On the satisfiability of dependency constraints in ER diagrams

7/27/2019 On the satisfiability of dependency constraints in ER diagrams

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hformarion Swems Vol. 15, No. 4, pp. 453461, 1990 0306-4379/90 s3.00 + 0.00Printed in Great Britain. All rights reserved Copyright (s 1990Pergamon Press plc

UN THE SATISFIABILITY OF DEPENDENCYCONSTRAINTS IN ENTITY-RELATIONSHIP SCHEMATAt

MAURIZIO LENZERINInd FAO~ONOBILI*Universit$ di Roma La Sapienza, Ripartimento di Infarmatica e Sistemistica, via Salaria 113,oOi9g Roma, ItalyIstituto cli Anal&i dei Sistemi ed Info~atica, Consigho Nazionale de&a Ricerche, Viale Manzoni 30,00185 Roma, Italy

Abstract-The satisfiability problem for a specific class of integrity constraints in data bases, namely thedependency constraints, is studied. An entity-relationship model is used for expressing data schemata. Inthis model suitable types of dependency constraints, called cardinality ratio constraints, allow one toimpose restrictions on the mappings between entities and relationships. We show that, as far as such aclass of constraints is concerned, the usual notion of ~tis~abjlity is not sufficiently meaningfuf. For thisreason we introduce the notion of strong satisfiability, ensuring that no entity or ~lationship is compelledto be empty in all of the legal instances of the schema. We propose to model the cardinality ratioconstraints of a schema by means of a suitable linear inequality system and we show that a schema isstrongly satisfiable if and only if the associated system admits positive solutions. Furthermore, we describea method for discovering which are the sets of constraints that prevent a schema from being stronglysatisfiable.

1. rNTR~DUCTrONSemantic integrity constraints in data bases are usedto specify the rules which data have to satisfy in orderto reflect the properties of the represented objects inthe modeled real world.

Great attention has generally been devoted to aparticutar class of integrity constraints, the so-calleddependency ~~~~~r~~ncs, that are used to specifyrestrictions on the mappings between the data classesof a schema. They represent a very important andcommonly occurring class of constraints [l, 23:functional and numerical dependencies (see [3] and[4]) in the relational model, as well as many types ofexistence constraints expressible in semantic datamodels (for example [5], [6] and f7]), are meaningfulexamples of such a kind of constraints.

In the data base community, dependencyconstraints have mainly been studied from theperspective of data design, where the goal is to obtaina good schema with respect to the efficiency of database operations. In such a context, the major issuesthat have been addressed are related to the impii-cation of data dependencies, i.e. the problem offinding sound and complete inference systems for agiven class of dependency constraints.

Dependency constraints are also used in manyknowledge representation languages based on the___.___ .~.__ ____. -I- __... _~~--?A previous version of this paper appeared in the Proceed-

i ngs qf the 13th Cmference on Very Large Du?u Bases,19X-f.

concept of frame (see [7]). In such languages, framesare used to modeI classes of objects, and a sort ofdependency assertion is provided for indicating theminimum and the maximum number of values for anattribute of a frame.

In this paper we deal with one important prop-erty of dependency constraints, namely their sat-isfiabiiity. We remind the reader that the set ofintegrity constraint of a schema is said to be sat-isfiable if some instance of the schema (i.e. database state) exists which satisfies them (in thiscase the schema itself is said to be satisfiable).Although satisfiability is a crucial issue in verify-ing the correctness of the representation, it hasnot been deeply addressed in the literature, neitherin the data base field, nor in knowledge represen-tation.

We shall see in Section 3 that, when depen-dency constraints are considered, the usual notionof satisfiability is not sufficient for capturingsignificant properties of a schema. In fact, althoughseveral instances of the schema may exist thatsatisfy a set of dependency constraints, it mayhappen that in all of such instances, some ofthe classes of the schema is invariably empty. Forthis reason we introduce a different property,called strong satisfiability, ensuring that each classis non-empty in at least one instance of theschema.Our work is carried out in the context of the~~t~#y-~eiut~~~hi~ (ER) approach to data modelingf9]. Such an approach has largely influenced many

IS t-4.. E 453


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454 MAURIZIOLENZERINI nd PAOLONOBILImethodological proposals for enhancing the effective-ness and correctness of information system design[lo, Ill.ER-based formalisms are now widely adopted inthe so-called conceptual phase of data base design,whose goal is to obtain a complete, precise andimplementation independent description of the objectsto be represented in the data base. Although the ERmodel has become very popular from the practicalpoint of view, there are only few works attempting toprovide a sound theoretical foundation for this model.Our work can also be seen as a contribution to thedevelopment of a formal framework for the ER model.

