1. Normalization First Normal Form (1NF) Boyce-Codd Normal Form (BCNF)
Third Normal Form (3NF)
Canonical Cover of FDs

2. Big Picture DB design may be bad and has problems
Insert/Update/Delete anomaly
Inconsistent data
When a FD violates a normalization rule
Then, one or more of the problems above exist
The decomposition will solve the problem

3. Third Normal Form: Motivation
There are some situations where
BCNF is not dependency preserving
Solution: Define a weaker normal form, called Third Normal Form (3NF)
Allows some redundancy (we will see examples later)
But all FDs are preserved
There is always a lossless, dependency-preserving decomposition in 3NF

4. R.H.S consists of prime attributes
Normal Form : 3NF
Relation R is in 3NF if, for every FD in F+
α  β,
where α ⊆ R and β ⊆ R, at least one of the following holds:
α → β is trivial (i.e.,β⊆α)
α is a superkey for R
Each attribute in β-α is part of a candidate key (prime attribute)
L.H.S is superkey OR
R.H.S consists of prime attributes

5. Testing for 3NF
Use attribute closure to check for each dependency α → β, if α is a superkey
If α is not a superkey, we have to verify if each attribute in (β- α) is contained in a candidate key of R

6. 3NF: Example Is relation Lot in 3NF ?
Lot (ID, county, lotNum, area, price, taxRate)
Primary key: ID
Candidate key: <county, lotNum>
FDs:
county  taxRate
area  price
Is relation Lot in 3NF ? NO
Decomposition based on county  taxRate
Lot (ID, county, lotNum, area, price)
County (county, taxRate)
Are relations Lot and County in 3NF ?
Lot is not

7. 3NF: Example (Cont’d) Is every relation in 3NF ?
Lot (ID, county, lotNum, area, price)
County (county, taxRate)
Candidate key for Lot: <county, lotNum>
FDs:
county  taxRate
area  price
Decompose Lot based on area  price
Lot (ID, county, lotNum, area)
County (county, taxRate)
Area (area, price)
Is every relation in 3NF ?
YES

8. Comparison between 3NF & BCNF ?
If R is in BCNF, obviously R is in 3NF
If R is in 3NF, R may not be in BCNF
3NF allows some redundancy and is weaker than BCNF
3NF is a compromise to use when BCNF with good constraint enforcement is not achievable
Important: Lossless, dependency-preserving decomposition of R into a collection of 3NF relations always possible ! 24

9. Normalization First Normal Form (1NF) Boyce-Codd Normal Form (BCNF)
Third Normal Form (3NF)
Canonical Cover of FDs

10. Canonical Cover of FDs

11. Canonical Cover of FDs Given set of FDs (F) with functional closure F+
Canonical Cover (Minimal Cover) = G
Is the smallest set of FDs that produce the same F+
There are no extra attributes in the L.H.S or R.H.S of and dependency in G
Given set of FDs (F) with functional closure F+
Canonical cover of F is the minimal subset of FDs (G), where
G+ = F+
Every FD in the canonical cover is needed, otherwise some dependencies are lost 8

12. Example : Canonical Cover
Given F:
A  B, ABCD  E, EF  GH, ACDF  EG
Then the canonical cover G:
A  B, ACD  E, EF  GH
The smallest set (minimal) of FDs that can generate F+ 8

13. Computing the Canonical Cover
Given a set of functional dependencies F, how to compute the canonical cover G

14. Example : Canonical Cover (Lets Check L.H.S)
Given F = {A  B, ABCD  E, EF  G, EF  H, ACDF  EG}
Union Step: {A  B, ABCD  E, EF  GH, ACDF  EG}
Test ABCD  E
Check A:
{BCD}+ = {BCD}  A cannot be deleted
Check B:
{ACD}+ = {A B C D E}  Then B can be deleted
Now the set is: {A  B, ACD  E, EF  GH, ACDF  EG}
Test ACD  E
Check C:
{AD}+ = {ABD}  C cannot be deleted
Check D:
{AC}+ = {ABC}  D cannot be deleted 8

15. Example: Canonical Cover (Lets Check L.H.S-Cont’d)
Now the set is: {A  B, ACD  E, EF  GH, ACDF  EG}
Test EF  GH
Check E:
{F}+ = {F}  E cannot be deleted
Check F:
{E}+ = {E}  F cannot be deleted
Test ACDF  EG
None of the H.L.S can be deleted 8

16. Example: Canonical Cover (Lets Check R.H.S)
Now the set is: {A  B, ACD  E, EF  GH, ACDF  EG}
Test EF  GH
Check G:
{EF}+ = {E F H}  G cannot be deleted
Check H:
{EF}+ = {E F G}  H cannot be deleted
Test ACDF  EG
Check E:
{ACDF}+ = {A B C D F E G}  E can be deleted
Now the set is: {A  B, ACD  E, EF  GH, ACDF  G} 8

17. Example: Canonical Cover (Lets Check R.H.S-Cont’d)
Now the set is: {A  B, ACD  E, EF  GH, ACDF  G}
Test ACDF  G
Check G:
{ACDF}+ = {A B C D F E G}  G can be deleted
Now the set is: {A  B, ACD  E, EF  GH}
The canonical cover is:
{A  B, ACD  E, EF  GH} 8

18. Canonical Cover
Used to find the smallest (minimal) set of FDs that have the same closure as the original set.
Used in the decomposition of relations to be in 3NF
The resulting decomposition is lossless and dependency preserving

19. Done with Normalization
First Normal Form (1NF)
Boyce-Codd Normal Form (BCNF)
Third Normal Form (3NF)
Canonical Cover of FDs

20. Summary (I) Functional Dependencies Decomposition
How to derive more FDs
FDs   Keys
Functional closure and Attribute closure
Decomposition
Lossy vs. Lossless
How to make the decomposition and how to check
Dependency Preservation
Whether all dependencies are preserved or some FDs are lost

21. Summary (II) Normalization Canonical Cover
Check whether a relation satisfies BCNF or 3rd NF
If there are violations, then how to decompose
Canonical Cover
Given a set of FDs, how to find its canonical cover

22. Questions ?