Randomness, lowness and degrees

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 , oracle B can also find patterns on . 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 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.

• Type: Journal
• Year: 2008
• Where: JSYML
• Authors: George Barmpalias, Andrew E. M. Lewis, Mariya Ivanova Soskova
• Keywords: B-random Number, JSYML 2008, Lr Degrees, Oracle