Circuit Depth Relative to a Random Oracle

The study of separation of complexity classes with respect to random oracles was initiated by Bennett and Gill [ll and continued by many researchers. Computation relative to an oracle A, where A c (0, l}* is a computation which is allowed to ask queries of the form y E A? and get answers. A random oracle A is simply a random (with respect to Lebesgue measure) subset of IO, l)*, i.e. for each x E (0, l)*, x E A with probability 4 and all these events are mutually independent. Wilson [6,7] defined relativized circuit depth and constructed various oracles A for which PA # NCA, NC,A + NC;+E, AC; # AC;+E, AC; $Z NC:+,-, and NC; g AC:_, for all positive rational k and E, thus separating those classes for which no trivial argument shows inclusion (the definition of NC; and AC: is restricted to rational k for constructibility reasons). In this note we show that as a consequence of a single lemma, these separations (or improvements of them) hold with respect to a random oracle A with probability 1.