A Relational Model of Data for Large Shared Data Banks-Codd.pdf ...

SIGMOD Record, March 2009 (Vol. 38, No. 1)

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SIGMOD Record, March 2009 (Vol. 38, No. 1)

SIGMOD Record, March 2009 (Vol. 38, No. 1)

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E, F. Codd ResearchDivision San Jose, California

REDUNDANCY AND CONSISTENCY OF RELATIONS STORED IN LARGE DATA BANKS

RJ 599(# 12343) August 19, 1969

ABSTRACT: The large, integrated data banks of the future will contain many relations of various degrees in stored form. It will not be unusual for this set of stored relations to be redundant. Two types of redundancy are defined and discussed. One type may be employed to improve accessibility of certain kinds of information which happen to be in great demand. When either type of redundancy exists, those responsible for control of the data bank should know about it and have some means of detecting any “logical” inconsistencies in the total set of stored relations. Consistency checking might be helpful in tracking down unauthorized (and possibly fraudulent) changes in the data bank contents.

DERIVABILITY,

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SIGMOD Record, March 2009 (Vol. 38, No. 1)

Copies may be requestedfrom IBM ThomasJ. WatsonResearchCenter, PostOffice Box 218, Yorktown Heights, New York 10598

LIMITED DISTRIBUTION NOTICE - This report has been submitted for publicatiori elsewhere and has been issued as a Research Report for early dissemination of tts contents. As a courtesy t6 the intended publisher, it should not be widely distributed until after the dote of outside publication.

SIGMOD Record, March 2009 (Vol. 38, No. 1)

21

Some Linguistic Operations

2. 3.

Aspects

View of Data

Data Bank Control

6.

Redundancy

Derivability,

5.

Composition

3.4 Relations

and Consistency

Named and Stored

Join

3.3

Expressible,

Projection

3.2

4.

Permutation

3.1

on Relations

A Relational

1.

CONTENTS

19

11

10

9

6

5

5

5

4

1

22

SIGMOD Record, March 2009 (Vol. 38, No. 1)

SIGMOD Record, March 2009 (Vol. 38, No. 1)

23

model

a high

data

[l,

for

of

of the

information

evaluation

management

data

within

system.

is

the

is,

of

merits

of present

a clearer

is

has

derivation

of which

hand,

part

of

and consist-

forms

A a

maximal

a basis

second

it

other.

representations

the

that

and machine

yield

relative

permits

for

least

other

in the

limitations

of competing

that

redundancy,

view

the

and also

and logical

will

on the

on the

connections

relational

of

not

model,

discussed

view

for

only:

a means

graph

machine

to the

of

an explanamodel)

provides

(or

with

provides

one hand,

which

it

structure

It

respects

view

structure

of data

derivability, are

This

concerned

vogue.

on the

relational

systems,

the

standpoint)

a single

natural

in

language

confusions,

scope

Finally,

a logical

these

The network

treating

of the

relations.

(from

programs

is

several

Accordingly,

and organization

derivation

the

mistaking

paper.

a number

this

of relations,--

basis

advantage

its

in

any additional

retrieval

purposes.

with

paper of data.

21 presently

between

level

spawned

of

ency

sound

further

representation

independence

for

representation

view

of this

to be superior

superimposing

describing

without

of

network

appears

data

part

a relational

of

first

tion

The

INTRODUCTION

or

All

The ordering

(3) (4)

(5)

The

(2)

R

rows

corresponding

conveyed

domain.

by labeling

The significance

is

is

R

ordering

it

it

with

it

of

S 2,

the

partially

. ..)

part represents

sn of

- it

which

name of the

is

defined;

Sl,

significant

R;

properties:

of be remembered an essential

must

make use

2 binary,

degree

Rela-

of

domain

j th n.

element

of n-tuples,

distinct),

mathematical

second as the

its

a set

degree

An array

not

but

column

is

Sj

sl’

is

frequently

immaterial;

is

accepted

necessarily

its

to have

to

from

if

an n-tuple

of each

on which

domains

in

unary,

following

columns

the

of

distinct;

of

to the

are

we shall

expounded.

