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.

翻译：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对照。为克服传统上对FDs过于严格的满足条件，可将相等性替换为更有意义的二元谓词，并采用诸如$g_3$-误差的覆盖度量来评估FD与数据的匹配程度。已知在采用相等性时，$g_3$-误差可在多项式时间内计算，但遗憾的是，当依赖更通用的谓词时，问题变为NP完全。然而，尚未有研究分析哪类谓词或哪些属性会改变问题的复杂度，特别是从相等性过渡到更通用谓词的过程中。本文提供了此类分析。我们聚焦于常用谓词（如相等性、相似关系和偏序）的属性，包括自反性、传递性、对称性和反对称性。研究显示，对称性与传递性共同足以保证$g_3$-误差可在多项式时间内计算，但缺失其中任意一个属性都会导致问题变为NP完全。