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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.
HIGHER RANDOMNESS NOTIONS AND THEIR LOWNESS PROPERTIES
, 2007
"... Abstract. We study randomness notions given by higher recursion theory, establishing the relationships Π 1 1randomness ⊂ Π 1 1MartinLöf randomness ⊂ ∆ 1 1randomness = ∆ 1 1MartinLöf randomness. We characterize the set of reals that are low for ∆ 1 1 randomness as precisely those that are ∆ 1 ..."
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Cited by 2 (2 self)
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Abstract. We study randomness notions given by higher recursion theory, establishing the relationships Π 1 1randomness ⊂ Π 1 1MartinLöf randomness ⊂ ∆ 1 1randomness = ∆ 1 1MartinLöf randomness. We characterize the set of reals that are low for ∆ 1 1 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.