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The Algebra of Equality Proofs

  • Aaron Stump
  • Li-Yang Tan
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3467)

Abstract

Proofs of equalities may be built from assumptions using proof rules for reflexivity, symmetry, and transitivity. Reflexivity is an axiom proving x=x for any x; symmetry is a 1-premise rule taking a proof of x=y and returning a proof of y=x; and transitivity is a 2-premise rule taking proofs of x=y and y=z, and returning a proof of x=z. Define an equivalence relation to hold between proofs iff they prove a theorem in common. The main theoretical result of the paper is that if all assumptions are independent, this equivalence relation is axiomatized by the standard axioms of group theory: reflexivity is the unit of the group, symmetry is the inverse, and transitivity is the multiplication. Using a standard completion of the group axioms, we obtain a rewrite system which puts equality proofs into canonical form. Proofs in this canonical form use the fewest possible assumptions, and a proof can be canonized in linear time using a simple strategy. This result is applied to obtain a simple extension of the union-find algorithm for ground equational reasoning which produces minimal proofs. The time complexity of the original union-find operations is preserved, and minimal proofs are produced in worst-case time \(O(n^{\textit{log}_2 3})\), where n is the number of expressions being equated. As a second application, the approach is used to achieve significant performance improvements for the CVC cooperating decision procedure.

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References

  1. 1.
    Barrett, C., Dill, D., Stump, A.: Checking Satisfiability of First-Order Formulas by Incremental Translation to SAT. In: Brinksma, E., Larsen, K.G. (eds.) CAV 2002. LNCS, vol. 2404, p. 236. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  2. 2.
    Barrett, C., Berezin, S.: CVC Lite: A new implementation of the cooperating validity checker. In: Alur, R., Peled, D.A. (eds.) CAV 2004. LNCS, vol. 3114, pp. 515–518. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  3. 3.
    Burch, J.R., Dill, D.L.: Automatic verification of pipelined microprocessor control. In: Dill, D.L. (ed.) CAV 1994. LNCS, vol. 818, pp. 68–80. Springer, Heidelberg (1994)Google Scholar
  4. 4.
    Cormen, T., Leiserson, C., Rivest, R.: Introduction to Algorithms. MIT Press, Cambridge (1992)Google Scholar
  5. 5.
    de Moura, L., Rueß, H., Shankar, N.: Justifying Equality. In: Ranise, S., Tinelli, C. (eds.) 2nd International Workshop on Pragmatics of Decision Procedures in Automated Reasoning (2004)Google Scholar
  6. 6.
    Filliâtre, J., Owre, S., Rueß, H., Shankar, N.: ICS: integrated canonizer and solver. In: Berry, G., Comon, H., Finkel, A. (eds.) CAV 2001. LNCS, vol. 2102, p. 246. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  7. 7.
    Knuth, D., Bendix, P.: SimpleWord Problems in Universal Algebras. In: Leech, J. (ed.) Computational Problems in Abstract Algebra, pp. 263–297. Pergamon Press, Oxford (1970)Google Scholar
  8. 8.
    Lahiri, S., Bryant, R., Goel, A., Talupur, M.: Revisiting Positive Equality. In: Jensen, K., Podelski, A. (eds.) TACAS 2004. LNCS, vol. 2988, pp. 1–15. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  9. 9.
    Nelson, G., Oppen, D.: Fast decision procedures based on congruence closure. Journal of the Association for Computing Machinery 27(2), 356–364 (1980)zbMATHMathSciNetGoogle Scholar
  10. 10.
    Nieuwenhuis, R., Oliveras, A.: Union-Find and Congruence Closure Algorithms that Produce Proofs. In: Ranise, S., Tinelli, C. (eds.) 2nd International Workshop on Pragmatics of Decision Procedures in Automated Reasoning (2004) (short paper)Google Scholar
  11. 11.
    Nieuwenhuis, R., Oliveras, A.: Proof-producing Congruence Closure. In: Giesl, J. (ed.) 16th International Conference on Rewriting Techniques and Applications (2005) (under review)Google Scholar
  12. 12.
    Pnueli, A., Rodeh, Y., Shtrichman, O., Siegel, M.: Deciding Equality Formulas by Small Domains Instantiations. In: Halbwachs, N., Peled, D.A. (eds.) CAV 1999. LNCS, vol. 1633, pp. 455–469. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  13. 13.
    Ruess, H., Shankar, N.: Deconstructing Shostak. In: 16th IEEE Symposium on Logic in Computer Science (2001)Google Scholar
  14. 14.
    Shostak, R.: Deciding combinations of theories. Journal of the Association for Computing Machinery 31(1), 1–12 (1984)zbMATHMathSciNetGoogle Scholar
  15. 15.
    Stump, A.: Checking Validities and Proofs with CVC and flea. PhD thesis, Stanford University (2002), available from http://www.cs.wustl.edu/~stump/
  16. 16.
    Stump, A., Barrett, C., Dill, D.: CVC: a Cooperating Validity Checker. In: Brinksma, E., Larsen, K.G. (eds.) CAV 2002. LNCS, vol. 2404, p. 500. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  17. 17.
    Velev, M., Bryant, R.: Superscalar Processor Verification Using Efficient Reductions of the Logic of Equality with Uninterpreted Functions. In: Pierre, L., Kropf, T. (eds.) CHARME 1999. LNCS, vol. 1703, pp. 37–53. Springer, Heidelberg (1999)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Aaron Stump
    • 1
  • Li-Yang Tan
    • 1
  1. 1.Dept. of Computer Science and EngineeringWashington University in St. LouisSt. LouisUSA

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