Mller LSV ENS Cachan CNRS INRIA France LudwigMaximiliansUniversitat Munchen Germany Aalborg University Denmark Abstract We consider asynchronously composed IOPetri nets AIOPNs with builtin communication channels They are equipped with a compositio ID: 58668
Download Pdf The PPT/PDF document "Channel Properties of Asynchronously Com..." is the property of its rightful owner. Permission is granted to download and print the materials on this web site for personal, non-commercial use only, and to display it on your personal computer provided you do not modify the materials and that you retain all copyright notices contained in the materials. By downloading content from our website, you accept the terms of this agreement.
restrictionsonthesubnetsareproposedinordertopreserveglobalpropertieslikelivenessordeadlock-freeness.In[21]ageneralcompositionoperatorisproposedanditsassociativityisestablished.AcloselyrelatedconcepttocompositionistheoneofopenPetrinetswhichhasbeenusedindi erentcontextsliketheanalysisofwebservices[25].Numerouscompositionalapproacheshavebeenproposedforthemodellingofcomplexapplicationsbutmostofthemarebasedonhigh-levelPetrinets;see[11]foradetailedsurvey.Channelproperties.Withthedevelopmentofcomponent-basedapplications,oneisinterestedinverifyingbehaviouralpropertiesofthecommunicationand,intheasynchronouscase,inverifyingthepropertiesrelatedtocommunicationchannels.Channelpropertiesnaturallyoccurwhenreasonningaboutdistributedmechanisms,algorithmsandapplications(e.g.managementofsocketsinUNIX,maintainingunicityofatokeninaringbasedalgorithm,recoverypointswithemptychannelsforfaultmanagement,guaranteeofemailreading,etc.).Ourcontributions.Inthisworkweareinterestedingeneralchannelproper-tiesandnotinspecicsystempropertiesrelatedtoparticularapplications.TheFIFOrequirementforchannelspotentiallycandecreasetheperformanceoflargescaledistributedsystems.ThuswerestrictourselvestounorderedchannelswhichcanbenaturallymodelledbyplacesofPetrinets.WeproposeasynchronouslycomposedPetrinets(AIOPNs)by(1)explicitelyrepresentingchannelsforinter-nalcommunicationinsidethenetand(2)deningcommunicationcapabilitiestotheoutsideintermsof(open)inputandoutputlabelswithappropriatetran-sitions.Thenwedeneanasynchronouscompositionoperatorwhichintroducesnewchannelsforthecommunicationbetweenthecomposednets.AIOPNsareequippedwithasemanticsintermsofasynchronouslycomposedI/O-transitionsystems(AIOTS).Weshowthatthissemanticsisfullycompositional,i.e.itcommuteswithasynchronouscomposition.Inourstudytwokindsofchannelpropertiesareconsideredwhicharere-latedtoconsumptionrequirementsandtotheterminationofcommunication.Consumptionpropertiesdealwithrequirementsthatmessagessenttoacom-municationchannelshouldalsobeconsumed.Theycanbeclassiedw.r.t.twocriteria.Therstcriterionisthenatureoftherequirement:consumingmes-sages,decreasingthenumberofmessagesonachannel,andemptyingchannels.Thesecondcriterionexpressesthewaytherequirementisachieved:possiblyimmediately,possiblyaftersomedelay,ornecessarilyineachweaklyfairrun.Communicationterminationdealswith(immediateordelayed)closingofcom-municationchannelswhenthereceiverisnotreadytoconsumeanymore.Weestablishusefulrelationsbetweenthechannelpropertiesandprovethatallchan-nelpropertiesconsideredherearecompositional,i.e.preservedbyasynchronouscomposition,whichisanimportantprerequisiteforincrementaldesign.Fromavericationpointofview,westudythedecidabilityofpropertiesintheframeworkofAIOPN.Thankstoseveralcomplementaryworksondecidabil-ityforPetrinetproblems,weshowthatallchannelpropertiesconsideredinthisworkaredecidable,thoughwithahighcomputationalcomplexity.2 {W�(resp.W+)isamatrixindexedbyPTwithvaluesinN;itiscalledthebackward(resp.forward)incidencematrix,{:T!isatransitionlabellingfunction,and{m0isavectorindexedbyPandcalledtheinitialmarking.Thelabellingfunctionisextendedasusualtosequencesoftransitions.Theinput(outputresp.)vectorW�(t)(W+(t)resp.)ofatransitiontisthecolumnvectorofmatrixW�(W+resp.)withindext.Giventwovectors�!vand�!v0,onewrites�!v�!v0if�!viscomponentwisegreaterorequalthan�!v0.AmarkingisavectorindexedbyP.Atransitiont2Tisrablefromamarkingm,denotedbymt�!,ifmW�(t).Theringoftfrommleadstothemarkingm0,denotedbymt�!m0,anddenedbym0=m�W�(t)+W+(t).If(t)=awewritema�!m0.Theringofatransitionisextendedasusualtoringsequencesm�!m0with2T.Amarkingmisreachableifthereexistsaringsequence2Tsuchthatm0�!m.OurapproachisbasedonastatetransitionsystemsemanticsforPetrinets.Alabelledtransitionsystem(LTS)isatupleS=(;Q;q0;�!),suchthat{isanitesetoflabels,{Qisa(possiblyinnite)setofstates,{q02Qistheinitialstate,and{�!QQisalabelledtransitionrelation.Wewillwriteqa�!q0for(q;a;q0)2�!,andwewriteqa�!ifthereexistsq02Qsuchthatqa�!q0.Letq12Q.AtraceofSstartinginq1isaniteorinnitesequence=q1a1�!q2a2�!q3a3�!.Fora2wewritea2,ifthereexistsaiinthesequencesuchthatai=a,and](a)denotesthe(possiblyinnite)numberofoccurrencesofain.Forq2Qwewriteq2,ifthereexistsqiinthesequencesuchthatqi=q.For=a1a2an2andq;q02Qwewriteq�!q0ifthereexistsa(nite)traceqa1�!q2a2�!an�!q0.Oftenweneedtoreasonaboutthesuccessorstatesreachablefromagivenstateq2Qwithasubsetoflabels.WedenePost(q;)=fq02Qj9a2:qa�!q0gandwewritePost(q)forPost(q;).FurtherwedenePost(q;)=fq02Qj92:q�!q0gandwewritePost(q)forPost(q;).ThesemanticsofalabelledPetrinetN=(P;T;;W�;W+;;m0)isgivenbyitsassociatedlabelledtransitionsystemlts(N)=(;Q;q0;�!)whichrep-resentsthereachabilitygraphofthenetandisdenedby{QNPisthesetofreachablemarkingsofN,{�!