Fundamentals of Database Systems

Transcript Fundamentals of Database Systems

Chapter 7 Functional Dependencies

Outline

  – – – – Informal Design Guidelines for Relational Databases Semantics of the Relation Attributes Redundant Information in Tuples and Update Anomalies Null Values in Tuples Spurious Tuples – – – – Functional Dependencies (FDs) Definition of FD Inference Rules for FDs Equivalence of Sets of FDs Minimal Sets of FDs

Elmasri/Navathe, Fundamentals of Database Systems, Fourth Edition

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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?

Elmasri/Navathe, Fundamentals of Database Systems, Fourth Edition

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Informal Design Guidelines for Relational Databases (2)

 We first discuss informal guidelines for good relational design  Then we discuss formal concepts of functional dependencies and normal forms - 1NF (First Normal Form) - 2NF (Second Normal Form) - 3NF (Third Normal Form) - BCNF (Boyce-Codd Normal Form)  Additional types of dependencies, further normal forms, relational design algorithms by synthesis are discussed in Chapter 16

Elmasri/Navathe, Fundamentals of Database Systems, Fourth Edition

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Semantics of the Relation Attributes

GUIDELINE 1:

Informally, each tuple in a relation should represent one entity or relationship instance. (Applies to individual relations and their attributes).

 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.

Elmasri/Navathe, Fundamentals of Database Systems, Fourth Edition

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A simplified COMPANY relational database schema

Elmasri/Navathe, Fundamentals of Database Systems, Fourth Edition

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Redundant Information in Tuples and Update Anomalies

 Mixing attributes of multiple entities may cause problems – – Information is stored redundantly wasting storage Data inconsistency  Problems with update anomalies – Insertion anomalies – Deletion anomalies – Modification anomalies

Elmasri/Navathe, Fundamentals of Database Systems, Fourth Edition

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Elmasri/Navathe, Fundamentals of Database Systems, Fourth Edition

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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.

Elmasri/Navathe, Fundamentals of Database Systems, Fourth Edition

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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.

Elmasri/Navathe, Fundamentals of Database Systems, Fourth Edition

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Elmasri/Navathe, Fundamentals of Database Systems, Fourth Edition

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Elmasri/Navathe, Fundamentals of Database Systems, Fourth Edition

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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

Elmasri/Navathe, Fundamentals of Database Systems, Fourth Edition

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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

Elmasri/Navathe, Fundamentals of Database Systems, Fourth Edition

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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. Avoid relations that contain matching attributes that are not (foreign key, primary key) combinations because joining on such attributes may produce spurious tuples

Elmasri/Navathe, Fundamentals of Database Systems, Fourth Edition

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Elmasri/Navathe, Fundamentals of Database Systems, Fourth Edition

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Elmasri/Navathe, Fundamentals of Database Systems, Fourth Edition

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Elmasri/Navathe, Fundamentals of Database Systems, Fourth Edition

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Spurious Tuples (2)

There are two important possleroperties of decompositions: (a) non-additive or lssness of the corresponding join (b) preservation of the functional dependencies. Note that property (a) is extremely important and

cannot

be sacrificed. Property (b) is less stringent and may be sacrificed. (See more in chapter 16 [1]).

Elmasri/Navathe, Fundamentals of Database Systems, Fourth Edition

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Functional Dependencies (FDs)

 Definition of FD  Direct, indirect, partial dependencies  Inference Rules for FDs  Equivalence of Sets of FDs  Minimal Sets of FDs

Elmasri/Navathe, Fundamentals of Database Systems, Fourth Edition

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Functional Dependencies (1)

 Functional dependencies (FDs) are used to specify

formal measures

designs of the "goodness" of relational  FDs and keys are used to define relations

normal forms

for  FDs are

meaning

attributes

constraints

that are derived from the and

interrelationships

of the data  A set of attributes X

functionally determines

of attributes Y if the value of X determines a unique value for Y a set

Elmasri/Navathe, Fundamentals of Database Systems, Fourth Edition

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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): t1[X]=t2[X],

then

t1[Y]=t2[Y]

