FUNCTIONAL DEPENDENCIES

Chapter Outline

1 Informal Design Guidelines for Relational Databases1.1Semantics of the Relation Attributes1.2 Redundant Information in Tuples and Update Anomalies1.3 Null Values in Tuples1.4 Spurious Tuples

2 Functional Dependencies (FDs)2.1 Definition of FD2.2 Inference Rules for FDs2.3 Equivalence of Sets of FDs2.4 Minimal Sets of FDs

Chapter Outline(contd.)

3 Normal Forms Based on Primary Keys3.1 Normalization of Relations

3.2 Practical Use of Normal Forms

3.3 Definitions of Keys and Attributes Participating in Keys

3.4 First Normal Form

3.5 Second Normal Form

3.6 Third Normal Form

4 General Normal Form Definitions (For Multiple Keys)

5 BCNF (Boyce-Codd Normal Form)

1 Informal Design Guidelines for

Relational Databases (1)

What is relational database design?

The grouping of attributes to form "good" relation schemas

 Two levels of relation schemas The logical "user view" level The storage "base relation" level

 Design is concerned mainly with base relations

 What are the criteria for "good" base relations? 

1.1 Semantics of the Relation Attributes

GUIDELINE 1: Informally, each tuple in a relation should represent one entity or relationship instance.

Attributes of different entities (EMPLOYEEs, DEPARTMENTs, PROJECTs) should not be mixed in the same relation

Only foreign keys should be used to refer to other entities

 Entity and relationship attributes should be kept apart as much as possible.

Bottom Line: Design a schema that can be explained easily relation by relation. The semantics of attributes should be easy to interpret.

Figure 10.1 A simplified COMPANY relational database schema

Note: The above figure is now called Figure 10.1 in Edition 4

1.2 Redundant Information in Tuples and Update

Anomalies

Mixing attributes of multiple entities may cause problems

Information is stored redundantly wasting storage

Problems with update anomalies Insertion anomalies Deletion anomalies Modification anomalies

EXAMPLE OF AN UPDATE ANOMALY (1) Consider the relation:EMP_PROJ ( Emp#, Proj#, Ename, Pname,

No_hours)

  Update Anomaly: Changing the name of

project number P1 from “Billing” to “Customer-Accounting” may cause this update to be made for all 100 employees working on project P1.

EXAMPLE OF AN UPDATE ANOMALY (2)

Insert Anomaly: Cannot insert a project unless an employee is assigned to .

Inversely - Cannot insert an employee unless an he/she is assigned to a project.

 Delete Anomaly: When a project is deleted, it will result in deleting all the employees who work on that project. Alternately, if an employee is the sole employee on a project, deleting that employee would result in deleting the corresponding project.

Figure 10.3 Two relation schemas suffering from update anomalies

Figure 10.4 Example States for EMP_DEPT and EMP_PROJ

Note: The above figure is now called Figure 10.4 in Edition 4

Guideline to Redundant Information in Tuples and Update Anomalies

GUIDELINE 2: Design a schema that does not suffer from the insertion, deletion and update anomalies.

If there are any present, then note them so that applications can be made to take them into account

1.3 Null Values in Tuples

GUIDELINE 3: Relations should be designed such that their tuples will have as few NULL values as possible

 Attributes that are NULL frequently could be placed in separate relations (with the primary key)

 Reasons for nulls: attribute not applicable or invalid attribute value unknown (may exist) value known to exist, but unavailable

1.4 Spurious Tuples

Bad designs for a relational database may result in erroneous results for certain JOIN operations

The "lossless join" property is used to guarantee meaningful results for join operations

GUIDELINE 4: The relations should be designed to satisfy the lossless join condition.

No spurious tuples should be generated by doing a natural-join of any relations.

Consider EMP_PROJ relation of Case II, and its relation state

Let us split EMP_PROJ into 2 relation EMP_PROJ1 and EMP_LOCS as shown

below.

