Journal of Automated Reasoning

, Volume 31, Issue 2, pp 129–168 | Cite as

Abstract Congruence Closure

  • Leo Bachmair
  • Ashish Tiwari
  • Laurent Vigneron


We describe the concept of an abstract congruence closure and provide equational inference rules for its construction. The length of any maximal derivation using these inference rules for constructing an abstract congruence closure is at most quadratic in the input size. The framework is used to describe the logical aspects of some well-known algorithms for congruence closure. It is also used to obtain an efficient implementation of congruence closure. We present experimental results that illustrate the relative differences in performance of the different algorithms. The notion is extended to handle associative and commutative function symbols, thus providing the concept of an associative-commutative congruence closure. Congruence closure (modulo associativity and commutativity) can be used to construct ground convergent rewrite systems corresponding to a set of ground equations (containing AC symbols).

term rewriting congruence closure associative-commutative theories 


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  1. 1.
    Bachmair, L.: Canonical Equational Proofs, Birkhäuser, Boston, 1991.MATHGoogle Scholar
  2. 2.
    Bachmair, L. and Dershowitz, N.: Completion for rewriting modulo a congruence, Theoret. Comput. Sci. 67(2 & 3) (Oct. 1989), 173-201.MATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Bachmair, L. and Dershowitz, N.: Equational inference, canonical proofs, and proof orderings, J. ACM 41 (1994), 236-276.MATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Bachmair, L., Ramakrishnan, I., Tiwari, A. and Vigneron, L.: Congruence closure modulo Associativity-Commutativity, in H. Kirchner and C. Ringeissen (eds), Frontiers of Combining Systems, Third International Workshop, FroCoS 2000, Nancy, France, March 2000, Lecture Notes in Artificial Intelligence 1794, Springer, Berlin, 2000, pp. 245-259.Google Scholar
  5. 5.
    Bachmair, L. and Tiwari, A.: Abstract congruence closure and specializations, in D. McAllester (ed.), Conference on Automated Deduction, CADE 2000, Pittsburgh, PA, June2000, Lecture Notes in Artificial Intelligence 1831, Springer, Berlin, 2000, pp. 64-78.Google Scholar
  6. 6.
    Bachmair, L. and Tiwari, A.: Congruence closure and syntactic unification, in C. Lynch and D. Narendran (eds), 14th International Workshop on Unification, 2000.Google Scholar
  7. 7.
    Ballantyne, A. M. and Lankford, D. S.: New decision algorithms for finitely presented commutative semigroups, Comp. Math. Appl. 7 (1981), 159-165.MathSciNetCrossRefGoogle Scholar
  8. 8.
    Becker, T. and Weispfenning, V.: Gröbner Bases: A computational Approach to Commutative Algebra, Springer-Verlag, Berlin, 1993.MATHGoogle Scholar
  9. 9.
    Cardozo, E., Lipton, R. and Meyer, A.: Exponential space complete problems for petri nets and commutative semigroups, in Proc. 8th Ann. ACM Symp on Theory of Computing, 1976, pp. 50-54.Google Scholar
  10. 10.
    Comon, H., Dauchet, M., Gilleron, R., Jacquemard, F., Lugiez, D., Tison, S. and Tommasi, M.: Tree automata techniques and applications. Available on:, 1997.Google Scholar
  11. 11.
    Cyrluk, D., Lincoln, P. and Shankar, N.: On Shostak’s decision procedure for combination of theories, in M. A. McRobbie and J. Slaney (eds), Proceedings of the 13th Int. Conference on Automated Deduction, Lecture Notes in Comput. Sci. 1104, Springer, Berlin, 1996, pp. 463-477.Google Scholar
  12. 12.
    Dershowitz, N. and Jouannaud, J. P.: Rewrite systems, in J. van Leeuwen (ed.), Handbook of Theoretical Computer Science (Vol. B: Formal Models and Semantics), North-Holland, Amsterdam, 1990.Google Scholar
  13. 13.