We shall refer to a particular ER model, called thesemantic entity-relationship model (SERM, in thefollowing), defined in [12], in which a specific kind ofconstraint, namely the cardinality ratio constraints, isprovided for expressing dependencies betweenentitites and relationships. Taking into account thegenerality of the notions of entity and relationship, itis easy to reinterpret these constraints in the contextof frame-based knowledge representation languages.In Section 2 we describe the main characteristics ofSERM that are useful for the subsequent sections.The goal of our work is twofold:

l To provide necessary and sufficient conditionsfor the strong satisfiability of cardinality ratioconstraints in SERM schemata. This aspect isdealt with in Section 3.

l To describe a method for analyzing SERMschemata in order to discover possible unsatisfi-able sets of cardinality ratio constraints. Such amethod is described in Section 4.

2. THE DATA MODELIn this section we briefly describe the characteristics

of the SERM data model that are useful for thesubsequent sections. We assume that the reader isfamiliar with the concepts and the terminology of theER model.

An entity (entity type in [9]) denotes a set ofindividuals, called its instances, representing real-world objects with common properties.

Relationships among entities are used to modelassociations among real-world objects. A re&ionship(relationship type in [9]) denotes a set of individ-uals, called its instances: each element of such a setrepresents an association among a different combi-nation of instances of the entities that are connectedto the relationship. In the following, we use the termclass to refer to an entity or a relationship. Since arelationship can be linked to the same entity morethan once, the concept of ro le is introduced todistinguish different connections of the same entitywith a relationship. More precisely, a role is a namethat is associated with a pair (entity, relationship)and that univocally determines the connection be-tween such a pair of classes.

For the purpose of this paper, a SERM schemaconsists of a set of entities, a set of relationships, a setof roles and a set of cardinality ratio constraints,which are defined later in this section.

Using the common conventions of representingER schemata by means of diagrams, we show anexample of schema in Fig. 1. Notice that in thediagram, roles are associated with the edges connect-ing the corresponding entities and relationships. Theexample deals with partnerships and financialsupports for bilateral research projects. Each researchinstitute may participate to bilateral projects eitheras a project leader or as a partner. For this reason,the entity Research-Institute is connected to therelationship Partnership through two differentroles, Project-Leader and Partner.

A subschema of a SERM schema S is aschema constituted by a subset of the entities andrelationships of S and satisfying the condition thatevery relationship is connected to the same set ofentities and through the same roles as in S. Forexample, the classes Research-Institute, Part-nership and Bilateral-Project, together withthe roles Project-Leader, Partner and Projectconstitute a subschema of the schema shown inFig. I.

In SERM, the concepts of attribute of entities andrelationships and subset relationship between classesare also considered; however, they are not dealt within the present work.

Reseatch-Ins t i tu teProject Bilateral-Project

Parmer

Fig. I. A SERM schema.


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Satisfiability of constraints in entity-relationship schemata 455An instance of a SERM schema S is a finitecollection of instances of the entities and the relation-

ships of S, satisfying a set of rules (inherentconstramts) to be described later. Each instance of arelationship R is linked to a combination of instancesof the entities that are connected to R in the schema.The roles are used also at the instance level foridentifying the links between relationship and entityinstances. For example, an instance of the relation-ship Partnership of the schema shown in Fig. I canbe linked respectively to the instance al of Research-Institute through the role Project-Leader, to theinstance a2 of Research-Leader through the rolePartner, and to the instance b, of Bilateral-Pro-ject through the role Project.

We assume that no limit exists for the number ofpossible instances of entities and relationships. Fur-thermore, the instance of a schema in which all theclasses have an empty set of instances is called empty.

