear-time algorithm for testing the truth of certain quantified Boolean formulas. Inform. Proc. Lett. 8, 3, 121--123.
[3]
Atserias, A. 2005. On digraph coloring problems and treewidth duality. In Proceedings of the LICS.
[4]
Bauland, M., Böhler, E., Creignou, N., Reith, S., Schnoor, H., and Vollmer, H. 2005. Quantified constraints: The complexity of decision and counting for bounded alternation. ECCC Tech. rep. TR05-24. http://eccc.uni-trier.de/zear/2005.
[5]
Bodirsky, M. 2004. Constraint satisfaction with infinite domains. Ph.D. dissertation, Humboldt-Universitat zu Berlin, Berlin, Germany.
[6]
Bodirsky, M. 2005. The core of a countably categorical structure. In Proceedings of the 22nd Annual Symposium on Theoretical Aspects of Computer Science (STACS'05), V. Diekert and B. Durand, Eds. Lecture Notes in Computer Science, vol. 3404. Springer-Verlag, Berlin Stuttgart (Germany), Heidelberg, Germany, 100--110.
[7]
Bodirsky, M. and Chen, H. 2006. Collapsibility in infinite-domain quantified constraint satisfaction. In Proceedings of CSL.
[8]
Bodirsky, M. and Chen, H. 2007. Quantified equality constraints. In Proceedings of LICS.
[9]
Bodirsky, M. and Dalmau, V. 2006. Datalog for constraint satisfaction with infinite domains. In Proceedings of STACS.
[10]
Bodirsky, M. and Kára, J. 2006. The complexity of equality constraint languages. In Proceedings of the International Computer Science Symposium in Russia (CSR).
[11]
Bodnarchuk, V. G., Kaluzhnin, L. A., Kotov, V. N., and Romov, B. A. 1969. Galois theory for post algebras. I, II. Cybernet. 5, 243--252, 531--539.
[12]
Böhler, E., Creignou, N., Reith, S., and Vollmer, H. 2003. Playing with Boolean blocks, part I: Post's lattice with applications to complexity theory. ACM SIGACT-Newslett. 34, 4, 38--52.
[13]
Böhler, E., Creignou, N., Reith, S., and Vollmer, H. 2004. Playing with Boolean blocks, part II: Constraint satisfaction problems. ACM SIGACT-Newslett. 35, 1, 22--35.
[14]
Börner, F., Bulatov, A., Krokhin, A., and Jeavons, P. 2003. Quantified constraints: Algorithms and complexity. In Proceedings of the International Conference on Computer Science Logic. Lecture Notes in Computer Science, vol. 2803. Springer, Berlin, Germany, 58--70.
[15]
Bulatov, A. 2003. Tractable conservative constraint satisfaction problems. In Proceedings of the 18th IEEE Symposium on Logic in Computer Science (LICS). 321--330.
[16]
Bulatov, A. 2004. A graph of a relational structure and constraint satisfaction problems. In Proceedings of the 19th IEEE Annual Symposium on Logic in Computer Science (LICS).
[17]
Bulatov, A. 2005. H-coloring dichotomy revisited. Theoret. Comput. Sci. 349, 1, 31--39.
[18]
Bulatov, A. A. 2006. A dichotomy theorem for constraint satisfaction problems on a 3-element set. J. ACM 53, 1, 66--120.
[19]
Bulatov, A. and Dalmau, V. 2006. A simple algorithm for mal'tsev constraints. SIAM J. Comput. 36, 1, 16--27.
[20]
Bulatov, A. and Jeavons, P. 2003. An algebraic approach to multi-sorted constraints. In Proceedings of CP.
[21]
Bulatov, A., Jeavons, P., and Krokhin, A. 2005. Classifying the complexity of constraints using finite algebras. SIAM J. Comput. 34, 3, 720--742.
[22]
Büning, H. K., Karpinski, M., and Flögel, A. 1995. Resolution for quantified boolean formulas. Inform. Computat. 117, 1, 12--18.
[23]
Burris, S. N. and Sankappanavar, H. 1981. A Course in Universal Algebra. www.math.uwaterloo.ca/~snburris/htclocs/ualg.html.
[24]
Chen, H. 2009. Existentially restricted quantified constraint satisfaction. Inform. Computat. 207, 3 (Mar.) 369--388.
[25]
Chen, H. 2008. The complexity of quantified constraint satisfaction: Collapsibility, sink algebras, and the three-element case. SIAM J. Comput. 37, 5, 1674--1701.
[26]
Chen, H. and Dalmau, V. 2005. From pebble games to tractability: An ambidextrous consistency algorithm for quantified constraint satisfaction. In Computer Science Logic. Lecture Notes in Computer Science, vol. 3634. Springer, Berlin/Heidelberg, Germany, 232--247.
[27]
Cohen, D. and Jeavons, P. 2006. The complexity of constraint languages. Handbook of Constraint Programming, Chapter 8. Elsevier, Amsterdam, The Netherlands.
[28]
Creignou, N., Khanna, S., and Sudan, M. 2001. Complexity Classification of Boolean Constraint Satisfaction Problems. SIAM Monographs on Discrete Mathematics and Applications. Society for Industrial and Applied Mathematics, Philadelphia, PA.
[29]
