Asymptotic density, immunity and randomness

TLDR: The notion of intrinsic asymptotic density was introduced in this article, with rich relations to both randomness and classical computability theory, and it has been used to define intrinsic density 0 as a new immunity property.

Abstract: In 2012, inspired by developments in group theory and complexity, Jockusch and Schupp introduced generic com- putability, capturing the idea that an algorithm might work correctly except for a vanishing fraction of cases. However, we observe that their definition of a negligible set is not computably invariant (and thus not well-defined on the 1-degrees), resulting in some failures of intuition and a break with standard expectations in computability theory. To strengthen their approach, we introduce a new notion of intrinsic asymptotic density, with rich relations to both randomness and classical computability theory. We then apply these ideas to propose alternative foundations for further development in (intrinsic) generic computability. Toward these goals, we classify intrinsic density 0 as a new immunity property, specifying its position in the standard hierar- chy from immune to cohesive for both general and � 0 sets, and identify intrinsic density 1 as the stochasticity corresponding to permutation randomness. We also prove that Rice's Theorem extends to all intrinsic variations of generic computability, demon- strating in particular that no such notion considers ∅ � to be "computable".