Appropriate Reasoning about Data Dependencies in Fixed and Undetermined Universes

We study inference systems for the combined class of functional and full hierarchical dependencies in relational databases. Two notions of implication are considered: the original version in which the underlying set of attributes is fixed, and the alternative notion in which this set is left undetermined. The first main result establishes a finite axiomatisation in fixed universes which clarifies the role of the complementation rule in the combined setting. In fact, we identify inference systems that are appropriate in the following sense: full hierarchical dependencies can be inferred without use of the complementation rule at all or with a single application of the complementation rule at the final step of the inference; and functional dependencies can be inferred without any application of the complementation rule. The second main result establishes a finite axiomatisation for functional and full hierarchical dependencies in undetermined universes.