Summary: Com S 631: Lower bounds and Separation Results
Lecture 13 Scribe: Aaron Sterling
1. Parity does not have low degree approximations
In the last lecture we showed the following theorem.
Theorem 1. If C is a size S depth d circuit (with Mod3 gates) then there is a polynomial p
of degree (2k)d
such that Pr[p(x) = C(x)] 1 - s
We will set the degree to
n so (2k)d
. Then k = n1/2d
, so 1 - s
This is not true about parity: there is no low-degree polynomial over GF(3) such that the
polynomial is a good approximation of parity.
Theorem 2. Let p be a