Specification of Products and Product Lines

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by  Ariel Gonzalez

Specification of Products and Product Lines
ArielGonzalez CarlosLuna
UniversidadNacionaldeRioCuarto UniversidadORT
RioCuarto,Argentina Montevideo,Uruguay
[email protected] [email protected]
Thestudyofvariabilityinsoftwaredevelopmenthasbecomeincreasinglyimportantinrecentyears.
Acommonmechanismtorepresentthevariabilityinaproductlineisbymeansoffeaturemodels.
However, the relationship between these models and UML design models is not straightforward.
UMLstatechartsareextendedintroducingvariabilityintheirmaincomponents,sothatthebehavior
of product lines can be specified. The contribution of this work is the proposal of a rule-based ap-
proachthatdefinesatransformationstrategyfromextendedstatechartstoconcreteUMLstatecharts.
Thisisaccomplishedviatheuseoffeaturemodels,inordertodescribethecommonandvariantcom-
ponents,insuchawaythat,startingfromdifferentfeatureconfigurationsandapplyingtherule-based
method,concretestatemachinescorrespondingtodifferentproductsofalinecanbeobtained.
1 Introduction
Aproductline(PL),alsocalledsystemfamily,isasetofsoftwaresystemssharingacommon,managed
set of features that satisfy the specific needs of a particular market segment or mission and that are
developedfromacommonsetofcoreassetsinaprescribedway[1,2,3].
Todevelopasystemfamilyasopposedtodevelopingasetofisolatedsystemshasimportantadvan-
tages. One of these is the possibility of building a kernel that includes common features, from which
desired products can be easily built as extensions. By adding distinguishing characteristics (variability)
to such a kernel, different products can be obtained [3, 4, 5, 6, 7, 8]. For example, today we observe
in the market a significant number of different types of mobile phones (MPs) that share a core of basic
featuresanddifferinothermorespecificcharacteristics: availabilityofadigitalcamera,internetaccess,
mp3player,amongothers.
The UML language [9] provides a graphical notation and has become the standard for modeling
different aspects of software systems. Statecharts and interaction diagrams are part of the set of tools
that UML provide so that the system behavior can be specified, which are specially suitable for the
software design phase. Statecharts are used to specify the behavior of the instances of a class (intra-
componentbehaviour),andthereforeconstituteanappropriatemechanismfordescribingthebehaviorof
certainproblemsbymeansofagraphicalrepresentation. InthelatestversionofUML2.0,thestatecharts
donotofferoperatorsand/orsublanguagesforspecifyingsystemfamilies.
InthispaperweproposeanextensionofUMLstatechartsformodelingPLs. Weusedfeaturemodels
so that both common and variant functionality of a system family can be described [10, 8], and we
incorporate variability in the essential components of the statecharts, in such a way that, starting from
different configurations of a feature model, concrete statecharts corresponding to different products of
a PL can be generated applying a rule-based method. The approach defines the transformation strategy
fromextendedstatechartstoconcreteUMLstatecharts.
Therestoftheworkisorganizedasfollows. Insection2and3webrieflyintroducestatechartsand
featuremodels,respectively. Section4presentsanextensionofstatechartswithvariantelements,which
M.Fernandez(Ed.):9thInt.WorkshoponReductionStrategies (cid:13)c A.Gonzalez&C.Luna
inRewritingandProgramming(WRS’09) Thisworkislicensedunderthe
EPTCS15,2010,pp.44–55,doi:10.4204/EPTCS.15.4 CreativeCommonsAttributionLicense.
A.Gonzalez&C.Luna 45
together with the use of feature models allow specifying PLs. In section 5 we detail the mechanisms
for obtaining products of a PL from distinct configurations of a feature model, via the use of rules and
a transformation strategy. Related work is discussed in section 6 and finally, we conclude and discuss
possible furtherwork in section 7. We exemplifythe proposed work bydeveloping part ofa case study
basedonmobilephonetechnology. Apreliminaryversionofthisworkis[11]. Asopposedto[11],this
paperpresentsarule-basedapproach. Wedefineanapplicationstrategyfortherulesinapropermanner,
in such a way that inconsistencies are avoided in the statechart obtained. The rules are organized in a
sequenceofrulesets,inwhicheachrulesetcanbeconsideredasalayer. Withinaruleset,therulesmay
beappliedinanon-deterministicorder[12]. Wealsoformalizedandaddedrulesthatin[11]areomitted
oronlydescribedinformally.
