Randomness and universal machines

Abstract

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. Furthermore, the notion ΩU [X] = ∑p : U (p) ↓ ∈ X 2- | 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. © 2006 Elsevier Inc. All rights reserved.