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38
NONCUPPING AND RANDOMNESS
, 2006
"... Let Y ∈ ∆0 2 be MartinLöfrandom. Then there is a promptly simple set A such that for each MartinLöfrandom set Z, Y ≤T A ⊕ Z ⇒ Y ≤T Z. When Y = Ω, one obtains a c.e. noncomputable set A which is not weakly MartinLöf cuppable. That is, for any MartinLöfrandom set Z, if ∅ ′ ≤T A ⊕ Z, then ∅ ′ ..."
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Cited by 158 (17 self)
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Let Y ∈ ∆0 2 be MartinLöfrandom. Then there is a promptly simple set A such that for each MartinLöfrandom set Z, Y ≤T A ⊕ Z ⇒ Y ≤T Z. When Y = Ω, one obtains a c.e. noncomputable set A which is not weakly MartinLöf cuppable. That is, for any MartinLöfrandom set Z, if ∅ ′ ≤T A ⊕ Z, then ∅ ′ ≤T Z.
Almost everywhere domination and superhighness
 Mathematical Logic Quarterly
"... Let ω denote the set of natural numbers. For functions f, g: ω → ω, we say that f is dominated by g if f(n) < g(n) for all but finitely many n ∈ ω. We consider the standard “fair coin ” probability measure on the space 2 ω of infinite sequences of 0’s and 1’s. A Turing oracle B is said to be almo ..."
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Cited by 28 (8 self)
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Let ω denote the set of natural numbers. For functions f, g: ω → ω, we say that f is dominated by g if f(n) < g(n) for all but finitely many n ∈ ω. We consider the standard “fair coin ” probability measure on the space 2 ω of infinite sequences of 0’s and 1’s. A Turing oracle B is said to be almost everywhere dominating if, for measure one many X ∈ 2 ω, each function which is Turing computable from X is dominated by some function which is Turing computable from B. Dobrinen and Simpson have shown that the almost everywhere domination property and some of its variant properties are closely related to the reverse mathematics of measure theory. In this paper we exposit some recent results of KjosHanssen, KjosHanssen/Miller/Solomon, and others concerning LRreducibility and almost everywhere domination. We also prove the following new result: If B is almost everywhere dominating, then B is superhigh, i.e., 0 ′′ is
KolmogorovLoveland randomness and stochasticity
 Annals of Pure and Applied Logic
, 2005
"... An infinite binary sequence X is KolmogorovLoveland (or KL) random if there is no computable nonmonotonic betting strategy that succeeds on X in the sense of having an unbounded gain in the limit while betting successively on bits of X. A sequence X is KLstochastic if there is no computable nonm ..."
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Cited by 27 (8 self)
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An infinite binary sequence X is KolmogorovLoveland (or KL) random if there is no computable nonmonotonic betting strategy that succeeds on X in the sense of having an unbounded gain in the limit while betting successively on bits of X. A sequence X is KLstochastic if there is no computable nonmonotonic selection rule that selects from X an infinite, biased sequence. One of the major open problems in the field of effective randomness is whether MartinLöf randomness is the same as KLrandomness. Our first main result states that KLrandom sequences are close to MartinLöf random sequences in so far as every KLrandom sequence has arbitrarily dense subsequences that are MartinLöf random. A key lemma in the proof of this result is that for every effective split of a KLrandom sequence at least one of the halves is MartinLöf random. However, this splitting property does not characterize KLrandomness; we construct a sequence that is not even computably random such that every effective split yields two subsequences that are 2random. Furthermore, we show for any KLrandom sequence A that is computable in the halting problem that, first, for any effective split of A both halves are MartinLöf random and, second, for any computable, nondecreasing, and unbounded function g
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 ..."
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Cited by 19 (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
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 14 (6 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.
Oscillation in the initial segment complexity of random reals
 Adv. Math
, 2010
"... Abstract. We study oscillation in the prefixfree complexity of initial segments of 1random reals. For upward oscillations, we prove that P n∈ω 2 −g(n) diverges iff (∃∞n) K(X n)> n+g(n) for every 1random X ∈ 2ω. For downward oscillations, we characterize the functions g such that (∃∞n) K(X n) ..."
