Advertisement

Rewriting with extensional polymorphic λ-calculus

  • Roberto Di Cosmo
  • Delia Kesner
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1092)

Abstract

We provide a confluent and strongly normalizing rewriting system, based on expansion rules, for the extensional second order typed lambda calculus with product and unit types: this system corresponds to the Intuitionistic Positive Calculus with implication, conjunction, quantification over proposition and the constant True. This result is an important step towards a new theory of reduction based on expansion rules, and gives a natural interpretation to the notion of second order η-long normal forms used in higher order resolution and unification, that are here just the normal forms of our reduction system.

Keywords

Normal Form Induction Hypothesis Basic Expansion Strong Normalization Lambda Calculus 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [Aka93]
    Yohji Akama. On Mints' reductions for ccc-Calculus. In Typed Lambda Calculus and Applications, number 664 in LNCS, pages 1–12. Springer Verlag, 1993.Google Scholar
  2. [Bar84]
    Henk Barendregt. The Lambda Calculus; Its syntax and Semantics (revised edition). North Holland, 1984.Google Scholar
  3. [BT88]
    Val Breazu-Tannen, Combining algebra and higher-order types. In Proceedings, Third Annual Symposium on Logic in Computer Science, pages 82–90, Edinburgh, Scotland, July 5–8 1988. IEEE Computer Society.Google Scholar
  4. [BTG94]
    Val Breazu-Tannen and Jean Gallier. Polymorphic rewiting preserves algebraic confluence. Information and Computation, 114:1–29, 1994.Google Scholar
  5. [BTM86]
    Val Breazu-Tannen and Albert R. Meyer. Polymorphism is conservative over simple types (preliminary report). In Proceedings, Symposium on Logic in Computer Science, pages 7–17, Cambridge, Massachusetts, June 16–18 1986. IEEE Computer Society.Google Scholar
  6. [CDC91]
    Pierre-Louis Curien and Roberto Di Cosmo. A confluent reduction system for the λ-calculus with surjective pairing and terminal object. In Leach, Monien, and Artalejo, editors, Intern. Conf. on Automata, Languages and Programming (ICALP), volume 510 of Lecture Notes in Computer Science, pages 291–302. Springer-Verlag, July 1991.Google Scholar
  7. [Cub92]
    Djordje Cubric. On free CCC. Distributed on the types mailing list, 1992.Google Scholar
  8. [DCK94a]
    Roberto Di Cosmo and Delia Kesner. Combining first order algebraic rewriting systems, recursion and extensional lambda calculi. In Serge Abite-boul and Eli Shamir, editors, Intern. Conf. on Automata, Languages and Programming (ICALP), volume 820 of Lecture Notes in Computer Science, pages 462–472. Springer-Verlag, July 1994.Google Scholar
  9. [DCK94b]
    Roberto Di Cosmo and Delia Kesner. Simulating expansions without expansions. Mathematical Structures in Computer Science, 4:1–48, 1994. A preliminary version is available as Technical Report LIENS-93-11/INRIA 1911.Google Scholar
  10. [DCP95]
    Roberto Di Cosmo and Adolfo Piperno. Expanding extensional polymorphism. In Mariangiola Dezani-Ciancaglini and Gordon Plotkin, editors, Typed Lambda Calculus and Applications, volume 902 of Lecture Notes in Computer Science, pages 139–153, April 1995.Google Scholar
  11. [Dou93]
    Daniel J. Dougherty. Some lambda calculi with categorical sums and products. In Proc, of the Fifth International Conference on Rewriting Techniques and Applications (RTA), 1993.Google Scholar
  12. [Ges90]
    Alfons Geser. Relative termination. Dissertation, Fakultät für Mathematik und Informatik, Universität Passau, Germany, 1990. Also available as: Report 91-03, Ulmer Informatik-Berichte, Universität Ulm, 1991.Google Scholar
  13. [Geu92]
    Herman Geuvers. The church-rosser property for βη-reduction in typed λ-calculi. In 7th Proceedings of the Symposium on Logic in Computer Science (LICS), pages 453–460, 1992.Google Scholar
  14. [Gha95]
    Neil Ghani. Extensionality and polymorphism. University of Edimburgh, Submitted, 1995.Google Scholar
  15. [GLT90]
    Jean-Yves Girard, Yves Lafont, and Paul Taylor. Proofs and Types. Cambridge University Press, 1990.Google Scholar
  16. [Jay92]
    Colin Barry Jay. Long βη normal forms and confluence (revised). Technical Report 44, LFCS — University of Edinburgh, August 1992.Google Scholar
  17. [JG92]
    Colin Barry Jay and Neil Ghani. The Virtues of Eta-expansion. Technical Report ECS-LFCS-92-243, LFCS, 1992. University of Edimburgh, preliminary version of [JG95].Google Scholar
  18. [JG95]
    Colin Barry Jay and Neil Ghani. The Virtues of Eta-expansion. Journal of Functional Programming, 5(2):135–154, April 1995.Google Scholar
  19. [JO91]
    Jean-Pierre Jouannaud and Mitsuhiro Okada. A computation model for executable higher-order algebraic specification languages. In Proceedings, Sixth Annual IEEE Symposium on Logic in Computer Science, pages 350–361, Amsterdam, The Netherlands, July 15–18 1991. IEEE Computer Society Press.Google Scholar
  20. [Klo80]
    Jan Willem Klop. Combinatory reduction systems. Mathematical Center Tracts, 27, 1980.Google Scholar
  21. [Nip90]
    Tobias Nipkow. A critical pair lemma for higher-order rewrite systems and its application to λ*. First Annual Workshop on Logical Frameworks, 1990.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1996

Authors and Affiliations

  • Roberto Di Cosmo
    • 1
  • Delia Kesner
    • 2
  1. 1.LIENS (CNRS) DMI Ecole Normale SupérieureParisFrance
  2. 2.CNRS and LRIUniversité de Paris-SudOrsay CedexFrance

Personalised recommendations