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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 {\it inner product} of two vectors of length $n$ over GF(2). More exactly we prove that $\Omega(n\log n)\leq \log w\log M$, where $w$ is the sum of the absolute values of the weights, and $M$ is the maximum fan--in of the AND gates on level 2. Setting all weights to 1, we got a trade--off between the logarithms of the top--fan--in and the maximum fan--in on level 2.
In contrast, with $n$ AND gates at the bottom and {\it a single} MOD 2 gate at the top one can compute the {\it 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 {\it random} evaluation protocol of a circuit with MOD $m$ gates.
