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1 VericationonInniteStructures LehrstuhlInformatikV UniversitatDortmund BaroperStrae301 OlafBurkart DidierCaucal D-44221Dortmund GERMANY CampusdeBeaulieu F-35042Rennes FRANCE IRISA FaronMoller UniversityofWalesSwansea DeptofComputerScience SingletonPark,Sketty SwanseaSA28PP LehrstuhlInformatikV BernhardSteen WALES UniversitatDortmund D-44221Dortmund BaroperStrae301 GERMANY Abstract.Inthischapter,wepresentahierarchyofinnite-statesystems inetheequivalenceandregularitycheckingproblemsfortheseclasses,withspe- cialemphasisonbisimulationequivalence,stressingthestructuraltechniques branching-timetemporallogics. whichhavebeendevisedforsolvingtheseproblems.finally,weexplorethe modelcheckingproblemovertheseclasseswithrespecttovariouslinear-and hierarchyincludesavarietyofcommonly-studiedclassesofsystemssuchas context-freeandpushdownautomata,andpetrinetprocesses.wethenexam- basedontheprimitiveoperationsofsequentialandparallelcomposition;the position/computation,automaticverication,decidability,complexity,be- haviouralequivalences,bisimulation,equivalencechecking,regularitychecking, Keywords:innite-staterewritetransitionsystems,sequential/parallelcom- linear/branching-timetemporallogics,modelchecking

2 Introduction Contents 1ATaxonomyofInniteStateProcesses 1.1RewriteTransitionSystems ExpressivityResults FurtherClassesofProcesses LanguagesandBisimilarity TheEquivalenceCheckingProblem 2.1DecidabilityResultsforBPAandBPP AFiniteBisimulationBaseforBPA CompositionandDecomposition PolynomialTimeAlgorithmsforBPAandBPP APolynomialTimeAlgorithmforNormedBPA AFiniteBisimulationBaseforBPP ComputingaFiniteBisimulationBaseforBPA UndecidabilityResultsforMSA SummaryofResults APolynomialTimeAlgorithmforNormedBPP TheModelCheckingProblem 3.1TemporalLogics Linear-TimeLogics Branching-TimeLogics AlgorithmsforBranching-TimeLogics BPA BPPandPetriNets PushdownProcessesandExtensions AlgorithmsforLinear-TimeLogics PA GeneralProcesses Summary BPAandPushdownProcesses BPPandPetriNets PA

3 Introduction Thestudyofautomated(sequential)programvericationhasaninherenttheoreticalbarrierintheguiseofthehaltingproblemandformalundecidability. Thesimplestprogramswhichmanipulatethesimplestinnitedatatypessuch asintegervariablesimmediatelyfallfoulofthesetheoreticallimitations.during execution,suchaprogrammayevolveintoanyofaninnitudeofstates,and correctness,andtherearenowelegantandacceptedtechniquesforthesemantic hasnotpreventedaverysuccessfulattackontheproblemofprovingprogram asthehaltingstate,willingeneralbeimpossibletodetermine.however,this knowingiftheexecutionoftheprogramwillleadtoanyparticularstate,such analysisofsoftware. forcompletelyandexplicitlyexploringthereachablestatesofanygivensystem, strictlyasnite-statesystems,andformalanalysistoolshavebeendeveloped waresystems,hasfollowedadierentcourse.here,systemshavebeenmodelled Thehistorybehindthemodellingofconcurrentsystems,inparticularhard- forinstancewiththegoalofdetectingwhetherornotahalting,i.e.,deadlocked, stateisaccessible.thisabstractionhasbeenwarranteduptoapoint.real hardwarecomponentsareindeedniteentities,andprotocolstypicallybehave inaregularfashionirrespectiveofthediversityofmessageswhichtheymaybe designedtodeliver. modellinglanguage.suchformalismsfordescribingconcurrentsystemsare statespacesofthevariouscomponents,forexamplebylistingoutthestates andtransitionfunction,butrathertospecifythemusingsomehigher-level Inspecifyingconcurrentsystems,itisnottypicaltoexplicitlypresentthe notusuallysorestrictiveintheirexpressivepower.forexample,thetypical processalgebracanencodeintegerregisters,andwiththemcomputearbitrary computablefunctions;andpetrinetsconstituteagraphicallanguagefornitely ofthesystembeingspeciedissemantically,ifnotsyntactically,nite.for presentingtypicallyinnite-statesystems.however,toolswhichemploysuch example,agivenprocessalgebratoolmightsyntacticallycheckthatnostatic formalismsgenerallyrelyontechniquesforrstassuringthatthestatespace operatorssuchasparallelcompositionappearwithinthescopeofarecursive denition;andagivenpetrinettoolmightcheckthatanetissafe,thatis, thatnoplacemayacquiremorethanonetoken.havingveriedtheniteness toprovidepracticalvericationtools,isofcoursethatofstatespaceexplosion. ofthesystemathand,thesearchalgorithmcan,atleastinprinciple,proceed. exponentialinthenumberofcomponentswhichmakeupthesystem.hencea Thenumberofreachablestatesofasystemwilltypicallybeontheorderof Theproblemwiththeblindsearchapproach,whichhasthwartedattempts greatdealofresearcheorthasbeenexpendedontamingthisstatespace,typicallybydevelopingintelligentsearchstrategies.variouspromisingtechniques ofbdd(binarydecisiondiagram)encodingsofautomata[20].however,such statespacesfeasible;onepopularapproachtothisproblemisthroughtheuse havebeendevelopedwhichmakefortheautomatedanalysisofextremelylarge approachesareinherentlyboundtotheanalysisofnite-statesystems. Recently,interestinaddressingtheproblemofanalysinginnite-statesys- 3

