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 Turán systems that might be of independent combinatorial interest.

超图 Horn 函数作为 Horn 函数的子类引入，可以用循环蕴涵规则的集合表示。这些函数具有独特的结构和计算特性。特别是，它们在隐含对偶性和闭包算子方面的表征分别提供了拟阵对偶性和拟阵闭包的 Mac Lane  –  Steinitz 交换性质的扩展。

在本文中，我们介绍了超图 Horn 函数的一个子类，我们称之为拟阵 Horn函数。我们根据其规范和完整的 CNF 表示提供了拟阵 Horn 函数的多个特征。我们还研究了此类的布尔最小化问题，其目标是找到由 CNF 表示给出的拟阵 Horn 函数的最小尺寸表示。虽然衡量 CNF 大小的方法有多种，但我们重点关注电路和电路子句的数量。我们确定二元拟阵的最佳表示的大小，并给出统一情况下的下限和上限。对于均匀拟阵，我们展示了我们的问题和图兰系统之间的紧密联系，这可能具有独立的组合兴趣。