Advertisement

Normal proofs and their grammar

  • M. Takahashi
  • Y. Akama
  • S. Hirokawa
Invited Talk 5
Part of the Lecture Notes in Computer Science book series (LNCS, volume 789)

Abstract

First we give a grammatical (or equational) description of the set {M normal form ¦ Γ ⊢ M: A} for a given basis Γ and a given type A in the simple type system, and give some applications of the description. Then we extend the idea to systems in λ-cube and more generally to normalizing pure type systems. The attempt resulted in derived (or ‘macro’) rules the totality of which is sound and complete for type assignments of normal terms. A feature of the derived rules is that they reflect the syntactic structure of legal terms in normal form, and thus they may give us more global view than the original definition of the systems.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [Aka91]
    Y. Akama. Number of proofs and tree automaton. manuscript, 1991.Google Scholar
  2. [Bar92]
    H Barendregt. Lambda calculi with types. In S. Abramsky, D.M. Gabbai, and T.S.E. Maibaum, editors, Handbook of Logic in Computer Science, volume II. Oxford University Press, 1992.Google Scholar
  3. [Ber93]
    S. Berardi. Personal communication, 1993.Google Scholar
  4. [BY79]
    C.-B. Ben-Yelles. Type-assignment in the Lambda-calculus. PhD thesis, University College, Swansea, 1979.Google Scholar
  5. [Geu93]
    H. Geuvers. Conservativity between logics and typed λ calculi. In H. Geuvers, editor, Informal Proceedings of the 1993 Workshop on Types for Proofs and Programs, pages 131–156, 1993.Google Scholar
  6. [GN91]
    H. Geuvers and M.J. Nederhof. Modular proof of strong normalisation for the calculus of constructions. Journal of Functional Programming, 1(2):155–189, 1991.Google Scholar
  7. [Hin]
    J.R Hindley. The Simple Theory of Type Assignment. Forthcoming.Google Scholar
  8. [Hir92]
    S. Hirokawa. The number of proofs for implicational formulas(abstract). In Proceedings of Logic Colloquium '92, volume 36. Janos Bolyai Mathematical Society, 1992.Google Scholar
  9. [HU79]
    J.E. Hopcroft and J.D. Ullman. Introduction to Automata Theory, Languages and Computation. Addison-Wesley, 1979.Google Scholar
  10. [Löb76]
    M. Löb. Embedding first order predicate logic in fragments of intuitionistic logic. Journal of Symbolic Logic, 41(4):705–718, 1976.Google Scholar
  11. [Pra65]
    D. Prawitz. Natural Deduction. Almqvist and Wiksell, 1965. Stockholm.Google Scholar
  12. [vBJ93]
    L.S. van Benthem Jutting. Typing in pure type systems. Information and Computation, 105:30–41, 1993.Google Scholar
  13. [Zai87]
    M. Zaionc. Word operation definable in the typed λ-calculus. Theoretical computer science, 52:1–14, 1987.Google Scholar
  14. [Zai91]
    M. Zaionc. λ-definability on free algebras. Annals of Pure and Applied Logic, 51:279–300, 1991.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1994

Authors and Affiliations

  • M. Takahashi
    • 1
  • Y. Akama
    • 1
  • S. Hirokawa
    • 2
  1. 1.Department of Information ScienceTokyo Institute of TechnologyJapan
  2. 2.Department of Computer Science College of General EducationKyushu UniversityJapan

Personalised recommendations