4 motivationforthishasbeenbothtoprovideforthestudyofparallelprogram temshasblossomedwithintheconcurrencytheorycommunity.thepractical verication,whereinnitedatatypesaremanipulated,aswellastoallowfor morefaithfulrepresentationsofconcurrentsystems.forexample,real-timeand probabilisticmodelshavecomeintovogueduringthelastdecadetoreectfor system'sstructure.suchenhancementstotheexpressivepowerofamodelling modelshavebeendevelopedwhichallowforthedynamicrecongurationofa instancethetemporalandnondeterministicbehaviourofasynchronoushardwarecomponentsrespondingtocontinuously-changinganaloguesignals;and languageimmediatelygiverisetoinnite-statemodels,andnewparadigms systemsexpressedinsuchformalisms. notbasedonstatespacesearchneedtobeintroducedtosuccessfullyanalyse equivalencechecking,whichaimsatestablishingsomesemanticequivalence plementationofthespecicationgivenbytheother;andmodelchecking, betweentwosystems,oneofwhichistypicallyconsideredtorepresenttheim- Inthissurvey,weexplorethetwomajorstreamsinsystemverication: whichaimsatdeterminingwhetherornotagivensystemsatisessomepropertywhichistypicallypresentedinsomemodalortemporallogic.inthesurvey weconcentrateexclusivelyondiscretesystems,andneglectinnite-statestructuresrelatedto,e.g.,timedandhybridsystems.thevarietyofprocessclasses whichweconsideriscataloguedinsection1,wherewedetailtherelativeexpressivepowersoftheseclasses.thetwoapproachestosystemvericationare summarizedhere,wherewealsosketchthecontentsofthetworelevantsections ofthesurvey. systemverication.indeed,suchquestionshavealongtraditionintheeld Equivalencechecking,thatis,determiningwhentwo(innite-state)systems EquivalenceChecking areinsomesemanticsenseequal,isclearlyaparticularlyrelevantproblemin guagetheoristshavebeenstudyingtheequivalenceproblemoverclassesof automatawhichexpresslanguageswhicharemoreexpressivethantheclassof of(theoretical)computerscience.sincetheproofbymoore[121]in1956of thedecidabilityoflanguageequivalencefornite-stateautomata,formallan- regularlanguagesgeneratedbynite-stateautomata.bar-hillel,perlesand Shamir[6]werethersttodemonstrate,in1961,thattheclassoflanguages denedbycontext-freegrammarswastoowidetoadmitadecidabletheoryfor languageequivalence.shortlyafterthis,korenjakandhopcroft[95]demonstratedthatlanguageequivalencebetweensimple(deterministic)grammarsis decidable.onlyrecentlyhasthelong-openproblemoflanguageequivalence betweendeterministicpush-downautomata(dpda)beensettled(positively) bysenizergues[132,133]. withthethesisofhack[63].however,ithasonlybeeninthemuchmorerecent interestdriveninpartbyanalogiesdrawnbetweenclassesofconcurrentsystem pastthatamoreconcertedeorthasbeenfocussedonsuchquestions,withthe DecidabilityquestionsforPetrinetswereaddressedalreadytwodecadesago, modelsandclassesofgeneratorsforfamiliesofformallanguages.in[114]milner 4

