# 数据库中的读取：关系数据库的要点

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1.Information Retrieval P. BAXENDALE, Editor A Relational Model of Data for The relational view (or model) of data described in Section 1 appears to be superior in several respects to the Large Shared Data Banks graph or network model [3, 4] presently in vogue for non- inferential systems. It provides a means of describing data with its natural structure only--that is, without superim- E. F. CODD posing any additional structure for machine representation I B M Research Laboratory, San Jose, California purposes. Accordingly, it provides a basis for a high level data language which will yield maximal independence be- tween programs on the one hand and machine representa- Future users of large data banks must be protected from tion and organization of data on the other. having to know how the data is organized in the machine (the A further advantage of the relational view is that it internal representation). A prompting service which supplies forms a sound basis for treating derivability, redundancy, such information is not a satisfactory solution. Activities of users and consistency of relations--these are discussed in Section at terminals and most application programs should remain 2. The network model, on the other hand, has spawned a unaffected when the internal representation of data is changed number of confusions, not the least of which is mistaking and even when some aspects of the external representation the derivation of connections for the derivation of rela- are changed. Changes in data representation will often be tions (see remarks in Section 2 on the "connection trap"). needed as a result of changes in query, update, and report Finally, the relational view permits a clearer evaluation traffic and natural growth in the types of stored information. of the scope and logical limitations of present formatted Existing noninferential, formatted data systems provide users data systems, and also the relative merits (from a logical with tree-structured files or slightly more general network standpoint) of competing representations of data within a models of the data. In Section 1, inadequacies of these models single system. Examples of this clearer perspective are are discussed. A model based on n-ary relations, a normal cited in various parts of this paper. Implementations of form for data base relations, and the concept of a universal systems to support the relational model are not discussed. data sublanguage are introduced. In Section 2, certain opera- 1.2. DATA DEPENDENCIES IN PRESENT SYSTEMS tions on relations (other than logical inference) are discussed The provision of data description tables in recently de- and applied to the problems of redundancy and consistency veloped information systems represents a major advance in the user's model. toward the goal of data independence [5, 6, 7]. Such tables KEY WORDS AND PHRASES: data bank, data base, data structure, data facilitate changing certain characteristics of the data repre- organization, hierarchies of data, networks of data, relations, derivability, sentation stored in a data bank. However, the variety of redundancy, consistency, composition, join, retrieval language, predicate data representation characteristics which can be changed calculus, security, data integrity CR CATEGORIES: 3.70, 3.73, 3.75, 4.20, 4.22, 4.29 without logically impairing some application programs is still quite limited. Further, the model of data with which users interact is still cluttered with representational prop- erties, particularly in regard to the representation of col- lections of data (as opposed to individual items). Three of the principal kinds of data dependencies which still need 1. Relational Model and Normal Form to be removed are: ordering dependence, indexing depend- ence, and access path dependence. In some systems these 1.1. INTRODUCTION dependencies are not clearly separable from one another. This paper is concerned with the application of ele- 1.2.1. Ordering Dependence. Elements of data in a mentary relation theory to systems which provide shared data bank may be stored in a variety of ways, some involv- access to large banks of formatted data. Except for a paper ing no concern for ordering, some permitting each element by Childs [1], the principal application of relations to data to participate in one ordering only, others permitting each systems has been to deductive question-answering systems. element to participate in several orderings. Let us consider Levein and Maron [2] provide numerous references to work those existing systems which either require or permit data in this area. elements to be stored in at least one total ordering which is In contrast, the problems treated here are those of data closely associated with the hardware-determined ordering independence--the independence of application programs of addresses. For example, the records of a file concerning and terminal activities from growth in data types and parts might be stored in ascending order by part serial changes in data representation--and certain kinds of data number. Such systems normally permit application pro- inconsistency which are expected to become troublesome grams to assume that the order of presentation of records even in nondeductive systems. from such a file is identical to (or is a subordering of) the V o l u m e 13 / N u m b e r 6 / J u n e , 1970 Communications o f t h e ACM 377

2.stored ordering. Those application programs which take Structure 1. Projects Subordinate to Parts advantage of the stored ordering of a file are likely to fail File Segment Pldds to operate correctly if for some reason it becomes necessary F PART part # to replace that ordering by a different one. Similar remarks part name part description hold for a stored ordering implemented by means of quantity-on-hand pointers. quantity-on-order It is unnecessary to single out any system as an example, PROJECT project # because all the well-known information systems that are project name marketed today fail to make a clear distinction between project description quantity committed 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. Structure 2. Parts Subordinate to Projects 1.2.2. Indexing Dependence. In the context of for- File Segment gidds matted data, an index is usually thought of as a purely F PROJECT project # performance-oriented component of the data representa- project name tion. It tends to improve response to queries and updates project description PART part # and, at the same time, slow down response to insertions part name and deletions. From an informational standpoint, an index part description is a redundant component of the data representation. If a quantity-on-hand system uses indices at all and if it is to perform well in an quantity-on-order environment with changing patterns of activity on the data quantity committed bank, an ability to create and destroy indices from time to time will probably be necessary. The question then arises: Structure 3. Parts and Projects as Peers Can application programs and terminal activities remain Commitment Relationship Subordinate to Projects invariant as indices come and go? File Segmen~ Fields Present formatted data systems take widely