{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,10,29]],"date-time":"2023-10-29T04:37:42Z","timestamp":1698554262084},"reference-count":16,"publisher":"Wiley","issue":"2","license":[{"start":{"date-parts":[[2006,11,13]],"date-time":"2006-11-13T00:00:00Z","timestamp":1163376000000},"content-version":"vor","delay-in-days":3238,"URL":"http:\/\/onlinelibrary.wiley.com\/termsAndConditions#vor"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Mathematical Logic Qtrly"],"published-print":{"date-parts":[[1998,1]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>We develop arithmetical measure theory along the lines of Lutz [10]. This yields the same notion of measure 0 set as considered before by Martin\u2010L\u00f6f, Schnorr, and others. We prove that the class of sets constructible by r.e.\u2010constructors, a direct analogue of the classes Lutz devised his resource bounded measures for in [10], is not equal to RE, the class of r.e. sets, and we locate this class exactly in terms of the common recursion\u2010theoretic reducibilities below <jats:italic>K<\/jats:italic>. We note that the class of sets that bounded truth\u2010table reduce to <jats:italic>K<\/jats:italic> has r.e.\u2010measure 0, and show that this cannot be improved to truth\u2010table. For \u0394<jats:sub>2<\/jats:sub>\u2010measure the borderline between measure zero and measure nonzero lies between weak truth\u2010table reducibility and Turing reducibility to <jats:italic>K<\/jats:italic>. It follows that there exists a Martin\u2010L\u00f6f random set that is tt\u2010reducible to <jats:italic>K<\/jats:italic>, and that no such set is btt\u2010reducible to <jats:italic>K<\/jats:italic>. In fact, by a result of Kautz, a much more general result holds.<\/jats:p>","DOI":"10.1002\/malq.19980440211","type":"journal-article","created":{"date-parts":[[2007,5,30]],"date-time":"2007-05-30T23:53:31Z","timestamp":1180569211000},"page":"277-286","source":"Crossref","is-referenced-by-count":1,"title":["Arithmetical Measure"],"prefix":"10.1002","volume":"44","author":[{"given":"Sebastiaan A.","family":"Terwijn","sequence":"first","affiliation":[]},{"given":"Leen","family":"Torenvliet","sequence":"additional","affiliation":[]}],"member":"311","published-online":{"date-parts":[[2006,11,13]]},"reference":[{"key":"e_1_2_1_2_2","doi-asserted-by":"publisher","DOI":"10.1016\/S0304-3975(95)00260-X"},{"key":"e_1_2_1_3_2","first-page":"227","volume-title":"The Universal Turing Machine: A Half Century Survey","author":"Bennett C. H.","year":"1988"},{"key":"e_1_2_1_4_2","doi-asserted-by":"publisher","DOI":"10.1007\/BF02007257"},{"key":"e_1_2_1_5_2","doi-asserted-by":"publisher","DOI":"10.1016\/0304-3975(94)00014-X"},{"key":"e_1_2_1_6_2","unstructured":"Kautz S. M. Degrees of random sets. PhD Thesis Cornell University Ithaca NY 1991."},{"key":"e_1_2_1_7_2","series-title":"Lecture Notes in Mathematics 1141","doi-asserted-by":"crossref","first-page":"245","DOI":"10.1007\/BFb0076224","volume-title":"Recursion Theory Week","author":"Ku\u010dera A.","year":"1985"},{"key":"e_1_2_1_8_2","unstructured":"Kurtz S. A. Randomness and genericity in the degrees of unsolvability. PhD Thesis University of Illinois1981."},{"key":"e_1_2_1_9_2","doi-asserted-by":"publisher","DOI":"10.2307\/2274821"},{"key":"e_1_2_1_10_2","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4757-3860-5"},{"key":"e_1_2_1_11_2","doi-asserted-by":"publisher","DOI":"10.1016\/0022-0000(92)90020-J"},{"key":"e_1_2_1_12_2","first-page":"158","volume-title":"Proc. 8th Structure in Complexity Theory Conference 1993","author":"Lutz J. H."},{"key":"e_1_2_1_13_2","doi-asserted-by":"publisher","DOI":"10.1016\/S0019-9958(66)80018-9"},{"key":"e_1_2_1_14_2","unstructured":"Mayordomo E. Contributions to the study of resource bounded measure. PhD Thesis Universitat Polit\u00e8cnica de Catalunya Barcelona1994."},{"key":"e_1_2_1_15_2","volume-title":"Classical Recursion Theory","author":"Odifreddi P.","year":"1989"},{"key":"e_1_2_1_16_2","series-title":"Lecture Notes in Mathematics 218","doi-asserted-by":"crossref","DOI":"10.1007\/BFb0112458","volume-title":"Zuf\u00e4lligkeit und Wahrscheinlichkeit","author":"Schnorr C. 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