Functional DependenciesFunctional Dependencies

Prof. Yin-Fu HuangProf. Yin-Fu Huang

CSIE, NYUST CSIE, NYUST

Chapter 11Chapter 11

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11.111.1 IntroductionIntroduction

A functional dependency (FD) is basically a many-to-one relationship from one set of attributes to another within a given relvar.

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11.211.2 Basic DefinitionsBasic Definitions

To distinguish between (a) the value of a given relvar at a given point in time and (b) the set of all possible values that the given relvar might assume at different times

Case (a):Let r be a relation, and let X and Y be arbitrary subsets of t

he set of attributes of r. Y is functionally dependent on X, X →Y, if and only if each X value in r has associated with it precisely one Y value in r.

Sample value (See Fig. 11.1)

{S#}→{City}

{S#,P#}→{S#,P#,City,Qty}

{S#}→{Qty}

{Qty}→{S#}

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11.211.2 Basic Definitions (Cont.)Basic Definitions (Cont.)

Case (b): Let R be a relation variable, and let X and Y be arbitrary subsets

of the set of attributes of R. Y is functionally dependent on X,

X →Y, if and only if in every possible legal value of R, each X value has associated with it precisely one Y value.

{S#}→{Qty}

and

{Qty}→{S#} do not hold “for all time” If relvar R satisfies the FD A →B and A is not a candidate key, then

R will necessarily involve some redundancy. To find some way of reducing the set of FDs to a manageable size. The reason is that FDs represent certain integrity constraints, and we

would thus like the DBMS to enforce them.

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11.311.3 Trivial and Nontrivial DependenciesTrivial and Nontrivial Dependencies

An FD is trivial if and only if the right-hand side is a subset of the left-hand side.

{S#,P#}→{S#} The trivial dependencies can be eliminated.

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11.411.4 Closure of a Set of DependenciesClosure of a Set of Dependencies

The set of all FDs that are implied by a given set S of FDs is called the closure of S, written S+ .

Armstrong‘s axioms:

1. Reflexivity: If B is a subset of A, A →B.

2. Augmentation: If A →B, then AC →BC.

3. Transitivity: If A →B and B →C, then A →C. The rules are complete and sound.

4. Self-determination: A →A.

5. Decomposition: If A →BC, then A →B and A →C.

6. Union: If A →B and A →C, then A →BC.

7. Composition: If A →B and C →D, then AC →BD.

8. General Unification Theorem: If A →B and C →D,

then A (C-B)→BD.∪

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11.411.4 Closure of a Set of Dependencies (Cont.)Closure of a Set of Dependencies (Cont.)

Example: R:{A,B,C,D,E,F} and the FDs

A →BC

B →E

CD →EF

AD →F is a member of the closure of the given set.

1. A →BC (given)

2. A →C (1, decomposition)

3. AD →CD (2, augmentation)

4. CD →EF (given)

5. AD →EF (3 and 4, transitivity)

6. AD →F (5, decomposition)

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11.511.5 Closure of a Set of AttributesClosure of a Set of Attributes

Given a relvar R, a set Z of attributes of R, and a set S of FDs that hold for R, we can determine the set of attributes of R that are functionally dependent on Z-the closure Z+ of Z under S.

A simple algorithm for computing the closure Z+

(See Fig. 11.2)

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11.511.5 Closure of a Set of Attributes (Cont.)Closure of a Set of Attributes (Cont.)

Example: R:{A,B,C,D,E,F} and the FDsA →BC

E →CF

B →E

CD →EF

Computing the closure {A,B} + ={A,B,C,E,F} Given a set S of FDs, we can easily tell whether a specific FD

X →Y follows from S, because that FD will follow if and only if Y is a subset of the closure X+ of X under S.

A superkey for a relvar R is a set of attributes of R that includes some candidate key of R as a subset.

K is a superkey if and only if the closure K+ of K under the given set of FDs is precisely the set of all attributes of R.

K is a candidate key if and only if it is an irreducible superkey.

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11.611.6 Irreducible Sets of Dependencies (1/3)Irreducible Sets of Dependencies (1/3)

If every FD implied by S1 is implied by S2, S2 is a cover for S1. Equivalence A set S of FDs to be irreducible if and only if it satisfies the follo

wing three properties:

1. The right side of every FD in S involves just one

attribute.

2. The left side of every FD in S is irreducible, meaning

that no attribute can be discarded from the determinant

without changing the closure S+ .

3. No FD in S can be discarded from S without changing the

closure S+ .

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11.611.6 Irreducible Sets of Dependencies (2/3)Irreducible Sets of Dependencies (2/3)

Example: 1. P#→{Pname,Color}

P#→Weight

P#→City

2. {P#,Pname}→Color

P#→Pname

P#→Weight

P#→City

3. P#→P#

P#→Pname

P#→Color

P#→Weight

P#→City

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11.611.6 Irreducible Sets of Dependencies (3/3)Irreducible Sets of Dependencies (3/3)

For every set of FDs, there exists at least one equivalent set that is irreducible.

Example: R:{A,B,C,D} and the FDs

A →BC

B →C A →B

A →B B →C

AB →C A →D

AC →D

A given set of FDs does not necessarily have a unique irreduci

ble equivalent.

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The End.