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A Critique of Proof Planning

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Computational Logic: Logic Programming and Beyond

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 2408))

Abstract

Proof planning is an approach to the automation of theorem proving in which search is conducted, not at the object-level, but among a set of proof methods. This approach dramatically reduces the amount of search but at the cost of completeness. We critically examine proof planning, identifying both its strengths and weaknesses. We use this analysis to explore ways of enhancing proof planning to overcome its current weaknesses.

The research reported in this paper was supported by EPSRC grant GR/M/45030. I would like to thank Andrew Ireland, Helen Lowe, Raul Monroy and two anonymous referees for helpful comments on this paper. I would also like to thank other members of the Mathematical Reasoning Group and the audiences at CIAO and Scottish Theorem Provers for helpful feedback on talks from which this paper arose.

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References

  1. Armando, A., Gallagher, J., Smaill, A. and Bundy, A. (3–5 January 1996). Automating the synthesis of decision procedures in a constructive metatheory. In Proceedings of the Fourth International Symposium on Artificial Intelligence and Mathematics, pages 5–8, Florida. Also in the Annals of Mathematics and Artificial Intelligence, 22, pp 259-79, 1998.

    Google Scholar 

  2. Benzmüller, C., Cheikhrouhou, L., Fehrer, D., Fiedler, A., Huang, X., Kerber, M., Kohlhase, K., Meier, A, Melis, E., Schaarschmidt, W., Siekmann, J. and Sorge, V. (1997). Wmega: Towards a mathematical assistant. In McCune, W., (ed.), 14th International Conference on Automated Deduction, pages 252–255. Springer-Verlag.

    Google Scholar 

  3. Bundy, A. (1988). The use of explicit plans to guide inductive proofs. In Lusk, R. and Overbeek, R., (eds.), 9th International Conference on Automated Deduction, pages 111–120. Springer-Verlag. Longer version available from Edinburgh as DAI Research Paper No. 349.

    Google Scholar 

  4. Bundy, Alan. (1991). A science of reasoning. In Lassez, J.-L. and Plotkin, G., (eds.), Computational Logic: Essays in Honor of Alan Robinson, pages 178–198. MIT Press. Also available from Edinburgh as DAI Research Paper 445.

    Google Scholar 

  5. Bundy, A., Smaill, A. and Hesketh, J. (1990a). Turning eureka steps into calculations in automatic program synthesis. In Clarke, S. L.H., (ed.), Proceedings of UK IT 90, pages 221–6. IEE. Also available from Edinburgh as DAI Research Paper 448.

    Google Scholar 

  6. Bundy, A., van Harmelen, F., Horn, C. and Smaill, A. (1990b). The Oyster-Clam system. In Stickel, M. E., (ed.), 10th International Conference on Automated Deduction, pages 647–648. Springer-Verlag. Lecture Notes in Artificial Intelligence No. 449. Also available from Edinburgh as DAI Research Paper 507.

    Google Scholar 

  7. Bundy, A., van Harmelen, F., Hesketh, J. and Smaill, A. (1991). Experiments with proof plans for induction. Journal of Automated Reasoning, 7:303–324. Earlier version available from Edinburgh as DAI Research Paper No 413.

    Article  MATH  MathSciNet  Google Scholar 

  8. Cantu, Francisco, Bundy, Alan, Smaill, Alan and Basin, David. (1996). Experiments in automating hardware verification using inductive proof planning. In Srivas, M. and Camilleri, A., (eds.), Proceedings of the Formal Methods for Computer-Aided Design Conference, number 1166 in Lecture Notes in Computer Science, pages 94–108. Springer-Verlag.

    Chapter  Google Scholar 

  9. Frank, I., Basin, D. and Bundy, A. (1992). An adaptation of proof-planning to declarer play in bridge. In Proceedings of ECAI-92, pages 72–76, Vienna, Austria. Longer Version available from Edinburgh as DAI Research Paper No. 575.