  X -> Y in R specifies a Written as X ->

constraint

on all relation instances r(R) 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

Elmasri/Navathe, Fundamentals of Database Systems, Fourth Edition

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Examples of FD constraints (1)

 social security number determines employee name SSN -> ENAME  project number determines project name and location PNUMBER -> {PNAME, PLOCATION}  employee ssn and project number determines the hours per week that the employee works on the project {SSN, PNUMBER} -> HOURS

Elmasri/Navathe, Fundamentals of Database Systems, Fourth Edition

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Examples of FD constraints (2)

 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])

Elmasri/Navathe, Fundamentals of Database Systems, Fourth Edition

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Functional Dependencies (3)

 Direct dependency (fully functional dependency): All attributes in a R must be fully functionally dependent on the primary key (or the PK is a determinant of all attributes in R) TicketID TicketName TicketType TicketLocation

Elmasri/Navathe, Fundamentals of Database Systems, Fourth Edition

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Functional Dependencies (4)

 Indirect dependency (transitive dependency): Value of an attribute is not determined directly by the primary key TicketID TicketName TicketType Price TicketLocation

Elmasri/Navathe, Fundamentals of Database Systems, Fourth Edition

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Functional Dependencies (5)

 Partial dependency –

Composite determinant

- more than one value is required to determine the value of another attribute, the combination of values is called a composite determinant

EMP_PROJ(SSN, PNUMBER, HOURS, ENAME, PNAME, PLOCATION) {SSN, PNUMBER} -> HOURS

–

Partial dependency

dependency - if the value of an attribute does not depend on an entire composite determinant, but only part of it, the relationship is known as the partial

SSN -> ENAME PNUMBER -> {PNAME, PLOCATION} Elmasri/Navathe, Fundamentals of Database Systems, Fourth Edition

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Functional Dependencies (6)

 Partial dependency TicketID TicketName TicketType TicketLocation Price Agent-id AgentName AgentLocation

Elmasri/Navathe, Fundamentals of Database Systems, Fourth Edition

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Inference Rules for FDs (1)

 An FD X → Y is inferred from a set of dependencies F specified on R if whenever r satisfies all the dependencies in F, X → Y also holds in r  F |=X → Y to denote that the functional dependency X→Y is inferred from the set of functional dependencies F  Exp: U = {ABC}, F = {A  B, B  C}, We can say F ⊨ A  C

Elmasri/Navathe, Fundamentals of Database Systems, Fourth Edition

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Inference Rules for FDs (1)

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) IR3. (

Transitive

) If X -> Y and Y -> Z, then X -> Z Lemma 1:  IR1, IR2, IR3 form a inference rules

sound

and

complete

set of

Elmasri/Navathe, Fundamentals of Database Systems, Fourth Edition

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Inference Rules for FDs (2)

Lemma 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)

Elmasri/Navathe, Fundamentals of Database Systems, Fourth Edition

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Sample Exercises

 F = {A  B, B  C} – A  C is inferred from F? (transitive)  F = {A  BC} – A  B , A  C are inferred from F?

Ans: A  BC và BC  B (reflexive) => A  B (transitive)  F = {A  B, B  C}, A  BC ?

 F = {A  B}, AC  B?

A   AC AC   AC  B is right A & A  B

Elmasri/Navathe, Fundamentals of Database Systems, Fourth Edition

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Inference Rules for FDs (3)



Closure

of a set F of FDs is the set F + (include F) that can be inferred from F of all FDs 

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

Elmasri/Navathe, Fundamentals of Database Systems, Fourth Edition

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Determining X+

 Example: Emp_Proj(Ssn, Ename,Pnumber, Pname, Plocation, Hours)

Elmasri/Navathe, Fundamentals of Database Systems, Fourth Edition

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Sample Exercises

R(ABCDEG) and FDs as follows: F = {AB→C, C →A, BC →D, ACD→B, D→EG, BE→C, CG→BD, CE→AG} X = {BD}, calculate X+ Result: X+ = R

