Journal of Symbolic Logic

Randomness, lowness and degrees

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

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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 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.

Article information

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

Dates
First available: 16 April 2008

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

Citation

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


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