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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
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