different F PART part # part name approaches to indexing. TDMS [7] unconditionally pro- part description vides indexing on all attributes. The presently released quantity-on-hand version of IMS [5] provides the user with a choice for each quantity-on-order file: a choice between no indexing at all (the hierarchic se- G PROJECT project # quential organization) or indexing on the primary key project name project description only (the hierarchic indexed sequential organization). In PART part # neither case is the user's application logic dependent on the quantity committed existence of the unconditionally provided indices. IDS [8], however, permits the file designers to select attributes Structure 4. Parts and Projects as Peers to be indexed and to incorporate indices into the file struc- Commitment Relationship Subordinate to Parts ture by means of additional chains. Application programs File Segment Fidds talcing advantage of the performance benefit of these in- P PART part # dexing chains must refer to those chains by name. Such pro- part description grams do not operate correctly if these chains are later quantity-on-hand quantity-on-order removed. PROJECT project # 1.2.3. Access Path Dependence. Many of the existing quantity committed formatted data systems provide users with tree-structured G PROJECT project # files or slightly more general network models of the data. project name Application programs developed to work with these sys- project description tems tend to be logically impaired if the trees or networks are changed in structure. A simple example follows. Structure 5. Parts, Projects, and Commitment Relationship as Peers Suppose the data bank contains information about parts F~¢ Segment Fields and projects. For each part, the part number, part name, part description, quantity-on-hand, and quantity-on-order F PART part # part name are recorded. For each project, the project number, project part description name, project description are recorded. Whenever a project quantity-on-hand makes use of a certain part, the quantity of that part com- quantity-on-order mitted to the given project is also recorded. Suppose that G PROJECT project # project name the system requires the user or file designer to declare or project description define the data in terms of tree structures. Then, any one H COMMIT part # of the hierarchical structures may be adopted for the infor- project # mation mentioned above (see Structures 1-5). quantity committed 378 C o m m u n i c a t i o n s o f t h e ACM V o l u m e 13 / N u m b e r 6 / J u n e , 1970

3. Now, consider the problem of printing out the p a r t ray which represents an n - a r y relation R has the following number, part name, and quantity committed for every p a r t properties: used in the project whose project name is "alpha." The (1) Each row represents an n-tuple of R. following observations m a y be made regardless of which (2) T h e ordering of rows is immaterial. available tree-oriented information system is selected to (3) All rows are distinct. tackle this problem. I f a program P is developed for this (4) T h e ordering of columns is significant--it corre- problem assuming one of the five structures a b o v e - - t h a t sponds to the ordering S~, $2, - . - , S~ of the do- is, P makes no test to determine which structure is in ef- mains on which R is defined (see, however, remarks f e c t - t h e n P will fail on at least three of the remaining below on domain-ordered and domain-unordered structures. More specifically, if P succeeds with structure 5, relations ). it will fail with all the others; if P succeeds with structure 3 (5) The significance of each column is partially con- or 4, it will fail with at least 1, 2, and 5; if P succeeds with veyed b y labeling it with the name of the corre- 1 or 2, it will fail with at least 3, 4, and 5. The reason is sponding domain. simple in each case. I n the absence of a test to determine The example in Figure 1 illustrates a relation of degree which structure is in effect, P fails because an a t t e m p t is 4, called supply, which reflects the shipments-in-progress made to execute a reference to a nonexistent file (available of parts from specified suppliers to specified projects in systems treat this as an error) or no a t t e m p t is made to specified quantities. execute a reference to a file containing needed information. T h e reader who is not convinced should develop sample supply (supplier part project quantity) programs for this simple problem. 1 2 5 17 Since, in general, it is not practical to develop applica- 1 3 5 23 2 3 7 9 tion programs which test for all tree structurings permitted 2 7 5 4 b y the system, these programs fail when a change in 4 1 1 12 structure becomes necessary. FIG. 1. A relation of degree 4 Systems which provide users with a network model of the data run into similar difficulties. I n both the tree and One might ask: I f the columns are labeled b y the name network cases, the user (or his program) is required to of corresponding domains, why should the ordering of col- exploit a collection of user access paths to the data. I t does umns matter? As the example in Figure 2 shows, two col- not m a t t e r whether these paths are in close correspondence umns m a y have identical headings (indicating identical with pointer-defined paths in the stored representation--in domains) but possess distinct meanings with respect to the I D S the correspondence is extremely simple, in T D M S it is relation. T h e relation depicted is called component. I t is a just the opposite. The consequence, regardless of the stored ternary relation, whose first two domains are called part representation, is t h a t terminal activities and programs be- and third domain is called quantity. The meaning of com- come dependent on the continued existence of the user ponent (x, y, z) is t h a t p a r t x is an immediate component access paths. (or subassembly) of p a r t y, and z units of p a r t x are needed One solution to this is to adopt the policy t h a t once a to assemble one unit of p a r t y. I t is a relation which plays user access p a t h is defined it will not be made obsolete un- a critical role in the parts explosion problem. til all application programs using t h a t p a t h have become obsolete. Such a policy is not practical, because the n u m b e r component (part part quantity) of access paths in the total model for the community of 1 5 9 users of a data bank would eventually become excessively 2 5 7 large. 3 5 2 2 6 12 1.3. A RELATIONAL VIEW OF DATA 3 6 3 The t e r m relation is used here in its accepted mathe- 4 7 1 matical sense. Given sets $1, $2, • • • , S~ (not necessarily 6 7 1 distinct), R is a relation on these n sets if it is a set of n- FiG. 2. A relation with two identical domains tuples each of which has its first element from S~, its second element from $2, and so on. 1 We shall refer to $i as I t is a remarkable fact t h a t several existing information the j t h domain of R. As defined above, R is said to have systems (chiefly those based on tree-structured files) fail degree n. Relations of degree 1 are often called unary, de- to provide d a t a representations for relations which have gree 2 binary, degree 3 ternary, and degree n n-ary. two or more identical domains. T h e present version of For expository reasons, we shall frequently m a k e use of I M S / 3 6 0 [5] is an example of such a system. an array representation of relations, but it must be re- T h e totality of data in a d a t a bank m a y be viewed as a membered t h a t this particular representation is not an es- collection of time-varying relations. These relations are of sential p a r t of the relational view being expounded. An ar- assorted degrees. As time progresses, each n-ary relation m a y be subject to insertion of additional n-tuples, deletion i More concisely, R is a subset of the Cartesian product $1 X of existing ones, and alteration of components of any of its S~X "'" X S.. existing n-tuples. V o l u m e 13 / N u m b e r 6 / J u n e , 1970 C o m m u n i c a t i o n s o f t h e ACM 379

