{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T00:47:12Z","timestamp":1759970832323,"version":"build-2065373602"},"reference-count":0,"publisher":"Centre pour la Communication Scientifique Directe (CCSD)","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"accepted":{"date-parts":[[2025,6,27]]},"abstract":"<jats:p>None of the first-order modal logics between $\\mathsf{K}$ and $\\mathsf{S5}$ under the constant domain semantics enjoys Craig interpolation or projective Beth definability, even in the language restricted to a single individual variable. It follows that the existence of a Craig interpolant for a given implication or of an explicit definition for a given predicate cannot be directly reduced to validity as in classical first-order and many other logics. Our concern here is the decidability and computational complexity of the interpolant and definition existence problems. We first consider two decidable fragments of first-order modal logic $\\mathsf{S5}$: the one-variable fragment $\\mathsf{Q^1S5}$ and its extension $\\mathsf{S5}_{\\mathcal{ALC}^u}$ that combines $\\mathsf{S5}$ and the description logic$\\mathcal{ALC}$ with the universal role. We prove that interpolant and definition existence in $\\mathsf{Q^1S5}$ and $\\mathsf{S5}_{\\mathcal{ALC}^u}$ is decidable in coN2ExpTime, being 2ExpTime-hard, while uniform interpolant existence is undecidable. These results transfer to the two-variable fragment $\\mathsf{FO^2}$ of classical first-order logic without equality. We also show that interpolant and definition existence in the one-variable fragment $\\mathsf{Q^1K}$ of first-order modal logic $\\mathsf{K}$ is non-elementary decidable, while uniform interpolant existence is again undecidable.<\/jats:p>","DOI":"10.46298\/lmcs-21(4:6)2025","type":"journal-article","created":{"date-parts":[[2025,10,8]],"date-time":"2025-10-08T11:10:08Z","timestamp":1759921808000},"source":"Crossref","is-referenced-by-count":0,"title":["Deciding the Existence of Interpolants and Definitions in First-Order Modal Logic"],"prefix":"10.46298","volume":"Volume 21, Issue 4","author":[{"given":"Agi","family":"Kurucz","sequence":"first","affiliation":[]},{"given":"Frank","family":"Wolter","sequence":"additional","affiliation":[]},{"given":"Michael","family":"Zakharyaschev","sequence":"additional","affiliation":[]}],"member":"25203","published-online":{"date-parts":[[2025,10,8]]},"container-title":["Logical Methods in Computer Science"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/arxiv.org\/pdf\/2303.04598v4","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/arxiv.org\/pdf\/2303.04598v4","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,8]],"date-time":"2025-10-08T11:10:08Z","timestamp":1759921808000},"score":1,"resource":{"primary":{"URL":"https:\/\/lmcs.episciences.org\/13733"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,10,8]]},"references-count":0,"URL":"https:\/\/doi.org\/10.46298\/lmcs-21(4:6)2025","relation":{"has-preprint":[{"id-type":"arxiv","id":"2303.04598v2","asserted-by":"subject"}],"is-same-as":[{"id-type":"arxiv","id":"2303.04598","asserted-by":"subject"},{"id-type":"doi","id":"10.48550\/arXiv.2303.04598","asserted-by":"subject"}]},"ISSN":["1860-5974"],"issn-type":[{"value":"1860-5974","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,10,8]]},"article-number":"13733"}}