Information Retrieval P. BAXENDALE, Editor

A Relational Model of Data

Large Shared Data Banks

E. F. CODD

IBM Research Laboratory, San Jose, California

for

Future users of large data banks must be protected from

having to know how the data is organized in the machine (the

internal representation). A prompting service which supplies

such information is not a satisfactory solution. Activities of users

at terminals and most application programs should remain

unaffected when the internal representation of data is changed

and even when some aspects of the external representation

are changed. Changes in data representation will often be

needed as a result of changes in query, update, and report

traffic and natural growth in the types of stored information.

Existing noninferential, formatted data systems provide users

with tree-structured files or slightly more general network

models of the data. In Section 1, inadequacies of these models

are discussed. A model based on n-ary relations, a normal

form for data base relations, and the concept of a universal

data sublanguage are introduced. In Section 2, certain operations

on relations (other than logical inference) are discussed

and applied to the problems of redundancy and consistency

in the user’s model.

KEY WORDS AND PHRASES: data bank, data base, data structure, data

organization, hierarchies of data, networks of data, relations, derivability,

redundancy, consistency, composition, join, retrieval language, predicate

calculus, security, data integrity

CR CATEGORIES: 3.70, 3.73, 3.75, 4.20, 4.22, 4.29

1. Relational Model and Normal Form

1.1. INTRODUCTION

This paper is concerned with the application of elementary

relation theory to systems which provide shared

access to large banks of formatted data. Except for a paper

by Childs [1], the principal application of relations to data

systems has been to deductive question-answering systems.

Levein and Maron [2] provide numerous references to work

in this area.

In contrast, the problems treated here are those of data

independence–the independence of application programs

and terminal activities from growth in data types and

changes in data representation–and certain kinds of data

inconsistency which are expected to become troublesome

even in nondeductive systems.

Volume 13 / Number 6 / June, 1970

The relational view (or model) of data described in

Section 1 appears to be superior in several respects to the

graph or network model [3, 4] presently in vogue for noninferential

systems. It provides a means of describing data

with its natural structure only–that is, without superimposing

any additional structure for machine representation

purposes. Accordingly, it provides a basis for a high level

data language which will yield maximal independence between

programs on the one hand and machine representation

and organization of data on the other.

A further advantage of the relational view is that it

forms a sound basis for treating derivability, redundancy,

and consistency of relations–these are discussed in Section

2. The network model, on the other hand, has spawned a

number of confusions, not the least of which is mistaking

the derivation of connections for the derivation of relations

(see remarks in Section 2 on the “connection trap”).

Finally, the relational view permits a clearer evaluation

of the scope and logical limitations of present formatted

data systems, and also the relative merits (from a logical

standpoint) of competing representations of data within a

single system. Examples of this clearer perspective are

cited in various parts of this paper. Implementations of

systems to support the relational model are not discussed.

1.2. DATA DEPENDENCIES IN PRESENT SYSTEMS

The provision of data description tables in recently developed

information systems represents a major advance

toward the goal of data independence [5, 6, 7]. Such tables

facilitate changing certain characteristics of the data representation

stored in a data bank. However, the variety of

data representation characteristics which can be changed

without logically impairing some application programs is

still quite limited. Further, the model of data with which

users interact is still cluttered with representational properties,

particularly in regard to the representation of collections

of data (as opposed to individual items). Three of

the principal kinds of data dependencies which still need

to be removed are: ordering dependence, indexing dependence,

and access path dependence. In some systems these

dependencies are not clearly separable from one another.

1.2.1. Ordering Dependence. Elements of data in a

data bank may be stored in a variety of ways, some involving

no concern for ordering, some permitting each element

to participate in one ordering only, others permitting each

element to participate in several orderings. Let us consider

those existing systems which either require or permit data

elements to be stored in at least one total ordering which is

closely associated with the hardware-determined ordering

of addresses. For example, the records of a file concerning

parts might be stored in ascending order by part serial

number. Such systems normally permit application programs

to assume that the order of presentation of records

from such a file is identical to (or is a subordering of) the

Communications of the ACM 377

stored ordering. Those application programs which take

advantage of the stored ordering of a file are likely to fail

to operate correctly if for some reason it becomes necessary

to replace that ordering by a different one. Similar remarks

hold for a stored ordering implemented by means of

pointers.

It is unnecessary to single out any system as an example,

because all the well-known information systems that are

marketed today fail to make a clear distinction between

order of presentation on the one hand and stored ordering

on the other. Significant implementation problems must be

solved to provide this kind of independence.

1.2.2. Indexing Dependence. In the context of formatted

data, an index is usually thought of as a purely

performance-oriented component of the data representation.

It tends to improve response to queries and updates

and, at the same time, slow down response to insertions

and deletions. From an informational standpoint, an index

is a redundant component of the data representation. If a

system uses indices at all and if it is to perform well in an

environment with changing patterns of activity on the data

bank, an ability to create and destroy indices from time to

time will probably be necessary. The question then arises:

Can application programs and terminal activities remain

invariant as indices come and go?

Present formatted data systems take widely different

approaches to indexing. TDMS [7] unconditionally provides

indexing on all attributes. The presently released

version of IMS [5] provides the user with a choice for each

file: a choice between no indexing at all (the hierarchic sequential

organization) or indexing on the primary key

only (the hierarchic indexed sequential organization). In

neither case is the user’s application logic dependent on the

existence of the unconditionally provided indices. IDS

[8], however, permits the file designers to select attributes

to be indexed and to incorporate indices into the file structure

by means of additional chains. Application programs

talcing advantage of the performance benefit of these indexing

chains must refer to those chains by name. Such programs

do not operate correctly if these chains are later

removed.

1.2.3. Access Path Dependence. Many of the existing

formatted data systems provide users with tree-structured

files or slightly more general network models of the data.

