Counting the changes of random Δ20 sets

Figueira, Santiago; Hirschfeldt, Denis R.; Miller, Joseph S.; Ng, Keng Meng; Nies, André

doi:10.1093/logcom/exs083

Cited by 15 publications

(31 citation statements)

References 13 publications

(10 reference statements)

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“…Balanced randomness, introduced in [6], interpolates between weak Demuth and ML-randomness. The current version of the m-th component of a monotonic test can change at most O(2 m ) times.…”

Section: Overview Of Notions Between 2-randomness and 1-randomnessmentioning

confidence: 99%

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Demuth randomness and computational complexity

Kučera

Nies²

2011

Annals of Pure and Applied Logic

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a b s t r a c t Demuth tests generalize Martin-Löf tests (G m ) m∈N in that one can exchange the m-th component a computably bounded number of times. A set Z ⊆ N fails a Demuth test if Z is in infinitely many final versions of the G m . If we only allow Demuth tests such that G m ⊇ G m+1 for each m, we have weak Demuth randomness.We show that a weakly Demuth random set can be high and ∆ 0 2 , yet not superhigh. Next, any c.e. set Turing below a Demuth random set is strongly jump-traceable.We also prove a basis theorem for non-empty Π 0 1 classes P. It extends the Jockusch-Soare basis theorem that some member of P is computably dominated. We use the result to show that some weakly 2-random set does not compute a 2-fixed point free function.

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“…Balanced randomness, introduced in [6], interpolates between weak Demuth and ML-randomness. The current version of the m-th component of a monotonic test can change at most O(2 m ) times.…”

Section: Overview Of Notions Between 2-randomness and 1-randomnessmentioning

confidence: 99%

“…As noted above, balanced randomness implies being Turing incomplete. The authors in [6] show, for instance, that each superlow ML-random set is balanced random.…”

Section: Overview Of Notions Between 2-randomness and 1-randomnessmentioning

confidence: 99%

Demuth randomness and computational complexity

Kučera

Nies²

2011

Annals of Pure and Applied Logic

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“…They defined a notion of randomness, called Oberwolfach randomness. They were motivated by work by Figueira, Hirschfeldt, Miller, Ng and Nies [14], who investigated the randomness strength of a ∆ 0 2 random set Z by counting the number of changes required in any computable approximation of Z. This was linked with Demuth's idea, mentioned earlier, of accepting changes in components of tests.…”

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confidence: 99%

“…The question whether such a random set exists was a strong variant of the covering problem, which was also posed by Stephan in 2004. Using work from [14], they also concluded that the smart K-trivial set is not computable from both halves of a random set, negatively solving another strong variant of the covering problem (Problem 4.7 in [27]). …”

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confidence: 99%

COMPUTINGK-TRIVIAL SETS BY INCOMPLETE RANDOM SETS

Bienvenu¹,

Day²,

Greenberg³

et al. 2014

Bull. symb. log

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Abstract. Every K-trivial set is computable from an incomplete Martin-Löf random set, i.e., a Martin-Löf random set that does not compute H 1 .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. Computably enumerable sets are each definable and, in a sense, they are as effective as any noncomputable set can be. As a consequence, 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, Martin-Löf randomness, is not strong enough to support this argument, and in fact, significant interactions between Martin-Löf 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 Martin-Löf random sets and the class of K-trivial sets. Interestingly, the significant interaction occurs "at the boundaries": the Martin-Löf random sets in question are close to being non-random (in that they fail fairly simple statistical tests), and K-trivial c.e. sets are close to being computable.The following theorem resolves one of the main open questions in algorithmic randomness, and further strengthens the relationship between the Martin-Löf random sets and the K-trivial sets. Theorem 1. There is an incomplete Martin-Löf random set that computes every K-trivial set.This theorem is essentially a corollary of two recent results, both proved in 2012: the first by Bienvenu, Greenberg, Kučera, Nies and Turetsky [3]; and the second by Day and Miller [10]. In the remainder of this announcement, we will explain the background to the problem behind this theorem, and indicate the main ideas used in the proof.In this announcement, by "random" we will henceforth mean Martin-Löf random. We will give the full definition shortly, but essentially, an element X of Cantor space is Martin-Löf random if it is not an element of a particular kind of effectively

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“…Bienvenu, et al [2] introduce Oberwolfach randomness and show that every Oberwolfach random real is a full density-one point. Based on earlier work by Figueira, Hirschfeldt, Miller, Ng, and Nies [7], they observe that one "half " of every Martin-Löf random real is always Oberwolfach random, hence full density-one 3 :…”

Section: Letmentioning

confidence: 99%

Lebesgue Density and Classes

Khan¹

2016

J. symb. log.

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Analyzing the effective content of the Lebesgue density theorem played a crucial role in some recent developments in algorithmic randomness, namely, the solutions of the ML-covering and MLcupping problems. Two new classes of reals emerged from this inquiry: the positive density points with respect to effectively closed (or Π 0 1 ) sets of reals, and a proper subclass, the density-one points. Bienvenu, Hölzl, Miller, and Nies have shown that the Martin-Löf random positive density points are exactly the ones that do not compute the halting problem. Treating this theorem as our starting point, we present several new results that shed light on how density, randomness, and computational strength interact.

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Counting the changes of random Δ20 sets

Cited by 15 publications

References 13 publications

Demuth randomness and computational complexity

Demuth randomness and computational complexity

COMPUTINGK-TRIVIAL SETS BY INCOMPLETE RANDOM SETS

Lebesgue Density and Classes

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