Journal of Symbolic Logic

Randomness, lowness and degrees

George Barmpalias, Andrew E. M. Lewis, and Mariya Soskova

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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 20 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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J. Symbolic Logic, Volume 73, Issue 2 (2008), 559-577.

First available in Project Euclid: 16 April 2008

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Barmpalias, George; Lewis, Andrew E. M.; Soskova, Mariya. Randomness, lowness and degrees. J. Symbolic Logic 73 (2008), no. 2, 559--577. doi:10.2178/jsl/1208359060.

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