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Reductions, intersection types, and explicit substitutions

Published online by Cambridge University Press:  06 March 2003

DAN DOUGHERTY
Affiliation:
Department of Mathematics and Computer Science, Wesleyan University, Middletown, CT 06459 USA
PIERRE LESCANNE
Affiliation:
Laboratoire de l'Informatique du Parallélisme, École Normale Supérieure de Lyon, 46, Allée d'Italie, 69364 Lyon 07, France

Abstract

This paper is part of a general programme of treating explicit substitutions as the primary λ-calculi from the point of view of foundations as well as applications. We work in a composition-free calculus of explicit substitutions and an augmented calculus obtained by adding explicit garbage-collection, and explore the relationship between intersection-types and reduction.

We show that the terms that normalise by leftmost reduction and the terms that normalise by head reduction can each be characterised as the terms typable in a certain system. The relationship between typability and strong normalisation is subtly different from the classical case: we show that typable terms are strongly normalising but give a counterexample to the converse. Our notions of leftmost and head reduction are non-deterministic, and our normalisation theorems apply to any computations obeying these strategies. In this way we refine and strengthen the classical normalisation theorems. The proofs require some new techniques in the presence of reductions involving explicit substitutions. Indeed, our proofs do not rely on results from classical λ-calculus, which in our view is subordinate to the calculus of explicit substitution.

Type
Research Article
Copyright
© 2003 Cambridge University Press

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Footnotes

A preliminary abstract of this paper was presented at the Fifth International Conference on Typed Lambda Calculi and Applications (TLCA '2001).

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