Horn functions and submodular boolean functions

Abstract

After providing a simple characterization of Horn functions (i.e., those Boolean functions that have a Horn DNF), we study in detail the special class of submodular functions. Every prime implicant of such a function involves at most one complemented and at most one uncomplemented variable, and based on this we give a one-to-one correspondence between submodular functions and partial preorders (reflexive and transitive binary relations), and in particular between the nondegenerate acyclic submodular functions and the partially ordered sets. There is a one-to-one correspondence between the roots of a submodular function and the ideals of the associated partial preorder. There is also a one-to-one correspondence between the prime implicants of the dual of the submodular function and the maximal antichains of the associated partial preorder. Based on these results, we give graph-theoretic characterizations for all minimum prime DNF representations of a submodular function. The problem of recognizing submodular functions in DNF representation is coNP-complete.