A generalized polymorphism of a predicate $P \subseteq \{0,1\}^m$ is a tuple of functions $f_1,\dots,f_m\colon \{0,1\}^n \to \{0,1\}$ satisfying the following property: If $x^{(1)},\dots,x^{(m)} \in \{0,1\}^n$ are such that $(x^{(1)}_i,\dots,x^{(m)}_i) \in P$ for all $i$, then also $(f_1(x^{(1)}),\dots,f_m(x^{(m)})) \in P$.

We show that if $f_1,\dots,f_m$ satisfy this property for most ... more >>>

A pair of sources $\mathbf{X},\mathbf{Y}$ over $\{0,1\}^n$ are $k$-indistinguishable if their projections to any $k$ coordinates are identically distributed. Can some $\mathit{AC^0}$ function distinguish between two such sources when $k$ is big, say $k=n^{0.1}$? Braverman's theorem (Commun. ACM 2011) implies a negative answer when $\mathbf{X}$ is uniform, whereas Bogdanov et ... more >>>

Håstad showed that any De Morgan formula (composed of AND, OR and NOT gates) shrinks by a factor of $O(p^{2})$ under a random restriction that leaves each variable alive independently with probability $p$ [SICOMP, 1998]. Using this result, he gave an $\widetilde{\Omega}(n^{3})$ formula size lower bound for the Andreev function, ... more >>>

The seminal result of Kahn, Kalai and Linial shows that a coalition of $O(\frac{n}{\log n})$ players can bias the outcome of *any* Boolean function $\{0,1\}^n \to \{0,1\}$ with respect to the uniform measure. We extend their result to arbitrary product measures on $\{0,1\}^n$, by combining their argument with a completely ... more >>>