Reductio ad Absurdum: Planning Proofs by Contradiction

  • Erica Melis
  • Martin Pollet
  • Jörg Siekmann
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4155)


Sometimes it is pragmatically useful to prove a theorem by contradiction rather than finding a direct proof. Some reductio ad absurdum arguments have made mathematical history and the general issue if and how a proof by contradiction can be replaced by a direct proof touches upon deep foundational issues such as the legitimacy of tertium non datur arguments in classical vs. intuitionistic foundations.

In this paper we are interested in the pragmatic issue when and how to use this proof strategy in everyday mathematics in general and in particular in automated proof planning. Proof planning is a general technique in automated theorem proving that captures and makes explicit proof patterns and mathematical search control. So, how can we proof plan an argument by reductio ad absurdum and when is it useful to do so? What are the methods and decision involved?


Theorem Prove Control Rule Natural Deduction Proof Schema Automate Theorem Prove 
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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  1. 1.
    Bartle, R.G., Sherbert, D.R.: Introduction to Real Analysis. John Wiley & Sons, New York (1982)MATHGoogle Scholar
  2. 2.
    Beeson, M.J.: Automatic generation of epsilon-delta proofs of continuity. In: Calmet, J., Plaza, J. (eds.) AISC 1998. LNCS (LNAI), vol. 1476, pp. 67–83. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  3. 3.
    Beeson, M.J.: Automatic generation of a proof of the irrationality of e. Journal of Symbolic Computation 32(4), 333–349 (2001)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Bledsoe, W.W.: Non-resolution theorem proving. Artificial Intelligence 9, 1–35 (1977)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Bundy, A.: The use of explicit plans to guide inductive proofs. In: Lusk, E., Overbeek, R. (eds.) CADE 1988. LNCS, vol. 310, pp. 111–120. Springer, Heidelberg (1988)CrossRefGoogle Scholar
  6. 6.
    Church, A.: A formulation of the simple theory of types. Journal of Symbolic Logic 5, 56–68 (1940)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Dummett, M.: Elements of Intuitionism, 2nd edn., Oxford (2000)Google Scholar
  8. 8.
    Gelernter, H.: Realization of a geometry theorem-proving machine. In: Proceedings of the International Conference on Information Processing, UNESCO (1959)Google Scholar
  9. 9.
    Lenat, D.B.: AM: an AI approach to discovery in mathematics. In: Davis, R., Lenat, D.B. (eds.) Knowledge-Based Systems in Artificial Intelligence. Mc-Graw Hill, New York (1981)Google Scholar
  10. 10.
    Melis, E.: AI-techniques in proof planning. In: European Conference on Artificial Intelligence, pp. 494–498. Kluwer, Brighton (1998)Google Scholar
  11. 11.
    Melis, E.: The “limit” domain. In: Simmons, R., Veloso, M., Smith, S. (eds.) Proceedings of the Fourth International Conference on Artificial Intelligence in Planning Systems, pp. 199–206 (1998)Google Scholar
  12. 12.
    Melis, E.: Automated Epsilon-Delta Proofs - A Research Story. Journal of Symbolic Computation (to be submitted, 2006)Google Scholar
  13. 13.
    Melis, E., Meier, A.: Proof planning with multiple strategies. In: Loyd, J., Dahl, V., Furbach, U., Kerber, M., Lau, K., Palamidessi, C., Pereira, L.M., Sagivand, Y., Stuckey, P. (eds.) CL 2000. Lecture Notes on Artificial Intelligence, vol. 1861, pp. 644–659. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  14. 14.
    Melis, E., Siekmann, J.H.: Knowledge-based proof planning. Artificial Intelligence 115(1), 65–105 (1999)MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Melis, E., Meier, A., Siekmann, J.: Proof planning with multiple strategies. Artificial Intelligence (submitted, 2006)Google Scholar
  16. 16.
    Meier, A., Melis, E.: Proof Planning Limit Problems with Multiple Strategies. SEKI Technical Report SR-2004-04, FR Informatik, Universitaet des Saarlandes (2004)Google Scholar
  17. 17.
    Polya, G.: How to Solve it. Princeton University Press, Princeton (1945)MATHGoogle Scholar
  18. 18.
    Robinson, J.A.: A machine-oriented logic based on the resolution principle. JACM 12 (1965)Google Scholar
  19. 19.
    Robinson, A., Voronkov, A.: Handbook of Automated Reasoning, vol. 1. Elsevier, Amsterdam (2001)MATHGoogle Scholar
  20. 20.
    Schoenfeld, A.H.: Mathematical Problem Solving. Academic Press, New York (1985)MATHGoogle Scholar
  21. 21.
    Sieg, W., Byrnes, J.: Normal natural deduction proofs (in classical logic). Studia Logica 60, 67–106 (1998)MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Siekmann, J., Benzmüller, C., Autexier, S.: Computer supported mathematics with Omega. Journal of Applied Logic (in press, 2006)Google Scholar
  23. 23.
    Siekmann, J., Benzmüller, C., Fiedler, A., Meier, A., Normann, I., Pollet, M.: Proof Development in OMEGA: The Irrationality of Square Root of 2. In: Kamareddine, F. (ed.) Thirty Five Years of Automating Mathematics. Kluwer Applied Logic series. Kluwer Academic Publishers, Dordrecht (2003)Google Scholar
  24. 24.
    Wiedijk, F. (ed.): The Seventeen Provers of the World (the first volume in the AI-Systems subseries). LNCS (LNAI), vol. 3600. Springer, Heidelberg (2006)Google Scholar
  25. 25.
    Zimmer, J., Melis, E.: Constraint solving for proof planning. Journal of Automated Reasoning 33(1), 51–88 (2004)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Erica Melis
    • 1
  • Martin Pollet
    • 1
  • Jörg Siekmann
    • 1
  1. 1.German Research Center for Artificial Intelligence (DFKI)Universität des SaarlandesSaarbrückenGermany

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