As can be deduced from prior postings (cf., e.g., Nr. 87 and 93), I'm excited about any progress regarding derandomization of AC0. The current work reduces the seed length from a level that seems best within the Nisan-Wigderson construction paradigm modulo the best known lower bound techniques to a level that seems best only modulo the latter.
We describe a new pseudorandom generator for AC0. Our generator $\epsilon$-fools circuits of depth $d$ and size $M$ and uses a seed of length $\tilde O( \log^{d+4} M/\epsilon)$. The previous best construction for $d \geq 3$ was due to Nisan, and had seed length $O(\log^{2d+6} M/\epsilon)$. A seed length of $O(\log^{2d + \Omega(1)} M)$ is best possible given Nisan-type generators and the current state of circuit lower bounds; Seed length $\Omega(\log^d M/\epsilon)$ is a barrier for any pseudorandom generator construction given the current state of circuit lower bounds. For $d=2$, a pseudorandom generator of seed length $\tilde O(\log^2 M/\epsilon)$ was known.
Our generator is based on a ``pseudorandom restriction'' generator which outputs restrictions that satisfy the conclusions of the Hastad Switching Lemma and that uses a seed of polylogarithmic length.