带谓词的函数依赖：是什么让$g_3$-误差易于计算？
Functional Dependencies with Predicates: What Makes the $g_3$-error Easy to Compute?
AI总结：
本研究分析了不同谓词性质对带谓词的函数依赖$g_3$-误差计算复杂度的影响，证明对称性加传递性可保证其多项式时间可计算，缺一则变为NP完全问题。
摘要：
函数依赖（FDs）的概念可被数据科学家和领域专家用于将背景知识与数据进行比对。为了克服经典的、过于严格的函数依赖满足性要求，可以用更具意义的二元谓词替换相等关系，并使用诸如$g_3$-误差之类的覆盖度度量来估计一条函数依赖与数据的匹配程度。已知在使用相等关系时，$g_3$-误差可以在多项式时间内计算，但遗憾的是，当转而依赖更通用的谓词时，该问题会变为NP完全问题。然而，目前尚未有研究分析哪类谓词或哪些性质会改变该问题的复杂度，尤其是从相等关系过渡到更通用谓词的过程中。\n在本研究中，我们开展了此类分析。我们聚焦于常用谓词的性质，例如相等关系、相似关系以及偏序关系，涉及的性质包括：自反性、传递性、对称性和反对称性。我们证明，对称性与传递性共同作用足以保证$g_3$-误差可在多项式时间内计算。但只要去掉其中任意一个性质，该问题就会变为NP完全问题。
英文摘要：
The notion of functional dependencies (FDs) can be used by data scientists and domain experts to confront background knowledge against data. To overcome the classical, too restrictive, satisfaction of FDs, it is possible to replace equality with more meaningful binary predicates, and use a coverage measure such as the $g_3$-error to estimate the degree to which a FD matches the data. It is known that the $g_3$-error can be computed in polynomial time if equality is used, but unfortunately, the problem becomes NP-complete when relying on more general predicates instead. However, there has been no analysis of which class of predicates or which properties alter the complexity of the problem, especially when going from equality to more general predicates. In this work, we provide such an analysis. We focus on the properties of commonly used predicates such as equality, similarity relations, and partial orders. These properties are: reflexivity, transitivity, symmetry, and antisymmetry. We show that symmetry and transitivity together are sufficient to guarantee that the $g_3$-error can be computed in polynomial time. However, dropping either of them makes the problem NP-complete.
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