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

Concept Flow - Canonical cover

Start with set of FDs

↓

Step 1: Decompose RHS

↓

Step 2: Remove extraneous attributes from LHS

↓

Step 3: Remove redundant FDs

↓

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.