A Chaitin $$\Upomega$$ number based on compressible strings

• Authors: Kohtaro Tadaki
• Venue: Natural Computing: an international journal
• Year: 2012

In 1975 Chaitin introduced his $$\Upomega$$ number as a concrete example of random real. The real $$\Upomega$$ 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 $$\Upomega$$ to be random by discovering the property that the first n bits of the base-two expansion of $$\Upomega$$ solve the halting problem of U for all binary inputs of length at most n. In this article, we introduce a new variant $$\Uptheta$$ of Chaitin $$\Upomega$$ number. The real $$\Uptheta$$ is defined based on the set of all compressible strings. We investigate the distribution of compressible strings and show that $$\Uptheta$$ is random. In addition, we generalize $$\Uptheta$$ to $$\Uptheta(Q, R)$$ with reals Q, R 0 and study its properties. In particular, we show that the computability of the real $$\Uptheta(T,\,1)$$ gives a sufficient condition for a real T 驴 (0, 1) to be a fixed point for partial randomness, i.e., to satisfy that the compression rate of T equals to T.