=f(m;a;m0)ja2andma�!m0g,and{q0=m0.2.2AsynchronousI/O-PetriNetsandTheirCompositionInthispaperweconsidersystemswhichmaybeopenforcommunicationwithothersystemsandmaybecomposedtolargersystems.Boththebehaviourof4 TwoI/O-alphabetsarecomposableiftherearenonamecon ictsbetweenlabelsandchannelsand,following[1],ifsharedlabelsareeitherinputlabelsofonealphabetandoutputlabelsoftheotherorconversely.Forthecompositioneachsharedlabelagivesrisetoanewcommunicationchannel,alsocalleda,andhencetonewcommunicationlabelsaBforputtingandBaforremovingmessages.Theinputandoutputlabelsofthealphabetcompositionarethenon-sharedinputandoutputlabelsoftheunderlyingalphabets.Denition2(Alphabetcomposition).LetS=inS]outS]comSandT=inT]outT]comTbetwoI/O-alphabetsoverchannelsCSandCTresp.SandTarecomposableif(S[T)\(CS[CT)=;andS\T=(inS\outT)[(inT\outS).ThecompositionofSandTistheI/O-alphabet=in]out]comoverthecomposedsetofchannelsC=CS]CT]CST,withnewchannelsCST=S\T,suchthat{in=(inSnoutT)](inTnoutS),{out=(outSninT)](outTninS),and{com=faB;Baja2Cg4}TwoAIOPNscanbe(asynchronously)composed,iftheirunderlyingI/O-alphabetsarecomposable.Thecompositionisconstructedbytakingthedisjointunionoftheunderlyingnetsandaddinganewchannelplaceforeachsharedlabel.EverytransitionwithsharedoutputlabelabecomesatransitionwiththecommunicationlabelaBthatproducesatokenonthe(new)channelplaceaand,similarly,anytransitionwithsharedinputlabelabecomesatransitionwiththecommunicationlabelBathatconsumesatokenfromthe(new)channelplacea.Forinstance,theAIOPNN3inFig.1cistheresultoftheasynchronouscompositionofthetwoAIOPNsN1andN2inFig.1aandFig.1bresp.Thenewlyintroducedchannelplaceistheplacemsg.OurapproachlooksverysimilartoopenPetrinets,seee.g.[17],whichusein-terfaceplacesforcommunication.Buttherearetwoimportantdi erences:First,weexplicitelydistinguishchannelplacesthusbeingabletoreasononthecom-municationbehaviourbetweencomposedcomponents;seeSect.4.Theseconddi erenceisquiteimportantfromthesoftwareengineer'spointofview.Wedonotuseinterfaceplacestoindicatecommunicationabilitiesofacomponentbutweusedistinguishedinputandoutputlabelsinstead.Webelievethatthishasanimportantadvantagetoachieveseparationofconcerns:Thedesignerofacompo-nenthasnottotakecarewhetherthecomponentwillbeusedinasynchronousorinanasynchronousenvironmentlateron;thisshouldbethedecisionofthesystemarchitect.IndeedopenPetrinetsalreadyrelyonasynchronouscomposi-tionwhileourformalismwouldalsosupportsynchronouscomposition,see[18],andmixedarchitectures.Sincesynchronouscompositionreliesonmatchingoftransitionsratherthancommunicationchannelswehavenotelaboratedthiscase 4=in]out]comisindeedadisjointunion,sinceforalla2CSTthecommunicationlabelsaB;BaarenewnamesduetothegeneralassumptionthatinputandoutputlabelsarenotoftheformxB;Bx.6 val(q)[a++](x)=(val(q)(a)+1ifx=a;val(q)(x)otherwise.Theupdatedmapval(q)[a��]isdenedsimilarly.Insteadofval(q)(a)wewilloftenwriteval(q;a).Denition4(AsynchronousI/O-transitionsystem).AnasynchronousI/O-transitionsystem(AIOTS)isatupleS=(C;;Q;q0;�!;val),suchthat{(;Q;q0;�!)isalabelledtransitionsystem,{Cisanitesetofchannels,{=in]out]comisanI/O-alphabetoverC,{val:Q�!NCisafunction,suchthatforalla2C;q;q02Q:val(q0;a)=0,qaB�!q0=)val(q0)=val(q)[a++],qBa�!q0=)val(q;a)�0andval(q0)=val(q)[a��],andforallx2(in[out);qx�!q0=)val(q0)=val(q).