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Cited by 11 (3 self)
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Abstract. We study oscillation in the prefixfree complexity of initial segments of 1random reals. For upward oscillations, we prove that P n∈ω 2 −g(n) diverges iff (∃∞n) K(X n)> n+g(n) for every 1random X ∈ 2ω. For downward oscillations, we characterize the functions g such that (∃∞n) K(X n) < n+g(n) for almost every X ∈ 2ω. The proof of this result uses an improvement of Chaitin’s counting theorem—we give a tight upper bound on the number of strings σ ∈ 2n such that K(σ) < n+K(n)−m. The work on upward oscillations has applications to the Kdegrees. Write X ≤K Y to mean that K(X n) ≤ K(Y n) + O(1). The induced structure is called the Kdegrees. We prove that there are comparable (∆02) 1random Kdegrees. We also prove that every lower cone and some upper cones in the 1random Kdegrees have size continuum. Finally, we show that it is independent of ZFC, even assuming that the Continuum Hypothesis fails, whether all chains of 1random Kdegrees of size less than 2ℵ0 have a lower bound in the 1random Kdegrees. “Although this oscillatory behaviour is usually considered to be a nasty feature, we believe that it illustrates one of the great advantages of complexity: the possibility to study degrees of randomness.” Michiel van Lambalgen, Ph.D. Dissertation [31, p. 145]. 1.
KOLMOGOROV COMPLEXITY OF INITIAL SEGMENTS OF SEQUENCES AND ARITHMETICAL DEFINABILITY
"... Abstract. The structure of the Kdegrees provides a way to classify sets of natural numbers or infinite binary sequences with respect to the level of randomness of their initial segments. In the Kdegrees of infinite binary sequences, X is below Y if the prefixfree Kolmogorov complexity of the firs ..."
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Cited by 8 (6 self)
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Abstract. The structure of the Kdegrees provides a way to classify sets of natural numbers or infinite binary sequences with respect to the level of randomness of their initial segments. In the Kdegrees of infinite binary sequences, X is below Y if the prefixfree Kolmogorov complexity of the first n bits of X is less than the complexity of the first n bits of Y, for each n. Identifying infinite binary sequences with subsets of N, we study the Kdegrees of arithmetical sets and explore the interactions between arithmetical definability and prefix free Kolmogorov complexity. We show that in the Kdegrees, for each n> 1 there exists a Σ0 n nonzero degree which does not bound any ∆0 n nonzero degree. An application of this result is that in the Kdegrees there exists a Σ0 2 degree which forms a minimal pair with all Σ0 1 degrees. This extends work of Csima/Montalbán [CM06] and Merkle/Stephan [MS07]. Our main result is that, given any ∆0 2 family C of sequences, there is a ∆0 2 sequence of nontrivial initial segment complexity which is not larger than the initial segment complexity of any nontrivial member of C. This general theorem has the following surprising consequence. There is a 0 ′computable sequence of nontrivial initial segment complexity which is not larger than the initial segment complexity of any nontrivial computably enumerable set. Our analysis and results demonstrate that, examining the extend to which arithmetical definability interacts with the K reducibility (and in general any ‘weak reducibility’) is a fruitful way of studying the induced structure. 1.
Eliminating concepts
 Proceedings of the IMS workshop on computational prospects of infinity
, 2008
"... Four classes of sets have been introduced independently by various researchers: low for K, low for MLrandomness, basis for MLrandomness and Ktrivial. They are all equal. This survey serves as an introduction to these coincidence results, obtained in [24] and [10]. The focus is on providing backdo ..."
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Cited by 7 (2 self)
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Four classes of sets have been introduced independently by various researchers: low for K, low for MLrandomness, basis for MLrandomness and Ktrivial. They are all equal. This survey serves as an introduction to these coincidence results, obtained in [24] and [10]. The focus is on providing backdoor access to the proofs. 1. Outline of the results All sets will be subsets of N unless otherwise stated. K(x) denotes the prefix free complexity of a string x. A set A is Ktrivial if, within a constant, each initial segment of A has minimal prefix free complexity. That is, there is c ∈ N such that ∀n K(A ↾ n) ≤ K(0 n) + c. This class was introduced by Chaitin [5] and further studied by Solovay (unpublished). Note that the particular effective epresentation of a number n by a string (unary here) is irrelevant, since up to a constant K(n) is independent from the representation. A is low for MartinLöf randomness if each MartinLöf random set is already MartinLöf random relative to A. This class was defined in Zambella [28], and studied by Kučera and Terwijn [17]. In this survey we will see that the two classes are equivalent [24]. Further concepts have been introduced: to be a basis for MLrandomness (Kučera [16]), and to be low for K (Muchnik jr, in a seminar at Moscow State, 1999). They will also be eliminated, by showing equivalence with Ktriviality. All