Abstract
In this paper, we investigate analogy-driven proof plan construction in inductive theorem proving. The intention is to produce a plan for a target theorem that is similar to a given source theorem. We identify second-order mappings from the source to the target that preserve induction-specific proof- relevant abstractions dictating whether the source plan can be replayed. We replay the planning decisions taken in the source if the reasons or justifications for these decisions still hold in the target. If the source and target plan differ significantly at some isolated point, additional reformulations are invoked to add, delete, or modify planning steps. These reformulations are not ad hoc but are triggered by peculiarities of the mappings and by failed justifications. Employing analogy on top of the proof planner CLAM has extended the problem-solving horizon of CLAM: With analogy, some theorems could be proved automatically that neither CLAM nor NQTHM could prove automatically.
Similar content being viewed by others
References
Basin, D. and Walsh, T.: Termination orderings for rippling, in Proceedings of the 12th International Conference on Automated Deduction (CADE-12), Lecture Notes in Artificial Intelligence 814, Springer, 1994.
Boyer, R. S. and Moore, J. S.: A Computational Logic, Academic Press, 1979.
Bundy, A., Stevens, A., Van Harmelen, F., Ireland, A. and Smaill, A.: A heuristic for guiding inductive proofs, Artificial Intelligence 63 (1993), 185-253.
Bundy, A., van Harmelen, F., Hesketh, J. and Smaill, A.: Experiments with proof plans for induction, Journal of Automated Reasoning 7 (1991), 303-324.
Carbonell, J. G.: Learning by analogy: Formulating and generalizing plans from past experience, in R. S. Michalsky, J. G. Carbonell, and T. M. Mitchell (eds.), Machine Learning: An Artificial Intelligence Approach, Tioga, 1983, pp. 137-162.
Carbonell, J. G.: Derivational analogy: A theory of reconstructive problem solving and expertise acquisition, in R. S. Michalsky, J. G. Carbonell, and T. M. Mitchell (eds.), Machine Learning: An Artificial Intelligence Approach, Morgan Kaufmann Publ., 1986, pp. 371-392.
Curien, R.: Outils pour la Preuve par Analogie, Ph.D. thesis, Universite Henri Poincare, Nancy, 1995.
Gordon, M., Milner, R. and Wadsworth, C. P.: Edinburgh LCF: A Mechanized Logic of Computation, Lecture Notes in Computer Science 78, Springer, 1979.
Hutter, D.: Guiding inductive proofs, in M. E. Stickel (ed.), Proc. of 10th International Conference on Automated Deduction (CADE-10), Lecture Notes in Artificial Intelligence 449, Springer, 1990.
Hutter, D.: Synthesis of induction orderings for existence proofs, in A. Bundy (ed.), Proc. of 12th International Conference on Automated Deduction (CADE-12), Lecture Notes in Artificial Intelligence 814, Springer, 1994, pp. 29-41.
Ireland, A. and Bundy, A.: Productive use of failure in inductive proof, Journal of Automated Reasoning 16(1-2) (1996), 79-111.
Kapur, D. and Subramaniam, M.: Lemma discovery in automating induction, in M. A. McRobbie and J. K. Slaney (eds.), Automated Deduction CADE-13, Lecture Notes in Artificial Intelligence 1104, Springer, 1996, pp. 538-552.
Kolbe, Th. and Walther, Ch.: Reusing proofs, in Proceedings of ECAI-94, Amsterdam, 1994.
Kolbe, Th. and Walther, Ch.: Second-order matching modulo evaluation - A technique for reusing proofs, Proceedings of the 14th International Joint Conference on Artificial Intelligence, Morgan Kaufmann, Montreal, 1995, pp. 190-195.
Kraan, I.: Proof Planning for Logic Program Synthesis, Ph.D. thesis, University of Edinburgh, 1994.
Melis, E.: Analogy in CLAM, Technical Report DAI Research Paper 766, University of Edinburgh, Dept. of Artificial Intelligence, Edinburgh, 1995. Available from http://jswww.cs.unisb.de/~melis/.
Melis, E.: A model of analogy-driven proof-plan construction, in Proceedings of the 14th International Joint Conference on Artificial Intelligence, Montreal, 1995, pp. 182-189.
Melis, E. and Whittlem, J.: Internal analogy in inductive theorem proving, in M. A. McRobbie and J. K. Slaney (eds.), Proceedings of the 13th Conference on Automated Deduction (CADE-96), Lecture Notes in Artificial Intelligence 1104, Springer, 1996, pp. 92-105.
Munyer, J. C.: Analogy as a Means of Discovery in Problem Solving and Learning, Ph.D. thesis, University of California, Santa Cruz, 1981.
Owen, S.: Analogy for Automated Reasoning, Academic Press, 1990.
Polya, G.: How to Solve It, Princeton University Press, 1945.
Boy de la Tour, T. and Caferra, R.: Proof analogy in interactive theorem proving: A method to express and use it via second order pattern matching, in Proceedings of the AAAI-87, 1987, pp. 95-99.
Vadera, S.: Proof by analogy in mural, Formal Aspects of Computing 7 (1995), 183-206.
van Harmelen, F., Ireland, A., Negrete, S., Stevens, A. and Smaill, A.: The CLAM proof planner, user manual and programmers manual, Technical Report version 2.0, University of Edinburgh, Edinburgh, 1993.
Whittle, J.: Analogy in CLAM, Technical Report MSc. thesis, University of Edinburgh, Dept. of Artificial Intelligence, Edinburgh, 1995. Available from http://www.dai.ed.ac.uk/daidb/students/jonathw/publications.html.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Melis, E., Whittle, J. Analogy in Inductive Theorem Proving. Journal of Automated Reasoning 22, 117–147 (1999). https://doi.org/10.1023/A:1005936130801
Issue Date:
DOI: https://doi.org/10.1023/A:1005936130801