4. In many commercial, governmental, and scientific data names, and part numbers are. We shall call the set of banks, however, some of the relations are of quite high de- values represented at some instant the active domain at t h a t gree (a degree of 30 is not at all uncommon). Users should instant. not normally be burdened with remembering the domain Normally, one domain (or combination of domains) of a ordering of any relation (for example, the ordering supplier, given relation has values which uniquely identify each ele- then part, then project, then quantity in the relation supply). ment (n-tuple) of that relation. Such a domain (or com- Accordingly, we propose t h a t users deal, not with relations bination) is called a primary key. In the example above, which are domain-ordered, but with relationships which are part number would be a primary key, while part color their domain-unordered counterparts. 2 To accomplish this, would not be. A primary key is nonredundant if it is either domains must be uniquely identifiable at least within any a simple domain (not a combination) or a combination given relation, without using position. Thus, where there such that none of the participating simple domains is are two or more identical domains, we require in each case superfluous in uniquely identifying each element. A rela- that the domain name be qualified by a distinctive role tion may possess more than one nonredundant primary name, which serves to identify the role played b y that key. This would be the case in the example if different parts domain in the given relation. For example, in the relation were always given distinct names. Whenever a relation component of Figure 2, the first domain part might be has two or more nonredundant primary keys, one of them qualified by the role name sub, and the second b y super, so is arbitrarily selected and called the primary key of that re- t h a t users could deal with the relationship component and lation. its domains--sub.part super.part, quantity--without regard A common requirement is for elements of a relation to to any ordering between these domains. cross-reference other elements of the same relation or ele- To sum up, it is proposed that most users should interact ments of a different relation. Keys provide a user-oriented with a relational model of the data consisting of a collection means (but not the only means) of expressing such cross- of time-varying relationships (rather than relations). Each references. We shall call a domain (or domain combina- user need not know more about any relationship than its tion) of relation R a foreign key if it is not the primary key name together with the names of its domains (role quali- of R but its elements are values of the primary key of some fied whenever necessary).3 Even this information might be relation S (the possibility that S and R are identical is not offered in menu style by the system (subject to security excluded). In the relation supply of Figure 1, the combina- and privacy constraints) upon request b y the user. tion of supplier, part, project is the primary key, while each There are usually m a n y alternative ways in which a re- of these three domains taken separately is a foreign key. lational model may be established for a data bank. In In previous work there has been a strong tendency to order to discuss a preferred way (or normal form), we treat the data in a data bank as consisting of two parts, one must first introduce a few additional concepts (active part consisting of entity descriptions (for example, descrip- domain, primary key, foreign key, nonsimple domain) tions of suppliers) and the other part consisting of rela- and establish some links with terminology currently in use tions between the various entities or types of entities (for in information systems programming. In the remainder of example, the supply relation). This distinction is difficult this paper, we shall not bother to distinguish between re- to maintain when one may have foreign keys in any rela- lations and relationships except where it appears advan- tion whatsoever. In the user's relational model there ap- tageous to be explicit. pears to be no advantage to making such a distinction Consider an example of a data bank which includes rela- (there m a y be some advantage, however, when one applies tions concerning parts, projects, and suppliers. One rela- relational concepts to machine representations of the user's tion called part is defined on the following domains: set of relationships). (1) part number So far, we have discussed examples of relations which are (2) part name defined on simple domains--domains whose elements are (3) part color atomic (nondecomposable) values. Nonatomic values can (4) part weight be discussed within the relational framework. Thus, some (5) quantity on hand domains may have relations as elements. These relations (6) quantity on order may, in turn, be defined on nonsimple domains, and so on. and possibly other domains as well. Each of these domains For example, one of the domains on which the relation em- is, in effect, a pool of values, some or all of which m a y be ployee is defined might be salary history. An element of the represented in the data bank at any instant. While it is salary history domain is a binary relation defined on the do- conceivable that, at some instant, all part colors are pres- main date and the domain salary. The salary history domain ent, it is unlikely that all possible part weights, part is the set of all such binary relations. At any instant of time In mathematical terms, a relationship is an equivalence class of there are as many instances of the salary history relation those relations that are equivalent under permutation of domains in the data bank as there are employees. In contrast, there (see Section 2.1.1). is only one instance of the employee relation. Naturally, as with any data put into and retrieved from a com- puter system, the user will normally make far more effective use The terms attribute and repeating group in present data of the data if he is aware of its meaning. base terminology are roughly analogous to simple domain 380 C o m m u n i c a t i o n s o f t h e ACM V o l u m e 13 / N u m b e r 6 / J u n e , 1970