Application programs developed to work with these systems

tend to be logically impaired if the trees or networks

are changed in structure. A simple example follows.

Suppose the data bank contains information about parts

and projects. For each part, the part number, part name,

part description, quantity-on-hand, and quantity-on-order

are recorded. For each project, the project number, project

name, project description are recorded. Whenever a project

makes use of a certain part, the quantity of that part committed

to the given project is also recorded. Suppose that

the system requires the user or file designer to declare or

define the data in terms of tree structures. Then, any one

of the hierarchical structures may be adopted for the information

mentioned above (see Structures 1-5).

Structure 1.

File Segment

F PART

PROJECT

Projects Subordinate to Parts

Pldds

part #

part name

part description

quantity-on-hand

quantity-on-order

project #

project name

project description

quantity committed

Structure 2.

File Segment

F PROJECT

PART

Parts Subordinate to Projects

gidds

project #

project name

project description

part #

part name

part description

quantity-on-hand

quantity-on-order

quantity committed

Structure 3. Parts and Projects as Peers

Commitment Relationship Subordinate to Projects

File Segmen~ Fields

F PART part #

part name

part description

quantity-on-hand

quantity-on-order

G PROJECT project #

project name

project description

PART part #

quantity committed

Structure 4. Parts and Projects as Peers

Commitment Relationship Subordinate to Parts

File Segment Fidds

P PART part #

part description

quantity-on-hand

quantity-on-order

PROJECT project #

quantity committed

G PROJECT project #

project name

project description

Structure 5. Parts, Projects, and

Commitment Relationship as Peers

F~¢ Segment Fields

F PART part #

part name

part description

quantity-on-hand

quantity-on-order

G PROJECT project #

project name

project description

H COMMIT part #

project #

quantity committed

378 Communications of the ACM Volume 13 / Number 6 / June, 1970

Now, consider the problem of printing out the part

number, part name, and quantity committed for every part

used in the project whose project name is “alpha.” The

following observations may be made regardless of which

available tree-oriented information system is selected to

tackle this problem. If a program P is developed for this

problem assuming one of the five structures above–that

is, P makes no test to determine which structure is in effect-then

P will fail on at least three of the remaining

structures. More specifically, if P succeeds with structure 5,

it will fail with all the others; if P succeeds with structure 3

or 4, it will fail with at least 1, 2, and 5; if P succeeds with

1 or 2, it will fail with at least 3, 4, and 5. The reason is

simple in each case. In the absence of a test to determine

which structure is in effect, P fails because an attempt is

made to execute a reference to a nonexistent file (available

systems treat this as an error) or no attempt is made to

execute a reference to a file containing needed information.

The reader who is not convinced should develop sample

programs for this simple problem.

Since, in general, it is not practical to develop application

programs which test for all tree structurings permitted

by the system, these programs fail when a change in

structure becomes necessary.

Systems which provide users with a network model of

the data run into similar difficulties. In both the tree and

network cases, the user (or his program) is required to

exploit a collection of user access paths to the data. It does

not matter whether these paths are in close correspondence

with pointer-defined paths in the stored representation–in

IDS the correspondence is extremely simple, in TDMS it is

just the opposite. The consequence, regardless of the stored

representation, is that terminal activities and programs become

dependent on the continued existence of the user

access paths.

One solution to this is to adopt the policy that once a

user access path is defined it will not be made obsolete until

all application programs using that path have become

obsolete. Such a policy is not practical, because the number

of access paths in the total model for the community of

users of a data bank would eventually become excessively

large.

1.3. A RELATIONAL VIEW OF DATA

The term relation is used here in its accepted mathematical

sense. Given sets $1, $2, • • • , S~ (not necessarily

distinct), R is a relation on these n sets if it is a set of ntuples

each of which has its first element from S~, its

second element from $2, and so on. 1 We shall refer to $i as

the jth domain of R. As defined above, R is said to have

degree n. Relations of degree 1 are often called unary, degree

2 binary, degree 3 ternary, and degree n n-ary.

For expository reasons, we shall frequently make use of

an array representation of relations, but it must be remembered

that this particular representation is not an essential

part of the relational view being expounded. An ari

More concisely, R is a subset of the Cartesian product $1 X

S~X “‘” X S..

ray which represents an n-ary relation R has the following

properties:

(1) Each row represents an n-tuple of R.

(2) The ordering of rows is immaterial.

(3) All rows are distinct.

(4) The ordering of columns is significant–it corresponds

to the ordering S~, $2, -.-, S~ of the domains

on which R is defined (see, however, remarks

below on domain-ordered and domain-unordered

relations ).

(5) The significance of each column is partially conveyed

by labeling it with the name of the corresponding

domain.

The example in Figure 1 illustrates a relation of degree

4, called supply, which reflects the shipments-in-progress

of parts from specified suppliers to specified projects in

specified quantities.

supply (supplier part project quantity)

1 2 5 17

1 3 5 23

2 3 7 9

2 7 5 4

4 1 1 12

FIG. 1. A relation of degree 4

One might ask: If the columns are labeled by the name

of corresponding domains, why should the ordering of columns

matter? As the example in Figure 2 shows, two columns

may have identical headings (indicating identical

domains) but possess distinct meanings with respect to the

relation. The relation depicted is called component. It is a

ternary relation, whose first two domains are called part

and third domain is called quantity. The meaning of component

(x, y, z) is that part x is an immediate component

(or subassembly) of part y, and z units of part x are needed

to assemble one unit of part y. It is a relation which plays

a critical role in the parts explosion problem.

component (part part quantity)

1 5 9

2 5 7

3 5 2

2 6 12

3 6 3

4 7 1

6 7 1

FiG. 2. A relation with two identical domains

It is a remarkable fact that several existing information

systems (chiefly those based on tree-structured files) fail

to provide data representations for relations which have

two or more identical domains. The present version of

IMS/360 [5] is an example of such a system.

