On noncommutative extensions of linear logic

In the seminal work [12] Retoré introduced Pomset logic, an extension of linear logic with a self-dual noncommutative connective. Pomset logic is defined by means of proof-nets, later a deep inference system BV [8] was designed for this extension, 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. This will be shown in a forthcoming paper. Motivated by these semantic considerations, in the current work we define 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 [1, 2]). We develop a syntax of proof-nets and show how this logic degenerates to Pomset logic. However, a more important problem than just finding yet another noncommutative logic is finding 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 (sort of) sequent calculus formulation for Pomset logic, which is one of the key results of the paper.