}Therstconditionforvalassumesthatinitiallyallcommunicationchannelsareempty.ThesecondconditionstatesthattransitionswithlabelsaBandBahavethedesirede ectofputtingonemessageonachannel(consumingonemessagefromachannelresp.).Thelastconditionrequiresthattheinputandoutputactionsofanopensystemdonotchangethevaluationofanychannel.SometimesweneedtoreasonaboutthenumberofmessagesonasubsetBCofthechannelsinastateq2Q.Wedeneval(q;B)=Pa2Bval(q;a).ThesemanticsofanasynchronousI/O-PetrinetNisgivenbyitsassociatedasynchronousI/O-transitionsystemaiots(N).Itisbasedonthetransitionsys-temsemanticsofalabelledPetrinet(seeSect.2.1)suchthatmarkingsbecomestates,butadditionallywedenethevaluationofachannelinacurrentstatembythenumberoftokensonthechannelunderthemarkingm.Denition5(AssociatedasynchronousI/O-transitionsystem).LetN=(C;P;T;;W�;W+;;m0)beanAIOPN.TheAIOTSassociatedwithNisgivenbyaiots(N)=(C;;Q;q0;�!;val),suchthat{(;Q;q0;�!)=lts(P;T;;W�;W+;;m0),{foralla2Candm2Q;val(m;a)=m(a):}Example6.ThetransitionsystemsassociatedwiththeAIOPNsN1andN2inFig.1aand1bhavetworeachablestatesandthetransitionsbetweenthemcorresponddirectlytotheirPetrinetrepresentations.Thesituationisdi erentfortheAIOPNN3inFig.1c.IthasinnitlymanyreachablemarkingsandhenceitsassociatedAIOTShasinnitelymanyreachablestates.Fig.2showsanexcerptofit.Thestatesindicatethenumberoftokensineachplaceintheorderp0;p1;msg;p2;p3.Theinitialstateisunderlined.8 4.1:ifqSa�!Sq0Sthen(qS;qT;v)aB�!(q0S;qT;v[a++])and(q0S;qT;v[a++])2Q;4.2:ifqTa�!Tq0Tandv(a)�0then(qS;qT;v)Ba�!(qS;q0T;v[a��])and(qS;q0T;v[a��])2Q.{Forall(qS;qT;v)2Qanda2C,val((qS;qT;v);a)=8�&#x]TJ ;� -1;.93; Td;&#x [00;:valS(qS;a)ifa2CSvalT(qT;a)ifa2CTv(a)ifa2CSTFortherules(1),(3.1)and(4.1),wesaythattheresultingtransitioninthecompositionistriggeredbyS.LetbeatraceofS Tstartingfromastateq=(qS;qT;v)2Q.TheprojectionoftoS,denotedbyjS,isthesequenceoftransitionsofS,startingfromqS,whichhavetriggeredcorrespondingtransitionsin.}Thefollowingtheoremshowsthatthetransitionsystemsemanticsofasyn-chronousI/O-Petrinetsiscompositional.Theproofisgivenin[12].Theorem8.LetNandMbetwocomposableAIOPNs.Thenitholdsthataiots(N pnM)=aiots(N) aiots(M)(uptobijectionbetweenstatespaces).4ChannelPropertiesandTheirCompositionalityInthissectionweconsidervariouspropertiesconcerningtheasynchronouscom-municationviachannels.Wegiveaclassicationoftheproperties,showtheirrelationshipsandprovethattheyarecompositionalw.r.t.asynchronouscompo-sition,aprerequisiteforincrementaldesign.4.1ChannelPropertiesWeconsidertwoclassesofchannelproperties.Therstclassdealswiththerequirementsthatmessagessenttoacommunicationchannelshouldalsobeconsumed;thesecondclassconcernstheterminationofcommunicationinthesensethatifconsumptionfromachannelhasbeenstoppedthenalsoproductiononthischannelwillstop.ThechannelpropertieswillbedenedbyconsideringthesemanticsofAIOPNs,i.e.theywillbeformulatedforAIOTSs.Someoftheproperties,preciselythe\necessarilyproperties"oftype(c)inDef.10below,relyontheconsiderationofsystemruns.Inprincipleasystemrunisamaximalexecutiontrace;itcanbeinnitebutalsoniteifnofurtheractionsareenabled.Itisimportanttoremember,thatwedealwithopensystemswhosepossiblebehavioursarealsodeterminedbytheenvironment.Hence,thedenitionofasystemrunmusttakeintoaccountthepossibilitythatthesystemmaystopinastatewheretheenvironmentdoesnotserveanyo