Horn functions and their DNFs

Abstract

In this paper we introduce the concept of Horn functions and investigate the structure of their disjunctive normal forms. Horn functions arise in a variety of applications, including in particular the analysis of production rule knowledge bases of propositional expert systems. The goal of this paper is to reduce the study of the irredundant prime disjunctive normal forms of Horn functions to the study of the irredundant prime disjunctive normal forms of pure Horn functions. This reduction is achieved by proving that every prime irredundant disjunctive normal form of a Horn function consists of a prime irredundant disjunctive normal form of its “pure Horn component”, and of a “positive restriction” of the function. We provide a constructive characterization of all the positive restrictions, and present an efficient algorithm for decomposing any Horn function into its pure Horn component and its positive restriction. Finally, we reduce in quadratic time the minimization of a Horn function to the minimization of its pure Horn component.