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Hausdorff dimension in exponential time
TLDR
In this paper we investigate effective versions of Hausdorff dimension which have been recently introduced by Lutz. Expand
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Kolmogorov Complexity and the Recursion Theorem
TLDR
We introduce the concepts of complex and autocomplex sets, where a set A is complex if there is a recursive, nondecreasing and unbounded lower bound on the Kolmogorov complexity of the prefixes (of the characteristic sequence) of A, and Autocomplex is defined likewise with recursive replaced by A-recursive. Expand
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Reconciling Data Compression and Kolmogorov Complexity
TLDR
Kolmogorov complexity defined via decidable machines yields characterizations in terms of the intial segment complexity of sequences of the concepts of Martin-Lof randomness, Schnorr randomness and Kurtz randomness. Expand
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Kolmogorov-Loveland randomness and stochasticity
TLDR
We show for any KL-random sequence A that is computable in the halting problem that, first, for any effective split of A both halves are Martin-Lof random and, second, for every computable, nondecreasing, and unbounded function g and almost all n , the prefix of A of length n has prefix-free Kolmogorov complexity. Expand
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Kolmogorov complexity and the Recursion Theorem
Several classes of diagonally nonrecursive (DNR) functions are characterized in terms of Kolmogorov complexity. In particular, a set of natural numbers A can wtt-compute a DNR function iff there is aExpand
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On C-Degrees, H-Degrees and T-Degrees
  • W. Merkle, F. Stephan
  • Mathematics, Computer Science
  • Twenty-Second Annual IEEE Conference on…
  • 13 June 2007
TLDR
We analyse the relations between Turing reducibility, defined in terms of computation power, and C-reducibility and H-Reducibility, and show that there are pairs of sets that have a meet neither in the C- Degrees nor in the H-degrees. Expand
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Effective Randomness for Computable Probability Measures
TLDR
We determine all the implications that hold between the equivalence relations induced by Martin-Lof Randomness, computable randomness, Schnorr randomness and weak randomness. Expand
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On the Autoreducibility of Random Sequences
TLDR
A binary sequence $A=A(0)A(1)\ldots$ is called infinitely often (i.o.)~Turing-au\-to\-re\-duc\-ible if $A$~is reducible to itself via an oracle Turing machine that never queries its oracle at the current input, outputs either $A(x)$ or a don't-know symbol on any given input~$x$. Expand
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On the Construction of Effective Random Sets
TLDR
We give a direct and rather simple construction of Martin-Lof random and rec-random sets with certain additional properties. First, reviewing the result of Gacs and Ku?era, given any set X we construct a Martin- Lof random set R from which X can be decoded effectively. Expand
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Some Results on Effective Randomness
TLDR
We investigate the characterizations of effective randomness concepts in terms of Martin-Löf tests and martingales. Expand
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