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andGrkemKõlõn1,21FakulttfrMathematikundInformatik,FernUniversittinHagen,GermanyUniversitdegliStudidiMilano-Bicocca,ItalyAbstract.WhereasthetraditionallivenesspropertyforPetrinetsguar-anteesthateachtransitioncanalwaysoccuragain,observablelivenessrequiresthat,fromanyreachablemarking,eachobservabletransitioncanbeforcedtoÞrebychoosingappropriatecontrollabletransitions;henceitisdeÞnedforPetrinetswithdistinguishedobservableandcontrol-labletransitions.Weintroduceobservablelivenessandshowthisnewnotiongeneralizeslivenessinthefollowingsense:livenessofanetim- 2TheSettingWhendeÞningobservableliveness,severaldesigndecisionshadtobemade.Wehadaparticularsettingofamodeledsysteminmind,thatmotivatedourchoices.Thissectionaimsatexplicatingthissettingandmotivatingourdesigndecisions.Thegenericsoftwaresystemtobemodeledconsistsofamachine(orseveralmachines),auserinterfacetothismachine,andperhapsofactivitiesandcondi-tionswhichdonotbelongtothemachine.Theusercanobserveandcontrolallactivitiesoutsidethemachine,hecanneithercontrolnorobserveanyactivities "#m(p)+1 '¥t,p/'t¥m(p)otherwiseWedenotethisbymt(&m!.ThesetofreachablemarkingsofthenetN,R(N),isthesmallestsetofmarkingsthatcontainstheinitialmarkingm0andsatisÞes[m'R(N))mt(& 2t2((&m3t3((& 1!(& ...isanoccurrencesequence.AninÞniteoccurrencesequencet1t2t3...enabledatsomemarkingmiscalledweaklyunfairw.r.t.sometransitiontif,forsomek'N,t1t2...tktisenabledatmand,foreach�jk,wehave¥tj!¥t="(aftersomeÞniteinitialphase,tispersistentlyenabledandnotinstructuralconßictwithanyoccurringtransition).NoticethatthisdeÞnitionisslightlyweakerthantheusualdeÞnitionofweakfairnesswhichonlydemandsthattispersistentlyenabled.Theoccurrencesequenceisweaklyfairw.r.t.tifitisnotweaklyunfairw.r.t.t.BythisdeÞnition,everyÞniteoccurrencesequenceisweaklyfairw.r.t.toalltransitions.Therearemanydi!erentfairnessnotionsforPetrinets(andpreviouslyforothermodels).Ournotion-oftenalsocalledprogressassumption-wasÞrstmentionedin[12].Itisparticularlyobviousforpartiallyorderedbehaviornotionssuchasoccurrencenetsandcannowbeviewedasastandardnotion.4ObservableLiveness m,thenetÞrstproceedsarbitrarilyandautonomously,i.e.,someoccurrencesequence!1withoutcontrollabletransitionsoccur.Thissequencecanbea) )weintroducearesponsefunction#,whichdeliversasetofpossiblecontrollabletransitionsasaresponseoftheagenttothesequenceobservedsofar.Noticethatanobservedsequencedoesnotdeterminethereachedmarkingbecauseunobservabletransitionsmightoccur,changingthemarkingbutnote!ectingtheobservedsequence.Inturn,di!erentobservedsequencesmightleadtothesamemarking.Wecallthetransitiontobservablyliveif,forsomesuchresponsefunction,weeventuallyobservetinthesequencecreatedthisway. areÞniteandcanbeempty, #(!0!!)foronlyÞnitelymanypreÞxes!!of!.e) leadsfromm0toamarkingmandisa#-maximaloccurrencesequenceenabledatm.If!=!1!2and suchthat,foreachm0!0((&m,each#t-maximaloccurrencesequenceenabledatmcontainsanoccurrenceoft.Anobservableplace/transitionnetisobservablyliveifallitsobservabletransitionsareobservablylive. !0andmbym!). isanobservablylivetransitionthereisaresponsefunction#tsuchthatforeachm0!0((&m,each#t-maximaloccurrencesequenceenabledatmincludest.ByLemma2thereexistsa#t-maximaloccurrencesequence.Thisimpliesthat,foreachreachablemarkingm,thereexistsanoccurrencesequencewhichenablest,andsotislive./0Corollary1.Anobservablylivenetisliveifalltransitionsareobservable./0NoticethatCor.1doesnotholdwithouttheassumptionthatalltransitions ,each#t-maximaloccurrencesequenceenabledatmcontainsanoccurrenceoft.Sincetislive,thereexists