在分级凸几何的表示之间转换
Translating between the representations of a ranked convex geometry
摘要：
每个闭包系统都可以用蕴涵基或交不可约元的集合来表示。在Horn逻辑中，这些分别被称为Horn表达式和特征模型。本文研究了非循环凸几何中两种表示之间的转换问题。令人惊讶的是，我们发现，在这种情况下的问题已经比分配格的二元论，一个众所周知的超图二元论问题的存在的输出准多项式时间算法是开放的推广是困难的。根据这个结果，我们考虑一个适当的子类的非循环凸几何，即排名凸几何，因为那些承认一个排名的蕴涵基类似于排名偏序集。对于这个类，我们提供了输出准多项式时间算法的基础上超图对偶之间的转换表示。这提高了对一个长期未决问题的理解。
英文摘要：
It is well known that every closure system can be represented by an implicational base, or by the set of its meet-irreducible elements. In Horn logic, these are respectively known as the Horn expressions and the characteristic models. In this paper, we consider the problem of translating between the two representations in acyclic convex geometries. Quite surprisingly, we show that the problem in this context is already harder than the dualization in distributive lattices, a generalization of the well-known hypergraph dualization problem for which the existence of an output quasi-polynomial time algorithm is open. In light of this result, we consider a proper subclass of acyclic convex geometries, namely ranked convex geometries, as those that admit a ranked implicational base analogous to that of ranked posets. For this class, we provide output quasi-polynomial time algorithms based on hypergraph dualization for translating between the two representations. This improves the understanding of a long-standing open problem.
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