Testing containment of conjunctive queries under functional and inclusion dependencies

We consider the problem of optimizing conjunctive queries in the presence of inclusion and functional dependencies. We show that the problem of containment (and hence those of equivalence and non-minimality) is in NP when either (a) there are no functional dependencies or (b) the set of dependencies is what we call key-based. These results assume that infinite databases are allowed. If only finite databases are allowed, new containments may arise, as we illustrate by an example. We also prove a "compactness" theorem that shows that no such examples can exist for case (b).