• A tuple in EMP_LOCS means that the employee whose name is ENAME

works on some project whose location is PLOCATION

• A tuple in EMP_PROJ1 means that the employee whose social security number is SSN works HOURS per week on the project whose name, number, and location are PNAME, PNUMBER, and PLOCATION

Now produce tuples for EMP_LOCS by applying project operation to

EMP_PROJ relation state , So we get tuples as shown below

Tuples for EMP_LOCS obtained by applying Project operation to EMP_PROJ

Now produce tuples for EMP_PROJ1 by applying project operation to

EMP_PROJ relation state , So we get tuples as shown below

Tuples for EMP_PROJ1 obtained by applying Project operation to EMP_PROJ

Now apply natural join operation (*) on EMP_PROJ1 and EMP_LOCS to get the original relation EMP_PROJ. EMP_PROJ1 * EMP_LOCS

• Additional tuples that were not in EMP_PROJ are called spurious tuples

because they represent spurious or wrong information that is not

valid. The spurious tuples are marked by asterisks (*).

• Decomposing EMP_PROJ into EMP_LOCS and EMP_PROJ! is undesirable

because, when we JOIN them back using NATURAL JOIN, we do not

get the correct original information. This is because PLOCATION

is neither a primary key nor a foreign key in either EMP_LOCS or

EMP_PROJ1

Spurious Tuples (2)

 There are two important properties of decompositions:

(a) non-additive or losslessness of the corresponding join

(b) preservation of the functional dependencies.

Note :property (a) is extremely important and cannot be sacrificed.

Property (b) is less stringent and may be sacrificed.

Functional Dependencies

2.1 Functional Dependencies (1)

Functional dependencies (FDs) are used to specify formal measures of the "goodness" of relational designs

FDs and keys are used to define normal forms for relations

FDs are constraints that are derived from the meaning and interrelationships of the data attributes

X->Y : A set of attributes X functionally determines a set of attributes Y if the value of X determines a unique value for Y.

Functional Dependencies (2)

X -> Y holds if whenever two tuples have the same value for X, they must have the same value for Y

For any two tuples t1 and t2 in any relation instance r(R): If t1[X]=t2[X], then t1[Y]=t2[Y]

X -> Y in R specifies a constraint on all relation instances r(R)

Written as X -> Y; can be displayed graphically on a relation schema as in Figures. ( denoted by the arrow->).

FDs are derived from the real-world constraints on the attributes

Examples of FD constraints (1)

social security number determines employee nameSSN -> ENAME

project number determines project name and locationPNUMBER -> {PNAME, PLOCATION}

employee ssn and project number determines the hours per week that the employee works on the project{SSN, PNUMBER} -> HOURS

From the semantics of the attributes, we know that the following functional dependencies should hold:

FD1 : {SSN, PNUMBER} HOURS

FD2 : SSN ENAME

FD3 : PNUMBER {PNAME, PLOCATION}

DIAGRAMMATIC NOTATION FOR DISPLAYING FDs

A FD cannot be inferred automatically from a given relation state r; but must be defined

explicitly by someone who knows the semantics of the attributes of R.

Example:

TEXT(Author) COURSE(Subject) . Is it true for all legal states of TEACH?

YES

But we can definitely say TEACHER COURSE is not true.

FDs – NOT A PROPERTY OF A RELATION STATE

Important….

An FD is a property of the attributes in the schema R The constraint must hold on every relation instance r(R) If K is a key of R, then K functionally determines all

attributes in R (since we never have two distinct tuples with t1[K]=t2[K])

2.2 Inference Rules for FDs (1)

Given a set of FDs F, we can infer additional FDs that hold whenever the FDs in F hold

 Armstrong's inference rules:

IR1. (Reflexive) If Y subset-of X, then X -> Y

IR2. (Augmentation) If X -> Y, then XZ -> YZ

(Notation: XZ stands for X U Z or {X, Z})

IR3. (Transitive) If X -> Y and Y -> Z, then X -> Z

 IR1, IR2, IR3 form a sound and complete set of inference rules

Inference Rules for FDs (2)

Some additional inference rules that are useful:(Decomposition) If X -> YZ, then X -> Y and X -> Z(Union) If X -> Y and X -> Z, then X -> YZ(Psuedotransitivity) If X -> Y and WY -> Z, then WX -> Z

 The last three inference rules, as well as any other inference rules, can be deduced from IR1, IR2, and IR3 (completeness property)

Given R = (A, B, C, G, H, I) F = { A B A C

CG HCG I

B H }

Based on Inference Rules, some members of F+ would be

A H (transitivity from A B and B H)

AG I (augmenting A C with G, to get AG CG and then transitivity with CG I )

CG HI (adding CG I and CG H to infer CG HI)

