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

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

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

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

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

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

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

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

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

Canonical Cover of FDs

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

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

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

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

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

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

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

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

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

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

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

Questions ?