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  1. Strengthening Prompt Simplicity.David Diamondstone & Keng Meng Ng - 2011 - Journal of Symbolic Logic 76 (3):946 - 972.
    We introduce a natural strengthening of prompt simplicity which we call strong promptness, and study its relationship with existing lowness classes. This notion provides a ≤ wtt version of superlow cuppability. We show that every strongly prompt c.e. set is superlow cuppable. Unfortunately, strong promptness is not a Turing degree notion, and so cannot characterize the sets which are superlow cuppable. However, it is a wtt-degree notion, and we show that it characterizes the degrees which satisfy a wtt-degree notion very (...)
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  • Reductions Between Types of Numberings.Ian Herbert, Sanjay Jain, Steffen Lempp, Manat Mustafa & Frank Stephan - 2019 - Annals of Pure and Applied Logic 170 (12):102716.
    This paper considers reductions between types of numberings; these reductions preserve the Rogers Semilattice of the numberings reduced and also preserve the number of minimal and positive degrees in their semilattice. It is shown how to use these reductions to simplify some constructions of specific semilattices. Furthermore, it is shown that for the basic types of numberings, one can reduce the left-r.e. numberings to the r.e. numberings and the k-r.e. numberings to the k+1-r.e. numberings; all further reductions are obtained by (...)
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  • Computing From Projections of Random Points.Noam Greenberg, Joseph S. Miller & André Nies - 2019 - Journal of Mathematical Logic 20 (1):1950014.
    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 characterize 1/2-bases as the sets computable from both halves of Chaitin’s Ω, and as the sets that obey the cost function c(x,s)=Ωs−Ωx−−−−−−−√. Generalizing these results yields a dense hierarchy of (...)
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  • Computing K-Trivial Sets by Incomplete Random Sets.Laurent Bienvenu, Adam R. Day, Noam Greenberg, Antonín Kučera, Joseph S. Miller, André Nies & Dan Turetsky - 2014 - Bulletin of Symbolic Logic 20 (1):80-90.
    EveryK-trivial set is computable from an incomplete Martin-Löf random set, i.e., a Martin-Löf random set that does not compute the halting problem.
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  • Demuth Randomness and Computational Complexity.Antonín Kučera & André Nies - 2011 - Annals of Pure and Applied Logic 162 (7):504-513.
    Demuth tests generalize Martin-Löf tests in that one can exchange the m-th component a computably bounded number of times. A set fails a Demuth test if Z is in infinitely many final versions of the Gm. If we only allow Demuth tests such that GmGm+1 for each m, we have weak Demuth randomness.We show that a weakly Demuth random set can be high and , yet not superhigh. Next, any c.e. set Turing below a Demuth random set is strongly jump-traceable.We (...)
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  • A Random Set Which Only Computes Strongly Jump-Traceable C.E. Sets.Noam Greenberg - 2011 - Journal of Symbolic Logic 76 (2):700 - 718.
    We prove that there is a ${\mathrm{\Delta }}_{2}^{0}$ , 1-random set Y such that every computably enumerable set which is computable from Y is strongly jump-traceable. We also show that for every order function h there is an ω-c.e. random set Y such that every computably enumerable set which is computable from Y is h-jump-traceable. This establishes a correspondence between rates of jump-traceability and computability from ω-c.e. random sets.
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  • Upper Bounds on Ideals in the Computably Enumerable Turing Degrees.George Barmpalias & André Nies - 2011 - Annals of Pure and Applied Logic 162 (6):465-473.
    We study ideals in the computably enumerable Turing degrees, and their upper bounds. Every proper ideal in the c.e. Turing degrees has an incomplete upper bound. It follows that there is no prime ideal in the c.e. Turing degrees. This answers a question of Calhoun [2]. Every proper ideal in the c.e. Turing degrees has a low2 upper bound. Furthermore, the partial order of ideals under inclusion is dense.
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  • Computably Enumerable Sets Below Random Sets.André Nies - 2012 - Annals of Pure and Applied Logic 163 (11):1596-1610.
    We use Demuth randomness to study strong lowness properties of computably enumerable sets, and sometimes of Δ20 sets. A set A⊆N is called a base for Demuth randomness if some set Y Turing above A is Demuth random relative to A. We show that there is an incomputable, computably enumerable base for Demuth randomness, and that each base for Demuth randomness is strongly jump-traceable. We obtain new proofs that each computably enumerable set below all superlow Martin-Löf random sets is strongly (...)
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  • Strong Jump-Traceability.Noam Greenberg & Dan Turetsky - 2018 - Bulletin of Symbolic Logic 24 (2):147-164.
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