5 exploitstherelationshipbetweenregular(nite-state)automataasdiscussed bysalomaain[130]andregularbehaviourstopresentthedecidabilityanda completeaxiomatisationofbisimulationequivalencefornite-statebehaviours, whilstinhistextbook[115]hedemonstratesthatthehaltingproblemforturing machinescanbeencodedasabisimulationquestionforthefullccscalculus thusdemonstratingundecidabilityingeneral.thisnalfeatiscarriedout Theseresultsareasexpected;however,realinterestwasgeneratedwiththe elegantlyusingniterepresentationsofcountersinthethesisoftaubner[141]. discoverybybaeten,bergstraandklop[4,5]thatbisimulationequivalenceis decidableforafamilyofinnite-stateautomatageneratedbyageneralclassof context-freegrammars. lence,focussingonthevarioustechniquesexploitedineachcase.ourinterestin decidabilityandcomplexity,withparticularemphasisonbisimulationequiva- bisimulationequivalencestemspredominantlyfromitsmathematicaltractability.apartfrombeingthefundamentalnotionofequivalenceforseveralprocess algebraicformalisms,bisimulationequivalencepossessesseveralpleasingmathematicalproperties,notleastofwhichbeing asweshalldiscover thatitis undecidable,inparticularovertheclassofprocessesdenedbycontext-free decidableoverprocessclassesforwhichallothercommonequivalencesremain InSection2,wepresentanoverviewofvariousresultsobtainedregarding whenanalysingpropertiesofanotherequivalence.inparticular,bystudying soitissensibletoconcentrateonthemostmathematicallytractableequivalence grammars.furthermoreinaparticularlyinterestingclassofprocesses namely bisimulationequivalencewecanrediscoverstandardtheoremsaboutthedecidabilityoflanguageequivalence,aswellasprovidemoreecientalgorithmsfor thestructuraltechniqueswhichcanbeexploitedinthestudyofbisimulation thesedecidabilityresultsthanhavepreviouslybeenpresented.weexpectthat equivalencewillprovetobeusefulintacklingvariousotherlanguagetheoreticproblems.indeed,whilesenizergues'proofofthedecidabilityofdpdais extremelylongandcomplex,developedover70pagesofa166pagejournalsubmission,stirling[138]hassincepresentedafarsimplerproofofthisresultusing questionsregardingregularitychecking,thatis,determiningifagivensys- variationsonsomeofthestructuralanalysistechniquesexploredinsection2. temissemanticallynite-state.withinatypicalapplicationdomainsuchas themodellingofprotocols,wemightanalyseinnite-statesystemswhichwe Apartfromthebasicequivalencecheckingproblem,wealsosurveyrelated thenormeddeterministicprocesses allofthestandardequivalencescoincide, systemisnite-state,butthestatespaceoftheimplementationis(syntactically)innite.apositiveanswerfortheregularitycheckingproblemwould allowtheequivalencecheckingalgorithmtoexploitwell-developedtechniques foranalysingnite-statesystems(assumingthattheequivalentnite-statesystemcouldbeextractedfromfromthisanswer). intendtosemanticallyrepresentnite-statebehaviours:ourspecicationofthe 5

