{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,2,7]],"date-time":"2023-02-07T14:56:27Z","timestamp":1675781787366},"reference-count":27,"publisher":"Cambridge University Press (CUP)","issue":"3","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":2018,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[2008,9]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Much previous study has been done on the degree spectra of prime models of a complete atomic decidable theory. Here we study the analogous questions for homogeneous models. We say a countable model <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200004321_inline1\" \/> has a <jats:italic><jats:bold>d<\/jats:bold>-basis<\/jats:italic> if the types realized in <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200004321_inline1\" \/> are all computable and the Turing degree <jats:italic><jats:bold>d<\/jats:bold><\/jats:italic> can list <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200004321_inline3\" \/>-indices for all types realized in <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200004321_inline1\" \/>. We say <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200004321_inline1\" \/> has a <jats:bold>d<\/jats:bold>-decidable copy if there exists a model <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200004321_inline2\" \/> \u2245 <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200004321_inline1\" \/> such that the elementary diagram of <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200004321_inline2\" \/> is <jats:bold>d<\/jats:bold>-computable. Goncharov, Millar, and Peretyat'kin independently showed there exists a homogeneous <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200004321_inline1\" \/> with a <jats:bold>0<\/jats:bold>-basis but no decidable copy.<\/jats:p><jats:p>We prove that any homogeneous <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200004321_inline1\" \/> with a <jats:bold>0<\/jats:bold>\u2032-basis has a low decidable copy. This implies Csima's analogous result for prime models. In the case where all types of the theory <jats:italic>T<\/jats:italic> are computable and <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200004321_inline1\" \/> is a homogeneous model with a <jats:bold>0<\/jats:bold>-basis, we show <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200004321_inline1\" \/> has copies decidable in every nonzero degree. A degree <jats:bold>d<\/jats:bold> is <jats:bold>0<\/jats:bold>-<jats:italic>homogeneous bounding<\/jats:italic> if any automorphically nontrivial homogeneous <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200004321_inline1\" \/> with a <jats:bold>0<\/jats:bold>-basis has a <jats:bold>d<\/jats:bold>-decidable copy. We show that the nonlow<jats:sub>2<\/jats:sub><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200004321_inline4\" \/> degrees are <jats:bold>0<\/jats:bold>-homogeneous bounding.<\/jats:p>","DOI":"10.2178\/jsl\/1230396762","type":"journal-article","created":{"date-parts":[[2008,12,27]],"date-time":"2008-12-27T16:52:45Z","timestamp":1230396765000},"page":"1009-1028","source":"Crossref","is-referenced-by-count":2,"title":["The degree spectra of homogeneous models"],"prefix":"10.1017","volume":"73","author":[{"given":"Karen","family":"Lange","sequence":"first","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200004321_ref026","volume-title":"Theory of algorithms and its applications","volume":"129","author":"Tusupov"},{"key":"S0022481200004321_ref023","doi-asserted-by":"publisher","DOI":"10.1007\/BF01674780"},{"key":"S0022481200004321_ref022","doi-asserted-by":"publisher","DOI":"10.1007\/BF02757002"},{"key":"S0022481200004321_ref021","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1982-0648078-8"},{"key":"S0022481200004321_ref019","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(78)90030-X"},{"key":"S0022481200004321_ref017","doi-asserted-by":"publisher","DOI":"10.1305\/ndjfl\/1172787551"},{"key":"S0022481200004321_ref011","unstructured":"Harris K. , Bounding saturated models, in preparation."},{"key":"S0022481200004321_ref009","first-page":"3","volume-title":"Handbook of recursive mathematics","volume":"138\u2013139","author":"Harizanov","year":"1998"},{"key":"S0022481200004321_ref007","doi-asserted-by":"publisher","DOI":"10.1007\/BF01674776"},{"key":"S0022481200004321_ref005","first-page":"1117","volume":"69","author":"Csima","year":"2004","journal-title":"Bounding prime models"},{"key":"S0022481200004321_ref004","first-page":"305","volume":"72","author":"Csima","year":"2007","journal-title":"Bounding homogeneous models"},{"key":"S0022481200004321_ref002","volume-title":"Model theory","volume":"73","author":"Chang","year":"1990"},{"key":"S0022481200004321_ref025","volume-title":"Computability theory and applications","author":"Soare"},{"key":"S0022481200004321_ref020","doi-asserted-by":"publisher","DOI":"10.2140\/pjm.1980.91.407"},{"key":"S0022481200004321_ref027","first-page":"301","volume-title":"Infinitistic Methods (Proceedings of the Symposium on Foundations of Mathematics, Warsaw, 1959)","author":"Vaught","year":"1961"},{"key":"S0022481200004321_ref018","volume-title":"Model theory: an introduction","volume":"277","author":"Marker","year":"2002"},{"key":"S0022481200004321_ref016","unstructured":"Lange K. 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A. , The atomic model theorem, submitted."},{"key":"S0022481200004321_ref014","first-page":"1034","volume":"51","author":"Knight","year":"1986","journal-title":"Degrees coded in jumps of orderings"},{"key":"S0022481200004321_ref001","volume-title":"Computable structures and the hyperarithmetical hierarchy","volume":"144","author":"Ash","year":"2000"},{"key":"S0022481200004321_ref006","unstructured":"Epstein R. , Computably enumerable degrees of Vaught's models, submitted."}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200004321","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,4,30]],"date-time":"2019-04-30T22:33:12Z","timestamp":1556663592000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200004321\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2008,9]]},"references-count":27,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2008,9]]}},"alternative-id":["S0022481200004321"],"URL":"https:\/\/doi.org\/10.2178\/jsl\/1230396762","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[2008,9]]}}}