Abstract
We discuss a pragmatic approach to integrate computer algebra into proof planning. It is based on the idea to separate computation and verification and can thereby exploit the fact that many elaborate symbolic computations are trivially checked. In proof planning the separation is realized by using a powerful computer algebra system during the planning process to do non-trivial symbolic computations. Results of these computations are checked during the refinement of a proof plan to a calculus level proof using a small, self-implemented system that gives us protocol information on its calculation. This protocol can be easily expanded into a checkable low-level calculus proof ensuring the correctness of the computation. We demonstrate our approach with the concrete implementation in the ΩMEGA system.
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
Adams, A., Gottliebsen, H., Linton, S., Martin, U.: VSDITLU: a verifiable symbolic definite integral table look-up. In: Ganzinger, H. (ed.) CADE 1999. LNCS (LNAI), vol. 1632, pp. 112–126. Springer, Heidelberg (1999)
Andrews, P., Bishop, M., Issar, S., Nesmith, D., Pfenning, F., Xi, H.: TPS: A Theorem Proving System for Classical Type Theory. J. of Autom. Reasoning 16(3), 321–353 (1996)
Ballarin, C., Homann, K., Calmet, J.: Theorems and Algorithms: An Interface between Isabelle and Maple. In: Proc. of ISSAC 1995, pp. 150–157. ACM Press, New York (1995)
Bauer, A., Clarke, E., Zhao, X.: Analytica: an Experiment in Combining Theorem Proving and Symbolic Computation. J. of Autom. Reasoning 21(3), 295–325 (1998)
Bundy, A.: The Use of Explicit Plans to Guide Inductive Proofs. In: Lusk, E.‘., Overbeek, R. (eds.) CADE 1988. LNCS, vol. 310. Springer, Heidelberg (1988)
Cannon, J., Playoust, C.: Algebraic Programming with Magma. Springer, Heidelberg (1998)
Cheikhrouhou, L., Sorge, V.: PDS — A Three-Dimensional Data Structure for Proof Plans. In: Proc. of ACIDCA 2000 (2000)
Church, A.: A Formulation of the Simple Theory of Types. J. of Symbolic Logic 5, 56–68 (1940)
Clément, D., Montagnac, F., Prunet, V.: Integrated Software Components: a Paradigm for Control Integration. In: Endres, A., Weber, H. (eds.) SDE 1991. LNCS, vol. 509. Springer, Heidelberg (1991)
Fikes, R., Nilsson, N.: STRIPS: A new approach to the application of theorem proving to problem solving. Artificial Intelligence 2, 189–208 (1971)
Franke, A., Hess, S.: Agent-Oriented Integration of Distributed Mathematical Services. J. of Universal Computer Science 5(3), 156–187 (1999)
The GAP Group: GAP — Groups, Algorithms, and Programming, Version 4, Aachen, St Andrews (1998), http://www-gap.dcs.st-and.ac.uk/~gap
Gentzen, G.: Untersuchungen über das Logische Schlieβen I und II. Mathematische Zeitschrift 39, 176–210, 405–431 (1935)
Gordon, M., Melham, T.: Introduction to HOL. Cambridge Univ. Press, Cambridge (1993)
Gordon, M., Wadsworth, C.P., Milner, R. (eds.): Edinburgh LCF. LNCS, vol. 78. Springer, Heidelberg (1979)
The Ωmega Group: Ωmega: Towards a Mathematical Assistant. In: McCune, W. (ed.) CADE 1997. LNCS (LNAI), vol. 1249, pp. 252–255. Springer, Heidelberg (1997)
Harrison, J., Théry, L.: A Skeptic’s Approach to Combining HOL and Maple. J. of Autom. Reasoning 21(3), 279–294 (1998)
Kerber, M., Kohlhase, M., Sorge, V.: Integrating Computer Algebra Into Proof Planning. J. of Autom. Reasoning 21(3), 327–355 (1998)
Melis, E.: The “Limit” Domain. In: Proc. of the Fourth International Conference on Artificial Intelligence in Planning Systems, pp. 199–206 (1998)
Melis, E., Sorge, V.: Specialized External Reasoners in Proof Planning. Seki Report SR-00-01, Computer Science Department, Universität des Saarlandes (2000)
Owre, S., Rajan, S., Rushby, J., Shankar, N., Srivas, M.: PVS: Combining Specification, Proof Checking, and Model Checking. In: Alur, R., Henzinger, T.A. (eds.) CAV 1996. LNCS, vol. 1102, pp. 411–414. Springer, Heidelberg (1996)
Redfern, D.: The Maple Handbook: Maple V Release 5. Springer, Heidelberg (1998)
Sorge, V.: Integration eines Computeralgebrasystems in eine logische Beweisumgebung. Master’s thesis, Universität des Saarlandes (November 1996)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2000 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Sorge, V. (2000). Non-trivial Symbolic Computations in Proof Planning. In: Kirchner, H., Ringeissen, C. (eds) Frontiers of Combining Systems. FroCoS 2000. Lecture Notes in Computer Science(), vol 1794. Springer, Berlin, Heidelberg. https://doi.org/10.1007/10720084_9
Download citation
DOI: https://doi.org/10.1007/10720084_9
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-67281-4
Online ISBN: 978-3-540-46421-1
eBook Packages: Springer Book Archive