Discuss a or El Act Erna Rio

7/29/2019 Discuss a or El Act Erna Rio

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SemanticsoftheMinimumMultiplicityinTernary

AssociationsinUML

GonzaloGnova,JuanLlorens,PalomaMartnez

ComputerScienceDepartment,CarlosIIIUniversityofMadrid,

Avda.Universidad30,28911Legans(Madrid),Spain{ggenova,llorens,pmf}@inf.uc3m.es

http://www.inf.uc3m.es/

Abstract.TheconceptofmultiplicityinUMLderivesfromthatofcardinalityin entity-relationshipmodeling techniques. TheUMLdocumentation defines

thisconceptbutatthesametimeacknowledgessomelackofobviousnessinthe

specification of multiplicities for n-ary associations. This paper shows an

ambiguity in the definition given by UML documentation and proposes a

clarification to this definition, as well as a simple extension to the current

notation to represent other multiplicity constraints, such as participation

constraints,thatareequallyvaluableinunderstandingn-aryassociations.

Introduction

The entity-relationship model [4] has been widely used in structured analysis andconceptualmodeling,andithasevolvedintoobjectorientedclassdiagramssuchas

those oftheUnifiedModelingLanguage[29].Thisapproach iseasytounderstand,

powerfultomodelreal-worldproblemsandreadilytranslatedintoadatabaseschema,

although other forms of implementation, such as object-oriented programming

languages,are notso simple andstraight [23].Both inentity-relationshipdiagrams

andinclassdiagramsthemain constructsare theentityand therelationshipamong

entities(classandassociation,inUMLterminology).Inthissense,importantauthors

strongly reject the entity-relationship approach, since they consider that the very

distinctionbetweenentityandrelationshipsuffersoflackofprecision[6,7].

Formanyanalysts,oneofthemostproblematicaspectsofsystemsmodelingisthe

correctunderstandingofternaryassociationsand,ingeneral,n-aryassociations(we

use this term, as usual, to refer to associationswith three ormore roles).Ternary

associations represent often a complex situation which modelers find speciallydifficulttounderstand,regardingbothstructuralandbehavioralmodeling.Fromthe

structuralpoint of view, thesedifficulties are very ofteninterlinkedwith the fourth

and fifth normal form issue [13]. From the behavioral point of view, atomic

interactions that involve more than two objects are another source of conceptual

complexity (these interactions are denoted by some authors as "joint actions" or

"atomicmultiwaysynchronousinteractions"[12]).

The cardinality of a relationship (multiplicity of an association, in UML

terminology)isconsideredbysomeexpertsasthemoststructuralpropertyofamodel


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[17]. However, the multiplicity values typically specified for n-ary associations

provideonlypartialunderstandingoftheobjectstructure.Additionalconditionsmay

be included within written descriptions that accompany the models, but a better

integration,asfarasitispossible,isalwaysdesirable.Worstofall,oftenthevery

meaningof themultiplicitiesisbadlyunderstood.Aswewillsee,themultiplicityofbinary associations is rather simple to specify and understand in UML, but,

unfortunately,thisisnotthecaseforn-aryassociations,forwhichUMLhasdefined

incompleteandunclearmultiplicities.

The purpose of this paper is, on the one hand, to clarify the meaning ofn-ary

multiplicityvalues,whichisacknowledgedbyUMLtobenotveryobvious[29,p.3-

73], and on the otherhand topropose an extension to the notationofUML n-arymultiplicities,oneextensionthatprovidesmorecompleteandprecisedefinitionsfor

n-aryassociationswiththesmallestnotationalburden.Althoughourmainconcernis

UML, the core of our exposition isgeneral enough to beuseful inothermethods

basedon the entity-relationship approach. Thiswork forms part ofa moregeneral

researchthatisaimedtowardsabetterunderstandingoftheconceptof"association"

among classes. We hope that this better understanding will improve the modelsconstructedbothbynovelandexpertanalysts.