6 promisingapproachtotheformalvericationofdistributed,reactivesystems. ModelChecking Apartfromequivalencechecking,modelcheckingprovidesperhapsthemost nite-statesystemsmodelcheckingis,atleasttheoretically,alwaysapplicable, propertiesofasystem,anda(semi-)decisionprocedurethencheckswhether thegivensystemisamodeloftheformulaathand.inparticular,inthecaseof Inthisapproach,oneusesformulaeofatemporallogictospecifythedesired underconsiderationcaneectivelyprovideenoughinformationtosolvetheveri- sinceanexhaustivetraversalthroughthereachablestatespaceofthesystem prominentindustrialsuccessstories,despitetheomni-presentstatespaceexplosionthreat,makingitanindispensabletoolforhardwaredesign.thepractical cationproblem.infact,usingbdds,nite-statemodelcheckinghashadsome newly-publishedtextbookbyclarke,grumbergandpeled[41]. successofmodelcheckingcanbewitnessedbytheimmediatepopularityofthe techniques.thenewtechniqueswhichhavebeendevelopedformodelcheckinghavemainlybeeninuencedbyformallanguagetheory,wherethereexists alongtraditionofreasoningaboutnitely-presentedinniteobjects,inthis caseformallanguagesdescribedbyautomataorgrammars.consequently,notionsfromformallanguagetheoryformarecurringthemeinthevericationohaustivetraversalofthestatespace,theyareinherentlyincapableofverifying innite-statesystems,andmustbereplacedbyalgorithmsbasedondierent Again,asalgorithmsfornite-statemodelcheckingtypicallyinvolvetheex- decidability.expressivemodelsofcomputation,inconjunctionwithpowerfulspecicationformalisms,inevitablyleadtoanundecidablemodelchecking incharacterizingtheexpressivepowerofcertaintemporallogics. innite-statesystems,e.g.,asameansforthedescriptionofsuchsystems,or problem.however,takingweakermodelsand/orformalismsinordertorecover Onegeneralproblemis,however,thetrade-obetweenexpressivenessand decidabilityoftengreatlydiminishesitspracticalvalue.thegoalofthissurvey istoprovideasystematicpresentationoftheresultsobtainedsofarconcerningdecidabilityandcomplexityissuesinmodelcheckinginnite-statesystems, therebyidentifyingfeasiblecombinationsofprocessclassesandtemporallogics. consider,togetherwiththeirbranchingtimeandlineartimeclassications.we checkingforvariousclassesofinnite-statesystems,rstforbranchingtime thenproceedbypresentingdecidabilityandcomplexityresultsaboutmodel InSection3.1weprovideabriefintroductiontothetemporallogicswe logicsinsection3.2,andthenforlineartimelogicsinsection3.3.duetothe wealthoftechniqueswhichhavebeendevisedforuseinconjunctionwiththe oftherelevantresultsinthissurvey.thisgivesaconceptualandeasytoread widevarietyoflogicswhichonemayconsider,weonlysketchthemainideas overview,whichiscomplementedbytightlinkstotheoriginalliterature,thus AcknowledgementsSections1and2arebasedonthesurveyInniteResults[118]presentedatCONCUR'96,andtheoriginsofSection3arefound enablingtheinterestedreaderstoaccessallthetechnicaldetails. inthesurveymoreinniteresults[25]presentedatinfinity'96.however, 6

7 therehasbeenaoodofnewresultsintheinterveningyears,andtheseearlier surveysarevastlyextendedhere.wewishtothankrichardmayrandmarkus Mueller-Olmfordetailedcommentsonearlierdraftsofthissurvey. 7

8 1Concurrentsystemsaremodelledsemanticallyinavarietyofways.Theymay 1.1RewriteTransitionSystems ATaxonomyofInniteStateProcesses bedenedforexamplebytheinnitetracesorexecutionswhichtheymayperform,orbytheentiretyofthepropertieswhichtheysatisfyinsomeparticular processlogic,orasaparticularalgebraicmodelofsomeequationalspecication.inanycase,afundamentalunifyingviewistointerpretsuchsystemsitionemanates;thelabelonatransitionrepresentsaneventcorresponding systemoriginatinginthestaterepresentedbythenodefromwhichthetran- asedge-labelleddirectedgraphs,whosenodesrepresentthestatesinwhicha systemmayexist,andwhosetransitionsrepresentthepossiblebehaviourofthe totheexecutionofthattransition,whichwilltypicallyrepresentaninteractionwiththeenvironment.thestartingpointforourstudywillthusbesuch graphs,whichwillforusrepresentprocesses. Denition1AlabelledtransitionsystemisatuplehS;;?!;0;Fiwhere?!SSisatransitionrelation,writtena Sisasetofstates. isanitesetoflabels. FSisanitesetofnalstateswhichareterminal,meaningthatfor 02Sisadistinguishedstartstate.?!forh;a;i2?!. Thisnotionofalabelledtransitionsystemdiersfromthestandarddenitionof anite-stateautomaton(asforexamplegivenin[74])inthatthesetofstates each2fthereisnoa2and2ssuchthata?!. neednotbenite,andnalstatesmustnothaveanyoutgoingtransitions. Thislastrestrictionismildandjustiedinthatanalstatereferstothe successfulterminationofaconcurrentsystem.thiscontrastswithunsuccessful automatawhichacceptonemptystack.(analternativeapproachcouldbe whichcharacterisesawideclassoflabelledtransitionsystemsaspush-down Wecouldremovethisrestriction,butonlyattheexpenseofTheorem4below termination(ie,deadlock)whichisrepresentedbyallnon-nalterminalstates. donotpursuethisalternativehere.) takentorecovertheorem4basedonpdawhichacceptbynalstate,butwe belledtransitionsystemsdenedbyvariousrewritesystems.suchanapproach andtransitionsystemgenerators,alinkwhichisofparticularinterestwhenit providesuswithaclearlinkbetweenwell-studiedclassesofformallanguages WefollowtheexamplesetbyCaucal[32]andconsiderthefamiliesofla- Denition2Asequentiallabelledrewritetransitionsystemisatuple comestoexploitingprocess-theoretictechniquesinsolvingproblemsinclassical formallanguagetheory. hv;;p;0;fiwhere 8

