Exponentials as Substitutions and the Cost of Cut Elimination in Linear Logic

TL;DR

This paper presents the exponential substitution calculus (ESC), a novel approach to cut elimination in linear logic that models exponentials as explicit substitutions, enabling polynomial cost analysis and alternative proof methods.

Contribution

It introduces ESC, pushing the idea of exponentials as substitutions to a new level, and provides the first polynomial cost model for cut elimination with unconstrained exponentials.

Findings

ESC models the sub-term property of proof systems.

Provides a polynomial cost model for cut elimination.

Proves confluence and strong normalization of ESC.

Abstract

This paper introduces the exponential substitution calculus (ESC), a new presentation of cut elimination for IMELL, based on proof terms and building on the idea that exponentials can be seen as explicit substitutions. The idea in itself is not new, but here it is pushed to a new level, inspired by Accattoli and Kesner's linear substitution calculus (LSC). One of the key properties of the LSC is that it naturally models the sub-term property of abstract machines, that is the key ingredient for the study of reasonable time cost models for the $\lambda$-calculus. The new ESC is then used to design a cut elimination strategy with the sub-term property, providing the first polynomial cost model for cut elimination with unconstrained exponentials. For the ESC, we also prove untyped confluence and typed strong normalization, showing that it is an alternative to proof nets for an advanced study of cut elimination.