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Abstract

We describe proof planning: a technique for both describing the hierarchical structure of proofs and then using this structure to guide proof attempts. When such a proof attempt fails, these failures can be analyzed and a patch formulated and applied. We also describe rippling: a powerful proof method used in proof planning. We pose and answer a number of common questions about proof planning and rippling.

Keywords

Automate Reasoning Induction Rule Inductive Proof Automate Theorem Prove Automate Deduction 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [Basin & Walsh, 1993]
    Basin, D.A., Walsh, T.: Difference unification. In: Bajcsy, R. (ed.) Proc. 13th Intern. Joint Conference on Artificial Intelligence (IJCAI 1993), vol. 1, pp. 116–122. Morgan Kaufmann, San Francisco (1993); Also available as Technical Report MPI-I-92-247, Max-Planck-Institut für InformatikGoogle Scholar
  2. [Basin & Walsh, 1996]
    Basin, D., Walsh, T.: A calculus for and termination of rippling. Journal of Automated Reasoning 16(1-2), 147–180 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  3. [Benzmüller et al, 1997]
    Benzmüller, C., Cheikhrouhou, L., Fehrer, D., Fiedler, A., Huang, X., Kerber, M., Kohlhase, K., Meier, A., Melis, E., Schaarschmidt, W., Siekmann, J., Sorge, V.: Ωmega: Towards a mathematical assistant. In: McCune, W. (ed.) 14th International Conference on Automated Deduction, pp. 252–255. Springer, Heidelberg (1997)Google Scholar
  4. [Boulton et al, 1998]
    Boulton, R., Slind, K., Bundy, A., Gordon, M.: An interface between CLAM and HOL. In: Grundy, J., Newey, M. (eds.) TPHOLs 1998. LNCS, vol. 1479, pp. 87–104. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  5. [Bundy & Green, 1996]
    Bundy, A., Green, I.: An experimental comparison of rippling and exhaustive rewriting. Research paper 836, Dept. of Artificial Intelligence, University of Edinburgh (December 1996)Google Scholar
  6. [Bundy & Lombart, 1995]
    Bundy, A., Lombart, V.: Relational rippling: a general approach. In: Mellish, C. (ed.) Proceedings of IJCAI, pp. 175–181 (1995)Google Scholar
  7. [Bundy, 1988]
    Bundy, A.: The use of explicit plans to guide inductive proofs. In: Lusk, R., Overbeek, R. (eds.) 9th International Conference on Automated Deduction, pp. 111–120. Springer, Heidelberg (1988); Longer version available from Edinburgh as DAI Research Paper No. 349CrossRefGoogle Scholar
  8. [Bundy, 1991]
    Bundy, A.: A science of reasoning. In: Lassez, J.-L., Plotkin, G. (eds.) Computational Logic: Essays in Honor of Alan Robinson, pp. 178–198. MIT Press, Cambridge (1991); Also available from Edinburgh as DAI Research Paper 445Google Scholar
  9. [Bundy, 2001]
    Bundy, A.: The automation of proof by mathematical induction. In: Robinson, A., Voronkov, A. (eds.) Handbook of Automated Reasoning, vol. 1, Elsevier, Amsterdam (2001)CrossRefGoogle Scholar
  10. [Bundy, 2002]
    Bundy, A.: A Critique of Proof Planning, pp. 160–177. Springer, Heidelberg (2002)Google Scholar
  11. [Bundy et al, 1989]
    Bundy, A., van Harmelen, F., Hesketh, J., Smaill, A., Stevens, A.: A rational reconstruction and extension of recursion analysis. In: Sridharan, N.S. (ed.) Proceedings of the Eleventh International Joint Conference on Artificial Intelligence, pp. 359–365. Morgan Kaufmann, San Francisco (1989); Also available from Edinburgh as DAI Research Paper 419Google Scholar
  12. [Bundy et al, 1990]
    Bundy, A., van Harmelen, F., Horn, C., Smaill, A.: The Oyster-Clam system. In: Stickel, M.E. (ed.) CADE 1990. LNCS (LNAI), vol. 449, pp. 647–648. Springer, Heidelberg (1990); Also available from Edinburgh as DAI Research Paper 507Google Scholar
  13. [Bundy et al, 1993]
    Bundy, A., Stevens, A., van Harmelen, F., Ireland, A., Smaill, A.: Rippling: A heuristic for guiding inductive proofs. Artificial Intelligence 62, 185–253 (1993); Also available from Edinburgh as DAI Research Paper No. 567zbMATHCrossRefMathSciNetGoogle Scholar
  14. [Bundy et al, 2005]
    Bundy, A., Basin, D., Hutter, D., Ireland, A.: Rippling: Meta-level Guidance for Mathematical Reasoning. Cambridge University Press, Cambridge (2005)zbMATHCrossRefGoogle Scholar
  15. [Dennis et al, 2000]
    Dennis, L., Bundy, A., Green, I.: Making a productive use of failure to generate witness for coinduction from divergent proof attempts. Annals of Mathematics and Artificial Intelligence 29, 99–138 (2000); Also available as paper No. RR0004 in the Informatics Report SerieszbMATHCrossRefMathSciNetGoogle Scholar
  16. [Desimone, 1989]
    Desimone, R.V.: Explanation-Based Learning of Proof Plans. In: Kodratoff, Y., Hutchinson, A. (eds.) Machine and Human Learning, Kogan Page (1989); Also available as DAI Research Paper 304. Previous version in proceedings of EWSL 1986 (1986)Google Scholar