5.and nonsimple domain, respectively. Much of the confusion If normalization as described above is to be applicable, in present terminology is due to failure to distinguish be- the unnormalized collection of relations must satisfy the tween type and instance (as in "record") and between following conditions: components of a user model of the data on the one hand (1) The graph of interrelationships of the nonsimple and their machine representation counterparts on the domains is a collection of trees. other hand (again, we cite "record" as an example). (2) No primary key has a component domain which is 1.4. NORMALFORM nonsimple. A relation whose domains are all simple can be repre- The writer knows of no application which would require sented in storage by a two-dimensional column-homo- any relaxation of these conditions. Further operations of a geneous array of the kind discussed above. Some more normalizing kind are possible. These are not discussed in complicated data structure is necessary for a relation with this paper. one or more nonsimple domains. For this reason (and others The simplicity of the array representation which becomes to be cited below) the possibility of eliminating nonsimple feasible when all relations are cast in normal form is not domains appears worth investigating.4 There is, in fact, a only an advantage for storage purposes but also for com- very simple elimination procedure, which we shall call munication of bulk data between systems which use widely normalization. different representations of the data. The communication Consider, for example, the collection of relations ex- form would be a suitably compressed version of the array hibited in Figure 3 (a). Job history and children are non- representation and would have the following advantages: simple domains of the relation employee. Salary history is a (1) It would be devoid of pointers (address-valued or nonsimple domain of the relation job history. The tree in displacement-valued). Figure 3 (a) shows just these interrelationships of the non- (2) It would avoid all dependence on hash addressing simple domains. schemes. (3) It would contain no indices or ordering lists. employee 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 jobhistory children would take a form such as I salaryhistory R (g).r.d employee (man#, name, birthdate, jobhistory, children) where R is a relational name; g is a generation identifier jobhistory (jobdate, title, salaryhistory) (optional); r is a role name (optional); d is a domain name. salaryhistory (salarydate, salary) Since g is needed only when several generations of a given children (childname, birthyear) relation exist, or are anticipated to exist, and r is needed FIG. 3(a). Unnormalized set only when the relation R has two or more domains named d, the simple form R.d will often be adequate. employee' (martS, name, birthdate) jobhistory' (man#, iobdate, title) 1.5. SOME LINGUISTIC ASPECTS salaryhistory' (man#, iobdate, salarydate, salary) The adoption of a relational model of data, as described children' (man#, childname, birthyear) above, permits the development of a universal data sub- Fro. 3(b). Normalized set language based on an applied predicate calculus. A first- order predicate calculus suffices if the collection of relations Normalization proceeds as follows. Starting with the re- is in normal form. Such a language would provide a yard- lation at the top of the tree, take its primary key and ex- stick of linguistic power for all other proposed data lan- pand each of the immediately subordinate relations by guages, and would itself be a strong candidate for embed- inserting this primary key domain or domain combination. ding (with appropriate syntactic modification) in a variety The primary key of each expanded relation consists of the of host languages (programming, command- or problem- primary key before expansion augmented by the primary oriented). While it is not the purpose of this paper to key copied down from the parent relation. Now, strike out describe such a language in detail, its salient features from the parent relation all nonsimple domains, remove the would be as follows. top node of the tree, and repeat the same sequence of Let us denote the data sublanguage by R and the host operations on each remaining subtree. language by H. R permits the declaration of relations and The result of normalizing the collection of relations in their domains. Each declaration of a relation identifies the Figure 3 (a) is the collection in Figure 3 (b). The primary primary key for that relation. Declared relations are added key of each relation is italicized to show how such keys to the system catalog for use by any members of the user are expanded by the normalization. community who have appropriate authorization. H per- ' M. E. Sanko of IBM, San Jose, independently recognized the mits supporting declarations which indicate, perhaps less desirability of eliminating nonsimple domains. permanently, how these relations are represented in stor- V o l u m e 13 / Number 6 / J u n e , 1970 C o m m u n i c a t i o n s o f t h e ACM 381

6. age. R permits the specification for retrieval of any subset 4-ary relation supply of Figure 1, which entails 5 names in of data from the data bank. Action on such a retrieval re- n-ary notation, would be represented in the form quest is subject to security constraints. The universality of the data sublangnage lies in its P (supplier, Q (part, R (project, quantity))) descriptive ability (not its computing ability). In a large in nested binary notation and, thus, employ 7 names. data bank each subset of the data has a very large number A further disadvantage of this kind of expression is its of possible (and sensible) descriptions, even when we as- asymmetry. Although this asymmetry does not prohibit sume (as we do) that there is only a finite set of function symmetric exploitation, it certainly makes some bases of subroutines to which the system has access for use in interrogation very awkward for the user to express (con- qualifying data for retrieval. Thus, the class of qualification sider, for example, a query for those parts and quantities expressions which can be used in a set specification must related to certain given projects via Q and R). have the descriptive power of the class of well-formed 1.6. EXPRESSIBLE, NAMED, AND STORED RELATIONS formulas of an applied predicate calculus. I t is well known Associated with a data bank are two collections of rela- t h a t to preserve this descriptive power it is unnecessary to tions: the named set and the expressible set. The named set express (in whatever syntax is chosen) every formula of is the collection of all those relations that the community of the selected predicate calculus. For example, just those in users can identify b y means of a simple name (or identifier). prenex normal form are adequate [9]. A relation R acquires membership in the named set when a Arithmetic functions may be needed in the qualification suitably authorized user declares R; it loses membership or other parts of retrieval statements. Such functions can when a suitably authorized user cancels the declaration of be defined in H and invoked in R. R. A set so specified may be fetched for query purposes The expressible set is the total collection of relations that only, or it may be held for possible changes. Insertions take can be designated b y expressions in the data language. Such the form of adding new