A new representation of Chaitin \Omega number based on compressible strings

Abstract

In 1975 Chaitin introduced his \Omega number as a concrete example of random real. The real \Omega is defined based on the set of all halting inputs for an optimal prefix-free machine U, which is a universal decoding algorithm used to define the notion of program-size complexity. Chaitin showed \Omega to be random by discovering the property that the first n bits of the base-two expansion of \Omega solve the halting problem of U for all binary inputs of length at most n. In this paper, we introduce a new representation \Theta of Chaitin \Omega number. The real \Theta is defined based on the set of all compressible strings. We investigate the properties of \Theta and show that \Theta is random. In addition, we generalize \Theta to two directions \Theta(T) and \bar{\Theta}(T) with a real T>0. We then study their properties. In particular, we show that the computability of the real \Theta(T) gives a sufficient condition for a real T in (0,1) to be a fixed point on partial randomness, i.e., to satisfy the condition that the compression rate of T equals to T.