Size, Depth and Energy of Threshold Circuits Computing Parity Function.

Abstract

We investigate relations among the size, depth and energy of threshold circuits computing the n-variable parity function PAR_n, where the energy is a complexity measure for sparsity on computation of threshold circuits, and is defined to be the maximum number of gates outputting 1 over all the input assignments. We show that PAR_n is hard for threshold circuits of small size, depth and energy: - If a depth-2 threshold circuit C of size s and energy e computes PAR_n, it holds that 2^{n/(elog ^e n)} ≤ s; and - if a threshold circuit C of size s, depth d and energy e computes PAR_n, it holds that 2^{n/(e2^{e+d}log ^e n)} ≤ s. We then provide several upper bounds: - PAR_n is computable by a depth-2 threshold circuit of size O(2^{n-2e}) and energy e; - PAR_n is computable by a depth-3 threshold circuit of size O(2^{n/(e-1)} + 2^{e-2}) and energy e; and - PAR_n is computable by a threshold circuit of size O((e+d)2^{n-m}), depth d + O(1) and energy e + O(1), where m = max (((e-1)/(d-1))^{d-1}, ((d-1)/(e-1))^{e-1}). Our lower and upper bounds imply that threshold circuits need exponential size if both depth and energy are constant, which contrasts with the fact that PAR_n is computable by a threshold circuit of size O(n) and depth 2 if there is no restriction on the energy. Our results also suggest that any threshold circuit computing the parity function needs depth to be sparse if its size is bounded.