An approximation theorem for topological lambda models and the topological incompleteness of lambda calculus

Abstract

It is well known that a reflexive object in the Cartesian closed category of complete partial orders and Scott-continuous functions is a model of λ-calculus (briefly a topological model). A topological model, through the interpretation function, induces a λ-theory, i.e., a congruence relation on λ-terms closed under α- and β-reduction. It is natural to ask if all possible λ-theories are induced by a topological model, i.e., if topological models are complete w.r.t. λ-calculus. The authors prove an Approximation Theorem, which holds in all topological models. Using this theorem, they analyze some topological models and their induced λ-theories, and they exhibit a λ-theory which cannot be induced by a topological model. So they prove that topological models are not complete w.r.t. λ-calculus.