Lower bounds for constant depth circuits in the presence of help bits

The problem of how many extra bits of `help´ a constant depth circuit needs in order to compute m functions is considered. Each help bit can be an arbitrary Boolean function. An exponential lower bound on the size of the circuit computing m parity functions in the presence of m-1 help bits is proved. The proof is carried out using the algebraic machinery of A. Razborov (1987) and R. Smolensky (1987). A by-product of the proof is that the same bound holds for circuits with mod_p gates for a fixed prime p>2. The lower bound implies a random oracle separation for PH and PSPACE, which is optimal in a technical sense