Polynomial interpolation, threshold circuits, and the polynomial hierarchy

"Abstract: "Toda [13] has shown that the polynomial hierarchy is contained in P[superscript PP]. It is natural to ask whether the polynomial hierarchy is in fact contained in PP. Along these lines, it has been shown [2] that P[superscript NP[log]] is contained in PP. However, a lower bound of Minsky and Papert [8] implies that [sigma] p/2 is not contained in PP relative to an oracle [5]. Thus we ask how much of the polynomial hierarchy is contained in PP. We construct an oracle relative to which P[superscript NP[f(n)]] is contained in PP if and only if f(n) = O(log n), so the results of [2] are optimal in a relativized world. In particular, relative to this oracle, [delta] p/2 is not contained in PP."@en
"Our oracle is also the first relative to which [formula] is properly contained of [formula], or in the terminology of Wagner's refined polynomial hierarchy [15], [theta] p/2 is properly contained in [delta] p/2. Our construction depends on a new lower bound for perceptrons, which is interesting in its own right. We construct a predicate that is computable by a small perceptron, but which requires exponentially large weights. This lower bound depends in turn on a fundamental property of polynomials: if p is bounded on the domain [1 ..., m] then the coefficients of p must be small as a function of m.""@en