Cleveland State University
CIS 611 – Relational Databases
Prepared by Victor Matos
Functional Dependencies
Source:
The Theory of Relational Databases
D. Maier, Ed. Computer Science Press
Available at: http://www.dbis.informatik.hu-berlin.de/~freytag/Maier/
1
Functional Dependencies
• Two primary purposes of databases are to
– attenuate data redundancy and
– enhance data reliability.
• Any a priori knowledge of restrictions or constraints on
permissible sets of data has considerable usefulness in
reaching these goals.
• Data dependencies are one way to formulate such
advance knowledge.
2
Example1
• Consider the relation
assign (Pilot, Flight, Date, Departs)
PILOT
FLIGHT
DATE
DEPARTS
Cushing
Cushing
Clark
Clark
Clark
Chin
Chin
Copely
Copely
Copely
83
116
281
301
83
83
116
281
281
412
9 Aug
IO Aug
8 Aug
12 Aug
11 Aug
13 Aug
12 Aug
9 Aug
13 Aug
15 Aug
10: 15a
1:25p
5:50a
6:35p
10: 15a
10: 15a
1:25p
5:50a
5:50a
1:25p
3
Example1 - Observations
•
•
•
The relation assign tells which pilot flies a given flight on a given day, and what
time the flight leaves.
Not every combination of pilots, flights, dates, and times is allowable in assign.
The following restrictions apply, among others:
1.
2.
3.
For each flight there is exactly one time.
For any given pilot, date, and time, there is only one flight.
For a given flight and date, there is only one pilot.
•
These restrictions are examples of functional dependencies.
•
Informally, a functional dependency occurs when the values of a tuple on one set
of attributes uniquely determine the values on another set of attributes.
•
Our restrictions can be phrased as
1.
2.
3.
TIME functionally depends on FLIGHT,
FLIGHT functionally depends on {PILOT, DATE, TIME}, and
PILOT functionally depends on {FLIGHT, DATE}.
4
FD Definition
Def. Let r be a relation on scheme R, with X and Y
subsets of R. Relation r satisfies the functional
dependency (FD) X  Y if for every X-value x,
y( X=x(r)) has at most one tuple.
One way to interpret this expression is to look at
pairs of tuples, t1 and t2, in r.
If t1(X) = t2(X), then t1(Y) = t2(Y).
In the FD X  Y the portion X is called the left
side and Y is called the right side.
5
FD Satisfies
Algorithm 4.1 SATISFIES
Input: A relation r and an FD X  Y.
Output: true if T satisfies X  Y, false otherwise.
SATISFIES(r, X  Y);
1. Sort the relation r on its X columns to bring tuples with
equal X-values together.
2. If each set of tuples with equal X-values has equal Yvalues, return true. Otherwise, return false.
SATISFIES tests if a relation r satisfies an FD X  Y.
6
Algorithm: Satisfies
Using algorithm satisfies to test if FLIGHT  DEPARTS
PILOT
Cushing
Clark
Chin
Cushing
Chin
Clark
Copely
Copely
Clark
Copely
FLIGHT
83
83
83
116
116
281
281
281
301
412
DATE
9-Aug
11-Aug
13-Aug
IO Aug
12-Aug
8-Aug
9-Aug
13-Aug
12-Aug
15-Aug
DEPART
10: 15a
10: 15a
10: 15a
1:25p
1:25p
5:50a
5:50a
5:50a
6:35p
1:25p
Question:
DEPARTS  FLIGHT ???
7
Inference Axioms
• The number of FDs that can apply to a relation
r(R) is finite, since there is only a finite
number of subsets of R.
• Thus it is always possible to find all the FDs
that r satisfies, by trying all possibilities using
the algorithm SATISFIES.
• This brute-force approach is time-consuming.
8
Inference Axioms
• Finding F requires semantic knowledge of the
relation r.
• After knowing some members of F, it is often
possible to infer other members of F.
