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29
Lowness Properties and Randomness
 ADVANCES IN MATHEMATICS
"... The set A is low for MartinLof random if each random set is already random relative to A. A is Ktrivial if the prefix complexity K of each initial segment of A is minimal, namely K(n)+O(1). We show that these classes coincide. This implies answers to questions of AmbosSpies and Kucera [2 ..."
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Cited by 95 (24 self)
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The set A is low for MartinLof random if each random set is already random relative to A. A is Ktrivial if the prefix complexity K of each initial segment of A is minimal, namely K(n)+O(1). We show that these classes coincide. This implies answers to questions of AmbosSpies and Kucera [2], showing that each low for MartinLof random set is # 2 . Our class induces a natural intermediate # 3 ideal in the r.e. Turing degrees (which generates the whole class under downward closure). Answering
Trivial Reals
"... Solovay showed that there are noncomputable reals ff such that H(ff _ n) 6 H(1n) + O(1), where H is prefixfree Kolmogorov complexity. Such Htrivial reals are interesting due to the connection between algorithmic complexity and effective randomness. We give a new, easier construction of an Htrivi ..."
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Cited by 60 (31 self)
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Solovay showed that there are noncomputable reals ff such that H(ff _ n) 6 H(1n) + O(1), where H is prefixfree Kolmogorov complexity. Such Htrivial reals are interesting due to the connection between algorithmic complexity and effective randomness. We give a new, easier construction of an Htrivial real. We also analyze various computabilitytheoretic properties of the Htrivial reals, showing for example that no Htrivial real can compute the halting problem. Therefore, our construction of an Htrivial computably enumerable set is an easy, injuryfree construction of an incomplete computably enumerable set. Finally, we relate the Htrivials to other classes of &quot;highly nonrandom &quot; reals that have been previously studied.
Randomness and reducibility
 J. Comput. System Sci
, 2001
"... How random is a real? Given two reals, which is more random? If we partition reals into equivalence classes of reals of the “same degrees of randomness”, what does the resulting structure look like? The goal of this paper is to look at questions like these, specifically by studying the properties of ..."
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Cited by 28 (4 self)
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How random is a real? Given two reals, which is more random? If we partition reals into equivalence classes of reals of the “same degrees of randomness”, what does the resulting structure look like? The goal of this paper is to look at questions like these, specifically by studying the properties of reducibilities that act as
KolmogorovLoveland randomness and stochasticity
 Annals of Pure and Applied Logic
, 2005
"... An infinite binary sequence X is KolmogorovLoveland (or KL) random if there is no computable nonmonotonic betting strategy that succeeds on X in the sense of having an unbounded gain in the limit while betting successively on bits of X. A sequence X is KLstochastic if there is no computable nonm ..."
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Cited by 23 (8 self)
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An infinite binary sequence X is KolmogorovLoveland (or KL) random if there is no computable nonmonotonic betting strategy that succeeds on X in the sense of having an unbounded gain in the limit while betting successively on bits of X. A sequence X is KLstochastic if there is no computable nonmonotonic selection rule that selects from X an infinite, biased sequence. One of the major open problems in the field of effective randomness is whether MartinLöf randomness is the same as KLrandomness. Our first main result states that KLrandom sequences are close to MartinLöf random sequences in so far as every KLrandom sequence has arbitrarily dense subsequences that are MartinLöf random. A key lemma in the proof of this result is that for every effective split of a KLrandom sequence at least one of the halves is MartinLöf random. However, this splitting property does not characterize KLrandomness; we construct a sequence that is not even computably random such that every effective split yields two subsequences that are 2random. Furthermore, we show for any KLrandom sequence A that is computable in the halting problem that, first, for any effective split of A both halves are MartinLöf random and, second, for any computable, nondecreasing, and unbounded function g
On the Autoreducibility of Random Sequences
, 2001
"... A binary sequence A = A(0)A(1) ... is called i.o. Turingautoreducible 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'tknow symbol on any given input x, and outputs A(x) for infinitely many x. If in addi ..."
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Cited by 14 (1 self)
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A binary sequence A = A(0)A(1) ... is called i.o. Turingautoreducible 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'tknow symbol on any given input x, and outputs A(x) for infinitely many x. If in addition the oracle Turing machine terminates on all inputs and oracles, A is called i.o. truthtableautoreducible.
Lowness Properties of Reals and Randomness
 Advances in Mathematics
, 2002
"... We investigate three properties of the set of natural numbers which have been discovered independently by different... ..."
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Cited by 10 (3 self)
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We investigate three properties of the set of natural numbers which have been discovered independently by different...
The Complexity of Stochastic Sequences
 In Conference on Computational Complexity 2003
, 2003
"... We observe that known results on the Kolmogorov complexity of pre xes of eectively stochastic sequences extend to corresponding random sequences. First, there are recursively random random sequences such that for any nondecreasing and unbounded computable function f and for almost all n, the unifor ..."
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Cited by 9 (4 self)
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We observe that known results on the Kolmogorov complexity of pre xes of eectively stochastic sequences extend to corresponding random sequences. First, there are recursively random random sequences such that for any nondecreasing and unbounded computable function f and for almost all n, the uniform complexity of the length n pre x of the sequence is bounded by f(n). Second, a similar result with bounds of the form f(n) log n holds for partialrecursively random sequences.
The KolmogorovLoveland stochastic sequences are not closed under selecting subsequences
 Journal of Symbolic Logic
, 2002
"... It is shown that the class of KolmogorovLoveland stochastic sequences is not closed under selecting subsequences by monotonic computable selection rules. This result gives a strong negative answer to the notorious open problem whether the KolmogorovLoveland stochastic sequences are closed unde ..."
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Cited by 9 (5 self)
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It is shown that the class of KolmogorovLoveland stochastic sequences is not closed under selecting subsequences by monotonic computable selection rules. This result gives a strong negative answer to the notorious open problem whether the KolmogorovLoveland stochastic sequences are closed under selecting subsequences by KolmogorovLoveland selection rules, i.e., by not necessarily monotonic partially computable selection rules. As a corollary, we obtain an easy proof for the previously known result that the KolmogorovLoveland stochastic sequences form a proper subclass of the MisesWaldChurch stochastic sequences.
On the construction of effective random sets
 MATHEMATICAL FOUNDATIONS OF COMPUTER SCIENCE 2002, LECTURE NOTES IN COMPUTER SCIENCE
, 2002
"... We give a direct and rather simple construction of MartinLöf random and recrandom sets with certain additional properties. First, reviewing the result of Gacs and Kucera, given any set X we construct a MartinLöf random set R from which X can be decoded effectively. Second, by essentially the same ..."
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Cited by 9 (0 self)
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We give a direct and rather simple construction of MartinLöf random and recrandom sets with certain additional properties. First, reviewing the result of Gacs and Kucera, given any set X we construct a MartinLöf random set R from which X can be decoded effectively. Second, by essentially the same construction we obtain a MartinLöf random set R that is computably enumerable selfreducible. Alternatively, using the observation that a set is computably enumerable selfreducible if and only if its associated real is computably enumerable, the existence of such a set R follows from the known fact that every Chaitin real is MartinLöf random and computably enumerable. Third, by a variant of the basic construction we obtain a recrandom set that is weak truthtable autoreducible. The mentioned results on self and autoreducibility complement work of Ebert, Merkle, and Vollmer [79], from which it follows that no MartinLöf random set is Turingautoreducible and that no recrandom set is truthtable autoreducible.