On the Logical Implication of Multivalued Dependencies with Null Values

The implication of multivalued dependencies (MVDs) in relational databases has originally been defined in the context of some fixed finite universe (Fagin 1977, Zaniolo 1976). While axiomatisability, implication problem and many design 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 (Biskup 1980). A milestone in the advancement of database systems was the permission of null values in databases. In particular, many achievements on MVDs have been extended to encompass incomplete information. Multivalued dependencies with null values (NMVDs) were defined and axiomatised in (Lien 1982). The definition of NMVDs is again based on a fixed underlying universe of attributes, and any complete set of inference rules requires therefore some version of the complementation rule. In this paper we show that the axiomatisation in (Lien 1982) does not reflect the fact that the complementation rule is merely a means to achieve database normalisation. Moreover, we provide an alternative axiomatisation for NMVDs that does reflect this property. We also suggest an alternative notion for the implication of NMVDs in which the underlying universe is left undetermined, and propose several sound and complete sets of inference rules for this notion. Moreover, a correspondence between (minimal) axiomatisations in fixed universes that do reflect the property of complementation and (minimal) axiomatisations in undetermined universes is shown.