corresponds

rows

ordering

said

n n-ary.

called

is

refer

relations,

has the

being

sets

element

n

representation

row represents

Each

(1)

view

relation

relational

the

particular

of

reasons,

and degree

often

R

here

. . . . Sn (not

.I\‘e shall

first

above,

its

an n-ary

this

S2,

is used

of Data

on these

Sl,

1 are

representation

that

an array

expository

3 ternary,

As defined

degree For

has

sets

relation

View

and so on.

of degree

R.

S2,

of which

tions

of

from

each

a relation

is

R

term Given

The

A Relational

sense.

1.

1.

24

SIGMOD Record, March 2009 (Vol. 38, No. 1)

ship

3 7 1

A Relation

2

2

4

FIGURE 1:

subassembly)

of part

y.

is that

y)

(x,

component

part

is

respect

part.

component

(or

relation,

The

but

The meaning of

It is a binary

x is an immediate

called

domains),

to the relation.

identical

component.

with

(indicating

is called

two domains

depicted

relation

meanings

each of whose

distinct

possess

headings

may

of columns

by the name of

)

two columns

the ordering

2 shows,

why should

in Figure

domains,

As the example

have identical

matter?

the corresponding

are labeled

12

4

9

23

17

quantity

projects

of Degree 4

1

5

7

5

5

project

If the columns

3

1

ask:

2

part

to specified

of parts

of degree

in specified

a relation

the shipments-in-progress

1 illustrates

1

( supplier

One might

ship

quantities.

reflects

in Figure

suppliers

which

The example

from specified

called

2. 4

2:

deletion

part defined on the following

domains:

there might be an entity

considerat ion,

type relation.

an entity

such a relation

part)

call

type relation

called

In the example under

and we shall

of a

for a class The set of entities

description

type can be viewed as a relation,

w.

The individual (such as a particular The prototype

object

and suppliers.

of

informa-

ones, and alteration n-tuples.

of existing existing

is called an entity

[3].

for an individual an entity

given entity

of objects

is called

description

projects,

As time

may be subject to insertion

degrees.

of time-varying

Domains

for example, a data bank which contains tion about parts,

Consider,

Two Identical

7

7

are of assorted

each n-ary relation n-tuples,

with

6

4

6

6

2 3

5

5

5

part)

3

2

1

part

a data bank is a collection

These relations

that

A Relation

(

of components of any of its

additional

progresses,

relations.

We now assert

Figure

component

-

‘..

SIGMOD Record, March 2009 (Vol. 38, No. 1)

25

quantity

quantity

(5)

(6)

numbers

part

are.

that

(not

names.

the example,

if

number would

different

parts

key.

each entity.

were

This

is

it

always

in

such

given

distinct

be the case in

type may possess

is superfluous

would

part

a x.

is either

or a combination

if

while

called

each

of

to what

names, and

type part.

identify

be a key,

An entity

attributes

a combination)

more than one non-redundant

identifying

uniquely

is

is,

are present,

it

of attributes)

of the entity

(or combination)

which

part

colors

above correspond

weights,

part

(or combination

listed

part

all

A key is non-redundant

part

none of the participating

uniquely

that

not be.

attribute

would

a simple

color

above,

Such an attribute

entity.

type has values

one attribute

the attributes

entity

In the example

possible

The domains

all

a given

Normally,

are commonly called

is unlikely

it

at some instant,

that,

domains may be

While

of which

Each of these

some or all

as well.

conceivable

of values,

domains

on order

on hand

weight

in the data bank at any instant.

a pool

other

part

(4)

color

represented

in effect,

and possibly

part

(3)

name

part

(2)

number

part

(1)

or refer

two keys

exhibited

exhibited

type part,

The

which are

are inter-entity

we have discussed examples of relations

Both of these relations

to serving to identify

two keys referring

project.

relations.

such binary

of all

history

domain is the set

defined on the domain date and The salary

relation

eli(pIoyee is defined An element of the salary history

type relation

the domain salary.

domain is a binary

might be salary history.

domains on which the entity

be defined For example, one of the

may, in turn,

Thus, somedomains may have These relations

framework.