2 Statecharts
UMLStateCharts(SCs)constituteawell-knownspecificationlanguageformodelingthedynamicsystem
behavior. SCswereintroducedbyD.Harel[13]andlaterincorporatedindifferentversionsoftheUML
withsomevariations. Inthissection,wepresentdefinitionsofSCsbasedon[14]. Foradditionaldetails,
thereaderisreferredto[11,14].
SCsconsistessentiallyofstatesandtransitionsbetweenstates. ThemainfeatureofSCsisthatstates
canberefined,defininginthiswayastatehierarchy. Astatedecompositioncanbesequentialorparallel.
In the first case, a state is decomposed into an automata (Or-state). The second case is supported by a
complexstatechartcomposedofseveralactivesub-statecharts(And-state),runningsimultaneously.
LetS,TR,ΠandA(Π⊆A)becountablesetsofstatenames,transitionnames,eventsandactionsofa
SC,respectively. Also,letusdefines∈Saseitherabasictermoftheforms=[n](Simple-state),asaterm
Or of the form s=[n,(s ,...,s ),l,T] (Or-state), or as a term And of the form s=[n,(s ,...,s )] (And-
1 k 1 k
state),wherename(s)= nisthenameofthestates. Here(s ,...,s )arethesubterms(substates)ofs,
def 1 k
alsodenotedbysub est(s)= (s ,...,s ). Likewise,inicial(s)= s istheinitialstateofs,T ⊆TRis
def 1 k def 1
thesetofinternaltransitionsofs,andltheactivestateindexofs. Atransitionisrepresentedasatuplet=
(t(cid:48),s ,e,c,α,s ,ht),wherename(t)= t(cid:48)isthetransitionname,source(t)= s andtarget(t)= s
o d def def o def d
are called source and target oft, respectively, ev(t)= e the trigger event, cond(t)= c the trigger
def def
condition,andacc(t)= α isthesequenceofactionsthatarecarriedoutwhenatransitionistriggered.
def
Inaddition,hist(t)= ht isthehistorytypeofthetargetstateoft [14]. Thegraphicalnotationusedin
def
thetransitionsist :e,c/α.
3 Feature Models
Feature Models (FMs) are used to describe properties or functionalities of a domain. A functionality
is a distinctive characteristic of a product or object, and depending of the context it may refer to, it is
a requirement or component inside an architecture, and even code pieces, among others. FMs allow
us to describe both commonalities and differences of all products of a PL and to establish relationships
between common and variant features of the line. There are multiple notations for describing FMs. In
thisworkwewillusetheproposalofCzarnecki[10].
AtreestructureinstanceisaFMconfiguration(FMConf)thatdescribesthemodelandthatrespects
the semantics of their relations. That is, a FM allows one to identify common and variant features
betweenproductsofaPL,whileaFMconfigurationcharacterizesthefunctionalitiesofaspecificproduct.
Formally,theconceptsofFMsaredefinedasfollows:
46 SpecificationofProductsandProductLines
Definition1. AFM isdefinedasatreestructurerepresentedbyatuple(Funcs,f ,Mand,Opt,Alt,
0
Or-rel), where Funcs is a set of functionalities of a domain (nodes of the tree), f ∈Funcs is the root
0
functionality of the tree and, Mand,Opt,Alt,Or-rel ⊆ Funcs×(γ(Funcs)−{0/}) the mandatory, op-
tional,alternativeanddisjunctrelationsofthemodel,respectively. If(f,sf)∈Mand∪Opt,#sf=1.
Definition 2. A FM configuration corresponding to a FM (Funcs, f ,Mand,Opt,Alt,Or-rel) is a
0
tree(F,R)whereF isthesetofnodesandRthesetofedges;F⊆FuncsandR⊆{(f,sf)∈F×(γ(F)−{0/})
|∃sf’∈γ(Funcs): sf⊆sf’∧(f,sf’)∈Mand∪Opt∪Alt∪Or-rel}. Moreover,thefollowingconditions
mustbefulfilledby(F,R): (1) f ∈F; (2)forevery(f,sf)∈Mand: if f ∈F then(f,sf)∈R; (3)if(f,sf)
0
∈Alt ∧ f ∈F then ∃! sf’∈γ(F): sf’⊆sf ∧ (f,sf’) ∈R ∧ #sf’=1; (4) if (f, sf) ∈Or-rel ∧ f ∈F then ∃!
sf’∈γ(F): sf’⊆sf∧(f,sf’)∈R∧#sf’≥1.
