Advertisement

Proof by induction using test sets

  • Deepak Kapur
  • Paliath Narendran
  • Hantao Zhang
Term Rewriting Systems
Part of the Lecture Notes in Computer Science book series (LNCS, volume 230)

Abstract

A new method for proving an equational formula by induction is presented. This method is based on the use of the Knuth-Bendix completion procedure for equational theories, and it does not suffer from limitations imposed by the inductionless induction methods proposed by Musser and Huet and Hullot. The method has been implemented in RRL, a Rewrite Rule Laboratory. Based on extensive experiments, the method appears to be more practical and efficient than a recently proposed method by Jouannaud and Kounalis. Using ideas developed for this method, it is also possible to check for sufficient completeness of equational axiomatizations.

Key Words

Inductionless Induction Proof by Induction Equational Theory Knuth-Bendix Completion Procedure Consistency Sufficient-Completeness Induction 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

8. References

  1. [1]
    Dershowitz, N., “Applications of the Knuth-Bendix Completion Procedure,” Laboratory Operation, Aerosapce Corporation, Aerospace Report No. ATR-83(8478)-2, May 15, 1983.Google Scholar
  2. [2]
    Goguen, J., “How to Prove Algebraic Inductive Hypotheses without Induction,” Proc. of the Fifth Conference on Automated Deduction, Les Arces, France, LNCS 87, Springer Verlag, pp. 356–372, July 1980.Google Scholar
  3. [3]
    Huet, G., “Confluent Reductions: Abstract Properties and Applications to Term Rewriting Systems,” JACM 27, 4, pp. 797–821, October 1980.CrossRefGoogle Scholar
  4. [4]
    Huet, G., “A Complete Proof of Correctness of the Knuth-Bendix Completion Procedure,” JCSS 23, 1, 1981.Google Scholar
  5. [5]
    Huet, G., and Hullot, J.M., “Proof by Induction in Equational Theories with Constructors,” JCSS 25, 2, 1982.Google Scholar
  6. [6]
    Jouannaud, J.-P., and Kirchner, H., “Completion of a Set of Rules modulo a Set of Equations,” Proc. of the 11th Symp. on Principles of Programming Languages, 1984.Google Scholar
  7. [7]
    Jouannaud, J.-P., and Kounalis, E., “Proofs by Induction in Equational Theories Without Constructors,” CRIN, University of Nancy, France, May 1985. To appear in the Proc. of IEEE Conf. on Logic in Computer Science, Cambridge, MA, June 1986.Google Scholar
  8. [8]
    Kapur, D., and Musser, D.R., “Proof by Consistency,” Proc. of an NSF Workshop on the Rewrite Rule Laboratory, Sept. 4–6, 1983, Schenectady, G.E. R&D Center Report GEN84008, April 1984. To appear in the AI Journal.Google Scholar
  9. [9]
    Kapur, D., Narendran, P., and Zhang, H., “On Sufficient Completeness and Related Properties of Term Rewriting Systems,” Unpublished Manuscript, General Electric R&D Center, Schenectady, NY, Oct. 1985. Submitted to Acta Informatica.Google Scholar
  10. [10]
    Kapur, D., and Sivakumar, G., “Experiments with and Architecture of RRL, a Rewrite Rule Laboratory,” Proc. of An NSF Workshop on the Rewrite Rule Lab., Sept. 1983, General Electric R&D Center Report 84GEN008, pp. 33–56, April 1984.Google Scholar
  11. [11]
    Kapur, D., Sivakumar, G., and Zhang, H., “RRL: A Rewrite Rule Laboratory,” to appear in the Proc. of the 8th International Conf. on Automated Deduction (CADE-8), Oxford, England, July 1986.Google Scholar
  12. [12]
    Knuth, D., and Bendix, P., “Simple Word Problems in Universal Algebras,” in Computational Problems in Abstract Algebra (ed. Leech), Pergamon Press, pp. 263–297, 1970.Google Scholar
  13. [13]
    Kounalis, E., and Zhang, H., “A General Completeness Test for Equational Specifications,” CRIN [85-R-05], Nancy, France, November 1985.Google Scholar
  14. [14]
    Musser, D.R., “On Proving Inductive Properties of Abstract Data Types,” Proc. of the 7th Symp. on Principles of Programming Languages, Las Vegas, Jan. 1980.Google Scholar
  15. [15]
    Musser, D.R., and Kapur, D., “Rewrite Rule Theory and Abstract Data Type Analysis,” Computer Algebra, EUROSAM 1982, LNCS 144 (ed. Calmet), Springer Verlag, pp. 77–90, April 1982.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1986

Authors and Affiliations

  • Deepak Kapur
    • 1
  • Paliath Narendran
    • 1
  • Hantao Zhang
    • 2
  1. 1.Corporate Research & DevelopmentGeneral Electric Co.SchenectadyUSA
  2. 2.Dept. of Computer ScienceRensselaer Polytechnic InstituteTroyUSA

Personalised recommendations