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Mathematical Vernacular and Conceptual Well-Formedness in Mathematical Language

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Logical Aspects of Computational Linguistics (LACL 1997)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 1582))

Abstract

This paper investigates the semantics of mathematical concepts in a type theoretic framework with coercive subtyping. The type-theoretic analysis provides a formal semantic basis in the design and implementation of Mathematical Vernacular (MV), a natural language suitable for interactive development of mathematics with the support of the current theorem provingtec hnology.

The idea of semantic well-formedness in mathematical language is motivated with examples. A formal system based on a notion of conceptual category is then presented, showing how type checking supports our notion of well-formedness. The power of this system is then extended by incorporating a notion of subcategory, using ideas from a more general theory of coercive subtyping, which provides the mechanisms for modelling conventional abbreviations in mathematics. Finally, we outline how this formal work can be used in an implementation of MV.

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This work is supported partly by the Durham Mathematical Vernacular project funded by the Leverhulme Trust (see http://www.dur.ac.uk/~dcs7ttg/mv.html) and partly by the project on Subtyping, Inheritance, and Reuse funded by UK EPSRC (GR/K79130, see http://www.dur.ac.uk/~dcs7ttg/sir.html).

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References

  1. A. Bailey. The Machine-checked Literate Formalisation of Algebra in Type Theory. PhD thesis, University of Manchester, 1998.

    Google Scholar 

  2. R. Burstall and J. McKinna. Deliverables: a categorical approach to program development in type theory. LFCS report ECS-LFCS-92-242, Dept of Computer Science, University of Edinburgh, 1992.

    Google Scholar 

  3. A. Copestake and T. Briscoe. Semi-productive polysemy and sense extension. In J. Pustejovsky and B. Boguraev, editors, Lexical Semantics: The Problem of Polysemy. Clarendon, 1996.

    Google Scholar 

  4. Y. Coscoy, G. Kahn, and L. Théry. Extractingtexts from proofs. Technical Report 2459, INRIA, Sophia-Antipolis, 1995.

    MATH  Google Scholar 

  5. P. Callaghan and Z. Luo. Mathematical vernacular in type theory-based proof assistants. In R. Backhouse, editor, User Interfaces for Theorem Proving, UITP '98, July 1998.

    Google Scholar 

  6. Coq. The Coq Proof Assistant Reference Manual (version 6.1). INRIA-Rocquencourt and CNRS-ENS Lyon, 1996.

    Google Scholar 

  7. N.G. de Bruijn. A survey of the project AUTOMATH. In J. Hindley and J. Seldin, editors, To H. B. Curry: Essays on Combinatory Logic, Lambda Calculus and Formalism. Academic Press, 1980.

    Google Scholar 

  8. N. G. de Bruijn. The mathematical vernacular, a language for mathematics with typed sets. In R. P. Nederpelt, J. H. Geuvers, and R. C. de Vrijer, editors, Selected Papers on Automath. North Holland, 1994.

    Google Scholar 

  9. R. Jackendoff. The Architecture of the Language Faculty. MIT, 1997.

    Google Scholar 

  10. A. Jones, Z. Luo, and S. Soloviev. Some proof-theoretic and algorithmic aspects of coercive subtyping. Proc. of the Annual Conf on Types and Proofs (TYPES'96), 1997. To appear.

    Google Scholar 

  11. A. Jones. The formalization of linear algebra in LEGO: The decidable dependency theorem. Master's thesis, University of Manchester, 1995.

    Google Scholar 

  12. Z. Luo and R. Pollack. LEGO Proof Development System: User's Manual. LFCS Report ECS-LFCS-92-211, Department of Computer Science, University of Edinburgh, 1992.

    Google Scholar 

  13. Z. Luo. A higher-order calculus and theory abstraction. Information and Computation, 90(1), 1991.

    Google Scholar 

  14. Z. Luo. Program specification and data refinement in type theory. Proc. of the Fourth Inter. Joint Conf. on the Theory and Practice of Software Development (TAPSOFT), LNCS 493, 1991. Also as LFCS report ECSLFCS-91-131, Dept. of Computer Science, Edinburgh University.

