Bennett and Gill [1981] showed that P
A
≠ NP
A
≠ coNP
A
for a random oracle
A
, with probability 1. We investigate whether this result extends to individual polynomial-time random oracles. We consider two notions of random oracles: p-random oracles in the sense of martingales and resource-bounded measure [Lutz 1992; Ambos-Spies et al. 1997], and p-betting-game random oracles using the betting games generalization of resource-bounded measure [Buhrman et al. 2000]. Every p-betting-game random oracle is also p-random; whether the two notions are equivalent is an open problem.
(1) We first show that P
A
≠ NP
A
for every oracle
A
that is p-betting-game random.
Ideally, we would extend (1) to p-random oracles. We show that answering this either way would imply an unrelativized complexity class separation:
(2) If P
A
≠ NP
A
relative to every p-random oracle
A
, then BPP ≠ EXP.
(3) If P
A
≠ NP
A
relative to some p-random oracle
A
, then P ≠ PSPACE.
Rossman, Servedio, and Tan [2015] showed that the polynomial-time hierarchy is infinite relative to a random oracle, solving a longstanding open problem. We consider whether we can extend (1) to show that PH
A
is infinite relative to oracles
A
that are p-betting-game random. Showing that PH
A
separates at even its first level would also imply an unrelativized complexity class separation:
(4) If NP
A
≠ coNP
A
for a p-betting-game measure 1 class of oracles
A
, then NP ≠ EXP.
(5) If PH
A
is infinite relative to every p-random oracle
A
, then PH ≠ EXP.
We also consider random oracles for time versus space, for example:
(6) L
A
≠ P
A
relative to every oracle
A
that is p-betting-game random.
With probability one, a random oracle separates PSPACE from the polynomial-time hierarchy