• A set F of FDs implies the FD X  Y, written
F  X  Y, if every relation that satisfies all
the FDs in F also satisfies
X  Y.
• An inference axiom is a rule that states if a
relation satisfies certain FDs, it must satisfy
certain other FDs.
9
Inference Axioms
F={
...}
Set of functional
dependencies

XY
Set of all
relations r(R)
satisfying FDs in
F
A set F of FDs implies the FD X  Y,
written F  X  Y, if every relation that
satisfies all the FDs in F also satisfies X  Y.
10
Example - Inference Axioms
F = { A  B, B  C }
Set of functional
dependencies

AC
Set of all
relations r(R)
satisfying FDs in
F
A set F of FDs implies the FD X  Y,
written F  X  Y, if every relation that
satisfies all the FDs in F also satisfies X  Y.
11
Inference Axioms
The Armstrong-Set of Inference Axioms
• Axioms will implement the “intelligence” needed to
prove (or disprove) a sequence of derivations.
• Inference Machines are used to determine whether or
not the application of the axioms on some ‘basic
knowledge’ produces a ‘new’ valid piece of knowledge
not there in the basic set.
• The first set we will consider is called the A-set
proposed by W. Armstrong1.
1 William
Armstrong: Dependency Structures of Data Base Relationships, page 580-583. IFIP Congress, 1974.
12
A-Axioms
A1. Reflexivity
XX
A2. Augmentation
If (Z
A3. Additivity
If { (X Y) and (X  Z)} then X  YZ
A4. Projectivity
If (X  YZ) then X  Y
A5. Transitivity
If (X  Y) and (Y  Z) then (X  Z)
W; X  Y) then XW  YZ
A6. Pseudotransitivity If (X  Y) and (YZ  W) then XZ  W
13
Inference Machine
INPUT:
Relation schema R
Set F of FDs on R
YES
A-Axioms
A1
A2
...
A6
INFERENCE
MACHINE
Output
Is the “new” rule XY
derived from what is
known (R, F) by using
the intelligence provided
by the A-Axioms ?
NO
INPUT:
A “new” rule of the form X
With X and Y in schema(R)
Y
If NO we must conclude
that (F  XY) is
not true
14
Example1 - Using the A-Axioms
Consider R = (Street, Zip, City) ; and the dependencies
F = { City Street  Zip, Zip  City }
We want to show: Street Zip  Street Zip City
Proof:
15
Example1 - Using the A-Axioms
Consider R = (Street, Zip, City) ; and the dependencies
F = { City Street  Zip, Zip  City }
We want to show: Street Zip  Street Zip City
Proof:
1. Zip  City
2. Street Zip  Street City
3. City Street  Zip
4. City Street  City Street Zip
– Given
– Augmentation of (1) by Street
– Given
– Augmentation of (3) by
City Street
5. Street Zip  City Street Zip – Transitivity of (2) and (4)
16
Example2 – Using A-Axioms
Consider the relation schema <R,F> where R = (ABCDEGHI) and dependencies
F = { ABE AGJ BE  I E  G GI  H }
Show that
AB  GH
is derived by F.
If YES give a proof
If NO provide a counter-example
17
Example2A – Using A-Axioms
Consider the relation schema <R,F> where R = (ABCDEGHI) and dependencies
F = { ABE AGJ BE  I E  G GI  H }
Show that
AB  GH
is derived by F.
Q.E.D.
quod erat
demonstrandum
Step
Statement
Explanation
1
AB  E
Given
2
E G
Given
3
AB  G
Transitivity on (1) and (2)
4
AB  BE
Augmentation (1) by B
5
BE  I
Given
6
AB  I
Transitivity on (4) and (5)
7
AB  GI
Additivity on (6) and (3)
8
GI  H
Given
9
AB  H
Transitivity on (7) and (8)
10
AB  GH
Additivity on (3) and (11)
18
Example2B – Using A-Axioms
Consider the relation schema <R,F> where R = (ABCDEGHI) and dependencies
F = { ABE AGJ BE  I E  G GI  H }
Show that
AB  GH
is derived by F.
again!
quod erat
demonstrandum
Q.E.D.
