Abstract
The underlying principles and main original techniques used in a running generic logic-based theorem prover are presented. The system (a prototype) is called HOARDATINF (Human Oriented Automated Reasoning on your Desk) and has been specialized in this work to proof learning through geometry. It is based on a new calculus, particularly suited to the class of problems we deal with. The calculus allows treatment of equality and automatic model building. HOARDATINF has some other original characteristics such as proving by analogy (using matching techniques), some possibilities of discovering lemmata (using diagrams), handling standard theories in geometry such as commutativity and symmetry (by encoding them in the unification algorithm used by the calculus), and proof verification in a rather large sense (by using capabilities of the calculus).
As this work is intended to set theoretical bases of a new logic-based approach to geometry theorem proving, a comparison of features of our system with respect to those of other important, representative logic-based systems is given. Some running examples give a good taste of the HOARDATINF capabilities. One of these examples allows us to compare qualitatively our approach with that of a powerful prover described in a recent paper [8]. Some directions of future research are mentioned.
Keywords
Acknowledgements
We thank an anonymous referee for a lot of detailed and constructive remarks that helped us to improve the presentation of this paper.
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References
N. Balacheff. Apprendre la preuve. In J. Sallantin and J.-J. Szczeniarz (eds.), Le concept de preuve à la lumière de l’intelligence artificielle, pages 197–236. PUF, Paris, 1999.
W. W. Bledsoe. Non-resolution theorem proving. Artificial Intelligence, 9:1–35, 1977.
C. Bourely, G. Défourneaux, and N. Peltier. Building proofs or counterexamples by analogy in a resolution framework. In Proceedings of JELIA 96, LNAI 1126, pages 34–49. Springer, 1996.
R. Caferra and M. Herment. A generic graphic framework for combining inference tools and editing proofs and formulae. Journal of Symbolic Computation, 19(2):217–243, 1995.
R. Caferra, M. Herment, and N. Zabel. User-oriented theorem proving with the ATINF graphic proof editor. In Fundamentals of Artificial Intelligence Research, LNCS 535, pages 2–10. Springer, 1991.
R. Caferra and N. Peltier. Extending semantic resolution via automated model building: Applications. In Proceeding of IJCAI’95, pages 328–334. Morgan Kaufman, 1995.
R. Caferra and N. Peltier. Disinference rules, model building and abduction. Logic at Work. Essays dedicated to the memory of Helena Rasiowa (Part 5: Logic in Computer Science, Chap. 20). Physica-Verlag, 1998.
S.-C. Chou, X.-S. Gao, and J.-Z. Zhang. A deductive database approach to automated geometry theorem proving and discovering. Journal of Automated Reasoning, 25(3):219–246, 2000.
S.-C. Chou. Mechanical Geometry Theorem Proving. Mathematics and its Applications. D. Reidel, 1988.
H. Coelho and L. Moniz Pereira. Automated reasoning in geometry theorem proving with Prolog. Journal of Automated Reasoning, 2(4):329–390, 1986.
G. Défourneaux, C. Bourely, and N. Peltier. Semantic generalizations for proving and disproving conjectures by analogy. Journal of Automated Reasoning, 20(1-2):27–45, 1998.
G. Défourneaux and N. Peltier. Analogy and abduction in automated reasoning. In M. E. Pollack (ed.), Proceedings of IJCAI’97, pages 216–225. Morgan Kaufmann, 1997.
G. Défourneaux and N. Peltier. Partial matching for analogy discovery in proofs and counter-examples. In W. McCune (ed.), Proceedings of CADE-14, LNAI 1249, pages 431–445. Springer, 1997.
C. Fermüller and A. Leitsch. Decision procedures and model building in equational clause logic. Journal of the IGPL, 6(1):17–41, 1998.
H. Gelernter, J. Hansen, and D. Loveland. Empirical explorations of the geometry theorem-proving machine. In J. Siekmann and G. Wrightson (eds.), Automation of Reasoning, vol. 1, pages 140–150. Springer, 1983. Originally published in 1960.
