A lower bound for perceptrons and an oracle separation of the PP PH hierarchy

We show that there are functions computable by linear size boolean circuits of depth k that require superpolynomial size perceptrons of depth k-1, for k<log n/(6 log log n). This result implies the existence of an oracle A such that Σ_k^p,A⊄PP^Σ(_k-2 ^p,A) and in particular this oracle separates the levels in the PP^PH hierarchy. Using the same ideas, we show a lower bound for another function, which makes it possible to strengthen the oracle separation to Δ_k^p,A⊄PP^Σ(_k-2 ^p,A)