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The http://ars. els-cdn. com/content/image/http://origin-ars. els-cdn. com/content/image/1-s2. 0-S0168007205001429-si1. gif"/> degrees of computably enumerable sets are not dense [Book Review]

George Barmpalias & Andrew Em Lewis

Annals of Pure and Applied Logic 141 (1):51-60 (2006)

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The computable Lipschitz degrees of computably enumerable sets are not dense.Adam R. Day - 2010 - Annals of Pure and Applied Logic 161 (12):1588-1602.details The computable Lipschitz reducibility was introduced by Downey, Hirschfeldt and LaForte under the name of strong weak truth-table reducibility [6]). This reducibility measures both the relative randomness and the relative computational power of real numbers. This paper proves that the computable Lipschitz degrees of computably enumerable sets are not dense. An immediate corollary is that the Solovay degrees of strongly c.e. reals are not dense. There are similarities to Barmpalias and Lewis’ proof that the identity bounded Turing degrees of c.e. (...) Download Export citation Bookmark 4 citations
Π 1 0 classes, L R degrees and Turing degrees.George Barmpalias, Andrew E. M. Lewis & Frank Stephan - 2008 - Annals of Pure and Applied Logic 156 (1):21-38.details We say that A≤LRB if every B-random set is A-random with respect to Martin–Löf randomness. We study this relation and its interactions with Turing reducibility, classes, hyperimmunity and other recursion theoretic notions. Download Export citation Bookmark 10 citations
Kolmogorov complexity and computably enumerable sets.George Barmpalias & Angsheng Li - 2013 - Annals of Pure and Applied Logic 164 (12):1187-1200.details Download Export citation Bookmark
The partial orderings of the computably enumerable ibT-degrees and cl-degrees are not elementarily equivalent.Klaus Ambos-Spies, Philipp Bodewig, Yun Fan & Thorsten Kräling - 2013 - Annals of Pure and Applied Logic 164 (5):577-588.details We show that, in the partial ordering of the computably enumerable computable Lipschitz degrees, there is a degree a>0a>0 such that the class of the degrees which do not cup to a is not bounded by any degree less than a. Since Ambos-Spies [1] has shown that, in the partial ordering of the c.e. identity-bounded Turing degrees, for any degree a>0a>0 the degrees which do not cup to a are bounded by the 1-shift a+1a+1 of a where a+1 (...) Download Export citation Bookmark
Maximal pairs of c.e. reals in the computably Lipschitz degrees.Yun Fan & Liang Yu - 2011 - Annals of Pure and Applied Logic 162 (5):357-366.details Computably Lipschitz reducibility , was suggested as a measure of relative randomness. We say α≤clβ if α is Turing reducible to β with oracle use on x bounded by x+c. In this paper, we prove that for any non-computable real, there exists a c.e. real so that no c.e. real can cl-compute both of them. So every non-computable c.e. real is the half of a cl-maximal pair of c.e. reals. Download Export citation Bookmark 1 citation

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