On Strong Separations from AC

1 SUMMARY As part of a study of almost-everywhere complex sets, we i n vestigate sets that are immune to AC 0 ; that is, sets with no innnite subset in AC 0. W e show that such sets exist in P PP and in DSPACElog n log n. Our main result is an oracle construction indicating that any improvement in these immunity results will represent a signiicant advance, in that we show that any answer to the question: Are there sets in NP that are immune to AC 0 ? will provide non-relativizable proof techniques suitable for attacking the Ntime vs Dtime question. That is, we show that the existence or nonexistence of AC 0-immune sets in NP has consequences concerning the complexity of sets in deterministic and nondeterministic exponential time.