*Fourth Normal Form*

The redundancy that we found in "Multivalued Dependencies" to be caused by MVD's can be removed if we use these dependencies in a new decomposition algorithm for relations. In this section we shall introduce a new normal form, called "fourth normal form". In this normal form, all "nontrivial" (in a sense to be defined below) MVD's are eliminated, as are all FD's that violate BCNF. Consequently, the decomposed relations have neither the redundancy from FD's that we discussed in "Design of Relational Database Schemas / Anomalies" nor the redundancy from MVD's that we discussed in "Multivalued Dependencies".

A MVD for a relation R is nontrivial if:

1. None of the B's is among the A's.

2. Not all the attributes of R are among the A's and B's

The "fourth normal form" condition is basically the BCNF condition, but applied to MVD's instead of FD's. Formally:

● A relation R is in fourth normal form (4NF) if whenever

is a nontrivial MVD, {A_{1},A_{2}, . . . , A_{n}} is a superkey.

That is, if a relation is in 4NF, then every nontrivial MVD is really a FD with a superkey on the left. Note that the concepts of keys and superkeys depend on FD's only; adding MVD's does not change the definition of "key".**Example :** The relation of "Multivalued Dependencies" figure, violates the 4NF condition. For instance,

is a nontrivial MVD, yet name by itself is not a superkey. Actually, the only key for this relation is all the attributes.

Fourth normal form is truly a generalization of BCNF. Recall from "Reasoning About Multivalued Dependencies" that every FD is also a MVD. Therefore, every BCNF violation is also a 4NF violation. Put another way, every relation that is in 4NF is therefore in BCNF.

On the other hand, there are some relations that are in BCNF but not 4NF. "Multivalued Dependencies" figure is a good example. The only key for this relation is all five attributes, and there are no nontrivial FD's. Thus it is surely in BCNF. However, as we observed in above example, it is not in 4NF.

### Tags

- algorithm
- superkey
- key
- Relationships Among Normal Forms
- Decomposition into Fourth Normal Form
- Recovering Information from a Decomposition
- Decomposition into BCNF
- Why the Closure Algorithm Works
- Superkeys
- Keys of Relations
- Summary of The Entity-Relationship Data Model
- Weak Entity Sets
- Representing Keys in the E/R Model