Dualization Problem over the Product of Chains: Asymptotic Estimates for the Number of Solutions

Abstract

A key intractable problem in logical data analysis, namely, dualization over the product of partial orders, is considered. The important special case where each order is a chain is studied. If the cardinality of each chain is equal to two, then the considered problem is to construct a reduced disjunctive normal form of a monotone Boolean function defined by a conjunctive normal form, which is equivalent to the enumeration of irreducible coverings of a Boolean matrix. The asymptotics of the typical number of irreducible coverings is known in the case where the number of rows in the Boolean matrix has a lower order of growth than the number of columns. In this paper, a similar result is obtained for dualization over the product of chains when the cardinality of each chain is higher than two. Deriving such asymptotic estimates is a technically complicated task, and they are required, in particular, for proving the existence of asymptotically optimal algorithms for the problem of monotone dualization and its generalizations.