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Theorem proving by analogy — A compelling example

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Progress in Artificial Intelligence (EPIA 1995)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 990))

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Abstract

This paper shows how a new approach to theorem proving by analogy is applicable to real maths problems. This approach works at the level of proof-plans and employs reformulation that goes beyond symbol mapping. The Heine-Borel theorem is a widely known result in mathematics. It is usually stated in R1 and similar versions are also true in R2, in topology, and metric spaces. Its analogical transfer was proposed as a challenge example and could not be solved by previous approaches to theorem proving by analogy. We use a proof-plan of the Heine-Borel theorem in R1 as a guide in automatically producing a proof-plan of the Heine-Borel theorem in R2 by analogy-driven proof-plan construction.

This work was supported in part by the HC&M grant CHBICT930806 and by a research grant of the Deutsche Forschungsgemeinschaft.

On leave from University Saarbrücken, Germany

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References

  1. W.W. Bledsoe. The use of analogy in automatic proof discovery. Tech.Rep. AI-158-86, Microelectronics and Computer Technology Corporation, Austin, TX, 1986.

    Google Scholar 

  2. W.W. Bledsoe. Heine-Borel theorem analogy example. Technical Report Memo ATP 124, University of Texas Computer Science Dept, Austin, TX, August 1994.

    Google Scholar 

  3. A. Bundy. The use of explicit plans to guide inductive proofs. In E. Lusk and R. Overbeek, editors, Proc. 9th International Conference on Automated Deduction (CADE), volume 310 of Lecture Notes in Computer Science, pages 111–120, Argonne, 1988. Springer.

    Google Scholar 

  4. J.G. Carbonell. Derivational analogy: A theory of reconstructive problem solving and expertise acquisition. In R.S. Michalsky, J.G. Carbonell, and T.M. Mitchell, editors, Machine Learning: An Artificial Intelligence Approach, pages 371–392. Morgan Kaufmann Publ., Los Altos, 1986.

    Google Scholar 

  5. P. Deussen. Halbgruppen und Automaten, volume 99 of Heidelberger Taschenbücher. Springer, 1971.

    Google Scholar 

  6. X. Huang, M. Kerber, M. Kohlhase, E. Melis, D. Nesmith, J. Richts, and J. Siekmann. Omega-MKRP: A Proof Development Environment. In Proc. 12th International Conference on Automated Deduction (CADE), Nancy, 1994.

    Google Scholar 

  7. X. Huang, M. Kerber, M. Kohlhase, and J. Richts. Methods — the basic units for planning and verifying proofs. In Proceedings of Jahrestagung für Künstliche Intelligenz, Saarbrücken, 1994. Springer.

    Google Scholar 

  8. Th. Kolbe and Ch. Walther. Patching proofs for reuse. In N. Lavrac and S. Wrobel, editors, Proceedings of the 8th European Conference on Machine Learning 1995, Kreta, 1995.

    Google Scholar 

  9. P. Madden. Automated Program Transformation Through Proof Transformation. PhD thesis, University of Edinburgh, 1991.

    Google Scholar 

  10. W.W. McCune. Otter 2.0 users guide. Technical Report ANL-90/9, Argonne National Laboratory, Maths and CS Division, Argonne, Illinois, 1990.

    Google Scholar 

  11. E. Melis. Change of representation in theorem proving by analogy. SEKI-Report SR-93-07, Universität des Saarlandes, Saarbrücken, 1993.

    Google Scholar 

  12. E. Melis. How mathematicians prove theorems. In Proceedings of the Sixteenth Annual Conference of the Cognitive Science Society, pages 624–628, Atlanta, Georgia U.S.A., 1994.

    Google Scholar 

  13. E. Melis. Analogy-driven proof-plan construction. Technical Report DAI Research Paper No 735, University of Edinburgh, AI Dept, Dept. of Artificial Intelligence, Edinburgh, 1995.

    Google Scholar 

  14. E. Melis. A model of analogy-driven proof-plan construction. In Proceedings of the International Conference on Artificial Intelligence, Toronto, 1995. Morgan Kaufmann.

    Google Scholar 

  15. E. Melis and M.M. Veloso. Analogy makes proofs feasible. In D. Aha, editor, AAAI-94 Workshop on Case Based Reasoning, pages 13–17, Seattle, 1994.

    Google Scholar 

  16. G. Polya. How to Solve it. 2nd ed. Doubleday, New York, 1957.

    Google Scholar 

  17. S.J. Russell. Analogy by similarity. In D. Helman, editor, Analogical Reasoning, pages 251–269. Kluwer Academic Publisher, 1988.

    Google Scholar 

  18. B.L. van der Waerden. Wie der Beweis der Vermutung von Baudet gefunden wurde. Abh.Math.Sem. Univ.Hamburg, 28, 1964.

    Google Scholar 

  19. M.M. Veloso. Flexible strategy learning: Analogical replay of problem solving episodes. In Proc. of the twelfth National Conference on Artificial Intelligence 1994, Seattle, WA, 1994.

    Google Scholar 

  20. L. Wos. Automated Reasoning: 33 Basic Research Problems. Prentice-Hall, Englewood Cliffs, 1988.

    Google Scholar 

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Carlos Pinto-Ferreira Nuno J. Mamede

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© 1995 Springer-Verlag Berlin Heidelberg

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Melis, E. (1995). Theorem proving by analogy — A compelling example. In: Pinto-Ferreira, C., Mamede, N.J. (eds) Progress in Artificial Intelligence. EPIA 1995. Lecture Notes in Computer Science, vol 990. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-60428-6_22

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  • DOI: https://doi.org/10.1007/3-540-60428-6_22

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  • Print ISBN: 978-3-540-60428-0

  • Online ISBN: 978-3-540-45595-0

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