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Randomness, lowness and degrees

Published online by Cambridge University Press:  12 March 2014

George Barmpalias ,
Andrew E. M. Lewis  and
Mariya Soskova
George Barmpalias
Affiliation:
University of Leeds, School of Mathematics, Leeds. LS2 9JT, UK, E-mail: georgeb@maths.leeds.ac.uk
Andrew E. M. Lewis
Affiliation:
Universita Degli Studi Di Siena, Dipartimento Di Scienze Matematiche Ed Informatiche, Via Del Capitano 15, 53100 Siena, Italy, E-mail: andy@aemlewis.co.uk
Mariya Soskova
Affiliation:
University of Leeds, School of Mathematics, Leeds, LS2 9JT, UK, E-mail: mariya@maths.leeds.ac.uk
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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 ∝ is not GL2 the LR degree of ∝ bounds 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.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 73 , Issue 2 , June 2008 , pp. 559 - 577
Copyright
Copyright © Association for Symbolic Logic 2008

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