A relational model of data for large shared data banks (in communications of the ACM 1970)

Information Retrieval P. BAXENDALE, Editor
A Relational Model of Data
Large Shared Data Banks
IBM Research Laboratory, San Jose, California
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
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.
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
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
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
Projects Subordinate to Parts
part #
part name
part description
project #
project name
project description
quantity committed
Structure 2.
File Segment
Parts Subordinate to Projects
project #
project name
project description
part #
part name
part description
quantity committed
Structure 3. Parts and Projects as Peers
Commitment Relationship Subordinate to Projects
File Segmen~ Fields
F PART part #
part name
part description
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
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
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
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
(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
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
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).
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
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
jobhistory I
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
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
(2) It would avoid all dependence on hash addressing
(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.
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).
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
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
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
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
~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
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
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
(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
FIG. 9.
U (8 p j) U’ (8 p i)
lad lad
2ae 2ad
2bd 2ae
2be 2bd
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
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.
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
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
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
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.
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
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
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.
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.
1. CHILDS, D.L. Feasibility of a set-theoretical data structure
–a general structure based on a reconstituted definition of
relation. Proc. IFIP Cong., 1968, North Holland Pub. Co.,
Amsterdam, p. 162-172.
2. LEVEIN, R. E., AND MARON, M. E. A computer system for
inference execution and data retrieval. Comm. ACM 10,
11 (Nov. 1967), 715–721.
3. BACHMAN, C. W. Software for random access processing.
Datamation (Apr. 1965), 36–41.
4. McGEE, W. C. Generalized file processing. In Annual Review
in Automatic Programming 5, 13, Pergamon Press,
New York, 1969, pp. 77-149.
5. Information Management System/360, Application Description
Manual H20-0524-1. IBM Corp., White Plains, N. Y.,
July 1968.
6. GIS (Generalized Information System), Application Description
Manual H20-0574. IBM Corp., White Plains, N. Y.,
7. BLEIER, R.E. Treating hierarchical data structures in the
SDC time-shared data management system (TDMS).
Proc. ACM 22nd Nat. Conf., 1967, MDI Publications,
Wayne, Pa., pp. 41—49.
8. IDS Reference Manual GE 625/635, GE Inform. Sys. Div.,
Pheonix, Ariz., CPB 1093B, Feb. 1968.
9. CHURCH, A. An Introduction to Mathematical Logic I. Princeton
U. Press, Princeton, N.J., 1956.
10. FELDMAN, J. A., AND ROVNEa, P.D. An Algol-based associative
language. Stanford Artificial Intelligence Rep. AI-66,
Aug. 1, 1968.
Volume 13 / Number 6 / June, 1970 Communications of the ACCM 387

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