Randomness, Lowness and Degrees

Journal of Symbolic Logic 73 (2):559 - 577 (2008)

Abstract
We say that A ≤LR B if every B-random number is A-random. Intuitively this means that if oracle A can identify some patterns on some real 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 enumberable degrees) and their relationship with the Turing degrees. Among other results we show that whenever α in not GL₂ the LR degree of α bounds $2^{\aleph _{0}}$ 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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DOI 10.2178/jsl/1208359060
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References found in this work BETA

Almost Everywhere Domination and Superhighness.Stephen G. Simpson - 2007 - Mathematical Logic Quarterly 53 (4):462-482.

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Citations of this work BETA

Almost Everywhere Domination and Superhighness.Stephen G. Simpson - 2007 - Mathematical Logic Quarterly 53 (4):462-482.
Mass Problems and Hyperarithmeticity.Joshua A. Cole & Stephen G. Simpson - 2007 - Journal of Mathematical Logic 7 (2):125-143.
Low Upper Bounds in the LR Degrees.David Diamondstone - 2012 - Annals of Pure and Applied Logic 163 (3):314-320.
Randomness and Lowness Notions Via Open Covers.Laurent Bienvenu & Joseph S. Miller - 2012 - Annals of Pure and Applied Logic 163 (5):506-518.

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