Congruence Closure with Integer Offsets

  • Robert Nieuwenhuis
  • Albert Oliveras
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2850)


Congruence closure algorithms for deduction in ground equational theories are ubiquitous in many (semi-)decision procedures used for verification and automated deduction. They are also frequently used in practical contexts where some interpreted function symbols are present. In particular, for the verification of pipelined microprocessors, in many cases it suffices to be able to deal with integer offsets, that is, instead of only having ground terms t built over free symbols, all (sub)terms can be of the form t+k for arbitrary integer values k.

In this paper we first give a different very simple and clean formulation for the standard congruence closure algorithm which we believe is of interest on itself. It builds on ideas from the abstract algorithms of [Kap97, BT00], but it is easily shown to run in the best known time, O(n log n), like the classical algorithms [DST80, NO80, Sho84].

After that, we show how this algorithm can be smoothly extended to deal with integer offsets without increasing this asymptotic complexity.


Lookup Table Function Symbol Ground Term Constant Symbol Automate Deduction 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. [BD94]
    Burch, J.R., Dill, D.L.: Automatic verification of pipelined microprocessor control. In: Proc. 6th International Computer Aided Verification Conference, pp. 68–80 (1994)Google Scholar
  2. [BLS02]
    Bryant, R., Lahiri, S., Seshia, S.: Modeling and verifying systems using a logic of counter arithmetic with lambda expressions and uninterpreted functions. In: Brinksma, E., Larsen, K.G. (eds.) CAV 2002. LNCS, vol. 2404, p. 78. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  3. [BT00]
    Bachmair, L., Tiwari, A.: Abstract congruence closure and specializations. In: McAllester, D. (ed.) CADE 2000. LNCS, vol. 1831, pp. 64–78. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  4. [BTV]
    Bachmair, L., Tiwari, A., Vigneron, L.: Abstract congruence closure. Journal of Automated Reasoning (to appear)Google Scholar
  5. [DLL62]
    Davis, M., Logemann, G., Loveland, D.: A machine program for theorem-proving. Communications of the ACM 5(7), 394–397 (1962)zbMATHCrossRefMathSciNetGoogle Scholar
  6. [DP60]
    Davis, M., Putnam, H.: A computing procedure for quantification theory. Journal of the ACM 7, 201–215 (1960)zbMATHCrossRefMathSciNetGoogle Scholar
  7. [DP01]
    Dershowitz, N., Plaisted, D.: Rewriting. In: Robinson, J.A., Voronkov, A. (eds.) Handbook of Automated Reasoning. Elsevier Science Publishers/ MIT Press (2001)Google Scholar
  8. [DST80]
    Downey, P.J., Sethi, R., Tarjan, R.E.: Variations on the common subexpressions problem. J. of the Association for Computing Machinery 27(4), 758–771 (1980)zbMATHMathSciNetGoogle Scholar
  9. [Kap97]
    Kapur, D.: Shostak’s congruence closure as completion. In: Comon, H. (ed.) RTA 1997. LNCS, vol. 1232. Springer, Heidelberg (1997)Google Scholar
  10. [NO80]
    Nelson, G., Oppen, D.C.: Fast decision procedures bases on congruence closure. Journal of the Association for Computing Machinery 27(2), 356–364 (1980)zbMATHMathSciNetGoogle Scholar
  11. [PSK96]
    Plaisted, D.A., Sattler-Klein, A.: Proof lengths for equational completion. Information and Computation 125(2), 154–170 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  12. [Sho84]
    Shostak, R.E.: Deciding combinations of theories. Journal of the ACM 31(1), 1–12 (1984)zbMATHCrossRefMathSciNetGoogle Scholar
  13. [Sny89]
    Snyder, W.: An O(n log n) algorithm for generating reduced sets of ground rewrite rules equivalent to a set of ground equations E. In: Dershowitz, N. (ed.) RTA 1989. LNCS, vol. 355. Springer, Heidelberg (1989)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Robert Nieuwenhuis
    • 1
  • Albert Oliveras
    • 1
  1. 1.Technical University of CataloniaBarcelonaSpain

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