Canonical cover in DBMS Theory - Step-by-Step Execution

Concept Flow - Canonical cover

Start with set of FDs

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Step 1: Decompose RHS

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Step 2: Remove extraneous attributes from LHS

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Step 3: Remove redundant FDs

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Result: Canonical cover (minimal FD set)

The process starts with a set of functional dependencies (FDs), then breaks down complex dependencies, removes unnecessary parts, and finally removes redundant dependencies to get the simplest equivalent set called the canonical cover.

Execution Sample

DBMS Theory

FDs = {AB->C, A->C, C->D}
Decompose RHS: AB->C stays, A->C stays, C->D stays
Remove extraneous in LHS: Check if B in AB->C is extraneous
Remove redundant FDs: Check if AB->C is redundant
Result: Canonical cover = {A->C, C->D}

This example shows how to simplify a set of functional dependencies to its canonical cover by decomposing, removing extraneous attributes, and eliminating redundant dependencies.

Analysis Table

Step	Action	FD Set	Explanation
1	Start with original FDs	{AB->C, A->C, C->D}	Initial set of functional dependencies
2	Decompose RHS if needed	{AB->C, A->C, C->D}	All RHS have single attributes, no change
3	Check extraneous attribute B in AB->C	{AB->C, A->C, C->D}	Test if A->C holds without B; it does, so B is extraneous
4	Remove B from AB->C	{A->C, A->C, C->D}	AB->C becomes A->C; duplicate A->C exists
5	Remove duplicate A->C	{A->C, C->D}	Duplicates removed
6	Check if any FD is redundant	{A->C, C->D}	Neither A->C nor C->D can be derived from the other, so none removed
7	Final canonical cover	{A->C, C->D}	Minimal equivalent set of FDs

💡 No more extraneous attributes or redundant FDs; canonical cover found

State Tracker

Variable	Start	After Step 3	After Step 4	After Step 5	Final
FD Set	{AB->C, A->C, C->D}	{AB->C, A->C, C->D}	{A->C, A->C, C->D}	{A->C, C->D}	{A->C, C->D}

Key Insights - 3 Insights

Why do we check if attribute B in AB->C is extraneous?

Because if removing B from the left side still allows us to derive the same dependency (A->C), then B is not needed. This is shown in step 3 of the execution_table.

Why do we remove duplicate functional dependencies?

Duplicates do not add new information and make the set unnecessarily large. Removing duplicates simplifies the set, as shown in step 5.

How do we know when to stop removing FDs?

When no FD can be derived from others and no extraneous attributes remain, the set is minimal. This is confirmed in step 6 and the exit_note.

Visual Quiz - 3 Questions

Look at the execution_table at step 3. What is the reason for checking attribute B in AB->C?

A To add more attributes to the FD

B To see if B can be removed without changing the FD

C To check if C is extraneous

D To duplicate the FD

💡 Hint

Refer to step 3 in execution_table where extraneous attribute B is tested

At which step does the FD set become free of duplicates?

A Step 3

B Step 4

C Step 5

D Step 6

💡 Hint

Look at step 5 in execution_table where duplicate A->C is removed

If attribute B was not extraneous in AB->C, how would the canonical cover change?

A It would include AB->C instead of A->C

B It would remove C->D

C It would have duplicate FDs

D It would be empty

💡 Hint

Check variable_tracker and execution_table steps 3 and 4 for attribute removal effects

Concept Snapshot

Canonical cover is a minimal set of functional dependencies.
Steps: Decompose RHS to single attributes.
Remove extraneous attributes from LHS.
Remove redundant FDs.
Result: Simplest equivalent FD set.

Full Transcript

Canonical cover is a simplified set of functional dependencies that represents the same constraints as the original set but with no extra parts. The process starts with the original FDs, breaks down any dependencies with multiple attributes on the right side, removes unnecessary attributes from the left side, and finally removes any redundant dependencies. This results in a minimal set called the canonical cover. For example, if we have FDs {AB->C, A->C, C->D}, we check if B in AB->C is needed. Since A->C already exists, B is extraneous and can be removed. We then remove duplicates and check for redundancy. The final canonical cover is {A->C, C->D}. This process helps in database design to avoid unnecessary complexity.

