{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,4,3]],"date-time":"2022-04-03T21:27:55Z","timestamp":1649021275682},"reference-count":20,"publisher":"Elsevier BV","issue":"2","license":[{"start":{"date-parts":[[2002,7,1]],"date-time":"2002-07-01T00:00:00Z","timestamp":1025481600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.elsevier.com\/tdm\/userlicense\/1.0\/"},{"start":{"date-parts":[[2013,7,17]],"date-time":"2013-07-17T00:00:00Z","timestamp":1374019200000},"content-version":"vor","delay-in-days":4034,"URL":"https:\/\/www.elsevier.com\/open-access\/userlicense\/1.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Theoretical Computer Science"],"published-print":{"date-parts":[[2002,7]]},"DOI":"10.1016\/s0304-3975(01)00107-4","type":"journal-article","created":{"date-parts":[[2002,10,9]],"date-time":"2002-10-09T19:39:34Z","timestamp":1034192374000},"page":"499-518","source":"Crossref","is-referenced-by-count":1,"title":["The closure properties on real numbers under limits and computable operators"],"prefix":"10.1016","volume":"284","author":[{"given":"Xizhong","family":"Zheng","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"78","reference":[{"key":"10.1016\/S0304-3975(01)00107-4_BIB1","doi-asserted-by":"crossref","unstructured":"K. Ambos-Spies, K. Weihranch, X. Zheng, Weakly computable real numbers, J. Complexity 6 (2000) 676\u2013690.","DOI":"10.1006\/jcom.2000.0561"},{"key":"10.1016\/S0304-3975(01)00107-4_BIB2","unstructured":"C. Calude, A characterization of c.e. random reals. CDMTCS Research Report Series 095, March 1999."},{"key":"10.1016\/S0304-3975(01)00107-4_BIB3","doi-asserted-by":"crossref","unstructured":"C. Calude, P. Hertling, B. Khoussainov, Y. Wang, Recursive enumerable reals and Chaintin's \u03a9-number, STACS\u201998, 1998, pp. 596\u2013606.","DOI":"10.1007\/BFb0028594"},{"key":"10.1016\/S0304-3975(01)00107-4_BIB4","unstructured":"G.S. Ce\u0131̆tin, A pseudofundamental sequence that is not equivalent to a monotone one, Zap. Nau\u010dn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 20 (1971) 263\u2013271, 290 (in Russian)."},{"key":"10.1016\/S0304-3975(01)00107-4_BIB5","doi-asserted-by":"crossref","first-page":"329","DOI":"10.1145\/321892.321894","article-title":"A theory of program size formally identical to information theory","volume":"22","author":"Chaitin","year":"1975","journal-title":"J. ACM"},{"key":"10.1016\/S0304-3975(01)00107-4_BIB6","doi-asserted-by":"crossref","first-page":"61","DOI":"10.4064\/fm-44-1-61-71","article-title":"On the definitions of recursive real continuous functions","volume":"44","author":"Grzegorczyk","year":"1957","journal-title":"Fund. Math."},{"key":"10.1016\/S0304-3975(01)00107-4_BIB7","doi-asserted-by":"crossref","first-page":"101","DOI":"10.1016\/0304-3975(84)90129-4","article-title":"Reducibilities of real numbers","volume":"31","author":"Ko","year":"1984","journal-title":"Theoret. Comp. Sci."},{"key":"10.1016\/S0304-3975(01)00107-4_BIB8","series-title":"Complexity Theory of Real Functions","author":"Ko","year":"1991"},{"key":"10.1016\/S0304-3975(01)00107-4_BIB9","series-title":"Computability in Analysis and Physics","author":"Pour-El","year":"1989"},{"key":"10.1016\/S0304-3975(01)00107-4_BIB10","unstructured":"T.A. Slaman, Randomness and recursive enumerability, preprint, 1999."