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  1. A Weakly 2-Generic Which Bounds a Minimal Degree.Rodney G. Downey & Satyadev Nandakumar - forthcoming - Journal of Symbolic Logic:1-22.
    Jockusch showed that 2-generic degrees are downward dense below a 2-generic degree. That is, if a is 2-generic, and $0 < {\bf{b}} < {\bf{a}}$, then there is a 2-generic g with $0 < {\bf{g}} < {\bf{b}}.$ In the case of 1-generic degrees Kumabe, and independently Chong and Downey, constructed a minimal degree computable from a 1-generic degree. We explore the tightness of these results. We solve a question of Barmpalias and Lewis-Pye by constructing a minimal degree computable from a weakly (...)
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  • Arithmetical Sacks Forcing.Rod Downey & Liang Yu - 2006 - Archive for Mathematical Logic 45 (6):715-720.
    We answer a question of Jockusch by constructing a hyperimmune-free minimal degree below a 1-generic one. To do this we introduce a new forcing notion called arithmetical Sacks forcing. Some other applications are presented.
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  • Almost Weakly 2-Generic Sets.Stephen A. Fenner - 1994 - Journal of Symbolic Logic 59 (3):868-887.
    There is a family of questions in relativized complexity theory--weak analogs of the Friedberg Jump-Inversion Theorem--that are resolved by 1-generic sets but which cannot be resolved by essentially any weaker notion of genericity. This paper defines aw2-generic sets. i.e., sets which meet every dense set of strings that is r.e. in some incomplete r.e. set. Aw2-generic sets are very close to 1-generic sets in strength, but are too weak to resolve these questions. In particular, it is shown that for any (...)
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  • A 1-Generic Degree with a Strong Minimal Cover.Masahiro Kumabe - 2000 - Journal of Symbolic Logic 65 (3):1395-1442.
  • Dynamic Notions of Genericity and Array Noncomputability.Benjamin Schaeffer - 1998 - Annals of Pure and Applied Logic 95 (1-3):37-69.
    We examine notions of genericity intermediate between 1-genericity and 2-genericity, especially in relation to the Δ20 degrees. We define a new kind of genericity, dynamic genericity, and prove that it is stronger than pb-genericity. Specifically, we show there is a Δ20 pb-generic degree below which the pb-generic degrees fail to be downward dense and that pb-generic degrees are downward dense below every dynamically generic degree. To do so, we examine the relation between genericity and array noncomputability, deriving some structural information (...)
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  • Generic Degrees Are Complemented.Masahiro Kumabe - 1993 - Annals of Pure and Applied Logic 59 (3):257-272.
  • 1-Generic Splittings of Computably Enumerable Degrees.Guohua Wu - 2006 - Annals of Pure and Applied Logic 138 (1):211-219.
    Say a set Gω is 1-generic if for any eω, there is a string σG such that {e}σ↓ or τσ↑). It is known that can be split into two 1-generic degrees. In this paper, we generalize this and prove that any nonzero computably enumerable degree can be split into two 1-generic degrees. As a corollary, no two computably enumerable degrees bound the same class of 1-generic degrees.
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