elements to declared relations with- expressions are constructed from simple names of relations out regard to any ordering that may be present in their in the named set; names of generations, roles and domains; machine representation. Deletions which are effective for logical connectives; the quantifiers of the predicate calcu- the community (as opposed to the individual user or sub- lus; 6 and certain constant relation symbols such as = , > . communities) take the form of removing elements from de- The named set is a subset of the expressible set--usually a clared relations. Some deletions and updates may be trig- very small subset. gered b y others, if deletion and update dependencies be- Since some relations in the named set m a y be time-inde- tween specified relations are declared in R. pendent combinations of others in that set, it is useful to One important effect that the view adopted toward data consider associating with the named set a collection of has on the language used to retrieve it is in the naming of statements t h a t define these time-independent constraints. data elements and sets. Some aspects of this have been dis- We shall postpone further discussion of this until we have cussed in the previous section. With the usual network introduced several operations on relations (see Section 2). view, users will often be burdened with coining and using One of the major problems confronting the designer of a more relation names than are absolutely necessary, since data system which is to support a relational model for its names are associated with paths (or path types) rather users is that of determining the class of stored representa- than with relations. tions to be supported. Ideally, the variety of permitted Once a user is aware that a certain relation is stored, he data representations should be just adequate to cover the will expect to be able to exploit 5 it using any combination spectrum of performance requirements of the total col- of its arguments as "knowns" and the remaining argu- lection of installations. Too great a variety leads to un- ments as "unknowns," because the information (like necessary overhead in storage and continual reinterpreta- Everest) is there. This is a system feature (missing from tion of descriptions for the structures currently in effect. many current information systems) which we shall call For any selected class of stored representations the data (logically) symmetric exploitation of relations. Naturally, system must provide a means of translating user requests symmetry in performance is not to be expected. expressed in the data language of the relational model into To support symmetric exploitation of a single binary re- corresponding--and efficient--actions on the current lation, two directed paths are needed. For a relation of de- stored representation. For a high level data language this gree n, the number of paths to be named and controlled is presents a challenging design problem. Nevertheless, it is a n factorial. problem which must be solved--as more users obtain con- Again, if a relational view is adopted in which every n- current access to a large data bank, responsibility for pro- ary relation (n > 2) has to be expressed b y the user as a viding efficient response and throughput shifts from the nested expression involving only binary relations (see individual user to the data system. Feldman's L E A P System [10], for example) then 2n -- 1 names have to be coined instead of only n -b 1 with direct s Because each relation in a practical data bank is a finite set at n-ary notation as described in Section 1.2. For example, the every instant of time, the existential and universal quantifiers can be expressed in terms of a function that counts the number of 5 Exploiting a relation includes query, update, and delete. elements in any finite set. 382 C o m m u n i c a t i o n s o f t h e ACM V o l u m e 13 / N u m b e r 6 / J u n e , 1970

7.2. Redundancy and Consistency ternary relation which preserves all of the information in the given relations? 2.1. OPERATIONS ON RELATIONS The example in Figure 5 shows two relations R, S, which Since relations are sets, all of the usual set operations are are joinable without loss of information, while Figure 6 applicable to them. Nevertheless, the result may not be a shows a join of R with S. A binary relation R is joinable relation; for example, the union of a binary relation and a ternary relation is not a relation. with a binary relation S if there exists a ternary relation U such that 7r12(U) = R and ~'23(U) -- S. Any such ternary The operations discussed below are specifically for rela- relation is called a join of R with S. If R, S are binary rela- tions. These operations are introduced because of their key tions such that v2 (R) = ~i (S), then R is joinable with S. role in deriving relations from other relations. Their One join that always exists in such a case is the natural principal application is in noninferential information sys- join of R with S defined b y t e m s - s y s t e m s which do not provide logical inference services--although their applicability is not necessarily R*S = {(a,b,c):R(a,b) A S(b,c)} destroyed when such services are added. where R (a, b) has the value true if (a, b) is a member of R and similarly for S (b, c). I t is immediate that Most users would not be directly concerned with these operations. Information systems designers and people con- • "12 ( R , S ) = R cerned with data bank control should, however, be thor- and oughly familiar with them. 2.1.1. Permutation. A binary relation has an array ~23 ( R , S ) = S. representation with two columns. Interchanging these col- umns yields the converse relation. More generally, if a Note that the join shown in Figure 6 is the natural join permutation is applied to the columns of an n-ary relation, of R with S from Figure 5. Another join is shown in Figure 7. the resulting relation is said to be a permutation of the given relation. There are, for example, 4! -- 24 permuta- tions of the relation supply in Figure 1, if we include the II31(supply) (project supplier) identity permutation which leaves the ordering of columns 5 1 unchanged. 5 2 Since the user's relational model consists of a collection 1 4 of relationships (domain-unordered relations), permuta- 7 2 tion is not relevant to such a model considered in isolation. FIG. 4. A permuted projection of the relation in Figure 1 I t 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 R (supplier part) S (part project) answerable b y a stored relation is identical to the set 1 1 1 1 answerable by any permutation of that relation. Although 2 1 1 2 2 2 2 1 it is logically unnecessary to store both a relation and some permutation of it, performance considerations could make FIG. 5. Two joinable relations it advisable. 2.1.2. Projection. Suppose now we select certain col- umns of a relation (striking out the others) and then re- R*S (supplier part project) move from the resulting array any duplication in the rows. 1 1 1 The final array represents a relation which is said to be a 1 1 2 2 1 1 projection of the given relation. 2 1 2 A selection operator 7r is used to obtain any desired 2 2 1 permutation, projection, or combination of the two opera- FIG. 6. The natural join of R, with S (from Figure 5) tions. 