9 Visanitesetofvariables;theelementsofVarereferredtoasstates. isanitesetoflabels. PVVisanitesetofrewriterules,writtena?!for h;a;i2p,whichareextendedbytheprexrewritingrule:ifa?! thena?!. 02Visadistinguishedstartstate. FVisanitesetofnalstateswhichareterminal. Aparallellabelledrewritetransitionsystemisdenedpreciselyasabove,exceptthattheelementsofVarereadmodulocommutativityofconcatenation,whichisthusinterprettedasparallel,ratherthansequential,composition. WecanthusconsiderstatesasmonomialsXk1 1Xk2 2Xkn noverthevariables V=fX1;X2;;Xng.Withthisinmind,weshallbeabletoexploitthefollowingresultduetoDickson[48]whichiseasilyprovedbyinductiononn. Lemma3(Dickson'sLemma)Givenaninnitesequenceofvectorsofnaturalnumbers~x1;~x2;~x3;:::2Nnwecanalwaysndindicesiandjwithi<j suchthat~xi~xj(whereisconsideredpointwise). ThestateXk1 1Xk2 2Xkn ncanbeviewedasthevector(k1;k2;:::;kn)2nn. Hence,Dickson'sLemmasaysthat,givenanyinnitesequence1;2;3;::: ofsuchstates,wecanalwaysndtwoofthese,iandjwithi<j,suchthat thenumberofoccurrencesofeachvariablexinjisatleastasgreatasini. Weshallfreelyextendthetransitionrelation?!homomorphicallytonite sequencesofactionsw2soastowrite"?!andaw?!whenevera?! w?!.also,werefertothesetofstatesintowhichtheinitialstatecanbe rewritten,thatis,suchthat0w?!forsomew2,asthereachablestates. Althoughwedonotinsistthatallstatesbereachable,weshallassumethatall variablesinvareaccessiblefromtheinitialstate,thatis,thatforallx2v thereissomew2and;2vsuchthat0w?!x. Thisdenitionisslightlymoregeneralthanthatgivenin[32],whichdoes nottakeintoaccountnalstatesnorthepossibilityofparallelrewritingasan alternativetosequentialrewriting.bydoingthis,weexpandthestudyofthe classesoftransitionsystemswhicharedened,andextendsomeoftheresults givenbycaucal,notablyinthecharacterisationofarbitrarysequentialrewrite systemsaspush-downautomata. Thefamiliesoftransitionsystemswhichcanbedenedbyrestrictedrewrite systemscanbeclassiedusingaformofchomskyhierarchy.thishierarchy providesanelegantclassicationofseveralimportantclassesoftransitionsystemswhichhavebeendenedandstudiedindependentoftheirappearanceas particularrewritesystems.thisclassicationispresentedinfigure1.(type1 rewritesystems,correspondingtocontext-sensitivegrammars,donotfeaturein thishierarchysincetherewriterulesbydenitionareonlyappliedtotheprex ofacomposition.)intheremainderofthissection,weexplaintheclassesof 9