The totality of data in a data bank may be viewed as a

collection of time-varying relations. These relations are of

assorted degrees. As time progresses, each n-ary relation

may be subject to insertion of additional n-tuples, deletion

of existing ones, and alteration of components of any of its

existing n-tuples.

Volume 13 / Number 6 / June, 1970 Communications of the ACM 379

In many commercial, governmental, and scientific data

banks, however, some of the relations are of quite high degree

(a degree of 30 is not at all uncommon). Users should

not normally be burdened with remembering the domain

ordering of any relation (for example, the ordering supplier,

then part, then project, then quantity in the relation supply).

Accordingly, we propose that users deal, not with relations

which are domain-ordered, but with relationships which are

their domain-unordered counterparts. 2 To accomplish this,

domains must be uniquely identifiable at least within any

given relation, without using position. Thus, where there

are two or more identical domains, we require in each case

that the domain name be qualified by a distinctive role

name, which serves to identify the role played by that

domain in the given relation. For example, in the relation

component of Figure 2, the first domain part might be

qualified by the role name sub, and the second by super, so

that users could deal with the relationship component and

its domains–sub.part super.part, quantity–without regard

to any ordering between these domains.

To sum up, it is proposed that most users should interact

with a relational model of the data consisting of a collection

of time-varying relationships (rather than relations). Each

user need not know more about any relationship than its

name together with the names of its domains (role qualified

whenever necessary).3 Even this information might be

offered in menu style by the system (subject to security

and privacy constraints) upon request by the user.

There are usually many alternative ways in which a relational

model may be established for a data bank. In

order to discuss a preferred way (or normal form), we

must first introduce a few additional concepts (active

domain, primary key, foreign key, nonsimple domain)

and establish some links with terminology currently in use

in information systems programming. In the remainder of

this paper, we shall not bother to distinguish between relations

and relationships except where it appears advantageous

to be explicit.

Consider an example of a data bank which includes relations

concerning parts, projects, and suppliers. One relation

called part is defined on the following domains:

(1) part number

(2) part name

(3) part color

(4) part weight

(5) quantity on hand

(6) quantity on order

and possibly other domains as well. Each of these domains

is, in effect, a pool of values, some or all of which may be

represented in the data bank at any instant. While it is

conceivable that, at some instant, all part colors are present,

it is unlikely that all possible part weights, part

In mathematical terms, a relationship is an equivalence class of

those relations that are equivalent under permutation of domains

(see Section 2.1.1).

Naturally, as with any data put into and retrieved from a computer

system, the user will normally make far more effective use

of the data if he is aware of its meaning.

names, and part numbers are. We shall call the set of

values represented at some instant the active domain at that

instant.

Normally, one domain (or combination of domains) of a

given relation has values which uniquely identify each element

(n-tuple) of that relation. Such a domain (or combination)

is called a primary key. In the example above,

part number would be a primary key, while part color

would not be. A primary key is nonredundant if it is either

a simple domain (not a combination) or a combination

such that none of the participating simple domains is

superfluous in uniquely identifying each element. A relation

may possess more than one nonredundant primary

key. This would be the case in the example if different parts

were always given distinct names. Whenever a relation

has two or more nonredundant primary keys, one of them

is arbitrarily selected and called the primary key of that relation.

A common requirement is for elements of a relation to

cross-reference other elements of the same relation or elements

of a different relation. Keys provide a user-oriented

means (but not the only means) of expressing such crossreferences.

We shall call a domain (or domain combination)

of relation R a foreign key if it is not the primary key

of R but its elements are values of the primary key of some

relation S (the possibility that S and R are identical is not

excluded). In the relation supply of Figure 1, the combination

of supplier, part, project is the primary key, while each

of these three domains taken separately is a foreign key.

In previous work there has been a strong tendency to

treat the data in a data bank as consisting of two parts, one

part consisting of entity descriptions (for example, descriptions

of suppliers) and the other part consisting of relations

between the various entities or types of entities (for

example, the supply relation). This distinction is difficult

to maintain when one may have foreign keys in any relation

whatsoever. In the user’s relational model there appears

to be no advantage to making such a distinction

(there may be some advantage, however, when one applies

relational concepts to machine representations of the user’s

set of relationships).

So far, we have discussed examples of relations which are

defined on simple domains–domains whose elements are

atomic (nondecomposable) values. Nonatomic values can

be discussed within the relational framework. Thus, some

domains may have relations as elements. These relations

may, in turn, be defined on nonsimple domains, and so on.

For example, one of the domains on which the relation employee

is defined might be salary history. An element of the

salary history domain is a binary relation defined on the domain

date and the domain salary. The salary history domain

is the set of all such binary relations. At any instant of time

there are as many instances of the salary history relation

in the data bank as there are employees. In contrast, there

is only one instance of the employee relation.

The terms attribute and repeating group in present data

base terminology are roughly analogous to simple domain

380 Communications of the ACM Volume 13 / Number 6 / June, 1970

and nonsimple domain, respectively. Much of the confusion

in present terminology is due to failure to distinguish between

type and instance (as in “record”) and between

components of a user model of the data on the one hand

and their machine representation counterparts on the

other hand (again, we cite “record” as an example).

1.4. NORMAL FORM

A relation whose domains are all simple can be represented

in storage by a two-dimensional column-homogeneous

array of the kind discussed above. Some more

complicated data structure is necessary for a relation with

one or more nonsimple domains. For this reason (and others

to be cited below) the possibility of eliminating nonsimple

domains appears worth investigating. 4 There is, in fact, a

very simple elimination procedure, which we shall call

normalization.

