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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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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is c ∈ N such that ∀n K(A ↾ n) ≤ K(0 n) + c. This class was introduced by Chaitin [5] and further studied by Solovay (unpublished).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
Higher randomness notions and their lowness properties. Israel journal of mathematics
, 2008
"... Abstract. We study randomness notions given by higher recursion theory, establishing the relationships Π11randomness ⊂ Π11MartinLöf randomness ⊂ ∆11randomness = ∆11MartinLöf randomness. We characterize the set of reals that are low for ∆11 randomness as precisely those that are ∆ 1 1tracea ..."
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Abstract. We study randomness notions given by higher recursion theory, establishing the relationships Π11randomness ⊂ Π11MartinLöf randomness ⊂ ∆11randomness = ∆11MartinLöf randomness. We characterize the set of reals that are low for ∆11 randomness as precisely those that are ∆ 1 1traceable. We prove that there is a perfect set of such reals. 1.
HIGHER KURTZ RANDOMNESS
"... Abstract. A real x is ∆ 1 1Kurtz random (Π 1 1Kurtz random) if it is in no closed null ∆ 1 1 set (Π 1 1 set). We show that there is a cone of Π 1 1Kurtz random hyperdegrees. We characterize lowness for ∆ 1 1Kurtz randomness as being ∆ 1 1dominated and ∆ 1 1semitraceable. 1. ..."
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Abstract. A real x is ∆ 1 1Kurtz random (Π 1 1Kurtz random) if it is in no closed null ∆ 1 1 set (Π 1 1 set). We show that there is a cone of Π 1 1Kurtz random hyperdegrees. We characterize lowness for ∆ 1 1Kurtz randomness as being ∆ 1 1dominated and ∆ 1 1semitraceable. 1.
RANDOMNESS AND COMPUTABILITY: OPEN QUESTIONS  ANNOTATED VERSION JUNE 2007
, 2007
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Randomness Zoo
, 2013
"... P olyR dim 1 compR s < 1 s < 1 Cdim s R dim s compR s ′ < s W R s ′ < s ..."
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P olyR dim 1 compR s < 1 s < 1 Cdim s R dim s compR s ′ < s W R s ′ < s