    Google Scholar 

  10. Gallagher, J. K. (1993). The Use of Proof Plans in Tactic Synthesis. Unpublished Ph.D. thesis, University of Edinburgh.

    Google Scholar 

  11. Gordon, M. J., Milner, A. J. and Wadsworth, C. P. (1979). Edinburgh LCF-A mechanised logic of computation, volume 78 of Lecture Notes in Computer Science. Springer-Verlag.

    Google Scholar 

  12. Gow, J. (1997). The Diagonalization Method in Automatic Proof. Undergraduate project dissertation, Dept of Artificial Intelligence, University of Edinburgh.

    Google Scholar 

  13. Hesketh, J., Bundy, A. and Smaill, A. (June 1992). Using middle-out reasoning to control the synthesis of tail-recursive programs. In Kapur, Deepak, (ed.), 11th International Conference on Automated Deduction, volume 607 of Lecture Notes in Artificial Intelligence, pages 310–324, Saratoga Springs, NY, USA.

    Google Scholar 

  14. Horn, Ch. (1992). Oyster-2: Bringing type theory into practice. Information Processing, 1:49–52.

    Google Scholar 

  15. Huang, X., Kerber, M. and Cheikhrouhou, L. (1995). Adapting the diagonalization method by reformulations. In Levy, A. and Nayak, P., (eds.), Proc. of the Symposium on Abstraction, Reformulation and Approximation (SARA-95), pages 78–85. Ville d’Esterel, Canada.

    Google Scholar 

  16. Ireland, A. and Bundy, A. (1996a). Extensions to a Generalization Critic for Inductive Proof. In McRobbie, M. A. and Slaney, J. K., (eds.), 13th International Conference on Automated Deduction, pages 47–61. Springer-Verlag. Springer Lecture Notes in Artificial Intelligence No. 1104. Also available from Edinburgh as DAI Research Paper 786.

    Google Scholar 

  17. Ireland, A. and Bundy, A. (1996b). Productive use of failure in inductive proof. Journal of Automated Reasoning, 16(1–2):79–111. Also available from Edinburgh as DAI Research Paper No 716.

    Article  MATH  MathSciNet  Google Scholar 

  18. Ireland, A. and Stark, J. (1997). On the automatic discovery of loop invariants. In Proceedings of the Fourth NASA Langley Formal Methods Workshop. NASA Conference Publication 3356. Also available as Research Memo RM/97/1 from Dept of Computing and Electrical Engineering, Heriot-Watt University.

    Google Scholar 

  19. Ireland, A. (1992). The Use of Planning Critics in Mechanizing Inductive Proofs. In Voronkov, A., (ed.), International Conference on Logic Programming and Automated Reasoning-LPAR 92, St. Petersburg, Lecture Notes in Artificial Intelligence No. 624, pages 178–189. Springer-Verlag. Also available from Edinburgh as DAI Research Paper 592.

    Google Scholar 

  20. Ireland, A., Jackson, M. and Reid, G. (1999). Interactive Proof Critics. Formal Aspects of Computing: The International Journal of Formal Methods, 11(3):302–325. A longer version is available from Dept. of Computing and Electrical Engineering, Heriot-Watt University, Research Memo RM/98/15.

    Article  Google Scholar 

  21. Jackson, M. (1999). Interacting with Semi-automated Theorem Provers via Interactive Proof Critics. Unpublished Ph.D. thesis, School of Computing, Napier University.

    Google Scholar 

  22. Kerber, Manfred and Sehn, Arthur C. (1997). Proving ground completeness of resolution by proof planning. In Dankel II, Douglas D., (ed.), FLAIRS-97, Proceedings of the 10th International Florida Artificial Intelligence Research Symposium, pages 372–376, Daytona, Florida, USA. Florida AI Research Society, St. Petersburg, Florida, USA.

    Google Scholar 

  23. Kerber, Manfred. (1998). Proof planning: A practical approach to mechanized reasoning in mathematics. In Bibel, Wolfgang and Schmitt, Peter H., (eds.), Automated Deduction, a Basis for Application-Handbook of the German Focus Programme on Automated Deduction, chapter III.4, pages 77–95. Kluwer Academic Publishers, Dordrecht, The Netherlands.