Consider, for example, the collection of relations exhibited

in Figure 3 (a). Job history and children are nonsimple

domains of the relation employee. Salary history is a

nonsimple domain of the relation job history. The tree in

Figure 3 (a) shows just these interrelationships of the nonsimple

domains.

jobhistory I

salaryhistory

employee

children

employee (man#, name, birthdate, jobhistory, children)

jobhistory (jobdate, title, salaryhistory)

salaryhistory (salarydate, salary)

children (childname, birthyear)

FIG. 3(a). Unnormalized set

employee’ (martS, name, birthdate)

jobhistory’ (man#, iobdate, title)

salaryhistory’ (man#, iobdate, salarydate, salary)

children’ (man#, childname, birthyear)

Fro. 3(b). Normalized set

Normalization proceeds as follows. Starting with the relation

at the top of the tree, take its primary key and expand

each of the immediately subordinate relations by

inserting this primary key domain or domain combination.

The primary key of each expanded relation consists of the

primary key before expansion augmented by the primary

key copied down from the parent relation. Now, strike out

from the parent relation all nonsimple domains, remove the

top node of the tree, and repeat the same sequence of

operations on each remaining subtree.

The result of normalizing the collection of relations in

Figure 3 (a) is the collection in Figure 3 (b). The primary

key of each relation is italicized to show how such keys

are expanded by the normalization.

‘ M. E. Sanko of IBM, San Jose, independently recognized the

desirability of eliminating nonsimple domains.

If normalization as described above is to be applicable,

the unnormalized collection of relations must satisfy the

following conditions:

(1) The graph of interrelationships of the nonsimple

domains is a collection of trees.

(2) No primary key has a component domain which is

nonsimple.

The writer knows of no application which would require

any relaxation of these conditions. Further operations of a

normalizing kind are possible. These are not discussed in

this paper.

The simplicity of the array representation which becomes

feasible when all relations are cast in normal form is not

only an advantage for storage purposes but also for communication

of bulk data between systems which use widely

different representations of the data. The communication

form would be a suitably compressed version of the array

representation and would have the following advantages:

(1) It would be devoid of pointers (address-valued or

displacement-valued).

(2) It would avoid all dependence on hash addressing

schemes.

(3) It would contain no indices or ordering lists.

If the user’s relational model is set up in normal form,

names of items of data in the data bank can take a simpler

form than would otherwise be the case. A general name

would take a form such as

R (g).r.d

where R is a relational name; g is a generation identifier

(optional); r is a role name (optional); d is a domain name.

Since g is needed only when several generations of a given

relation exist, or are anticipated to exist, and r is needed

only when the relation R has two or more domains named

d, the simple form R.d will often be adequate.

1.5. SOME LINGUISTIC ASPECTS

The adoption of a relational model of data, as described

above, permits the development of a universal data sublanguage

based on an applied predicate calculus. A firstorder

predicate calculus suffices if the collection of relations

is in normal form. Such a language would provide a yardstick

of linguistic power for all other proposed data languages,

and would itself be a strong candidate for embedding

(with appropriate syntactic modification) in a variety

of host languages (programming, command- or problemoriented).

While it is not the purpose of this paper to

describe such a language in detail, its salient features

would be as follows.

Let us denote the data sublanguage by R and the host

language by H. R permits the declaration of relations and

their domains. Each declaration of a relation identifies the

primary key for that relation. Declared relations are added

to the system catalog for use by any members of the user

community who have appropriate authorization. H permits

supporting declarations which indicate, perhaps less

permanently, how these relations are represented in storVolume

13 / Number 6 / June, 1970 Communications of the ACM 381

age. R permits the specification for retrieval of any subset

of data from the data bank. Action on such a retrieval request

is subject to security constraints.

The universality of the data sublangnage lies in its

descriptive ability (not its computing ability). In a large

data bank each subset of the data has a very large number

of possible (and sensible) descriptions, even when we assume

(as we do) that there is only a finite set of function

subroutines to which the system has access for use in

qualifying data for retrieval. Thus, the class of qualification

expressions which can be used in a set specification must

have the descriptive power of the class of well-formed

formulas of an applied predicate calculus. It is well known

that to preserve this descriptive power it is unnecessary to

express (in whatever syntax is chosen) every formula of

the selected predicate calculus. For example, just those in

prenex normal form are adequate [9].

Arithmetic functions may be needed in the qualification

or other parts of retrieval statements. Such functions can

be defined in H and invoked in R.

A set so specified may be fetched for query purposes

only, or it may be held for possible changes. Insertions take

the form of adding new elements to declared relations without

regard to any ordering that may be present in their

machine representation. Deletions which are effective for

the community (as opposed to the individual user or subcommunities)

take the form of removing elements from declared

relations. Some deletions and updates may be triggered

by others, if deletion and update dependencies between

specified relations are declared in R.

One important effect that the view adopted toward data

has on the language used to retrieve it is in the naming of

data elements and sets. Some aspects of this have been discussed

in the previous section. With the usual network

view, users will often be burdened with coining and using

more relation names than are absolutely necessary, since

names are associated with paths (or path types) rather

than with relations.

Once a user is aware that a certain relation is stored, he

will expect to be able to exploit 5 it using any combination

of its arguments as “knowns” and the remaining arguments

as “unknowns,” because the information (like

Everest) is there. This is a system feature (missing from

many current information systems) which we shall call

(logically) symmetric exploitation of relations. Naturally,

symmetry in performance is not to be expected.

To support symmetric exploitation of a single binary relation,

two directed paths are needed. For a relation of degree

n, the number of paths to be named and controlled is

n factorial.

Again, if a relational view is adopted in which every nary

relation (n > 2) has to be expressed by the user as a

nested expression involving only binary relations (see

Feldman’s LEAP System [10], for example) then 2n — 1

names have to be coined instead of only n -b 1 with direct

n-ary notation as described in Section 1.2. For example, the

5 Exploiting a relation includes query, update, and delete.

4-ary relation supply of Figure 1, which entails 5 names in

n-ary notation, would be represented in the form

P (supplier, Q (part, R (project, quantity)))

in nested binary notation and, thus, employ 7 names.

