Canonical cover in DBMS Theory - Mini Project: Build & Apply

Canonical Cover for Functional Dependencies

📖 Scenario: You are working with a database design team. They have a set of functional dependencies (FDs) for a relation, and they want to simplify these FDs to a minimal form called the canonical cover. This helps in designing efficient database schemas.

🎯 Goal: Build a step-by-step process to find the canonical cover of a given set of functional dependencies by simplifying and minimizing them.

📋 What You'll Learn

Create a dictionary called fds with given functional dependencies as keys and values

Create a list called attributes containing all attributes involved

Implement a function called remove_extraneous to remove extraneous attributes from the left side of FDs

Implement the final step to combine and minimize the FDs into a canonical cover stored in canonical_cover

💡 Why This Matters

🌍 Real World

Database designers use canonical covers to simplify functional dependencies, which helps in normalizing database schemas and avoiding redundancy.

💼 Career

Understanding canonical covers is essential for roles like database administrator, data engineer, and software developer working with relational databases.

1

Set up the initial functional dependencies

Create a dictionary called fds with these exact functional dependencies: {'AB': 'C', 'A': 'B', 'B': 'C'}.

DBMS Theory

# Create the dictionary fds with the given functional dependencies
# Your code here

Hint

Use a Python dictionary with strings as keys and values representing the functional dependencies.

2

List all attributes involved

Create a list called attributes containing the unique attributes 'A', 'B', and 'C'.

DBMS Theory

fds = {'AB': 'C', 'A': 'B', 'B': 'C'}
# Create the list attributes with 'A', 'B', and 'C'
# Your code here

Hint

List all attributes as strings inside a Python list.

3

Define a function to remove extraneous attributes

Define a function called remove_extraneous that takes a left side string and returns a simplified string by removing extraneous attributes. Use the fds dictionary inside the function.

DBMS Theory

fds = {'AB': 'C', 'A': 'B', 'B': 'C'}
attributes = ['A', 'B', 'C']
# Define the function remove_extraneous(left) below
# Your code here

Hint

Check each attribute in the left side. Remove it temporarily and check if the right side is still implied by the closure. If yes, remove it permanently.

4

Create the canonical cover by minimizing FDs

Create a dictionary called canonical_cover by applying remove_extraneous to each left side in fds and combining FDs with the same left side.

DBMS Theory

fds = {'AB': 'C', 'A': 'B', 'B': 'C'}
attributes = ['A', 'B', 'C']

def remove_extraneous(left):
    for attr in left:
        temp = left.replace(attr, '')
        closure = set(temp)
        changed = true
        while changed:
            changed = false
            for l, r in fds.items():
                if set(l).issubset(closure) and r not in closure:
                    closure.add(r)
                    changed = true
        if all(c in closure for c in fds[left]):
            left = temp
    return left

# Create canonical_cover dictionary by minimizing fds
# Your code here

Hint

For each FD, simplify the left side and merge right sides if left sides match.

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