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.