# Typed categorical combinatory logic

## Abstract

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 λ*c*-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

- [BeSy]G. Berry, Some Syntactic and Categorical Constructions of Lambdacalculus models, Rapport INRIA 80 (1981).Google Scholar
- [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
- [CuTh3]P-L. Curien, Algorithmes Séquentiels sur Structures de Données Concrètes, Thèse de Troisième Cycle, Université Paris VII (Mars 1979).Google Scholar
- [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
- [CuCCL]P-L. Curien, Categorical Combinatory Logic, submitted to ICALP 85.Google Scholar
- [CuEq]Syntactic Equivalences Inducing Semantic Equivalences, submitted to EUROCAL 1985.Google Scholar
- [Dyb]P. Dybjer, Category-Theoretic Logics and Algebras of Programs, PhD Thesis, Chalmers University of Technology, Goteborg (1983).Google Scholar
- [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
- [LamSco]J. Lambek and P.J. Scott, Introduction to Higher Order Categorical Logic, to be published by Cambridge University Press (1984).Google Scholar
- [Lan]P.J. Landin, The Mechanical Evaluation of Expressions, Computer Journal 6, 308–320 (1964).Google Scholar
- [PaGhoTh]K. Parsaye-Ghomi, Higher Order Abstract Algebras, PhD Thesis, UCLA (1981).Google Scholar
- [Poi]A. Poigné, Higher Order Data Structures, Cartesian Closure Versus λ-calculus, STACS 84, Lect. Notes in Comput. Sci.Google Scholar
- [Pot]G. Pottinger, The Church-Rosser Theorem for the Typed λ-calculus with Extensional Pairing, preprint, Carnegie-Mellon University, Pittsburgh (March 1979).Google Scholar
- [Sco4]D. Scott, Relating Theories of the Lambda-calculus, cf. [Lam].Google Scholar