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On strongly jump traceable reals

2008
*
Annals of Pure and Applied Logic
*

The uniformity of the proof method allows us to adapt the technique to showing that the index set of the c.e.

doi:10.1016/j.apal.2007.11.014
fatcat:copmnhys5rcnzj7pdsnsv5w43y
*strongly**jump**traceables*is Π 0 4 -complete. ... In particular, there is no single order function such that strong*jump**traceability*is equivalent to*jump**traceability*for that order. ... This is equivalent to making A*strongly**jump**traceable*, and noth-*jump**traceable*for someh. ...##
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Strong jump-traceability II: K-triviality

2011
*
Israel Journal of Mathematics
*

We show that every

doi:10.1007/s11856-011-0217-z
fatcat:c63qkfmf4vdldgfqymuf4ewkxy
*strongly**jump*-*traceable*set is K-trivial. Unlike other results, we do not assume that the sets in question are computably enumerable. ... A*real*A is h-*jump*-*traceable*if every A-partial computable function has a c.e. h-trace. A*real*is*strongly**jump*-*traceable*if it is h-*jump*-*traceable*for every order function h. ... Every*strongly**jump*-*traceable*set is computable from some c.e.,*strongly**jump*-*traceable*set. ...##
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Strong jump-traceability I: The computably enumerable case

2008
*
Advances in Mathematics
*

We show that the combinatorial class of computably enumerable,

doi:10.1016/j.aim.2007.09.008
fatcat:62dgru3agfbe3e5hwlfapqygfe
*strongly**jump*-*traceable**reals*, defined via the*jump*operator by Figueira, Nies and Stephan [Santiago Figueira, André Nies, Frank Stephan, ... .,*strongly**jump*-*traceable*set is not Martin-Löf cuppable, thus giving a combinatorial property which implies non-ML cuppability. incomplete. ... They then showed that*jump*-*traceability*and strong*jump*-*traceability*differ*on*the computably enumerable*reals*. ...##
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Lowness for Demuth Randomness
[chapter]

2009
*
Lecture Notes in Computer Science
*

We show that every

doi:10.1007/978-3-642-03073-4_17
fatcat:5pt2tuhlkzae5jn7aol3acsipm
*real*low for Demuth randomness is of hyperimmune-free degree. ... They defined A to be*strongly**jump**traceable*, if A is*jump**traceable*with respect to all order functions. ... In fact, Greenberg and Downey [8] showed that if*one*got down to a level of log log n, then every*jump**traceable**real*was ∆ 0 2 . ...##
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$K$-trivial degrees and the jump-traceability hierarchy

2009
*
Proceedings of the American Mathematical Society
*

For every order h such that n 1/h(n) is finite, every K-trivial degree is h-

doi:10.1090/s0002-9939-09-09761-5
fatcat:v3hlxpuf4jfdxg6ikz375zgc2m
*jump*-*traceable*. ... We show however that the K-trivial degrees are properly contained in those that are h-*jump*-*traceable*for every convergent order h. ... Cholak, Downey and Greenberg [2] , however, showed that while all c.e.*strongly**jump*-*traceable**reals*are K-trivial, there are K-trivial*reals*which are not*strongly**jump*-*traceable*, so this attempt at ...##
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Lowness Properties and Approximations of the Jump

2006
*
Electronical Notes in Theoretical Computer Science
*

Informally, a set A is

doi:10.1016/j.entcs.2005.05.025
fatcat:tb2bqz5fezddrdlvtf7it6lt6q
*strongly**jump*-*traceable*if for each order function h, for each input e*one*may effectively enumerate a set T e of possible values for the*jump*J A (e), and the number of values enumerated ... We prove that there is a*strongly**jump*-*traceable*set which is not computable, and that if A is well-approximable then A is*strongly**jump*-*traceable*. For r.e. sets, the converse holds as well. ... Definition 4 A set A is*strongly**jump*-*traceable*iff for each order function h, A is*jump**traceable*via h. Clearly, strong*jump*-*traceability*implies*jump*-*traceability*. ...##
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Lowness properties and approximations of the jump

2008
*
Annals of Pure and Applied Logic
*

Informally, a set A is

doi:10.1016/j.apal.2007.11.002
fatcat:xhfbk5gyz5dzbo4fyzmiek3mt4
*strongly**jump*-*traceable*if for each order function h, for each input e*one*may effectively enumerate a set T e of possible values for the*jump*J A (e), and the number of values enumerated ... We prove that there is a*strongly**jump*-*traceable*set which is not computable, and that if A is well-approximable then A is*strongly**jump*-*traceable*. For r.e. sets, the converse holds as well. ... A set A is*strongly**jump*-*traceable*iff for each order function h, A is*jump*-*traceable*via h.Clearly, strong*jump*-*traceability*implies*jump*-*traceability*. ...##
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Superhighness and Strong Jump Traceability
[chapter]

