The perceptron strikes back

It is shown that every AC⁰ predicate is computed by a low-degree probabilistic polynomial over the reals. It is demonstrated that circuits composed of a symmetric gate at the root with AND-OR subcircuits of constant depth can be simulated by probabilistic depth-2 circuits with essentially the same symmetric gate at the root and AND gates of small fanin at the bottom. In particular, every language recognized by a depth-d AC⁰ circuit is decidable by a probabilistic perceptron of size 2 to the power O(log^4d n) and of order O(log^4d n) that uses O(log³ n) probabilistic bits. As a corollary, the authors present a new proof that depth-d AND-OR circuits computing the parity of n binary inputs require size 2 to the power n^Ω(1/d)