In Fig. 2 we give a representation of an instance ofthe schema shown in Fig. 1.We write:

to denote the relationship instance r connected to theentity Instances e,. ez, . , em, respectively throughroles Cl,. Uz, . U,,,.The pair (e,, U,) is said to bethe component of r corresponding to the role U,.Every instance I of a SERM schema must satisfythe followmg set of rules, called inherent constraints:1. For each relationship R, for each instance

r = {(e,, U,), . . . % e,, Urn)] of R, for eachif1 < i , , )sl = {, }sz = {, }Fig 2. An instance of a schema shown in Fig. I.

We shall write cardinality ratio constraints in the form:

where:&%)I)E(U) - R,

l E is an entity, R a relationship, and U a role; inparticular, E is connected to R by means of roleu;

l x is a non-negative integer, called the minilnumcardinality of R with respect to E in the role U;

l r is either a positive integer or cc, called themaximum cardinality of R with respect to E inthe role U.0 y 2:s.

In order to characterize precisely the meaning ofcardinality ratio constraints, we now state the con-ditions under which an instance of a schema satisfies(i.e. makes true) a given constraint.Definition l-Let ir.1)E(U)- Rbe a cardinality ratio constraint of a schema S, andlet Z be an instance of S. 1 satifies

E(U)=+ Rif for every instance e of E, the number M of Rinstances connected to e by means of role U verifies:

Notice that the value 0 for the minimum cardinal-ity and the value a; for the maximum cardinality donot represent real constraints; however, for thepurpose of this paper, they are treated like any othervalues.

In the following, when we need to refer to thevalue of the minimum and the maximum cardinalityof the relationship R with respect to the entity E inthe role U, we write MIN-CARD(R, E, U) andMAX-CARD(R, E, U), respectively. If no cardinalityratio constraint is defined for a triple R. E and U.we assume that MIN-CARD(R, E, U) = 0 andMAX-CARD(R, E, U) = 8x1.The set of cardinalityratio constraints of a schema S will be denoted by r $.Every instance of S in which all the cardinality ratioconstraints m f are satisfied will be called legal.

Referring again to the example of Fig 1, thefollowing is a possible set of cardinahty ratioconstraints defined in the schema:

Research-Institute(Project-Leader)(0.:)- Partnership(I 11Bilateral-Project(Project) - Partnership

Bilateral-Project(Financed-Project)(11)- Supported-by(l,%,lInstitution(Sponsor) - Supported-by


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456 MAURIZIO ENZERINInd PAOLOOBILIThe first constraint represents the rule whichrestricts research institutes to be project leaders of at

most two bilateral projects. The second one imposesthat each bilateral project is associated with exactlyone project leader and one partner. By the thirdconstraint, bilateral projects are supported by oneand only one institution. Finally, the fourth con-straint represents the fact that only the institutionssupporting some projects are meaningful for theapplication.It is easy to verify that the instance of Fig. 2 satis-fies all of the above constraints and, therefore, is legal.

3. STRONG SATISFIABILITY OF SERMSCHEMATA

As we said in the introduction, the usual notion ofsatisfiability is not sufficient for capturing interestingproperties of a set of cardinality ratio constraints. Infact, since at least the empty instance of a schema Ssatisfies all the constraints in Ts, it follows that everySERM schema S is satisfiable with respect to Ts. Onthe other hand, it may happen that the cardinalityratio constraints of a SERM schema interact in sucha way that no legal instance of the schema other thanthe empty one exists. Consider, for example, theschema shown in Fig. 3 (in the diagram, the minimumand the maximum cardinalities of relationships areassociated with the corresponding roles).

It is easy to verify that only the empty instance islegal for such a schema: in fact, the constraintsdefined on relationship R impose that entity A cannothave more instances than entity B, whereas theconstraints defined on relationship Q impose that thenumber of instances of A is at least two times thenumber of instances of B.

In the general case, the cardinality ratio constraintscompel only some classes to be invariably empty inall the legal instances of the schema. When thishappens, we say that such classes cannot bepopulated. Since a class has to be consideredmeaningful only if the corresponding instances can berepresented in the data base, we look for a newproperty, ensuring that all of the classes of theschema can be populated. We call such a propertystrong sati sj i abi l i t y .

Fig. 3. A SERM schema which is not strongly satisfiable. for each entity E (relationship R) of S.

Definition 2-A SERMA schema S is strongly satisfi-able if for each class C of S, there exists at least onelegal instance of S in which the set of instances of Cis not empty.

When a schema S is not strongly satisfiable, weshall say that both S and the set of cardinality ratioconstraints Ts are unsatisfiable.

