{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T15:40:19Z","timestamp":1753890019408,"version":"3.41.2"},"reference-count":1,"publisher":"Centre pour la Communication Scientifique Directe (CCSD)","license":[{"start":{"date-parts":[[2013,12,30]],"date-time":"2013-12-30T00:00:00Z","timestamp":1388361600000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/arxiv.org\/licenses\/nonexclusive-distrib\/1.0"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"abstract":"<jats:p>Tarski initiated a logic-based approach to formal geometry that studies\nfirst-order structures with a ternary betweenness relation \\beta, and a\nquaternary equidistance relation \\equiv. Tarski established, inter alia, that\nthe first-order (FO) theory of (R^2,\\beta,\\equiv) is decidable. Aiello and van\nBenthem (2002) conjectured that the FO-theory of expansions of (R^2,\\beta) with\nunary predicates is decidable. We refute this conjecture by showing that for\nall n&gt;1, the FO-theory of the class of expansions of (R^2,\\beta) with just one\nunary predicate is not even arithmetical. We also define a natural and\ncomprehensive class C of geometric structures (T,\\beta), and show that for each\nstructure (T,\\beta) in C, the FO-theory of the class of expansions of (T,\\beta)\nwith a single unary predicate is undecidable. We then consider classes of\nexpansions of structures (T,\\beta) with a restricted unary predicate, for\nexample a finite predicate, and establish a variety of related undecidability\nresults. In addition to decidability questions, we briefly study the\nexpressivities of universal MSO and weak universal MSO over expansions of\n(R^n,\\beta). While the logics are incomparable in general, over expansions of\n(R^n,\\beta), formulae of weak universal MSO translate into equivalent formulae\nof universal MSO.<\/jats:p>","DOI":"10.2168\/lmcs-9(4:26)2013","type":"journal-article","created":{"date-parts":[[2014,7,15]],"date-time":"2014-07-15T09:37:51Z","timestamp":1405417071000},"source":"Crossref","is-referenced-by-count":1,"title":["Undecidable First-Order Theories of Affine Geometries"],"prefix":"10.46298","volume":"Volume 9, Issue 4","author":[{"given":"Antti","family":"Kuusisto","sequence":"first","affiliation":[]},{"given":"Jeremy","family":"Meyers","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1582-3718","authenticated-orcid":false,"given":"Jonni","family":"Virtema","sequence":"additional","affiliation":[]}],"member":"25203","published-online":{"date-parts":[[2013,12,30]]},"reference":[{"key":"841:not-found"}],"container-title":["Logical Methods in Computer Science"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/lmcs.episciences.org\/728\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/lmcs.episciences.org\/728\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,4,11]],"date-time":"2023-04-11T19:54:44Z","timestamp":1681242884000},"score":1,"resource":{"primary":{"URL":"https:\/\/lmcs.episciences.org\/728"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2013,12,30]]},"references-count":1,"URL":"https:\/\/doi.org\/10.2168\/lmcs-9(4:26)2013","relation":{"is-same-as":[{"id-type":"arxiv","id":"1310.8200","asserted-by":"subject"},{"id-type":"doi","id":"10.48550\/arXiv.1310.8200","asserted-by":"subject"}],"is-part-of":[{"id-type":"doi","id":"10.4230\/lipics.csl.2012","asserted-by":"subject"}]},"ISSN":["1860-5974"],"issn-type":[{"type":"electronic","value":"1860-5974"}],"subject":[],"published":{"date-parts":[[2013,12,30]]},"article-number":"728"}}