Local and global relational consistency

  • Rina Dechter
  • Peter van Beek
Constraint Satisfaction Problems 2
Part of the Lecture Notes in Computer Science book series (LNCS, volume 976)


Local consistency has proven to be an important concept in the theory and practice of constraint networks. In this paper, we present a new definition of local consistency, called relational consistency. The new definition is relationbased, in contrast with the previous definition of local consistency, which we characterize as variable-based. It allows the unification of known elimination operators such as resolution in theorem proving, joins in relational databases and variable elimination for solving linear inequalities. We show the usefulness and conceptual power of the new definition in characterizing relationships between four properties of constraints — domain tightness, row-convexity, constraint tightness, and constraint looseness — and the level of local consistency needed to ensure global consistency. As well, algorithms tor enforcing relational consistency are introduced and analyzed.


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  1. 1.
    S. Arnborg, D. G. Corneil, and A. Proskurowski. Complexity of finding an embedding in k-trees. SIAM Journal of Algebraic Discrete Methods, 8:177–184, 1987.Google Scholar
  2. 2.
    C. Beeri, R. Fagin, D. Maier, and M. Yannakakis. On the desirability of acyclic database schemes. J. ACM, 30:479–513, 1983.CrossRefGoogle Scholar
  3. 3.
    M. C. Cooper. An optimal k-consistency algorithm. Artif. Intell., 41:89–95, 1989.CrossRefGoogle Scholar
  4. 4.
    M. Davis and H. Putnam. A computing procedure for quantification theory. J. ACM, 7:201–215, 1960.CrossRefGoogle Scholar
  5. 5.
    R. Dechter. From local to global consistency. Artif. Intell., 55:87–107, 1992.CrossRefGoogle Scholar
  6. 6.
    R. Dechter, I. Meiri, and J. Pearl. Temporal constraint networks. Artif. Intell., 49:61–95, 1991.MathSciNetGoogle Scholar
  7. 7.
    R. Dechter and J. Pearl. Network-based heuristics for constraint satisfaction problems. Artif. Intell., 34:1–38, 1988.CrossRefGoogle Scholar
  8. 8.
    R. Dechter and J. Pearl. Tree clustering for constraint networks. Artif. Intell., 38:353–366, 1989.CrossRefGoogle Scholar
  9. 9.
    R. Dechter and J. Pearl. Directed constraint networks: A relational framework for causal modeling. In Proc. of the 12th Int'l Joint Conf. on AI, pages 1164–1170, 1991.Google Scholar
  10. 10.
    R. Dechter and I. Rish. Directional resolution: The Davis-Putnam procedure, revisited. In Proc. of the 4th Int'l Conf. on Principles of KR&R, 1994.Google Scholar
  11. 11.
    E. C. Freuder. Synthesizing constraint expressions. Comm. ACM, 21:958–966, 1978.CrossRefGoogle Scholar
  12. 12.
    E. C. Freuder. A sufficient condition for backtrack-free search. J. ACM, 29:24–32, 1982.CrossRefGoogle Scholar
  13. 13.
    M. L. Ginsberg, M. Frank, M. P. Halpin, and M. C. Torrance. Search lessons learned from crossword puzzles. In Proc. of the 8th Nat'l Conf. on AI, pages 210–215, 1990.Google Scholar
  14. 14.
    P. Jégou. On the consistency of general constraint satisfaction problems. In Proc. of the 11th National Conf. on AI, pages 114–119, 1993.Google Scholar
  15. 15.
    L. M. Kirousis. Fast parallel constraint satisfaction. Artif. Intell., 64:147–160, 1993.CrossRefGoogle Scholar
  16. 16.
    J-L Lassez and M. Mahler, “On Fourier's algorithm for linear constraints” Journal of Automated Reasoning, Vol 9, 1992.Google Scholar
  17. 17.
    A. K. Mackworth. Consistency in networks of relations. Artif. Intell., 8:99–118, 1977.CrossRefGoogle Scholar
  18. 18.
    A. K. Mackworth and E. C. Freuder. The complexity of some polynomial network consistency algorithms for constraint satisfaction problems. Artif. Intell., 25:65–74, 1985.CrossRefGoogle Scholar
  19. 19.
    D. Maier. The Theory of Relational Databases. Computer Science Press, 1983.Google Scholar
  20. 20.
    U. Montanari. Networks of constraints: Fundamental properties and applications to picture processing. Inform. Sci., 7:95–132, 1974.CrossRefGoogle Scholar
  21. 21.
    B. A. Nadel. Constraint satisfaction algorithms. Comput. Intell., 5:188–224, 1989.Google Scholar
  22. 22.
    P. van Beek. On the minimality and decomposability of constraint networks. In Proc. of the 10th National Conf. on AI, pages 447–452, 1992.Google Scholar
  23. 23.
    P. van Beek. On the inherent level of local consistency in constraint networks. In Proc. of the 12th National Conf. on AI, pages 368–373, 1994.Google Scholar
  24. 24.
    P. van Beek and R. Dechter. Constraint tightness versus global consistency. In Proc. of the 4th Int'l Conf. on Principles of KR&R, pages 572–582, 1994.Google Scholar
  25. 25.
    P. van Beek and R. Dechter. On the minimality and global consistency of rowconvex constraint networks. To appear in J. ACM, 1995.Google Scholar
  26. 26.
    P. Van Hentenryck, Y. Deville, and C.-M. Teng. A generic arc consistency algorithm and its specializations. Artif. Intell., 57:291–321, 1992.CrossRefGoogle Scholar
  27. 27.
    M. Vilain and H. Kautz. Constraint propagation algorithms for temporal reasoning. In Proc. of the 5th National Conf. on AI, pages 377–382, 1986.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1995

Authors and Affiliations

  • Rina Dechter
    • 1
  • Peter van Beek
    • 2
  1. 1.Department of Information and Computer ScienceUniversity of CaliforniaIrvine IrvineUSA
  2. 2.Department of Computing ScienceUniversity of AlbertaEdmontonCanada

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