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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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Cited by 7 (2 self)
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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
Π 0 1 CLASSES, LR DEGREES AND TURING DEGREES
"... Abstract. We say that A ≤LR B if every Brandom set is Arandom with respect to MartinLöf randomness. We study this reducibility and its interactions with the Turing reducibility, Π0 1 classes, hyperimmunity and other recursion theoretic notions. A natural variant of the Turing reducibility from th ..."
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Cited by 6 (5 self)
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Abstract. We say that A ≤LR B if every Brandom set is Arandom with respect to MartinLöf randomness. We study this reducibility and its interactions with the Turing reducibility, Π0 1 classes, hyperimmunity and other recursion theoretic notions. A natural variant of the Turing reducibility from the point of view of MartinLöf randomness is the LR reducibility which was introduced in [12]. We say that A ≤LR B for two sets A, B if every Brandom real is Arandom (throughout this paper randomness means MartinLöf randomness). Intuitively this means that whenever A can derandomize a real, B also has this ability. This reducibility naturally induces an equivalence relation ≡LR which defines a partition of Cantor space into the LR degrees. Two reals A, B belong to the same LR degree iff the Arandom reals and Brandom reals coincide. The LR degrees were first introduced by André Nies [12] and were further studied by Barmpalias, Lewis, Soskova [2] and Simpson [15]. In this paper we study ≤LR and its interactions with ≤T. In Section 1 we lay out the basic framework and facts which are used throughout the rest of
Kolmogorov complexity and computably enumerable sets
, 2011
"... Abstract. We study the computably enumerable sets in terms of the: (a) Kolmogorov complexity of their initial segments; (b) Kolmogorov complexity of finite programs when they are used as oracles. We present an extended discussion of the existing research on this topic, along with recent developments ..."
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Abstract. We study the computably enumerable sets in terms of the: (a) Kolmogorov complexity of their initial segments; (b) Kolmogorov complexity of finite programs when they are used as oracles. We present an extended discussion of the existing research on this topic, along with recent developments and open problems. Besides this survey, our main original result is the following characterization of the computably enumerable sets with trivial initial segment prefixfree complexity. A computably enumerable set A is Ktrivial if and only if the family of sets with complexity bounded by the complexity of A is uniformly computable from the halting problem. 1.
COMPUTING KTRIVIAL SETS BY INCOMPLETE RANDOM SETS
"... Abstract. Every Ktrivial set is computable from an incomplete MartinLöf random set, i.e., a MartinLöf random set that does not compute the halting problem. A major objective in algorithmic randomness is to understand how random sets and computably enumerable (c.e.) sets interact within the Turi ..."
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Abstract. Every Ktrivial set is computable from an incomplete MartinLöf random set, i.e., a MartinLöf random set that does not compute the halting problem. A major objective in algorithmic randomness is to understand how random sets and computably enumerable (c.e.) sets interact within the Turing degrees. At some level of randomness all interesting interactions cease. The lower and upper cones of noncomputable c.e. sets are definable null sets, and thus if a set is “sufficiently” random, it cannot compute, nor be computed by, a noncomputable c.e. set. However, the most studied notion of algorithmic randomness, MartinLöf randomness, is not strong enough to support this argument, and in fact, significant interactions between MartinLöf random sets and c.e. sets occur. The study of these interactions has lead to a number of surprising results that show a remarkably robust relationship between MartinLöf random sets and the class of Ktrivial sets. Interestingly, the significant interaction occurs “at the boundaries”: the MartinLöf random sets in question are close to being nonrandom (in that they fail fairly simple statistical