Definition3. ThekernelN ofaFM(Funcs, f ,Mand,Opt,Alt,Or-rel)isthesetoffunctionalities,
0
whicharepresentinallconfigurations,inductivelycharacterizedbythefollowingrules: (1) f ∈N; (2)
0
if f ∈N ∧(f1,{f2})∈Mand then f ∈N.
1 2
4 SCs with Variabilities
InthissectionweextendtheSCswithoptional(variant)elementsandlateronweestablishthebinding
betweentheseelementswithfunctionalitiesofaFM,inordertomodelthebehaviorofaPL.Wewillcall
ourproposedextendedmachinesStateCharts∗ (SCs∗).
4.1 GraphicalRepresentationofaSC∗
TherepresentationoftheoptionalelementsthatextendthesystemkernelinaSC∗aredepictedinfigure1.
Weusedashedlinestographicallydenotebothoptionalstatesaswellastransitions.
Figure1: OptionalstateandOptionaltransition.
4.2 AbstractSyntaxofaSC∗
Let S∗, TR∗,Π∗ and A∗ be set of states, transitions, events and actions of a SC∗, respectively. Now
the terms that define a state have an additional component S ∈{optional,no optional} that we will
op
call StateType(s), which indicates whether the state s is optional or not. Similarly, we add component
t ∈{optional,no optional} to the transitions, and we denote it by TransType(t). We also define the
op
followingsetsofSCs∗ optionalelements: SOp⊆S∗,TOp⊆TR∗ andVarElem=SOp∪TOp.
We will refer to states directly by their names, when these are unique for every state in all the SC∗;
otherwise, we will use the dot (.) as separator between state and substate names. A transition name is
builtbythetriggereventnamefollowedbysourceandtargetstatenames,respectively.
4.3 Casestudy: MPs
WeconsideredhereafamilyofMPswhichsharesomefunctionalities,suchas,forexample,thecapacity
of reproducing monophonic sounds and vibration. Optionally, we could incorporate into the kernel of
A.Gonzalez&C.Luna 47
functionalitiesthecapacitytomakecallsbymeansofquick-marked,towritetextmessages,toadminis-
termultimediacontents,andcombinationsofthese,suchasmessageswithmultimediacontent(images,
polyphonic sounds and videos). In order to exhibit an example in the development of this article, we
formulateinfigure3aFM,usingthenotationproposedbyCzarnecki,thatrelatestheinvolvedfunction-
alitiesinthepartiallydescribedMPoffigure2.
4.4 RelationbetweenFMsandSCs∗
FMs and SCs∗ are complementary. Both model different aspects of a system and in our proposal, will
not be treated independently, since SC∗ elements model behaviors of present functionalities in the FM.
Ingeneral,afunctionalityisdescribedbymorethanoneSC∗ element. Duetothis,wedefineafunction
Imp, which represents the association between the SC∗ variant elements and the functionalities of the
FM.ThiswayweestablishwhatvariantelementsoftheSC∗ implementthecharacteristicsofthesystem
described in the FM. Given a FM (Funcs, f , Mand, Opt, Alt, Or-rel) and a SC∗ (S∗, TR∗, Π∗ and
0
A∗), the type of function Imp is as follows: Imp : Funcs → Set(VarElem), where VarElem is the set
SOp∪TOp,SOp⊆S∗ andTOp⊆TR∗.
Takingintoaccountthatthemandatoryfunctionalitiesarealwayspresentinallproductsoftheline,
itisnotnecessarytodefinetheSC∗ syntacticelementsthattheseimplement. However,itisnecessaryto
doitforthosefunctionalitiesthatcannotbelongtoFMconfiguration. Impwillbethenapartialfunction
definedonFMelementswhichdonotbelongtothekernel. Therefore,thebehaviorofaPLisdefinedby
aFM,aSC∗,andafunctionofimplementationthatbindsthem.
