1. Chapter 7 Functional Dependencies
4/25/2017
Chapter 7 Functional Dependencies
Copyright © 2004 Pearson Education, Inc.

2. 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

3. 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? 

4. 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

5. 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.

6. A simplified COMPANY relational database schema

7. 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

9. 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.

10. 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.

13. 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

14. 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

15. 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

19. There are two important possleroperties of decompositions:
Spurious Tuples (2)
 There are two important possleroperties of decompositions:
non-additive or lssness of the corresponding join
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]).

20. Functional Dependencies (FDs)
Definition of FD
Direct, indirect, partial dependencies
Inference Rules for FDs
Equivalence of Sets of FDs
Minimal Sets of FDs

21. 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
A set of attributes X functionally determines a set of attributes Y if the value of X determines a unique value for Y

22. 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

23. 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

24. 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])

25. 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)
TicketName
TicketID
TicketType
TicketLocation

26. Functional Dependencies (4)
Indirect dependency (transitive dependency): Value of an attribute is not determined directly by the primary key
TicketID
TicketName
TicketType
TicketLocation
Price

27. 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 - 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 dependency
SSN -> ENAME
PNUMBER -> {PNAME, PLOCATION}

28. Functional Dependencies (6)
Partial dependency
TicketID
TicketName
TicketType
TicketLocation
Price
Agent-id
AgentName
AgentLocation

29. 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

30. 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 sound and complete set of inference rules

31. 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)

32. Sample Exercises F = {AB, BC} F = {ABC}
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A & AB
 ACB is right

33. Inference Rules for FDs (3)
Closure of a set F of FDs is the set F+ of all FDs (include F) 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

34. Determining X+
Example: Emp_Proj(Ssn, Ename,Pnumber, Pname, Plocation, Hours)

35. 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

36. 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

37. 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  DH
3. CEG  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.

38. 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

39. Algorithm for Finding a Key
Note: the algorithm determines only one key out of the possible candidate keys for R;

40. 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

41. Finding Keys using FDs We have:
UL and UR are the set of LHS and RHS attributes
N = U – UR 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 = UR – UL is the set of attributes which only appears in RHS  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  X}+. If {N  X}+ = R so it is a key

42. 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

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

44. 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}

45. Find all the candidate keys of R
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}

46. 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
There is an algorithm for checking equivalence of sets of FDs
Home Ex:

47. 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.
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. (X  A also is called a complete functional dependency)

48. 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

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

50. 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  AB (IR2) => B  AB (However, AB  D), so we have B  D
So replace AB  D by B  D. We have E’ = {B  A, D A, B  D)
Step 4: We also derive from E’ B  A is redundant