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On strongly jump traceable reals

Keng Meng Ng
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. 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.  ... 
doi:10.1016/j.apal.2007.11.014 fatcat:copmnhys5rcnzj7pdsnsv5w43y

Strong jump-traceability II: K-triviality

Rod Downey, Noam Greenberg
2011 Israel Journal of Mathematics  
We show that every 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.  ... 
doi:10.1007/s11856-011-0217-z fatcat:c63qkfmf4vdldgfqymuf4ewkxy

Strong jump-traceability I: The computably enumerable case

Peter Cholak, Rod Downey, Noam Greenberg
2008 Advances in Mathematics  
We show that the combinatorial class of computably enumerable, 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.  ... 
doi:10.1016/j.aim.2007.09.008 fatcat:62dgru3agfbe3e5hwlfapqygfe

Lowness for Demuth Randomness [chapter]

Rod Downey, Keng Meng Ng
2009 Lecture Notes in Computer Science  
We show that every 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 .  ... 
doi:10.1007/978-3-642-03073-4_17 fatcat:5pt2tuhlkzae5jn7aol3acsipm

$K$-trivial degrees and the jump-traceability hierarchy

George Barmpalias, Rod Downey, Noam Greenberg
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-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  ... 
doi:10.1090/s0002-9939-09-09761-5 fatcat:v3hlxpuf4jfdxg6ikz375zgc2m

Lowness Properties and Approximations of the Jump

Santiago Figueira, André Nies, Frank Stephan
2006 Electronical Notes in Theoretical Computer Science  
Informally, a set A is 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.  ... 
doi:10.1016/j.entcs.2005.05.025 fatcat:tb2bqz5fezddrdlvtf7it6lt6q

Lowness properties and approximations of the jump

Santiago Figueira, André Nies, Frank Stephan
2008 Annals of Pure and Applied Logic  
Informally, a set A is 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.  ... 
doi:10.1016/j.apal.2007.11.002 fatcat:xhfbk5gyz5dzbo4fyzmiek3mt4

Superhighness and Strong Jump Traceability [chapter]

André Nies
2009 Lecture Notes in Computer Science  
Then A is 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.  ... 
doi:10.1007/978-3-642-02927-1_60 fatcat:quzpuqeaefhbdpndwbar4puq2a

Characterizing the strongly jump-traceable sets via randomness [article]

Noam Greenberg, Denis Hirschfeldt, Andre Nies
2011 arXiv   pre-print
We show that if a set A is computable from every superlow 1-random set, then A is 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.  ... 
arXiv:1109.6749v1 fatcat:2i3bgtix3rctnmvaxz6zte62ai

Characterizing the strongly jump-traceable sets via randomness

Noam Greenberg, Denis R. Hirschfeldt, André Nies
2012 Advances in Mathematics  
We also prove the analogous result for superhighness: a c.e. set is 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.  ... 
doi:10.1016/j.aim.2012.06.005 fatcat:62prazncfjhllisvoxkqvwmaoi

Computably enumerable sets below random sets

André Nies
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 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. .  ... 
doi:10.1016/j.apal.2011.12.011 fatcat:wwrz4p4uonelhffmzbc2yjrzwi

CHARACTERIZING LOWNESS FOR DEMUTH RANDOMNESS

LAURENT BIENVENU, ROD DOWNEY, NOAM GREENBERG, ANDRÉ NIES, DAN TURETSKY
2014 Journal of Symbolic Logic (JSL)  
We fully characterize lowness for Demuth randomness using an appropriate notion of 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.  ... 
doi:10.1017/jsl.2013.21 fatcat:y3ujz3qs35ecrm4gofm24dlj4a

Inherent enumerability of strong jump-traceability [article]

David Diamondstone, Noam Greenberg, Daniel Turetsky
2011 arXiv   pre-print
Moreover, we show that every 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.  ... 
arXiv:1110.1435v1 fatcat:bkm7323ju5fplpechztx3dcuju

Strong jump traceability and Demuth randomness [article]

Noam Greenberg, Daniel Turetsky
2011 arXiv   pre-print
We show that 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.  ... 
arXiv:1109.6128v1 fatcat:5vpjrxw45fcg7frqlzj746ye64

Studying Randomness Through Computation [chapter]

André Nies
2011 Randomness Through Computation  
I thank Christopher Porter for comments 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.  ... 
doi:10.1142/9789814327756_0017 fatcat:4vyju3t7nrathd77qtmwn5nqs4
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