Approximate Reduction and Lambda Calculus Models

Abstract

This paper gives the technical details and proofs for the notion of approximate reduction introduced in an earlier paper. The main theorem asserts that every lambda expression determines a set of approximate normal forms of which it is the limit in the lambda calculus models discovered by Scott in 1969. The proof of this theorem rests on the introduction of a notion of type assignments for the lambda calculus corresponding to the projections present in Scott’s models; the proof is then achieved by a series of lemmas providing connections between the type-free lambda calculus and calculations with these type assignments. As motivation for these semantic properties, we derive also some relations between the computational behavior of lambda expressions and their approximate normal forms, and we establish a syntactic analogue of the general considerations motivating the continuity of functions in Scott’s lattice theoretic approach.