Journal of Automated Reasoning

, Volume 7, Issue 3, pp 303–324 | Cite as

Experiments with proof plans for induction

  • Alan Bundy
  • Frank van Harmelen
  • Jane Hesketh
  • Alan Smaill


The technique of proof plans is explained. This technique is used to guide automatic inference in order to avoid a combinatorial explosion. Empirical research is described to test this technique in the domain of theorem proving by mathematical induction. Heuristics, adapted from the work of Boyer and Moore, have been implemented as Prolog programs, called tactics, and used to guide an inductive proof checker, Oyster. These tactics have been partially specified in a meta-logic, and the plan formation program, CLAM, has been used to reason with these specifications and form plans. These plans are then executed by running their associated tactics and, hence, performing an Oyster proof. Results are presented of the use of this technique on a number of standard theorems from the literature. Searching in the planning space is shown to be considerably cheaper than searching directly in Oyster's search space. The success rate on the standard theorems is high.

Key words

Theorem proving mathematical induction search combinatorial explosion proof plans tactics planning program synthesis 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Aubin, R., ‘Some generalization heuristics in proofs by induction’. In Actes du Colloque Construction: Amelioration et verification de Programmes, Institut de recherche d'informatique et d'automatique, 1975.Google Scholar
  2. 2.
    Boyer, R. S. and Moore, J S., A Computational Logic, Academic Press, 1979, ACM monograph series.Google Scholar
  3. 3.
    Bundy, A. and Sterling, L. S., ‘Meta-level inference: two application’, Journal of Automated Reasoning 4(1), 15–27 (1988). Also available from Edinburgh, Dept. of AI, as Research Paper No. 273.Google Scholar
  4. 4.
    Bundy, A., ‘The use of explicit plans to guide inductive proofs’. In 9th Conference on Automated Deduction, pages 111–120, Springer-Verlag, 1988. Longer version available as DAI Research Paper No. 349.Google Scholar
  5. 5.
    Bundy, A., van Harmelen, F., Hesketh, J., Smaill, A. and Stevens, A., ‘A Rational Reconstruction and Extension of Recursion Analysis’, Proc. of IJCAI-89 (Sridharan, ed.), Morgan Kaufmann, 1989, 359–365.Google Scholar
  6. 6.
    Burstall, R. M. and Darlington, J., ‘A transformation system for developing recursive programs’. Journal of the ACM 24(1), 44–67 (1977).Google Scholar
  7. 7.
    Constable, R. L., Allen, S. F., Bromley, H. M. et al. Implementing Mathematics with the Nuprl Proof Development System. Prentice-Hall, 1986.Google Scholar
  8. 8.
    Desimone, R. V., ‘Explanation-based learning of proof plans’. In Y. Kodratoff (Ed.) Machine and Human Learning, Ellis Horwood, 1987. Also available as DAI Research Paper 304. Previous version in proceedings of EWSL-86.Google Scholar
  9. 9.
    Gallaire, H. and Lasserre, C., ‘Metalevel control for logic programs’. In Logic Programming, p. 173–185, Academic Press, 1982.Google Scholar
  10. 10.
    Giunchiglia, F. and Walsh, T., ‘Abstract Theorem Proving’. Proc. of IJCAI-89 (Shridharan, ed.), Morgan Kaufmann, 1989, pp. 372–377.Google Scholar
  11. 11.
    Gordon, M. J., Milner, A. J. and Wadsworth, C. P., Edinburgh LCF-A mechanised logic of computation. Volume 78 of Lecture Notes in Computer Science, Springer Verlag, 1979.Google Scholar
  12. 12.
    Horn, C., The Nur PRL Proof Development System. Working paper 214, Dept. of Artificial Intelligence, Edinburgh, 1988. The Edinburgh version of NurPRL has been renamed Oyster.Google Scholar
  13. 13.
    Kanamori, T. and Fujita, H., ‘Formulation of induction formulas in verification of Prolog programs. In 8th Conference on Automated Deduction, p. 281–299, Springer-Verlag, 1986. Springer Lecture Notes in Computer Science No. 230.Google Scholar
  14. 14.
    Knoblock, T. B. and Constable, R. L., ‘Formalized metareasoning in type theory’. In Proceedings of LICS, p. 237–248, IEEE, 1986.Google Scholar
  15. 15.
    Martin-Löf, Per, ‘Constructive mathematics and computer programming’. In 6th International Congress for Logic, Methodology and Philosophy of Science, p. 153–175, Hanover, August 1979. Published by North Holland, Amsterdam, 1982.Google Scholar
  16. 16.
    Paulson, L., ‘Experience with Isabelle: a generic theorem prover’. In COLOG 88, Institute of Cybernetics of the Estonian SSR, 1988.Google Scholar
  17. 17.
    Sacerdoti, E. D., ‘Planning in a hierarchy of a abstraction spaces’. Artificial Intelligence 5, 115–135 (1974).Google Scholar
  18. 18.
    Silver, B., Meta-Level Inference: Representing and Learning Control Information in Artificial Intelligence. North Holland, 1985. Revised version of the author's PhD thesis, DAI 1984.Google Scholar

Copyright information

© Kluwer Academic Publishers 1991

Authors and Affiliations

  • Alan Bundy
    • 1
  • Frank van Harmelen
    • 1
  • Jane Hesketh
    • 1
  • Alan Smaill
    • 1
  1. 1.Department of Artificial IntelligenceUniversity of EdinburghEdinburghScotland

Personalised recommendations