 
Summary: Com S 631: Lower bounds and Separation Results
Lecture 12 Scribe: Adam Case
1. Introduction
We have shown that Parity can not be computed by constant depth, polynomialsize
circuits. Now present an alternate proof. We will show something stonger: Parity does not
have polynomialsize constant depth circuits using Mod3 gates. Mod3 on a series of bits is
defined as:
Mod3(x1x2 . . . xn) = 0 if xi is divisible by 3
= 1 else
This will be shown by using the RazborovSmolensky Polynomial Method. This method
has two steps. Show that every constant depth, polynomialsize circuit can be approximated
by a low degree polynomial. Show tha Parity can not be approximated by a low degree
polynomial. Now we discuss what it means for a boolean function to be represented by a
polynomial.
Note: For the rest of this lecture, we will be referring to finite fields. The notation for
a finite field is GF(p) = {0, · · · , p  1} where p is a prime. A field uses two operations
(addition and multiplication) where (mod p) is applied to the result of each operation in
order to satisfy closure. For example: GF(3) = {0, 1, 2}. If we use addition on the elements
1 and 2, we will receive 0 since 3 (mod 3) is 0.
2. Polynomial Representation of Boolean Functions