The remainder of this paper is organized as follows. Section 1 recalls some

definitionsfromUML:multiplicity,association,andmultiplicityforbinaryandn-ary

associations. Section 2 searches the roots of these definitions in data modeling

techniquesthatderivemainlyfromtheentity-relationshipapproachofChen.Section3 reveals anambiguity, orat leastuncertainty, inthe definitionofUMLminimum

multiplicityforn-aryassociations;threealternativeinterpretationsarepresented,each

onewithitsownproblemsandunexpectedconsequences.Finally, section4 tries to

understand theroot of theseproblemsby payingmore attentionto theparticipationconstraint;anotationcompatiblewiththethreealternativeinterpretationsofsection3

isproposedtorecoveraplaceforthisconstraintinn-aryassociationsinUML,andits

semanticsiscarefullyexplained.

1 DefinitionofMultiplicityinUML

The Unified Modeling Language defines "multiplicity" as the range of allowable

cardinalitiesthatasetmayassume[29,p.3-68],where"cardinality"isthenumberof

elementsinaset[29,p.B-4].Acardinalityisaspecificvalue,whereasmultiplicityis

the range of possible cardinalities a set may hold [22, p. 182]. Multiplicity

specificationsaregivenmostlyforassociationends,buttheyareusedalso forother

purposes like repetitions of messages, etc. An "association" is the semanticrelationship between two ormore classes that involves connections (links) among

their instances [22, p. 152; 29, p. 2-19]. These are the definitions within UML,

although many authors, probably coming from the field of entity-relationship

modeling,willusethetermcardinalitytomeanmultiplicity,andthetermrelationship

tomeanassociation.Inthispaperweuseoneterminologyortheotherdependingon

thecontext.


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A binary association is drawn in UML as a solid path connecting two class

symbols.Themultiplicityofabinaryassociation,placedonanassociationend(the

target end), specifies the numberof target instances thatmaybeassociatedwith a

singlesourceinstanceacrossthegivenassociation,inotherwords,howmanyobjects

ofoneclass(thetargetclass)maybeassociatedwithagivensingleobjectfromtheotherclass(thesourceclass)[22,p.348;29,p.2-22].

Theclassicalexample inFigure1 illustratesbinarymultiplicity.Eachinstanceof

Personmaywork fornoneoroneinstanceofCompany(0..1),whileeachcompany

maybelinkedtooneormorepersons(1..*).Forthosereaderslessfamiliarizedwith

UMLnotation,thesymbol(*)standsfor"many"(unboundednumber),andtheranges

(1..1)and(0..*)maybeabbreviatedrespectivelyas(1)and(*).

Person Companyworksfor

1..* 0..1

Fig.1.Aclassicalexampleofbinaryassociationwiththeexpressionofmultiplicities

(Notethatthisassociationisintendedtomeanonlythepresentsituation:aperson

is working for 0..1 companies, but not "a person has worked or works for 0..1companies". In this paper we are going to avoid all issues of history, since this

concernwoulddepartusfromourmainobjective.)

An n-aryassociation isan association among threeor moreclasses, shownasa

diamondwithapathfromthediamondtoeachparticipantclass.Eachinstanceofthe

associationisann-tupleofvaluesfromtherespectiveclasses (a3-tupleortriplet,in

thecaseofternaryassociations).Multiplicityforn-aryassociationsmaybespecified,but is less obvious than binarymultiplicity. Themultiplicityonanassociationend

representsthepotentialnumberofvaluesattheend,whenthevaluesattheothern-1

endsarefixed[29,p.3-73].Thisdefinitioniscompatiblewithbinarymultiplicity[22,

p.350].

TheexampleinFigure2,takenfromtheUMLReferenceManual[22,p.351]andtheUMLStandard[29,p. 3-74],showstherecordofa teamineachseasonwitha

particular goalkeeper. It isassumedthatthe goalkeepermightbe traded during the

seasonandcanappearwithdifferentteams.Thatis,foragivenplayerandyear,there

maybemanyteams,andsoonfortheothermultiplicitiesstatedinthediagram.


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Team

Year

Player* *

*

Fig.2.Ternaryassociationwithmany-many-manymultiplicities

However,asweshallsee,asubtleparadoxhidesbehindtheapparentclearnessof

thesemultiplicityspecifications.

