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

Abstract

We consider the problem of how many extra bits of “help” a constant-depth Boolean circuit needs in order to compute m functions of the same input. Each help bit can be an arbitrary Boolean function of the input. We prove an exponential lower bound on the size of the circuit computing m parity functions in the presence of m - 1 help bits. The proof is carried out using the algebraic machinery of Razborov and Smolensky. 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.