The additivity Problem for Data Dependencies in Incomplete Relational Databases

Functional dependencies (FDs) and inclusion dependencies (INDs) are the most fundamental integrity constraints that arise in practice in relational databases. We introduce null functional dependencies (NFDs) and null inclusion dependencies (NINDs) to cater for the situation when a database is incomplete and thus contains null values. If a NFD or NIND is weakly satisfied in a database, then there exists a possible world of this database in which the NFD or NIND is satisfied in the standard way. Additivity is the property of the equivalence of the weak satisfaction of a set of NFDs and NINDs, e, with the individual weak satisfaction of each member of e in the said database. We show that in general satisfaction of NFDs and NINDs is not additive. The problem that arises is: under what conditions is weak satisfaction of NFDs and NINDs additive. We solve this problem for the following cases: when E is a set of NFDs, when E is a set of unary NINDs and when E is a set of NFDs and unary NINDs. We show that, when the set of NINDs is unary, checking whether E is additive can be done in time polynomial in the size of E.

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