Non-atomic values can be discussed

on non-simple domains, and so on.

as elements.

the relational relations

within

(non-decomposable) values.

defined on simple domains - domains whose elements are atomic

So far,

relations.

component.

this. three keys, one

help clarify

part,

key

its

An

to distinct

an assembly containing

the first

in Figure 2 involves

refer

is that

relations.

entity

type serving distinct

either

1 involves

types supplier,

in Figure

a component, the second to identify

the commonentity

relation

for each of the entity

that

which

relation

inter-entity

inter-entity

called

in a data bank are between

to a commonentity

at least

of every

therefore,

relations

The examples in Figures 1 and 2 will

types

include

property

and are,

The relation

roles.

entity

domains

essential

types,

The remaining

3.

26

SIGMOD Record, March 2009 (Vol. 38, No. 1)

by

functions

calculus

normal in

form

of well-formed

needed can be defined

in prenex

the class

is in a precisely

expressions

request

which

H

[4].

to security

of the

and invoked

Any arith-

formulas

one-to-one

can be used

subject

specified

is

of data from the data

the

permanently,

R permits

less

H permits

together

*The second-order predicate calculus (rather than first-order) is needed because the domains on which relations are defined may themselves have relations as elements (see section 1 ) .

metic

predicate

with

specification

correspondence

in a set

The class

constraints.

of qualification

on such a retrieval

Action

perhaps

be as

R and the host

domains.

in storage.

indicate,

on those

of any subset

bank.

retrieval

for

relations

which

degrees

the declaration

by

would

paper to describe

command or problem

of domains,

features

sublanguage

salient

are represented

of various

R permits

specification

how these

H.

its

all

be a strong

for

Such a

sub-

modification)

power itself

syntactic

of this

(programming,

appropriate

and would

of linguistic

retrieval

as described

calculus.*

of data,

predicate

is not the purpose

the retrieval

declarations

relations

supporting

with

language

Let us denote

follows.

it

in detail,

While

(with

view of a universal

languages,

languages

embedding

retrieval

of host

such a language

oriented).

in a variety

for

proposed

candidate

other

a yardstick

would

language

provide

based on the second-order

the development

of a relational

Aspects

language

permits

The adoption

Some Linguistic

above,

2.

4.

between

often

be burdened

and sets.

directed

paths

For a relation

a relational

view

data

view,

users more relation

Two such paths

every

relation is stored, he using any combination remaining arguments (like Everest) (missing from many we shall call (logically)

as a nested

in which

n. n-ary

n, the number of

of a single

by the user

adopted

is factorial

*Once a user is aware that a certain will expect to be able to exploit it of its arguments as “knowns” and the as “unknowns,” because the information is there. This is a system feature current information systems), which symmetric exploitation of relations.

R.

depen-

names are associated relations.

since

and using

network

of degree

is

toward

in

deletion

relations.

is in the naming of

adopted

exploitation*

than with

necessary,

(n > 2) has to be expressed

if

it

the view

coining

symmetric

rather

with

if

are declared

With the usual

to be named and controlled

relation.

