Polynomial-Time Random Oracles and Separating Complexity Classes

Bennett and Gill [1981] showed that PA ≠ NPA ≠ coNPA 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 PA ≠ NPA 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 PA ≠ NPA relative to every p-random oracle A, then BPP ≠ EXP. (3) If PA ≠ NPA 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 PHA is infinite relative to oracles A that are p-betting-game random. Showing that PHA separates at even its first level would also imply an unrelativized complexity class separation: (4) If NPA ≠ coNPA for a p-betting-game measure 1 class of oracles A, then NP ≠ EXP. (5) If PHA is infinite relative to every p-random oracle A, then PH ≠ EXP. We also consider random oracles for time versus space, for example: (6) LA ≠ PA relative to every oracle A that is p-betting-game random.