{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,28]],"date-time":"2026-02-28T18:47:56Z","timestamp":1772304476249,"version":"3.50.1"},"reference-count":17,"publisher":"World Scientific Pub Co Pte Lt","issue":"06","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Int. J. Bifurcation Chaos"],"published-print":{"date-parts":[[2007,6]]},"abstract":"<jats:p> A Chaitin Omega number is the halting probability of a universal prefix-free Turing machine. Every Omega number is simultaneously computably enumerable (the limit of a computable, increasing, converging sequence of rationals), and algorithmically random (its binary expansion is an algorithmic random sequence), hence uncomputable. The value of an Omega number is highly machine-dependent. In general, no more than finitely many scattered bits of the binary expansion of an Omega number can be exactly computed; but, in some cases, it is possible to prove that no bit can be computed. <\/jats:p><jats:p> In this paper, we will simplify and improve both the method and correctness of the proof proposed in an earlier paper, and we will compute the exact approximations of two Omega numbers of the same prefix-free Turing machine, which is universal when used with data in base 16 or base 2: we compute 43 exact bits for the base 16 machine and 40 exact bits for the base 2 machine. <\/jats:p>","DOI":"10.1142\/s0218127407018130","type":"journal-article","created":{"date-parts":[[2007,9,4]],"date-time":"2007-09-04T11:33:08Z","timestamp":1188905588000},"page":"1937-1954","source":"Crossref","is-referenced-by-count":13,"title":["EXACT APPROXIMATIONS OF OMEGA NUMBERS"],"prefix":"10.1142","volume":"17","author":[{"given":"CRISTIAN S.","family":"CALUDE","sequence":"first","affiliation":[{"name":"Department of Computer Science, University of Auckland, Private Bag 92019, Auckland, New Zealand"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"MICHAEL J.","family":"DINNEEN","sequence":"additional","affiliation":[{"name":"Department of Computer Science, University of Auckland, Private Bag 92019, Auckland, New Zealand"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"219","published-online":{"date-parts":[[2011,11,20]]},"reference":[{"key":"rf1","doi-asserted-by":"publisher","DOI":"10.1007\/3-540-58131-6_37"},{"key":"rf2","doi-asserted-by":"publisher","DOI":"10.1016\/S0304-3975(99)00159-0"},{"key":"rf3","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-04978-5"},{"key":"rf4","doi-asserted-by":"publisher","DOI":"10.1016\/S0304-3975(01)00027-5"},{"key":"rf5","doi-asserted-by":"publisher","DOI":"10.1016\/S0304-3975(01)00068-8"},{"key":"rf6","first-page":"285","volume":"12","author":"Calude C. S.","journal-title":"J. Multiple-Valued Logic and Soft Computing"},{"key":"rf7","first-page":"369","volume":"11","author":"Calude C. S.","journal-title":"Experim. Math."},{"key":"rf8","volume":"284","author":"Calude C. S.","journal-title":"CDMTCS Res. Rep."},{"key":"rf9","doi-asserted-by":"publisher","DOI":"10.1145\/321892.321894"},{"key":"rf10","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511608858"},{"key":"rf11","volume-title":"The Limits of Mathematics","author":"Chaitin G. J.","year":"1997"},{"key":"rf12","doi-asserted-by":"publisher","DOI":"10.1142\/S0218127407018129"},{"key":"rf15","doi-asserted-by":"publisher","DOI":"10.1137\/S0097539799357441"},{"key":"rf16","volume-title":"Computation: Finite and Infinite Machines","author":"Minsky M.","year":"1967"},{"key":"rf17","volume-title":"Classical Recursion Theory","volume":"1","author":"Odifreddi P.","year":"1999"},{"key":"rf18","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-02460-7"},{"key":"rf19","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4471-0751-4_21"}],"container-title":["International Journal of Bifurcation and Chaos"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.worldscientific.com\/doi\/pdf\/10.1142\/S0218127407018130","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,8,7]],"date-time":"2019-08-07T14:57:52Z","timestamp":1565189872000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.worldscientific.com\/doi\/abs\/10.1142\/S0218127407018130"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2007,6]]},"references-count":17,"journal-issue":{"issue":"06","published-online":{"date-parts":[[2011,11,20]]},"published-print":{"date-parts":[[2007,6]]}},"alternative-id":["10.1142\/S0218127407018130"],"URL":"https:\/\/doi.org\/10.1142\/s0218127407018130","relation":{},"ISSN":["0218-1274","1793-6551"],"issn-type":[{"value":"0218-1274","type":"print"},{"value":"1793-6551","type":"electronic"}],"subject":[],"published":{"date-parts":[[2007,6]]}}}