Circuit Lower Bounds for Parity Using Polynomials

Before we talk about bounds on the size of a circuit, let us first clarify what we mean by circuit depth and circuit size. The depth of a circuit is defined as the length of the longest path from the input to output. The size of a circuit is the number of AND and OR gates in the circuit. Note that, for our purpose, we assume all the gates have unlimited fan-in and fan-out. We define AC0 to be the class of decision problems solvable by circuits of polynomial size and constant depth. We want to prove the result that PARITY is not in AC0. There are two known techniques to prove this result. In this class, we will talk about a proof which uses polynomials; in the next class we will look at a different proof which uses random restrictions.