Example1. TakingtheFMoffigure3andtheSC∗ ofcasestudyoffigure2,wedefineSC∗ elements
thatimplementsfunctionalitiesofMPasfollows:
Imp(PolyphonicSounds)={SelectPolSound, TRightMultimediaType-SelectPolSound, ToChoosePol
Sound, TRightSoundType, ToChoosePolSound, TLeftToChoosePolSound-SoundType, TRightToChoo
sePolSound-PhoneFuncionality, ...}; Imp(Multimedia)={Multimedia, AdmMultimedia, TRight
Multimedia.Selecct-SelectContact}∪Imp(Images)∪Imp(PolyphonicSounds)∪Imp(Videos);
Imp(MessagesAdm)={MessagesCenter, TMessage-MainDisplay-IncomingMess, TMessage-MainDis
play-MessagesCenter, TLeft-MessagesCenter-MainDisplay, TRight-OptionsMenu-MessagesCen
ter}∪Imp(AlarmNewMessages); Imp(AlarmNewMessages)={MessagesState, TMessageMainDis
play-IncomingMess}.
5 Instantiation of StateCharts with Variabilities
A FM configuration defines a product given a set of selected characteristics. Given a FM configuration
andtheSC∗correspondingtotheFM(linkedviaafunctionImp),wedefineaninstantiationfunctionthat
returnsaSC,whichspecifiesthedefinedproductbehavior,Inst :SC∗×FMConf →SC.
We eliminate of the SC∗ both states and transitions which implement functionalities not present in
FMconfiguration,viatheuseofthefunctionImpdefinedintheprevioussection. Thedirectelimination
ofstatesaswellastransitionsoftheSC∗ isnottrivial. ThesuppressionofSC∗ componentswithoutes-
tablishingacontrolcanreturninconsistentresults,suchas,forexample,unreachablestatesortransitions
withouttarget. AcontrolandrebuildmechanismofSCsstartingfromaSC∗isdefinedinsuchawaythat
aconcreteproductisobtained.
Insection5.1wepresentthecasesandrulesofrebuildingthatconstitutethebaseoftheinstantiation
methodwhichweincludedinsection5.2. Later,insection5.3weanalyzeourcasestudy: MPs.
48 SpecificationofProductsandProductLines
Figure2: SC∗ ofMPs.
Figure3: FMofSC∗ offigure2.
A.Gonzalez&C.Luna 49
5.1 CasesandRebuildingRules
Case1. Whenastateisdeleted
Case1.1. Whenasimplestateisdeleted
Ifasimplestates=[E]isdeleted,thentheirentryandexitoptionaltransitionsaredeleted,whilethe
mandatorytransitionsarecomposedusingthefollowingrebuildingmethod.
LetE ∈SOpbethestatetoeliminate,A ,...,A predecessorstatesofE(i.e.,statesfromwhichthere
1 n
arenon-optionaltransitionswithtargetE:t ,...,t ),andS ,...,S successorstatesofE(i.e.,target
AE 1 AE n 1 m
statesofnon-optionaltransitionswithsourceE:t ,...,t ). WhenthevariantstateE isdeleted,all
ES 1 ES m
entry and exit transitions linked to E are deleted. Simultaneously, new transitions are generated by the
compositionofthenon-optionalentrytransitions(t ,...,t )withthenon-optionalexittransitions
AE 1 AE n
(t ,...,t ).
ES 1 ES m
Figure4: ResultingSC∗ afterthedeletionoftheoptionalstate.
Thecompositionoftwotransitionst =(t ,so ,e ,c ,α ,sd ,ht ,no optional)andt =(t ,so ,
1 1 1 1 1 1 1 1 2 2 2
e ,c ,α ,sd ,ht ,no optional)defineanewtransitionasfollows: comp(t1,t2)=(t ,so ,e ::e ,
2 2 2 2 2 12 1 1 2
c ∧c ,α ::a ,sd ,ht ,no optional),where::isthesequentialcompositionofeventsandactions,and
1 2 1 2 2 2
∧ the conjunction of conditions. Both operations must be associative in order to make the instantiation
methoddeterministic.
Letsc=(S∗,TR∗,Π∗,A∗)beSC∗ (infuture, wewillomitthecomponentsΠ∗ yA∗), wedefinethe
setofentryandexitnon-optionaltransitionspairsofastateEofscasfollows:
T (E)={(t ,t )∈TR∗×TR∗|target(t )=E∧source(t )=E∧TransType(t )=TransType(t )=
e s e s e s e s
no optional}.
