An Abstract Approach to Stratification in Linear Logic

TL;DR

This paper introduces an abstract, categorical approach to stratification in linear logic, decoupling it from exponential modalities and revealing new complexity insights and reformulations of light linear logic.

Contribution

It presents a new logical system where stratification is handled by a separate modality derived from a categorical construction, independent of exponential modalities.

Findings

Stratification can be formulated independently of exponential modalities.

Connecting stratification to exponential modalities yields interesting complexity properties.

Provides three reformulations of Baillot and Mazza's linear logic by levels.

Abstract

We study the notion of stratification, as used in subsystems of linear logic with low complexity bounds on the cut-elimination procedure (the so-called light logics), from an abstract point of view, introducing a logical system in which stratification is handled by a separate modality. This modality, which is a generalization of the paragraph modality of Girard's light linear logic, arises from a general categorical construction applicable to all models of linear logic. We thus learn that stratification may be formulated independently of exponential modalities; when it is forced to be connected to exponential modalities, it yields interesting complexity properties. In particular, from our analysis stem three alternative reformulations of Baillot and Mazza's linear logic by levels: one geometric, one interactive, and one semantic.