The strong weak truth table (sw) reducibility was suggested by Downey, Hirschfeldt, and LaForte as a measure of relative randomness, alternative to the Solovay reducibility. It also occurs naturallyâ€¦ (More)

We prove that every diagonally noncomputable function computes a set A which is bi-immune, meaning that neither A nor its complement has an infinite computably enumerable subset.

For any P âŠ† 2Ï‰, define S(P), the degree spectrum of P, to be the set of all Turing degrees a such that there exists A âˆˆ P of degree a. We prove a number of basic properties of the structure which isâ€¦ (More)

We say that A â‰¤LR B if every B-random number isA-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 isâ€¦ (More)

We show that there exists a real Î± such that, for all reals Î², if Î± is linear reducible to Î² (Î± â‰¤` Î², previously denoted Î± â‰¤sw Î²) then Î² â‰¤T Î±. In fact, every random real satisfies thisâ€¦ (More)

Lipschitz continuity is used as a tool for analyzing the relationship between incomputability and randomness. Having presented a simpler proof of one of the major results in this areaâ€”the theorem ofâ€¦ (More)

The Turing degree of a real measures the computational difficulty of producing its binary expansion. Since Turing degrees are tailsets, it follows from Kolmogorovâ€™s 0-1 law that for any propertyâ€¦ (More)