To minimize the number of functional dependencies that need to be tested
in case of an update we may restrict to a canonical cover .
A canonical cover for is a set of dependencies such that logically
implies all dependencies in , and vice versa.
must also have the following properties:
Every functional dependency in
contains no extraneous attributes in (ones that can be
removed from without changing ).
So is extraneous in if and
logically implies .
Every functional dependency in
contains no extraneous attributes in (ones that can be
removed from without changing ).
So is extraneous in if and
logically implies .
Each left side of a functional dependency in is unique.
That is there are no two dependencies
and in such that
.
To compute a canonical cover for ,
Use the union rule to replace dependencies of the form
and
with
.
Test each functional dependency to see
if there is an extraneous attribute in .
Test each functional dependency to see
if there is an extraneous attribute in .
Continue until there are no changes occurring in the loop.
An example: for the relational scheme , and the set of
functional dependencies
we will compute .
We have two dependencies with the same left hand side:
We can replace these two with just A BC .
is extraneous in AB C because
B C logically implies AB C .
Then our set is
We still have an extraneous attribute on the right-hand side
of the first dependency.
is extraneous in A BC because
A B and B C logically imply that
A BC .