2 DefinitionofCardinalityinEntity-RelationshipModels

Thedefinition ofmultiplicityof anassociation inUMLfollows that ofOMT[24],

whichisgenerallyacknowledged[2,20]toderivefromthedefinitionofcardinalityof

arelationshipintheentity-relationshipmodel[4].Infact,Chendoesnotusetheterm

"cardinality"in hisproposal:heusestheexpressions"1:1mapping","1:nmapping"and"m:nmapping",andheexplainsthemeaningofeachone,buthedoesnotgive

any formaldefinitionoftheconceptof"type ofmapping".Moreover, hedrawsanexampleofanM:N:PternaryrelationshipSUPPLIER-PROJECT-PART,buthedoes

notexplainat allhow these "cardinalities"are tobeunderstood.Note,too, that he

deals only with "maximum cardinality", in a close relation to the concept of

"functionaldependency".

Manydatamodelingtechniqueshavefollowed,formalizedandextendedtheChen

styleofcardinality values[8,10,18,21,28].Others,followingtheFrenchmethod

Merise [27], invert the placement of cardinality values [1, 3, 5]. It has beenwell

established that the semantics of both conventions are equivalent for binary

relationships,butdiffersubstantiallywhentheyareappliedtorelationshipsofhigherdegree[2,19,20,26].Thisissuewillbedealtwithmoredetailinsection4.

Insomeof thesemethodswe findtheexplicitandusefuldistinctionbetweenthe

conceptsofacardinalityconstraintandaparticipationconstraint[10,18,26]:

Acardinalityconstraintspecifiesthenumberofrelationshipinstancesinwhichanentitycanparticipate.Theyareintheformof1:1,1:N,orM:N,toexpress

thetwoconstraintsinabinaryrelationship,and1:1:1,1:1:N,1:N:M,orM:N:P,to express the three constraints in a ternary relationship. These constraints

correspondtomaximum cardinality constraintsin somenotations. In theChen

style, theentitywitha cardinality constraintof1 isfunctionallydependenton

theotherentity(orentities,inann-aryrelationship).IntheexampleinFigure1,

CompanyisfunctionallydependentonPerson.


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Aparticipationconstraintspecifieswhetheranentityinstancecanexistwithout

participatinginarelationshipwithanotherentity.Thisconstraintcorrespondsto

minimum cardinality constraints in some notations. Total and partial

participationare thetwotypesofparticipation,alsodenotedasmandatoryand

optional participation.Mandatory participation existswhen an entity instancecannotexistwithoutparticipatinginarelationshipwithanotherentityinstance.

Optional participation exists when the entity instance can exist without

participating in a relationship with another entity instance. In the example in

Figure 1, Company has mandatory participation, while Person has optional

participation.

Other authors define in a more general way a co-occurrence constraint, whichspecifies howmany objects (orn-tuplesofobjects)may co-occur in a relationship

with another object (or m-tuple of objects) [11]: for example, how many pairsproduct-prize may co-occur with a particular pair seller-buyer in the 4-ary

relationship sale. Thisgeneralized conceptof cardinality is studiedmoredeeplyin

section4.

Some methods combine cardinality and participation constraints and representthem usingminimum andmaximumconstraints in theformof(min,max) notation.

This is the case of UML. However, the concept of minimummultiplicity is not

equivalent to the concept of participation constraint. For example, a minimum

multiplicityof2impliesmandatoryparticipation,butmandatoryparticipationimplies

only a minimum multiplicity of 1. Moreover, the potential multiplicities in UMLextendtoanysubsetofnonnegativeintegers[22,p.346],notonlyasingleintervalas

(2..*), not even to a comma-separated list of integer intervals as (1..3, 7..10, 15,

19..*): specifications ofmultiplicity like {prime numbers} or {squares of positive

integers}arealsovalid,althoughthereisnostandardnotationforthem.Nevertheless,inUML as in other modeling techniques, the most usual multiplicities are (0..1),

(1..*),(0..*)and(1..*).Wearegoingtorestrictouranalysistothesecombinationsofmultiplicityvalues.