    Google Scholar 

  15. Z. Luo. Program specification and data refinement in type theory. Mathematical Structures in Computer Science, 3(3), 1993. An earlier version appears as [Luo91b].

    Google Scholar 

  16. Z. Luo. Computation and Reasoning: A Type Theory for Computer Science. Oxford University Press, 1994.

    Google Scholar 

  17. Z. Luo. Coercive subtypingin type theory. Proc. of CSL'96, the 1996 Annual Conference of the European Association for Computer Science Logic, Utrecht. LNCS 1258, 1997.

    Google Scholar 

  18. Z. Luo. Coercive subtyping. Journal of Logic and Computation, 1998. To appear.

    Google Scholar 

  19. J. Lyons. Linguistic Semantics. Cambridge University Press, 1995.

    Google Scholar 

  20. J. McKinna. Deliverables: a categorical approach to program development in type theory. PhD thesis, Department of Computer Science, University of Edinburgh, 1992.

    Google Scholar 

  21. Mizar. Mizar home page. http://mizar.uw.bialystok.pl/.

  22. L. Magnusson and B. Nordström. The ALF proof editor and its proof engine. In Types for Proof and Programs, LNCS, 1994.

    Google Scholar 

  23. B. Nordström, K. Petersson, and J. Smith. Programming in Martin-Löf's Type Theory: An Introduction. Oxford University Press, 1990.

    Google Scholar 

  24. J. Pustejovsky. The Generative Lexicon. MIT, 1995.

    Google Scholar 

  25. A. Ranta. Type-theoretical Grammar. Oxford University Press, 1994.

    Google Scholar 

  26. A. Ranta. Type-theoretical interpretation and generalization of phrase structure grammar. Bulletin of the IGPL, 1995.

    Google Scholar 

  27. Aarne Ranta. Context-relative syntactic categories and the formalization of mathematical text. In S. Berardi and M. Coppo, editors, Types for Proofs and Programs. Springer-Verlag, Heidelberg, 1996.

    MATH  Google Scholar 

  28. A. Ranta. A grammatical framework (some notes on the source files), 1997.

    Google Scholar 

  29. Mark Ruys. Formalizing Mathematics in Type Theory. PhD thesis, Computing Science Institute, University of Nijmegen. (to be submitted).

    Google Scholar 

  30. A. Saibi. Typing algorithm in type theory with inheritance. Proc of POPL'97, 1997.

    Google Scholar 

  31. S. Soloviev and Z. Luo. Coercive subtyping: coherence and conservativity, 1998. In preparation.

    Google Scholar 

  32. L. van Benthem Jutting, James McKinna, and Robert Pollack. Type checking in pure type systems. submitted for publication, 1993.

    Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Computer Science, University of Durham, South Road, Durham, DH1 3LE, UK

    Zhaohui Luo & Paul Callaghan

Authors
  1. Zhaohui Luo
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  2. Paul Callaghan
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Editor information

Editors and Affiliations

  1. Université Pierre Mendès France (Grenoble 2), 1251 avenue Centrale, BP 47, F-38040, Grenoble Cedex, France

    Alain Lecomte

  2. LORIA (Nancy), 615 rue de Jardin Botanique, BP 101, F-54602, Villers lès Nancy Cedex, France

    François Lamarche  & Guy Perrier  & 

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© 1999 Springer-Verlag Berlin Heidelberg

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Luo, Z., Callaghan, P. (1999). Mathematical Vernacular and Conceptual Well-Formedness in Mathematical Language. In: Lecomte, A., Lamarche, F., Perrier, G. (eds) Logical Aspects of Computational Linguistics. LACL 1997. Lecture Notes in Computer Science(), vol 1582. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-48975-4_12

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  • DOI: https://doi.org/10.1007/3-540-48975-4_12

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-65751-4

  • Online ISBN: 978-3-540-48975-7

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