The interplay between computability and randomness has been an active area of research in recent years, reflected by ample funding in the USA, numerous workshops, and publications on the subject. The… Expand

Abstract The set A is low for (Martin-Lof) randomness if each random set is already random relative to A . A is K -trivial if the prefix complexity K of each initial segment of A is minimal, namely ∀… Expand

We show that a set is 2-random if and only if there is a constant c such that infinitely many initial segments x of the set are c-incompressible.Expand

AbstractLet R be a notion of algorithmic randomness for individual subsets of ℕ. A set B is a base for R randomness if there is a Z ≥ T B such that Z is R random relative to B. We show that the bases… Expand

We characterize some major algorithmic randomness notions via differentiability of effective functions.
(1) As the main result we show that a real number z in [0,1] is computably random if and only… Expand

We investigate definability in R, the recursively enumerable Turing degrees, using codings of standard models of arithmetic (SMA’s) as a tool. First we show that an SMA can be interpreted in R… Expand

We prove that any Chaitin Ω number (i.e., the halting probability of a universal self-delimiting Turing machine) is wtt-complete, but not tt-complete.Expand