Randomness, lowness and degrees

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.

Almost everywhere domination and superhighness

“…[18,19] have constructed a B such that {A | A ≤ LR B} is uncountable. Recently Barmpalias/Lewis/Soskova [1] have shown that this holds for any B which is generalized superhigh. We now prove the following theorem due to Kjos-Hanssen [13].…”

“…In fact, Barmpalias/Lewis/Soskova [1] have shown that this holds for any B which is generalized superhigh.…”

“…In particular 0 ′ = {e ∈ ω | ϕ (1) e (0) ↓} = a Turing oracle for the Halting Problem. For A, B ⊆ ω we write A ⊕ B = {2n | n ∈ A} ∪ {2n + 1 | n ∈ B}, the Turing join of A and B.…”

Randomness notions and partial relativization

“…Hence, although the degree structures differ as partially ordered sets, the actual degrees as equivalence classes coincide. Barmpalias/Lewis/Soskova [BLS08a] proved that there are continuum many sets ≤ LR ∅ . We finish Section 4 with a similar result, proving that there are continuum many sets ≤ W 2R ∅ .…”

“…6 The final value of T (ρ) is only fixed at the end of stage e. 7 The method in this case is the same as in the proof in [BLS08a] that the class of sets ≤ LR ∅ is uncountable.…”