Randomness, lowness and degrees
George Barmpalias, Andrew E. M. Lewis, and Mariya Soskova
Source: J. Symbolic Logic Volume 73, Issue 2
(2008), 559-577.
Abstract
We say that A≤LRB if every B-random number is A-random. Intuitively this means that if oracle A can identify some patterns on some real γ, oracle B can also find patterns on γ. In other words, B is at least as good as A for this purpose. We study the structure of the LR degrees globally and locally (i.e., restricted to the computably enumerable degrees) and their relationship with the Turing degrees. Among other results we show that whenever α is not GL2 the LR degree of α bounds 2ℵ0 degrees (so that, in particular, there exist LR degrees with uncountably many predecessors) and we give sample results which demonstrate how various techniques from the theory of the c.e. degrees can be used to prove results about the c.e. LR degrees.
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Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.jsl/1208359060
Digital Object Identifier: doi:10.2178/jsl/1208359060
Mathematical Reviews number (MathSciNet): MR2414465
Zentralblatt MATH identifier: 1145.03020
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