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 j t h column is column ij of R (j = 1, 2, . . . , k) U (supplier part project) except that duplication in resulting rows is removed. Con- 1 1 2 sider the relation supply of Figure 1. A permuted projection 2 1 1 of this relation is exhibited in Figure 4. Note that, in this 2 2 1 particular case, the projection has fewer n-tuples than the FIG. 7. Another join of R with S (from Figure 5) relation from which it is derived. 2.1.3. Join. Suppose we are given two binary rela- tions, which have some domain in common. Under what Inspection of these relations reveals an element (ele- circumstances can we combine these relations to form a ment 1 ) of the domain part (the domain on which the join 7 When dealing with relationships, we use domain names (role- is to be made) with the property that it possesses more qualified whenever necessary) instead of domain positions. than one relative under R and also under S. I t is this ele- V o l u m e 13 / N u m b e r 6 / J u n e , 1970 Communications o f t h e ACMM 383

8.m e n t which gives rise to the plurality of joins. Such an ele- y), and T with R (say z), and, furthermore, y must be a ment in the joining domain is called a point of ambiguity relative of x under S, z a relative of y under T, and x a with respect to the joining of R with S. relative of z under R. N o t e t h a t in Figure 8 the points I f either ~r21(R) or S is a function, 8 no point of ambiguity x = a; y = d; z = 2 have this property. can occur in joining R with S. I n such a case, the natural T h e natural linear 3-join of three binary relations R, S, join of R with S is the only join of R with S. Note t h a t the T is given b y reiterated qualification "of R with S " is necessary, because S might be joinable with R (as well as R with S), and this R.S.T = {(a,b,c,d):R(a,b) A S(b,c) A T(c,d)} join would be an entirely separate consideration. I n Figure where parentheses are not needed on the left-hand side be- 5, none of the relations R, ml (R), S, ~r21(S) is a function. cause the natural 2-join (*) is associative. To obtain the Ambiguity in the joining of R with S can sometimes be cyclic counterpart, we introduce the operator .y which pro- resolved b y means of other relations. Suppose we are given, duces a relation of degree n - 1 from a relation of degree n or can derive from sources independent of R and S, a rela- b y tying its ends together. Thus, if R is an n - a r y relation tion T on the domains project and supplier with the follow- (n >_ 2), the tie of R is defined b y the equation ing properties: (1) ~-~(T) = ~-2(S), "I(R) = { (al, a2, . . . , a , _ l ) : R ( a l , a2, . . . , a~-l, an) A al = an}. (2) w2(T) = ~ ( R ) , (3) T ( j , s) - - + 3 p ( R ( S , p) A S ( p , j ) ) , We m a y now represent the natural cyclic 3-join of R, S, T b y the expression (4) R(s, p) ---->3j(S(p,j) A T ( j , s)), (5) S ( p , j ) ---->3s(T(j, s) A R(s, p ) ) , (R.S.T). then we m a y form a three-way join of R, S, T; t h a t is, a Extension of the notions of linear and cyclic 3-join and ternary relation such t h a t their natural counterparts to the joining of n binary rela- ~(v) = R, ~(U) = S, ~(V) = T. tions (where n > 3) is obvious. A few words m a y be ap- propriate, however, regarding the joining of relations which Such a join will be called a cyclic 3-join to distinguish it are not necessarily binary. Consider the case of two rela- from a linear 3-join which would be a quaternary relation tions R (degree r), S (degree s) which are to be joined on V such t h a t p of their domains (p < r, p < s). For simplicity, sup- pose these p domains are the last p of the r domains of R, ~r12(V) = R, ~r~3(V) = Z, 7r34(V) = T. and the first p of the s domains of S. I f this were not so, we While it is possible for more than one cyclic 3-join to exist could always apply appropriate permutations to m a k e it (see Figures 8, 9, for an example), the circumstances under so. Now, take the Cartesian product of the first r-p do- which this can occur entail much more severe constraints mains of R, and call this new domain A. T a k e the Car- tesian product of the last p domains of R, and call this B. R (8 p) s (p i) T ~ 8) T a k e the Cartesian product of the last s-p domains of S la ad dl and call this C. 2a ae d2 We can t r e a t R as if it were a binary relation on the 2b bd e2 domains A, B. Similarly, we can t r e a t S as if it were a bi- be e2 n a r y relation on the domains B, C. The notions of linear FIG. 8. B i n a r y r e l a t i o n s w i t h a p l u r a l i t y o f c y c l i c 3 ~ o i n s and cyclic 3-join are now directly applicable. A similar ap- proach can be t a k e n with the linear and cyclic n-joins of n U (8 p j) U' (8 p i) relations of assorted degrees. 2.1.4. Composition. T h e reader is probably familiar lad lad 2ae 2ad with the notion of composition applied to functions. We 2bd 2ae shall discuss a generalization of t h a t concept and apply it 2be 2bd first to binary relations. Our definitions of composition 2be and composability are based very directly on the definitions FIG. 9. Two cyclic 3-joins of the relations in Figure 8 of join and joinability given above. Suppose we are given two relations R, S. T is a com- t h a n those for a plurality of 2-joins. To be specific, the re- position of R with S if there exists a join U of R with S such lations R, S, T m u s t possess points of ambiguity with t h a t T = 7913(U). Thus, two relations are composable if respect to joining R with S (say point x), S with T (say and only if they are j oinable. However, the existence of more t h a n one join of R with S does not imply the existence 8 A function is a binary relation, which is one-one or many-one, of more t h a n one composition of R with S. but not one-many. Corresponding to the natural join of R with S is the 384 Communications o f the ACM Volume 13 / Number 6 / June, 1970

9.natural composition 9 of R with S defined b y 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 R.S = ~'i~(R*S). generate a subset of R is through the operation restriction Taking the relations R, S from Figure 5, their natural com: of R by S. This operation is a generalization of the restric- position is exhibited in Figure 10 and another composition tion of a function to a subset of its domain, and is defined is exhibited in Figure 11 (derived from the join exhibited as follows. in Figure 7). Let L, M be equal-length lists of indices such that R. S (project supplier) L = /1,/2, " " , ik, M = j l , j 2 , " " ,jk where k ~ degree 1 1 of R and k ~ degree of S. Then the L, M restriction of R by 1 2 S denoted R d M S is the maximal subset R' of R such that 2 1 2 2 ~L ( R ' ) = ~M (S). Fro. 10. The natural composition of R with S (from Figure 5) The operation is defined only if equality is applicable be- tween elements of ~'~h(R) on the one hand and ~'~h(S) on T (project supplier) the other for all h = 1, 2, • • •, k. 1 2 The three relations R, S, R' of Figure 13 satisfy the equa- 2 1~: tion R ' = R¢~,3)[¢i,2iS. Fro. 11. Another composition of R with S (from Figure 5) R 0 p J) S (p j) R' (s p j) When two or more joins exist, the number of distinct 1 a A a A 1 a A compositions may be as few as one or as many as the num- 2 a A c B 2 a A ber of distinct joins. Figule 12 shows an example of two 2 a B b B 2 b B 2 b A relations which have several joins but only one composition. 