Boolean circuit complexity is the combinatorics of computer science and involves many
intriguing problems that are easy to state and explain, even for the layman. This book is a …

Monotone Boolean functions are an important object in discrete mathematics and
mathematical cybernetics. Topics related to these functions have been actively studied for …

For any unsatisfiable CNF formula F that is hard to refute in the Resolution proof system, we
show that a gadget-composed version of F is hard to refute in any proof system whose lines …

Query-to-communication lifting theorems translate lower bounds on query complexity to
lower bounds for the corresponding communication model. In this paper, we give a …

We survey lower-bound results in complexity theory that have been obtained via newfound
interconnections between propositional proof complexity, boolean circuit complexity, and …

Monotone functions and data sets arise in a variety of applications. We study the
interpolation problem for monotone data sets: The input is a monotone data set with $ n …

We present lower and upper bounds showing that the average-case complexity of the k-
Clique problem on monotone circuits is n^k/4+O(1). Similar bounds for AC^0 circuits were …

For a universal constant α> 0 we prove size lower bounds of 2α (n) for an explicit function in
monotone NP in the following models of computation: monotone formulas, monotone …

Query-to-communication lifting theorems, which connect the query complexity of a Boolean
function to the communication complexity of an associatedlifted'function obtained by …

Valiant (1980) showed that general arithmetic circuits with negation can be exponentially
more powerful than monotone ones. We give the first improvement to this classical result: we …