A further disadvantage of this kind of expression is its

asymmetry. Although this asymmetry does not prohibit

symmetric exploitation, it certainly makes some bases of

interrogation very awkward for the user to express (consider,

for example, a query for those parts and quantities

related to certain given projects via Q and R).

1.6. EXPRESSIBLE, NAMED, AND STORED RELATIONS

Associated with a data bank are two collections of relations:

the named set and the expressible set. The named set

is the collection of all those relations that the community of

users can identify by means of a simple name (or identifier).

A relation R acquires membership in the named set when a

suitably authorized user declares R; it loses membership

when a suitably authorized user cancels the declaration of

R.

The expressible set is the total collection of relations that

can be designated by expressions in the data language. Such

expressions are constructed from simple names of relations

in the named set; names of generations, roles and domains;

logical connectives; the quantifiers of the predicate calculus;

6 and certain constant relation symbols such as =, >.

The named set is a subset of the expressible set–usually a

very small subset.

Since some relations in the named set may be time-independent

combinations of others in that set, it is useful to

consider associating with the named set a collection of

statements that define these time-independent constraints.

We shall postpone further discussion of this until we have

introduced several operations on relations (see Section 2).

One of the major problems confronting the designer of a

data system which is to support a relational model for its

users is that of determining the class of stored representations

to be supported. Ideally, the variety of permitted

data representations should be just adequate to cover the

spectrum of performance requirements of the total collection

of installations. Too great a variety leads to unnecessary

overhead in storage and continual reinterpretation

of descriptions for the structures currently in effect.

For any selected class of stored representations the data

system must provide a means of translating user requests

expressed in the data language of the relational model into

corresponding–and efficient–actions on the current

stored representation. For a high level data language this

presents a challenging design problem. Nevertheless, it is a

problem which must be solved–as more users obtain concurrent

access to a large data bank, responsibility for providing

efficient response and throughput shifts from the

individual user to the data system.

s Because each relation in a practical data bank is a finite set at

every instant of time, the existential and universal quantifiers

can be expressed in terms of a function that counts the number of

elements in any finite set.

382 Communications of the ACM Volume 13 / Number 6 / June, 1970

2. Redundancy and Consistency

2.1. OPERATIONS ON RELATIONS

Since relations are sets, all of the usual set operations are

applicable to them. Nevertheless, the result may not be a

relation; for example, the union of a binary relation and a

ternary relation is not a relation.

The operations discussed below are specifically for relations.

These operations are introduced because of their key

role in deriving relations from other relations. Their

principal application is in noninferential information systems-systems

which do not provide logical inference

services–although their applicability is not necessarily

destroyed when such services are added.

Most users would not be directly concerned with these

operations. Information systems designers and people concerned

with data bank control should, however, be thoroughly

familiar with them.

2.1.1. Permutation. A binary relation has an array

representation with two columns. Interchanging these columns

yields the converse relation. More generally, if a

permutation is applied to the columns of an n-ary relation,

the resulting relation is said to be a permutation of the

given relation. There are, for example, 4! — 24 permutations

of the relation supply in Figure 1, if we include the

identity permutation which leaves the ordering of columns

unchanged.

Since the user’s relational model consists of a collection

of relationships (domain-unordered relations), permutation

is not relevant to such a model considered in isolation.

It is, however, relevant to the consideration of stored

representations of the model. In a system which provides

symmetric exploitation of relations, the set of queries

answerable by a stored relation is identical to the set

answerable by any permutation of that relation. Although

it is logically unnecessary to store both a relation and some

permutation of it, performance considerations could make

it advisable.

2.1.2. Projection. Suppose now we select certain columns

of a relation (striking out the others) and then remove

from the resulting array any duplication in the rows.

The final array represents a relation which is said to be a

projection of the given relation.

A selection operator 7r is used to obtain any desired

permutation, projection, or combination of the two operations.

Thus, if L is a list of lc indices 7 L = /1,/2, • • • , ik

and R is an n-ary relation (n _> k), then 7rL (R) is the k-ary

relation whose jth column is column ij of R (j = 1, 2, … , k)

except that duplication in resulting rows is removed. Consider

the relation supply of Figure 1. A permuted projection

of this relation is exhibited in Figure 4. Note that, in this

particular case, the projection has fewer n-tuples than the

relation from which it is derived.

2.1.3. Join. Suppose we are given two binary relations,

which have some domain in common. Under what

circumstances can we combine these relations to form a

7 When dealing with relationships, we use domain names (rolequalified

whenever necessary) instead of domain positions.

ternary relation which preserves all of the information in

the given relations?

The example in Figure 5 shows two relations R, S, which

are joinable without loss of information, while Figure 6

shows a join of R with S. A binary relation R is joinable

with a binary relation S if there exists a ternary relation U

such that 7r12(U) = R and ~’23(U) — S. Any such ternary

relation is called a join of R with S. If R, S are binary relations

such that v2 (R) = ~i (S), then R is joinable with S.

One join that always exists in such a case is the natural

join of R with S defined by

R*S = {(a,b,c):R(a,b) A S(b,c)}

where R (a, b) has the value true if (a, b) is a member of R

and similarly for S (b, c). It is immediate that

• “12 (R,S) = R

and

~23 (R,S) = S.

Note that the join shown in Figure 6 is the natural join

of R with S from Figure 5. Another join is shown in Figure

7.

FIG. 4.

II31 (supply) (project supplier)

5 1

5 2

1 4

7 2

A permuted projection of the relation in Figure 1

R (supplier part) S (part project)

1 1 1 1

2 1 1 2

2 2 2 1

FIG. 5. Two joinable relations

R*S

FIG. 6.