3 ParadoxesandAmbiguitiesofTernaryMultiplicities

Let's recall the definition of UML multiplicity in an n-ary association: "themultiplicity onanassociation end represents the potential number ofvalues at the

end, when the values at the other n-1 ends are fixed". Now consider the ternary

association"AworksinBusingC",whichisaclassicalexampleintheliterature[20],

definedamongemployees,projectsandskills:anemployeeworksinacertainproject

usingacertainskill.Table1illustratespossiblesetsofinstancesforthethreeclasses,

whileTable2illustratesapossiblesetofinstances(triplets)fortheassociation.

Table1.ThreepossiblesetsofinstancesforthethreeclassesEmployee,Project,Skill

Employee Project Skill

Albert Kitchen Welding

Benedict Laboratory Painting

Claire Basement Foreman


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Table2.Possiblesetofinstancesfortheternaryassociation"works-in-using"

worksinusing

Employee Project Skill

Albert Kitchen Welding

Albert Laboratory Welding

Benedict Kitchen Foreman

Benedict Basement Foreman

Claire Kitchen Painting

Figure3showsadiagramforthisternaryassociation,withmultiplicityconstraints

that,accordingto thedefinitiongivenabove,areconsistentwiththevaluesinTables

1and2:

MultiplicityforclassProjectinassociation"works-in-using"is0..*,sinceann-

tupleofinstancesofEmployee-Skillmaybelinkedtoaminimumof0andan

unboundedmaximumofinstancesofProject:tupleAlbert-Weldingis linkedto

two different instancesofProject, Kitchen and Laboratory, and the same for

Benedict-Foreman.

MultiplicityforclassEmployee,also0..*,isconsistentaswell:althoughthereis

nopairProject-SkilllinkedtotwodifferentinstancesofEmployee,thediagram

states that a tuple such asClaire-Laboratory-Welding,which would duplicate

thepairLaboratory-Welding,maybeaddedtotheexistingsetoftuples.

Finally,multiplicity for class Skill, in this case 0..1, states that for each pair

Employee-Projecttheremaybenoneoroneskill:thatis,anemployeeusesat

mostoneskillineachproject,whichisconsistentwiththegivenvalues,butthe

constraintalsoforbidsaddingatuplesuchasClaire-Kitchen-Weldingunlessthe

tuple Claire-Kitchen-Painting is deleted first. In other words, Skill is

functionallydependentonEmployee-Project.

Employee Project

Skill

worksin

using

0..* 0..*

0..1

Fig.3. Theternary association"works-in-using",accordingto theinterpretationofpotential

tuples

Now,everythingseemsworkingwellbutthat'snotthateasy.Sofar,inapplying

thedefinition to thisexamplewe haveconsideredonlymaximummultiplicity.Let's

concentrate now on minimum multiplicity, and we will see that there is some


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ambiguity in its definition. We are going to propose and examine three different

interpretations of the phrase "each pair Employee-Project", inviting the reader to

check which one he or she has accepted until now, probably in an unconscious

manner.Wewillshowthateachoneoftheseinterpretationshasalsoitsownproblems

and unexpected consequences. As far as we know, nobody has made this point

beforehand.

First interpretation (actual tuples). "Each pair Employee-Project" may be

understoodasan"actuallyexistingpair",oranactualpair,thatis,apairofinstances

that are linked by some ternary link within the ternary association. Pairs Albert-

KitchenandBenedict-Basementareactualpairs,sincethereareinfactsometriplets

thatcontainthem,whilepairsAlbert-BasementandClaire-Laboratoryarenot.

Thisinterpretationoftheruleseemsratherintuitive,butnotethatforanactual

pairEmployee-Projecttheremustbe alwaysat leastoneSkill:ifitisanactualpair,

thereisanactualtripletthatcontainsit,thereforethereisaninstanceofSkillthatis

also in the triplet. There cannot be an actual pairthat is not connected to a third

element,becauseaternarylinkisbydefinitiona tripletofvaluesfromtherespective

classes;a ternary linkhas three"legs", and noneofthemmaybeempty:"limping"

linksarenotallowed.