Step
Statement
Explanation
1
AB  E
Given
2
AB  AB
Reflexivity
3
AB  B
Projectivity on (2)
4
AB  BE
Additivity on (1) and (3)
5
BE  I
Given
6
AB  I
Transitivity on (4) and (5)
7
EG
Given
8
AB  G
Transitivity on (1) and (7)
9
AB  GI
Additivity on (6) and (8)
10
GI  H
Given
11
AB  H
Transitivity on (9) and (10)
12
AB  GH
Additivity on (8) and (11)19
Example3 – Using A-Axioms
Consider the relation schema <R,F> where R = (ABCDEGHI) and dependencies
F = { ABE AGJ BE  I E  G GI  H }
Step
Show that
AEI  H
is derived by F.
Statement
Explanation
1
2
3
4
Your turn!
5
6
7
8
9
10
11
12
20
Reducing the A-Axioms
The set of A-Axioms is not minimal, therefore some of its rules could be
eliminated.
Observations
• Rule A5 (transitivity) is a special case of rule A6 (pseudo-transitivity).
• Rules A3 (additivity) and A4 (projectivity) can be derived from A1
(reflexivity), A2 (augmentation), A6 (pseudo-transitivity).
Proof
(a) First observation is trivial (just make Z= Ø)
(b) Axiom A3 (Additivity) states that two rules, say X Y and X Z, can be
combined in one X YZ. Lets use A2 on X Y to produce XZ YZ. Repeat
A2 this time on X Z to produce X XZ. Now apply A5 on X XZ and
XZ YZ; we get X YZ. Therefore, we conclude that X YZ without using
the rule A3 itself (see next page)
21
Reducing the A-Axioms
The set of A-Axioms is not minimal, therefore some of its rules could be
eliminated.
Statement
Axiom A3 is redundant. Rule A3 (Additivity) states that two rules, say X Y
and X Z, can be combined in one X YZ.
Proof
We can prove that this fact is true without using A3
1.
2.
3.
4.
5.
X
XZ
X
X
X
Y
YZ
Z
XZ
YZ
Given
(A2) Augmenting (1) by Z
Given
(A2) Augmenting (3) by Z
(A5) Transitivity on (4) and (2)
22
Characterizing the A-Axioms
•
The set of A-Axioms is complete
Therefore every FD that is implied by a set F of FDs can be derived from the
FDs in F and one or more applications of the A-Axioms ( FA XY )
•
A-Axioms are correct
Applying the axioms to FDs in a set F can only produce FDs that are implied
by F.
•
The set of A-axioms is not minimal
Some rules are added for convenience but they can be removed without
diminishing the expressive power of the A-axioms
23
Correctness of the A-Axioms
The axioms can not be used to prove a false derivation. In such a case showing a
counter-example is sufficient to establish the falsity of a statement.
Example
Assume schema R(XYZW). Does ( XY ZW ) A X Z ?
The correct answer is NO. To show support for our argument we produce a
counter-example. For instance:
X
Y
Z
W
1
2
3
4
1
5
6
7
On the example table there are no violations to the fact that XY implies a unique ZW
(12 34 and 15 67). However X=1 determines two different Z values, 3 and 6.
Therefore X Z is not a valid dependency as shown in the counter-example.
24
Closure F+
•
Let F be a set of FDs for a relation r(R). The closure of F, denoted F+,
is the smallest set containing F such that the A-axioms cannot be
applied to the set to produce a new rule not included in the set
already
•
Since F+ must be finite, we can compute it by starting with F, applying
A1, A2, and A6, and adding the derived FDs to F until no new FDs can
be derived.
•
•
The closure of F depends
on the scheme R.
