Moore and Shannon have shown that relays with arbitrarily high reliability can be built from relays with arbitrarily poor reliability. Valiant used similar methods to construct monotone read-once formulae of size O(n +2) (where = log p 5?1 2 ' 3:27) that amplify (? 1 n ; + 1 n) (where = p 5?1 2 ' 0:62) to (2 ?n ; 1 ? 2 ?n) and deduced as a consequence the existence of monotone formulae of the s...

Moore and Shannon have shown that relays with arbitrarily high reliability can be built from relays with arbitrarily poor reliability. Valiant used similar methods to construct monotone read-once formulas of size O(n) (where α = log√ 5−1 2 ≃ 3.27) that amplify (ψ− 1 n , ψ+ 1 n ) (where ψ = ( √ 5− 1)/2 ≃ 0.62) to (2−n, 1− 2−n) and deduced as a consequence the existence of monotone formulas of th...

In the last lecture we looked at lower bounds for constant-depth circuits, proving that PARITY cannot be computed by constant-depth circuits, i.e. PARITY / ∈ AC0. General circuit lower bounds for explicit functions are quite weak: the best we can prove after years of effort is that there is a function, which requires circuits of size 5n − o(n). In this lecture we will examine what happens if we...

We extend Valiant's construction of monotone formulae for the majority function to obtain an eecient probabilistic construction of small monotone formulae for arbitrary positive linear threshold functions. We show that any positive linear threshold function on n boolean variables which has weight complexity q(n) can be computed by a monotone boolean formula of size O(q(n) 3:3 n 2): Our techniqu...

Motivated by results for the HCIZ integral in Part I of this paper, we study the structure of monotone Hurwitz numbers, which are a desymmetrized version of classical Hurwitz numbers. We prove a number of results for monotone Hurwitz numbers and their generating series that are striking analogues of known results for the classical Hurwtiz numbers. These include explicit formulas for monotone Hu...

In the paper we present a new notion of stochastic monotone measure and its application to image processing. By definition, a stochastic monotone measure is a random value with values in the set of monotone measures and it can describe a choice of random features in image processing. In this case, a monotone measure describes uncertainty in the problem of choosing the set of features with the h...