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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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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.
Lowness for Kurtz randomness
 J. Symbolic Logic
"... Abstract. We prove that degrees that are low for Kurtz randomness cannot be diagonally nonrecursive. Together with the work of Stephan and Yu [16], this proves that they coincide with the hyperimmunefree nonDNR degrees, which are also exactly the degrees that are low for weak 1genericity. We als ..."
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Abstract. We prove that degrees that are low for Kurtz randomness cannot be diagonally nonrecursive. Together with the work of Stephan and Yu [16], this proves that they coincide with the hyperimmunefree nonDNR degrees, which are also exactly the degrees that are low for weak 1genericity. We also consider Low(M,Kurtz), the class of degrees a such that every element of M is aKurtz random. These are characterised when M is the class of MartinLöf random, computably random, or Schnorr random reals. We show that Low(ML,Kurtz) coincides with the nonDNR degrees, while both Low(CR,Kurtz) and Low(Schnorr,Kurtz) are exactly the nonhigh, nonDNR degrees. 1.
Five Lectures on Algorithmic Randomness
 in Computational Prospects of Infinity, ed. C.T. Chong, Proc. 2005 Singapore meeting
, 2007
"... This paper follows on from the author’s Five Lectures on Algorithmic Randomness. It is concerned with material not found in that long paper, concentrating on MartinLöf lowness and triviality. We present a hopefully userfriendly account of the decanter method, and discuss recent results of the auth ..."
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This paper follows on from the author’s Five Lectures on Algorithmic Randomness. It is concerned with material not found in that long paper, concentrating on MartinLöf lowness and triviality. We present a hopefully userfriendly account of the decanter method, and discuss recent results of the author with Peter Cholak and Noam Greenberg concerning the class of strongly jump traceable reals introduced by
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.
Jump inversions inside effectively closed sets and applications to randomness
 J. Symbolic Logic
"... Abstract. We study inversions of the jump operator on Π0 1 classes, combined with certain basis theorems. These jump inversions have implications for the study of the jump operator on the random degrees—for various notions of randomness. For example, we characterize the jumps of the weakly 2random ..."
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Abstract. We study inversions of the jump operator on Π0 1 classes, combined with certain basis theorems. These jump inversions have implications for the study of the jump operator on the random degrees—for various notions of randomness. For example, we characterize the jumps of the weakly 2random sets which are not 2random, and the jumps of the weakly 1random relative to 0 ′ sets which are not 2random. Both of the classes coincide with the degrees above 0 ′ which are not 0 ′dominated. A further application is the complete solution of [Nie09, Problem 3.6.9]: one direction of van Lambalgen’s theorem holds for weak 2randomness, while the other fails. Finally we discuss various techniques for coding information into incomplete randoms. Using these techniques we give a negative answer to [Nie09, Problem 8.2.14]: not all weakly 2random sets are array computable. In fact, given any oracle X, there is a weakly 2random which is not array computable relative to X. This contrasts with the fact that all 2random sets are array computable. 1.
Reconciling Data Compression and . . .
"... While data compression and Kolmogorov complexity are both about effective coding of words, the two settings differ in the following respect. A compression algorithm or compressor, for short, has to map a word to a unique code for this word in one shot, whereas with the standard notions of Kolmogor ..."
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While data compression and Kolmogorov complexity are both about effective coding of words, the two settings differ in the following respect. A compression algorithm or compressor, for short, has to map a word to a unique code for this word in one shot, whereas with the standard notions of Kolmogorov complexity a word has many different codes and the minimum code for a given word cannot be found effectively. This gap is bridged by introducing decidable Turing machines and a corresponding notion of Kolmogorov complexity, where compressors and suitably normalized decidable machines are essentially the same concept. Kolmogorov complexity defined via decidable machines yields characterizations in terms of the intial segment complexity of sequences of the concepts of MartinLöf randomness, Schnorr randomness, Kurtz randomness, and computable dimension. These results can also be reformulated in terms of timebounded Kolmogorov complexity. Other applications of decidable machines are presented, such as a simplified proof of the MillerYu theorem (characterizing MartinLöf randomness by the plain complexity of the initial segments) and a new characterization of computably traceable sequences via a natural lowness notion for decidable machines.
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.
AN ANALOGY BETWEEN CARDINAL CHARACTERISTICS AND HIGHNESS PROPERTIES OF ORACLES
"... Abstract. We develop an analogy between cardinal characteristics from set theory and highness properties from computability theory, which specify a sense in which a Turing oracle is computationally strong. We focus on characteristics from Cichoń’s diagram. 1. ..."
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Abstract. We develop an analogy between cardinal characteristics from set theory and highness properties from computability theory, which specify a sense in which a Turing oracle is computationally strong. We focus on characteristics from Cichoń’s diagram. 1.