Functional completeness of the mixed λ-calculus and combinatory logic

Abstract

Functional completeness of the combinatory logic means that every λ-expression may be translated into an equivalent combinator expression and this is the theoretical basis for the implementation of functional languages on combinator-based abstract machines. To obtain efficient implementations it is important to distinguish between early and late binding times. i.e. to distinguish between compile-time and run-time computations. We therefore introduce a two-level version of the λ-calculus where this distinction is made in an explicit way. Turning to the combinatory logic we only wish to generate combinator-code for the run-timecomputations. The two-level version of the combinatory logic therefore will be a mixed λ-calculus and combinatory logic. A previous paper has shown that (a natural formulation of) the mixed λ-calculus and combinatory logic is not functionally complete but only corresponds to a strict subset of the two-level λ-calculus. In this paper we extend the mixed λ-calculus and combinatory logic with a new combinator, Ψ, and show that this suffices for the mixed λ-calculus and combinatory logic to be functionally complete. However, the new combinator may not always be implementable and we therefore discuss conditions under which it can be dispensed with.