Dynamic notions of genericity and array noncomputability Original Research Article

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 about the Δ20 degrees in the process.