2009
*
Lecture Notes in Computer Science
*

Then A is

doi:10.1007/978-3-642-02927-1_60
fatcat:quzpuqeaefhbdpndwbar4puq2a
*strongly**jump**traceable*if and only if A is Turing below each superhigh Martin-Löf random set. The proof combines priority with measure theoretic arguments. ...*One*wants to restrict the changes of A to the extent that A is*strongly**jump**traceable*. To this end,*one*attempts to define a "naughty set" Y ∈ H ∩ MLR. ... [4] built a promptly simple set that is*strongly**jump**traceable*. ...##
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Characterizing the strongly jump-traceable sets via randomness
[article]

2011
*
arXiv
*
pre-print

We show that if a set A is computable from every superlow 1-random set, then A is

arXiv:1109.6749v1
fatcat:2i3bgtix3rctnmvaxz6zte62ai
*strongly**jump*-*traceable*. ... This theorem shows that the computably enumerable (c.e.)*strongly**jump*-*traceable*sets are exactly the c.e. sets computable from every superlow 1-random set. ... Then A is*strongly**jump*-*traceable*. ...##
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Characterizing the strongly jump-traceable sets via randomness

2012
*
Advances in Mathematics
*

We also prove the analogous result for superhighness: a c.e. set is

doi:10.1016/j.aim.2012.06.005
fatcat:62prazncfjhllisvoxkqvwmaoi
*strongly**jump*-*traceable*if and only if it is computable from every superhigh 1-random set. ... Symbolic Logic 76 (1) (2011) 289-312], this theorem shows that the computably enumerable (c.e.)*strongly**jump*-*traceable*sets are exactly the c.e. sets computable from every superlow 1-random set. ... The class of*jump*-*traceable*sets is much larger than the class of*strongly**jump*-*traceable*sets. ...##
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Computably enumerable sets below random sets

2012
*
Annals of Pure and Applied Logic
*

We show that there is an incomputable, computably enumerable base for Demuth randomness, and that each base for Demuth randomness is

doi:10.1016/j.apal.2011.12.011
fatcat:wwrz4p4uonelhffmzbc2yjrzwi
*strongly**jump*-*traceable*. (2) We obtain new proofs that each computably ... enumerable set below all superlow (superhigh) Martin-Löf random sets is*strongly**jump**traceable*, using Demuth tests. (3) The sets Turing below each ω 2 -computably approximable Martin-Löf random set form ... We say that A is*strongly**jump**traceable*(s.j.t.) [9] if A is*jump**traceable*with bound h for each order function h. The class of c.e.,*strongly**jump**traceable*sets is denoted by SJT c.e. . ...##
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CHARACTERIZING LOWNESS FOR DEMUTH RANDOMNESS

2014
*
Journal of Symbolic Logic (JSL)
*

We fully characterize lowness for Demuth randomness using an appropriate notion of

doi:10.1017/jsl.2013.21
fatcat:y3ujz3qs35ecrm4gofm24dlj4a
*traceability*. ... This shows that in some sense, the computably dominated Demuth*traceable*sets are an analogue of the*strongly**jump**traceable*sets, outside the ∆ 0 2 degrees. ...*Jump**traceability*. An oracle A is*jump**traceable*In the rest of this section we will show that Demuth*traceability*strictly implies*jump**traceability*. However, they agree*on*the c.e. degrees. ...##
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Inherent enumerability of strong jump-traceability
[article]

2011
*
arXiv
*
pre-print

Moreover, we show that every

arXiv:1110.1435v1
fatcat:bkm7323ju5fplpechztx3dcuju
*strongly**jump*-*traceable*set is computable from a computably enumerable*strongly**jump*-*traceable*set. ... We show that every*strongly**jump*-*traceable*set obeys every benign cost function. ... are all*strongly**jump*-*traceable*. ...##
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Strong jump traceability and Demuth randomness
[article]

2011
*
arXiv
*
pre-print

We show that

arXiv:1109.6128v1
fatcat:5vpjrxw45fcg7frqlzj746ye64
*on*the other hand, the class of sets which form a base for Demuth randomness is a proper subclass of the class of*strongly**jump*-*traceable*sets. ... We solve the covering problem for Demuth randomness, showing that a computably enumerable set is computable from a Demuth random set if and only if it is*strongly**jump*-*traceable*. ... set is*strongly**jump*-*traceable*. ...##
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Studying Randomness Through Computation
[chapter]

2011
*
Randomness Through Computation
*

I thank Christopher Porter for comments

doi:10.1142/9789814327756_0017
fatcat:4vyju3t7nrathd77qtmwn5nqs4
*on*earlier drafts of this paper, and the editor Hector Zenil for getting me to write this. ... For background*on*the next variant of Question 4.1, see Subsection 3.2. Question 4.3 [18] Is every*strongly**jump**traceable*(c.e.) set Turing below a Demuth random? ... Cholak, Downey and Greeberg [7] showed that the c.e.*strongly**jump**traceable*sets form a proper subclass of the c.e. K-trivials. ...
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