Matroid Horn functions

TL;DR

This paper introduces matroid Horn functions, a subclass of hypergraph Horn functions, and explores their structural properties, characterizations, and the complexity of minimizing their CNF representations, especially for binary and uniform matroids.

Contribution

The paper defines matroid Horn functions, provides multiple characterizations, and analyzes the Boolean minimization problem focusing on circuit-based size measures.

Findings

Optimal representation size for binary matroids determined.

Lower and upper bounds established for uniform matroids.

Connection identified between uniform matroid minimization and Turán systems.

Abstract

Hypergraph Horn functions were introduced as a subclass of Horn functions that can be represented by a collection of circular implication rules. These functions possess distinguished structural and computational properties. In particular, their characterizations in terms of implicate-duality and the closure operator provide extensions of matroid duality and the Mac Lane-Steinitz exchange property of matroid closure, respectively. In the present paper, we introduce a subclass of hypergraph Horn functions that we call matroid Horn functions. We provide multiple characterizations of matroid Horn functions in terms of their canonical and complete CNF representations. We also study the Boolean minimization problem for this class, where the goal is to find a minimum size representation of a matroid Horn function given by a CNF representation. While there are various ways to measure the size of a CNF, we focus on the number of circuits and circuit clauses. We determine the size of an optimal representation for binary matroids, and give lower and upper bounds in the uniform case. For uniform matroids, we show a strong connection between our problem and Tur\'an systems that might be of independent combinatorial interest.