On Inferences of Full Hierarchical Dependencies

Full hierarchical dependencies (FHDs) constitute a large class of relational dependencies. A relation exhibits an FHD precisely when it can be decomposed into at least two of its projections without loss of information. Therefore, FHDs generalise multivalued dependencies (MVDs) in which case the number of these projections is precisely two. The implication of FHDs has been defined in the context of some fixed finite universe. This paper identifies a sound and complete set of inference rules for the implication of FHDs. This axiomatisation is very reminiscent of that for MVDs. Then, an alternative notion of FHD implication is introduced in which the underlying set of attributes is left undetermined. The main result proposes a finite axiomatisation for FHD implication in undetermined universes. Moreover, the result clarifies the role of the complementation rule as a mere means of database normalisation. In fact, an axiomatisation for FHD implication in fixed universes is proposed which allows to infer any FHDs either without using the complementation rule at all or only in the very last step of the inference. This also characterises the expressiveness of an incomplete set of inference rules in fixed universes. The results extend previous work on MVDs by Biskup.