Lecture 13

Today we covered: Parity has unbounded fan-in circuits of depth-$d$ size-$2^{\Theta(n^{1/(d-1)})}$. Hastad's Theorem statement: This is also a lower bound for constant $d$. Proof in the case $d = 2$ (CNF, DNF). Razborov-Smolensky proof with $1/(4d)$ rather than $1/(d-1)$. Its main tool: For any $k$, $\epsilon$, there is a probability distribution on $k$-variable polynomials of degree $O(\log k \log(1/\epsilon))$ such that for every $k$-bit input $x$ we have $\Pr[p(x) = OR(x)] \geq 1 - \epsilon$ when $p$ is chosen from the distribution. Expanding functions $f : \{-1,1\}^n \to \mathbb{R}$ in the Fourier basis.