Size-depth trade-offs for threshold circuits

The following size{depth tradeo for threshold circuits is obtained: any threshold circuit of depth d that computes the parity function on n variables must have at least n1+c d edges, where c> 0 and 3 are constants independent of n and d. Previously known constructions show that up to the choice of c and this bound is best possible. In particular, the lower bound implies an armative answer to the conjecture of Paturi and Saks that a bounded-depth threshold circuit that computes parity requires a superlinear number of edges. This is the rst superlinear lower bound for an explicit function that holds for any xed depth and the rst that applies to threshold circuits with unrestricted weights. The tradeo is obtained as a consequence of a general restriction theorem for threshold circuits with a small number of edges: For any threshold circuit with n inputs, depth d, and at most kn edges, there exists a partial assignment to the inputs that xes the output of the circuit to a constant while leavingbn=(c1k)c2 d c variables unxed, where c1;c 2>0 and 3 are constants independent of n, k, and d. A tradeo between the number of gates and depth is also proved: any threshold circuit of depth d that computes the parity of n variables has at least (n=2) 1=2(d 1) gates. This tradeo, which is essentially the best possible, was proved previously (with a better constant in the exponent) for the case of threshold circuits with polynomially bounded weights in (K. Siu, V. Roychowdury, and T. Kailath, IEEE Trans. Inform. Theory, 40 (1994), pp. 455{466); the result in the present paper holds for unrestricted weights.