TheresultofeliminatinganoptionalSimple-stateEofsccorrespondstothefollowingSC∗:
Delete simple state(E,(S∗,TR∗))=(S∗−{E}, TR∗∪{comp(t ,t )| (t ,t )∈T (E)}−{t∈TR∗|t∈
e s e s e s
Domain(T (E)) ∨ t ∈ Range(Te s(E))}−{t ∈ TR∗|TransType(t) = optional∧(source(t) = E∨
e s
target(t)=E)})WecallthisruleDelete simple state(E,(S∗,TR∗)). Figure4showstheresultoftheir
application.
Case1.2. WhenaOr-stateisdeleted
If an Or-state s=[E, (s , ..., s ), l, T] is deleted, then their entry and exit optional transitions are
1 k
deleted,whilethemandatorytransitionsarecomposedusingthefollowingrebuildingmethod.
The proposal consists of applying the previous transition composition method of case 1.1 on E,
considering certain conditions and affectations to SC∗. Let sc = (S∗, TR∗) be a SC∗, we previously
definethesetofalltheentryandexitnon-optionaltransitionpairsofanOr-stateEofsc∗ asfollows:
T (E)={(t ,t )∈TR∗×TR∗|target(t )∈sub states(E)∧source(t )∈sub states(E)∧TransType(t )
e s e s e s e
= TransType(t ) = no optional}
s
WeestablishthateachentrytransitiontoEiscomposedwithoneexittransitionifthesourcestateof
the exit transition is reached from the target state of the entry transition. We define Reachable(E, A) as
50 SpecificationofProductsandProductLines
the set of reachable substates of E from the substate A. Formally, we define TComp (E)={(t ,t )∈
e s e s
Te s(E)|source(t )∈Reachable(E,target(t ))} as the set of transition pairs that must be composed by
s e
means of case 1.1, previous modification of these transitions as is indicated as follows. For each entry
transitiont ∈Domain(TComp (E)) its target state is now E, i.e., target(t )=E. Also, for each exit
e e s e
transition t ∈Range(TComp (E)), source(t )=E. The result of eliminating the optional Or-state E
s e s s
ofsc∗ correspondstotheSC∗ following:
Delete Or state(E,(S∗,TR∗))=(S∗−({E}∪sub states(E)), TR∗ ∪ {comp(change target(t ,E),
e
change source(t ,E))|(t ,t )∈TComp (E)}−{t∈TR∗|source(t)∈({E}∪sub states(E))∨target(t)
s e s e s
∈({E}∪sub states(E))})
change target(t,E)changethetargetoftransitiont,suchthattarget(t)=E.Likewise,change source(t,
E) change the source of transition t, such that source(t)=E. We call this rule Delete Or state(E, (S∗,
TR∗)).
Case1.3. WhenanAnd-stateisdeleted
IfanoptionalAnd-stateE isdeleted,thentheirentryandexitoptionaltransitionsaredeleted,while
themandatorytransitionsarecomposedusingasimilarrebuildingmethodtocase1.2.
Twopossiblerelationsofdependencyorsynchronizationbetweenparallelstatesexist. Oneofthem
referstotheoccurrenceofaneventthatproducesthetriggeroftwoormoretransitionsbelongingtoeach
oneoftheparallelsubstates. Thesecondrelationcorrespondstousingconditionsoftype‘inE’(seecase
2.2). The latter forces to redefine the concept of reachability, since it is not valid to apply the previous
definitionofreachabilityinawayindependentineachoneoftheorthogonalstates.
Let E be an And-state with n orthogonal states. We define now Reachable(E,(E ,E ,...,E )) as
1 2 n
the set of n-tuples of reachable states from (E ,E ,...,E ). In this way, maintaining the definition of
1 2 n
T (E)ofthepreviouscaseandredefiningTComp (E),itispossibletosolvethemethodoftransition
e s e s
eliminationandcomposition(Delete And state)inananalogousformtocase1.2.
TComp (E)={(t ,t )∈T (E)|(∃n-tupla init,n-tuple end∈(S∗×S∗×...×S∗) | n-tupla end∈
e s e s e s n
Reachable(E,n-tupla init)∧(∃i,j1≤i,j≤n|n-tupla end[i]=source(t )∧n-tupla init[j]=target(t )∧
s e
cond(t )))},beingntheamountoforthogonalstates. WecalltothisruleDelete And state(E,(S∗,TR∗)).
s
Case2. Consequencesoftheeliminationofastate
Somesituationscanappearasaconsequenceofapplyingthecasesdescribedpreviously,whichmust
beconsideredinordertoreestablishtheSC.Thesesituationsareanalyzedinthefollowingcases.