In the following, every instance of a schema inwhich no class is empty will be called fufly populat ed.The next theorem shows that we check for the strongsatisfiability of a schema by looking for the existenceof fully populated legal instances.Theorem I -A SERM schema S is strongly sati sfiabl eif and only i f there is at east one ull y popul ated I egaIi nstance of S.Proof. I f -par t - I t is evident that the existence of afully populated legal instance of S implies that S isstrongly satisfiable.

Only-$-part- I f S is strongly satisfiable, then, forevery class C, of S, there exists a legal instance J, ofS in which C, has a non-empty set of instances.Consider the instance J of S obtained from all the J, sby means of the following rules: (1) the set ofinstances of the generic class C in J is the union ofthe instances of C in all the J, s; (2) instances of classescoming from different J,s are considered different.Clearly, J is fully populated. Furthermore, since allthe J,s are legal, J is legal too. 0

In order to characterize the strong satisfiability ofSERM schemata we propose to model the cardinalityratio constraints of a schema by means of an associ-ated linear inequality system. This system is definedin such a way that the existence of a fully populatedlegal instance of the schema (and, by Theorem 1, thestrong satisfiability of the schema) is reflected into theexistence of some solutions for the correspondingsystem.Definition &Given a SERM schema S, we associatewith it an inequality system Ys whose unknowns andinequalities are defined as follows.

Unknowns of Ys:l one unknown k for each entity E of S;l one unknown 8 for each relationship R of S.Inequalities of Y:l one inequality of the form:

ri>X._&for each cardinality ratio constraintE(U) -+(.)R in S, with x # 0;

l one inequality of the form:ri 0 (ri >O)


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Satisfiability of constraints in entity-relationship schemata 457Notice that, according with the above definition,

Y s is homogeneous (i.e. all of its constant terms areequal to zero) and has integer coefficients.

A preliminary result can now be proved regard-ing a sufficient condition for the strong satisfiabil-ity of a simple SERM schema, constituted by onerelationship R connected to a collection of entities,each entity being possibly connected through severalroles.

instances of R linked to the same instance of E, bymeans of role U,, has at most:

elements. After the second execution of Step 2b, thelargest group of instances of R having the same setof component has at most:

Lemma l-Let S be a schema constituted by one re-lationship R connected to m dtyerent entities E, , E,,,Assume that each E, is connected to R through rolesU,,,U,, ,..., V,,f, with ql> 1, for each i (1


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458 MAURIZIOLENZERINI nd PAOLONOBILIDuring the first iteration of Step 2a (for i = 1 andj = I), the following assignments are performed:

ro+-{)rl+-{(ell, uu >Irz4>rS4I

During the execution of the first iteration of Step2b, the instances of R are renumbered as follows:

Notice that the instances of R with the same set ofassociated components are now contiguous. SinceStep 2a assigns different components (wheneverpossible) to contiguous instances of R, such a renum-bering is crucial in order to satisfy the abovecondition /I.

At the end of the second iteration of Step 2, i.e. theiteration corresponding to i = 1 and j = 2, we have:rO+{, 1rI+-{* , Ir2 = i, h, U,2>, Q22, b>>

r3 = {, (e,ov U12 >, >

r4 = {Cell 9 U,, >, , (e2,, U2, >I

r5 = {(e,, 9 UI I >, 63, t U12 >, >We are now ready to relate the strong satisfiabilityof a SERM schema S to the existence of solutions forthe associated system Y . The following theorem

gives a necessary and sufficient condition for the

strong satisfiability of SERM schemata. Its proofmakes use of the following lemma:Lemma 2-3 a l i near homogeneous inequali t y systemH w it h ati onal coeficients admits a posit ive soluti on,then it also admits an i nteger posit iv e solut ion.Proof-Let 0, , v2, . . . , v, be the (possibly irrational)values assigned to the unknowns xi, x2, . . , x, of Hby a positive solution X0. By adding the following setof inequalities to H:

x,>b, (1


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Satisfiability of constraints in entity-relationship schemata 459In other words, the positive integers Q and p satisfy

Y s when substituted for the unknowns i and A. Bygenerahzation. it is easy to see that we can constructa positive solution of Yys by assigning to eachunknown c the value corresponding to the number ofinstances of C in I. qThe above result ensures that the problem ofverifying the strong satisfiability of a SERM schemaScan be solved in polynomial time with respect to thenumber of classes of S. In effect. by Theorem 4. sucha problem can be reduced to the one of testing thepolyhedral cone defined by Ys (i.e. its solution space)for non-emptyness, which can be done in polynomialtime (see. for example [13], pp. 170-185).

