{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,8,12]],"date-time":"2022-08-12T01:26:45Z","timestamp":1660267605090},"reference-count":21,"publisher":"Cambridge University Press (CUP)","issue":"1","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":5855,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1998,3]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>A set <jats:italic>X<\/jats:italic> of nonnegative integers is <jats:italic>computably enumerable<\/jats:italic> (<jats:italic>c.e<\/jats:italic>.), also called <jats:italic>recursively enumerable<\/jats:italic> (<jats:italic>r.e<\/jats:italic>.), if there is a computable method to list its elements. Let \u03b5 denote the structure of the computably enumerable sets under inclusion, \u03b5 = ({<jats:italic>W<jats:sub>e<\/jats:sub><\/jats:italic>}<jats:sub><jats:italic>e<\/jats:italic>\u03f5\u03c9<\/jats:sub>,\u2286). We previously exhibited a first order \u03b5-definable property <jats:italic>Q(X)<\/jats:italic> such that <jats:italic>Q(X)<\/jats:italic> guarantees that <jats:italic>X<\/jats:italic> is not Turing complete (i.e., does not code complete information about c.e. sets).<\/jats:p><jats:p>Here we show first that <jats:italic>Q(X)<\/jats:italic> implies that <jats:italic>X<\/jats:italic> has a certain \u201cslowness\u201d property whereby the elements must enter <jats:italic>X<\/jats:italic> slowly (under a certain precise complexity measure of speed of computation) even though <jats:italic>X<\/jats:italic> may have high information content. Second we prove that every <jats:italic>X<\/jats:italic> with this slowness property is computable in some member of any nontrivial orbit, namely for any noncomputable <jats:italic>A<\/jats:italic> \u03f5 \u03b5 there exists <jats:italic>B<\/jats:italic> in the orbit of <jats:italic>A<\/jats:italic> such that <jats:italic>X<\/jats:italic> \u2264<jats:sub><jats:italic>T<\/jats:italic><\/jats:sub><jats:italic>B<\/jats:italic> under relative Turing computability (\u2264<jats:sub><jats:italic>T<\/jats:italic><\/jats:sub>). We produce <jats:italic>B<\/jats:italic> using the <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200015280_inline1\" \/>-automorphism method we introduced earlier.<\/jats:p>","DOI":"10.2307\/2586583","type":"journal-article","created":{"date-parts":[[2006,4,18]],"date-time":"2006-04-18T18:43:03Z","timestamp":1145385783000},"page":"1-28","source":"Crossref","is-referenced-by-count":7,"title":["Codable sets and orbits of computably enumerable sets"],"prefix":"10.1017","volume":"63","author":[{"given":"Leo","family":"Harrington","sequence":"first","affiliation":[]},{"given":"Robert I.","family":"Soare","sequence":"additional","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200015280_ref002","doi-asserted-by":"publisher","DOI":"10.1090\/memo\/0541"},{"key":"S0022481200015280_ref019","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-02460-7"},{"key":"S0022481200015280_ref015","doi-asserted-by":"publisher","DOI":"10.1002\/malq.19660120125"},{"key":"S0022481200015280_ref016","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9904-1944-08111-1"},{"key":"S0022481200015280_ref018","doi-asserted-by":"publisher","DOI":"10.2307\/1970842"},{"key":"S0022481200015280_ref001","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1984-0719661-8"},{"key":"S0022481200015280_ref021","doi-asserted-by":"publisher","DOI":"10.1112\/plms\/s2-45.1.161"},{"key":"S0022481200015280_ref006","volume-title":"Computability, enumerability, unsolvability: Directions in recursion theory, proceedings of the recursion theory conference, University of Leeds, July, 1994","author":"Harrington","year":"1996"},{"key":"S0022481200015280_ref010","doi-asserted-by":"publisher","DOI":"10.2307\/1970579"},{"key":"S0022481200015280_ref014","doi-asserted-by":"publisher","DOI":"10.1007\/BF02760850"},{"key":"S0022481200015280_ref017","first-page":"695","volume":"41","author":"Shoenfield","year":"1976","journal-title":"Degrees of classes of r.e. sets"},{"key":"S0022481200015280_ref008","first-page":"431","volume":"33","author":"Lachlan","year":"1968","journal-title":"Degrees of recursively enumerable sets which have no maximal superset"},{"key":"S0022481200015280_ref011","doi-asserted-by":"publisher","DOI":"10.2140\/pjm.1980.87.135"},{"key":"S0022481200015280_ref004","unstructured":"Harrington L. and Soare R. I. , Martin's invariance conjecture and low sets, in preparation."},{"key":"S0022481200015280_ref009","doi-asserted-by":"publisher","DOI":"10.1215\/S0012-7094-68-03513-8"},{"key":"S0022481200015280_ref007","doi-asserted-by":"publisher","DOI":"10.1090\/S0894-0347-96-00181-6"},{"key":"S0022481200015280_ref005","doi-asserted-by":"publisher","DOI":"10.1073\/pnas.88.22.10242"},{"key":"S0022481200015280_ref020","doi-asserted-by":"publisher","DOI":"10.2307\/420992"},{"key":"S0022481200015280_ref013","first-page":"138","volume":"50","author":"Maass","year":"1985","journal-title":"Variations on promptly simple sets"},{"key":"S0022481200015280_ref012","first-page":"809","volume":"47","author":"Maass","year":"1982","journal-title":"Recursively enumerable generic sets"},{"key":"S0022481200015280_ref003","volume-title":"Proceedings of the 1996 Oberwolfach conference on computability theory","author":"Harrington"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200015280","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,11]],"date-time":"2019-05-11T19:04:00Z","timestamp":1557601440000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200015280\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1998,3]]},"references-count":21,"journal-issue":{"issue":"1","published-print":{"date-parts":[[1998,3]]}},"alternative-id":["S0022481200015280"],"URL":"https:\/\/doi.org\/10.2307\/2586583","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1998,3]]}}}