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Kolmogorov complexity and the Recursion Theorem. Manuscript, submitted for publication
, 2005
"... Abstract. Several classes of diagonally nonrecursive (DNR) functions are characterized in terms of Kolmogorov complexity. In particular, a set of natural numbers A can wttcompute a DNR function iff there is a nontrivial recursive lower bound on the Kolmogorov complexity of the initial segments of ..."
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Cited by 46 (11 self)
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Abstract. Several classes of diagonally nonrecursive (DNR) functions are characterized in terms of Kolmogorov complexity. In particular, a set of natural numbers A can wttcompute a DNR function iff there is a nontrivial recursive lower bound on the Kolmogorov complexity of the initial segments of A. Furthermore, A can Turing compute a DNR function iff there is a nontrivial Arecursive lower bound on the Kolmogorov complexity of the initial segements of A. A is PAcomplete, that is, A can compute a {0, 1}valued DNR function, iff A can compute a function F such that F (n) is a string of length n and maximal Ccomplexity among the strings of length n. A ≥T K iff A can compute a function F such that F (n) is a string of length n and maximal Hcomplexity among the strings of length n. Further characterizations for these classes are given. The existence of a DNR function in a Turing degree is equivalent to the failure of the Recursion Theorem for this degree; thus the provided results characterize those Turing degrees in terms of Kolmogorov complexity which do no longer permit the usage of the Recursion Theorem. 1.
RANDOMNESS NOTIONS AND PARTIAL RELATIVIZATION
"... Abstract. We study weak 2 randomness, weak randomness relative to ∅ ′ and Schnorr randomness relative to ∅ ′. One major theme is characterizing the oracles A such that ML[A] ⊆ C, where C is a randomness notion and ML[A] denotes the MartinLöf random reals relative to A. We discuss the connections ..."
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Cited by 2 (2 self)
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Abstract. We study weak 2 randomness, weak randomness relative to ∅ ′ and Schnorr randomness relative to ∅ ′. One major theme is characterizing the oracles A such that ML[A] ⊆ C, where C is a randomness notion and ML[A] denotes the MartinLöf random reals relative to A. We discuss the connections with LRreducibility and also study the reducibility associated with weak 2randomness. 1.
Centre for Discrete Mathematics and
, 2008
"... Abstract. A real x is ∆ 1 1Kurtz random (Π 1 1Kurtz random) if it 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 by 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 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 by being ∆ 1 1dominated and ∆ 1 1semitraceable. 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.
CHARACTERIZING LOWNESS FOR DEMUTH RANDOMNESS
"... Abstract. We show the existence of noncomputable oracles which are low for Demuth randomness, answering a question in [15] (also Problem 5.5.19 in [35]). We fully characterize lowness for Demuth randomness using an appropriate notion of traceability. Central to this characterization is a partial rel ..."
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Abstract. We show the existence of noncomputable oracles which are low for Demuth randomness, answering a question in [15] (also Problem 5.5.19 in [35]). We fully characterize lowness for Demuth randomness using an appropriate notion of traceability. Central to this characterization is a partial relativization of Demuth randomness, which may be more natural than the fully relativized version. We also show that an oracle is low for weak Demuth randomness if and only if it is computable. 1.