New Non-Uniform Lower Bounds for Uniform Classes

We strengthen the nondeterministic hierarchy theorem for non-deterministic polynomial time to show that the lower bound holds against sub-linear advice. More formally, we show that for any constants d and d′ such that 1 6 d < d′, and for any time-constructible bound t = o(n), there is a language in NTIME(n) which is not in NTIME(t)/n1/d . The best known earlier separation of Fortnow, Santhanam and Trevisan could only handle o(log(n)) bits of advice in the lower bound, and was not tight with respect to the time bounds. We generalize our hierarchy theorem to work for other syntactic complexity measures between polynomial time and polynomial space, including alternating polynomial time with any fixed number of alternations. We also use our technique to derive an almost-everywhere hierarchy theorem for non-deterministic classes which use a sub-linear amount of non-determinism, i.e., the lower bound holds on all but finitely many input lengths rather than just on infinitely many. As one application of our main result, we derive a new lower bound for NP against NP-uniform non-deterministic circuits of size O(n) for any fixed k. This result is a significant strengthening of a result of Kannan, which states that not all of NP can be solved with P-uniform circuits of size O(n) for any fixed k. As another application, we show strong non-uniform lower bounds for the complexity class RE of languages decidable in randomized linear exponential time with