On noncommutative extensions of linear logic
Abstract
Pomset logic introduced by Retoré is an extension of linear logic with a selfdual noncommutative connective. The logic is defined by means of proofnets, 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 ongoing work on semantics we discovered a system similar to Pomset logic, where a noncommutative connective is no longer selfdual. 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 AbrusciRuet noncommutative logic). We develop a syntax of proofnets 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 cutfree, 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.
 Publication:

arXiv eprints
 Pub Date:
 March 2017
 arXiv:
 arXiv:1703.10092
 Bibcode:
 2017arXiv170310092S
 Keywords:

 Computer Science  Logic in Computer Science;
 Mathematics  Logic
 EPrint:
 Logical Methods in Computer Science, Volume 15, Issue 3 (September 20, 2019) lmcs:5774