Again, relation

paths

binary

are needed to support

with

that

relations

used to retrieve

names than are absolutely

will

data elements

has on the language

effect

specified

the com-

machine

without

or sub-communities)

for

in their

from declared

user

are effective

by others,

elements

may be triggered

One important

dencies

which

may be present

relations

take

purposes

Insertions

for query

changes.

to declared

possible

to the individual

the form of removing

(as opposed

Some deletions

take

munity

that

Deletions

to any ordering

representation.

regard

for

may be fetched

new elements

may be held

A set so specified

or it

R.

the form of adding

only

in

SIGMOD Record, March 2009 (Vol. 38, No. 1)

27

for

with

if

these

the resulting

a permutation

relation

is applied

yields

with

two

is said

to the

the converse

operations.

systems

not

in

relations.

should,

Information

would

key role

for

and a ternary

Most users

representation

these

operations

) ) )

nota-

may not be a rela-

set

data bank control

two columns

has an array

relation,

More generally,

relation.

columns of an n-ary

Interchanging

relation

familiar

with

operations.

relations.

of their

are specifically

because

below

relation

the result

of the usual

of a binary

these

concerned

columns.

A binary

Permutation

be thoroughly

however,

3.1

and people

with

from other

concerned

relations

discussed

are introduced

designers

be directly

deriving

These operations

all

Nevertheless,

the union

is not a relation.

example,

The operations

relation

tion;

to them.

are sets,

on Relations

employ 7 names.

R ( project,

quantity

5 names in n-ary

in the form

entails

the 4-ary

n-ary

then 2n-1 names direct

For example,

n+l with

relations,

and, thus,

Q ( part,

notation

Since relations

Operations

binary

1.

1, which

be represented

of Figure

( supplier,

would

ship

are applicable

3.

in nested

P

notation,

relation

as described

in Section

of only

binary

tion

only

instead

involving

have to be coined

expression

R

except

relation

and

if

that

i

operator

i

th

k

.[:, i (n z k),

i2,

indiceb

in resulting

j rows

is the k-ary

Consider

R (j = 1, 2, . . . . k)

lIL(R)

is removed.

of

then

ik

column is column-~~,ii duplication

j

L = il,

of

relation

is a list

any desired

relation.

any

(striking

a relation

array

*

of the two operations.

is used to obtain

relaboth a

of that

is

of rela-

considerations

represents

1,

the ordering

of a relation

of the given

array

or combination

II

to be a projection

projection,

whose

columns

performance

for

in Figure

relation

to store

and then remove from the resulting

is an n-ary

L

of it,

unnecessary

in the TOWS. The final

A selection

is said

permutation, Thus,

advisable.

by a stored

by any permutation

certain

is logically

Suppose now we select

Projection

make it

duplication which

it

and some permutation

Although

leaves

ship

There are,

exploitation

which

symmetric

answerable

provides

of queries

which

permutation

to the set answerable

the set

out the others)

3.2

could

relation

tion.

identical

tions,

In a system

unchanged.

the identity

relation.

of the relation

of the given

4! = 24 permutations

we include

of columns

if

example,

to be a permutation

5.

28

SIGMOD Record, March 2009 (Vol. 38, No. 1)

2 4 2

5 1 7

has fewer

in Figure

)

relation

1

which have some

is derived.

relations,

from which it

case, the projection

of the Relation

1

supplier

of this

5

( project

A projection

Suppose we are given two binary

Join

1.

Projection

in this particular

A Permuted

than the relation

3:

II31 (ship)

3.

of Figure

in Figure

ship

of

A binary

a join of

R with

S.

If

R, S are binary

Any such ternary

relation

R is joinable

rela-

U

with

while Figure 5 shows

a ternary

relation

S if there exists

S.

loss of information,

of the

R, S, which

which preserves all

lI12(U) = R and II23(U) = S.

relation

R with

without

is called

such that

a binary

a join

are joinable

tion

relation

in the given relations?

to form a ternary

The example in Figure 4 shows two relations

information

relations

domain in common. Under what circumstances can we combine these

3.3

n-tuples

Note that,

Figure

is exhibited

the relation

6.

of

always exists

b, c):R(a,

R with

= R

It

if

is immediate that

(a, b) is a member of

1 2

2 2

Figure 4:

1

part 1

R ( supplier

with

1

2

1

join

join

is not the

project)

Relations

2

1

1

S (part

Two Joinable

)

join

Figure 6 shows another possible in Figure 4.

