Com S 631: Lower bounds and Separation Results Lecture 12 Scribe: Adam Case 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, polynomial-size circuits. Now present an alternate proof. We will show something stonger: Parity does not have polynomial-size 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 Razborov-Smolensky Polynomial Method. This method has two steps. Show that every constant depth, polynomial-size 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 Collections: Computer Technologies and Information Sciences