So,inthisinterpretation, theminimummultiplicityisalwaysatleast1 ,sincethe

value0hasnosense.This"zero-forbiddeneffect"isnotconsistentwiththefrequent

assigningofminimum multiplicity 0 in ternary associations (and, first of all, with

UMLdocumentation,as intheexampleinFigure2).Infact,thediagraminFigure3

wouldbeincorrect,althoughitcouldbesubstitutedbytheoneinFigure4.

Employee Project

Skill

worksin

using

1..* 1..*

1..1

Fig.4.Theternaryassociation"works-in-using",accordingtotheinterpretationofactualtuples

Secondinterpretation(potentialtuples). "ForeachpairEmployee-Project"maybe

understoodasa"merelypossiblepair",orapotentialpair,thatis,apairofinstances

that belongs to the Cartesian product of Employee and Project. There are three

employees and three projects, so there are nine potential pairs. For some of these

pairs, like Benedict-Basement or Claire-Kitchen, there is a related skill; for some

others, like Albert-Basement or Claire-Laboratory, there is none. So, minimum

multiplicity0 isallowed inTables1 and 2,and the diagram inFigure 3would be

correctandconsistentwiththem.


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But,whatwouldbethemeaningofminimummultiplicity1?Considermultiplicity

1..1assignedtoclassSkill,asinFigure4.Itwouldmeanthat,foreachpotentialpair

Employee-Product,theremightbeoneSkill,butnotzero;thatis,anypotentialpair

notlinkedtoaskillwouldbeforbidden.Inotherwords,anypotentialpairmustbe

linked toat leastoneskill,and thereforeanypotentialpairEmployee-Productmustexistatleastoncewithinonetripletintheassociation:everyemployeemustbelinked

toevery project atleast once.Thisruleand the diagramin Figure 4would not be

consistentwiththevaluesinTables1and2,sincewewouldneedthefullCartesian

productEmployee-ProducttobepresentinTable2(infact,weshouldhaveexactly

ninelinesandnomore,duetomaximummultiplicity1ofSkill).

So,inthisinterpretation,aminimummultiplicity1assignedtooneclassforcesallpotential pairs of instances of the remaining classes to actually existwithin some

triplet.Thiswouldbea"bouncingeffectoftheone"thatisprobablyunexpectedby

mostmodelers. Nevertheless, this interpretation seems valid, as it is implicitly in

agreement with UML documentation and some works on the formalization of

multiplicities[20].

Third interpretation (limping links). We could try an entirely different kind of

interpretation, bymeans of allowing theexistence of limping links, that is, ternary

linksthatlinkonlytwoinstancesandleaveablankforthethirdone.Thus,wewould

readthemultiplicityforclassSkillinFigure3as"eachactualpairEmployee-Project

maybelinkedtononeoroneinstanceofSkill",thatis,eachlinkwouldbeactuallyeitherapairEmployee-Project(belongingtoahiddenbinaryassociation)ora triplet

Employee-Project-Skill(belongingtothetrueternaryassociation).Onthecontrary,if

multiplicitywere1..1, then limpinglinkswouldnotbe allowedonthesideof class

Skill:everyactualcombinationofemployeeandprojectshouldhaveaskilllinkedtoit.Withinthisinterpretation, thediagraminFigure3wouldbeequivalenttotheone

in Figure 5, in which the limping links have been removed from the ternary

association,andare represented in a superimposedexplicit binaryassociation (this

couldbefurtherdonewiththeothertwozero-multiplicities).


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Employee Project

Skill

worksin

using

0..* 0..*

1..1

0..* 0..*

Fig. 5. The ternary association "works-in-using", according to the interpretation of limpinglinks,inwhichthelimpinglinksonthesideofSkillareshownasanexplicitbinaryassociation

In this interpretation, the ternary association symbol isused asan abbreviated

form to represent a genuine ternary association together with a hidden binary

association. Probablymanymodelersuse the ternary symbol as ifit were tomean

this, but, in general, the procedure ofmerging a binary association with a ternary

associationby usinglimping linksis not advisable.Ontheotherhand,UML states

thateachinstanceofann-aryassociationisann-tupleofvaluesfromtherespective

classes(recallsection1ofthispaper).

