Immunity and Hyperimmunity for Sets of Minimal Indices

Abstract

We extend Meyer's 1972 investigation of sets of minimal indices. Blum showed that minimal index sets are immune, and we show that they are also immune against high levels of the arithmetic hierarchy. We give optimal immunity results for sets of minimal indices with respect to the arithmetic hierarchy, and we illustrate with an intuitive example that immunity is not simply a refinement of arithmetic complexity. Of particular note here are the fact that there are three minimal index sets located in Π₃ − Σ₃ with distinct levels of immunity and that certain immunity properties depend on the choice of underlying acceptable numbering. We show that minimal index sets are never hyperimmune; however, they can be immune against the arithmetic sets. Lastly, we investigate Turing degrees for sets of random strings defined with respect to Bagchi's size-function s.