If R = (A B) then F+ will always
contain B  B, but if R = (A C),
F+ never contains B  B.
F+
F
25
Closure F+
• The set F derives an FD X  Y if X  Y is in F+.
• Since our inference axioms are correct,
if F derives X  Y, then F implies X  Y ( F
A
X Y)
• Note that F+ = (F+)+
• It is desirable to determine whether F
computing F+
A
X Y without
• Computing the entire set F+ is time-consuming and
tedious
26
Closure F+
Example: Consider the relation schema <R,F>
where R = (A B C) and F = { AB  C, C  B }.
By the use of brute-force we produce all rules out of F.
F+ is the set of rules listed below
A
B
C
C
A
B
C
B
AB
AC
BC
AB
AB
AB
…
BC
AB
AC
BC
C
A
B
C
ABC
ABC
ABC
ABC
ABC
…
ABC
ABC
A
B
C
AB
BC
F+
F = { ABC, CB }
27
Closure F+
Example: Consider the relation schema <R,F> where R = (ABC)
and F = { AB  C, C  B }.
Question: Does F  B  C ?
Answer: F+ is the set of rules listed below and B  C is not in
the set; therefore the rule B  C is not implied by F.
A
B
C
C
A
B
C
B
AB
AC
BC
AB
AB
AB
…
BC
AB
AC
BC
C
A
B
C
ABC
ABC
ABC
ABC
ABC
…
ABC
ABC
A
B
C
AB
BC
This rule is NOT reachable
from F
BC
F+
F = { ABC, CB }
28
Closure F+
Aside: How many FDs are there in <R,F>
n
An upper bound is
r 1
n
r
n
r 1
n
r
(2n 1)2
Each sum term represents the possible combinations of r
attributes made out of the total n domains for each of the m
X  Y rule in F.
for n=3 there are (23-1)2 = 49 possibilities, however for R
holding 10 attributes there are over a million possible FDs
29
Closure F+
Definition.
An FD X  Y is trivial if X Y.
• If F is a set of FDs over R and X is a subset of R,
then there is a FD X  Y in F+ such that Y is
maximal: for any other FD X  Z in F+, Y Z.
– This result follows from additivity.
– The right side Y is called the closure of X and is
denoted by X+.
• The closure of X always contains X, by reflexivity.
30
Derivations and DDAGs
•
If F XY, then either X Y is in F, or a series of applications of the
inference A-axioms to F will yield X Y.
•
This sequence of axiom applications and resulting FDs is called a
derivation of X  Y from F.
•
More formally, let F be a set of FDs over scheme R . A sequence P of
FDs over R is a derivation sequence on F if every FD in P either
– is a member of F, or
– follows from previous FDs in P by an application of one of the inference
axioms A1 to A6.
•
P is a derivation sequence for XY if X Y is one of the FDs in P.
•
Definition
Let P be a derivation sequence on F. The use set of P is the collection of
all FDs (originally) in F that appear in P.
31
Derivations and DDAGs
EXAMPLE
Consider schema r(ABCDEG) and functional dependencies
F = { A BC, BD G, C ED }
A derivation sequence for A E is
Step
Explanation
1
2
3
4
5
Try…
32
Derivations and DDAGs
EXAMPLE
Consider schema r(ABCDEG) and functional dependencies
F = { A BC, BD G, C ED }
A derivation sequence for A E is
Step
1
2
3
4
5
Explanation
A
A
C
C
A
BC
C
ED
E
E
(given)
(Projectivity [A4] on 1)
(given)
(Projectivity[A4] on 3)
(Transitivity[A5] on 2 and 4)
The set P for AE (five rules written above) is a derivation sequence
on F. The Use_Set_Of_P is = {A BC, C ED }
33
Derivations and DDAGs
Example. Consider schema <R, F> where R= { A B C D E G H I J }
and F = { ABE, AG J, BE  I, E  G, GI  H}
The following sequence is a derivation sequence for A B  G H.
