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On the Degeneracy of Σ-Types in Presence of Computational Classical Logic

  • Hugo Herbelin
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3461)

Abstract

We show that a minimal dependent type theory based on Σ-types and equality is degenerated in presence of computational classical logic. By computational classical logic is meant a classical logic derived from a control operator equipped with reduction rules similar to the ones of Felleisen’s \({\mathcal C}\) or Parigot’s μ operators. As a consequence, formalisms such as Martin-Löf’s type theory or the (Set-predicative variant of the) Calculus of Inductive Constructions are inconsistent in presence of computational classical logic. Besides, an analysis of the role of the η-rule for control operators through a set-theoretic model of computational classical logic is given.

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References

  1. 1.
    The Coq development team: The Coq Proof Assistant Reference Manual, Version 8.0 (2004), Available at http://coq.inria.fr/doc
  2. 2.
    Coquand, T.: Metamathematical investigations of a calculus of constructions. In: Odifreddi, P. (ed.) Logic and Computer Science. Apic Series, vol. 31, pp. 91–122. Academic Press, London (1990); Also INRIA Research Report number 1088 (september 1989)Google Scholar
  3. 3.
    Crolard, T.: A confluent lambda-calculus with a catch/throw mechanism. Journal of Functional Programming 9(6), 625–647 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Felleisen, M., Friedman, D.P., Kohlbecker, E., Duba, B.F.: Reasoning with continuations. In: First Symposium on Logic and Computer Science, pp. 131–141 (1986)Google Scholar
  5. 5.
    Griffin, T.G.: The formulae-as-types notion of control. In: Conf. Record 17th Annual ACM Symp. on Principles of Programming Languages, POPL 1990, San Francisco, CA, USA, January 17-19, pp. 47–57. ACM Press, New York (1990)Google Scholar
  6. 6.
    Martin-Löf, P.: Intuitionistic Type Theory. Bibliopolis (1984)Google Scholar
  7. 7.
    Parigot, M.: Lambda-mu-calculus: An algorithmic interpretation of classical natural deduction. In: Voronkov, A. (ed.) LPAR 1992. LNCS, vol. 624, pp. 190–201. Springer, Heidelberg (1992)CrossRefGoogle Scholar
  8. 8.
    Pottinger, G.: Definite descriptions and excluded middle in the theory of constructions, Communication to the TYPES electronic mailing list (1989)Google Scholar
  9. 9.
    Prawitz, D.: Natural Deduction - A Proof-Theoretical Study. Almqvist & Wiksell, Stockholm (1965)zbMATHGoogle Scholar
  10. 10.
    Smith, J.M.: The independence of Peano’s fourth axiom from Martin-Löf’s type theory without universes. Journal of Symbolic Logic 53, 840–845 (1988)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Spector, C.: Provably recursive functionals of analysis: a consistency proof of analysis by an extension of principles in current intuitionistic mathematics. In: Recursive Function Theory: Proc. Symposia in Pure Mathematics, vol. 5, pp. 1–27. American Mathematical Society, Providence (1962)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Hugo Herbelin
    • 1
  1. 1.École PolytechniqueLIX – INRIA-Futurs – PCRIPalaiseau Cedex

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