A Generalisation of Entity and Referential Integrity in Relational Databases

Entity and referential integrity are the most fundamental constraints that any relational database should satisfy. We re-examine these fundamental constraints in the context of incomplete relations, which may have null values of the types “value exists but is unknown” and “value does not exist”. We argue that in practice the restrictions that these constraints impose on the occurrences of null values in relations are too strict. We justify a generalisation of the said constraints wherein we use key families, which are collections of attribute sets of a relation schema, rather than keys, and foreign key families which are collections of pairs of attribute sets of two relation schemas, rather than foreign keys. Intuitively, a key family is satisfied in an incomplete relation if each one of its tuples is uniquely identifiable on the union of the attribute sets of the key family, in all possible worlds of the incomplete relation, and, in addition, is distinguishable from all the other tuples in the incomplete relation by its nonnull values on some element in the key family. Our proposal can be viewed as an extension of Thalheim’s key set, which only deals with null values of type “value exists but is unknown”. The intuition behind the satisfaction of a foreign key family in an incomplete database is that each pair (Fi,Ki) of attribute sets in the foreign key family takes the foreign key attribute values over Fi of a tuple in one incomplete relation referencing the key attribute values over Ki of a tuple in another incomplete relation. Such referencing is defined only in the case when the foreign key attribute values do not have any null values of the type “value does not exist”; we insist that the referencing be defined for at least one such pair. We also investigate some combinatorial properties of key families, and show that they are comparable to the standard combinatorial properties of keys.