Relative to a Random Oracle<i>A</i>, ${\bf P}^A \ne {\bf NP}^A \ne \text{co-}{\bf NP}^A $ with Probability 1

Abstract

Let A be a language chosen randomly by tossing a fair coin for each string x to determine whether x belongs to A. With probability 1, each of the relativized classes ${\textbf{LOGSPACE}}^A $, ${\bf P}^A $, ${\bf NP}^A $, ${\bf PP}^A $, and ${\textbf{PSPACE}}^A $ is properly contained in the next. Also, ${\bf NP}^A \ne {\text{co-}} {\bf NP}^A $ with probability 1. By contrast, with probability 1 the class ${\bf P}^A $ coincides with the class ${\bf BPP}^A $ of languages recognized by probabilistic oracle machines with error probability uniformly bounded below $\tfrac{1}{2}$. ${\bf NP}^A $ is shown, with probability 1, to contain a ${\bf P}^A $-immune set, i.e., a set having no infinite subset in ${\bf P}^A $. The relationship of ${\bf P}^A $-immunity to p-sparseness and ${\bf NP}^A $-completeness is briefly discussed: ${\bf P}^A $-immune sets in ${\bf NP}^A $ can be sparse or moderately dense, but not co-sparse. Relativization with respect to a random length-preserving permutation $\pi $, instead of a random oracle A, yields analogous results and in addition the proper containment, with probability 1, of ${\bf P}^\pi $ in ${\bf NP}^\pi \cap {\text{co-}}{\bf NP}^\pi $, which we have been unable to decide for a simple random oracle. Most of these results are shown by straightforward counting arguments, applied to oracle-dependent languages designed not to be recognizable without a large number of oracle calls. It is conjectured that all $p^A $-invariant statements that are true with probability 1 of subrecursive language classes uniformly relativized to a random oracle are also true in the unrelativized case.