Elmasri/Navathe, Fundamentals of Database Systems, Fourth Edition

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Inference Rules for FDs (3)

 Lemma 3: X  Y is inferred from F based on Armstrong’s rules if and only if Y is a subset of X+ with respect to F F ⊢ Arm (X  Y)  Y  XF+  Note: – We can check whether X is a key of R by calculating X+. If X+ = R then X is a key

Elmasri/Navathe, Fundamentals of Database Systems, Fourth Edition

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Checking if an FD Holds on F Using the Closure

 Let R(ABCDEFGH) satisfy the following functional dependencies: {A->B, CH->A, B->E, BD->C, EG->H, DE->F}  Which of the following FD is also guaranteed to be satisfied by R?

1. BFG  AE 2. ACG 3. CEG   DH AB

Hint:

Compute the closure of the LHS of each FD that you get as a choice. If the RHS of the candidate FD is contained in the closure, then the candidate follows from the given FDs, otherwise not.

Elmasri/Navathe, Fundamentals of Database Systems, Fourth Edition

Checking for Keys Using the Closure

 Which of the following could be a key for R(A,B,C,D,E,F,G) with functional dependencies {AB  C, CD  E, EF  G, FG  E, DE  C, and BC  A} 1. BDF 2. ACDF 3. ABDFG 4. BDFG

Elmasri/Navathe, Fundamentals of Database Systems, Fourth Edition

Algorithm for Finding a Key

Note: the algorithm determines only one key out of the possible candidate keys for R;

Elmasri/Navathe, Fundamentals of Database Systems, Fourth Edition

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Finding Keys using FDs



Tricks for finding the key:

 If an attribute never appears on the

RHS

of any FD, it

must be part of the key

 If an attribute never appears on the

LHS

of any FD, but appears on the

RHS

of any FD, it

must not be part of any key

Elmasri/Navathe, Fundamentals of Database Systems, Fourth Edition

Finding Keys using FDs

   We have:    U L and U R are the set of LHS and RHS attributes N = U – U R is the set of independent attributes and those which only appear on LHS  N must be a part of keys If N+ = R, then N is a minimal key 

Stop here

  Otherwise: D = U R RHS  – U L is the set of attributes which only appears in D cannot be a part of key L = U – (N  D) is the set of attributes which may or may not be a part of keys For each combination X in L, we calculate {N  {N  X}+ = R so it is a key X}+. If

Elmasri/Navathe, Fundamentals of Database Systems, Fourth Edition

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Finding Keys using FDs

Consider R = {A, B, C, D, E, F, G, H} with a set of FDs F = {CD→A, EC→H, GHB→AB, C→D, EG→A, H→B, BE→CD, EC→B}

Find all the candidate keys of R Elmasri/Navathe, Fundamentals of Database Systems, Fourth Edition

Finding Keys using FDs

F = {CD→A, EC→H, GHB→AB, C→D, EG→A, H→B, BE→CD, EC→B}   U R = {AHBDC} = {ABCDH} N = U – U R = {EFG} but EFG+ = EFGA ≠ R     U L = {CDEGHB} = {BCDEGH} D = U R – U L L = U – (N  = {A} D) = {BCDH} We have combinations such as {B, C, D, H, BC, BD, BH, CD, CH, DH, BCD, BCH, CDH, BCDH}

Elmasri/Navathe, Fundamentals of Database Systems, Fourth Edition

Finding Keys using FDs

  For each combination X, calculate {X  N}+ BEFG+ = ABCDEFGH = R; it’s a key [BE→CD, EG→A, EC→H]  CEFG+ = ABCDEFGH = R; it’s a key [EG→A, EC→H, H→B, BE→CD]  DEFG+ = ADEFG ≠ R; it’s not a key [EG→A]  EFGH+ = ABCDEFGH = R; it’s a key [EG→A, H→B, BE→CD]  If we add any further attribute(s), they will form the superkey. Therefore, we can stop here searching for candidate key(s).  So, candidate keys are: {BEFG, CEFG, EFGH}

