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Every 2random real is Kolmogorov random
 J. Symbolic Logic
, 2004
"... Abstract. We study reals with infinitely many incompressible prefixes. Call A ∈ 2 ω Kolmogorov random if ( ∃ ∞ n) C(A ↾ n)> n − O(1), where C denotes plain Kolmogorov complexity. This property was suggested by Loveland and studied by MartinLöf, Schnorr and Solovay. We prove that 2random reals are ..."
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Cited by 13 (3 self)
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Abstract. We study reals with infinitely many incompressible prefixes. Call A ∈ 2 ω Kolmogorov random if ( ∃ ∞ n) C(A ↾ n)> n − O(1), where C denotes plain Kolmogorov complexity. This property was suggested by Loveland and studied by MartinLöf, Schnorr and Solovay. We prove that 2random reals are Kolmogorov random. 1 Together with the converse—proved by Nies, Stephan and Terwijn [11]—this provides a natural characterization of 2randomness in terms of plain complexity. We finish with a related characterization of 2randomness. §1. Introduction. This paper is part of an ongoing program to understand randomness for real numbers, which we take to be elements of 2 ω, by investigating the complexity of their initial segments. Solomonoff [13] and Kolmogorov [4] independently defined a measure of the information content of finite strings. Intuitively, a complex string should be difficult to compress. The Kolmogorov
Quantum randomness and value indefiniteness
 Advanced Science Letters
"... As computability implies value definiteness, certain sequences of quantum outcomes cannot be computable. 1. CONCEPTUALISATION It certainly would be fascinating to pinpoint the time of the emergence of the notion that certain quantum processes, such as the decay of an excited quantum state, occurs pr ..."
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Cited by 13 (6 self)
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As computability implies value definiteness, certain sequences of quantum outcomes cannot be computable. 1. CONCEPTUALISATION It certainly would be fascinating to pinpoint the time of the emergence of the notion that certain quantum processes, such as the decay of an excited quantum state, occurs principally and irreducibly at random; and howlong it took to become the dominant way of thinking about them after almost two centuries of quasirationalistic dominance. Bohr’s and Heisenberg’s influence has been highly recognised and has prevailed, even against the strong rationalistic and philosophic objections raised by, for instance, by Einstein and Schrödinger. 1 � 2 Of course, one of the strongest reasons for this growing acceptance of quantum randomness has been the factual inability to go “beyond ” the quantum in any manner which would encourage new phenomenology and might result in any hope for a progressive quasiclassical research program. 3
Algorithmic randomness, quantum physics, and incompleteness
 Proceedings of the Conference “Machines, Computations and Universality” (MCU’2004), number 3354 in Lecture Notes in Computer Science
, 2006
"... When a distinguished but elderly scientist states that something is possible, he is almost certainly right. When he states that something is impossible, he is almost certainly wrong. Arthur C. Clarke ..."
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Cited by 12 (2 self)
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When a distinguished but elderly scientist states that something is possible, he is almost certainly right. When he states that something is impossible, he is almost certainly wrong. Arthur C. Clarke
Measures and their random reals
 IN PREPARATION
"... We study the randomness properties of reals with respect to arbitrary probability measures on Cantor space. We show that every nonrecursive real is nontrivially random with respect to some measure. The probability measures constructed in the proof may have atoms. If one rules out the existence of ..."
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Cited by 11 (2 self)
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We study the randomness properties of reals with respect to arbitrary probability measures on Cantor space. We show that every nonrecursive real is nontrivially random with respect to some measure. The probability measures constructed in the proof may have atoms. If one rules out the existence of atoms, i.e. considers only continuous measures, it turns out that every nonhyperarithmetical real is random for a continuous measure. On the other hand, examples of reals not random for a continuous measure can be found throughout the hyperarithmetical Turing degrees.
A statistical mechanical interpretation of algorithmic information theory III: Composite systems and fixed points
, 2009
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Is Complexity a Source of Incompleteness?
 IS COMPLEXITY A SOURCE OF INCOMPLETENESS
, 2004
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EXTRACTING INFORMATION IS HARD: A TURING DEGREE OF NONINTEGRAL EFFECTIVE HAUSDORFF DIMENSION
"... Abstract. We construct a ∆0 2 infinite binary sequence with effective Hausdorff dimension 1/2 that does not compute a sequence of higher dimension. Introduced by Lutz, effective Hausdorff dimension can be viewed as a measure of the information density of a sequence. In particular, the dimension of A ..."
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Cited by 8 (0 self)
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Abstract. We construct a ∆0 2 infinite binary sequence with effective Hausdorff dimension 1/2 that does not compute a sequence of higher dimension. Introduced by Lutz, effective Hausdorff dimension can be viewed as a measure of the information density of a sequence. In particular, the dimension of A ∈ 2ω is the lim inf of the ratio between the information content and length of initial segments of A. Thus the main result demonstrates that it is not always possible to extract information from a partially random source to produce a sequence that has higher information density. 1.
KOLMOGOROV COMPLEXITY AND SOLOVAY FUNCTIONS
, 2009
"... Solovay [19] proved that there exists a computable upper bound f of the prefixfree Kolmogorov complexity function K such that f(x) = K(x) for infinitely many x. In this paper, we consider the class of computable functions f such that K(x) ≤ f(x)+O(1) for all x and f(x) ≤ K(x) + O(1) for infinit ..."
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Cited by 8 (3 self)
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Solovay [19] proved that there exists a computable upper bound f of the prefixfree Kolmogorov complexity function K such that f(x) = K(x) for infinitely many x. In this paper, we consider the class of computable functions f such that K(x) ≤ f(x)+O(1) for all x and f(x) ≤ K(x) + O(1) for infinitely many x, which we call Solovay functions. We show that Solovay functions present interesting connections with randomness notions such as MartinLöf randomness and Ktriviality.
Lowness for weakly 1generic and Kurtzrandom
 in Theory and Applications of Models of Computation: Third Internationa l Conference, TAMC 2006
, 2006
"... Abstract. We prove that a set is low for weakly 1generic iff it has neither dnr nor hyperimmune Turing degree. As this notion is more general than being recursively traceable, we refute a recent conjecture on the characterization of these sets. Furthermore, we show that every set which is low for w ..."
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Cited by 7 (2 self)
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Abstract. We prove that a set is low for weakly 1generic iff it has neither dnr nor hyperimmune Turing degree. As this notion is more general than being recursively traceable, we refute a recent conjecture on the characterization of these sets. Furthermore, we show that every set which is low for weakly 1generic is also low for Kurtzrandom. 1
Effectively closed sets of measures and randomness
 Ann. Pure Appl. Logic
"... We show that if a real x ∈ 2ω is strongly Hausdorff Hhrandom, where h is a dimension function corresponding to a convex order, then it is also random for a continuous probability measure µ such that the µmeasure of the basic open cylinders shrinks according to h. The proof uses a new method to con ..."
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Cited by 7 (1 self)
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We show that if a real x ∈ 2ω is strongly Hausdorff Hhrandom, where h is a dimension function corresponding to a convex order, then it is also random for a continuous probability measure µ such that the µmeasure of the basic open cylinders shrinks according to h. The proof uses a new method to construct measures, based on effective (partial) continuous transformations and a basis theorem for Π0 1classes applied to closed sets of probability measures. We use the main result to give a new proof of Frostman’s Lemma, to derive a collapse of randomness notions for Hausdorff measures, and to provide a characterization of effective Hausdorff dimension similar to Frostman’s Theorem. 1