# On noncommutative extensions of linear logic

## TL;DR

This paper introduces semicommutative linear logic with noncommutative connectives, develops proof-net syntax, and provides a decorated sequent calculus that is sound, complete, and degenerates to Pomset logic, addressing the lack of sequent calculus formulations.

## Contribution

It defines a new semicommutative linear logic with noncommutative connectives and presents a decorated sequent calculus that is cut-free, sound, and complete, including for Pomset logic.

## Key findings

- Developed a syntax of proof-nets for semicommutative logic
- Proved the decorated sequent calculus is cut-free, sound, and complete
- Demonstrated degeneration to Pomset logic in specific cases

## Abstract

Pomset logic introduced by Retor\'e is an extension of linear logic with a self-dual noncommutative connective. The logic is defined by means of proof-nets, rather than a sequent calculus. Later a deep inference system BV was developed with an eye to capturing Pomset logic, but equivalence of system has not been proven up to now. As for a sequent calculus formulation, it has not been known for either of these logics, and there are convincing arguments that such a sequent calculus in the usual sense simply does not exist for them. In an on-going work on semantics we discovered a system similar to Pomset logic, where a noncommutative connective is no longer self-dual. Pomset logic appears as a degeneration, when the class of models is restricted. Motivated by these semantic considerations, we define in the current work a semicommutative multiplicative linear logic}, which is multiplicative linear logic extended with two nonisomorphic noncommutative connectives (not to be confused with very different Abrusci-Ruet noncommutative logic). We develop a syntax of proof-nets and show how this logic degenerates to Pomset logic. However, a more interesting problem than just finding yet another noncommutative logic is to find a sequent calculus for this logic. We introduce decorated sequents, which are sequents equipped with an extra structure of a binary relation of reachability on formulas. We define a decorated sequent calculus for semicommutative logic and prove that it is cut-free, sound and complete. This is adapted to "degenerate" variations, including Pomset logic. Thus, in particular, we give a variant of sequent calculus formulation for Pomset logic, which is one of the key results of the paper.