Abstract
Proof planning is a form of theorem proving in which the proving procedure is viewed as a planning process. The plan operators in proof planning are called methods. In this paper we propose a strategy for heuristically restricting the set of methods to be applied in proof search. It is based on the idea that the plausibility of a method can be estimated by comparing the model class of proof lines newly generated by the method with that of the assumptions and of the theorem. For instance, in forward reasoning when a method produces a new assumption whose model class is not a superset of the model class of the given premises, the method will lead to a situation which is semantically not justified and will not lead to a valid proof in later stages. A semantic restriction strategy is to reduce the search space by excluding methods whose application results in a semantic mismatch. A semantic selection strategy heuristically chooses the method that is likely to make most progress towards filling the gap between the assumptions and the theorem. Each candidate method is evaluated with respect to the subset and superset relation with the given premises. All models considered are taken from a finite reference subset of the full model class. In this contribution we present the model-guided approach as well as first experiments with it.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Benzmüller, C., Cheikhrouhou, L., Fehrer, D., Fiedler, A., Huang, X., Kerber, M., Kohlhase, M., Konrad, K., Meier, A., Melis, E., Schaarschmidt, W., Siekmann, J., and Sorge, V., 1997, ΩMEGA: Towards a mathematical assistant, in: Proceedings of the 14th International Conference on Automated Deduction, W. McCune, ed., Springer, Berlin, pp. 252–255.
Braine, M., 1978, On the relation between the natural logic of reasoning and standard logic, Psychological Review 85(1): 1–21.
Bundy, A., Stevens, A., van Harmelen, F., Ireland, A., and Smaill, A., Rippling: A heuristic for guiding inductive proofs, Artificial Intelligence 62:185–253.
Bundy, A., 1988, The use of explicit plans to guide inductive proofs, in: Proceedings of the 9th International Conference on Automated Deduction, E. Lusk and R. Overbeek, eds., Springer, Berlin, pp. 111–120.
Bundy, A., van Harmelen, F., Hesketh, J., and Smaill, A., 1991, Experiments with proof plans for induction, JAR 7(3):303–324.
Bundy, A., van Harmelen, F., Smaill, A., and Ireland, A., 1990, Extensions to the rippling-out tactic for guiding inductive proofs, in: Proceedings of the 10th International Conference on Automated Deduction, M.E. Stickel, ed., Springer, Berlin, pp. 132–146.
Chu, H. and Plaisted, D. A., 1994, Semantically guided first-order theorem proving using hyper-linking, in: Proceedings of the 12th International Conference on Automated Deduction, A. Bundy, ed., Springer, Berlin, pp. 192–206.
Eisinger, N. and Ohlbach, H. J., 1986, The Markgraf Karl Refutation Procedure (MKRP), in: Proceedings of the 8th International Conference on Automated Deduction, J. Siekmann, ed., Springer, Berlin, pp. 681–682.
Gelernter, H., 1959, Realization of a geometry theorem-proving machine, in: Proceedings of the International Conference on Information Processing, UNESCO.
Hadamard, H., 1944, The Psychology of Invention in the Mathematical Field, Dover Publications, New York.
Hodgson, K. and Slaney, J., 2001, Development of a semantically guided theorem prover, in: Proceedings of the International Joint Conference on Automated Reasoning, Springer, Berlin, pp. 443–447.
Huang, X., Kerber, M., and Kohlhase, M., 1992, Methods — the basic units for planning and verifying proofs, SEKI Report, SR-92-20, Fachbereich Informatik, Universität des Saarlandes, Saarbrücken, Germany.
Huang, X., Kerber, M., Richts, J., and Sehn, A., 1994, Planning mathematical proofs with methods, Journal of Information Processing and Cybernetics, EIK 30(5-6):277–291.
Ireland, A. and Bundy, A., 1996, Productive use of failure in inductive proof, JAR 16(l-2):79–111.
Johnson-Laird, P.N. and Byrne, R., 1991, Deduction, Lawrence Erlbaum Associates Publishers, Hove.
Kerber, M., Melis, E., and Siekmann, J., 1993, Analogical reasoning based on typical instances, in: Proceedings of the IJCAI-Workshop on Principles of Hybrid Reasoning and Representation, Chambéry, France, pp. 85–95.
Kerber, M. and Choi, S., 2000, The semantic clause graph procedure, in: Proceedings of CADE-17 Workshop on Model Computation–Principles, Algorithms, and Applications, P. Baumgartner, C. Fermüller, N. Peltier and H. Zhang, eds., pp. 29–37.
Kerber, M., 1998, Proof planning — a practical approach to mechanised reasoning in mathematics, in: Automated Deduction — A Basis for Applications, W. Bibel and P. H. Schmitt, eds., vol. III, Springer Science+Business Media New York, pp. 77–95.
Letz, R., Schumann, J., Bayerl, S., and Bibel, W., 1992, SETHEO: A high-performance theorem prover, JAR 8:183–212.
McCune, W., 1994, OTTER 3.0 Reference Manual and Guide, Mathematics and Computer Science Division, Argonne National Laboratory, Argonne, IL.
McCune, W., 1997, Solution of the Robbins problem, JAR 19:263–276.
Pólya, G., 1945, How to Solve It, Princeton University Press, Princeton, NJ.
Pólya, G., 1954, Mathematics and Plausible Reasoning, Princeton University Press, Princeton, NJ.
Pólya, G., 1962, Mathematical Discovery — On Understanding, Learning, and Teaching Problem Solving, Princeton University Press, Princeton.
Robinson, J.A., 1965, A machine-oriented logic based on the resolution principle, JACM 12(1):23–41.
Riazanov, A. and Voronkov, A., 2001, Vampire 1.1, in: Proceedings of the International Joint Conference on Automated Reasoning, R. Goré, A. Leitsch, T. Nipkow, eds., Springer, Berlin, pp. 376–380.
Slaney, J., 1995, FINDER — Finite Domain Enumerator Version 3.0 Notes and Guide, Centre for Information Science Research, Australian National University, Canberra, Australia.
Slaney, J., Lusk, E., and McCune, W., 1994, SCOTT: Semantically Constrained Otter, in: Proceedings of the 12th International Conference on Automated Deduction, A. Bundy, ed., Springer, Berlin, pp. 764–768.
Smullyan, R.M., 1968, First-Order Logic, Springer, Berlin.
van der Waerden, B., 1964, Wie der Beweis der Vermutung von Baudet gefunden wurde, Abh. Math. Sem. Univ. Hamburg 28:6–15.
Weidenbach, C., Bernd, G., and Georg, R., 1996, SPASS & FLOTTER, Version 0.42, in: Proceedings of the 13th International Conference on Automated Deduction, M.A. McRobbie and J.K. Slaney, eds., Springer, Berlin, pp. 141–145.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2002 Springer Science+Business Media Dordrecht
About this chapter
Cite this chapter
Choi, S., Kerber, M. (2002). Model-Guided Proof Planning. In: Magnani, L., Nersessian, N.J., Pizzi, C. (eds) Logical and Computational Aspects of Model-Based Reasoning. Applied Logic Series, vol 25. Springer, Dordrecht. https://doi.org/10.1007/978-94-010-0550-0_7
Download citation
DOI: https://doi.org/10.1007/978-94-010-0550-0_7
Publisher Name: Springer, Dordrecht
Print ISBN: 978-1-4020-0791-0
Online ISBN: 978-94-010-0550-0
eBook Packages: Springer Book Archive