eredinputofthesystemwhileatthesametimethesystemhasnoenabledautonomousaction,i.e.10 a2B,thatifinanarbitraryreachablestateqofSthereisamessageavail-ableona,thenScanconsumethemessagepossiblyaftertheexecutionofsomeautonomousactions.Letuscommentontheroleoftheenvironmentfortheformulationofthisproperty.First,weconsiderarbitraryreachablestatesq2Post(q0)withq0beingtheinitialstateofS.ThismeansthatwetakeintoaccounttheworstenvironmentwhichcanletSgoeverywherebyproviding(non-deterministically)allinputsthatScanaccept.Then,atsomepointatwhichamessageisavailableonchannela,theenvironmentcanstoptoprovidefurtherinputsandwaitswhetherScanautonomouslyreachastateq02Post(q;nin)inwhichitcanconsumefroma,i.e.executeBa.Toallowautonomousactionsbeforeconsumptionisinspiredbythepropertyof\outputcompatibility"stud-iedforsynchronouslycomposedtransitionsystemsin[14].Property(P1.b)doesnotallowautonomousactionsbeforeconsumption.ItrequiresthatScanim-mediatelyconsumethemessageinstateq,similartothepropertyofspeciedreceptionin[4].Property(P1.c)requiresthatthemessagewilldenitelybecon-sumedoneachweaklyfairrunofSstartingfromqand,duetothedenitionofasystemrun,thatthiswillhappeninanyenvironmentwhateverinputsareprovided.AsanexampleconsidertheAIOTSS=aiots(N3)associatedwiththeAIOPNN3inFig.1canditsreachablestate01101suchthatonemessageisonchannelmsg.InthisstateScanautonomouslyperformtheoutputout!reachingstate01110andthenitcanconsumethemessagebyperformingBmsg.SincealsoinallotherreachablestatesinwhichthechannelisnotemptyScanautonomouslyreachastateinwhichitcanconsumefromthechannel,Ssatisesproperty(P1.a)(foritsonlychannelmsg).However,Sisnotstronglyconsuming(P1.b).Forinstanceinstate01101,Scannotimmediatelyconsumethemessage.Ontheotherhand,Sisnecessarilyconsuming(P1.c).Wheneverinareachablestateqthechannelisnotemptyanautonomousaction,eitherBmsgorout!,isenabled.Henceqisnotapureinputstateand,duetotheweakfairnesscondition,eventuallyBmsgorout!mustbeperformedinanyweaklyfairrunstartingfromq.IfBmsgisperformedwearedone.Ifout!isperformedwereachastatewhereBmsgisenabledandwiththesamereasoningeventuallyBmsgwillbeperformed.ThiscanbeeasilydetectedbyconsideringFig.2.Theothergroupsofproperties(P2)-(P4)expresssuccessivelystronger(orequivalent)requirementsonthekindofconsumption.Forinstance,(P3)requiresthattheconsumptionwillleadtoastateinwhichthechannelisempty.Againwedistinguishifthiscanbeachievedaftersomeautonomousactions(P3.a),canbeachievedimmediately(P3.b),ormustbeachievedinanyweaklyfairrun(P3.c).Denition10(Consumptionproperties).LetS=(C;;Q;q0;�!;val)beanAIOTSwithI/O-alphabet=in]out]comandletBCbeasubsetofitschannels.P1:(Consuming)a)SisB-consuming,ifforalla2Bandallq2Post(q0),val(q;a)�0=)9q02Post(q;nin):q0Ba�!:12 