S.

However, this

the join shown in Figure 5 is the natural

R with

of the relations

R is joinable

b) A S(b, c))

I123(R*S) = S

lIl,(R*S)

for S(b, c).

S from Figure 4.

Note that

only one of

of

and

R and similarly

then

in such a case is the natural

I12(R) = II,(S),

S defined by

R*S = ((a,

R with

One join that

such that

where R(a, b) has the value true

join

S.

relations

SIGMOD Record, March 2009 (Vol. 38, No. 1)

29

1

2

2

S.

that

it is this

S.

*A function

R21(R)

or relation.

S is a function*,

no point

of

with respect to the joining

is a many-one binary

If either

R with

to

of

domain

element which gives rise

Such an element in the joining

It

under

is to be made)

an element (element 1)

(from Figure 4)

possesses more than one relative

is called a point of ambiguity

of joins.

R and also under

rJith the property

reveals

S

)

S (from Figure 4)

)

(the domain on which the join

of these relations

R with

2

2

Another Join of

1

1

2

1

2

project

1

part

1

U ( supplier

of the domain part

the plurality

2

1

2

R with

1

1

2

Join of

2

1

1

The Natural

Inspection

Figure 6:

Figure 5:

1

project

1

part

1

R*S ( supplier

R with

qualification

join would be an entirely

in the joining

of

P) * 3j(S(p,

relation

such that = R

nsl(U)

= T.

“23uJ) = s

“12W)

s))

R, S, T;

that

is,

with the following

a ternary

or

S can sometimes be

R, gZ1(R),

separate

R and S, a relation

s) A R(s, p)),

j) A T(j,

s) * 3p(R(s, P) A S(p, j))

S(P, j) * 3s(T(j,

Rb,

T(j,

then we may form a three-way join of

(5)

(4)

(3)

f12(T) = n,(R)

(2)

= n,(s)

n,(n

(1)

:

and supplier

S”

R (as well

R with

Suppose we are given,

R with

from sources independent of

on the domains project

properties

T

with

“of

R with

In such a case,

In Figure 4, none of the relations

S), and this

resolved by means of other relations. can derive

S.

S is the only join of

R with

because S might be joinable

is a function.

Ambiguity

S, $1(S)

consideration.

as

R with

Note that the reiterated

join of

can occur in joining

is necessary,

s .

the natural

ambiguity

7.

30

SIGMOD Record, March 2009 (Vol. 38, No. 1)

under

Figure

2

Binary

a

b

2

7:

a

FL)

S(p

Relations

y

x),

e

b with

d

e

d

i)

b

a

a

y=d;

T

a Plurality

2=2

x

of

and

e

d

d

2

2

1

~1

of

J-Joins

x

T with

to join-

a relative

of Cyclic

T(L

7 the points

T, and

than

under

to

the relations respect

(say y),

with

must be a relative

under

y

S with

of ambiguity

in Figure

of

x = a;

Note that

property.

R.

a relative

1

R (2

(say point

21, and, furthermore,

S, z

(say

S

points

To be specific,

constraints

the circumstances

much more severe

an example),

of 2-joins.

entail

7, 8 for

J-join

r’elation

to distinguish

more than one cyclic

= T.

“34(V) for

=s

3-join

be a quaternary

a cyclic would

33(v)

must possess

a plurality

R with

have this

z

under

R

ing

R, S, T

for

those

can occur

which

be called

“12W) = R

3-join

will

is possible

this

it

which

While

such that

(see Figures

V

from a --linear

Such a join

exist

it

8.

of degree

a2,

Thus,

n-l

y(R*S*T).

the natural

R

a b b

2 2 2

cyclic

a2,

the cyclic

. . . . an-l,

of

by tying

R, S, T

by

an) A al = an).

relation

n

a

because

produces of degree

which

3-join

d))

7

R, S,

in Figure

hand side To obtain y

e

d

e

d

d

relations

c) A T(c,

is an n-ary

from a relation

the operator

if

a

2

binary

b) A S(b,

of three

is associative.

