Algorithmic information theory
Algorithmic information theory
A Theory of Program Size Formally Identical to Information Theory
Journal of the ACM (JACM)
Recursively enumerable reals and Chaitin &OHgr; numbers
Theoretical Computer Science
Randomness and Recursive Enumerability
SIAM Journal on Computing
Partial Randomness and Dimension of Recursively Enumerable Reals
MFCS '09 Proceedings of the 34th International Symposium on Mathematical Foundations of Computer Science 2009
On oscillation-free chaitin h-random sequences
WTCS'12 Proceedings of the 2012 international conference on Theoretical Computer Science: computation, physics and beyond
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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 Θ(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 ∈ (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.