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The strong anticupping property for recursively enumerable degrees

Journal of Symbolic Logic 54 (2):527-539 (1989)

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Dynamic notions of genericity and array noncomputability.Benjamin Schaeffer - 1998 - Annals of Pure and Applied Logic 95 (1-3):37-69.details 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 (...) Download Export citation Bookmark 3 citations
Computability and the Symmetric Difference Operator.Uri Andrews, Peter M. Gerdes, Steffen Lempp, Joseph S. Miller & Noah D. Schweber - 2022 - Logic Journal of the IGPL 30 (3):499-518.details Combinatorial operations on sets are almost never well defined on Turing degrees, a fact so obvious that counterexamples are worth exhibiting. The case we focus on is the symmetric-difference operator; there are pairs of degrees for which the symmetric-difference operation is well defined. Some examples can be extracted from the literature, e.g. from the existence of nonzero degrees with strong minimal covers. We focus on the case of incomparable r.e. degrees for which the symmetric-difference operation is well defined. Download Export citation Bookmark
Intervals containing exactly one c.e. degree.Guohua Wu - 2007 - Annals of Pure and Applied Logic 146 (1):91-102.details Cooper proved in [S.B. Cooper, Strong minimal covers for recursively enumerable degrees, Math. Logic Quart. 42 191–196] the existence of a c.e. degree with a strong minimal cover . So is the greastest c.e. degree below . Cooper and Yi pointed out in [S.B. Cooper, X. Yi, Isolated d.r.e. degrees, University of Leeds, Dept. of Pure Math., 1995. Preprint] that this strongly minimal cover cannot be d.c.e., and meanwhile, they proposed the notion of isolated degrees: a d.c.e. degree is isolated (...) Download Export citation Bookmark

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