Practice

1. What is the main purpose of a canonical cover in database design?

easy

A. To simplify and minimize the set of functional dependencies

B. To increase the number of functional dependencies

C. To add redundant dependencies for safety

D. To create more complex database schemas

Solution

1. Step 1: Understand the definition of canonical cover

   A canonical cover is a minimal set of functional dependencies that preserves the original dependencies without redundancy.

2. Step 2: Identify the purpose in database design

   It simplifies the design by removing redundant dependencies and splitting dependencies with multiple attributes on the right side.

3. Final Answer:

   To simplify and minimize the set of functional dependencies -> Option A

4. Quick Check:

   Canonical cover = minimal functional dependencies [OK]

Hint: Canonical cover means minimal and simple dependencies [OK]

Common Mistakes:

• Thinking it adds more dependencies
• Confusing it with normalization
• Assuming it complicates the schema

2. Which of the following is the correct step when creating a canonical cover?

easy

A. Remove extraneous attributes from dependencies

B. Add extra attributes to the left side

C. Combine all dependencies into one

D. Ignore dependencies with single attributes

Solution

1. Step 1: Recall the process of forming a canonical cover

   One key step is to remove extraneous attributes from the left or right side of functional dependencies to simplify them.

2. Step 2: Evaluate the options

   Removing extraneous attributes is correct, while adding extra attributes or ignoring dependencies is incorrect.

3. Final Answer:

   Remove extraneous attributes from dependencies -> Option A

4. Quick Check:

   Remove extraneous attributes = canonical cover step [OK]

Hint: Remove unnecessary parts to simplify dependencies [OK]

Common Mistakes:

• Adding attributes instead of removing
• Merging all dependencies incorrectly
• Ignoring single attribute dependencies

3. Given the functional dependencies:
F = { A -> BC, B -> C, A -> B }
What is the canonical cover of F?

medium

A. { A -> B, A -> C }

B. { A -> BC, B -> C }

C. { A -> B, B -> C }

D. { A -> B, B -> C, A -> C }

Solution

1. Step 1: Split dependencies with multiple attributes on right

   Split A -> BC into A -> B and A -> C.

2. Step 2: Remove redundant dependencies

   Since B -> C exists, and A -> B exists, A -> C is implied and redundant.

3. Final Answer:

   { A -> B, B -> C } -> Option C

4. Quick Check:

   Canonical cover removes redundant dependencies [OK]

Hint: Split right side, then remove implied dependencies [OK]

Common Mistakes:

• Not splitting multi-attribute dependencies
• Keeping redundant dependencies
• Confusing implied dependencies

4. Identify the error in this canonical cover step:
F = { A -> BC, B -> C }
Step: Remove C from A -> BC because B -> C exists.
Is this step correct?

medium

A. No, because C is not extraneous in A -> BC

B. Yes, because C is extraneous in A -> BC

C. Yes, because B -> C covers C

D. No, because B -> C is unrelated

Solution

1. Step 1: Check if C is extraneous in A -> BC

   Since B -> C exists, C can be derived from B, so C is extraneous on the right side of A -> BC.

2. Step 2: Confirm correctness of removal

   Removing C from A -> BC to get A -> B is valid and simplifies the dependencies.

3. Final Answer:

   Yes, because C is extraneous in A -> BC -> Option B

4. Quick Check:

   Extraneous attribute removal is valid [OK]

Hint: If attribute can be derived, it's extraneous [OK]

Common Mistakes:

• Thinking all right-side attributes are essential
• Ignoring dependency chains
• Confusing extraneous with essential attributes

5. You have the dependencies:
F = { AB -> C, A -> B, B -> C }
What is the canonical cover of F?

hard

A. { A -> B, AB -> C }

B. { AB -> C, A -> B }

C. { A -> B, B -> C, AB -> C }

D. { A -> B, B -> C }

Solution

1. Step 1: Check for extraneous attributes on left side

   In AB -> C, check if A or B is extraneous. Since B -> C exists, A is extraneous.

2. Step 2: Simplify AB -> C to B -> C and remove redundant dependencies

   Now F becomes { A -> B, B -> C }. This set is minimal and covers all dependencies.

3. Final Answer:

   { A -> B, B -> C } -> Option D

4. Quick Check:

   Remove extraneous left attributes and redundant dependencies [OK]

Hint: Remove extraneous left attributes to simplify [OK]

Common Mistakes:

• Not checking left side for extraneous attributes
• Keeping redundant dependencies
• Assuming all attributes on left are needed