{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,16]],"date-time":"2026-02-16T18:00:41Z","timestamp":1771264841138,"version":"3.50.1"},"reference-count":13,"publisher":"Oxford University Press (OUP)","issue":"3","license":[{"start":{"date-parts":[[2024,6,11]],"date-time":"2024-06-11T00:00:00Z","timestamp":1718064000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/academic.oup.com\/pages\/standard-publication-reuse-rights"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2025,3,11]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>In this paper, we investigate the numerical expressive power of various logical languages, encompassing fragments of Presburger Arithmetic (PbA), monadic second-order logic with counting with respect to finite domains (MSO$^{\\phi }(\\#)$) and shallow second-order graded modal logic with counting with respect to image-finite frames (SOGML$^{\\textsf{s},\\phi }$(#)). We show that in their respective existential fragments, the $1$-free fragment of PbA, the =-free fragment of MSO$^{\\phi }(\\#)$ and the graded modality-free fragment of SOGML$^{\\textsf{s},\\phi }$(#) possess equivalent numerical expressive power, specifically defining strongly semilinear sets. When adding universal quantifiers or adding $1$, = and graded modality to these three languages, the resulting definable sets become semilinear sets.<\/jats:p>","DOI":"10.1093\/logcom\/exae028","type":"journal-article","created":{"date-parts":[[2024,6,12]],"date-time":"2024-06-12T01:24:30Z","timestamp":1718155470000},"source":"Crossref","is-referenced-by-count":2,"title":["Numerical expressive power of logical languages with cardinality comparison"],"prefix":"10.1093","volume":"35","author":[{"given":"Xiaoxuan","family":"Fu","sequence":"first","affiliation":[{"name":"School of Humanities , China University of Political Science and Law, Beijing, China, 100088"}]},{"given":"Zhiguang","family":"Zhao","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics , Taishan University, Tai'an, Shandong, 271000,China"}]}],"member":"286","published-online":{"date-parts":[[2024,6,11]]},"reference":[{"key":"2025042210120152100_ref1","first-page":"161","article-title":"Numerical abstraction via the Frege quantifier","volume":"51","author":"Aldo Antonelli","year":"2010","journal-title":"Notre Dame Journal of Formal Logic"},{"key":"2025042210120152100_ref2","doi-asserted-by":"crossref","DOI":"10.1017\/CBO9781107050884","volume-title":"Modal Logic","author":"Blackburn","year":"2001"},{"key":"2025042210120152100_ref3","doi-asserted-by":"crossref","first-page":"972","DOI":"10.1017\/jsl.2019.67","article-title":"The logic of comparative cardinality","volume":"85","author":"Ding","year":"2020","journal-title":"Journal of Symbolic Logic"},{"key":"2025042210120152100_ref4","doi-asserted-by":"crossref","first-page":"285","DOI":"10.2140\/pjm.1966.16.285","article-title":"Semigroups, Presburger formulas, and languages","volume":"16","author":"Ginsburg","year":"1966","journal-title":"Pacific Journal of Mathematics"},{"key":"2025042210120152100_ref5","doi-asserted-by":"crossref","first-page":"67","DOI":"10.1145\/3242953.3242964","article-title":"A survival guide to Presburger arithmetic","volume":"5","author":"Haase","year":"2018","journal-title":"ACM SIGLOG News"},{"key":"2025042210120152100_ref6","doi-asserted-by":"crossref","first-page":"1153","DOI":"10.2307\/2275466","article-title":"The H\u00e4rtig quantifier: a survey","volume":"56","author":"Heinrich Herre","year":"1991","journal-title":"Journal of Symbolic Logic"},{"key":"2025042210120152100_ref7","volume-title":"Model Theory for Graded Modal Languages","author":"Ma","year":"2011"},{"key":"2025042210120152100_ref8","doi-asserted-by":"crossref","DOI":"10.1017\/9781316716878","volume-title":"Bounded Variable Logics and Counting: A Study in Finite Models","author":"Otto","year":"2017"},{"key":"2025042210120152100_ref9","article-title":"Uber die vollstandigkeit eines gewissen systems der arithmetik ganser 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Hoek","year":"1996","journal-title":"International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems"}],"container-title":["Journal of Logic and Computation"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/academic.oup.com\/logcom\/article-pdf\/35\/3\/exae028\/58196918\/exae028.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"https:\/\/academic.oup.com\/logcom\/article-pdf\/35\/3\/exae028\/58196918\/exae028.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,4,22]],"date-time":"2025-04-22T14:53:15Z","timestamp":1745333595000},"score":1,"resource":{"primary":{"URL":"https:\/\/academic.oup.com\/logcom\/article\/doi\/10.1093\/logcom\/exae028\/7690669"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,6,11]]},"references-count":13,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2025,3,11]]}},"URL":"https:\/\/doi.org\/10.1093\/logcom\/exae028","relation":{},"ISSN":["0955-792X","1465-363X"],"issn-type":[{"value":"0955-792X","type":"print"},{"value":"1465-363X","type":"electronic"}],"subject":[],"published-other":{"date-parts":[[2025,4]]},"published":{"date-parts":[[2024,6,11]]},"article-number":"exae028"}}