4. ANALYSIS OF UNSATISFIABLESCHEMATA

As already noticed in Section 3, a SERM schemaS which is not strongly satisfiable includes one ormore sets of cardinality ratio constraints (i.e. subsetsof f ) that are unsatisfiable. As a result, some classesof S are compelled to be invariably empty in all of thelegal instances of the schema. The goal of this sectionIS to provide a method for discovering each unsatisfi-able set of cardmality ratio constraints of a schema.Such a method can be very helpful for identifyingthose dependency constraints whose specification iserroneous.

In what follows we make use of graph concepts torepresent SERM schemata. In particular, we showthat. for a schema S which is not strongly satisfiable,information about the sets of unsatisfiable cardinalityconstraints can be obtained from a suitable analysison a particular graph associated with S.Definition &Given a SERM schema S, the associ-ated graph GS is a directed multigraph (N, A) la-beled in the arcs. where:

l the set of nodes N is in one-to-one correspon-dence with the set of classes of S;

l the set of arcs A is determined by the followingrules: for each connection in S between an entityE and a relationship R through role U, two arcse, and e, are in A; e, is directed from the node

Fig 4. An example of a graph associated with a SERMschema.

corresponding to E to the node corresponding toR and is labeled with max(R, E, U); ez is directedfrom the node corresponding to R to the nodecorresponding to E and is labeled either withl/min(R, E, U), if min(R, E, U) # 0 or with 00, ifmin(R, E, U) = 0.

As an example, we show in Fig. 4 the graphassociated with the schema of Fig. 3.Every graph associated with a SERM schema will

be called the ER-graph. The label of an arc e will bedenoted by LABEL(e). Moreover, if x is a path (ora cycle) of an ER-graph GS, then the weight of 7t[denoted by WEIGHT(n)] is defined as follows:

WEIGHT(n) = n LABEL(e).eEnIf y is a cycle, and WEIGHT(y) < 1, then we saythat y is critical. It follows from the definition ofER-graph, that a critical cycle contains at least one

arc e such that LABEL(e) < 1, which corresponds toa minimum cardinality constraint whose value isgreater than 1.An assignment C#Jor an ER-graph G = (N, A) isa mapping:

4: N+R+,associating positive rational numbers with its nodes.

An assignment is said to be correct if for each arce = (n,, n,) in A, the following condition holds:

rb(nd- < LABEL(e).+(4)An ER-graph is said to be inconsistent if no correctassignment exists for it, consistent otherwise.We observe that, if a is a path from n, to nb

(not necessarily distinct) of an ER-graph G., thenWEIGHT(n) represents an upper bound forCpn,,)/4(n,), for each correct assignment 4 of G, i.e.:

4(%)- < WEIGHT(n).ddn,)

Notice that the assignment obtained by multiplyinga correct assignment by a rational number IS alsocorrect. Therefore, whenever a correct assignmentexists for a graph, an integer assignment (I.e. anassignment associating integer numbers to the nodes)also exists which is correct.

By the above definitions and by Theorem 2, onecan easily verify that the problem of checking aschema for strong satisfiability is isomorphic to theone of finding a correct assignment for the associatedgraph.

The goal of this section is to show that there existsa strict correspondence between sets of unsatisfiablecardinality ratio constraints in a schema and criticalcycles in the associated graph. In particular, it will beshown by Theorem 3 that a critical cycle is aninconsistent ER-graph and, on the converse, any


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460 MAURIZIO ENZERINInd PAOLONOBILIinconsistent ER-graph contains a critical cycle. The assignment 4 for G and a path II from nb to n, suchfollowing lemma introduces such a theorem: that:Lemma 3-Let G be a consistent ER-graph and n, andn,, two of its nodes. If 9 is the collection of all thecorrect assignments for G, and ll the collection of allpaths from nh to n, in G, then it holds that:

$j$ = WEIGHT(n).Consider now the cycle y constituted by 71 nd e, andsuppose that WEIGHT(y) 2 1, i.e.

