Exponential lower bounds for depth three Boolean circuits

Abstract

We consider the class $ \sum^{k}_{3} $ of unbounded fan-in depth three Boolean circuits, for which the bottom fan-in is limited by k and the top gate is an OR. It is known that the smallest such circuit computing the parity function has $ \Omega(2^{\varepsilon n/k}) $ gates (for k = O(n 1/2)) for some $ \varepsilon > 0 $ , and this was the best lower bound known for explicit (P-time computable) functions. In this paper, for k = 2, we exhibit functions in uniform NC 1 that require $ 2^{n-o(n)} $ size depth 3 circuits. The main tool is a theorem that shows that any $ \sum {2\over3} $ circuit on n variables that accepts a inputs and has size s must be constant on a projection (subset defined by equations of the form x i = 0, x i = 1, x i = x j or x i = $ \bar{x}_i $ ) of dimension at least log(a/s)log n.