Hypergraph Horn functions

TL;DR

This paper introduces hypergraph Horn functions, a new subclass of Boolean functions generalizing matroids and equivalence relations, with characterizations and polynomial-time algorithms for recognition and key realization.

Contribution

It defines hypergraph Horn functions, explores their properties, and provides polynomial-time algorithms for recognition and key realization tasks.

Findings

Recognition of hypergraph Horn functions is polynomial-time decidable.

Key realization problem can be solved in polynomial time.

Impicate sets can be generated with polynomial delay.

Abstract

Horn functions form a subclass of Boolean functions possessing interesting structural and computational properties. These functions play a fundamental role in algebra, artificial intelligence, combinatorics, computer science, database theory, and logic. In the present paper, we introduce the subclass of hypergraph Horn functions that generalizes matroids and equivalence relations. We provide multiple characterizations of hypergraph Horn functions in terms of implicate-duality and the closure operator, which are respectively regarded as generalizations of matroid duality and Mac Lane-Steinitz exchange property of matroid closure. We also study algorithmic issues on hypergraph Horn functions, and show that the recognition problem (i.e., deciding if a given definite Horn CNF represents a hypergraph Horn function) and key realization (i.e., deciding if a given hypergraph is realized as a key set by a hypergraph Horn function) can be done in polynomial time, while implicate sets can be generated with polynomial delay.