Randomness and Universal Machines

The 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. It is shown that there are universal machines for which ΩU is just ∑ x 2 1−H(x). For such a universal machine there exists a co-r.e. set X such that ΩU [X] = ∑ p:U(p)↓∈X 2 −|p| 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 hyperimmunefree Turing degree of a given Π1-class are not low for Ω unless this class contains a recursive set.