Lecture 13

Today we covered: Parity has unbounded fan-in circuits of depth-$d$ size-$2^{\Theta \left(n^{1/(d-1)}\right)}$. 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 $1/\left(4d\right)$ 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 $\mathrm{Pr}[p\left(x\right)=OR\left(x\right)]\ge 1-\epsilon$ when $p$ is chosen from the distribution. Expanding functions $f:\{-1,1{\}}^n\to ℝ$ in the Fourier basis.