Thelimping linksinterpretation for incomplete associationsis avariationof the

actual pairs interpretation, in which minimum multiplicity 0 means that a lack of

informationisallowed.Nevertheless,therearesomedifficultiesleft:Howmanylegs

may be lacking in an n-ary link? One, two, up to n-2? How is the maximummultiplicityconstrainttobeinterpretedwhenalegislackingonanoppositeend?

So,thisinterpretationmayseemsimpleandusefulatfirstglance,buttherearestill

somepointsthatarenotatallclear,firstofalltheverydefinitionofn-aryassociation

inUML.Theseunsolvedsemanticdifficultiesareenough,inouropinion,todiscard

theconceptofa limpinglinkasamisleadingone,inspiteofitsapparentadvantages,

in favor of the more rigorous approach of the potential pairs interpretation. In

consequence, in the rest of this paper we adopt this interpretation for clearness,

althoughtheissuesaddressedareuptoapointindependentofthischoice.

4 APlacefortheParticipationConstraint

We have seen three different interpretations that could solve the ambiguity in the

definition of minimum multiplicity of n-ary associations in UML. The first one,

actual pairs, implies that minimum multiplicity must be always 1, which is not

consistentwith documentation and practice; the second one,potential pairs, seems

correctbuthas a strange effectwhen the valueis 1;thethirdone, limpinglinks,is

semantically weak and contradicts the definition of n-ary association. Why is

minimummultiplicityinn-aryassociationssoelusive?


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McAllister offers a good formalization of the concept of cardinality (or

multiplicity,inUMLterminology)[20].Beingaandbtwonon-null,non-overlapping

setsofroles inann-aryrelationshipR,thecardinalityC(a,b)=(min,max)specifies

thatanygivensetofentityinstancesofamustbeassociatedbyRwithbetweenmin

andmaxuniquesetsofentityinstancesforb (forsimplicity,werestrictourselvestothesimplestformofcardinality,asingleintegerinterval,althoughthisdoesnotaffect

the reasoning).Note that this corresponds to the co-occurrence constraint defined

above.IntheexampleofFigure3,ifa={Employee,Skill}andb={Project},then

C(a,b)= (0..*).McAllisterdemonstrates that thetotalnumberofcardinalityvalues

thatmaybedefinedina relationshipwithNrolesisgivenby3N-2N+1+1,andapplies

thiscalculationforNfrom2to5,givingtheresultsinTable3.

Table3.TotalnumberofcardinalityvaluesthatmaybedefinedinarelationshipwithNroles

N=numberofrolesinR numberofC(a,b)forR

2 2

3 12

4 50

5 180

AsN increases, there isa rapidincrease inthe numberofcardinalityvaluesthat

shouldbeanalyzedifthenatureoftherelationshipistobefullyunderstood.Thismay

beonefactorwhymanydatamodelingpractitionersencounterdifficultiesindealing

with n-ary relationships, especially if only a small number of the applicable

cardinalitiesareconsideredforeachsuchrelationship.

ForN=3,aternaryrelationshipsuchas"works-for-using",thetwelvevaluestobe

consideredare: the three Chen/UML style values, the threeMerise values,andsixvalues for the three embedded (that is, implicit) binary relationships Employee-

Project, Employee-Skill andProject-Skill (remembertheyarenottruly independent

relationships).McAllisterfurtherdefinesasetofrulesforconsistencychecking,since

thesevaluesarenotcompletelyindependent[20].Tofullyunderstandthestructureof

ann-aryrelationship,themodelershouldspecifyallthese co-occurrenceconstraints,

butmostoftimesitisenoughwiththeNChenvaluesandtheNMerisevalues(theothersbeingusuallymany-to-many[15]).Whenthesevaluesarespecifiedassimple

