What is the minimal set of functional dependencies or canonical cover of FD?

A minimal cover of a set of functional dependencies (FD) E is a minimal set of dependencies F that is equivalent to E.

The formal definition is: A set of FD F to be minimal if it satisfies the following conditions −

• Every dependency in F has a single attribute for its right-hand side.

• We cannot replace any dependency X->A in F with a dependency Y->A, where Y is a proper subset of X, and still have a set of dependencies that is equivalent to F.

• We cannot remove any dependency from F and still have a set of dependencies that are equivalent to F.

Canonical cover is called minimal cover which is called the minimum set of FDs. A set of FD FC is called canonical cover of F if each FD in FC is a −

• Simple FD.
• Left reduced FD.
• Non-redundant FD.

Simple FD − X->Y is a simple FD if Y is a single attribute.

Left reduced FD − X->Y is a left reduced FD if there are no extraneous attributes in X. {extraneous attributes: Let XA->Y then, A is a extraneous attribute if X_>Y}

Non-redundant FD − X->Y is a Non-redundant FD if it cannot be derived from F- {X->y}.

Example

Consider an example to find canonical cover of F.

The given functional dependencies are as follows −

A -> BC

B -> C

A -> B

AB -> C

• Minimal cover: The minimal cover is the set of FDs which are equivalent to the given FDs.

• Canonical cover: In canonical cover, the LHS (Left Hand Side) must be unique.

First of all, we will find the minimal cover and then the canonical cover.

First step − Convert RHS attribute into singleton attribute.

A -> B

A -> C

B -> C

A -> B

AB -> C

Second step − Remove the extra LHS attribute

Find the closure of A.

A+ = {A, B, C}

So, AB -> C can be converted into A -> C

A -> B

A -> C

B -> C

A -> B

A -> C

Third step − Remove the redundant FDs.

A -> B

B -> C

Now, we will convert the above set of FDs into canonical cover.

The canonical cover for the above set of FDs will be as follows −

A -> BC

B -> C