    Dershowitz, N. and Manna, Z.: Proving termination with multiset orderings, Comm. ACM 22(8) (1979), 465-476.MATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    Domenjoud, E. and Klay, F.: Shallow AC theories, in Proceedings of the 2nd CCL Workshop, La Escala, Spain, Sept. 1993.Google Scholar
  15. 15.
    Downey, P. J., Sethi, R. and Tarjan, R. E.: Variations on the common subexpressions problem, J. ACM 27(4) (1980), 758-771.MATHMathSciNetCrossRefGoogle Scholar
  16. 16.
    Evans, T.: The word problem for abstract algebras, J. London Math. Soc. 26 (1951), 64-71.MATHMathSciNetGoogle Scholar
  17. 17.
    Evans, T.: Word problems, Bull. Amer. Math. Soc. 84(5) (1978), 789-802.MATHMathSciNetCrossRefGoogle Scholar
  18. 18.
    Kapur, D.: Shostak’s congruence closure as completion, in H. Comon (ed.), Rewriting Techniques and Applications, RTA 1997, Sitges, Spain, July 1997, Lecture Notes in Comput. Sci. 1103, Springer, Berlin, pp. 23-37.Google Scholar
  19. 19.
    Koppenhagen, U. and Mayr, E.W.: An optimal algorithm for constructing the reduced Gröbner basis of binomial ideals, in Y. D. Lakshman (ed.), Proceedings of the International Symposium on Symbolic and Algebraic Computation, 1996, pp. 55-62.Google Scholar
  20. 20.
    Marche, C.: On ground AC-completion, in R. V. Book (ed.), 4th International Conference on Rewriting Techniques and Applications, Lecture Notes in Comput. Sci. 488, Springer, Berlin, 1991, pp. 411-422.Google Scholar
  21. 21.
    Mayr, E. W. and Meyer, A. R.: The complexity of the word problems for commutative semigroups and polynomial ideals, Adv. in Math. 46 (1982), 305-329.MATHMathSciNetCrossRefGoogle Scholar
  22. 22.
    Narendran, P. and Rusinowitch, M.: Any ground associative-commutative theory has a finite canonical system, in R. V. Book (ed.), 4th International Conference on Rewriting Techniques and Applications, Lecture Notes in Comput. Sci. 488, Springer, Berlin, 1991, pp. 423-434.Google Scholar
  23. 23.
    Nelson, G. and Oppen, D.: Fast decision procedures based on congruence closure, J. Assoc. Comput. Mach. 27(2) (Apr. 1980), 356-364.MATHMathSciNetGoogle Scholar
  24. 24.
    Peterson, G. E. and Stickel, M. E.: Complete sets of reductions for some equational theories, J. ACM 28(2) (Apr. 1981), 233-264.MATHMathSciNetCrossRefGoogle Scholar
  25. 25.
    Plaisted, D. and Sattler-Klein, A.: Proof lengths for equational completion, Inform. and Comput. 125 (1996), 154-170.MATHMathSciNetCrossRefGoogle Scholar
  26. 26.
    Rubio, A. and Nieuwenhuis, R.: A precedence-based total AC-compatible ordering, in C. Kirchner (ed.), Proceedings of the 5 Intl. Conference on Rewriting Techniques and Applications, Lecture Notes in Comput. Sci. 960, Springer, Berlin, 1993, pp. 374-388.Google Scholar
  27. 27.
    Sherman, D. J. and Magnier, N.: Factotum: Automatic and systematic sharing support for systems analyzers, in Proc. TACAS, Lecture Notes in Comput. Sci. 1384, 1998.Google Scholar
  28. 28.
    Shostak, R. E.: Deciding combinations of theories, J. ACM 31(1) (1984), 1-12.MATHMathSciNetCrossRefGoogle Scholar
  29. 29.
    Snyder, W.: A fast algorithm for generating reduced ground rewriting systems from a set of ground equations, J. Symbolic Comput. 15(7) (1993).Google Scholar
  30. 30.
    Tiwari, A.: Decision procedures in automated deduction, Ph.D. thesis, State University of New York at Stony Brook, New York, 2000.Google Scholar

Copyright information

© Kluwer Academic Publishers 2003

Authors and Affiliations

  • Leo Bachmair
    • 1
  • Ashish Tiwari
    • 2
  • Laurent Vigneron
    • 3
  1. 1.Department of Computer ScienceState University of New YorkStony BrookU.S.A.
  2. 2.SRI InternationalMenlo ParkU.S.A. e-mail
  3. 3.LORIA – Université Nancy 2Vandœuvre-lès-Nancy CedexFrance

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