(supplier part project)

1 1 1

1 1 2

2 1 1

2 1 2

2 2 1

The natural join of R, with S (from Figure 5)

U (supplier part project)

1 1 2

2 1 1

2 2 1

FIG. 7. Another join of R with S (from Figure 5)

Inspection of these relations reveals an element (element

1 ) of the domain part (the domain on which the join

is to be made) with the property that it possesses more

than one relative under R and also under S. It is this eleVolume

13 / Number 6 / June, 1970 Communications of the ACMM 383

ment which gives rise to the plurality of joins. Such an element

in the joining domain is called a point of ambiguity

with respect to the joining of R with S.

If either ~r21 (R) or S is a function, 8 no point of ambiguity

can occur in joining R with S. In such a case, the natural

join of R with S is the only join of R with S. Note that the

reiterated qualification “of R with S” is necessary, because

S might be joinable with R (as well as R with S), and this

join would be an entirely separate consideration. In Figure

5, none of the relations R, ml (R), S, ~r21 (S) is a function.

Ambiguity in the joining of R with S can sometimes be

resolved by means of other relations. Suppose we are given,

or can derive from sources independent of R and S, a relation

T on the domains project and supplier with the following

properties:

(1) ~-~(T) = ~-2(S),

(2) w2(T) = ~(R),

(3) T(j, s) –+3p(R(S, p) A S(p,j)),

(4) R(s, p) —->3j(S(p,j) A T(j, s)),

(5) S(p,j) —->3s(T(j, s) A R(s, p)),

then we may form a three-way join of R, S, T; that is, a

ternary relation such that

~(v) = R, ~(U) = S, ~(V) = T.

Such a join will be called a cyclic 3-join to distinguish it

from a linear 3-join which would be a quaternary relation

V such that

~r12(V) = R, ~r~3(V) = Z, 7r34(V) = T.

While it is possible for more than one cyclic 3-join to exist

(see Figures 8, 9, for an example), the circumstances under

which this can occur entail much more severe constraints

FIG. 8.

R (8 p) s (p i) T ~ 8)

la ad dl

2a ae d2

2b bd e2

be e2

Binaryrelationswithapluralityofcyclic3~oins

FIG. 9.

U (8 p j) U’ (8 p i)

lad lad

2ae 2ad

2bd 2ae

2be 2bd

2be

Two cyclic 3-joins of the relations in Figure 8

than those for a plurality of 2-joins. To be specific, the relations

R, S, T must possess points of ambiguity with

respect to joining R with S (say point x), S with T (say

8 A function is a binary relation, which is one-one or many-one,

but not one-many.

y), and T with R (say z), and, furthermore, y must be a

relative of x under S, z a relative of y under T, and x a

relative of z under R. Note that in Figure 8 the points

x = a; y = d; z = 2 have this property.

The natural linear 3-join of three binary relations R, S,

T is given by

R.S.T = {(a,b,c,d):R(a,b) A S(b,c) A T(c,d)}

where parentheses are not needed on the left-hand side because

the natural 2-join (*) is associative. To obtain the

cyclic counterpart, we introduce the operator .y which produces

a relation of degree n – 1 from a relation of degree n

by tying its ends together. Thus, if R is an n-ary relation

(n >_ 2), the tie of R is defined by the equation

“I(R) = { (al, a2, … , a,_l):R(al, a2, … , a~-l, an)

A al = an}.

We may now represent the natural cyclic 3-join of R, S, T

by the expression

(R.S.T).

Extension of the notions of linear and cyclic 3-join and

their natural counterparts to the joining of n binary relations

(where n > 3) is obvious. A few words may be appropriate,

however, regarding the joining of relations which

are not necessarily binary. Consider the case of two relations

R (degree r), S (degree s) which are to be joined on

p of their domains (p < r, p < s). For simplicity, suppose

these p domains are the last p of the r domains of R,

and the first p of the s domains of S. If this were not so, we

could always apply appropriate permutations to make it

so. Now, take the Cartesian product of the first r-p domains

of R, and call this new domain A. Take the Cartesian

product of the last p domains of R, and call this B.

Take the Cartesian product of the last s-p domains of S

and call this C.

We can treat R as if it were a binary relation on the

domains A, B. Similarly, we can treat S as if it were a binary

relation on the domains B, C. The notions of linear

and cyclic 3-join are now directly applicable. A similar approach

can be taken with the linear and cyclic n-joins of n

relations of assorted degrees.

2.1.4. Composition. The reader is probably familiar

with the notion of composition applied to functions. We

shall discuss a generalization of that concept and apply it

first to binary relations. Our definitions of composition

and composability are based very directly on the definitions

of join and joinability given above.

Suppose we are given two relations R, S. T is a composition

of R with S if there exists a join U of R with S such

that T = 7913 (U). Thus, two relations are composable if

and only if they are j oinable. However, the existence of

more than one join of R with S does not imply the existence

of more than one composition of R with S.

Corresponding to the natural join of R with S is the

384 Communications of the ACM Volume 13 / Number 6 / June, 1970

natural composition 9 of R with S defined by

R.S = ~’i~(R*S).

Taking the relations R, S from Figure 5, their natural com:

position is exhibited in Figure 10 and another composition

is exhibited in Figure 11 (derived from the join exhibited

in Figure 7).

Fro. 10.

R. S (project supplier)

1 1

1 2

2 1

2 2

The natural composition of R with S (from Figure 5)

T (project supplier)

1 2

2 1~:

Fro. 11. Another composition of R with S (from Figure 5)

When two or more joins exist, the number of distinct

compositions may be as few as one or as many as the number

of distinct joins. Figule 12 shows an example of two

relations which have several joins but only one composition.

Note that the ambiguity of point c is lost in composing R

with S, because of unambiguous associations made via the

points a, b, d, e.

