A Chaitin Omega number based on compressible strings
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 article, we introduce a new variant Theta of Chaitin Omega number. The real Theta is defined based on the set of all compressible strings. We investigate the distribution of compressible strings and show that Theta is random. In addition, we generalize Theta to Theta(Q,R) with reals Q, R > 0 and study its properties. In particular, we show that the computability of the real Theta(T, 1) gives a sufficient condition for a real T is an element of (0, 1) to be a fixed point for partial randomness, i.e., to satisfy that the compression rate of T equals to T.
chaitin omega number compressible string program-size complexity fixed point partial randomness algorithmic information theory algorithmic randomness