Preservation of Strong Normalisation modulo permutations for the structural lambda-calculus

TL;DR

This paper introduces a new untyped structural lambda-calculus called lambda j, which combines actions at a distance with exponential rules, and proves it preserves strong normalization properties.

Contribution

It presents the lambda j calculus with properties like confluence and strong normalization preservation, and extends it with bisimulation and equations maintaining these properties.

Findings

Lambda j is confluent and preserves beta-strong normalization.

Adding bisimulation and equations maintains strong normalization.

The calculus offers a new framework inspired by linear logic for lambda calculus.

Abstract

Inspired by a recent graphical formalism for lambda-calculus based on linear logic technology, we introduce an untyped structural lambda-calculus, called lambda j, which combines actions at a distance with exponential rules decomposing the substitution by means of weakening, contraction and derelicition. First, we prove some fundamental properties of lambda j such as confluence and preservation of beta-strong normalisation. Second, we add a strong bisimulation to lambda j by means of an equational theory which captures in particular Regnier's sigma-equivalence. We then complete this bisimulation with two more equations for (de)composition of substitutions and we prove that the resulting calculus still preserves beta-strong normalization. Finally, we discuss some consequences of our results.