. . . . a,,l):R(al,

may now represent

= ((al,

the expression

We

y(R)

(*)

a

1

E i>

of the Relations

U’(s

are not needed on the left

we introduce

ends together.

relation

counterpart,

J-join

3-Joins

e

d

e

d

b, c, d):R(a,

linear

Two Cyclic

2-join

parentheses

the natural

its

by

R*S*T = {(a,

is given

where

T

8:

The natural

Figure

b

2 b

a

2 2

a

E i)

1

u (2

,

I

..

SIGMOD Record, March 2009 (Vol. 38, No. 1)

31

counterparts

We

tion

3.4

applied

were

to functions.

is probably We shall

familiar

as if

discuss

a

a generalization

of

of composi-

of

of linear

were

n-joins

the notion

this

of

on the

A similar

it

relation

and cyclic

with

Take the

product

domains

so.

not so,

S and call

B.

The notions

S

applicable.

C.

the linear

degrees.

with

are now directly

B,

of

a binary

s-p domains

this

these

of

R

R, and

were

of

p

to make it r-p

this

on suppose

Take the Cartesian

we can treat

it

on the domains

Similarly,

R as if

of assorted

The reader

of

of the last

domains

A.

If

of the first

R, and call

product

S.

domains

permutations

domains of

r

For simplicity, of the

appropriate

s

p

p < s).

new domain

can be taken

J-join

Composition

n relations

approach

A, B.

can treat

relation

and cyclic

binary

domains

C.

Cartesian

p

this

product

R, and call

of the last

apply

of the

the Cartesian

always

we could

Now, take

(p < r,

are the last

p

domains

domains

the first

p

their

are to be joined

the case of two relations

s) which

Consider

are not

may be appropriate, which

and

relations

3-join

of n binary

and cyclic

of relations

A few words

the joining

(degree

S

(degree

r),

binary.

necessarily

regarding

of linear

to the joining

of the notions

n 2 3) is obvious.

natural

however,

(where

their

Extension

if

and only

S such that if

they

Figure

9:

in Figure

is exhibited

Thus,

there

of

R with

R-S

The Natural

6).

4, their

2 1 2

1 2 2

R with

A

of

supplier

)

from the join

S

S. is

S

(from

exhibited

the

Figure

composition

/’

R

the exist-

natural

by

R with

R with

S defined

of

of

1

Composition

of

are com-

U

is a

given

S does not imply

However,

9 and another

Figure in Figure 10 (derived

/

( project

in Figure

is exhibited:

R, Si from

R-5 =,,.h13(R*S).

of : R with

join

T

a join

R, S.

two relations

exists

Our

are based very

relations.

and joinability

are joinable.

to the natural

composition

the relations

composition

Taking

the natural

S if

two relations

of join

of more than one composition

Corresponding

existence

to binary

and composability

first

T = II13(U).

R with

ence of more than one join

posable

with

of

Suppose we are given composition

above.

directly

it

of composition

and apply

on the definitions

concept

definitions

that

9.

4)

32

SIGMOD Record, March 2009 (Vol. 38, No. 1)

10:

Another

joins

c

several

of point

4)

com-

Figure

d

e

2

2

binary

what

of relations

relations

are

)

need to be actually

operations

to

degrees)

which

g f

g

f

g f

project

joining

to make use of these

of pairwise

may be of different

as extension

now proceed

in considering

We

e

d

C

of

a, b, d, e.

Only One Composition

to pairs

(and which

of composition

Many Joins,

C

2

11:

C

1 C

b

b

(part

have

the ambiguity

which

S, because

1

s

the points

R with

a

part)

made via

in composing

the same pattern

such relations.

stored.

lost

Note that

of two relations

one composition.

an example

a

Figure

Extension

on relations

(from

the number of distinct

S

1

( supplier

not necessarily

follows

is

but only

11 shows

associations

Figure

R

exist,

R with

)

may be as few as one or as many as the number of distinct

joins .

unambiguous

of

1

2

Composition

2

supplier

1

T ( project

When two or more joins

positions

Figure

10.

