Asymptotic density, immunity, and randomness

Abstract

In 2012, inspired by developments in group theory and complexity, Jockusch and Schupp introduced generic computability, 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 hierarchy from immune to cohesive for both general and $\Delta^0_2$ sets, and identify intrinsic density $\frac{1}{2}$ as the stochasticity corresponding to permutation randomness. We also prove that Rice's Theorem extends to all intrinsic variations of generic computability, demonstrating in particular that no such notion considers $\emptyset'$ to be "computable".