Case2.1. Whenaninitialstateisdeleted
If an initial state of a state E =[s, (s , ..., s ), l, T] is eliminated, then anyone of their successors
1 k
that belongs to E becomes the new initial substate, i.e. if ∃t,t = (t , s , e, c, α, s , ht) ∈
name 1 new initial
T|s ∈(s ,...,s )theninitial(cid:48)(E)=s . WecallthisruleDelete Initial state(E,(S∗,TR∗)).
new initial 2 k new initial
Case2.2. Whensomesubstateinaparalleldecompositionisdeleted
The conditions of transitions in a parallel decomposition of type ‘in E’ are deleted when the state
E is eliminated via some FM configuration, i.e. if E is deleted and t = (t , s , e, c, α, s , ht) ∈
name o d
TR|‘inE’∈c,thent(cid:48)=(t ,s ,e,delete expresion in condition(‘inE’,c),α,s ,ht),wherethefunc-
name o d
tion delete expresion in condition(expr, c) eliminates the logic subexpression of c. We call this rule
Delete condition(E,(S∗,TR∗)).
Case3. Whenatransitionisdeleted
If a transition t of a SC∗ disappears, it does not produce alterations in the SC∗.We call this rule
Delete transition(t,(S∗,TR∗)).
A.Gonzalez&C.Luna 51
Case4. Changingtheoptionalelementstonon optionalelements
Thisrulewillbeusedinthereductionstrategy,inordertomakepartofthefinalproductalloptional
elements remainders. That is to say, ∀s∈SOp∗,StateType(s(cid:48))=non optional. In a same way, ∀t ∈
TR∗,TransType(t(cid:48))=non optional. WecallthisruleChanging optional to non optional(S∗,TR∗).
5.2 InstantiationMethod
Given a FM and its configuration, we will call NSF the set of non-selected functionalities of the model
as consequence of the configuration. Formally, for the FM fm=(Funcs, f ,Mand,Opt,Alt,Or-rel)
0
and a configuration conf =(F, R) of fm, NSF = Funcs − F. We define also the set of non-selected
fm
components(NSC)bytheconfigurationconf ofSC∗ sc, whichwillnotbe partoftheresultingSC,as
fm
follows: NSC(conf ,sc)={x∈VarElem|∃f ∈NSF :x∈Imp(f)∧¬∃f(cid:48)∈F : f(cid:48)(cid:54)= f ∧x∈Imp(f(cid:48))},
fm i i i i i i
with VarElem e Imp defined for sc according to section 4.4. This is, the states and transitions that do
not implement selected functionalities by a configuration will be excluded, through the rules, from the
behaviorofresultingSC.
5.2.1 ApplicationStrategy
The rules should be executed in a certain order for the result to be deterministic. We will assume that
rules are organized in rule sets, which are then organized in a sequence of rule sets in which each rule
setcanbeconsideredasalayer[12]. Withinaruleset,rulesmaybeappliedinanon-deterministicorder.
Syntacticallyweexpresslayersofrulesetsasshowninthefollowingexample: giventhreerulesp1,p2,
andp3,({p1,p2},p3↓)specifiestwolayers. Thefirstonecontainsp1,p2,andthesecondcontainsp3.
Thismeansthatfirstanyruleinthefirstlayerisapplied,andthentheoneinthesecondlayerisapplied.
Thesymbol p↓denotesthattherule p↓isiterateduntilitcannotbeappliedanymore.
The order of application of the rules will determine the order of selection of the elements in NSC
(conf , sc). Let sc ∈ SC∗, fmc ∈ FMConf, and E,t ∈ NSC(fmc, sc). We establish the following
fm
applicationorderofrules:
S =({Delete Initial state(E,sc),Delete condition(E,sc)}↓;
R
{Delete simple state(E,sc),Delete Or state(E,sc),Delete And state(E,sc)}↓;
Delete transition(t,sc)↓;Changing optional to non optional(sc))
The strategy should be applied only once. It is important to consider that the application of a rule
producestheeliminationofelementsinNSC(fmc,sc). ThestrategyiscompletedwhenNSC(fmc,sc)=
0/. Note that the implementation of the first two rules does not eliminate states, but only change initial
states and deleting conditions like ‘in E’. Later the affected states will be eliminated by the successor
rules. The rule Changing optional to non optional(sc) convert optional elements, which remain in sc,
tonon optionalelements.
