A journey through resource control lambda calculi and explicit substitution using intersection types (an account)

TL;DR

This paper explores three resource-controlled lambda calculi with explicit substitution, developing intersection type systems to characterize strong normalization through advanced proof techniques.

Contribution

It introduces intersection type assignment systems for three resource control lambda calculi and characterizes their strong normalization using novel proof methods.

Findings

Successful characterization of strong normalization for all three calculi.

Development of intersection type systems tailored to different variable types.

Application of reducibility and head subject expansion techniques.

Abstract

In this paper we invite the reader to a journey through three lambda calculi with resource control: the lambda calculus, the sequent lambda calculus, and the lambda calculus with explicit substitution. All three calculi enable explicit control of resources due to the presence of weakening and contraction operators. Along this journey, we propose intersection type assignment systems for all three resource control calculi. We recognise the need for three kinds of variables all requiring different kinds of intersection types. Our main contribution is the characterisation of strong normalisation of reductions in all three calculi, using the techniques of reducibility, head subject expansion, a combination of well-orders and suitable embeddings of terms.