Response to “Remarks on two new theorems of Date and Fagin”

In [DF92], we present simple conditions, which we now describe, for guaranteeing higher normal forms for relational databases. A key is simple if it consists of a single attribute. We show in [DF92] that if a relation schema is in third normal form (3NF) and every key is simple, then it is in projection-join normal form (sometimes called fifth normal form), the ultimate normal form with respect to projections and joins. We also show in [DF92] that if a relation schema is in Boyce-Codd normal form (BCNF) and some key is simple, then it is in fourth normal form (4NF). These results give the database designer simple sufficient conditions, defined in terms of functional dependencies alone, that guarantee that the schema being designed is automatically in higher normal forms. In [Bu93], Buff gives a nice generalization of the second of these results. He defines a set C of attributes to be a cut of a relation schema if every key of the schema intersects both C and its complement. He proves that if a relation schema is in BCNF and has no cut, then it is in 4NF. Buff’s Theorem is an immediate consequence of the lemma (Lemma 5.1) that we used in [DF92] to prove our second result. Therefore, we should have discovered it (but we didn’t!