Hesse pencil

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:

1. No functional dependency in $F_c$ contains an extraneous attribute.
2. 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 $a=c$.

Algorithm for computing a canonical cover

1. $F_c=F$
2. 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 $F_c$
3. ... until $F_c$ does not change

References

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