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The notion of functional dependencies (FDs) can be used by data scientistsand domain experts to confront background knowledge against data. To overcomethe classical, too restrictive, satisfaction of FDs, it is possible to replaceequality with more meaningful binary predicates, and use a coverage measuresuch 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 equalityis used, but unfortunately, the problem becomes NP-complete when relying onmore general predicates instead. However, there has been no analysis of whichclass 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 ofcommonly used predicates such as equality, similarity relations, and partialorders. These properties are: reflexivity, transitivity, symmetry, andantisymmetry. We show that symmetry and transitivity together are sufficient toguarantee that the $g_3$-error can be computed in polynomial time. However,dropping either of them makes the problem NP-complete.

Functional Dependencies with Predicates: What Makes the $g_3$-error Easy to Compute?