We show that unbounded fan-in boolean formulas of depth $d+1$ and size $s$ have average sensitivity $O(\frac{1}{d}\log s)^d$. In particular, this gives a tight $2^{\Omega(d(n^{1/d}-1))}$ lower bound on the size of depth $d+1$ formulas computing the PARITY function. These results strengthen the corresponding $2^{\Omega(n^{1/d})}$ and $O(\log s)^d$ bounds for circuits ... more >>>

A recent work of Chen, Kabanets, Kolokolova, Shaltiel and Zuckerman (CCC 2014, Computational Complexity 2015) introduced the Compression problem for a class $\mathcal{C}$ of circuits, defined as follows. Given as input the truth table of a Boolean function $f:\{0,1\}^n \rightarrow \{0,1\}$ that has a small (say size $s$) circuit from ... more >>>

We introduce and study the notion of read-$k$ projections of the determinant: a polynomial $f \in \mathbb{F}[x_1, \ldots, x_n]$ is called a {\it read-$k$ projection of determinant} if $f=det(M)$, where entries of matrix $M$ are either field elements or variables such that each variable appears at most $k$ times in ... more >>>