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85
A Tight Upper Bound on Kolmogorov Complexity by Hausdorff Dimension and Uniformly Optimal Prediction
 Theory of Computing Systems
, 1995
"... The present paper links the concepts of Kolmogorov complexity (in Complexity theory) and Hausdorff dimension (in Fractal geometry) for a class of recursive (computable) !languages. ..."
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Cited by 43 (5 self)
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The present paper links the concepts of Kolmogorov complexity (in Complexity theory) and Hausdorff dimension (in Fractal geometry) for a class of recursive (computable) !languages.
Recursively Enumerable Reals and Chaitin Ω Numbers
"... A real is called recursively enumerable if it is the limit of a recursive, increasing, converging sequence of rationals. Following Solovay [23] and Chaitin [10] we say that an r.e. real dominates an r.e. real if from a good approximation of from below one can compute a good approximation of from b ..."
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Cited by 34 (3 self)
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A real is called recursively enumerable if it is the limit of a recursive, increasing, converging sequence of rationals. Following Solovay [23] and Chaitin [10] we say that an r.e. real dominates an r.e. real if from a good approximation of from below one can compute a good approximation of from below. We shall study this relation and characterize it in terms of relations between r.e. sets. Solovay's [23]like numbers are the maximal r.e. real numbers with respect to this order. They are random r.e. real numbers. The halting probability ofa universal selfdelimiting Turing machine (Chaitin's Ω number, [9]) is also a random r.e. real. Solovay showed that any Chaitin Ω number islike. In this paper we show that the converse implication is true as well: any Ωlike real in the unit interval is the halting probability of a universal selfdelimiting Turing machine.
Randomness Space
 Automata, Languages and Programming, Proceedings of the 25th International Colloquium, ICALP’98
, 1998
"... MartinL#of de#ned in#nite random sequences over a #nite alphabet via randomness tests which describe sets having measure zero in a constructive sense. In this paper this concept is generalized to separable topological spaces with a measure, following a suggestion of Zvonkin and Levin. After stud ..."
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Cited by 21 (4 self)
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MartinL#of de#ned in#nite random sequences over a #nite alphabet via randomness tests which describe sets having measure zero in a constructive sense. In this paper this concept is generalized to separable topological spaces with a measure, following a suggestion of Zvonkin and Levin. After studying basic results and constructions for such randomness spaces a general invariance result is proved which gives conditions under which a function between randomness spaces preserves randomness. This corrects and extends a result bySchnorr. Calude and J#urgensen proved that the randomness notion for real numbers obtained by considering their bary representations is independent from the base b. We use our invariance result to show that this notion is identical with the notion which one obtains by viewing the real number space directly as a randomness space. Furthermore, arithmetic properties of random real numbers are derived, for example that every computable analytic function pres...
Computing a glimpse of randomness
 Experimental Mathematics
"... A Chaitin Omega number is the halting probability of a universal Chaitin (selfdelimiting Turing) machine. Every Omega number is both computably enumerable (the limit of a computable, increasing, converging sequence of rationals) and random (its binary expansion is an algorithmic random sequence). In ..."
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Cited by 20 (10 self)
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A Chaitin Omega number is the halting probability of a universal Chaitin (selfdelimiting Turing) machine. Every Omega number is both computably enumerable (the limit of a computable, increasing, converging sequence of rationals) and random (its binary expansion is an algorithmic random sequence). In particular, every Omega number is strongly noncomputable. The aim of this paper is to describe a procedure, which combines Java programming and mathematical proofs, for computing the exact values of the first 63 bits of a Chaitin Omega: 000000100000010000100000100001110111001100100111100010010011100. Full description of programs and proofs will be given elsewhere. 1
Chaitin Ω Numbers, Solovay Machines, and Incompleteness
 COMPUT. SCI
, 1999
"... Computably enumerable (c.e.) reals can be coded by Chaitin machines through their halting probabilities. Tuning Solovay's construction of a Chaitin universal machine for which ZFC (if arithmetically sound) cannot determine any single bit of the binary expansion of its halting probability, we show ..."
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Cited by 17 (15 self)
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Computably enumerable (c.e.) reals can be coded by Chaitin machines through their halting probabilities. Tuning Solovay's construction of a Chaitin universal machine for which ZFC (if arithmetically sound) cannot determine any single bit of the binary expansion of its halting probability, we show that every c.e. random real is the halting probability of a universal Chaitin machine for which ZFC cannot determine more than its initial block of 1 bitsas soon as you get a 0 it's all over. Finally, a constructive version of Chaitin informationtheoretic incompleteness theorem is proven.
