Charting the completeness frontier of inference systems for multivalued dependencies

The implication of multivalued dependencies in relational databases has originally been defined in the context of some fixed finite universe. While axiomatisability and implication problems have been intensely studied with respect to this notion almost no research has been devoted towards the alternative notion of implication in which the underlying universe of attributes is left undetermined. Based on a set of common inference rules we establish all axiomatisations in undetermined universes, and all axiomatisations in fixed universes that indicate the role of the complementation rule as a means of database normalisation. This characterises the expressiveness of several incomplete sets of inference rules. We also establish relationships between axiomatisations in fixed and undetermined universes, and study the time complexity of the implication problem in undetermined universes. The results of this paper establish a foundation for reasoning about multivalued dependencies without the assumption of a fixed underlying universe.