Boyce-Codd Normal Form (BCNF)

Boyce-Codd Normal Form (BCNF) is a stricter version of the Third Normal Form (3NF). It addresses certain anomalies that can still occur even in 3NF, particularly when there are multiple overlapping candidate keys. A relation is in BCNF if, for every non-trivial functional dependency X → Y, X is a superkey.

Definition of BCNF

A relation schema R is in BCNF if and only if for every functional dependency X → Y in R, at least one of the following is true:

X → Y is a trivial functional dependency (i.e., Y is a subset of X).
X is a superkey of R.

In simpler terms, for any dependency, the determinant (the left side of the arrow, X) must be a candidate key (or part of one).

Key Differences from 3NF

3NF requires that for every non-trivial functional dependency X → Y, either X is a superkey or Y is a prime attribute. BCNF removes the second condition, requiring X to always be a superkey for non-trivial dependencies.

BCNF Dependency: If a relation R is not in BCNF, it must contain a functional dependency X → Y such that X is not a superkey and X → Y is non-trivial.

When BCNF is Necessary

BCNF is particularly important when:

A relation has multiple candidate keys.
These candidate keys overlap significantly.
You need to eliminate all insertion, deletion, and update anomalies related to functional dependencies.

Example

Scenario: Student Course Enrollments

Consider a relation Enrollment with attributes:

StudentID
CourseID
InstructorID
InstructorRoom

Assume the following functional dependencies:

{StudentID, CourseID} → {InstructorID} (A student enrolls in a specific course, which determines the instructor for that enrollment.)
{InstructorID} → {InstructorRoom} (Each instructor is assigned a specific room.)

Candidate keys:

{StudentID, CourseID}

Let's analyze the dependencies:

Dependency 1: {StudentID, CourseID} → {InstructorID}

The determinant {StudentID, CourseID} is a candidate key. This dependency is fine for BCNF.

Dependency 2: {InstructorID} → {InstructorRoom}

The determinant {InstructorID} is not a superkey for the Enrollment relation.
This violates BCNF.

The relation Enrollment is not in BCNF due to dependency 2.

Decomposition to BCNF

To decompose a relation into BCNF, we break down the relation based on the violating functional dependency. For the dependency X → Y where X is not a superkey:

Create a new relation consisting of the attributes in X and Y.
Remove the attributes in Y from the original relation.

Applying this to our example:

The violating dependency is {InstructorID} → {InstructorRoom}.

Create a new relation, say InstructorInfo, with attributes {InstructorID, InstructorRoom}. This relation is now in BCNF.
Remove InstructorRoom from the original Enrollment relation, leaving {StudentID, CourseID, InstructorID}.

The decomposed relations are:

Enrollment (StudentID, CourseID, InstructorID) with dependency {StudentID, CourseID} → {InstructorID}. Here, {StudentID, CourseID} is a superkey.
InstructorInfo (InstructorID, InstructorRoom) with dependency {InstructorID} → {InstructorRoom}. Here, {InstructorID} is a superkey.

Both decomposed relations are now in BCNF.

Benefits of BCNF

Eliminates all data redundancies caused by functional dependencies.
Ensures that all data is stored in one place and logically consistent.
Minimizes the risk of update, insert, and delete anomalies.

Drawbacks of BCNF

The decomposition process might result in more tables than 3NF, potentially increasing the complexity of queries that require joining multiple tables.
Lossless-join decomposition is guaranteed, but dependency preservation is not always guaranteed. This means some functional dependencies might not be expressible within any single decomposed relation.

BCNF is generally the highest level of normalization considered for practical relational database design, providing a strong guarantee against functional dependency anomalies.