A Highly Random Number
, 2001
"... In his celebrated 1936 paper Turing defined a machine to be circular iff it performs an infinite computation outputting only finitely many symbols. We define ( as the probability that an arbitrary machine be circular and we prove that is a random number that goes beyond $2, the probability that ..."
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Cited by 15 (5 self)
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In his celebrated 1936 paper Turing defined a machine to be circular iff it performs an infinite computation outputting only finitely many symbols. We define ( as the probability that an arbitrary machine be circular and we prove that is a random number that goes beyond $2, the probability that a universal self alelimiting machine halts. The algorithmic complexity of c is strictly greater than that of $2, but similar to the algorithmic complexity of 2 , the halting probability of an oracle machine. What makes ( interesting is that it is an example of a highly random number definable without considering oracles.
Algorithmic randomness, quantum physics, and incompleteness
 PROCEEDINGS OF THE CONFERENCE “MACHINES, COMPUTATIONS AND UNIVERSALITY” (MCU’2004), LECTURES NOTES IN COMPUT. SCI. 3354
, 2004
"... ..."
Hypercomputability of quantum adiabatic processes: facts versus prejudices
 http://arxiv.org/quantph/0504101
, 2005
"... Abstract. We give an overview of a quantum adiabatic algorithm for Hilbert’s tenth problem, including some discussions on its fundamental aspects and the emphasis on the probabilistic correctness of its findings. For the purpose of illustration, the numerical simulation results of some simple Diopha ..."
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Cited by 12 (3 self)
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Abstract. We give an overview of a quantum adiabatic algorithm for Hilbert’s tenth problem, including some discussions on its fundamental aspects and the emphasis on the probabilistic correctness of its findings. For the purpose of illustration, the numerical simulation results of some simple Diophantine equations are presented. We also discuss some prejudicial misunderstandings as well as some plausible difficulties faced by the algorithm in its physical implementations. “To believe otherwise is merely to cling to a prejudice which only gives rise to further prejudices... ” 1
Measures of Temporal Pattern Complexity
"... In this study, three measures of temporal pattern complexity were compared as regards their perceptual validity. The first measure, based on the work of Tanguiane (1993), uses the idea that a temporal pattern can be described in terms of (elaborations of) more simple patterns, simultaneously at diff ..."
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Cited by 11 (0 self)
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In this study, three measures of temporal pattern complexity were compared as regards their perceptual validity. The first measure, based on the work of Tanguiane (1993), uses the idea that a temporal pattern can be described in terms of (elaborations of) more simple patterns, simultaneously at different levels. The second measure is based on the complexity measure for finite sequences proposed by Lempel and Ziv (1976), which is related to the number of steps in a selfdelimiting production process by which such a sequence is presumed to be generated. The third measure, newly developed here, is rooted in the theoretical framework of rhythm perception of Povel and Essens (1985). It takes into account the ease of coding a temporal pattern and the complexity of the segments resulting from this coding. The perceptual validity of the three measures was evaluated in an experiment in which subjects judged the complexity of 35 temporal patterns. Correlations between the three measures and the ...
Computable Approximations of Reals: An InformationTheoretic Analysis
 Fundamenta Informaticae
, 1997
"... How fast can one approximate a real by a computable sequence of rationals? We show that the answer to this question depends very much on the information content in the finite prefixes of the binary expansion of the real. Computable reals, whose binary expansions haveavery low information content, ca ..."
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Cited by 10 (3 self)
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How fast can one approximate a real by a computable sequence of rationals? We show that the answer to this question depends very much on the information content in the finite prefixes of the binary expansion of the real. Computable reals, whose binary expansions haveavery low information content, can be approximated (very fast) with a computable convergence rate. Random reals, whose binary expansions contain very much information in their prefixes, can be approximated only very slowly by computable sequences of rationals (this is the case, for example, for Chaitin's \Omega numbers) if they can be computably approximated at all. We show that one can computably approximate any computable real also very slowly, with a convergence rate slower than any computable function. However, there is still a large gap between computable reals and random reals: any computable sequence of rationals which converges (monotonically) to a random real converges slower than any computable sequence of rat...