  17. [Dixon & Fleuriot, 2003]
    Dixon, L., Fleuriot, J.D.: IsaPlanner: A prototype proof planner in Isabelle. In: Baader, F. (ed.) CADE 2003. LNCS (LNAI), vol. 2741, pp. 279–283. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  18. [Duncan et al, 2004]
    Duncan, H., Bundy, A., Levine, J., Storkey, A., Pollet, M. (2004). The use of data-mining for the automatic formation of tactics. In: Workshop on Computer-Supported Mathematical Theory Development. IJCAR 2004 (2004)Google Scholar
  19. [Frank & Basin, 1998]
    Frank, I., Basin, D.: Search in games with incomplete information: A case study using bridge card play. Artificial Intelligence 100(1-2), 87–123 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  20. [Frank et al, 1992]
    Frank, I., Basin, D., Bundy, A.: An adaptation of proofplanning to declarer play in bridge. In: Proceedings of ECAI, pp. 72–76, Vienna, Austria (1992); Longer Version available from Edinburgh as DAI Research Paper No. 575Google Scholar
  21. [Gow, 2004]
    Gow, J.: The Dynamic Creation of Induction Rules Using Proof Planning. Unpublished Ph.D. thesis, Division of Informatics, University of Edinburgh (2004)Google Scholar
  22. [Hutter & Kohlhase, 1997]
    Hutter, D., Kohlhase, M.: A colored version of the λ-Calculus. In: McCune, W. (ed.) 14th International Conference on Automated Deduction, pp. 291–305. Springer, Heidelberg (1997); Also available as SEKI-Report SR-95- 05Google Scholar
  23. [Ireland & Bundy, 1996]
    Ireland, A., Bundy, A.: Productive use of failure in inductive proof. Journal of Automated Reasoning 16(1-2), 79–111 (1996); Also available from Edinburgh as DAI Research Paper No 716zbMATHCrossRefMathSciNetGoogle Scholar
  24. [Ireland, 1992]
    Ireland, A.: The Use of Planning Critics in Mechanizing Inductive Proofs. In: Voronkov, A. (ed.) LPAR 1992. LNCS (LNAI), vol. 624, pp. 178–189. Springer, Heidelberg (1992); Also available from Edinburgh as DAI Research Paper 592CrossRefGoogle Scholar
  25. [Jackson & Lowe, 2000]
    Jackson, M., Lowe, H.: XBarnacle: Making theorem provers more accessible. In: McAllester, D. (ed.) CADE 2000. LNCS, vol. 1831, Springer, Heidelberg (2000)CrossRefGoogle Scholar
  26. [Kraan et al, 1996]
    Kraan, I., Basin, D., Bundy, A.: Middle-out reasoning for synthesis and induction. Journal of Automated Reasoning 16(1-2), 113–145 (1996); Also available from Edinburgh as DAI Research Paper 729.zbMATHCrossRefMathSciNetGoogle Scholar
  27. [Lowe & Duncan, 1997]
    Lowe, H., Duncan, D.: XBarnacle: Making theorem provers more accessible. In: McCune, W. (ed.) 14th International Conference on Automated Deduction, pp. 404–408. Springer, Heidelberg (1997)Google Scholar
  28. [Lowe et al, 1998]
    Lowe, H., Pechoucek, M., Bundy, A.: Proof planning for maintainable configuration systems. Artificial Intelligence in Engineering Design, Analysis and Manufacturing 12, 345–356 (1998); Special issue on configurationGoogle Scholar
  29. [Monroy et al, 1994]
    Monroy, R., Bundy, A., Ireland, A.: Proof Plans for the Correction of False Conjectures. In: Pfenning, F. (ed.) LPAR 1994. LNCS (LNAI), vol. 822, pp. 54–68. Springer, Heidelberg (1994); Also available from Edinburgh as DAI Research Paper 681Google Scholar
  30. [Silver, 1985]
    Silver, B.: Meta-level inference: Representing and Learning Control Information in Artificial Intelligence. North Holland, Amsterdam (1985); Revised version of the author’s PhD thesis, Department of Artificial Intelligence, U. of Edinburgh (1984)Google Scholar
  31. [Smaill & Green, 1996]
    Smaill, A., Green, I.: Higher-order annotated terms for proof search. In: Gunter, E.L., Felty, A.P. (eds.) TPHOLs 1997. LNCS, vol. 1275, pp. 399–414. Springer, Heidelberg (1997); Also available as DAI Research Paper 799Google Scholar
  32. [Stark & Ireland, 1998]
    Stark, J., Ireland, A.: Invariant discovery via failed proof attempts. In: Flener, P. (ed.) LOPSTR 1998. LNCS, vol. 1559, pp. 271–288. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  33. [Sterling et al, 1989]
    Sterling, L.: Bundy, Alan, Byrd, L., O’Keefe, R. and Silver, B. Solving symbolic equations with PRESS. J. Symbolic Computation 7, 71–84 (1989); Also available from Edinburgh as DAI Research Paper 171zbMATHCrossRefMathSciNetGoogle Scholar
  34. [Willmott et al, 2001]
    Willmott, S., Richardson, J.D.C., Bundy, A., Levine, J.M.: Applying adversarial planning techniques to Go. Journal of Theoretical Computer Science 252(1-2), 45–82 (2001); Special issue on algorithms, Automata, complexity and GameszbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Alan Bundy
    • 1
  1. 1.School of InformaticsUniversity of EdinburghEdinburghUK

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