A weight-size trade-off for circuits with MOD m gates

Abstract

We prove that any depth-3 circuit with MOD m gates of unbounded fan-in on the lowest level, AND gates on the second, and a weighted threshold gate on the top needs either exponential size or exponential weights to compute the inner product of two vectors of length n over GF(2). More exactly we prove that log(wikf) = Ω(n), where w is the sum of the absolute values of the weights, and &f is the maximum fan-in of the AND gates on level 2. Setting all weights to 1, we have got a trade-off between the numbers of the MOD m gates and the AND gates. By our knowledge, this is the first trade-off result involving hard-To-handle MOD m gates. In contrast, with n AND gates at the bottom and a single MOD 2 gate at the top one can compute the inner product function. The lower-bound proof does not use any monotonicity or uniformity assumptions, and all of our gates have unbounded fan-in. The key step in the proof is a random evaluation protocol of a circuit with MOD m gates.