Strong Jump-Traceability

Greenberg, Noam; Turetsky, Dan

doi:10.1017/bsl.2017.38

Cited by 6 publications

(6 citation statements)

References 47 publications

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“…if it can be computably approximated with a computably bounded number of changes; equivalently, A ď wtt H 1 . A set is strongly jump-traceable (SJT) if it is h-jump traceable for every order function h; see [23] for a survey. Every strongly jump traceable set is a 1{ω-base.…”

Section: {ω-Bases and Strong Jump-traceabilitymentioning

confidence: 99%

“…For background, there are several results characterising sub-ideals of the Ktrivials as those degrees computable from all random elements of some null Σ 0 3 class. One example is the class of strongly jump-traceable sets; they are precisely the sets computable from all superhigh random sequences [21,23]. Theorem 2.18 implies that the ideal of k{n-bases is such a class: the collection of sets computable from the n-columns of Ω.…”

Section: Being Computable From All Weakly Lr-hard Randomsmentioning

confidence: 99%

See 1 more Smart Citation

Computing from projections of random points

2019

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We study the sets that are computable from both halves of some (Martin-Löf) random sequence, which we call 1{2-bases. We show that the collection of such sets forms an ideal in the Turing degrees that is generated by its c.e. elements. It is a proper subideal of the K-trivial sets. We characterise 1{2-bases as the sets computable from both halves of Chaitin's Ω, and as the sets that obey the cost function cpx, sq "? Ωs´Ωx.Generalising these results yields a dense hierarchy of subideals in the Ktrivial degrees: For k ă n, let B k{n be the collection of sets that are below any k out of n columns of some random sequence. As before, this is an ideal generated by its c.e. elements and the random sequence in the definition can always be taken to be Ω. Furthermore, the corresponding cost function characterisation reveals that B k{n is independent of the particular representation of the rational k{n, and that Bp is properly contained in Bq for rational numbers p ă q. These results are proved using a generalisation of the Loomis-Whitney inequality, which bounds the measure of an open set in terms of the measures of its projections. The generality allows us to analyse arbitrary families of orthogonal projections. As it turns out, these do not give us new subideals of the K-trivial sets; we can calculate from the family which Bp it characterises.We finish by studying the union of Bp for p ă 1; we prove that this ideal consists of the sets that are robustly computable from some random sequence. This class was previously studied by Hirschfeldt, Jockusch, Kuyper, and Schupp [24], who showed that it is a proper subclass of the K-trivial sets. We prove that all such sets are robustly computable from Ω, and that they form a proper subideal of the sets computable from every (weakly) LR-hard random sequence. We also show that the ideal cannot be characterised by a cost function, giving the first such example of a Σ 0 3 subideal of the K-trivial sets.

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Section: {ω-Bases and Strong Jump-traceabilitymentioning

confidence: 99%

Section: Being Computable From All Weakly Lr-hard Randomsmentioning

confidence: 99%

Computing from projections of random points

2019

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“…There is more than one such characterisation. For example, a set is strongly jump-traceable if and only if it is computable from all superlow ML-random sequences [10], also if and only if it is computable from all superhigh random sequences [10,13]. Alternatively, a c.e.…”

Section: Thenmentioning

confidence: 99%

“…The strongly jump-traceable sets form an ideal in the Turing degrees [5,7] which is a proper sub-ideal of the K-trivials [8]. For more on strong jump-traceability, see the survey [13]. A characterisation that will concern us here is that a set is strongly jump-traceable if and only if it obeys all benign cost functions [12,7].…”

Section: Fragments Of ω and Strong Jump-traceabilitymentioning

confidence: 99%

Martin-Löf reducibility and cost functions

Greenberg¹,

Miller²,

Nies³

et al. 2017

Preprint

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Martin-Löf (ML)-reducibility compares K-trivial sets by examining the Martin-Löf random sequences that compute them. We show that every K-trivial set is computable from a c.e. set of the same ML-degree. We investigate the interplay between ML-reducibility and cost functions, which are used to both measure the number of changes in a computable approximation, and the type of null sets used to capture ML-random sequences. We show that for every cost function there is a c.e. set ML-above the sets obeying it (called a "smart" set for the cost function). We characterise the K-trivial sets computable from a fragment of the left-c.e. random real Ω. This leads to a new characterisation of strong jump-traceability. Contents 1. Introduction 1 2. Inherent enumerability of K -trivials 4 3. The most powerful set obeying a given cost function 6 4. The strongest cost function obeyed by a c.e. K -trivial set 8 5. The computational strength of fragments of Ω 11 6. Proof of Lemma 5.7 16 7. Fragments of Ω and strong jump-traceability 19 References 21

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“…sets tW f pnq | n P ωu and |W f pnq | ď hpnq such that if J A pnq Ó, then J A pnq P W f pnq . Then A is strongly jump traceable if A is jump traceable for all orders h. This is a fascinating class of reals associated with algorithmic randomness (see [24,36]). We can pursue the same forĴ in place of J.…”

mentioning

confidence: 99%