On the interplay between effective notions of randomness and genericity

Journal of Symbolic Logic 84 (1):393-407 (2019)
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Abstract

In this paper, we study the power and limitations of computing effectively generic sequences using effectively random oracles. Previously, it was known that every 2-random sequence computes a 1-generic sequence and every 2-random sequence forms a minimal pair in the Turing degrees with every 2-generic sequence. We strengthen these results by showing that every Demuth random sequence computes a 1-generic sequence and that every Demuth random sequence forms a minimal pair with every pb-generic sequence. Moreover, we prove that for every comeager${\cal G} \subseteq {2^\omega }$, there is some weakly 2-random sequenceXthat computes some$Y \in {\cal G}$, a result that allows us to provide a fairly complete classification as to how various notions of effective randomness interact in the Turing degrees with various notions of effective genericity.

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Christopher Porter
Drake University

Citations of this work

The Equivalence of Definitions of Algorithmic Randomness.Christopher Porter - 2021 - Philosophia Mathematica 29 (2):153–194.

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References found in this work

Demuth randomness and computational complexity.Antonín Kučera & André Nies - 2011 - Annals of Pure and Applied Logic 162 (7):504-513.
The degrees of bi-hyperhyperimmune sets.Uri Andrews, Peter Gerdes & Joseph S. Miller - 2014 - Annals of Pure and Applied Logic 165 (3):803-811.

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