Letm0!0((&mandassumethattisanobservablylivetransition.Thereisaresponsefunction#tsuchthateach#t-maximaloccurrencesequenceenabledatmcontainsanoccurrenceoft.SoaninÞniteweaklyfairoccurrencesequencewithoutcontrollabletransitions whereasforobservablelivenessthischoiceisonlypossibleforcontrollabletransitions(whicharenotinconßictwithunob-servableones)andthenetbehavesarbitrarilyelsewhere. ofanet,bothenabledatsome enablesuaswellasasequence!uwhereudoesnotappearin andv!!u.Letmv(&m!.Theinductionhypothesiscanbeappliedtothemarkingm!,enablinguand!!u,yieldingthesequenceu!!enabledatm!.Sovu!!isenabledatm.AgainsinceuandvareconcurrentandbyLemma3,malsoenablesuv uandsincemenablesbothuandv,thesetransitionsareconcurrent.Therefore,andbyLemma3,malsoenablesthesequencevu.Letmv(&m!.Theinductionhypothesiscanbeappliedtothemarkingm!,enablinguand!!,yieldingthesequence!!uenabledatm!.Sov!!uisenabledatm.Wehavev!!=!,whichÞnishestheproof./0Thefollowingtheoremconstitutesthemainresultofthispaper.Itappliesonlytonetswheretheonlypossibleconßictsoccurbetweencontrollabletran-sitions,i.e.,tonetswhichareconßict-freew.r.t.alluncontrollabletransitions.Thisrulesoutconßictsbetweentwouncontrollabletransitionsaswellasconßictsbetweencontrollableanduncontrollabletransitions.Asapreparation,weneedacoupleofdeÞnitionsandlemmas.DeÞnition4.Anoccurrencesequence!enabledatamarkingmiscalledmin- mighthavetooccurbeforebecauseitproducesthetokenconsumedbyu.Wethencalltheoccurrenceofvacausalpredecessoroftheoccurrenceofu.Aminimaloccurrencesequencetowardsatransitiontcontainsoneoccurrenceoft,itscausalpredecessors,thepredecessorsofthesepredecessorsetc.,andnothing yieldsasubsequenceof!.Itscomplementaryse-quenceisthesequenceobtainedfrom!bydeletingallelementsthatappearinthesubsequence.ThisdeÞnitioncapturesthecase ,...,t2n,thesequence istherightmosttransition(transitionoccurrence,respectively)inµforwhichsuchadivisionispossible,weobtainfrom .Sothissequenceisminimaltowardst.Byconstruction,itisasubsequenceof andinµ2.Weproceedindirectlyandassumethecontrary. suchthatµ!!1beginswiththeÞrstoccurrenceof andaftertheoccurrenceofµ!2.Byconßict-freenessandsincethesetsofoccurringtransitionsinµ!1andµ!2aredisjoint,wecanalsoÞreboth,i.e.µ!1µ!2,fromtheinitialmarking.Thisyieldsamarkingwithtwotokensontheplaces,contradicting1-boundedness./0Theproofoftheabovelemmaalsoshowsthatallminimalsequencestowardshavethesamelength,whencethesesequencesareexactlythesequenceswithminimallengthcontaininganoccurrenceoft.Nowwearereadyforthemainresult:livenessofa1-boundednetimpliesobservableliveness,providedtheonlyconßictthatcanappeararebetweencon-trollabletransitions.AlthoughthisresultmightseemobviousatÞrstsight,itsproofissurprisinglyinvolved.Thecoreargumentoftheproofisthat,inalivePetrinet,foreachtransitiont,everyreachablemarking tooccurbyonlyprovidingasuitablere-sponsefunction#twhichcontrolsthebehaviorwheneverthereisaconßict.SoanobviousideaistodeÞne#tinsuchawaythatalwaysthenexttransitionin!misresponded,ifthistransitioniscontrollable.However,#tdependsnotonmarkings,butonobservedsequences.Thatmeans,insteadoftheuseronlyknowsthesequenceofobservabletransitionsoftheinitiallyenabledoccurrencesequence!0thatleadstom.Forthisobservedsequence,theremightexistmanysequencesincludingunobservabletransitions,andhencemanydi!erentreachedmarkingsm,andsoalsomanydi!erentoccurrencesequences!m.Insteadoftheunknownoccurrencesequence!0weconsiderthesetofalloccurrencesequencesµ0satisfyingµ0=!0.Amongthesesequencesweconcentrateontheminimalones.Wewillshowthat,ifthenetis1-bounded,alltheseminimaloccurrencesequencesleadtothesamemarkingwhichwecallm!0.Wewillmoreovershowthatm,themarkingreachedbytheoccurrenceof!0isreachablefromm!0.However,theseresultsonlyholdforconßict-freenets,andourconsiderednetisnotnecessarilyconßict-free.Sinceuntilnowweonlyconsiderthebehaviorgivenbytheobservedtransitionsof!0,sinceallcontrollabletransitionsareobservableandsinceconßictsonlyappearamongcontrollabletransitions,wecantransformtheconsiderednetintoaconßict-freeone,withoutspoilingtherelevantbehavior.Byliveness(oftheoriginalnet),m!0enablesanoccurrencesequence!contain-ingt.First,welookattheÞrstobservabletransitionin