20 DB Formal Design

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Theory I: Database Foundations 20. Formal Design 20.

Theory I: Database Foundations

Jan-Georg Smaus (Georg Lausen)

2.8.2011

1. Formal DesignMotivationFunctional DependenciesDecomposition

Jan-Georg Smaus (Georg Lausen) 1
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Theory I: Database Foundations 20. Formal Design 20.

Formal Design

We want to distinguish good from bad database design.

What kind of additional information do we need?

Can we transform a bad into a good design?

(At which cost?)

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Theory I: Database Foundations 20. Formal Design 20.1. Motivation

20.1 Motivation

Relations and anomalies

City

CityNo CityName CCode CSurface

7 Freiburg D 357

9 Berlin D 357

40 Moscow RU 17075

43 St.Petersburg RU 17075

Continent

ConName CCode ConSurface Percent

Europe D 3234 100

Europe RU 3234 20

Asia RU 44400 80

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Theory I: Database Foundations 20. Formal Design 20.1. Motivation

Having removed anomalies

City

CityNo CityName CCode

7 Freiburg D

9 Berlin D

40 Moscow RU

43 St.Petersburg RU

Country

CCode CSurface

D 357

RU 17075

Location

CCode ConName Percent

D Europe 100RU Europe 20

RU Asia 80

Continent

ConName ConSurface

Europe 3234Asia 44400

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Theory I: Database Foundations 20. Formal Design 20.2. Functional Dependencies

20.2 Functional Dependencies

Definition

Let a relation schema be given by its format X and let Y,Z X.

Let r Rel(X). r fulfills a functional dependency Y Z, if for all , r:

[Y] = [Y] [Z] = [Z].

Let F be a set of functional dependencies over X. The set of all relationsr Rel(X), which fulfill all functional dependencies in F, is calledSat(X, F).

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20.2.2 Membership Test

The functional dependency Y Z, is implied by F, written F |= Y Z,

if for each relation r, whenever r Sat(X, F) then r fulfills Y Z.

The set F+ = {Y Z | F |= Y Z} is called closure of F.

The question Y Z F+? is called membership test.

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Key

Let X = {A1, . . . ,An}. Y X is called key ofX (wrt. F), if

Y A1 . . .An F+

,Z Y Z A1 . . .An / F

+.


Th I D t b F d ti 20 F l D i 20 2 F ti l D d i
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Armstrong axioms

Let r Sat(X, F).

(A1) Reflexivity: IfZ Y X, then r fulfills functional dependencyY Z.

(A2) Augmentation: IfY Z F, V X, then r fulfills functionaldependency YV ZV.

(A3) Transitivity: IfY Z,Z V F, then r fulfills functionaldependency Y V.

(A1): trivial functional dependencies.


Theory I: Database Foundations 20 Formal Design 20 2 Functional Dependencies
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Correctness and Completeness

Every functional dependency derivable by the Armstrong axioms is anelement of the closure (correctness).

Every functional dependency in F+ is derivable by the Armstrong axioms(completeness)


Theory I: Database Foundations 20 Formal Design 20 2 Functional Dependencies
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Membership test

Starting from F apply (A1)(A3) until Y Z is derived, or F+ is derived andY Z F+.

In practice this test is too complex and therefore other tests have beendeveloped.


Theory I: Database Foundations 20 Formal Design 20 3 Decomposition
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Theory I: Database Foundations 20. Formal Design 20.3. Decomposition

20.3 Decomposition

Let = {Y1, . . . ,Yk} a decomposition ofX, i.e., Y1 . . . Yk = X. Let F be aset of functional dependencies.

Let r Sat(X, F) and let ri = [Yi]r, 1 i k.

is called lossless, if for any r Sat(X, F) it holds that:

r = [Y1]r . . . [Yk]r.


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Example

X = {A, B, C} and F= {A B, A C}.

r Sat(X,F):

r =A B C

a1 b1 c1

a2 b1 c2

1 = {AB, BC} and 2 = {AB, AC}.

r [AB]r [BC]r,

r [AB]r [AC]r.


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y g p

TheoremLet a format X and set F of functional dependencies. Let = (Y1,Y2) be adecomposition ofX.

is lossless, iff

(Y1 Y2) (Y1 \ Y2) F+, or (Y1 Y2) (Y2 \ Y1) F+.

There is a similar notion called dependency-preserving.

The aim of good database design is to decompose relations to remove

redundancies.

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20 DB Formal Design

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