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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.
Low for random reals and positivemeasure domination
 Proceedings of the American Mathematical Society, 2007. Preprint
, 2005
"... Abstract. The low for random reals are characterized topologically, as well as in terms of domination of Turing functionals on a set of positive measure. 1. ..."
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Abstract. The low for random reals are characterized topologically, as well as in terms of domination of Turing functionals on a set of positive measure. 1.
Lowness for weakly 1generic and Kurtzrandom
 in Theory and Applications of Models of Computation: Third Internationa l Conference, TAMC 2006
, 2006
"... Abstract. We prove that a set is low for weakly 1generic iff it has neither dnr nor hyperimmune Turing degree. As this notion is more general than being recursively traceable, we refute a recent conjecture on the characterization of these sets. Furthermore, we show that every set which is low for w ..."
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Cited by 7 (2 self)
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Abstract. We prove that a set is low for weakly 1generic iff it has neither dnr nor hyperimmune Turing degree. As this notion is more general than being recursively traceable, we refute a recent conjecture on the characterization of these sets. Furthermore, we show that every set which is low for weakly 1generic is also low for Kurtzrandom. 1
KTRIVIAL DEGREES AND THE JUMPTRACEABILITY HIERARCHY
"... Abstract. For every order h such that P n 1/h(n) is finite, every Ktrivial degree is hjumptraceable. This motivated Cholak, Downey and Greenberg [2] to ask whether this traceability property is actually equivalent to Ktriviality, thereby giving the hoped for combinatorial characterisation of low ..."
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Abstract. For every order h such that P n 1/h(n) is finite, every Ktrivial degree is hjumptraceable. This motivated Cholak, Downey and Greenberg [2] to ask whether this traceability property is actually equivalent to Ktriviality, thereby giving the hoped for combinatorial characterisation of lowness for MartinLöf randomness. We show however that the Ktrivial degrees are properly contained in those that are hjumptraceable for every convergent order h. 1.
LOWNESS FOR COMPUTABLE MACHINES
"... Abstract. Two lowness notions in the setting of Schnorr randomness have been studied (lowness for Schnorr randomness and tests, by Terwijn and Zambella [19], and by KjosHanssen, Stephan, and Nies [7]; and Schnorr triviality, by Downey, Griffiths and LaForte [3, 4] and Franklin [6]). We introduce lo ..."
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Cited by 1 (1 self)
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Abstract. Two lowness notions in the setting of Schnorr randomness have been studied (lowness for Schnorr randomness and tests, by Terwijn and Zambella [19], and by KjosHanssen, Stephan, and Nies [7]; and Schnorr triviality, by Downey, Griffiths and LaForte [3, 4] and Franklin [6]). We introduce lowness for computable machines, which by results of Downey and Griffiths [3] is an analog of lowness for K. We show that the reals that are low for computable machines are exactly the computably traceable ones, and so this notion coincides with that of lowness for Schnorr randomness and for Schnorr tests. 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.
unknown title
, 2006
"... The low for random reals are characterized topologically, as well as in terms of domination of Turing functionals on a set of positive measure. 1 Low for random reals and positivemeasure domination ..."
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The low for random reals are characterized topologically, as well as in terms of domination of Turing functionals on a set of positive measure. 1 Low for random reals and positivemeasure domination