P-selectivity, immunity, and the power of one bit

Abstract

We prove that P-sel, the class of all P-selective sets, is EXPimmune, but is not EXP/1-immune. That is, we prove that some infinite P-selective set has no infinite EXP-time subset, but we also prove that every infinite P-selective set has some infinite subset in EXP/1. Informally put, the immunity of P-sel is so fragile that it is pierced by a single bit. of information. The above claims follow from broader results that we obtain about the immunity of the P-selective sets. In particular, we prove that for every recursive function f, P-sel is DTlME(f)-immune. Yet we also prove that P-sel is not ∏p2/1-immune. © Springer-Verlag Berlin Heidelberg 2006.