A new representation of Chaitin Ω number based on compressible strings

Abstract

In 1975 Chaitin introduced his Ω number as a concrete example of random real. The real Ω 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 Ω to be random by discovering the property that the first n bits of the base-two expansion of Ω solve the halting problem of U for all binary inputs of length at most n. In this paper, we introduce a new representation Θ of Chaitin Ω number. The real Θ is defined based on the set of all compressible strings. We investigate the properties of Θ and show that Θ is random. In addition, we generalize Θ to two directions Θ(T) and Θ̄(T) with a real T > 0. We then study their properties. In particular, we show that the computability of the real Θ(T) gives a sufficient condition for a real T ε (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. © Springer-Verlag Berlin Heidelberg 2010.