IF3111 Basis Data – Functional Dependency

I F3111 Basis Data – Functional Dependency

IF-ITB/WD dari Silberschatz, modifikasi TW /27 Okt 2003

IF3111 – Functional Dependency

Wikan Danar Departemen Teknik I nformatika I nstitut Teknologi Bandung

Functional Dependencies • Constraints on the set of legal relations

Require that the value for a certain set of attributes determines uniquely the value for another set of attributes.
A functional dependency is a generalization of the notion of a key.

IF-ITB/WD dari Silberschatz, modifikasi TW /27 Okt 2003

IF3111 – Functional Dependency

Let R be a relation schema
The functional dependency

α ]

4 1 5

→

B does NOT hold, but B

β ] • Example: Consider r(A,B) with the following instance of r.

2 [

β ] = t

1 [

⇒ t

2 [

α ] = t

1 [

β . That is, t

α , they also agree on the attributes

2 of r agree on the attributes

1 and t

α → β holds on R if and only if for any legal relations r(R), whenever any two tuples t

β ⊆ R

α ⊆ R and

Functional Dependencies (Cont.)

On this instance, A

A does hold

IF3111 – Functional Dependency

IF-ITB/WD dari Silberschatz, modifikasi TW /27 Okt 2003 Functional Dependencies (Cont.)

K is a superkey for relation schema R if and only if K R

→

K is a candidate key for R if and only if

– K R, and

→

– for no K, R

α ⊂ α →

Functional dependencies allow us to express constraints that cannot be expressed using superkeys. Consider the schema:

Loan -info-schema = (customer -name, loan -number, branch -name, amount).

We expect this set of functional dependencies to hold:

loan-number amount

→

loan-number branch -nam e

→ but would not expect the following to hold:

loan-number customer-nam e

→

IF-ITB/WD dari Silberschatz, modifikasi TW /27 Okt 2003

IF3111 – Functional Dependency

We use functional dependencies to:

– test relations to see if they are legal under a given set of functional dependencies.

I f a relation r is legal under a set F of functional dependencies, we say that r satisfies F.

– specify constraints on the set of legal relations

F.
Note: A specific instance of a relation schema may satisfy a functional dependency even if the functional dependency does not hold on all legal instances.
– For example, a specific instance of Loan-schema may, by chance, satisfy

IF-ITB/WD dari Silberschatz, modifikasi TW /27 Okt 2003

IF3111 – Functional Dependency

Use of Functional Dependencies

loan- number → customer-name.

A functional dependency is trivial if it is satisfied by all instances of a relation – E.g.

customer-name, loan- number
customer-name
– I n general,

IF-ITB/WD dari Silberschatz, modifikasi TW /27 Okt 2003

IF3111 – Functional Dependency

Functional Dependencies (Cont.)

→ customer- name

→

customer-name

α → β is trivial if

β ⊆ α

Given a set F set of functional dependencies, there are certain other functional dependencies that are logically implied by F.

– E.g. I f A

The set of all functional dependencies logically implied by F is the closure of F.

We denote the closure of F by F
+ .
+ by applying Armstrong’s Axioms:
We can find all of F
– if
– if
– if
These rules are
– sound (generate only functional dependencies that actually hold)

α → γ ( transitivity)

, then

β → γ

, and

α → β

γ α → γ β ( augmentation)

, t hen

α → β

α → β ( reflexivity)

β ⊆ α

, then

→

C, then we can infer that A

→

B and B

→

Closure of a Set of Functional Dependencies

IF3111 – Functional Dependency

IF-ITB/WD dari Silberschatz, modifikasi TW /27 Okt 2003

and – complete (generate all functional dependencies that hold).

R = (A, B, C, G, H, I )

– A

some members of F
by transitivity from A

– AG

by augmenting A

– CG

from CG

– definition of functional dependencies, or

– Augmentation of CG

C with G, to get AG

→

CG and then transitivity with CG

→

H and CG

→

I : “union rule” can be inferred from

→ I t o infer CG

→

CGI , augment at ion of
CG →
H t o infer CGI →
→
I
→
H CG →
IF-ITB/WD dari Silberschatz, modifikasi TW /27 Okt 2003
IF3111 – Functional Dependency
Example
F = { A →
B A →
C CG →
I B →
H
H}
→
H
→
B and B
→
HI , and t hen t ransit ivit y
To compute the closure of a set of functional dependencies F:
= F
apply reflexivity and augmentation rules on f add the resulting functional dependencies to F
for each
if f
in F
F
add the resulting functional dependency to F
then
can be combined using transitivity
2
and f
1
2
and f
1
pair of functional dependencies f
repeat for each functional dependency f in F
F
Procedure for Computing F
until
does not change any further
IF3111 – Functional Dependency
IF-ITB/WD dari Silberschatz, modifikasi TW /27 Okt 2003
NOTE: We will see an alternative procedure for this task later
We can further simplify manual computation of F
by using the following additional rules.
– If
– If
– If
IF-ITB/WD dari Silberschatz, modifikasi TW /27 Okt 2003
IF3111 – Functional Dependency
Closure of Functional Dependencies (Cont.)
α → β holds and
α → γ holds, then
α → β γ holds ( union)
α → β γ holds, then
α → β holds and
α → γ holds
( decomposition)
α → β holds and
γ β → δ holds, then
α γ → δ holds
( pseudotransitivity) The above rules can be inferred from Armstrong’s axioms. Closure of Attribute Sets
Given a set of attributes define the closure
α, of under F (denoted by
) as the set of attributes that are functionally determined by under F:
α α
α is in F
ó β ⊆ α
α → β
α α
Algorithm to compute , the closure of under F
result := ; α w hile (changes to result) do
F for each in do
β → γ begin if result then result := result
β ⊆ ∪ γ end
IF-ITB/WD dari Silberschatz, modifikasi TW /27 Okt 2003
IF3111 – Functional Dependency Example of Attribute Set Closure
R = (A, B, C, G, H, I ) •
B F = { A
→
A C
→
CG H
→
CG
I
→
B H}
→
(AG)
1. result = AG 2. result = ABCG (A C and A
B)
→ →
3. result = ABCGH (CG H and CG AGBC)
→ ⊆
4. result = ABCGHI (CG I and CG AGBCH)
→ ⊆
I s AG a candidate key?
1. I s AG a super key?
1. Does AG R? = = I s ( AG) R
→ ⊇
2. I s any subset of AG a superkey?
1. Does A R? = = I s (A) R
→ ⊇
2. Does G R? = = I s ( G) R
→ ⊇
IF-ITB/WD dari Silberschatz, modifikasi TW /27 Okt 2003
IF3111 – Functional Dependency Uses of Attribute Closure There are several uses of the attribute closure algorithm:
Testing for superkey:
,
– To test if is a superkey, we compute and check if
α α α contains all attributes of R.
Testing functional dependencies
– To check if a functional dependency holds (or, in
α → β β ⊆ α
other words, is in F+ ), just check if .
– That is, we compute
by using attribute closure, and then
α check if it contains . β
– I s a simple and cheap test, and very useful
Computing closure of F
– For each R, we find the closure , and for each S ,
γ ⊆ γ ⊆ γ we output a functional dependency S.

