The Price of Evolution in Temporal Databases

Temporal Functional Dependencies (TFDs for short) are functional dependencies that predicate on temporal databases characterized by a special temporal dimension called valid time (VT). In [1] Combi et al. proposed a uniform framework that subsumes many of the TFDs proposed in literature and, by the combination of them, allow us to express finer constraints. Some interesting constraints are the Temporally Mixed Functional Dependencies (TMFD for short) that allow one to write constraints on the evolution of the data in the database. The problem of checking a TMFD against an instance of a temporal schema is polynomial. We will show that when approximation comes into play (i.e., we look for TMFD holding for almost all database tuples) the problem turns out to be NP-Complete. Moreover we introduce a type of association rules build over TMFD called Temporally Mixed Association Rule (TMAR). We prove that verifying TMAR under approximation is still NP-Complete, by reducing it to a novel problem on directed acyclic graphs.