{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,28]],"date-time":"2026-02-28T01:27:24Z","timestamp":1772242044580,"version":"3.50.1"},"reference-count":0,"publisher":"Centre pour la Communication Scientifique Directe (CCSD)","license":[{"start":{"date-parts":[[2019,9,20]],"date-time":"2019-09-20T00:00:00Z","timestamp":1568937600000},"content-version":"am","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2019,9,20]],"date-time":"2019-09-20T00:00:00Z","timestamp":1568937600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2019,9,20]],"date-time":"2019-09-20T00:00:00Z","timestamp":1568937600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"accepted":{"date-parts":[[2025,3,31]]},"abstract":"<jats:p>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 &amp;quot;degenerate&amp;quot; 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.<\/jats:p>","DOI":"10.23638\/lmcs-15(3:30)2019","type":"journal-article","created":{"date-parts":[[2025,4,3]],"date-time":"2025-04-03T13:32:52Z","timestamp":1743687172000},"source":"Crossref","is-referenced-by-count":0,"title":["On noncommutative extensions of linear logic"],"prefix":"10.23638","volume":"Volume 15, Issue 3","author":[{"given":"Sergey","family":"Slavnov","sequence":"first","affiliation":[]}],"member":"25203","published-online":{"date-parts":[[2019,9,20]]},"container-title":["Logical Methods in Computer Science"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/arxiv.org\/pdf\/1703.10092v6","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/arxiv.org\/pdf\/1703.10092v6","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,4,3]],"date-time":"2025-04-03T13:32:52Z","timestamp":1743687172000},"score":1,"resource":{"primary":{"URL":"http:\/\/lmcs.episciences.org\/3765"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,9,20]]},"references-count":0,"URL":"https:\/\/doi.org\/10.23638\/lmcs-15(3:30)2019","relation":{"has-preprint":[{"id-type":"arxiv","id":"1703.10092v5","asserted-by":"subject"},{"id-type":"arxiv","id":"1703.10092v4","asserted-by":"subject"},{"id-type":"arxiv","id":"1703.10092v3","asserted-by":"subject"}],"is-same-as":[{"id-type":"arxiv","id":"1703.10092","asserted-by":"subject"},{"id-type":"doi","id":"10.48550\/arXiv.1703.10092","asserted-by":"subject"}],"is-cited-by":[{"id-type":"doi","id":"10.4204\/EPTCS.353.8","asserted-by":"object"}]},"ISSN":["1860-5974"],"issn-type":[{"value":"1860-5974","type":"electronic"}],"subject":[],"published":{"date-parts":[[2019,9,20]]},"article-number":"3765"}}