Journal of Automated Reasoning

, Volume 31, Issue 2, pp 129–168

Abstract Congruence Closure

  • Leo Bachmair
  • Ashish Tiwari
  • Laurent Vigneron

DOI: 10.1023/B:JARS.0000009518.26415.49

Cite this article as:
Bachmair, L., Tiwari, A. & Vigneron, L. Journal of Automated Reasoning (2003) 31: 129. doi:10.1023/B:JARS.0000009518.26415.49


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 rewritingcongruence closureassociative-commutative theories

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