γ →
IF-ITB/WD dari Silberschatz, modifikasi TW /27 Okt 2003
IF3111 – Functional Dependency Canonical Cover
Sets of functional dependencies may have redundant dependencies that can be inferred from the others
– Eg: A C is redundant in: { A
B, B
C, A C}
→ → → →
– Parts of a functional dependency may be redundant
E.g. on RHS: { A
B, B
C, A CD} can be simplified
→ → →
to { A
B, B
C, A D}
→ → →
E.g. on LHS: { A
B, B
C, AC D} can be simplified
→ → →
to { A
B, B
C, A D}
→ → →
I ntuitively, a canonical cover of F is a “ minimal” set of functional dependencies equivalent to F, having no redundant dependencies or redundant parts of dependencies
IF-ITB/WD dari Silberschatz, modifikasi TW /27 Okt 2003
IF3111 – Functional Dependency
Consider a set F of functional dependencies and the functional dependency
– Attribute A is extraneous in
– A)
– Attribute A is extraneous in
→
(
β – A)} logically implies F.
→
→ C }
→
C because { A

→
C, AB
C} logically implies A
{
→
C (I .e. the result of dropping B from AB
→ C).
→
→ CD}
→
CD since AB

→
α →
∪
C can be inferred even after deleting C Testing if an Attribute is Extraneous
α → β
IF-ITB/WD dari Silberschatz, modifikasi TW /27 Okt 2003
IF3111 – Functional Dependency
Extraneous Attributes
α → β in F.
α
if A
∈ α
and F logically implies (F – {
} )
} )

∪
{ (
α
→ β } .
β
if A
∈ β
and the set of functional dependencies ( F – {
α → β
Note: implication in the opposite direction is trivial in each of the cases above, since a “ stronger” functional dependency always implies a weaker one

Example: Given F = { A
– B is extraneous in AB
Example: Given F = { A
– C is extraneous in AB
Consider a set F of functional dependencies and the functional dependency in F.
α → β
To test if attribute A is extraneous in
∈ α α
1. compute ({ } – A) using the dependencies in F
α
2. check that ( { } – A) contains A; if it does, A is
α extraneous
To test if attribute A is extraneous in
∈ β β α
1. compute using only the dependencies in F’ = ( F – { } ) { ( – A)} ,
α → β ∪ α → β 2. check that
contains A; if it does, A is extraneous
α
IF-ITB/WD dari Silberschatz, modifikasi TW /27 Okt 2003
IF3111 – Functional Dependency Canonical Cover
A canonical cover for F is a set of dependencies F such
c that
– F logically implies all dependencies in F and
c,
– F logically implies all dependencies in F, and
c
– No functional dependency in F contains an extraneous
c
attribute, and – Each left side of functional dependency in F is unique.
c
To compute a canonical cover for F:
repeat
Use the union rule to replace any dependencies in F and with
α → β α → β α → β β
1
1
1
1
1
1
2 Find a functional dependency with an
α → β
extraneous attribute either in or in
α β
I f an extraneous attribute is found, delete it from
α → β until F does not change
IF-ITB/WD dari Silberschatz, modifikasi TW /27 Okt 2003
IF3111 – Functional Dependency Note: Union rule may become applicable after some extraneous attributes have been deleted, so it has to be re -applied
R = ( A, B, C)
Combine A
– Set is now { A
A is extraneous in AB
– Check if the result of deleting A from AB
Yes: in fact, B
– Set is now { A
C is extraneous in A
– Check if A
Yes: using transitivity on A
– Can use attribute closure of A in more complex cases
The canonical cover is: A
→
C

→
C is implied by the other dependencies
→ C is already present!
→
BC, B
→
C}
BC
→
→
C is logically implied by A
→
B and the other dependencies
→ B and B
→
C.
→
B B
→
C}
C, AB
→
B AB
IF-ITB/WD dari Silberschatz, modifikasi TW /27 Okt 2003
IF3111 – Functional Dependency
Example of Computing a Canonical Cover
F = { A
→
BC B
→
C A
→
→
→
C}
→
BC and A
→
B into A
→
BC
→
BC, B
C

IF3111 Basis Data – Functional Dependency