Step
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
Explanation
AB E
AB AB
AB B
AB BE
BE I
AB I
E G
AB G
AB GI
GI H
AB H
GI GI
G I I
ABG H
given)
(reflexivity)
(projectivity from 2)
(additivity from 1 and 3)
(given)
(transitivity from 4 and 5)
(given)
(transitivity from 1 and 7)
(additivity from 6 and 8)
(given)
(transitivity from 9 and 10)
(reflexivity)
(projectivity from 12)
(additivity from 8 and 11)
This P sequence contains unneeded FDs, such as 12 and 13, and
is also a derivation sequence for other FDs, such as A B  G I.
The Use_Set_Of_P is {C ED, BE I, E G, GI H }
34
B-Axioms
Definition:
The B-Axioms set is a small and complete collection of inference rules. It is not a
subset of A1 to A6, however it is equally expressive. For a relation (R), with W, X, Y,
and Z subsets of R, and C an attribute in R then:
B1. Reflexivity
X
X
B2. Accumulation
If (X
YZ) and (Z
CW) then X
B3. Projectivity
If (X
YZ) then X
Y
YZC
Motivation:
This is another approach to the problem of finding a sequence of derivations using
a smaller set of axioms.
Significance:
Since B-Axioms are complete, we can always find a derivation sequence using only
the three B-axioms to assert whether or not FB XY.
35
B-Axioms
Example:
Let R = (ABCDEGHI) and
F = {ABE
AG  J
BE  I
EG
GI  H}
Problem
Find a derivation sequence P
showing FB AB  GH using
only B-axioms.
Answer
See P sequence on the right
Step
Explanation
1
2
3
4
5
6
7
8
9
10
11
Comment
(a) Use_Set_Of_P contains
rules on lines 2, 5, 9, 11.
(b) Too many steps!
12
13
14
36
B-Axioms
Example:
Let R = (ABCDEGHI) and
F = {ABE
AG  J
BE  I
EG
GI  H}
Problem
Find a derivation sequence P
showing FB AB  GH using
only B-axioms.
Answer
See P sequence on the right
Comment
(a) Use_Set_Of_P contains
rules on lines 2, 5, 9, 11.
(b) Too many steps!
Step
Explanation
1
EI  EI
Reflexivity (B1)
2
EG
Given
3
EI  EIG
Accumulation (B2)
4
EI  GI
Projectivity (B3) from (3)
5
GI  H
Given
6
EI  GHI
Accumulation from (4) and (5)
7
EI  GH
Projectivity from (6)
8
AB  AB
Reflexivity
9
AB  E
Given
10
AB  ABE
Accumulation from (8) and (9)
11
BE  I
Given
12
AB  ABEI
Accumulation from (10) and (11)
13
AB  ABEIG
Accumulation from (4) and (12)
14
AB  ABEGHI
Accumulation from (7) and (13)
15
AB  GH
Projectivity from (14)
Ok, but useless
37
RAP-Derivation Sequence
RAP: Stands for: Reflexivity, Augmentation, Projectivity
Definition:
Consider derivation sequences for X Y on a set F of FDs
using the B-axioms that satisfy the following constraints:
1.
2.
3.
The first FD is X X
The last FD is X Y
Every FD other than the first and last is either an FD in F (given) or
and FD of the form X Z that was derived using axiom B2
(Accumulation).
Such a derivation is called a RAP-derivation sequence
38
RAP-Derivation Sequence
Example:
Let R = (ABCDEGHI) and
F = { ABE AGJ BE  I E  G GI  H }
Find a RAP-sequence for AB  GH
Step
Comments
1. The table contains a RAP sequence
for ABGH.
2. Each rule in P is either given in F
or the result of applying B2 on
previous rules in P.
3. First and Last lines agree with the
definition of RAP sequence.
4. Use_Set_Of_P contains rules in
lines 2, 4, 6, 8.