R (supplier part) S (part project)

1 a a g

1 b b f

1 c c f

2 c c g

2 d d g

2′ e e f

Fio. 12. Many joins, only one composition

Extension of composition to pairs of relations which are

not necessarily binary (and which may be of different degrees)

follows the same pattern as extension of pairwise

joining to such relations.

A lack of understanding of relational composition has led

several systems designers into what may be called the

connection trap. This trap may be described in terms of the

following example. Suppose each supplier description is

linked by pointers to the descriptions of each part supplied

by that supplier, and each part description is similarly

linked to the descriptions of each project which uses that

part. A conclusion is now drawn which is, in general, erroneous:

namely that, if all possible paths are followed from

a given supplier via the parts he supplies to the projects

using those parts, one will obtain a valid set of all projects

supplied by that supplier. Such a conclusion is correct

only in the very special case that the target relation between

projects and suppliers is, in fact, the natural composition

of the other two relations–and we must normally

add the phrase “for all time,” because this is usually implied

in claims concerning path-following techniques.

9 Other writers tend to ignore compositions other than the natural

one, and accordingly refer to this particular composition as

the composition–see, for example, Kelley’s “General Topology.”

2.1.5. Restriction. A subset of a relation is a relation.

One way in which a relation S may act on a relation R to

generate a subset of R is through the operation restriction

of R by S. This operation is a generalization of the restriction

of a function to a subset of its domain, and is defined

as follows.

Let L, M be equal-length lists of indices such that

L = /1,/2, “” , ik, M = jl,j2, “” ,jk where k ~ degree

of R and k ~ degree of S. Then the L, M restriction of R by

S denoted RdMS is the maximal subset R’ of R such that

~L (R’) = ~M (S).

The operation is defined only if equality is applicable between

elements of ~’~h (R) on the one hand and ~’~h (S) on

the other for all h = 1, 2, • • •, k.

The three relations R, S, R’ of Figure 13 satisfy the equation

R’ = R¢~,3)[¢i,2iS.

R 0 p J) S (p j) R’ (s p j)

1 a A a A 1 a A

2 a A c B 2 a A

2 a B b B 2 b B

2 b A

2 b B

FIG. 13. Example of restriction

We are now in a position to consider various applications

of these operations on relations.

2.2. REDUNDANCY

Redundancy in the named set of relations must be distinguished

from redundancy in the stored set of representations.

We are primarily concerned here with the former.

To begin with, we need a precise notion of derivability for

relations.

Suppose 0 is a collection of operations on relations and

each operation has the property that from its operands it

yields a unique relation (thus natural join is eligible, but

join is not). A relation R is O-derivable from a set S of relations

if there exists a sequence of operations from the collection

0 which, for all time, yields R from members of S.

The phrase “for all time” is present, because we are dealing

with time-varying relations, and our interest is in derivability

which holds over a significant period of time. For the

named set of relationships in noninferential systems, it appears

that an adequate collection 01 contains the following

operations: projection, natural join, tie, and restriction.

Permutation is irrelevant and natural composition need

not be included, because it is obtainable by taking a natural

join and then a projection. For the stored set of representations,

an adequate collection ~ of operations would include

permutation and additional operations concerned with subsetting

and merging relations, and ordering and connecting

their elements.

2.2.1. Stronq Redundancy. A set of relations is strongly

redundant if it contains at least one relation that possesses

a projection which is derivable from other projections of

relations in the set. The following two examples are intended

to explain why strong redundancy is defined this

way, and to demonstrate its practical use. In the first exVolume

13 / Number 6 / June, 1970 Communications of the ACM 385

ample the collection of relations consists of just the following

relation:

employee (serial #, name, manager#, managername )

with serial# as the primary key and manager# as a foreign

key. Let us denote the active domain by ,~, and suppose

that

and

At (manager#) c A, (serial#)

At (managername ) C At (name)

for all time t. In this case the redundancy is obvious: the

domain managername is unnecessary. To see that it is a

strong redundancy as defined above, we observe that

~34 (employee) = ~’12 (employee )l[lm (employee).

In the second example the collection of relations includes a

relation S describing suppliers with primary key s#, a relation

D describing departments with primary key d#, a

relation J describing projects with primary key j#, and the

following relations:

P (s#, d#, … ), Q (s#, j#, .-. ), R (d#, j#, … ),

where in each case .-. denotes domains other than s#, d#,

j#. Let us suppose the following condition C is known to

hold independent of time: supplier s supplies department

d (relation P) if and only if supplier s supplies some project

j (relation Q) to which d is assigned (relation R). Then, we

can write the equation

~’12 (P) = 71-12 (Q). 71″21 (R)

and thereby exhibit a strong redundancy.

An important reason for the existence of strong redundancies

in the named set of relationships is user convenience.

A particular case of this is the retention of semiobsolete

relationships in the named set so that old programs

that refer to them by name can continue to run correctly.

Knowledge of the existence of strong redundancies

in the named set enables a system or data base administrator

greater freedom in the selection of stored representations

to cope more efficiently with current traffic. If the

strong redundancies in the named set are directly reflected

in strong redundancies in the stored set (or if other strong

redundancies are introduced into the stored set), then, generally

speaking, extra storage space and update time are

consumed with a potential drop in query time for some

queries and in load on the central processing units.

2.2.2. Weak Redundancy. A second type of redundancy

may exist. In contrast to strong redundancy it is not

characterized by an equation. A collection of relations is

weakly redundant if it contains a relation that has a projection

which is not derivable from other members but is at

all times a projection of some join of other projections of

relations in the collection.

We can exhibit a weak redundancy by taking the second

example (cited above) for a strong redundancy, and assuming

now that condition C does not hold at all times.

The relations ~12 (P), ~r12 (Q), ‘~’12 (R) are complex l° relations

with the possibility of points of ambiguity occurring from

time to time in the potential joining of any two. Under

these circumstances, none of them is derivable from the

other two. However, constraints do exist between them,

since each is a projection of some cyclic join of the three of

them. One of the weak redundancies can be characterized

by the statement: for all time, ~12 (P) is some composition

of ~-12 (Q) with “~’21 (R). The composition in question might

be the natural one at some instant and a nonnatural one at

another instant.

