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Parigot’s Second Order λμ-Calculus and Inductive Types

  • Ralph Matthes
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2044)

Abstract

A new proof of strong normalization of Parigot’s (second order) λμ-calculus is given by a reduction-preserving embedding into system F (second order polymorphic λ-calculus). The main idea is to use the least stable supertype for any type. These non-strictly positive inductive types and their associated iteration principle are available in system F, and allow to give a translation vaguely related to CPS translations (corresponding to the Kolmogorov embedding of classical logic into intuitionistic logic). However, they simulate Parigot’s μ-reductions whereas CPS translations hide them.

As a major advantage, this embedding does not use the idea of reducing stability (¬¬¬ ¬ ¬) to that for atomic formulae. Therefore, it even extends to non-interleaving positive fixed-point types. As a non-trivial application, strong normalization of λμ-calculus, extended by primitive recursion on monotone inductive types, is established.

Keywords

Classical Logic Intuitionistic Logic Natural Deduction Elimination Rule Inductive Type 
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.
    Henk P. Barendregt. The Lambda Calculus: Its Syntax and Semantics. North-Holland, Amsterdam, second revised edition, 1984.zbMATHGoogle Scholar
  2. 2.
    Pierre-Louis Curien and Hugo Herbelin. The duality of computation. In Proceedings of the fifth ACM SIGPLAN International Conference on Functional Programming (ICFP’ 00), Montréal, pages 233–243. ACM Press, 2000.Google Scholar
  3. 3.
    Philippe de Groote. A CPS-translation of the λμ-calculus. In Sophie Tison, editor, Trees in Algebra and Programming — CAAP’94, 19th International Colloquium, volume 787 of Lecture Notes in Computer Science, pages 85–99, Edinburgh, 1994. Springer Verlag.CrossRefGoogle Scholar
  4. 4.
    Herman Geuvers. Inductive and coinductive types with iteration and recursion. In Bengt Nordström, Kent Pettersson, and Gordon Plotkin, editors, Proceedings of the 1992 Workshop on Types for Proofs and Programs, Bästad, Sweden, June 1992, pages 193–217, 1992. Only published electronically: ftp://ftp.cs.chalmers.se/pub/cs-reports/baastad.92/proc.dvi.Z
  5. 5.
    Jean-Yves Girard, Yves Lafont, and Paul Taylor. Proofs and Types, volume 7 of Cambridge Tracts in Theoretical Computer Science. Cambridge University Press, 1989.Google Scholar
  6. 6.
    Thierry Joly. Un plongement de la logique classique du 2nd ordre dans AF2. Unpublished manuscript. In French, 5 pp., January 1996.Google Scholar
  7. 7.
    Ralph Matthes. Extensions of System F by Iteration and Primitive Recursion on Monotone Inductive Types. Doktorarbeit (PhD thesis), University of Munich, 1998. Available via http://www.tcs.informatik.uni-muenchen.de/~matthes/.
  8. 8.
    Ralph Matthes. Monotone (co)inductive types and positive fixed-point types. The-oretical Informatics and Applications, 33(4/5):309–328, 1999.zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Ralph Matthes. Monotone fixed-point types and strong normalization. In Georg Gottlob, Etienne Grandjean, and Katrin Seyr, editors, Computer Science Logic, 12th International Workshop, Brno, Czech Republic, August 24–28, 1998, Proceedings, volume 1584 of Lecture Notes in Computer Science, pages 298–312. Springer-Verlag, 1999.Google Scholar
  10. 10.
    Nax P. Mendler. Recursive types and type constraints in second-order lambda calculus. In Proceedings of the Second Annual IEEE Symposium on Logic in Computer Science, Ithaca, N.Y., pages 30–36. IEEE Computer Society Press, 1987. Forms a part of [11].Google Scholar
  11. 11.
    Paul F. Mendler. Inductive definition in type theory. Technical Report 87-870, Cornell University, Ithaca, N.Y., September 1987. Ph.D. Thesis (Paul F. Mendler = Nax P. Mendler).Google Scholar
  12. 12.
    Michel Parigot. λμ-calculus: an algorithmic interpretation of classical natural deduction. In Andrei Voronkov, editor, Logic Programming and Automated Reasoning, International Conference LPAR’92, St. Petersburg, Russia, volume 624 of Lecture Notes in Computer Science, pages 190–201. Springer Verlag, 1992.CrossRefGoogle Scholar
  13. 13.
    Michel Parigot. Strong normalization for second order classical natural deduction. In Proceedings, Eighth Annual IEEE Symposium on Logic in Computer Science, pages 39–46, Montreal, Canada, 1993. IEEE Computer Society Press.Google Scholar
  14. 14.
    Michel Parigot. Proofs of strong normalisation for second order classical natural deduction. The Journal of Symbolic Logic, 62(4):1461–1479, 1997.zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Christine Paulin-Mohring. Définitions Inductives en Théorie des Types d’Ordre Supérieur. Habilitation á diriger les recherches, ENS Lyon, 1996.Google Scholar
  16. 16.
    Dag Prawitz. Natural Deduction. A Proof-Theoretical Study. Almquist and Wiksell, 1965.Google Scholar
  17. 17.
    Zdzisław Spławski and Paweł Urzyczyn. Type Fixpoints: Iteration vs. Recursion. SIGPLAN Notices, 34(9):102–113, 1999. Proceedings of the 1999 International Conference on Functional Programming (ICFP), Paris, France.CrossRefGoogle Scholar
  18. 18.
    Thomas Streicher and Bernhard Reus. Classical logic, continuation semantics and abstract machines. Journal of Functional Programming, 8(6):543–572, 1998.zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Ralph Matthes
    • 1
  1. 1.Institut für Informatik der Ludwig-Maximilians-Universität MünchenMünchenGermany

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