Explanation
1
AB  AB
B1
2
AB  E
Given
3
AB  ABE
B2
4
BE  I
Given
5
AB  ABEI
B2
6
EG
Given
7
AB  ABEIG
B2
8
GI  H
Given
9
AB  ABIGH
B2
10
AB  GH
B3
39
RAP-Derivation Sequence
Example:
Let R = (ABCDEGHI) and
F = { ABE AGJ BE  I E  G GI  H }
Find a RAP-sequence for BHE  GI
Step
Explanation
1
Your turn…
2
3
4
5
6
7
8
9
10
40
RAP-Derivation Sequence
Example:
Let R = (ABCDEG) and
F = { A BC, BD G, C ED }
Find a RAP-sequence for AD  GE
Step
Explanation
1
Your turn…
2
3
4
5
6
7
8
9
10
41
RAP-Derivation Sequence
Example:
Let R = (ABCDEI) and
F = { A D, AB E, BI E, CD I, E
Find a RAP-sequence for AE  DCI
C}
Step
Explanation
1
Your turn…
2
3
4
5
6
7
8
9
10
42
Derivation DAGs
A directed acyclic graph (DAG) is a directed graph with no
directed paths from any node to itself.
A labeled DAG is a DAG with an element from some labeling set L
associated with each node.
Valid DAG
(disconnected but OK)
Not a Valid DAG
(Path: A-D-A makes a cycle)
43
Derivation DAGs
• DAGS are a convenient way of graphically representing a
derivation sequence of the form F B X Y
• Whenever there is a RAP derivation sequence there is an
equivalent DDAG (and conversely)
44
Derivation DAGs
EXAMPLE
Consider schema r(ABCDEG) and functional dependencies
F = { A BC, BD G, C ED }. Show a DDAG for AD GE
D
G
B
A
C
E
NOTE:
The Use_Set of the derivation sequence is { A BC, BD G, C E }
45
Derivation DAGs
Rules for Constructing a DDAG
Rule 1. Any set of unconnected nodes with labels from r(R) is an F-based DDAG
A1
A2
Rule 2. Let H be a DDAG including nodes labeled A1 … Ak. Let rule A1…Ak B be
part of F. Form graph H’ by adding a new node labeled “B” and new edges
<A1,B>,…,<AK,B>.
A1
…
New edges
B
New node
AK
AN
Rule 3. Nothing else is an F-based derivation DDAG.
46
Derivation DAGs
Example
Consider the relation schema r(ABCDEGHIJ) subject to the dependencies
in F = { AB E, AG J, BE I, E G, GI H }.
Draw a DDAG for rule AB GH
A
B
E
G
I
H
Note: The Use_Set of the derivation sequence is { AB E, BE I, E G, GI H }
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Derivation DAGs
Example
Consider the relation schema r(ABCDEGHIJ) subject to the dependencies
in F = { AB E, AG J, BE I, E G, GI H }.
Draw a DDAG for the new rule BIG
JA
B
J
I
H
A
G
NOTE: No path from source to destination is possible, therefore the new rule
BIG AJ is not derivable from F.
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Derivation DAGs
Example
Consider the relation schema R(ABCDEGHIJ) subject to the dependencies
in F = { AB E, AG J, BE I, E G, GI H }.
Draw a derivation DDAG for the new rule AB
HC
G
A
B
E
C
I
H
NOTE: Node C is not reachable from the source. Therefore the rule AB CH cannot
be deduced from F.