5.2.2 TerminationandDeterministicImplementation
Terminationandconfluenceproblemscanoccurwheneverarule-basedapproachisused. Astermination
andconfluencearefundamentalforthecorrectnessofamodeltransformation,systematicallyvalidating
these properties is a main prerequisite for successful practical applications, such as transformations in
theMDAcontext,forexample.
In our context, the property of termination is clear, given that rules are applied on elements of
NSC(fmc, sc), and these are eliminated in each application until NSC(fmc, sc) = 0/ . Therefore the
52 SpecificationofProductsandProductLines
strategy is terminating because the rules decrease components of the statecharts (either states or transi-
tions).
Ingeneral,confluencemeansthat,whenthereisachoiceofapplyingdifferentrules,thechoicedoes
not affect the result. The role of the first two rules is to avoid ill-formed statecharts. The main rule is
Delete simple state(E, sc), since Delete Or state(E, sc) and Delete And state(E, sc) are based on this.
NotethatDelete simple state(E,sc)doesnotproduceill-formedstatecharts(seecase1.1).
The application of a rule r on a SC∗ is denoted by sc∗ →r sc∗ , and sc∗ (r→1,r2) sc∗ denotes the
i i+1 i i+2
applicationofr yr inthatorder.
1 2
Lemma1. (Applyingtworules.) Letr ,r ∈{Delete simple state(E,sc), Delete Or state(E,sc), Delete
1 2
And state(E,sc)}betworulestobeappliedtostatesE1andE2∈NSC(fmc,sc),respectively.
sc∗(r→1,r2)sc∗ ∧ sc∗(r→2,r1)sc∗ ⇒ sc∗=sc∗.
i j i t j t
Proof. LetE andE ∈NSC(fmc,sc)bethestatestoeliminate. Weconsiderbothindependentifthere
1 2
is no transition which connects, in this case it is clear that the elimination in any order produces the
sameSC.Theoppositecaseisdiscussed. Lett =(t ,so ,a ,c ,α ,E ,ht ,non optional),t =
E1 E1 1 1 1 1 1 1 E1 E2
(t ,E ,a ,c ,α ,E ,ht ,non optional)andt =(t ,E ,a ,c ,α ,sd ,ht ,non optional)be
E1 E2 1 12 12 12 12 12 E2 E2 2 2 2 2 2 2
entryandexittransitionsofE andE . Ontheonehand,weassumethattheruleDelete simple state(E ,
1 2 1
sc)isappliedbeforeruleDelete simple state(E ,sc),i.e. t iscomposedwitht andaftercompose
2 E1 E1 E2
witht .
E
2
comp(comp(t ,t ),t )=t =(t ,so ,(a ::a )::a ,(c ∧c )∧c ,(α ::
E1 E1 E2 E2 E1 (E1 E2) E2 E1 (E1 E2) E2 1 1 12 2 1 12 2 1
α )::α ,sd ,ht ,non optional).
12 2 2 2
On the other hand, if we assume now the inverse application of both rules, we can see that the
resulting composition has the same result as the previous case. Given that the operators ∧ and :: are
associative: comp(comp(t ,t ),t )=comp(t ,comp(t ,t )).
E E E E E E E E
1 1 2 2 1 1 2 2
TherulesDelete Or stateandDelete And statearebasedontheruleDelete simple state,therefore
wemayconsiderE andE (seeabove)ascompoundstates.
1 2
LetSC∗,RyS bearule-basedrewritingsystemonStatecharts(RS-SC),whereRisthesetofrules
R
definedinsection5.1andS theimplementationstrategyontherulesofR. LetValSeq S bethesetof
R R
allvalidsequencesofapplications.
An RS-SC execution starting from a SC∗ (sc∗) is a sequence s of rule applications of R such that
s∈ValSeq S . Wedenoteitbysc∗→s sc,wheresc∈SCisaconcretestatechart.
R
Theorem1. (Confluence.) TwodistinctexecutionsofRS-SCgeneratethesamestatechart.
sc∗→s1 sc∧sc∗→s2 sc(cid:48) ⇒ sc=sc(cid:48).