2 b B Note that the ambiguity of point c is lost in composing R FIG. 13. Example of restriction with S, because of unambiguous associations made via the points a, b, d, e. We are now in a position to consider various applications R (supplier part) S (part project) of these operations on relations. 1 a a g 2.2. REDUNDANCY 1 b b f Redundancy in the named set of relations must be dis- 1 c c f tinguished from redundancy in the stored set of representa- 2 c c g tions. We are primarily concerned here with the former. 2 d d g 2' e e f To begin with, we need a precise notion of derivability for relations. Fio. 12. Many joins, only one composition Suppose 0 is a collection of operations on relations and Extension of composition to pairs of relations which are each operation has the property that from its operands it not necessarily binary (and which may be of different de- yields a unique relation (thus natural join is eligible, but grees) follows the same pattern as extension of pairwise join is not). A relation R is O-derivable from a set S of rela- joining to such relations. tions if there exists a sequence of operations from the col- A lack of understanding of relational composition has led lection 0 which, for all time, yields R from members of S. several systems designers into what may be called the The phrase "for all time" is present, because we are dealing connection trap. This trap may be described in terms of the with time-varying relations, and our interest is in derivabil- following example. Suppose each supplier description is ity which holds over a significant period of time. For the linked by pointers to the descriptions of each part supplied named set of relationships in noninferential systems, it ap- by that supplier, and each part description is similarly pears that an adequate collection 01 contains the following linked to the descriptions of each project which uses that operations: projection, natural join, tie, and restriction. part. A conclusion is now drawn which is, in general, er- Permutation is irrelevant and natural composition need roneous: namely that, if all possible paths are followed from not be included, because it is obtainable b y taking a natural a given supplier via the parts he supplies to the projects join and then a projection. For the stored set of representa- using those parts, one will obtain a valid set of all projects tions, an adequate collection ~ of operations would include supplied b y that supplier. Such a conclusion is correct permutation and additional operations concerned with sub- only in the very special case that the target relation be- setting and merging relations, and ordering and connecting tween projects and suppliers is, in fact, the natural com- their elements. position of the other two relations--and we must normally 2.2.1. Stronq Redundancy. A set of relations is strongly add the phrase "for all time," because this is usually im- redundant if it contains at least one relation that possesses plied in claims concerning path-following techniques. a projection which is derivable from other projections of 9 Other writers tend to ignore compositions other than the na- relations in the set. The following two examples are in- tural one, and accordingly refer to this particular composition as tended to explain why strong redundancy is defined this the composition--see, for example, Kelley's "General Topology." way, and to demonstrate its practical use. In the first ex- Volume 13 / N u m b e r 6 / June, 1970 Communications of the ACM 385

10.ample the collection of relations consists of just the follow- The relations ~12 (P), ~r12(Q), '~'12(R) are complex l° relations ing relation: with the possibility of points of ambiguity occurring from time to time in the potential joining of any two. Under employee (serial #, name, manager#, managername ) these circumstances, none of them is derivable from the with serial# as the primary key and manager# as a foreign other two. However, constraints do exist between them, key. Let us denote the active domain by ,~, and suppose since each is a projection of some cyclic join of the three of that them. One of the weak redundancies can be characterized b y the statement: for all time, ~12 (P) is some composition At (manager#) c A, (serial#) of ~-12(Q) with "~'21(R). The composition in question might and be the natural one at some instant and a nonnatural one at another instant. At (managername ) C At (name) Generally speaking, weak redundancies are inherent in for all time t. In this case the redundancy is obvious: the the logical needs of the community of users. T h e y are not domain managername is unnecessary. To see that it is a removable b y the system or data base administrator. If strong redundancy as defined above, we observe that they appear at all, they appear in both the named set and ~34(employee) = ~'12(employee )l[lm (employee). the stored set of representations. In the second example the collection of relations includes a 2.3. CONSISTENCY relation S describing suppliers with primary key s#, a re- Whenever the named set of relations is redundant in lation D describing departments with primary key d#, a either sense, we shall associate with that set a collection of relation J describing projects with primary key j#, and the statements which define all of the redundancies which hold following relations: independent of time between the member relations. If the P (s#, d#, . . . ), Q (s#, j#, . - . ), R (d#, j#, . . . ), information system lacks--and it most probably will--de- tailed semantic information about each named relation, it where in each case . - . denotes domains other than s#, d#, cannot deduce the redundancies applicable to the named j#. Let us suppose the following condition C is known to set. I t might, over a period of time, make attempts to hold independent of time: supplier s supplies department induce the redundancies, but such attempts would be fal- d (relation P ) if and only if supplier s supplies some project lible. j (relation Q) to which d is assigned (relation R). Then, we Given a collection C of time-varying relations, an as- can write the equation sociated set Z of constraint statements and an instantaneous ~'12 ( P ) = 71-12( Q ) . 