{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,9]],"date-time":"2026-03-09T23:00:03Z","timestamp":1773097203131,"version":"3.50.1"},"reference-count":0,"publisher":"Centre pour la Communication Scientifique Directe (CCSD)","license":[{"start":{"date-parts":[[2019,8,1]],"date-time":"2019-08-01T00:00:00Z","timestamp":1564617600000},"content-version":"am","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2019,8,1]],"date-time":"2019-08-01T00:00:00Z","timestamp":1564617600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2019,8,1]],"date-time":"2019-08-01T00:00:00Z","timestamp":1564617600000},"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>A cyclic proof system, called CLKID-omega, gives us another way of representing inductive definitions and efficient proof search. The 2005 paper by Brotherston showed that the provability of CLKID-omega includes the provability of LKID, first order classical logic with inductive definitions in Martin-L\\&amp;quot;of's style, and conjectured the equivalence. The equivalence has been left an open question since 2011. This paper shows that CLKID-omega and LKID are indeed not equivalent. This paper considers a statement called 2-Hydra in these two systems with the first-order language formed by 0, the successor, the natural number predicate, and a binary predicate symbol used to express 2-Hydra. This paper shows that the 2-Hydra statement is provable in CLKID-omega, but the statement is not provable in LKID, by constructing some Henkin model where the statement is false.<\/jats:p>","DOI":"10.23638\/lmcs-15(3:10)2019","type":"journal-article","created":{"date-parts":[[2025,4,3]],"date-time":"2025-04-03T13:35:42Z","timestamp":1743687342000},"source":"Crossref","is-referenced-by-count":7,"title":["Classical System of Martin-Lof's Inductive Definitions is not Equivalent to Cyclic Proofs"],"prefix":"10.23638","volume":"Volume 15, Issue 3","author":[{"given":"Stefano","family":"Berardi","sequence":"first","affiliation":[]},{"given":"Makoto","family":"Tatsuta","sequence":"additional","affiliation":[]}],"member":"25203","published-online":{"date-parts":[[2019,8,1]]},"container-title":["Logical Methods in Computer Science"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/arxiv.org\/pdf\/1712.09603v5","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/arxiv.org\/pdf\/1712.09603v5","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,4,3]],"date-time":"2025-04-03T13:35:42Z","timestamp":1743687342000},"score":1,"resource":{"primary":{"URL":"http:\/\/lmcs.episciences.org\/4173"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,8,1]]},"references-count":0,"URL":"https:\/\/doi.org\/10.23638\/lmcs-15(3:10)2019","relation":{"has-preprint":[{"id-type":"arxiv","id":"1712.09603v3","asserted-by":"subject"},{"id-type":"arxiv","id":"1712.09603v2","asserted-by":"subject"},{"id-type":"arxiv","id":"1712.09603v1","asserted-by":"subject"}],"is-same-as":[{"id-type":"arxiv","id":"1712.09603","asserted-by":"subject"},{"id-type":"doi","id":"10.48550\/arXiv.1712.09603","asserted-by":"subject"}],"is-cited-by":[{"id-type":"doi","id":"10.4204\/EPTCS.342.11","asserted-by":"object"}]},"ISSN":["1860-5974"],"issn-type":[{"value":"1860-5974","type":"electronic"}],"subject":[],"published":{"date-parts":[[2019,8,1]]},"article-number":"4173"}}