We develop a new method to prove communication lower bounds for composed functions of the form $f\circ g^n$ where $f$ is any boolean function on $n$ inputs and $g$ is a sufficiently ``hard'' two-party gadget. Our main structure theorem states that each rectangle in the communication matrix of $f \circ ... more >>>

We investigate the space complexity of refuting $3$-CNFs in Resolution and algebraic systems. No lower bound for refuting any family of $3$-CNFs was previously known for the total space in resolution or for the monomial space in algebraic systems. We prove that every Polynomial Calculus with Resolution refutation of a ... more >>>

Let $P$ be a fixed graph (hereafter called a ``pattern''), and let
$Subgraph(P)$ denote the problem of deciding whether a given graph $G$
contains a subgraph isomorphic to $P$. We are interested in
$AC^0$-complexity of this problem, determined by the smallest possible exponent
$C(P)$ for which $Subgraph(P)$ possesses bounded-depth circuits ... more >>>