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Constant compression and random weights ∗
"... Omega numbers, as considered in algorithmic randomness, are by definition real numbers that are equal to the halting probability of a universal prefixfree Turing machine. Omega numbers are obviously leftr.e., i.e., are effectively approximable from below. Furthermore, among all leftr.e. real numb ..."
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Omega numbers, as considered in algorithmic randomness, are by definition real numbers that are equal to the halting probability of a universal prefixfree Turing machine. Omega numbers are obviously leftr.e., i.e., are effectively approximable from below. Furthermore, among all leftr.e. real numbers in the appropriate range between 0 and 1, the Omega numbers admit wellknown characterizations as the ones that are MartinLöf random, as well as the ones such that any of their effective approximation from below is slower than any other effective approximation from below to any other real, up to a constant factor. In what follows, we obtain a further characterization of Omega numbers in terms of Theta numbers. Tadaki considered for a given prefixfree Turing machine and some natural number a the set of all strings that are compressed by this machine by at least a bits relative to their length, and he introduced Theta numbers as the weight of sets of this form. He showed that in the case of a universal prefixfree Turing machine any Theta number is an Omega number and he asked whether this implication can be reversed. We answer his question in the affirmative and thus obtain a new characterization of Omega numbers. In addition to the onesided case of the set of all strings compressible by at least a certain number a of bits, we consider sets that comprise all strings that are compressible by at least a but no more than b bits, and we call the weight of such a set a twosided Theta number. We demonstrate that in the case of a universal prefixfree Turing machine, for given a and all sufficiently large b the corresponding twosided Theta number is again an Omega number. Conversely, any Omega number can be realized as twosided Theta number for any pair of natural numbers a and b> a.
Superbranching degrees
 Proceedings Oberwolfach 1989, Springer Verlag Lecture Notes in Mathematics
, 1990
"... Solovay ..."
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Partial Randomness and Dimension of Recursively Enumerable Reals
, 906
"... Abstract. A real α is called recursively enumerable (“r.e. ” for short) if there exists a computable, increasing sequence of rationals which converges to α. It is known that the randomness of an r.e. real α can be characterized in various ways using each of the notions; programsize complexity, Mart ..."
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Abstract. A real α is called recursively enumerable (“r.e. ” for short) if there exists a computable, increasing sequence of rationals which converges to α. It is known that the randomness of an r.e. real α can be characterized in various ways using each of the notions; programsize complexity, MartinLöf test, Chaitin Ω number, the domination and Ωlikeness of α, the universality of a computable, increasing sequence of rationals which converges to α, and universal probability. In this paper, we generalize these characterizations of randomness over the notion of partial randomness by parameterizing each of the notions above by a real T ∈ (0, 1], where the notion of partial randomness is a stronger representation of the compression rate by means of programsize complexity. As a result, we present ten equivalent characterizations of the partial randomness of an r.e. real. The resultant characterizations of partial randomness are powerful and have many important applications. One of them is to present equivalent characterizations of the dimension of an individual r.e. real. The equivalence between the notion of Hausdorff dimension and compression rate by programsize complexity (or partial randomness) has been established at present by a series of works of many researchers over the last two decades. We present ten equivalent characterizations of the dimension of an individual r.e. real. Key words: algorithmic randomness, recursively enumerable real, partial
unknown title
, 2004
"... An extension of Chaitin’s halting probability Ω to measurement operator in infinite dimensional quantum system ..."
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An extension of Chaitin’s halting probability Ω to measurement operator in infinite dimensional quantum system
UNIVERSALITY PROBABILITY OF A PREFIXFREE MACHINE
, 2012
"... We study the notion of universality probability of a universal prefixfree machine, as introduced by C.S. Wallace (see [Dow08, Section 0.2.2] and [Dow11, Section 2.5]). We show that it is random relative to the third iterate of the halting problem and determine its Turing degree and its place in th ..."
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We study the notion of universality probability of a universal prefixfree machine, as introduced by C.S. Wallace (see [Dow08, Section 0.2.2] and [Dow11, Section 2.5]). We show that it is random relative to the third iterate of the halting problem and determine its Turing degree and its place in the arithmetical hierarchy of complexity. Furthermore, we give a computational characterization of the real numbers which are universality probabilities of universal prefixfree machines.
A New Representation of Chaitin Ω Number Based on Compressible Strings
, 2010
"... 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 prefixfree machine U, which is a universal decoding algorithm used to define the notion of programsize complexity. Chaitin showed Ω to be rando ..."
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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 prefixfree machine U, which is a universal decoding algorithm used to define the notion of programsize complexity. Chaitin showed Ω to be random by discovering the property that the first n bits of the basetwo 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 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.