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Intuitive and Formal Representations: The Case of Matrices

  • Martin Pollet
  • Volker Sorge
  • Manfred Kerber
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3119)

Abstract

A major obstacle for bridging the gap between textbook mathematics and formalising it on a computer is the problem how to adequately capture the intuition inherent in the mathematical notation when formalising mathematical concepts. While logic is an excellent tool to represent certain mathematical concepts it often fails to retain all the information implicitly given in the representation of some mathematical objects. In this paper we concern ourselves with matrices, whose representation can be particularly rich in implicit information. We analyse different types of matrices and present a mechanism that can represent them very close to their textbook style appearance and captures the information contained in this representation but that nevertheless allows for their compilation into a formal logical framework. This firstly allows for a more human-oriented interface and secondly enables efficient reasoning with matrices.

Keywords

Formal Representation Mathematical Object Block Matrix Block Matrice Intermediate Representation 
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.
    Andrews, P.B.: An Introduction to Mathematical Logic and Type Theory: To Truth Through Proof, 2nd edn. Kluwer, Dordrecht (2002)CrossRefzbMATHGoogle Scholar
  2. 2.
    Bundy, A., Richardson, J.: Proofs about lists using ellipsis. In: Ganzinger, H., McAllester, D., Voronkov, A. (eds.) LPAR 1999. LNCS (LNAI), vol. 1705, pp. 1–12. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  3. 3.
    de Bruijn, N.G.: The mathematical vernacular, a language for mathematics with typed sets. In: Selected Papers on Automath, pp. 865–935. Elsevier, Amsterdam (1994)CrossRefGoogle Scholar
  4. 4.
    Elbers, H.: Connecting Informal and Formal Mathematics. PhD thesis. Eindhoven University of Technology (1998)Google Scholar
  5. 5.
    Kamareddine, F., Nederpelt, R.: A refinement of de Bruijn’s formal language of mathematics. Journal of Logic, Language and Information 13(3), 287–340 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Köcher, M.: Lineare Algebra und analytische Geometrie. Springer, Heidelberg (1992)CrossRefGoogle Scholar
  7. 7.
    Kutsia, T.: Unification with sequence variables and flexible arity symbols and its extension with pattern-terms. In: Calmet, J., Benhamou, B., Caprotti, O., Hénocque, L., Sorge, V. (eds.) AISC 2002 and Calculemus 2002. LNCS (LNAI), vol. 2385. Springer, Heidelberg (2002)Google Scholar
  8. 8.
    Lang, S.: Algebra, 2nd edn. Addison-Wesley, Reading (1984)zbMATHGoogle Scholar
  9. 9.
    NCAlgebra 3.7 - A Noncommutative Algebra Package for Mathematica, Available at: http://math.ucsd.edu/~ncalg/
  10. 10.
    Siekmann, J.H., Brezhnev, V., Cheikhrouhou, L., Fiedler, A., Horacek, H., Kohlhase, M., Meier, A., Melis, E., Moschner, M., Normann, I., Pollet, M., Sorge, V., Ullrich, C., Wirth, C.-P.: Proof development with OMEGA. In: Voronkov, A. (ed.) CADE 2002. LNCS (LNAI), vol. 2392, pp. 144–148. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  11. 11.
    Pollet, M., Sorge, V.: Integrating computational properties at the term level. In: Proc. of Calculemus 2002, pp. 78–83 (2003)Google Scholar
  12. 12.
    Wenzel, M., Wiedijk, F.: A Comparison of Mizar and Isar. J. of Automated Reasoning 29(3-4), 389–411 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Wolfram, S.: The Mathematica book, 5th edn. Wolfram Media, Inc. (2003)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Martin Pollet
    • 1
    • 2
  • Volker Sorge
    • 2
  • Manfred Kerber
    • 2
  1. 1.Fachbereich InformatikUniversität des SaarlandesGermany
  2. 2.School of Computer ScienceThe University of BirminghamEngland

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