satisÞesui=vj.WeapplyLemma6tobothsequencesandthusobtainminimalsubsequencestowardsui(vj,respectively).ByLemma7,bothsubsequencesleadtothesamemarking.Theinductionhypothesisappliestothetwocomplementarysequences.Thisendstheproofoftheclaim.Theunique(foragivenµ0)markingreachedbyaminimalsequence iscontrollable,and#t(µ)="ifµ!beginswithuanduisnotcontrollable.Noticethatµ!containstasitslasttransitionandishencenotempty.Wenowcomebacktothecoreofthisproofandconsideranarbitraryinitiallyenabledoccurrencesequence!0whichleadstoamarkingm.Wehavetoshowthateach#t-maximaloccurrencesequenceenabledatmeventuallycontainst.Weconsideraconßict-freevariantofthenetasbefore,butinsteadofconsid- positedirectionholds.Thispaperprovesthatfor1-boundedPetrinetswithtransitionsthatcanbeobservableoradditionallycontrollable,livenessimpliesobservableliveness,wherethelattermeansthatcontrolcanforceeverytransi-tiontoÞreeventuallyfromanarbitraryreachablemarkingÐprovidedthenetmodelbehavesdeterministicallyinitsuncontrollablepart.Thiscontrolcanonlyselectenabledcontrollabletransitionsandisbasedonlyonthesequenceoftran-sitionsobservedsofar.Thiswaytheresultgeneralizestheobviousobservation,thatinafullydeterministicnetatransitionisliveifandonlyifiteventuallyÞres.Afutureconsiderationreferstopossiblegeneralizationsofourresult.Itclearlystillholdswhenthereissomelimitednondeterminismintheuncon-trolledpart.Forexample,iftwoalternativeuncontrollabletransitionscausethesamemarkingtransformation,theresultisnotspoiled.Moregenerally,weaimatdeÞninganequivalencenotiononnets,basedontherespectiveobservedbe-havior,whichpreservesobservableliveness.Reductionrules,asdeÞnede.g.in[1],[6]and[4]butalsoinmanyotherpapers,couldbeappliedtotheuncon-trollablepartleadingtosimplerbutequivalentnets.However,thereareobviousadditionalrules.Forexample,arulethatdeletesadeadtransitionissoundw.r.t.theequivalencebecausedeaduncontrollabletransitionsdonotcontributetotheobservablelivenessornon-livenessoftheconsiderednet.Asafuturework,weplantoconsideranautomataapproachfortheim-plementationoftheresponsefunction.ThedomainoftheresponsefunctionisdeÞnedinÞnite.InordertodecidewhichcontrollabletransitionscanbeÞrednext,anarbitraryhistoryofobservedtransitionshastobeconsidered.Often,aÞniteamountofthehistoryisenoughforthisdecision.Ifthisisthecase,anau-tomatabasedapproachcanbeusedfortherealizationoftheresponsefunction:theresponsethenonlydependsonastate(ofÞnitelymany)ofthisautomaton.Concerningbehavior,eachrunhasanalternationbetweenfreechoicesofthemachine(whereinanalysisallpossibilitiesmustbeconsidered)andparticularchoicesoftheuser.Therefore,describingthebehaviorwithAND/OR-treesseemspromising,maybeincombinationwithunfoldingapproaches.Thepartialorderviewwouldhaveobviousadvantagestocapturetheprogressassumption(thatwecalledweakfairness)inanaturalway[5,14].AÞnalremarkconcernstherelationtoTemporalLogics.SincelivenessandallreachabilityquestionsintraditionalPetrinetsuseexistentialquantiÞcationonpaths(ofthereachabilitygraph),andthereforerequireBranchingTimeconcepts,ourapproachexplicatesreasonsfordesiredactivities,i.e.,transitionoccurrences.Moreprecisely,asinthediscussionoflivenessinthispaper,wedistinguishuncontrollablealternativesandcontrollablechoices,tobeabletoexpressthatacertainactivity(ofauser)leadstotheeventualoccurrenceofanevent,nomatter