Typed categorical combinatory logic

  • P-L. Curien
Colloquium On Trees In Algebra And Programming Algorithms And Combinatorics
Part of the Lecture Notes in Computer Science book series (LNCS, volume 185)


The subject of the paper is the connection between the typed λ-calculus and the cartesian closed categories, pointed out by several authors. Three languages and their theories, defined by equations, are shown to be equivalent: the typed λc-calculus (i.e. the λ-calculus with explicit products and projections) λc K , the free cartesian closed category CCC K , and a third intermediary language, the typed categorical combinatory logic CCL K , introduced by the author. In contrast to CCC K , CCL K has the same types as λc K , and roughly the terminal object in CCC K is replaced by the application and couple operators in CCL K . In CCL K β-reductions as well as evaluations w.r.t. environments (the basis of most practical implementations of λ-calculus based languages) may be simulated in the well-known framework of a same term rewriting system. Finally the introduction of CCL K allowed the author to understand the untyped underlying calculus, investigated in a companion paper. Another companion paper describes a general setting for equivalences between equational theories and their induced semantic equivalences, the equivalence between CCL K and CC K is an instance of which.

4. References

  1. [BeSy]
    G. Berry, Some Syntactic and Categorical Constructions of Lambdacalculus models, Rapport INRIA 80 (1981).Google Scholar
  2. [Bru]
    N.G. De Bruijn, Lambda-calculus Notation without Nameless Dummies, a Tool for Automatic Formula Manipulation, Indag Math. 34, 381–392 (1972).Google Scholar
  3. [CuTh3]
    P-L. Curien, Algorithmes Séquentiels sur Structures de Données Concrètes, Thèse de Troisième Cycle, Université Paris VII (Mars 1979).Google Scholar
  4. [CuTh]
    P-L. Curien, Combinateurs Catégoriques, Algorithmes Séquentiels et Programmation Applicative, Thèse d'Etat, Université Paris VII (Décembre 83), to be published in english as a monograph.Google Scholar
  5. [CuCCL]
    P-L. Curien, Categorical Combinatory Logic, submitted to ICALP 85.Google Scholar
  6. [CuEq]
    Syntactic Equivalences Inducing Semantic Equivalences, submitted to EUROCAL 1985.Google Scholar
  7. [Dyb]
    P. Dybjer, Category-Theoretic Logics and Algebras of Programs, PhD Thesis, Chalmers University of Technology, Goteborg (1983).Google Scholar
  8. [Lam]
    J. Lambek, From Lambda-calculus to Cartesian Closed Categories, in To H.B. Curry: Essays on Combinatory Logic, Lambda-calculus and Formalism, ed. J.P. Seldin and J.R. Hindley, Academic Press (1980).Google Scholar
  9. [LamSco]
    J. Lambek and P.J. Scott, Introduction to Higher Order Categorical Logic, to be published by Cambridge University Press (1984).Google Scholar
  10. [Lan]
    P.J. Landin, The Mechanical Evaluation of Expressions, Computer Journal 6, 308–320 (1964).Google Scholar
  11. [PaGhoTh]
    K. Parsaye-Ghomi, Higher Order Abstract Algebras, PhD Thesis, UCLA (1981).Google Scholar
  12. [Poi]
    A. Poigné, Higher Order Data Structures, Cartesian Closure Versus λ-calculus, STACS 84, Lect. Notes in Comput. Sci.Google Scholar
  13. [Pot]
    G. Pottinger, The Church-Rosser Theorem for the Typed λ-calculus with Extensional Pairing, preprint, Carnegie-Mellon University, Pittsburgh (March 1979).Google Scholar
  14. [Sco4]
    D. Scott, Relating Theories of the Lambda-calculus, cf. [Lam].Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1985

Authors and Affiliations

  • P-L. Curien
    • 1
  1. 1.CNRS-Université Paris VII, LITPParis Cedex 05

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