Randomness and universal machines

AbstractThe present work investigates several questions from a recent survey of Miller and Nies related to Chaitin's Ω numbers and their dependence on the underlying universal machine. Furthermore, the notion ΩU[X]=∑p:U(p)↓∈X2-|p| is studied for various sets X and universal machines U. A universal machine U is constructed such that for all x, ΩU[{x}]=21-H(x). For such a universal machine there exists a co-r.e. set X such that ΩU[X] is neither left-r.e. nor Martin-Löf random. Furthermore, one of the open problems of Miller and Nies is answered completely by showing that there is a sequence Un of universal machines such that the truth-table degrees of the ΩUn form an antichain. Finally, it is shown that the members of hyperimmune-free Turing degree of a given Π10-class are not low for Ω unless this class contains a recursive set