Dalmau, V. 1997. Some dichotomy theorems on constant-free quantified Boolean formulas. Tech. rep. LSI-97-43-R. Llenguatges i Sistemes Informàtics—Universitat Politècnica de Catalunya, Barcelona, Spain.
[30]
Dalmau, V. 2000. Computational complexity of problems over generalized formulas. Ph.D. dissertation. Department of Computer Science, Polytechnical University of Catalonia, Barcelona, Spain.
[31]
Dalmau, V. 2005a. Generalized majority-minority operations are tractable. In Proceedings of LICS.
[32]
Dalmau, V. 2005b. Linear datalog and bounded path duality of relational structures. Log. Meth. Comput. Sci. 1, 1.
[33]
Dalmau, V. and Pearson, J. 1999. Closure functions and width 1 problems. In Proceedings of CP. 159--173.
[34]
Feder, T. and Vardi, M. Y. 1993. Monotone monadic snp and constraint satisfaction. In Proceedings of STOC. 612--622.
[35]
Feder, T. and Vardi, M. Y. 1998. The computational structure of monotone monadic snp and constraint satisfaction: A study through datalog and group theory. SIAM J. Comput. 28, 1, 57--104.
[36]
Geiger, D. 1968. Closed systems of functions and predicates. Pac. J. Math. 27, 95--100.
[37]
Grädel, E. 1992. Capturing complexity classes by fragments of second order logic. Theoret. Comput. Sci. 101, 35--57.
[38]
Hell, P. and Nešetřil, J. 1990. On the complexity of H-coloring. J. Combin. Theor. Ser.B 48, 92--110.
[39]
Hemaspaandra, E. 2004. Dichotomy theorems for alternation-bounded quantified Boolean formulas. eprint. arXiv.cs/0406006.
[40]
Jeavons, P. 1998. On the algebraic structure of combinatorial problems. Theoret. Comput. Sci. 200, 185--204.
[41]
Jeavons, P., Cohen, D., and Cooper, M. 1998. Constraints, consistency, and closure. Art. Intell. 101, 1-2, 251--265.
[42]
Jeavons, P., Cohen, D., and Gyssens, M. 1997. Closure properties of constraints. J. ACM 44, 527--548.
[43]
Karpinski, M., Büning, H. K., and Schmitt, P. H. 1987. On the computational complexity of quantified horn clauses. In Proceedings of CSL. 129--137.
[44]
Kiss, E. and Valeriote, M. 2006. On tractability and congruence distributivity. In Proceedings of LICS.
[45]
Klíma, O., Tesson, P., and Thérien, D. 2004. Dichotomies in the complexity of solving systems of equations over finite semigroups. ECCC Tech rep. TR04-091. http://eccc.uni-trier.de/year/2004.
[46]
Kozen, D. 1981. Positive first-order logic is NP-complete. IBM J. Res. Devel. 25, 4, 327--332.
[47]
Krokhin, A., Bulatov, A., and Jeavons, P. 2003a. The complexity of constraint satisfaction: An algebraic approach. In Proceedings of SMS-NATO ASI. 181--213.
[48]
Krokhin, A., Bulatov, A., and Jeavons, P. 2003b. Functions of multiple-valued logic and the complexity of constraint satisfaction: A short survey. In Proceedings of the 33rd IEEE International Symposium on Multiple-Valued Logic (ISMVL). 343--351.
[49]
Larose, B., Loten, C., and Tardif, C. 2006. A characterisation of first-order constraint satisfaction problems. In Proceedings of LICS.
[50]
Larose, B. and Zádori, L. 2007. Bounded width problems and algebras. Alg. Universalis 56, 3-4 (June), 439--466.
[51]
Larose, B. and Zádori, L. 2006. Taylor terms, constraint satisfaction and the complexity of polynomial equations over finite algebras. Int. J. Alg. Comput. 16, 3, 563--581.
[52]
McKenzie, R., McNulty, G., and Taylor, W. 1987. Algebras, Lattices and Varieties. Vol. I. Wadsworth and Brooks/Cole, Belmount, CA.
[53]
Post, E. L. 1941. The Two-Valued Iterative Systems of Mathematical Logic. Princeton University Press, Princeton, NJ.
[54]
Reingold, O. 2004. Undirected ST-connectivity in log-space. ECCC Tech. rep. TR04-094. http:eccc.uni-trier.de/year/2004.
[55]
Rosenberg, I. 1986. Minimal clones I: The five types. In Lectures in Universal Algebra (Proceedings of the Conference on Szeged 1983). Colloq. Math. Soc. Janos Bolyai, vol. 43. North-Holland, Amsterdam, The Netherlands, 405--427.
[56]
Schaefer, T. J. 1978. The complexity of satisfiability problems. In Proceedings of the ACM Symposium on Theory of Computing (STOC). 216--226.
[57]
Stockmeyer, L. 1976. The polynomial-time hierarchy. Theoret. Comput. Sci. 3, 1, 1--22.
[58]
Stockmeyer, L. and Meyer, A. R. 1973. Word problems requiring exponential time. In Proceedings of the 5th ACM Symposium on the Theory of Computing. 1--10.
[59]
Szendrei, A. 1986. Clones in Universal Algebra. Seminaires de Mathematiques Superieures, vol. 99. University of Montreal, Montreal, P.Q., Canada.
[60]
Wrathall, C. 1976. Complete sets and the polynomial-time hierarchy. Theoret. Comput. Sci. 3, 1, 23--33.

