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Abstract

One of the main applications of computational techniques to pure mathematics has been the use of computer algebra systems to perform calculations which mathematicians cannot perform by hand. Because the data is produced within the computer algebra system, this becomes an environment for the exploration of new functions and the data produced is often analysed in order to make conjectures empirically. We add some automation to this by using the HR theory formation system to make conjectures about Maple functions supplied by the user. Experience has shown that HR produces too many conjectures which are easily proven from the definitions of the functions involved. Hence, we use the Otter theorem prover to discard any theorems which can be easily proven, leaving behind the more interesting ones which are empirically true but not trivially provable. By providing an application of HR’s theory formation in number theory, we show that using Otter to prune HR’s dull conjectures has much potential for producing interesting conjectures about standard computer algebra functions.

Keywords

Prime Number Concept Formation Theorem Prover Computer Algebra Computer Algebra System 
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. 1.
    B Buchberger. Theory exploration versus theorem proving. In Proceedings of Calculemus 99, Systems for Integrated Computation and Deduction, 1998.Google Scholar
  2. 2.
    S Chou. Proving and discovering geometry theorems using Wu’s method. Technical Report 49, Computing Science, University of Austin at Texas, 1985.Google Scholar
  3. 3.
    S Colton. Refactorable numbers-a machine invention. Journal of Integer Sequences, 2, 1999.Google Scholar
  4. 4.
    S Colton. Automated Theory Formation in Pure Mathematics. PhD thesis, Department of Artificial Intelligence, University of Edinburgh, 2000.Google Scholar
  5. 5.
    S Colton, A Bundy, and T Walsh. Automatic identification of mathematical concepts. In Machine Learning: Proceedings of the 17th International Conference, 2000.Google Scholar
  6. 6.
    S Colton, A Bundy, and T Walsh. Automatic invention of integer sequences. In Proceedings of the Seventeenth National Conference on Artificial Intelligence, 2000.Google Scholar
  7. 7.
    S Colton and I Miguel. Constraint generation via automated theory formation. In Proceedings of CP-01, 2001.Google Scholar
  8. 8.
    S Fajtlowicz. On conjectures of Graffiti. Discrete Mathematics 72, 23:113–118, 1988.CrossRefMathSciNetGoogle Scholar
  9. 9.
    Andreas Franke and Michael Kohlhase. System description: MathWeb, an agent-based communication layer for distributed automated theorem proving. In Proceedings of CADE-16, pages 217–221, 1999.Google Scholar
  10. 10.
    G Hardy and E Wright. The Theory of Numbers. Oxford University Press, 1938.Google Scholar
  11. 11.
    D Lenat. AM: Discovery in mathematics as heuristic search. In D Lenat and R Davis, editors, Knowledge-Based Systems in Artificial Intelligence. McGraw-Hill Advanced Computer Science Series, 1982.Google Scholar
  12. 12.
    R McCasland, M Moore, and P Smith. An introduction to Zariski spaces over Zariski topologies. Rocky Mountain Journal of Mathematics, 28:1357–1369, 1998.zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    W McCune. The OTTER user’s guide. Technical Report ANL/90/9, Argonne National Laboratories, 1990.Google Scholar
  14. 14.
    W McCune. Single axioms for groups and Abelian groups with various operations. Journal of Automated Reasoning, 10(1):1–13, 1993.zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    W McCune. A Davis-Putnam program and its application to finite first-order model search. Technical Report ANL/MCS-TM-194, Argonne National Laboratories, 1994.Google Scholar
  16. 16.
    D. Redfern. The Maple Handbook: Maple V Release 5. Springer Verlag, 1999.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Simon Colton
    • 1
  1. 1.Division of InformaticsUniversity of EdinburghUK

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