INFERENCE RULES - EXAMPLE

Inference Rules for FDs (3)

Closure of a set F of FDs is the set F+ of all FDs that can be inferred from F

Closure of a set of attributes X with respect to F is the set X + of all attributes that are functionally determined by X

X + can be calculated by repeatedly applying IR1, IR2, IR3 using the FDs in F

2.3 Equivalence of Sets of FDs

Two sets of FDs F and G are equivalent if:- every FD in F can be inferred from G, and

- every FD in G can be inferred from F Hence, F and G are equivalent if F + =G +

Definition: F covers G if every FD in G can be inferred from F (i.e., if G + subset-of F +)

F and G are equivalent if F covers G and G covers F

1. Consider the following two sets of functional dependencies:

F = {A C, AC D, E AD, E H}

G = {A CD, E AH}.

Check whether they are equivalent.

2. Consider two sets of FDs, F and G,

F = {A B, B C, AC D} and

G = {A B, B C, A D}

Are F and G equivalent?3. F = {A B, A C} G = {A B, B C} Are F and G equivalent?

PRACTICE - EQUIVALENCE OF SETS OF FUNCTIONAL DEPENDENCIES

Closure of a Set of Functional Dependencies

For a set F of functional dependencies, we call the closure of F, noted F+, the set of all the functional dependencies that can be derived from F (by the application of Armstrong’s axioms). Intuitively, F+ is equivalent to F, but it contains some additional

FDs that are only implicit in F.

Consider the relation scheme R(A,B,C,D) withF = {{A} {B},{B,C} {D}}F+ = {

{A} {A}, {B}{B}, {C}{C}, {D}{D}, {A,B}{A,B}, […], {A}{B}, {A,B}{B}, {A,D}{B,D}, {A,C}{B,C}, {A,C,D}{B,C,D}, {A} {A,B}, {A,D}{A,B,D}, {A,C}{A,B,C}, {A,C,D}{A,B,C,D}, {B,C} {D}, […], {A,C} {D}, […]}

Algorithm for Computing the Closure of a Set of Attributes

Input: R a relation scheme F a set of functional dependencies X R (the set of attributes for which we want to compute

the closure)Output:

X+ the closure of X w.r.t. F

X(0) := XRepeat

X(i+1) := X(i) Z, where Z is the set of attributes such that there exists YZ in F, and Y X(i)

Until X(i+1) := X(i)

Return X(i+1)

Closure of a Set of Attributes: Example

R = {A,B,C,D,E,G}

F = { {A,B}{C}, {C}{A}, {B,C}{D}, {A,C,D}{B}, {D}{E,G}, {B,E}{C}, {C,G}{B,D}, {C,E}{A,G}}

X = {B,D}

Find X+

X(0) = {B,D} {D}{E,G},

X(1) = {B,D,E,G}, {B,E}{C}

X(2) = {B,C,D,E,G}, {C,E}{A,G}

X(3) = {A,B,C,D,E,G}

X(4) = X(3)

Uses of Attribute Closure

There are several uses of the attribute closure algorithm: Testing for superkey

To test if X is a superkey, we compute X+, and check if X+ contains all attributes of R. X is a candidate key if none of its subsets is a key.

Testing functional dependencies To check if a functional dependency X Y holds (or, in other words, is in

F+), just check if Y X+. Computing the closure of F

For each subset X R, we find the closure X+, and for each Y X+, we output a functional dependency X Y.

Computing if two sets of functional dependencies F and G are equivalent, i.e., F+ = G+ For each functional dependency YZ in F

Compute Y+ with respect to G If Z Y+ then YZ is in G+

And vice versa

Redundancy of FDs

Sets of functional dependencies may have redundant dependencies that can be inferred from the others {A}{C} is redundant in: {{A}{B}, {B}{C},{A} {C}}

Parts of a functional dependency may be redundant Example of extraneous/redundant attribute on RHS: {{A}{B}, {B}{C}, {A}{C,D}} can be simplified to {{A}{B}, {B}{C}, {A}{D}} (because {A}{C} is inferred from {A} {B}, {B}{C})

Example of extraneous/redundant attribute on LHS: {{A}{B}, {B}{C}, {A,C}{D}} can be simplified to {{A}{B}, {B}{C}, {A}{D}} (because of {A}{C})

2.4 Minimal Sets of FDs (1)

A set of FDs is minimal if it satisfies the following conditions:

(1) Every dependency in F has a single attribute for its RHS.

