. Functional languages are based on the notion of application: programs may be applied to data or programs. By application one may define algebraic functions; and a programming language is functionally complete when any algebraic function f(x 1 ,...,x n ) is representable (i.e. there is a constant a such that f(x 1 ,...,x n ) = (a . x 1 . ... . x n ). Combinatory Logic is the simplest type-free language which is functionally complete. In a sound category-theoretic framework the constant a above may be considered as an "abstract gödel-number" for f, when gödel-numberings are generalized to "principal morphisms", in suitable categories. By this, models of Combinatory Logic are categorically characterized and their relation is given to lambda-calculus models within Cartesian Closed Categories. Finally, the partial recursive functionals in any finite higher type are shown to yield models of Combinatory Logic. ________________ (+) Theoretical Computer Science, 70 (2), 1990, pp.193-211. A p...

Machine by Curien [Cur86]. It is based upon a weak categorical combinatory logic, viz. lacking surjective pairing and extensionality, that arose as a direct semantic-to-syntactic translation of the lambda calculus of tuples. The computational mode was combinator term reduction through rewriting using a direct left-to-right parse algorithm, initially making the evaluation strategy inefficiently eager 1 . Application is therefore simply juxtaposition, losing the full expressiveness of-reduction that computes via substitution. Its overly strong bias towards the lambda calculus was another factor that limited its expressiveness. On one hand the CAM demanded the existence of categorical products but on the other it had no coproducts for developing many useful data structures. Nevertheless, the high acceptance and efficiency of the CAM-based ML compiler, CAML, gives significant encouragement towards developing a highly-programmable categorical computing paradigm. Some prominent workers in ...