SIZE-DEPTH TRADEOFFS FOR THRESHOLD CIRCUITS

The following size-depth tradeoff for threshold circuits is obtained: any threshold circuit of depth d that computes the parity function on n variables must have at least n(1+c theta-d) edges, where c > 0 and t heta less than or equal to 3 are constants independent of n and d. Pre viously known constructions show that up to the choice of c and a this bound is best possible. In particular, the lower bound implies an aff irmative answer to the conjecture of Paturi and Saks that a bounded-de pth threshold circuit that computes parity requires a superlinear numb er of edges. This is the first superlinear lower bound for an explicit function that holds for any fixed depth and the first that applies to threshold circuits with unrestricted weights. The tradeoff is obtaine d as a consequence of a general restriction theorem for threshold circ uits with a small number of edges: For any threshold circuit with n in puts, depth d, and at most kn edges, there exists a partial assignment to the inputs that fixes the output of the circuit to a constant whil e leaving [n/(c(1)k)(c2 theta d)] variables unfixed, where c(1),c(2) > 0 and theta less than or equal to 3 are constants independent of 72, k, and d. A tradeoff between the number of gates and depth is also pro ved: any threshold circuit of depth d that computes the parity of n va riables has at least (n/2)(1/2(d-1)) gates. This tradeoff, which is es sentially the best possible, was proved previously (with a better cons tant in the exponent) for the case of threshold circuits with polynomi ally bounded weights in [K. Siu, V. Roychowdury, and T. Kailath, IEEE Trans. Inform. Theory, 40 (1994), pp. 455-466]; the result in the pres ent paper holds for unrestricted weights.