Benign cost functions and lowness properties

Greenberg, Noam; Nies, André

doi:10.2178/jsl/1294171001

Cited by 29 publications

(75 citation statements)

References 20 publications

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“…If C is not null, then C˛consists of the computable sets; the Hirschfeldt-Miller argument shows that if C is a null Σ 0 3 class then C˛contains a noncomputable set. Greenberg and Nies [18] showed, for example, that the strongly jump-traceable c.e. sets form a diamond class.…”

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confidence: 99%

COMPUTINGK-TRIVIAL SETS BY INCOMPLETE RANDOM SETS

Bienvenu¹,

Day²,

Greenberg³

et al. 2014

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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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confidence: 99%

COMPUTINGK-TRIVIAL SETS BY INCOMPLETE RANDOM SETS

Bienvenu¹,

Day²,

Greenberg³

et al. 2014

Bull. symb. log

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“…Being ω-c.e. tracing is a highness property due to Greenberg and Nies [2] that is incompatible with superlowness (see Sect. 6).…”

Section: Introductionmentioning

confidence: 99%

“…8.2.3]), the bound 2 x can be replaced by any order function without changing the class. Greenberg and Nies [2] show that there is a single benign cost function such that each c.e. set obeying it is Turing below each ω-c.e.…”

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confidence: 99%

Counting the changes of random Δ20 sets

Figueira¹,

Hirschfeldt²,

Miller³

et al. 2013

J Logic Computation

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Abstract. Consider a Martin-Löf random ∆ 0 2 set Z. We give lower bounds for the number of changes of Zs n for computable approximations of Z. We show that each nonempty Π 0 1 class has a low member Z with a computable approximation that changes only o(2 n ) times. We prove that each superlow ML-random set already satisfies a stronger randomness notion called balanced randomness, which implies that for each computable approximation and each constant c, there are infinitely many n such that Zs n changes more than c2 n times.

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“…Kučera and Nies [15] showed that every c.e. set which is computable from a Demuth random set is strongly jump-traceable, relating such random sets with the "benign cost functions" which by work of Greenberg and Nies [11] characterise c.e., strong jump-traceability. Other attractive spin-offs in the arena of randomness include Nies's new work on the calculus of cost functions [22].…”

Section: Introductionmentioning

confidence: 99%