 
Summary: Com S 631: Lower bounds and Separation Results
Lecture 10 Scribe: Sudheer Vakati
In this lecture we will study about constant depth circuits.
Definition 1. A constant depth circuit is a circuit with a constant depth d and computes a
function f : n
.
In the definition of circuit, we require that "AND" and "OR" gates have fanin two. Thus
if a circuit with depth d computes a function, the function can depend on at most 2d
bits.
Thus any function, for example Parity, that depends on all its input bits can not be computed
by such circuits.
We now relax the restriction on the fanin. For rest of the lecture we study constant depth
circuits with unbounded fanin.
Definition 2. A boolean function f :
is in AC0 if there is a constant d, a
polynomial p and a family of circuits (C1, C2, · · · ) such that
(1) Cn computes f : n
(2) For all n, depth(Cn) d
(3) For all n, size(Cn) p(n)
