Canonical cover

A canonical cover $F_{c}$ for F (a set of functional dependencies on a relation scheme) is a set of dependencies such that F logically implies all dependencies in $F_{c}$ , and $F_{c}$ logically implies all dependencies in F.

The set $F_{c}$ has two important properties:

No functional dependency in $F_{c}$ contains an extraneous attribute.
Each left side of a functional dependency in $F_{c}$ is unique. That is, there are no two dependencies $a\to b$ and $c\to d$ in $F_{c}$ such that $a=c$ .

A canonical cover is not unique for a given set of functional dependencies, therefore one set F can have multiple covers $F_{c}$ .

Algorithm for computing a canonical cover [1]

$F_{c}=F$
Repeat:
1. Use the union rule to replace any dependencies in $F_{c}$ of the form $a\to b$ and $a\to d$ with $a\to bd$ ..
2. Find a functional dependency in $F_{c}$ with an extraneous attribute and delete it from $F_{c}$
... until $F_{c}$ does not change

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References

Database system concepts by Abraham Silberschatz et al

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[1] Database system concepts by Abraham Silberschatz et al