Cited By

View all
  • (2024)Estimating the Weight Enumerators of Reed-Muller Codes via Sampling2024 IEEE International Symposium on Information Theory (ISIT)10.1109/ISIT57864.2024.10619284(280-285)Online publication date: 7-Jul-2024
  • (2023)The computational complexity of concise hypersphere classificationProceedings of the 40th International Conference on Machine Learning10.5555/3618408.3618771(9060-9070)Online publication date: 23-Jul-2023
  • (2023)SDPs and Robust Satisfiability of Promise CSPProceedings of the 55th Annual ACM Symposium on Theory of Computing10.1145/3564246.3585180(609-622)Online publication date: 2-Jun-2023
  • Show More Cited By

Recommendations

Comments

Information & Contributors

Information

Published In

cover image ACM Computing Surveys
ACM Computing Surveys  Volume 42, Issue 1
December 2009
162 pages
ISSN:0360-0300
EISSN:1557-7341
DOI:10.1145/1592451
Issue’s Table of Contents
Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

Publisher

Association for Computing Machinery

New York, NY, United States

Publication History

Published: 14 December 2009
Accepted: 01 June 2008
Revised: 01 May 2008
Received: 01 September 2007
Published in CSUR Volume 42, Issue 1

Permissions

Request permissions for this article.

Check for updates

Author Tags

  1. Constraint satisfaction
  2. Schaefer's theorem
  3. polymorphisms
  4. propositional logic
  5. quantified formulas

Qualifiers

  • Research-article
  • Research
  • Refereed

Contributors

Other Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

  • Downloads (Last 12 months)46
  • Downloads (Last 6 weeks)8
Reflects downloads up to 20 Jan 2025

Other Metrics

Citations

Cited By

View all
  • (2024)Estimating the Weight Enumerators of Reed-Muller Codes via Sampling2024 IEEE International Symposium on Information Theory (ISIT)10.1109/ISIT57864.2024.10619284(280-285)Online publication date: 7-Jul-2024
  • (2023)The computational complexity of concise hypersphere classificationProceedings of the 40th International Conference on Machine Learning10.5555/3618408.3618771(9060-9070)Online publication date: 23-Jul-2023
  • (2023)SDPs and Robust Satisfiability of Promise CSPProceedings of the 55th Annual ACM Symposium on Theory of Computing10.1145/3564246.3585180(609-622)Online publication date: 2-Jun-2023
  • (2022)Ideal Membership Problem over 3-Element CSPs with Dual Discriminator PolymorphismSIAM Journal on Discrete Mathematics10.1137/21M139713136:3(1800-1822)Online publication date: 1-Jan-2022
  • (2022)Constraint Satisfaction Problems with Global Modular Constraints: Algorithms and Hardness via Polynomial RepresentationsSIAM Journal on Computing10.1137/19M129105451:3(577-626)Online publication date: 19-May-2022
  • (2022)Testability of relations between permutations2021 IEEE 62nd Annual Symposium on Foundations of Computer Science (FOCS)10.1109/FOCS52979.2021.00037(286-297)Online publication date: Feb-2022
  • (2021)Promise Constraint Satisfaction: Algebraic Structure and a Symmetric Boolean DichotomySIAM Journal on Computing10.1137/19M128212X50:6(1663-1700)Online publication date: 18-Nov-2021
  • (2021)Verified Model Checking for Conjunctive Positive LogicSN Computer Science10.1007/s42979-020-00417-32:5Online publication date: 19-Jun-2021
  • (2021)Finding and Counting Permutations via CSPsAlgorithmica10.1007/s00453-021-00812-z83:8(2552-2577)Online publication date: 1-Aug-2021
  • (2020)The complete set of minimal simple graphs that support unsatisfiable 2-CNFsDiscrete Applied Mathematics10.1016/j.dam.2019.12.017283(123-132)Online publication date: Sep-2020
  • Show More Cited By

View Options

Login options

Full Access

View options

PDF

View or Download as a PDF file.

PDF

eReader

View online with eReader.

eReader

Media

Figures

Other

Tables

Share

Share

Share this Publication link

Share on social media