AI 中文总结
研究有限集上有限闭包系统两种规范表示是否等价的问题，证明该等价性测试是coNP完全的，此结论对相关的特征模型识别、FD关系等价性等问题有重要影响，还排除了一些输出多项式算法。
AI 中文摘要
有限集\(U\)上的有限闭包系统是包含\(U\)且在交运算下封闭的子集族。它可以通过两种基本方式指定：蕴含规范列出规则\(A \to b\)，由满足每个规则的所有\(X \subseteq U\)组成；交规范列出子集\(M_1, \ldots, M_t\)，由该列表子族的所有交组成。我们研究这两种规范是否定义相同的族。这个问题以多种形式存在约三十年未解决：Khardon（1995）表明在Horn公式与其特征模型之间转换等同于判定所提议特征模型列表的完备性，确切复杂度未知；在ISAAC 2025上，从蕴含关系枚举不可约闭集仍被认为“广泛开放”，即使对于无环凸几何；形式概念分析中的伪意图和Duquenne - Guigues基以及数据库中的函数依赖和Armstrong关系也有密切相关问题。我们证明等价性测试是coNP完全的。对于前提大小至多为三的无环蕴含关系，即使每个列出的子集正确且不可移除，否则会改变列表生成的闭包系统，判定所需集合是否缺失也很难。除非\(P = NP\)，即使对于无环凸几何，也无法在输入加上总输出大小的多项式时间内生成完整规范列表。通过标准对应，该定理使特征模型识别和FD关系等价性成为coNP完全问题，并排除了用于Horn特征模型、给定形式背景的所有伪意图（等价于其Duquenne - Guigues基）以及最小函数依赖覆盖前提的输出多项式算法。
英文摘要
A finite closure system on a finite set $U$ is a family of subsets that contains $U$ and is closed under intersections. It can be specified in two elementary ways: an implicational specification lists rules $A \to b$ and consists of all $X \subseteq U$ satisfying every rule, while an intersection specification lists subsets $M_1, \ldots, M_t$ and consists of all intersections of subfamilies of that list. We ask whether one specification of each kind defines the same family. This question has remained open in several guises for about thirty years: Khardon (1995) showed that translating between Horn formulas and their characteristic models is equivalent to deciding completeness of a proposed list of characteristic models, leaving the exact complexity open; at ISAAC 2025, enumerating irreducible closed sets from implications was still described as "widely open," even for acyclic convex geometries; closely related questions concerned pseudo-intents and the Duquenne-Guigues basis in Formal Concept Analysis, and functional dependencies and Armstrong relations in databases. We prove that the equivalence test is coNP-complete. Hardness holds for acyclic implications with premises of size at most three, even when every listed subset is correct and none can be removed without changing the closure system generated by the list; the hard part is deciding whether a required set is missing. Unless $P = NP$, the complete canonical lists cannot be generated in time polynomial in the input plus the total output size, even for acyclic convex geometries. Through standard correspondences, the theorem makes Characteristic Models Identification and FD-Relation Equivalence coNP-complete and rules out output-polynomial algorithms for Horn characteristic models, all pseudo-intents of a given formal context (equivalently, its Duquenne-Guigues basis), and premises of minimum functional-dependency covers.
CommentsAlso available at Zenodo, doi:10.5281/zenodo.21431468