Associated

Expressible,

the stored

(3)

operators

of the predicate

relational quantifiers

set

The stored

This

included

set,

should in the named set.

then it

should be given

such a relation

on some unnamed but expressible that

whose

a public

be

relation

it

set would

relations

and we assume that

in the data bank.

of all

set

-

in the

and the

by means of simple

relations

connectives

of the expressible

of the named set,

stored

is the collection

in the stored name and thereby

is a subset

can identify

subset.

set

to such proportions included

grows

the traffic

is.

If

be a subset

are actually

small

This a very

names.

the user

normally

values

usually

public

data bank which

is the collection

calculus.

Such names of relations,

of all

from simple

of data to be retrieved.

for

which

of

language

of relations

collections

in the retrieval

such as =, logical

are constructed

expressions

sets

of defining

The named set

set

is the collection

by expressions

set

the named set

(2) set

the expressible

(1)

the purpose

can be designated

Relations

a data bank are three

Named and Stored with

The expressible

relations:

4.

.

SIGMOD Record, March 2009 (Vol. 38, No. 1)

33

and tie

mainly

regarding

R

is

any time

virtually

(for

relations

named

from

the

yields

yields

from

stored

of

we must

value

for

of R

S.

set

take

the

at times

This

natural

relations

that

loads,

exclude

projections, members

a set

S

and changes

interaction

stored

hand,

membership

in

of

named

in the list of operations, a join and a projection.

set,

a correct

R

of permutations,

derivable

and

other

of past

with

the

language

by

community

in the

investment

the

in the

of

On the belong

users,

set.

natural

together

retrieval

belong

the

and Consistency

of the

*We can omit natural composition because it is a combination of

of operations

which

needs

of

in

of time)

and not

Such definitions

ever-increasing

logical

set,

composition,

by name as a result

Redundancy

factors.

a sequence*

and ties

exists

A relation

Derivability,

in these

sequence

joins,

there

5.

place

on the

scope

y).

the

relations

transaction

the

which

in

requirements

mainly

are based

performance

regarding

decisions

relations

of these

relations

the

0, *,

on the

on the

in

set

(independent

named

stored

in the

natural

which

(II,

be within

and particularly

based

using

are

Decisions

must

are

by expressions

which

of relations

defined

operators

programs

users,

set

names

are

relations

permutation-projection,

set

expressions

join

the

involving

stored

Those

if

R.

Note

because

actually

strong

set,

is

base

the

on the

resulting

other

in both

or data

weak

of users.

redundancies

changes

removable

are

from

sets.

the

at

acceptable.

stored

Strong the

of

removable

in

natural

appear

not

at

of other

time. inherent

they

of

members

be the

They are

are

and stored

performance

are

named hand,

the

might

If

set

bank

of queries

and

data

some join

set storage

contains other

it

one at some other

question

administrator.

community

speaking,

in

if from of

the

load

of the to be stored

rest

R

there

of

contains

to save

the

stored

with

it

likely

the

if

insertions,

in the

derivable

values

specified,

a heavy

redundant

a projection

not

with

updates,

in order

is

from

convenience,

relations

interaction

weakly

The join

times

is

and an unnatural

set.

of

be justified

kinds

an environment

at all

of the

appear

providing

redundancies,

they

system

by the all,

needs

logical

Generally

some time

one at

is of the

but

user

is

redundant

derivable of

to perform

which

of relations

one relation set,

other

in

redundancy

members

the

least

Only

as time

to the

as well

A set

relations.

would

relative

-deletions.

space

for

be non-strongly-redundant

often

sense

will

this

in

is

join

made to the

to take.

strongly

named set

which

is

being natural join

redundant

the

one relation

of relations

as to which

that,

are

While

least

A set

no question

S).

changes

members.

at

is

and

at which

11.