Proof. Theproofisgivenbythefollowing:
Both executions, s and s , have the same length, since the number of applications of rules is equal to
1 2
#NSC. Moreover, the set of rules of sequence s is equal to set of sequence s , therefore s is a permu-
1 2 1
tation of s . Since the rules Delete Initial state, Delete condition andChanging optional to non op-
2
tional of S are clearly deterministic and do not remove states or transitions, we focus only on the
R
subsequences of s (sub s ) and s (sub s ) which include rules Delete simple state, Delete Or state
1 1 2 2
andDelete And state.
Sincesub s isapermutationofsub s ,theprooffollowsimmediatelyfromLemma1byswaprules
2 1
inanyofsequences.
A.Gonzalez&C.Luna 53
5.3 InstantiationoftheCasestudy: MPs
TheFMoffigure3canbeconfiguredtocharacterizedifferentMPs,accordingtothespecificationofthe
casestudyofsection4.3. Next,wepresentaconfigurationofaMPoffigure2andweproceedtowards
obtaining the corresponding SC (the MP wanted), according to the application order of rules defined in
section5.2.
AMPwithneitherthesupportforthemanagementofpolyphonicsoundsnorthecapacityofalerting
the user when new messages enter in the incoming mailbox, is defined by the configuration conf =
fm
(F,R)oftheFMoffigure3,where:
F ={MP,Display,Contacts,MessagesAdm,Multimedia,Images,Videos,Quick-Marking,Ringer in func
tions},and
R={(MP,{Multimedia}), (MP,{MessagesAdm}), (MP,{Quick-Marking}), (MP,{Display}),(MP,{Con
tacts}), (MP,{Ringer in functions}), (Multimedia,{Images}),(Multimedia,{Videos})}.
Taking the previous configuration and the function Imp described in example 1 of section 4.4, the
setsNSF andNSC(conf ,sc)aredefinedasfollows:
fm
NSF ={PolyphonicSounds,AlarmNewMessages},and
NSC(conf ,sc)={SelectPolSound, TRightMultimediaType-SelectPolSound, ToChoosePolSou
fm
nd, TRightSoundType-ToChoosePolSound, TLeftToChoosePolSound-SoundType, TRightToChoosePo
lSound-PhoneFuncionality, MessagesState, TMessage-MainDisplay-IncomingMess}.
Inaccordancewiththestrategydefinedinsection5.2,mustapplytherulesinthefollowingorder:
Delete condition(MessagesState,sc)→Delete simple state(SelectPolSound,sc)→Delete simple state(To
ChoosePolSound,sc)→Delete transition(TRightMultimedia-Type-SelectPolSound,sc)→Delete transition(
TRightSoundType-ToChoosePolSound,sc)→Delete transition(TLeftToChoosePolSound-SoundType,sc)→
Delete transition(TRightToChoosePolSound-PhoneFuncionality,sc)→Delete transition(TMessage-MainDis
play-IncomingMess,sc)→Changing optional to non optional(sc).
TheresultingSCisobservedinfigure5.
6 Related Works
Avarietyofexistingapproachesproposetheincorporationofvariabilityinsoftwaresystemsandinpar-
ticularonPLs. OneoftheseistheonedesignedbyJacobson[15],whoseweaknesseshavebeenanalyzed
by severalauthors. Von der Maen[4] proposes theuse of agraphical notation withnew relations. John
andMuthig[16]suggesttheapplicationofusecasetemplates,althoughtheydonotdistinguishbetween
optional, alternative or obligatory variants. However, Halman and Pohl propose in [17] to make use of
UML2packagemerge,basedon[6],asatoolforthevariabilityrepresentationandconfigurationinPLs.
As opposedto the mentioned proposals, oursolution is centeredin one behavior specification model of
aPL,asaretheSCs,introducingaclearlydefinedformalsustenance.
Also, to define PLs and to characterize their different products we use FMs, which admit a formal
definitionandallowustoconfigurethefunctionalcharacteristicsofaline. Analternativeapproachtothis
paperisdevelopedinparallelbyothermembersoftheresearchprojectinwhichthisworkissubsumed.
In[18]theauthors,inaformalframework,definefunctionsthatassociateSCs(notcomponentsofSCs,
as in our case) to functionalities of a FM, and analyze forms of combination between different SCs
which specify possible variants of a PL. Whereas under our method the behavior of a product into a
PL is obtained basically by a selection process, in [18] the focus is oriented towards a process of SC
combinations.