H. Hong, D. Wang, and F. Winkler. Algebraic Approaches to Geometric Reasoning. Special issue of the Annals of Mathematics and Artificial Intelligence 13 (1–2). Baltzer, Amsterdam, 1995.
J. Hsiang and M. Rusinowitch. Proving refutational completeness of theorem proving strategies: The transfinite semantic tree method. Journal of the ACM, 38(3):559–587, 1991.
D. Kapur and J. E. Mundy. Geometric Reasoning. MIT Press, 1989.
A. Leitsch. The Resolution Calculus. Texts in Theoretical Computer Science. Springer, 1997.
V. Luengo. A semi-empirical agent for learning mathematical proof. In S. P. Lajoie and M. Vivet (eds.), Artificial Intelligence and Education, pages 475–482. IOS Press, Amsterdam, 1999.
V. Luengo. Cabri-Euclide: Un micromonde de preuve intégrant la réfutation. Thèse de doctorat, I.N.P.G., France, Septembre 1997.
N. Peltier. Combining resolution and enumeration for finite model building. In P. Baumgartner and H. Zhang (eds.), FTP’00 (Third International Workshop on First-Order Theorem Proving), St-Andrews, Scotland, pages 170–181. Technical Report, Universität Koblenz-Landau, July 2000.
N. Peltier. On the decidability of the PVD class with equality. Technical report, LEIBNIZ Laboratory, 2000. To appear in the Logic Journal of the IGPL.
D. A. Plaisted and Y. Zhu. Ordered semantic hyperlinking. Journal of Automated Reasoning, 25(3):167–217, 2000.
G. Polya. How to Solve It: A New Aspect of Mathematical Method (second edition). Princeton University Press, 1973.
F. Puitg and J.-F. Dufourd. Formal specifications and theorem proving breakthroughs in geometric modelling. In Theorem Proving in Higher-Order Logics, LNCS 1479, pages 401–422. Springer, 1998.
F. Puitg and J.-F. Dufourd. Formalizing mathematics in higher-order logic: A case study in geometric modelling. Theoretical Computer Science, 234:1–57, 2000.
A. Quaife. Automated development of Tarski’s geometry. Journal of Automated Reasoning, 5:97–118, 1989.
T. Recio and M. Vélez. Automatic discovery of theorems in elementary geometry. Journal of Automated Reasoning, 23(1):63–82, 1999.
R. Reiter. A semantically guided deductive system for automatic theorem proving. IEEE Transactions on Computers, C-25(4):328–334, 1976.
J. Richter-Gebert and U. Kortenkamp. The Interactive Geometry Software Cinderella. Springer, 2000.
M. Rusinowitch. Démonstration automatique par des techniques de réécriture. Thèse d’état, Université Nancy 1, France, 1987. Also available as textbook, Inter Editions, Paris, 1989.
J. R. Slagle. Automatic theorem proving with renamable and semantic resolution. Journal of the ACM, 14(4):687–697, 1967.
J. Slaney. scott: A model-guided theorem prover. In Proceedings IJCAI-93, vol. 1, pages 109–114. Morgan Kaufmann, 1993.
L. Wos. Automated Reasoning: 33 Basic Research Problems. Prentice Hall, 1988.
L. Wos, R. Overbeek, E. Lush, and J. Boyle. Automated Reasoning: Introduction and Applications (second edition). McGraw-Hill, 1992.
L. Wos, G. Robinson, and D. Carson. Efficiency and completeness of the set of support strategy in theorem proving. Journal of the ACM, 12:536–541, 1965.
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Caferra, R., Peltier, N., Puitg, F. (2001). Emphasizing Human Techniques in Automated Geometry Theorem Proving: A Practical Realization. In: Richter-Gebert, J., Wang, D. (eds) Automated Deduction in Geometry. ADG 2000. Lecture Notes in Computer Science(), vol 2061. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45410-1_16
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