Principal Typings in a Restricted Intersection Type System for Beta Normal Forms with De Bruijn Indices

TL;DR

This paper introduces a de Bruijn index-based intersection type system for lambda calculus, characterising principal typings of beta-normal forms and providing algorithms for type inference and normal form reconstruction.

Contribution

It extends previous work by developing a de Bruijn version of an intersection type system for principal typings, including algorithms for type inference and normal form reconstruction.

Findings

Characterisation of principal typings for beta-normal forms.

Algorithms for type inference and normal form reconstruction.

Discussion on subject reduction property failure and potential solutions.

Abstract

The lambda-calculus with de Bruijn indices assembles each alpha-class of lambda-terms in a unique term, using indices instead of variable names. Intersection types provide finitary type polymorphism and can characterise normalisable lambda-terms through the property that a term is normalisable if and only if it is typeable. To be closer to computations and to simplify the formalisation of the atomic operations involved in beta-contractions, several calculi of explicit substitution were developed mostly with de Bruijn indices. Versions of explicit substitutions calculi without types and with simple type systems are well investigated in contrast to versions with more elaborate type systems such as intersection types. In previous work, we introduced a de Bruijn version of the lambda-calculus with an intersection type system and proved that it preserves subject reduction, a basic property of type systems. In this paper a version with de Bruijn indices of an intersection type system originally introduced to characterise principal typings for beta-normal forms is presented. We present the characterisation in this new system and the corresponding versions for the type inference and the reconstruction of normal forms from principal typings algorithms. We briefly discuss the failure of the subject reduction property and some possible solutions for it.