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

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

References

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Database theory
Mathematical concepts
Database algorithms