Exponential Sums and Circuits with a Single Threshold Gate and Mod-Gates

Consider circuits consisting of a threshold gate at the top, Mod _m gates at the next level (for a fixed m ), and polylog fan-in AND gates at the lowest level. It is a difficult and important open problem to obtain exponential lower bounds for such circuits. Here we prove exponential lower bounds for restricted versions of this model, in which each Mod _m -of-AND subcircuit is a symmetric function of the inputs to that subcircuit. We show that if q is a prime not dividing m , the Mod _q function requires exponential-size circuits of this type. This generalizes recent results and techniques of Cai et al. CGT] (which held only for q=2 ) and Goldmann (which held only for depth two threshold over Mod _m circuits). As a further generalization of the CGT] result, the symmetry condition on the Mod _m subcircuits can be relaxed somewhat, still resulting in an exponential lower bound. The basis of the proof is to reduce the problem to estimating an exponential sum, which generalizes the notion of ``correlation" studied by CGT]. This identifies the type of exponential sum that will be instrumental in proving the general case. Along the way we substantially simplify previous proofs. Received June 1997, and in final form October 1998.