Complexity of monotone networks for computing conjunctions

Let F₁, F₂…, F_m be a set of Boolean functions of the form F₁ = ^{x E X₁}, where ^ denotes conjuction and each X₁, is a subset of a set X of n Boolean variables. We study the size of monotone Boolean networks for computing such sets of functions. We exhibit anomalous sets of conjunctions whose smallest monotone networks contain disjunctions. We show that if |F|, is sufficiently small for all i, such anomalies cannot happen. We exhibit sets of m conjuctions in n unknowns which require c₂ma(m, n) binary conjunctions, where α(m, n) is a very slowly growing function related to a functional inverse of Ackermann's function. This class of examples shows that an algorithm given in [12] for computing functions defined on paths in trees is optimum to within a constant factor.