In this paper we study the Kolmogorov complexity for non-effective computations, that is, either halting or non-halting computations on Turing machines. This complexity function is defined as the length of the shortes...
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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 al...
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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.
It is shown that several natural, undecidable properties of grammars are such that the size of the smallest Turing machine which correctly answers questions of length n grows at a nearly maximal rate as n grows. Thus,...
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