},{"key":"10.1016\/S0304-3975(01)00107-4_BIB11","first-page":"271","article-title":"Recursion theory and Dedekind cuts","volume":"140","author":"Soare","year":"1969","journal-title":"Trans. Amer. Math. Soc."},{"issue":"1","key":"10.1016\/S0304-3975(01)00107-4_BIB12","doi-asserted-by":"crossref","first-page":"215","DOI":"10.2140\/pjm.1969.31.215","article-title":"Cohesive sets and recursively enumerable Dedekind cuts","volume":"31","author":"Soare","year":"1969","journal-title":"Pacific J. Math."},{"key":"10.1016\/S0304-3975(01)00107-4_BIB13","series-title":"Recursively Enumerable Sets and Degrees","author":"Soare","year":"1987"},{"key":"10.1016\/S0304-3975(01)00107-4_BIB14","unstructured":"R. Solovay, Draft of a paper (or series of papers) on Chaitin's work \u2026 done for the most part during the period of Sept.\u2013Dec. 1975, unpublished manuscript, IBM Thomas J. Watson Research Center, Yorktoen Heights, New York, May 1975, 215pp."},{"key":"10.1016\/S0304-3975(01)00107-4_BIB15","doi-asserted-by":"crossref","first-page":"145","DOI":"10.2307\/2267043","article-title":"Nicht konstruktive beweisbare S\u00e4tze der Analysis","volume":"14","author":"Specker","year":"1949","journal-title":"J. Symbolic Logic"},{"key":"10.1016\/S0304-3975(01)00107-4_BIB16","series-title":"Computability","volume":"Vol. 9","author":"Weihrauch","year":"1987"},{"key":"10.1016\/S0304-3975(01)00107-4_BIB17","series-title":"An Introduction to Computable Analysis","author":"Weihrauch","year":"2000"},{"key":"10.1016\/S0304-3975(01)00107-4_BIB18","doi-asserted-by":"crossref","unstructured":"K. Weihrauch, X. Zheng, A finite hierarchy of the recursively enumerable real numbers, MFCS\u201998, Brno, Czech Republic, August 1998, pp. 798\u2013806.","DOI":"10.1007\/BFb0055831"},{"key":"10.1016\/S0304-3975(01)00107-4_BIB19","doi-asserted-by":"crossref","unstructured":"K. Weihrauch, X. Zheng, Arithmetical hierarchy of real numbers, MFCS\u201999, Szklarska Poreba, Poland, September 1999, pp. 23\u201333.","DOI":"10.1007\/3-540-48340-3_3"},{"key":"10.1016\/S0304-3975(01)00107-4_BIB20","doi-asserted-by":"crossref","unstructured":"X. Zheng, Binary enumerability of real numbers, Computing and Combinatorics, Proc. COCOON\u201999, Tokyo, Japan, July 26\u201328, 1999, pp. 300\u2013309.","DOI":"10.1007\/3-540-48686-0_30"}],"container-title":["Theoretical Computer Science"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/api.elsevier.com\/content\/article\/PII:S0304397501001074?httpAccept=text\/xml","content-type":"text\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/api.elsevier.com\/content\/article\/PII:S0304397501001074?httpAccept=text\/plain","content-type":"text\/plain","content-version":"vor","intended-application":"text-mining"}],"deposited":{"date-parts":[[2020,3,10]],"date-time":"2020-03-10T15:55:42Z","timestamp":1583855742000},"score":1,"resource":{"primary":{"URL":"https:\/\/linkinghub.elsevier.com\/retrieve\/pii\/S0304397501001074"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2002,7]]},"references-count":20,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2002,7]]}},"alternative-id":["S0304397501001074"],"URL":"https:\/\/doi.org\/10.1016\/s0304-3975(01)00107-4","relation":{},"ISSN":["0304-3975"],"issn-type":[{"value":"0304-3975","type":"print"}],"subject":[],"published":{"date-parts":[[2002,7]]}}}