{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T15:37:58Z","timestamp":1753889878789,"version":"3.41.2"},"reference-count":0,"publisher":"Centre pour la Communication Scientifique Directe (CCSD)","license":[{"start":{"date-parts":[[2019,7,31]],"date-time":"2019-07-31T00:00:00Z","timestamp":1564531200000},"content-version":"am","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2019,7,31]],"date-time":"2019-07-31T00:00:00Z","timestamp":1564531200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2019,7,31]],"date-time":"2019-07-31T00:00:00Z","timestamp":1564531200000},"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>Let $b$ be an integer strictly greater than $1$. Each set of nonnegative integers is represented in base $b$ by a language over $\\{0, 1, \\dots, b - 1\\}$. The set is said to be $b$-recognisable if it is represented by a regular language. It is known that ultimately periodic sets are $b$-recognisable, for every base $b$, and Cobham's theorem implies the converse: no other set is $b$-recognisable in every base $b$.   We consider the following decision problem: let $S$ be a set of nonnegative integers that is $b$-recognisable, given as a finite automaton over $\\{0,1, \\dots, b - 1\\}$, is $S$ periodic? Honkala showed in 1986 that this problem is decidable. Later on, Leroux used in 2005 the convention to write number representations with the least significant digit first (LSDF), and designed a quadratic algorithm to solve a more general problem.   We use here LSDF convention as well and give a structural description of the minimal automata that accept periodic sets. Then, we show that it can be verified in linear time if a minimal automaton meets this description. In general, this yields a $O(b \\log(n))$ procedure to decide whether an automaton with $n$ states accepts an ultimately periodic set of nonnegative integers.<\/jats:p>","DOI":"10.23638\/lmcs-15(3:8)2019","type":"journal-article","created":{"date-parts":[[2025,4,3]],"date-time":"2025-04-03T17:34:09Z","timestamp":1743701649000},"source":"Crossref","is-referenced-by-count":0,"title":["An efficient algorithm to decide periodicity of $b$-recognisable sets using LSDF convention"],"prefix":"10.23638","volume":"Volume 15, Issue 3","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-2325-6004","authenticated-orcid":false,"given":"Victor","family":"Marsault","sequence":"first","affiliation":[]}],"member":"25203","published-online":{"date-parts":[[2019,7,31]]},"container-title":["Logical Methods in Computer Science"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/arxiv.org\/pdf\/1708.06228v6","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/arxiv.org\/pdf\/1708.06228v6","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,4,3]],"date-time":"2025-04-03T17:34:10Z","timestamp":1743701650000},"score":1,"resource":{"primary":{"URL":"http:\/\/lmcs.episciences.org\/3882"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,7,31]]},"references-count":0,"URL":"https:\/\/doi.org\/10.23638\/lmcs-15(3:8)2019","relation":{"has-preprint":[{"id-type":"arxiv","id":"1708.06228v5","asserted-by":"subject"},{"id-type":"arxiv","id":"1708.06228v4","asserted-by":"subject"},{"id-type":"arxiv","id":"1708.06228v3","asserted-by":"subject"},{"id-type":"arxiv","id":"1708.06228v1","asserted-by":"subject"}],"is-same-as":[{"id-type":"arxiv","id":"1708.06228","asserted-by":"subject"},{"id-type":"doi","id":"10.48550\/arXiv.1708.06228","asserted-by":"subject"}]},"ISSN":["1860-5974"],"issn-type":[{"type":"electronic","value":"1860-5974"}],"subject":[],"published":{"date-parts":[[2019,7,31]]},"article-number":"3882"}}