(min,max) intervals,consistencybetweenChenandMerisevaluesisdeterminedby

ensuring thateachminormax Chenvalue isless thanorequal totheminormax

Merisevaluesoftheotherclasses[20].Thisrulemaybecheckedagainsttheexample

inFigure6,whichshowsbothsetsofvalues:minChenforSkillis0,whichisequalto

minMerise for Employee (0)andless thanminMeriseforProject(1);maxChenfor

Skillis1,whichislessthanmaxMeriseforEmployee(*)andmaxMeriseforProject

(*). If, besides Chen and Merise values, the other values are also important,

McAllister'stabularrepresentationforrelationshipcardinalityisagoodchoice;these

cardinality constraints that are difficult to express in traditional entity-relationship

models are also very naturally expressed using assertions [17] ("assertions" aredeclarativespecificationsofwhathastobetrueinthe"stablestate"ofthesystem,that

is,outsideofatomicoperations;theyarealsoreferredtoas"invariants").


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0..* 1..*

0..*

Chen

Merise

Employee Project

Skill

worksinusing

0..* 0..*

0..1

Fig.6.Theternaryassociation"works-in-using",showingbothChenandMerisemultiplicity

values

We can better understand now the semantic problems ofminimum multiplicity

considered above. Minimum multiplicity is associated with the participation

constraint,butinthecaseofaternaryassociation,intheChenstyle,itdoesnotmean

theparticipationoftheclass,buttheparticipationofapairoftheothertwoclasses.A

value0forSkilldoesnotmeanoptionalparticipationforSkillintheassociation,but

optionalparticipation forinstancepairs ofEmployee-Projectin theassociationwith

aninstanceofSkill.Ifthisgoesagainstintuition,allthemorereasontobeclarified.

In fact, the participationofeach individual class remains unexpressed intheChen

style,whiletheMerisestylerepresentsitadequately.Ontheotherside,the functional

dependenciesremainunexpressedintheMerisestyle,whiletheyarerepresentedby

maximummultiplicity1intheChenstyle.ThisisprobablythereasonwhyOMTand

UML have chosen Chen instead ofMerise,although functional dependency is not

inherentlymoreimportantthanparticipation.

BothChenandMerisestylesarecorrectandcandescribethesameassociation,but

theystatedifferentfactsaboutthenatureoftheassociation.Thefactsrepresentedby

each stylearenot specifiedwhenusingtheother,norcantheybederivedfrom the

other(exceptinthecaseofbinaryassociations,wheretheysimplyinterchangetheir

placement).Therefore,if thetwostylesprovideusefulinformationtounderstandthe

association, why not represent both in the same diagram? Figure 6 repeats the

example of Figure 3, but adding a set of Merise values close to the association

diamond.ThisvaluesareconsistentwiththevaluesinTables1and2,andaddnew

and useful semanticsto the association:we note especially thatclassProjectis the

only one that has mandatory participation (minMerise for Project is 1), that is, a

projectcannotexistwithoutbeinglinkedtoapairemployee-skill,althoughtheremay

be many (potential) pairs employee-skill not linked to any project (minChen for

Projectis0);andwenotealsothatclassSkillmayparticipateinmultipleassociation

instances (maxMerise for Class is*),thatis, a certain skillmaybe linked tomany

differentpairsemployee-skill, although for each pair atmostone skillcan beused

(maxChenforClassis1).


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Thisnotationmayseem similar to thatof replacing the ternaryassociation bya

new entity and three binary associations that simulate the ternary association, as

shown inFigure 7. This new entity is usually referred to as intersection entity orassociativeentityorgerund[26].WenotethattheMerisevaluesofmultiplicityare

preservedinthistransformation,andplacedagaincloseto theassociativeentity,butallChenvalueshavebeenreplacedby1..1,sinceeveryinstanceofWorkislinkedto

oneandonlyoneinstanceoftheotherclasses(thisisthesameassayingthatevery

ternary link has "three legs"). In other words, the semantics of functional

dependenciesexpressedby theternaryassociationare lostwhensimulatingitwitha

gerund,but thesemanticsofparticipationarepreserved.Thereareotherdifferences

between binary and ternary associations and, in general, binary representations ofternaryassociationsarenotfunctional-dependencypreserving[14,16,25].