(2) We cannot remove any dependency from F and have a set of dependencies that is equivalent to F.

(3) We cannot replace any dependency X -> A in F with a dependency Y -> A, where Y proper-subset-of X ( Y subset-of X) and still have a set of dependencies that is equivalent to F.

Minimal Sets of FDs (2)

Every set of FDs has an equivalent minimal set

There can be several equivalent minimal sets

There is no simple algorithm for computing a minimal set of FDs that is equivalent to a set F of FDs

Canonical Cover

A canonical cover for F is a set of dependencies Fc such that F and Fc,are equivalent

Fc contains no redundancy

Each left side of functional dependency in Fc is unique.

For instance, if we have two FD XY, XZ, we convert them to XYZ.

Algorithm for canonical cover of F:repeat

Use the union rule to replace any dependencies in F X1 Y1 and X1 Y2 with X1 Y1 Y2

Find a functional dependency X Y with an extraneous attribute either in X or in Y

If an extraneous attribute is found, delete it from X Y until F does not change

Note: Union rule may become applicable after some extraneous attributes have been deleted, so it has to be re-applied

Example of Computing a Canonical Cover

R = (A, B, C)F = {A BC

B C A BAB C}

Combine A BC and A B into A BC Set is now {A BC, B C, AB C}

A is extraneous in AB C because of B C. Set is now {A BC, B C}

C is extraneous in A BC because of A B and B C. The canonical cover is: A B

B C

Keys and FDs

A FD is a generalization of the notion of a key.

For Student (sid, name, supervisor_id, specialization), we write:

{sid} {name, supervisor_id, specialization} The sid determines all attributes (i.e., the entire record) If two tuples in the relation student have the same sid,

then they must have the same values on all attributes. In other words they must be the same tuple (since the

relational model does not allow duplicate records)

Superkeys and Candidate Keys A set of attributes that determine the entire tuple is a

superkey {sid, name} is a superkey for the student table. Also {sid, name, supervisor_id} etc.

A minimal set of attributes that determines the entire tuple is a candidate key {sid, name} is not a candidate key because I can remove the

name. sid is a candidate key

If there are multiple candidate keys, the DB designer chooses designates one as the primary key.

Finding Keys

Example: Consider the relation scheme R(A,B,C,D) with functional dependencies {A}{C} and {B}{D}.

Is {A,B} a candidate key?

For {A,B} to be a candidate key, it must determine all attributes (i.e., be a superkey) be minimal

{A,B} is a superkey because: {A}{C} {A,B}{A,B,C} (augmentation by AB)

{B}{D} {A,B,C}{A,B,C,D} (augmentation by A,B,C)

We obtain {A,B}{A,B,C,D} (transitivity)

{A,B} is minimal because neither {A} nor {B} alone are candidate keys

Pitfalls in Relational Database Design

Functional dependencies can be used to refine ER diagrams or independently (i.e., by performing repetitive decompositions on a "universal" relation that contains all attributes).

Relational database design requires that we find a “good” collection of relation schemas. A bad design may lead to Repetition of Information. Inability to represent certain information.

Design Goals: Avoid redundant data Ensure that relationships among attributes are represented Facilitate the checking of updates for violation of database integrity

constraints.

Example of Bad Design Consider the relation schema: Lending-schema = (branch-name, branch-

city, assets, customer-name, loan-number, amount) where: {branch-name}{branch-city, assets}

Bad Design Wastes space. Data for branch-name, branch-city, assets are repeated

for each loan that a branch makes Complicates updating, introducing possibility of inconsistency of

assets value Difficult to store information about a branch if no loans exist. Can use

null values, but they are difficult to handle.

Usefulness of FDs

Use functional dependencies to decide whether a particular relation R is in “good” form.

In the case that a relation R is not in “good” form, decompose it into a set of relations {R1, R2, ..., Rn} such that each relation is in good form the decomposition is a lossless-join decomposition if possible, preserve dependencies

In our example the problem occurs because there FDs ({branch-name}{branch-city, assets}) where the LHS is not a key

Solution: decompose the relation schema Lending-schema into: Branch-schema = (branch-name, branch-city,assets) Loan-info-schema = (customer-name, loan-number, branch-name,

amount)