Generally speaking, weak redundancies are inherent in

the logical needs of the community of users. They are not

removable by the system or data base administrator. If

they appear at all, they appear in both the named set and

the stored set of representations.

2.3. CONSISTENCY

Whenever the named set of relations is redundant in

either sense, we shall associate with that set a collection of

statements which define all of the redundancies which hold

independent of time between the member relations. If the

information system lacks–and it most probably will–detailed

semantic information about each named relation, it

cannot deduce the redundancies applicable to the named

set. It might, over a period of time, make attempts to

induce the redundancies, but such attempts would be fallible.

Given a collection C of time-varying relations, an associated

set Z of constraint statements and an instantaneous

value V for C, we shall call the state (C, Z, V) consistent

or inconsistent according as V does or does not satisfy Z.

For example, given stored relations R, S, T together with

the constraint statement ‘%’12(T) is a composition of

7r~2 (R) with 7r~2 (S)”, we may check from time to time that

the values stored for R, S, T satisfy this constraint. An algorithm

for making this check would examine the first two

columns of each of R, S, T (in whatever way they are represented

in the system) and determine whether

(1) Try(T) = ~-i(R),

(2) ~(T) = ~(S),

(3) for every element pair (a, c) in the relation ~2 (T)

there is an element b such that (a, b) is in ~-12 (R)

and (b, c) is in 7r12(S).

There are practical problems (which we shall not discuss

here) in taking an instantaneous snapshot of a collection

of relations, some of which may be very large and highly

variable.

It is important to note that consistency as defined above

is a property of the instantaneous state of a data bank, and

is independent of how that state came about. Thus, in

particular, there is no distinction made on the basis of

whether a user generated an inconsistency due to an act of

omission or an act of commission. Examination of a simple

10 A binary relation is complex if neither it nor its converse is a

function.

386 Communications of the AMC Volume 13 / Number 6 / June, 1970

example will show the reasonableness of this (possibly unconventional)

approach to consistency.

Suppose the named set C includes the relations S, J, D,

P, Q, R of the example in Section 2.2 and that P, Q, R

possess either the strong or weak redundancies described

therein (in the particular case now under consideration, it

does not matter which kind of redundancy occurs). Further,

suppose that at some time t the data bank state is consistent

and contains no project j such that supplier 2 supplies

project j and j is assigned to department 5. Accordingly,

there is no element (2, 5) in ~-i2 (P). Now, a user introduces

the element (2, 5) into ~ri2 (P) by inserting some appropriate

element into P. The data bank state is now inconsistent.

The inconsistency could have arisen from an act of omission,

if the input (2, 5) is correct, and there does exist a

project j such that supplier 2 supplies j and j is assigned to

department 5. In this case, it is very likely that the user

intends in the near future to insert elements into Q and R

which will have the effect of introducing (2, j) into ~r12 (Q)

and (5, j) in 7rI2(R). On the other hand, the input (2, 5)

might have been faulty. It could be the case that the user

intended to insert some other element into P–an element

whose insertion would transform a consistent state into

a consistent state. The point is that the system will

normally have no way of resolving this question without

interrogating its environment (perhaps the user who created

the inconsistency).

There are, of course, several possible ways in which a

system can detect inconsistencies and respond to them.

In one approach the system checks for possible inconsistency

whenever an insertion, deletion, or key update occurs.

Naturally, such checking will slow these operations down.

If an inconsistency has been generated, details are logged

internally, and if it is not remedied within some reasonable

time interval, either the user or someone responsible for

the security and integrity of the data is notified. Another

approach is to conduct consistency checking as a batch

operation once a day or less frequently. Inputs causing the

inconsistencies which remain in the data bank state at

checking time can be tracked down if the system mainrains

a journal of all state-changing transactions. This

latter approach would certainly be superior if few nontransitory

inconsistencies occurred.

2.4. SUMMARY

In Section 1 a relational model of data is proposed as a

basis for protecting users of formatted data systems from

the potentially disruptive changes in data representation

caused by growth in the data bank and changes in traffic.

A normal form for the time-varying collection of relationships

is introduced.

In Section 2 operations on relations and two types of

redundancy are defined and applied to the problem of

maintaining the data in a consistent state. This is bound to

become a serious practical problem as more and more different

types of data are integrated together into common

data banks.

Many questions are raised and left unanswered. For

example, only a few of the more important properties of

the data sublanguage in Section 1.4 are mentioned. Neither

the purely linguistic details of such a language nor the

implementation problems are discussed. Nevertheless, the

material presented should be adequate for experienced

systems programmers to visualize several approaches. It

is also hoped that this paper can contribute to greater precision

in work on formatted data systems.

Acknowledgment. It was C. T. Davies of IBM Poughkeepsie

who convinced the author of the need for data

independence in future information systems. The author

wishes to thank him and also F. P. Palermo, C. P. Wang,

E. B. Altman, and M. E. Senko of the IBM San Jose Research

Laboratory for helpful discussions.

RECEIVED SEPTEMBER, 1969; REVISED FEBRUARY, 1970

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Amsterdam, p. 162-172.

2. LEVEIN, R. E., AND MARON, M. E. A computer system for

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3. BACHMAN, C. W. Software for random access processing.

Datamation (Apr. 1965), 36–41.

4. McGEE, W. C. Generalized file processing. In Annual Review

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5. Information Management System/360, Application Description

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6. GIS (Generalized Information System), Application Description

Manual H20-0574. IBM Corp., White Plains, N. Y.,

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7. BLEIER, R.E. Treating hierarchical data structures in the

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Volume 13 / Number 6 / June, 1970 Communications of the ACCM 387