Asymptotic estimates for the number of solutions of the dualization problem and its generalizations

Abstract

Asymptotic estimates for the typical number of irreducible coverings and the typical length of an irreducible covering of a Boolean matrix are obtained in the case when the number of rows is no less than the number of columns. As a consequence, asymptotic estimates are obtained for the typical number of maximal conjunctions and the typical rank of a maximal conjunction of a monotone Boolean function of variables defined by a conjunctive normal form of clauses. Similar estimates are given for the number of irredundant coverings and the length of an irredundant covering of an integer matrix (for the number of maximal conjunctions and the rank of a maximal conjunction of a two-valued logical function defined by its zero set). Results obtained previously in this area are overviewed.