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:

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 $a=c$.

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. $F_{c}=F$
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

Canonical cover example

In the following example, F_c is the canonical cover of F.

Given the following, we can find the canonical cover: R = (A, B, C, G, H, I), F = {A→BC, B→C, A→B, AB→C}

1. {A→BC, B→C, A→B, AB→C}
2. {A → BC, B →C, AB → C}
3. {A → BC, B → C}
4. {A → B, B →C}

F_c =  {A → B, B →C}

Extraneous attributes

An attribute is extraneous in a functional dependency if its removal from that functional dependency does not alter the closure of any attributes.

Extraneous determinant attributes

Given a set of functional dependencies $F$ and a functional dependency $A\rightarrow B$ in $F$, the attribute $a$ is extraneous in $A$ if $a\subset A$ and any of the functional dependencies in $(F-(A\rightarrow B)\cup {(A-a)\rightarrow B})$ can be implied by $F$ using Armstrong's Axioms.

Using an alternate method, given the set of functional dependencies $F$, and a functional dependency X → A in $F$, attribute Y is extraneous in X if $Y\subseteq X$, and $(X-Y)^{+}\supseteq A$.

For example:

• If F = {A → C, AB → C}, B is extraneous in AB → C because A → C can be inferred even after deleting B. This is true because if A functionally determines C, then AB also functionally determines C.
• If F = {A → D, D → C, AB → C}, B is extraneous in AB → C because {A → D, D → C, AB → C} logically implies A → C.

Extraneous dependent attributes

Given a set of functional dependencies $F$ and a functional dependency $A\rightarrow B$ in $F$, the attribute $a$ is extraneous in $A$ if $a\subset A$ and any of the functional dependencies in $(F-(A\rightarrow B)\cup \{A\rightarrow (B-a)\})$ can be implied by $F$ using Armstrong's axioms.

A dependent attribute of a functional dependency is extraneous if we can remove it without changing the closure of the set of determinant attributes in that functional dependency.

For example:

• If F = {A → C, AB → CD}, C is extraneous in AB → CD because AB → C can be inferred even after deleting C.
• If F = {A → BC, B → C}, C is extraneous in A → BC because A → C can be inferred even after deleting C.