Fourth Normal Form (4NF)

Definition

A relation R is in Fourth Normal Form (4NF) if, for every non-trivial multivalued dependency X →→ Y in R, the determinant X is a superkey of R. Fourth Normal Form builds on BCNF by additionally eliminating problems caused by multivalued dependencies (MVDs).

A relation can be in BCNF (no problematic functional dependencies) yet still contain significant redundancy caused by independent multi-valued facts stored in the same table.

Multivalued Dependency Recap

X →→ Y means: for a given X value, the set of Y values is determined independently of the other attributes (Z = R - X - Y).

Formal definition: In R(X, Y, Z), X →→ Y holds if whenever two tuples t1 and t2 agree on X, there exist tuples t3 and t4 such that:

• t3[X] = t1[X], t3[Y] = t1[Y], t3[Z] = t2[Z]
• t4[X] = t2[X], t4[Y] = t2[Y], t4[Z] = t1[Z]

In practice, this means: if you swap the Y values between any two tuples with the same X, the resulting tuples must also be in the relation.

4NF Decomposition

To convert a relation violating 4NF, decompose it so each MVD resides in its own binary relation:

4NF Decomposition Algorithm

Another Example — Teacher Subjects and Hobbies

When an MVD Does NOT Violate 4NF

If X is a superkey, the MVD is trivially satisfied and does not cause redundancy:

Detecting MVDs

Unlike functional dependencies, MVDs are harder to detect from data alone. You need to understand the business semantics:

Comparison: BCNF, 4NF, 5NF

Lossless Property of 4NF Decomposition

The decomposition using MVD X →→ Y is always lossless:

Summary

• 4NF addresses redundancy from multivalued dependencies that BCNF cannot detect
• An MVD X →→ Y means Y values are determined independently of other attributes given X
• 4NF requires every non-trivial MVD to have a superkey as its determinant
• Decomposition splits independent multi-valued facts into separate tables
• The result is always a lossless decomposition with dramatically less redundancy