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-bused. These results assume that infinite databases are allowed. If only finife 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).

Conjunctive queries are generalized so that inequality comparisons can be made between elements of the query. Algorithms for containment and equivalence of such "inequality queries" are given, under the assumption that the data domains are dense and totally ordered. In general, containment does not imply the existence of homomorphisms (containment mappings), but the homomorphism property does exist for subclasses of inequality queries. A minimization algorithm is defined using the equivalence algorithm. It is first shown that the constants appearing in a query can be divided into "essential" and "nonessential" subgroups. The minimum query can be nondetenninistically guessed using only the essential constants of the original query.

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This paper deals with the problem of determining which of a certain class of constraints hold on a given relational algebra expression where the base relations come from a given schema. The class of constraints includes functional dependencies, equality of domains, and constancy of domains. The relational algebra consists of projection, selection, restriction, cross product, union, and difference. The problem as given is undecidable, but if set difference is removed from the algebra, there is a solution. Operators specifying a closure function (similar to functional dependency closure on one relation) are defined; these will generate exactly the set of constraints valid on the given relational algebra expression. We prove that the operators are sound and complete.