Immunity and Hyperimmunity for Sets of Minimal Indices

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. 1 A short introduction to shortest programs The set of shortest programs is {e : (∀j < e) [φj 6= φe]}. (1.1) In 1967, Blum [4] showed that one can enumerate at most finitely many shortest programs. Five years later, Meyer [13] formally initiated the investigation of minimal index sets with questions on the Turing and truth-table degrees of (1.1). Meyer’s research parallels inquiry from Kolmogorov complexity where one searches for shortest programs generating single numbers or strings. The clearest confluence ∗This work was supported in part by NUS grant R252–000–212–112. The addresses are: Frank Stephan, National University of Singapore, Departments of Mathematics and Computer Science, Singapore 117543, [email protected]; Jason Teutsch, RAND Corporation, 1776 Main Street, P.O. Box 2138, Santa Monica, CA 90407-2138, USA, [email protected].