Employee Project

Skill

Work1..1 0..*

1..1

0..*

1..11..*

Fig.7.Theternaryassociation"works-in-using"substitutedbytheassociativeentity"Work".

OnlyMerisemultiplicityvaluesarepreservedinthetransformation

Conclusions

Inthispaperwehaveconsideredsomesemanticproblemsofminimummultiplicityin

n-ary associations,as it iscurrentlyexpressed inUML; nevertheless, our ideas are

generalenoughtobe applicable toothermodelingtechniquesmoreorlessbasedonthe entity-relationship approach. Minimum multiplicity is closely related to the

participationconstraint,althoughinthecaseofn-aryassociationsitdoesnotmeantheparticipationoftheclassintheassociation,buttheparticipationoftuplesoftheother

n-1 classes. Moreover, we discovered that this latter participation is defined with

uncertainty,allowingthree conflictive interpretations: participationof actual tuples,participationofpotentialtuples,andparticipationwithlimpinglinks.

The second interpretation seems more probable, as it is implicitly inagreement

withUMLdocumentation,inspiteofthebouncingeffectofminimummultiplicity1.

The Standard should clarify this question, without resigning itself to a lack of

obviousnessin thedefinition.Besides, ifthissecondinterpretationwerechosen, the


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Standard should also warn, since this result is not at all intuitive, that aminimum

multiplicity1orgreaterassignedtooneclassforcesallpotentialtuplesofinstancesof

the remaining classes to actually exist within some n-tuple; therefore, minimum

multiplicitywouldbe0innearlyeveryn-aryassociation.

The third interpretation,whichisavariationof thefirst one,seemsintuitiveand

hasalsosomepragmaticadvantages,althoughitisincontradictionwiththedefinition

of n-ary association inUML (maybe more with the letter thanwith the spirit). In

addition,ithassomeunsolvedsemanticdifficultiesthathaveleadustodiscardit,at

leastforthetimebeing.

The eventual clarification of this point leaves another problem unresolved: the

participation of each class remains unexpressed in the Chen style of representing

multiplicities (which is also the UML style), while the Merise style shows it

adequately. Both Chen and Merise styles are correct, but they describe different

characteristicsof thesame association,whichcannotbederived from eachother in

then-arycase,althoughtheyarerelatedbyasimpleconsistencyrule.

Being both styles useful to understand the nature ofassociations,weproposea

simple extension to the notation of UML n-ary multiplicities that enables the

representationof bothparticipation and functionaldependency(that is,Meriseand

Chen styles). Since this notation is compatible with the three alternative

interpretationsofChenmultiplicities,itsusedoesnotavoidbyitselftheambiguityof

the definition ofmultiplicity: they are independent problems. If this notationwere

accepted, the Standard should also modify the metamodel accordingly, since it

foresees only onemultiplicityspecification in theAssociationEndmetaclass. If this

werenotthecase,itcouldbeatleastrecognizedthatChenmultiplicitiesarenotthe

onlysensibleco-occurrenceconstraintsthatmaybedefinedinann-aryassociation.

Understandingn-aryassociationsisadifficultproblemin itself.If therulesofthelanguageusedtorepresentthemarenotclear,thistaskmaybecomeinaccessible.If

the interpretation of n-ary associations is uncertain, straight communication among

modelersbecomesimpossible.Ifthesemanticimplicationsofamodelareambiguous,

implementerswillhavetotakedecisionsthatdonotcorrespondtothem,andpossibly

wrong decisions. These reasons are more than enough to expect a more precise

definitionofUMLonthistopics.

Acknowledgements

Theauthorswould like to thankVicente Palacios and JosMiguel Fuentes for his

frequent conversations on the issues discussed in this paper; JorgeMorato for his

stylisticcorrectionsonthefirstdraft;AnaMaraIglesias,ElenaCastroandDolores

Cuadrafortheusefulbibliographicmaterialprovidedforthisresearch,andcriticism

onthefirstdraft;andGuyGenilloudforhismanysuggestionstoimprovethispaper.


14/15

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