A Faithful and Quantitative Notion of Distant Reduction for the Lambda-Calculus with Generalized Applications

TL;DR

This paper introduces a lambda-calculus with generalized applications and distant reduction, providing a faithful, quantitative characterization of strong normalization, and relating it to explicit substitution calculi.

Contribution

It presents a new lambda-calculus with distant reduction that unblocks beta-redexes without permutative conversions, and characterizes strong normalization via a quantitative typing system.

Findings

Strong normalization of simply-typed terms is proven.

A faithful translation to explicit substitution calculi is established.

The calculus and original system define equivalent strong normalization notions.

Abstract

We introduce a call-by-name lambda-calculus $\lambda Jn$ with generalized applications which is equipped with distant reduction. This allows to unblock $\beta$-redexes without resorting to the standard permutative conversions of generalized applications used in the original $\Lambda J$-calculus with generalized applications of Joachimski and Matthes. We show strong normalization of simply-typed terms, and we then fully characterize strong normalization by means of a quantitative (i.e. non-idempotent intersection) typing system. This characterization uses a non-trivial inductive definition of strong normalization --related to others in the literature--, which is based on a weak-head normalizing strategy. We also show that our calculus $\lambda Jn$ relates to explicit substitution calculi by means of a faithful translation, in the sense that it preserves strong normalization. Moreover, our calculus $\lambda Jn$ and the original $\Lambda J$-calculus determine equivalent notions of strong normalization. As a consequence, $\lambda J$ inherits a faithful translation into explicit substitutions, and its strong normalization can also be characterized by the quantitative typing system designed for $\lambda Jn$, despite the fact that quantitative subject reduction fails for permutative conversions.