    Google Scholar 

  24. Kraan, I., Basin, D. and Bundy, A. (1996). Middle-out reasoning for synthesis and induction. Journal of Automated Reasoning, 16(1–2):113–145. Also available from Edinburgh as DAI Research Paper 729.

    Article  MATH  MathSciNet  Google Scholar 

  25. Lowe, H. and Duncan, D. (1997). XBarnacle: Making theorem provers more accessible. In McCune, William, (ed.), 14th International Conference on Automated Deduction, pages 404–408. Springer-Verlag.

    Google Scholar 

  26. Lowe, Helen. (1991). Extending the proof plan methodology to computer configuration problems. Artificial Intelligence Applications Journal, 5(3). Also available from Edinburgh as DAI Research Paper 537.

    Google Scholar 

  27. Lowe, H., Pechoucek, M. and Bundy, A. (October 1996). Proof planning and configuration. In Proceedings of the Ninth Exhibition and Symposium on Industrial Applications of Prolog. Also available from Edinburgh as DAI Research Paper 859.

    Google Scholar 

  28. Lowe, H., Pechoucek, M. and Bundy, A. (1998). Proof planning for maintainable configuration systems. Artificial Intelligence in Engineering Design, Analysis and Manufacturing, 12:345–356. Special issue on configuration.

    Google Scholar 

  29. Melis, E. (1998). The “limit” domain. In Simmons, R., Veloso, M. and Smith, S., (eds.), Proceedings of the Fourth International Conference on Artificial Intelligence in Planning Systems, pages 199–206.

    Google Scholar 

  30. Melis, E., Zimmer, J. and Müller, T. (2000a). Extensions of constraint solving for proof planning. In Horn, W., (ed.), European Conference on Artificial Intelligence, pages 229–233.

    Google Scholar 

  31. Melis, E., Zimmer, J. and Müller, T. (2000b). Integrating constraint solving into proof planning. In Ringeissen, Ch., (ed.), Frontiers of Combining Systems, Third International Workshop, FroCoS’2000, number 1794 in Lecture Notes on Artificial Intelligence, pages 32–46. Springer.

    Google Scholar 

  32. Richardson, J. D. C, Smaill, A. and Green, I. (July 1998). System description: proof planning in higher-order logic with Lambda-Clam. In Kirchner, Claude and Kirchner, Hélène, (eds.), 15th International Conference on Automated Deduction, volume 1421 of Lecture Notes in Artificial Intelligence, pages 129–133, Lindau, Germany.

    Google Scholar 

  33. Willmott, S., Richardson, J., Bundy, A. and Levine, J. (1999). An adversarial planning approach to Go. In Jaap van den Herik, H. and Iida, H., (eds.), Computers and Games, pages 93–112. 1st Int. Conference, CG98, Springer. Lecture Notes in Computer Science No. 1558.

    Chapter  Google Scholar 

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Author information

Authors and Affiliations

  1. Division of Informatics, University of Edinburgh, UK

    Alan Bundy

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  1. Alan Bundy
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Editors and Affiliations

  1. Department of Computer Science, University of Cyprus, 75 Kallipoleos St., 1678, Nicosia, Cyprus

    Antonis C. Kakas

  2. Department of Computing, Imperial College of Science, Technology and Medicine, 180 Queen’s Gate, London, SW7 2BZ, UK

    Fariba Sadri

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Bundy, A. (2002). A Critique of Proof Planning. In: Kakas, A.C., Sadri, F. (eds) Computational Logic: Logic Programming and Beyond. Lecture Notes in Computer Science(), vol 2408. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45632-5_7

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  • DOI: https://doi.org/10.1007/3-540-45632-5_7

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  • Print ISBN: 978-3-540-43960-8

  • Online ISBN: 978-3-540-45632-2

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