Elmasri/Navathe, Fundamentals of Database Systems, Fourth Edition

Exercises

Consider R = {A, B, C, D, E, F} with a set of FDs F = {A  BC, B  D, AD  E, CD  A}

Find all the candidate keys of R

Consider R = {A, B, C, D, E, F, G} with a set of FDs F = {ABC→DE, AB→D, DE→ABCF, E→C}

Find all the candidate keys of R Elmasri/Navathe, Fundamentals of Database Systems, Fourth Edition

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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

from F (i.e., if G + G if every FD in G can be inferred

subset-of

F + )  F and G are equivalent if F covers G and G covers F  There is an algorithm for checking equivalence of sets of FDs 

Home Ex:

Elmasri/Navathe, Fundamentals of Database Systems, Fourth Edition

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Minimal Sets of FDs

(1)

 A minimal cover of a set of functional dependencies E is a minimal set of dependencies (in the standard

canonical

form and without redundancy) that is equivalent to E.

 (1) (2) (3) A set of FDs is

minimal

if it satisfies the following conditions: Every dependency in F has a single attribute for its RHS We cannot remove any dependency from F and have a set of dependencies that is equivalent to F.

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



A also is called a complete functional dependency)

Elmasri/Navathe, Fundamentals of Database Systems, Fourth Edition

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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  To synthesize a set of relations, we assume that we start with a set of dependencies that is a minimal set

Elmasri/Navathe, Fundamentals of Database Systems, Fourth Edition

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Finding a Minimal Cover F for a Set of Functional Dependencies E

Elmasri/Navathe, Fundamentals of Database Systems, Fourth Edition

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Finding a Minimal Cover F for a Set of Functional Dependencies E

 Let the given set of FDs be E :{B→A, D→A, AB→D}. Please find the minimal cover of E.

 Step 1: Set F = E  Step 2: All FDs in the canonical form  Step 3: Determine if AB  D has any redundant attribute on LHS – – Since B  A, we have BB (However, AB   AB (IR2) => B D), so we have B  D  AB So replace AB  D  A, B  D) D by B  D. We have E’ = {B  A,  Step 4: We also derive from E’ B  A is redundant

Elmasri/Navathe, Fundamentals of Database Systems, Fourth Edition

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Fundamentals of Database Systems

Transcript Fundamentals of Database Systems

Chapter 7 Functional Dependencies

Outline

Informal Design Guidelines for Relational Databases (1)

Informal Design Guidelines for Relational Databases (2)

Semantics of the Relation Attributes

A simplified COMPANY relational database schema

Redundant Information in Tuples and Update Anomalies

EXAMPLE OF AN UPDATE ANOMALY (1)

EXAMPLE OF AN UPDATE ANOMALY (2)

Guideline to Redundant Information in Tuples and Update Anomalies

Null Values in Tuples

Spurious Tuples

Spurious Tuples (2)

Functional Dependencies (FDs)

Functional Dependencies (1)

Functional Dependencies (2)

Examples of FD constraints (1)

Examples of FD constraints (2)

Functional Dependencies (3)

Functional Dependencies (4)

Functional Dependencies (5)

Functional Dependencies (6)

Inference Rules for FDs (1)

Inference Rules for FDs (1)

Inference Rules for FDs (2)

Sample Exercises

Inference Rules for FDs (3)

Determining X+

Sample Exercises

Inference Rules for FDs (3)

Checking if an FD Holds on F Using the Closure

Checking for Keys Using the Closure

Algorithm for Finding a Key

Finding Keys using FDs

Finding Keys using FDs

Finding Keys using FDs

Finding Keys using FDs

Finding Keys using FDs

Exercises

Equivalence of Sets of FDs

Minimal Sets of FDs

(1)

Minimal Sets of FDs

(2)

Finding a Minimal Cover F for a Set of Functional Dependencies E

Finding a Minimal Cover F for a Set of Functional Dependencies E

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