The Implication Problem for Functional and Inclusion Dependencies is Undecidable

The implication problem for a class of dependencies is the following: given a finite set of dependencies, determine if they logically imply another dependency. We show that the implication problem is undecidable for the class of functional and inclusion dependencies. This holds true even if the inclusion dependencies are restricted to be binary. It may be noted that the implication problem is known to be decidable for functional and unary inclusion dependencies and also for inclusion dependencies without functional dependencies.