4(n,)max - = min WEIGHT(n)64 4(nb) XEllwhere, if Il is empty, the right-hand side is to beinterpreted as co.Proof-It is obvious that:

6(n,)max - < min WEIGHT(K),due c%) Z6so that we can consider only the case where theleft-hand side is finite. Let 4 be a correct assignmentsuch that ~$(n,)/q%(n,) is maximum, and supposethat:

4(n,)- < yeF WEIGHT(n).4(nb)This means that every path rc, n I7 contains an arc e,(say from p to q) such that:

4(q)--c LABEL(e,) .4C.P)It is easy to verify that the set of all such e,sconstitutes a cut (N,, Nh) of G, with n,o N, andn,, E Nh. Hence, by multiplying every 4(n) (forn E N,) to a suitable positive number, we can obtaina correct assignment $ such that:

contradicting the hypothesis on 4. 0Theorem 3-An ER-graph G is inconsistent ifand onlyif it contains a critical cycle.Proof. If part-Assume that G contains a criticalcycle y with weight w, and suppose that 4 is a correctassignment for G. Then, for any node n in y, it wouldhold that:

4(n)mGWobtained by applying (4) to y. Since (5) cannot besatisfied, we can conclude that no correct assignmentexists for G.

OnIy-Q-part-Assume that G is inconsistent. LetG be a maximal consistent subgraph of G(notice that such a subgraph always exists, since atleast the graph obtained from G by eliminating all ofits arcs is consistent). Let e = (n,, nb) be any arc ofG - G. Lemma 5 ensures us that there is at least onepath from nb to no in G (otherwise we could find acorrect assignment for Gu{e}, contradicting thehypothesis that G is maximally consistent). FromLemma 3 again, it follows that there is a correct

1WEIGHT(z) < LABEL(e).

It follows that $(n,)/4(n,) < LABEL(e) and, there-fore, c5 is correct also for Gu{e}. Since this contra-dicts the hypothesis that G is maximally consistent,it follows that y is critical. 0

The above theorem shows that critical cycles areresponsible for the inconsistency of graph associatedwith a SERM schema. Taking into account thecorrespondence between the strong satisfiability of aSERM schema and the consistency of the associatedgraph, each critical cycle singles out an unsatisfiableset of cardinality ratio constraints, namely thosecorresponding to the labels of the component arcs.Notice that discovering critical cycles can be done inpolynomial time, for examples using a variant of theFloyd-Warshall algorithm for the determination ofthe shortest path between two nodes in a graph (see[13], pp. 1299133).

5. CONCLUSIONSIn this paper we have shown that significant

properties of a SERM schema can be recognized bymeans of suitable computations performed on anassociated inequality system and by an analysis on acorresponding graph. In particular, the resultsreported in Section 3 show that strong satisfiabilitycan be checked in polynomial time with respect tothe size of the schema by looking for solutions ofthe associated system; moreover, in Section 4 it isshown that, for a schema which is not stronglysatisfiable, information about the sets of unsatisfiablecardinality ratio constraints can be obtained bydiscovering critical cycles in the correspondinggraph.Since realistic schemata could lead to correspond-ing sizeable systems and graphs, it is worth notingthat some simplifications of the schemata can beadopted and should obviously be considered, for thesake of efficiency, when applying the describedtechniques. For example, we already noticed thatcritical cyles may appear in an ER-graph onlybecause of the existence of cardinality ratioconstraints with the value of the minimum cardinalitygreater than 1. This suggests that, in looking forcritical cycles in an ER-graph, one can ignore thosecycles in which all the minimum cardinality ratioconstraints have values 0 or 1, which are veryfrequent in practice. We notice, however, that acomplete analysis of the possibilities of improving the


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Satisfiability of constraints in entity-relationship schemata 461efficiency of the proposed methods is beyond thescope of this paper and is investigated in [14].

151In the same paper, we study the problem of

checking a schema for strong satisfiability when 161further capabilities of SERM are taken into account.In particular, we demonstrate that even small enrich- [71ments of the expressive power of the data model usedin this paper, may result in a dramatical increase of PIthe complexity of such a problem.Acknow ledgement-W e wish to thank Giorgio Ausiellofor his useful comments on an earlier version of this

[91

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