{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,6,8]],"date-time":"2023-06-08T11:15:08Z","timestamp":1686222908784},"reference-count":8,"publisher":"Cambridge University Press (CUP)","issue":"4","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":101,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[2013,12]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>We examine a definition of the mutual information of two reals proposed by Levin in [5]. The mutual information is<\/jats:p><jats:p><jats:disp-formula><jats:graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" orientation=\"portrait\" mime-subtype=\"gif\" mimetype=\"image\" position=\"float\" xlink:type=\"simple\" xlink:href=\"S0022481200127318_Ueqn1\" \/><\/jats:disp-formula><\/jats:p><jats:p>where<jats:italic>K<\/jats:italic>(\u00b7) is the prefix-free Kolmogorov complexity. A real<jats:italic>A<\/jats:italic>is said to have finite self-information if<jats:italic>I (A : A)<\/jats:italic>is finite. We give a construction for a perfect \u03a0<jats:sub arrange=\"stack\">1<\/jats:sub><jats:sup arrange=\"stack\">0<\/jats:sup>class of reals with this property, which settles some open questions posed by Hirschfeldt and Weber. The construction produces a perfect set of reals with<jats:italic>K(\u03c3)<\/jats:italic>\u2264<jats:sup>+<\/jats:sup><jats:italic>K<jats:sup>A<\/jats:sup>(\u03c3)<\/jats:italic>+<jats:italic>f (\u03c3)<\/jats:italic>for any given \u0394<jats:sub arrange=\"stack\">2<\/jats:sub><jats:sup arrange=\"stack\">0<\/jats:sup><jats:italic>f<\/jats:italic>with a particularly nice approximation and for a specific choice of f it can also be used to produce a perfect \u03a0<jats:sub arrange=\"stack\">1<\/jats:sub><jats:sup arrange=\"stack\">0<\/jats:sup>set of reals that are low for effective Hausdorff dimension and effective packing dimension. The construction can be further adapted to produce a single perfect set of reals that satisfy<jats:italic>K(\u03c3)<\/jats:italic>\u2264<jats:sup>+<\/jats:sup><jats:italic>K<jats:sup>A<\/jats:sup>(\u03c3)<\/jats:italic>+<jats:italic>f (\u03c3)<\/jats:italic>for all<jats:italic>f<\/jats:italic>in a \u2018nice\u2019 class of \u0394<jats:sub arrange=\"stack\">2<\/jats:sub><jats:sup arrange=\"stack\">0<\/jats:sup>functions which includes all \u0394<jats:sub arrange=\"stack\">2<\/jats:sub><jats:sup arrange=\"stack\">0<\/jats:sup>orders.<\/jats:p>","DOI":"10.2178\/jsl.7804130","type":"journal-article","created":{"date-parts":[[2014,1,5]],"date-time":"2014-01-05T20:33:25Z","timestamp":1388954005000},"page":"1229-1246","source":"Crossref","is-referenced-by-count":2,"title":["A Perfect Set of Reals with Finite Self-Information"],"prefix":"10.1017","volume":"78","author":[{"given":"Ian","family":"Herbert","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200127318_ref008","doi-asserted-by":"publisher","DOI":"10.1093\/acprof:oso\/9780199230761.001.0001"},{"key":"S0022481200127318_ref007","doi-asserted-by":"publisher","DOI":"10.1016\/j.aim.2004.10.006"},{"key":"S0022481200127318_ref006","doi-asserted-by":"publisher","DOI":"10.1016\/S0020-0190(02)00343-5"},{"key":"S0022481200127318_ref005","first-page":"30","article-title":"Laws on the conservation (zero increase) of information, and questions on the foundations of probability theory","volume":"10","author":"Levin","year":"1974","journal-title":"Problemy Pereda\u010di Informacii"},{"key":"S0022481200127318_ref002","doi-asserted-by":"publisher","DOI":"10.1007\/978-0-387-68441-3"},{"key":"S0022481200127318_ref001","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-540-24749-4_55"},{"key":"S0022481200127318_ref003","doi-asserted-by":"crossref","first-page":"85","DOI":"10.3233\/COM-2012-003","article-title":"Finite self-information","volume":"1","author":"Hirschfeldt","year":"2012","journal-title":"Computability"},{"key":"S0022481200127318_ref004","doi-asserted-by":"publisher","DOI":"10.1142\/S0219061314500111"}],"container-title":["The Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200127318","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,8,13]],"date-time":"2020-08-13T11:21:17Z","timestamp":1597317677000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200127318\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2013,12]]},"references-count":8,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2013,12]]}},"alternative-id":["S0022481200127318"],"URL":"https:\/\/doi.org\/10.2178\/jsl.7804130","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[2013,12]]}}}