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.
翻译:Horn 函数是作为Horn 函数的亚类引入的,它可以通过一系列循环隐含规则来体现,这些功能具有不同的结构性和计算性特性,特别是,这些功能在隐含性方面的特性和关闭操作员分别提供了超大型双体功能和Mac Lane-Steinitz 交换机密关闭特性的延伸。在本文件中,我们引入了一个超大型功能的亚类,我们称之为“机器人合角”功能。我们以光学和完整的 CNF 表示方式提供了对合角 功能的多重定性。我们还研究了这一类的Boolean 最小化问题,目的是找到由CNF 代表提供的配角功能的最小规模。虽然我们有多种方法来衡量CNF的大小,但我们侧重于电路和电路条款的数量。我们决定了双胞胎机的最佳代表规模,并在制服中给出下限和上限。对于统一的配型机器人,我们显示了我们的问题和Tur\'an系统之间的密切联系。