Intersection Types for the lambda-mu Calculus

TL;DR

This paper develops an intersection type system for the lambda-mu calculus, linking it to denotational models of continuations, and characterizes strongly normalising terms through typeability.

Contribution

It introduces an invariant intersection type system for lambda-mu calculus based on domain-theoretic models, and relates typeability to strong normalisation.

Findings

Type system is invariant under subject reduction and expansion.

Typeability characterizes strongly normalising lambda-mu terms.

Provides an alternative proof of strong normalisability for typed lambda-mu calculus.

Abstract

We introduce an intersection type system for the lambda-mu calculus that is invariant under subject reduction and expansion. The system is obtained by describing Streicher and Reus's denotational model of continuations in the category of omega-algebraic lattices via Abramsky's domain-logic approach. This provides at the same time an interpretation of the type system and a proof of the completeness of the system with respect to the continuation models by means of a filter model construction. We then define a restriction of our system, such that a lambda-mu term is typeable if and only if it is strongly normalising. We also show that Parigot's typing of lambda-mu terms with classically valid propositional formulas can be translated into the restricted system, which then provides an alternative proof of strong normalisability for the typed lambda-mu calculus.