Denition12(ChannelpropertiesofAIOPNs).LetPbeanarbitrarychannelpropertyasdenedabove.AnAIOPNNhaspropertyPw.r.t.somesubsetBofitschannelsifitsgeneratedAIOTSaiots(N)haspropertyPw.r.t.B;NhaspropertyPifaiots(N)haspropertyP.Relevanceofchannelproperties.Thegenericpropertiesthatwehavedenedtwellwithpropertiesrelatedtospecicdistributedmechanisms,algorithmsandapplications.Forinstance:{Whensendinganemailtheusermustbecondentthatitsmailwillbeeven-tuallyread.Suchapropertycanbeformalizedasthenecessarilyconsumingproperty.{Mostdistributedapplicationscanbedesignedwithanunderlyingtokencir-culationbetweentheprocessesoftheapplications.Thisrequiresthatatanytimethereisatmostonetokeninallchannelsandthatthistokencanbeim-mediatelyhandled.Suchapropertycanbeformalizedasthestrongwhollyemptyingproperty.{Recoverypointsareusefulforapplicationspronetofaults.Whilealgorithmsforbuildingrecoverypointscanhandlenon-emptychannels,theexistence(andidentication)ofstateswithemptychannelseasesthistask.Suchapropertycanbeformalizedasthenecessarilywhollyemptyingproperty.{InUNIX,oneoftenrequiresthataprocessshouldnotwriteinasocketwhennoreaderofthesocketisstillpresent(andthiscouldraiseasignal).Suchapropertycanbeformalizedasthestrongcommunicationstoppingproperty.4.2RelationshipsBetweenChannelPropertiesTable1showsrelationshipsbetweenthechannelpropertiesandpointerstoex-amplesofAIOPNsfromFig.1andFig.3whichhavetheindicatedproperties.Allthedownwardimplicationsinsidetheboxesaredirectconsequencesofthedenitions.Itistrivialtoseethatdownwardimplication3isanequivalence,sinceimmediateconsumptionleadstoadecreasingvaluation.Downwardimplications9and16areequivalences,sincerepeateddecreasingofmessagesonachannelwilleventuallyleadtoanemptychannel.Theimplications4,5,7,11-14and18areprovedin[12].Additionallywehavethatallpropertiesinboxb)ofTab.1implythestrongestpropertyinboxa),sinceifSisstronglyB-consumingwecanbyrepeatedconsumptionemptyallchannelsinB.Letusnowdiscusssomecounterexamples.AsdiscussedinSect.4.1,acoun-terexamplefortheconverseofimplication7istheAIOPNN3inFig.1c.Anobviouscounterexamplefortheconverseoftheimplications2,10,11,12,13isgivenbytheAIOPNN4showninFig.3a.N4isalsoacounterexampleforimpli-cation6.TheAIOPNN5inFig.3bwithchannelsaandbisacounterexamplefortheconverseofimplication15.Thenetcanemptyeachsinglechannelaandbbutitcanneverhavebothchannelsemptyatthesametime(aftertherstmessagehasbeenproducedonachannel).Acounterexamplefortheconverseofimplication14isshownbythenetN6inFig.3c.Thenetcanputatokenon14 aB a Ba (a)N4 aB Bb a b Ba bB 2 2 (b)N5 aB Ba p0 out! out! p1 a (c)N6Fig.3:ExamplesofAIOPNs.Lemma13.LetS=(CS;S;QS;q0S;�!S;valS);T=(CT;T;QT;q0T;�!T;valT)betwocomposableAIOTSs,andletS T=(C;;Q;q0;�!;val).Forall(qS;qT;v)2Post(q0)and2(SninS)itholdsthatqS�!Sq0S=)9v0:(qS;qT;v)�!