34

SIGMOD Record, March 2009 (Vol. 38, No. 1)

three relations

most probably

*A binary relation a function.

will

is complex if neither

about each named relation,

system lacks - and it

of time between the member relations.

information

of statements

sense,

it

semantic

it nor its

converse is

cannot deduce the -

- detailed

If the information

of the redundancies which hold independent

associate with that set a collection

which define all

we shall

is redundant in either

possesses

join

do exist

of some cyclic

However, constraints

Thus, this set of relations

Whenever a set of relations

a weak redundancy.

of the three of them.

from time to time in

with the possi-

Hence, none of them is

between them, since each is a projection

from the other two.

of any two.

of ambiguity occurring

joining

of points

are complex* relations

part by supplier

s

is supplied at least one kind of

j

project

All

derivable

p to at least one

j

s)

the potential

bility

supplies part

R, S, T

p is supplied by at least one supplier

s

relations

consider the case

to project

part

project

supplier

T(j,

S(P, j)

R(s, P)

with meanings as follows:

in which there are binary

As an example of a weak redundancy,

cited previously

12.

to the named set.

It might,

over

this

constraint.

= gl(R)

such that

there is an element

an instantaneous

large and highly

of relations, variable.

snapshot of a collection some of which may be very

taking

problems (which we shall not discuss here) in

II12(R) and (b, c) is in lI12(S).

II12(T) (a, b) is in

(a, c) in the relation b

R, S, T in the system) and

two columns of each of

for every element pair

HZ(T) = h,(S)

Rl(T)

There are practical

(3)

(2)

(1)

determine whether

R, for making this

the values stored for An algorithm

(in whatever way they are represented

check would examine the first

satisfy

statement

is a composition of IIl2(R) with l’I12(S)tt,

we may check from time to time that S, T

C consistent

For example, given stored relations

with the constraint

redundanci’es .

call

and an associated

as it does or does not comply with

we shall

C of relations

statements, according

together

“IIk2(T)

R, S, T

the stated

or inconsistent

set of constraint

Given a collection

but such attempts would be fallible.

a period of time, make attempts to -induce the redundancies,

redundancies applicable

‘\

SIGMOD Record, March 2009 (Vol. 38, No. 1)

35

Bank

S (part,

him that

wishes

different

Laboratory

for

helpful

E. B.

Altman

of rela-

discussions.

Dr. F. P. Palermo and Dr.

ACKNOWLEDGMENT

subcollections

by

of system

Ideally,

now need to be

selections

by

if

the system

the system

interval

so that

of an incon-

and deletions

in the collection.

data bank.

to thank

of the San Jose Research

The author

tions

for

R, T

time

Alternatively, insertions

to make different

consistency

to inconsistency

be possible

reasonable

internally,

such and such relations

in an individual

should

reaction

it

changed to restore

informing

officer.

in making

assist

could

the user

the security

notify

some

be logged

The generation

in the relations

within

say (2, 5) in the relation who supplies

due

input

may arise

An example of inadequate

of relations

2 has no supplier

section).

could

could

not remedied

kind

insertions

were

of this

5 (see previous

when part

appropriate

it

sistency

input.

in a collection

of a new element,

or faulty

project)

is the insertion

project

Control

Inconsistencies

Data

to inadequate

6.

4.

3.

2.

1.

Random

A. Church, Princeton,

“An Introduction 1956.

Access

to Mathematical

Logic

I,”

Fall

Annual Review Pergamon

Processing,”

Proceedings

File Processing,” 5, 13, pp. 77-149,

for

G. H. Mealy, “Another Look at Data,” Joint Computer Conference, 1967.

C. McGee, “Generalized in Automatic Programming Press, 1969.

W.

C. W. Bachman, YSoftware Datamation, April 1965.

REFERENCES

13.

36

SIGMOD Record, March 2009 (Vol. 38, No. 1)