71"21( R ) value V for C, we shall call the state (C, Z, V) consistent or inconsistent according as V does or does not satisfy Z. and thereby exhibit a strong redundancy. For example, given stored relations R, S, T together with An important reason for the existence of strong re- the constraint statement '%'12(T) is a composition of dundancies in the named set of relationships is user con- 7r~2(R) with 7r~2(S)", we m a y check from time to time t h a t venience. A particular case of this is the retention of semi- the values stored for R, S, T satisfy this constraint. An al- obsolete relationships in the named set so that old pro- gorithm for making this check would examine the first two grams that refer to them by name can continue to run cor- columns of each of R, S, T (in whatever way they are repre- rectly. Knowledge of the existence of strong redundancies sented in the system) and determine whether in the named set enables a system or data base adminis- trator greater freedom in the selection of stored representa- (1) Try(T) = ~-i(R), tions to cope more efficiently with current traffic. If the (2) ~ ( T ) = ~ ( S ) , strong redundancies in the named set are directly reflected (3) for every element pair (a, c) in the relation ~2 (T) in strong redundancies in the stored set (or if other strong there is an element b such t h a t (a, b) is in ~-12(R) redundancies are introduced into the stored set), then, gen- and (b, c) is in 7r12(S). erally speaking, extra storage space and update time are consumed with a potential drop in query time for some There are practical problems (which we shall not discuss queries and in load on the central processing units. here) in taking an instantaneous snapshot of a collection 2.2.2. Weak Redundancy. A second type of redun- of relations, some of which may be very large and highly dancy may exist. In contrast to strong redundancy it is not variable. characterized by an equation. A collection of relations is I t is important to note that consistency as defined above weakly redundant if it contains a relation t h a t has a projec- is a property of the instantaneous state of a data bank, and tion which is not derivable from other members but is at is independent of how that state came about. Thus, in all times a projection of some join of other projections of particular, there is no distinction made on the basis of relations in the collection. whether a user generated an inconsistency due to an act of We can exhibit a weak redundancy b y taking the second omission or an act of commission. Examination of a simple example (cited above) for a strong redundancy, and as- 10A binary relation is complex if neither it nor its converse is a suming now that condition C does not hold at all times. function. 386 C o m m u n i c a t i o n s o f t h e AMC V o l u m e 13 / N u m b e r 6 / J u n e , 1970

11.example will show the reasonableness of this (possibly un- In Section 2 operations on relations and two types of conventional) approach to consistency. redundancy are defined and applied to the problem of Suppose the named set C includes the relations S, J, D, maintaining the data in a consistent state. This is bound to P, Q, R of the example in Section 2.2 and that P, Q, R become a serious practical problem as more and more dif- possess either the strong or weak redundancies described ferent types of data are integrated together into common therein (in the particular case now under consideration, it data banks. does not matter which kind of redundancy occurs). Further, M a n y questions are raised and left unanswered. For suppose that at some time t the data bank state is consistent example, only a few of the more important properties of and contains no project j such that supplier 2 supplies the data sublanguage in Section 1.4 are mentioned. Neither project j and j is assigned to department 5. Accordingly, the purely linguistic details of such a language nor the there is no element (2, 5) in ~-i2(P). Now, a user introduces implementation problems are discussed. Nevertheless, the the element (2, 5) into ~ri2(P) by inserting some appropri- material presented should be adequate for experienced ate element into P. The data bank state is now inconsistent. systems programmers to visualize several approaches. I t The inconsistency could have arisen from an act of omis- is also hoped that this paper can contribute to greater pre- sion, if the input (2, 5) is correct, and there does exist a cision in work on formatted data systems. project j such that supplier 2 supplies j and j is assigned to Acknowledgment. I t was C. T. Davies of I B M Pough- department 5. In this case, it is very likely that the user keepsie who convinced the author of the need for data intends in the near future to insert elements into Q and R independence in future information systems. The author which will have the effect of introducing (2, j ) into ~r12(Q) wishes to thank him and also F. P. Palermo, C. P. Wang, and (5, j ) in 7rI2(R). On the other hand, the input (2, 5) E. B. Altman, and M. E. Senko of the I B M San Jose Re- might have been faulty. I t could be the case that the user search Laboratory for helpful discussions. intended to insert some other element into P - - a n element whose insertion would transform a consistent state into a consistent state. The point is that the system will RECEIVED SEPTEMBER, 1969; REVISED FEBRUARY, 1970 normally have no way of resolving this question without interrogating its environment (perhaps the user who cre- REFERENCES ated the inconsistency). 1. CHILDS,D.L. Feasibility of a set-theoretical data structure There are, of course, several possible ways in which a --a general structure based on a reconstituted definition of system can detect inconsistencies and respond to them. relation. Proc. IFIP Cong., 1968, North Holland Pub. Co., Amsterdam, p. 162-172. In one approach the system checks for possible inconsist- 2. LEVEIN,R. E., AND MARON, M. E. A computer system for ency whenever an insertion, deletion, or key update occurs. inference execution and data retrieval. Comm. ACM 10, Naturally, such checking will slow these operations down. 11 (Nov. 1967), 715--721. If an inconsistency has been generated, details are logged 3. BACHMAN,C. W. Software for random access processing. internally, and if it is not remedied within some reasonable Datamation (Apr. 1965), 36--41. 4. McGEE, W. C. Generalized file processing. In Annual Re- time interval, either the user or someone responsible for view in Automatic Programming 5, 13, Pergamon Press, the security and integrity of the data is notified. Another New York, 1969, pp. 77-149. approach is to conduct consistency checking as a batch 5. Information Management System/360, Application Descrip- operation once a day or less frequently. Inputs causing the tion Manual H20-0524-1. IBM Corp., White Plains, N. Y., inconsistencies which remain in the data bank state at July 1968. 6. GIS (Generalized Information System), Application Descrip- checking time can be tracked down if the system main- tion Manual H20-0574. IBM Corp., White Plains, N. Y., rains a journal of all state-changing transactions. This 1965. latter approach would certainly be superior if few non- 7. BLEIER, R . E . Treating hierarchical data structures in the transitory inconsistencies occurred. SDC time-shared data management system (TDMS). Proc. ACM 22nd Nat. Conf., 1967, MDI Publications, 2.4. SUMMARY Wayne, Pa., pp. 41---49. In Section 1 a relational model of data is proposed as a 8. IDS Reference Manual GE 625/635, GE Inform. Sys. Div., basis for protecting users of formatted data systems from Pheonix, Ariz., CPB 1093B, Feb. 1968. the potentially disruptive changes in data representation 9. CHURCH,A. An Introduction to Mathematical Logic I. Prince- ton U. Press, Princeton, N.J., 1956. caused b y growth in the data bank and changes in traffic. 10. FELDMAN,J. A., AND ROVNEa, P.D. An Algol-based associ- A normal form for the time-varying collection of relation- ative language. Stanford Artificial Intelligence Rep. AI-66, ships is introduced. Aug. 1, 1968. [] V o l u m e 13 / Number 6 / J u n e , 1970 C o m m u n i c a t i o n s of t h e ACCM 387