Abstract

Russell Impagliazzo , Ramamohan Paturi , Michael E. Saks y Abstract The following size--depth trade--off 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` \Gammad edges, where constants 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 affirmative answer to the conjecture of Paturi and Saks that a bounded depth threshold circuit that computes parity requires a super--linear number of edges. This is the first super-- linear lower bound for an explicit function that holds for any fixed depth, and the first that applies to threshold circuits with unrestricted weights. The trade-off 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, de...

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