Proof Theory of Skew Non-Commutative MILL

TL;DR

This paper introduces a sequent calculus for skew non-commutative MILL, providing a proof-theoretic framework that models skew monoidal closed categories and addresses coherence via focusing techniques.

Contribution

It presents the first cut-free sequent calculus for skew NMILL and adapts focusing to handle non-invertible structural laws, enabling proof search and coherence decision.

Findings

Developed a focused sequent calculus for skew NMILL

Provided a method to decide equality of maps in free skew monoidal closed categories

Enhanced proof search efficiency through tagging and focusing techniques

Abstract

Monoidal closed categories naturally model NMILL, non-commutative multiplicative intuitionistic linear logic: the monoidal unit and tensor interpret the multiplicative verum and conjunction; the internal hom interprets linear implication. In recent years, the weaker notion of (left) skew monoidal closed category has been proposed by Ross Street, where the three structural laws of left and right unitality and associativity are not required to be invertible, they are merely natural transformations with a specific orientation. A question arises: is it possible to find a logic which is naturally modelled by skew monoidal closed categories? We answer positively by introducing a cut-free sequent calculus for a skew version of NMILL that is a presentation of the free skew monoidal closed category. We study the proof-theoretic semantics of the sequent calculus by identifying a calculus of derivations in normal form, obtained from an adaptation of Andreoli's focusing technique to the skew setting. The resulting focused sequent calculus peculiarly employs a system of tags for keeping track of new formulae appearing in the antecedent and appropriately reducing non-deterministic choices in proof search. Focusing solves the coherence problem for skew monoidal closed categories by exhibiting an effective procedure for deciding equality of maps in the free such category.