Canonical Cover in DBMS

When working with a large set of functional dependencies (FDs) in a database, the dependency set may contain redundant dependencies or unnecessary attributes. This can make database design and dependency analysis more complicated than necessary.

A canonical cover provides a simplified version of a functional dependency set while preserving all the dependencies implied by the original set. In other words, it removes unnecessary elements without changing the meaning of the dependency set.

The main goal of finding a canonical cover is to obtain a smaller and easier-to-manage set of functional dependencies that is equivalent to the original set.

Understanding Canonical Cover

An attribute in a functional dependency is called extraneous if removing it does not change the implication represented by the functional dependency set.

Similarly, an entire functional dependency is redundant if it can be removed without changing the closure or the overall implications of the dependency set.

Definition

A canonical cover, also known as a minimal cover or irreducible cover, is a minimal set of functional dependencies that is equivalent to the original set of functional dependencies.

A canonical cover generally satisfies these conditions:

Every functional dependency has a single attribute on its right-hand side.
No attribute on the left-hand side of a dependency is extraneous.
No functional dependency is redundant.

Therefore, the canonical cover preserves the same functional dependency implications while eliminating unnecessary components.

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Why Do We Need a Canonical Cover?

A functional dependency set can contain several dependencies that are unnecessary because they can already be derived from other dependencies.

Finding a canonical cover helps us:

Reduce the number of functional dependencies.
Remove unnecessary attributes.
Simplify database design and normalization.
Make dependency analysis easier.
Reduce the complexity of checking functional dependencies.

Syntax of Functional Dependencies

Suppose we have two functional dependencies with the same left-hand side:

A → B
A → C

They can be combined into a single dependency:

A → BC

Similarly, a dependency with multiple attributes on the right-hand side can be decomposed using the decomposition rule.

A → BC

becomes:

A → B
A → C

Steps to Find the Canonical Cover

The canonical cover is obtained by systematically removing redundant dependencies and extraneous attributes while preserving equivalence with the original functional dependency set.

Step 1: Decompose Functional Dependencies

First, split every functional dependency that contains multiple attributes on the right-hand side.

For example:

A → BC

is decomposed into:

A → B
A → C

Step 2: Remove Extraneous Attributes from the Left-Hand Side

An attribute is extraneous on the left-hand side if it can be removed without changing the implications of the dependency set.

Suppose we have:

AB → C

To check whether A is extraneous, temporarily remove A and test whether:

B⁺

under the appropriate dependency set contains C.

If C can still be derived, then A is extraneous and the dependency can be reduced to:

B → C

The same process can be performed to check whether B is extraneous.

Step 3: Remove Extraneous Attributes from the Right-Hand Side

After decomposing the dependencies, each right-hand side normally contains a single attribute. If a dependency has not yet been decomposed, an attribute on its right-hand side can be tested for extraneousness.

For example:

A → BC

To check whether B is unnecessary, temporarily consider:

A → C

and determine whether B can still be derived from A using the remaining dependencies.

If B can be derived, then B is extraneous.

Step 4: Remove Redundant Functional Dependencies

Next, check whether any complete functional dependency can be removed.

For each dependency:

X → A

temporarily remove it from the dependency set and compute the closure of X using the remaining dependencies.

If A is still present in X⁺, the dependency is redundant and can be removed.

If A cannot be derived, the dependency must be retained.

Step 5: Verify the Canonical Cover

After completing the above steps, verify that:

Every right-hand side contains a single attribute.
No left-hand-side attribute is extraneous.
No functional dependency is redundant.
The resulting dependency set is equivalent to the original set.

Example 1: Canonical Cover

Consider the following relation and functional dependencies:

R(A, B, C, D)

F = {
    B → A,
    AD → BC,
    C → ABD
}

After decomposing the dependencies and removing extraneous attributes and redundant dependencies, a canonical cover can be represented as:

F_c = {
    B → A,
    AD → C,
    C → B,
    C → D
}

The dependencies are reduced while preserving the implications of the original functional dependency set.

Example 2: Canonical Cover

Consider:

R(W, X, Y, Z)

F = {
    W → X,
    Y → X,
    Z → WXY,
    WY → Z
}

First decompose:

W → X
Y → X
Z → W
Z → X
Z → Y
WY → Z

Since Z determines W and W determines X, the dependency Z → X is redundant.

Therefore, a canonical cover is:

F_c = {
    W → X,
    Y → X,
    Z → W,
    Z → Y,
    WY → Z
}

Example 3: Canonical Cover

Consider the following dependency set:

R(V, W, X, Y, Z)

F = {
    V → W,
    VW → X,
    Y → VXZ
}

Decomposing the dependencies gives:

V → W
VW → X
Y → V
Y → X
Y → Z

Since V already determines W, the dependency VW → X can be reduced to:

V → X

Also, because Y determines V and V determines W and X, the dependency Y → X is redundant.

Thus, one valid canonical cover is:

F_c = {
    V → W,
    V → X,
    Y → V,
    Y → Z
}

Canonical Cover Algorithm

The standard procedure for finding a canonical cover can be summarized as follows:

Decompose every functional dependency so that each dependency has a single attribute on the right-hand side.
Check every attribute on the left-hand side for extraneousness.
Remove any extraneous left-hand-side attributes.
Check every functional dependency for redundancy.
Remove dependencies that can be derived from the remaining dependencies.
Repeat the checks if necessary until no further reduction is possible.
Verify that the resulting set is equivalent to the original functional dependency set.

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Canonical Cover vs Functional Dependency Set

Feature	Original FD Set	Canonical Cover
Redundant dependencies	May contain them	Does not contain them
Extraneous attributes	May contain them	Removed
Right-hand side	May contain multiple attributes	Single attribute
Size	May be larger	Minimal
Implications	Original implications	Equivalent implications

Key Points to Remember

A canonical cover is a minimal equivalent set of functional dependencies.
It is also commonly called a minimal cover.
Right-hand sides are decomposed into individual attributes.
Extraneous attributes are removed from functional dependencies.
Redundant functional dependencies are eliminated.
The canonical cover must preserve the same functional dependency implications as the original set.
Closure computation is an important technique for testing extraneous attributes and redundant dependencies.

Conclusion

The canonical cover in DBMS is an important concept in functional dependency analysis and database normalization. It simplifies a large dependency set by removing redundant functional dependencies and extraneous attributes while preserving the original implications.

By decomposing dependencies, checking for extraneous attributes, removing redundant dependencies, and verifying equivalence, we can obtain a compact and manageable representation of the original functional dependency set.

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Canonical Cover in DBMS – A Complete Guide