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X+
Closure of a Set of Attributes
• In order to simplify the asserting of whether or not a
rule X Y follows from a set F of FDs, we will compute
X+ the closure of a set of attributes X
• The set X+ is the maximal set of attributes which can
be derived from X using a RAP derivation sequence
starting on X
• We will say that X Y is in F+ whenever Y is in X+
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Computing X +
The following algorithm to compute X+ has poor performance but is easy to
understand
Algorithm:
Input:
Output:
X-Closure
A set of attributes X and a set of FDs F
The closure of X under F denoted X+
function X-CLOSURE (X, F)
begin
OldDep = ; NewDep = X;
while ( NewDep OldDep ) do begin
OldDep = NewDep
for every FD A B in F do
if ( NewDep A ) then NewDep = NewDep
end while;
return ( NewDep )
end function;
B;
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Computing X +
EXAMPLES
Consider the relation schema r(ABCDEGHIJ) subject to the dependencies in
F = { AB E, AG J, BE I, E G, GI H }.
Compute closure of AB
(AB) + = A B
ABE
ABEI
ABEIG
A B E I G HJ
reflexivity
using AB E
BE I
E G
GI H
nothing else could be added to AB+
Note: Observe that AB
ABEIHG. This rule is a compact notation for the 27 FDs having AB as LHS.
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Computing X +
EXAMPLES
Consider the relation schema r(ABCDEGHIJ) subject to the dependencies in
F = { AB E, AG J, BE I, E G, GI H }.
Compute closure DEC
(DEC) + = D E C
DECG
using E
G
nothing else could be added
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Member Algorithm
Checking Membership
In order to verify whether or not a functional
dependency X Y could be derived from a set F of FDs
the following simple test could be applied
F
B
X Y if Y is part of X+
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Member Algorithm
Member Algorithm
Input: Rule X Y and functional dependencies F
Output: TRUE whenever the rule is derived from F
Method:
begin
if ( Xclosure (X, F)
return( True )
else
return( False );
Y ) then
end;
Example
Question: Does rule AB EH follow from F = { AB E, AG J, BE I, E G, GI H }
Answer: YES. Observe that (AB)+ = ABEIGHJ EH.
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Linear Closure – XF+
Input: A set of attributes X and a set of functional dependencies F
Output: The closure of X under F demote XF+
Procedure LINCLOSURE ( Attribute X, SetOfFDs F)
BEGIN
/* Initialization */
for each FD W Z in F do begin
COUNT[ W Z ] = lenghtOf(w);
for each attribute A in W do
add rule W Z into LIST[ A ];
end;
NEWDEP = X; UPDATE = X;
/* Computation */
while ( UPDATE  Ø) do begin
Choose an attribute A in UPDATE;
UPDATE = UPDATE - A;
for each FD W Z in LIST[A] do begin
COUNT[W Z] = COUNT[W Z] - 1;
if ( COUNT[w Z] = 0 ) then
ADD = Z - NEWDEP;
NEWDEP = NEWDEP  ADD;
UPDATE = UPDATE  ADD;
end if;
end for;
end while;
END
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Linear Closure – XF+
Example
Consider the schema r(ABCDEI) subject to the dependencies
F = { A D, AB E, BI E, CD I, E C}
Find the closure of AE using the Linear Closure algorithm
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Linear Closure – XF+
Example (continuation…) Tracing the execution of the linear time closure algorithm
UPDATE= AE
NEWDEP= AE
List[A]
Rule 1 A D Count[1] = 0
therefore add D to both strings
Rule 2 AB E Count[2] = 1
UPDATE= ED
NEWDEP= AED
List[E]
Rule 5 E C Count[5] = 0
therefore add C to both strings
UPDATE= DC
NEWDEP= AEDC
List[D]
Rule 4 CD I Count[5] = 1
UPDATE= C
NEWDEP= AEDC
List[C]
Rule 4 CD I Count[4] = 0
therefore add I to both strings
UPDATE= I
NEWDEP= AEDCI
List[I]
Rule 3 BI
UPDATE= Ø
NEWDEP= AEDCI
E Count[3] = 1
therefore (AE) + = AEDCI
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Linear Closure – XF+
Homework
You will create a CASE tool for designing ‘good’ databases.
The first step involves the implementation of the XLinearClosure() algorithm.
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