(q0S;qT;v0);with2(nin)obtainedfrombyreplacinganyoccurrenceofasharedlabela2outS\inTbythecommunicationlabelaB.Thenextlemmaiscrucialtoprovecompositionalityofthe\necessarily"propertiesoftype(c)inDef.10andthecommunicationstoppingpropertiesinDef.11.Itshowsthatprojectionsofweaklyfairrunsareweaklyfairrunsagain.Thisresultcanonlybeachievedintheasynchronouscontext.Lemma14.LetS,TbetwocomposableAIOTSs,andS T=(C;;Q;q0;�!;val).Letq=(qS;qT;v)2Qand2wfrunS T(q)beaweaklyfairrun.ThenjS2wfrunS(qS),isaweaklyfairrun.Proposition15(CompositionalityofChannelProperties).LetSandTbetwocomposableAIOTSssuchthatCSisthesetofchannelsofS.LetBCSandletPbeanarbitrarychannelpropertyasdenedinSec.4.1.IfShaspropertyPwithrespecttothechannelsB,thenS ThaspropertyPwithrespecttothechannelsB.ThisholdsanalogouslyforasynchronousI/O-Petrinets(duetothecompositionalsemanticsofAIOPNs;seeThm.8).Proposition15leadstothedesiredmodularvericationresultforallproper-tiesexceptwhollyemptying(P4):InordertocheckthatacompositionN pnMoftwoAIOPNshasachannelpropertyP,i.e.Pholdsforallchannelsofthecomposition,itissucienttoknowthatNandMhavepropertyPandtoprovethatN pnMhaspropertyPwithrespecttothenewchannelsintroducedbytheasynchronouscomposition.16 Thenexttheoremestablishesthedecidabilityofthestrongpropertiesoftype(b)ofDef.10.Observethattheirproofsgivenin[12]arecloselyrelatedandthattheyrelyonthedecidabilityofreachabilityandcoverabilityproblems.Theorem17.ThefollowingproblemsaredecidableforAIOPNs:IsanAIOPNNstronglyB-consuming,stronglyB-decreasing,stronglyB-emptying,stronglyB-whollyemptying?Thenexttheoremestablishesthedecidabilityofthepropertiesoftype(a)ofDef.10.Observethattheirproofsrelyon(1)thee ectivenessofbackwardanalysisforupwardclosedmarkingsets(2)thedecidabilityofreachabilityandhomespaceproblemsand(3)appropriatetransformationsofthenet.Theorem18.ThefollowingproblemsaredecidableforAIOPNs:IsanAIOPNNB-consuming,B-decreasing,B-emptying,B-whollyemptying?Proof.B-consuming.GivenanAIOPNNandBasubsetofitschannels,onedecideswhetherNisB-consumingasfollows.Leta2BandEabetheupwardclosedsetofmarkingsdenedby:Ea=fmj9t2Twith(t)=BaandmW�(t)gEaisthesetofmarkingsfromwhichonecanimmediatelyconsumesomemes-sagea.LetFabetheupwardclosedsetofmarkingsdenedby:Fa=fmj9m02Ea92T:()2(nin)^m�!m0gFaisthesetofmarkingsfromwhichonecanlater(withoutthehelpoftheen-vironment)consumesomemessagea.OnecomputesFabybackwardanalysis.LetGbedenedby:G=fmj9a2B:m(a)�0^m=2FagGisasemi-linearsetcorrespondingtothemarkingsfromwhichsomemessagea2Bwillneverbeconsumed.ThenNisnotB-consumingi Gisreachable.B-emptying(andB-decreasing).GivenanAIOPNNandBasubsetofitschannels,onedecideswhetherNisB-emptyingasfollows.FirstonebuildsanetN0:{P0=P]frung{T0=T]fstopg{8p2P8t2TW0�(p;t)=W�(p;t);W0+(p;t)=W+(p;t),m00(p)=m0(p){W0�(run;stop)=1;,W0+(run;stop)=0,m00(run)=1{8p2PW0�(p;stop)=W0+(p;stop)=0{8t2Tsuchthat(t)2inW0�(run;t)=W0+(run;t)=1{8t2Tsuchthat(t)=2inW0�(run;t)=W0+(run;t)=0N0behavesasNaslongasstopisnotred.Whenstopisredonlytransitionsnotlabelledbyinputsarereable.ThusNisB-emptyingi