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Flexible Encoding of Mathematics on the Computer

  • Fairouz Kamareddine
  • Manuel Maarek
  • J. B. Wells
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3119)

Abstract

This paper reports on refinements and extensions to the MathLang framework that add substantial support for natural language text. We show how the extended framework supports multiple views of mathematical texts, including natural language views using the exact text that the mathematician wants to use. Thus, MathLang now supports the ability to capture the essential mathematical structure of mathematics written using natural language text. We show examples of how arbitrary mathematical text can be encoded in MathLang without needing to change any of the words or symbols of the texts or their order. In particular, we show the encoding of a theorem and its proof that has been used by Wiedijk for comparing many theorem prover representations of mathematics, namely the irrationality of \(\sqrt{2}\) (originally due to Pythagoras). We encode a 1960 version by Hardy and Wright, and a more recent version by Barendregt.

Keywords

Natural Language Original Text Weak Type Grammatical Category Concrete Syntax 
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.
    Asperti, A., Padovani, L., Sacerdoti Coen, C., Guidi, F., Schena, I.: Mathematical Knowledge Management in HELM. AMAI 38(1-3), 27–46 (2003)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Audebaud, P., Rideau, L.: TEXMACS as authoring tool for publication and dissemination of formal developments. UITP (2003)Google Scholar
  3. 3.
    Barendregt, H.: Towards an interactive mathematical proof mode. In: Kamareddine (ed.) Thirty Five Years of Automating Mathematics. Applied Logic, vol. 28 (2003)Google Scholar
  4. 4.
    de Bruijn, N.G.: The Mathematical Vernacular, a language for mathematics with typed sets. In: Workshop on Programming Logic (1987)Google Scholar
  5. 5.
    Coscoy, Y.: A natural language explanation for formal proofs. In: Retoré, C. (ed.) LACL 1996. LNCS, vol. 1328, p. 149. Springer, Heidelberg (1997)CrossRefGoogle Scholar
  6. 6.
    Davenport, J.H.: MKM from Book to Computer: A Case Study. In: Asperti, A., Buchberger, B., Davenport, J.H. (eds.) MKM 2003. LNCS, vol. 2594, pp. 17–29. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  7. 7.
    Deach, S.: Extensible Stylesheet Language (XSL) Recommendation, World Wide Web Consortium (1999), http://www.w3.org/TR/xslt
  8. 8.
    Mathematics On the Web: Get it by Logic and Interfaces (MOWGLI), http://www.mowgli.cs.unibo.it/
  9. 9.
    Théry, L.: Formal Proof Authoring: an Experiment, UITP (2003)Google Scholar
  10. 10.
    Heath, The 13 Books of Euclid’s Elements, Dover (1956)Google Scholar
  11. 11.
    van Heijenoort (ed.): From Frege to Gödel: A Source Book in Mathematical Logic, pp. 1879–1931. Harvard University Press, Cambridge (1967)zbMATHGoogle Scholar
  12. 12.
    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
  13. 13.
    Kamareddine, F., Maarek, M., Wells, J.B.: MathLang: Experience-driven development of a new mathematical language. ENTCS 93, 138–160 (2004)zbMATHGoogle Scholar
  14. 14.
    Kohlhase, M.: OMDoc: An Open Markup Format for Mathematical Documents (Version 1.1), Technical report (2003)Google Scholar
  15. 15.
    Landau, E.: Foundations of Analysis, Chelsea (1951)Google Scholar
  16. 16.
    Luo, Z., Callaghan, P.: Mathematical vernacular and conceptual well-formedness in mathematical language. In: Lecomte, A., Perrier, G., Lamarche, F. (eds.) LACL 1997. LNCS, vol. 1582, p. 232. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  17. 17.
    Maarek, M., Prevosto, V.: FoCDoc: The documentation system of FoC, Calculemus (2003)Google Scholar
  18. 18.
    Ranta, A.: Grammatical Framework: A Type-Theoretical Grammar Formalism. Journal of Functional Programming (2003)Google Scholar
  19. 19.
    Rudnicki, P., Trybulec, A.: On equivalents of well-foundedness. Journal of Automated Reasoning 23, 197–234 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Wiedijk, F.: The Fifteen Provers of the World, University of NijmegenGoogle Scholar
  21. 21.
    Wiedijk, F.: Comparing Mathematical Provers. In: Asperti, A., Buchberger, B., Davenport, J.H. (eds.) MKM 2003. LNCS, vol. 2594, pp. 188–202. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  22. 22.
    Zimmer, J., Kohlhase, M.: System Description: The MathWeb Software Bus for Dis- tributed Mathematical Reasoning. In: Voronkov, A. (ed.) CADE 2002. LNCS (LNAI), vol. 2392. Springer, Heidelberg (2002)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Fairouz Kamareddine
    • 1
  • Manuel Maarek
    • 1
  • J. B. Wells
    • 1
  1. 1.Heriot-Watt UniversityUK

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