foralla2BthesetofmarkingsZa=fmjm(a)=0gisahomespaceforN0.B-whollyemptying.UsingthesameconstructionNisB-weaklywhollyemp-tyingifZB=fmjm(B)=0gisahomespaceforN0.ut18 6.H.Carstensen.DecidabilityquestionsforfairnessinPetrinets.4thAnnualSympo-siumonTheoreticalAspectsofComputerScience(STACS),LNCS247,396{407,1987.7.G.CeceandA.Finkel.Vericationofprogramswithhalf-duplexcommunication.InformationandComputation,202(2),166{190,2005.8.D.Elhog-Benzina,S.HaddadandR.Hennicker.RenementandasynchronouscompositionofmodalPetrinets.InTransactionsonPetriNetsandOtherModelsofConcurrency,V,LNCS6900,96{120,2012.9.D.FrutosandC.Johnen.Decidabilityofhomespaceproperty.LRI,report503,1989.10.S.GinsburgandE.H.Spanier.Semigroups,Presburgerformulasandlanguages.PacicJournalofMathematics,16(2),285{296,1966.11.L.GomesandJ.P.Barros.StructuringandcomposabilityissuesinPetriNetsmodeling.IEEETransactionsonIndustrialInformatics1(2),112{123,2005.12.S.Haddad,R.HennickerandM.H.M ller.ChannelPropertiesofAsynchronouslyComposedPetriNets.ResearchReportLSV-13-05,LaboratoireSpcicationetVrication,ENSCachan,France,13.R.Hennicker,S.JanischandA.Knapp.RenementofComponentsinConnection-SafeAssemblieswithSynchronousandAsynchronousCommunication.Proc.ofthe15thMontereyWorkshop,LNCS6028,154{180,2008.14.R.HennickerandA.Knapp.ModalInterfaceTheoriesforCommunication-SafeComponentAssemblies.Proc.ofthe8thInt.ColloquiumonTheoreticalAspectsofComputing(ICTAC'11),LNCS6916,135{153,2011.15.P.Jancar.DecidabilityofatemporallogicproblemforPetrinets.Theor.Comput.Sci.,74(1):71{93,1990.16.E.Kindler.AcompositionalpartialordersemanticsforPetrinetcomponents.ICATPN1997,LNCSvol.1248,235{252,1997.17.N.Lohmann,P.MassutheandK.Wolf.Operatingguidelinesfornite-stateser-vices.ICATPN2007,LNCSvol.4546,321{341,2007.18.J.L.Peterson.PetriNetTheoryandtheModelingofSystems.PrenticeHall,1981.19.E.Mayr.AnalgorithmforthegeneralPetrinetreachabilityproblem.InProc.ofthe13thAnnualACMSymposiumonTheoryofComputing(STOC'81)pages238{246,1981.20.C.Racko .Thecoveringandboundednessproblemsforvectoradditionsystems.TCS,6,223{231,1978.21.W.Reisig.Simplecompositionofnets.30thInt.Conf.onApplicationsandTheoryofPetriNets,LNCS5606,23{42,2009.22.Y.Souissi.Onlivenesspreservationbycompositionofnetsviaasetofplaces.11thInt.Conf.onApplicationsandTheoryofPetriNets,LNCS524,277{295,1990.23.Y.SouissiandG.Memmi.Compositionofnetsviaacommunicationmedium.10thInt.Conf.onApplicationsandTheoryofPetriNets,LNCS483,457{470,1989.24.C.StahlandW.Vogler.Atrace-basedservicesemanticsguaranteeingdeadlockfreedom.ActaInformatica,49(2):69{103,201225.C.StahlandK.Wolf.Decidingservicecompositionandsubstitutabilityusingextendedoperatingguidelines.DataKnowl.Eng.,68(9),819{833,2009.26.R.ValkandM.Jantzen.TheresidueofvectorsetswithapplicationstodecidabilityproblemsinPetrinetsAdvancesinPetriNets1984,LNCS188,234{258,1984.20