10 Restriction Type0 none ontherules a?!ofp none Restriction onf Sequential composition Parallel composition Type112 V=Q]? 2Q?where 2Q?and F=Q PDA MSA PN Type2 Type3 2V[f"g 2Vand F=f"g BPA FSA BPP Figure1:Ahierarchyoftransitionsystems. FSA transitionsystemswhicharerepresentedinthistable,workingupwardsstarting withthemostrestrictiveclass. thenitesetf2v:jjj0jg. restrictedtobeoftheformaa statesofboththesequentialandparalleltransitionsystemswillbeelementsof FSArepresentstheclassofnite-stateautomata.Clearlyiftherulesare?!BorAa '?!"witha;b2v,thenthereachable Example0Inthefollowingwepresenttwotype3(regular)rewritesystemsalongwiththeFSAtransitionsystemswhichtheinitialstatesXand A,respectively,denote. $ Yb Xa?!Y? Yc?!"?? "YX? a b c Aa?!B?!C?A Cc Bb?!" B= a a ~ " = ~ C & Thesetwoautomatabothrecognisethesame(regular)languagefab;acg. b c Klop[9],whicharethetransitionsystemsassociatedwithGreibachnormalform However,theyaresubstantiallydierentautomata. (GNF)context-freegrammarsinwhichonlyleft-mostderivationsarepermitted. BPArepresentstheclassofBasicProcessAlgebraprocessesofBergstraand% 10

11 ' statexdenotes. Example1Inthefollowingwepresentatype2(GNFcontext-freegrammar)rewritesystemalongwiththeBPAtransitionsystemwhichtheinitial$ Xc Xa?!XB R Bb X?!" & "? c a-xb a-xbb a- b B? cb BB? c b tensen[35]asaparallelanalogytobpa,andaredenedbythetransition Thisautomatonrecognisesthecontext-freelanguagefancbn:n0g. systemsassociatedwithgnfcontext-freegrammarsinwhicharbitrarygrammarderivationsarepermitted. ' followingbpptransitionsystemwithinitialstatex. Example2Thetype2rewritesystemfromExample1givesrisetothe$ Xc Xa?!XB R BPPrepresentstheclassofBasicParallelProcessesintroducedbyChris-% Bb X -XB -XBB -?!" "? c a a a b B? cbb BB? cb b & Thisautomatonrecognisesthelanguageconsistingofallstringsoftheform (a+b)cbwhichcontainanequalnumberofa'sandb'sinwhichnoprex containsmoreb'sthana's. fromtheinitalstate,thisimpliesthateachvariablesisonthelefthandsideof state.combinedwithourpreviousassumptionthateachvariableisreachable Weshallassumethatforalltype2rewritesystems,"istheonlyterminal% atleastonerule. stack.topresentsuchpdaasarestrictedformofrewritesystem,werst assumethatthevariablesetvispartitionedintodisjointsetsq(nitecontrol states)and?(stacksymbols).therewriterulesarethenoftheformpaa PDArepresentstheclassofpush-downautomatawhichacceptonempty withp;q2q,a2?and2?,whichrepresentstheusualpdatransition stack,youmayreadtheinputsymbola,moveintocontrolstateq,andreplace whichsaysthatwhileincontrolstatepwiththesymbolaatthetopofthe?!q thestackelementawiththesequence.finally,thesetofnalstatesisgiven byq,whichrepresentthepdacongurationsinwhichthestackisempty. 11

12 thetransitionsystemsareisomorphicuptothelabellingofstates.thestronger (type0)sequentialrewritesystemcanbepresentedasapda,inthesensethat result,inwhichnalstatesaretakenintoconsideration,actuallyholdsaswell. Caucal[32]demonstratesthat,disregardingnalstates,anyunrestricted Theideabehindtheencodingisasfollows.Givenanarbitraryrewritesystem hv;;p;0;fi,takensatisfying ThenletQ=fp:2Vandjj<ng; n>maxfjj:a nmaxfjj:a?!2pg;?!with2fg: and Everynaltransitionstate2FisrepresentedbythePDAstatep,thatis,?=fZ:2Vandjj=ng[fg: bythepdabeingincontrolstatepwithanemptystackdenotingacceptance; andeverynon-naltransitionsystemstate12k62fwithjj<nand bythepdabeingincontrolstatepwithwiththesequencez1z2zk jij=nfor1ik,isrepresentedinthepdabypz1z2zk,thatis, onitsstack.theneveryrewriteruleintroducesappropriatepdaruleswhich mimicitandrespectthisrepresentation.thuswearriveatthefollowingresult. Theorem4Everysequentiallabelledrewritetransitionsystemcanberepresented(uptothelabellingofstates)byaPDAtransitionsystem. ' Example3TheBPPtransitionsystemofExample2isgivenbythefollowingsequentialrewritesystem. $ statepx.(weomitrulescorrespondingtotheunreachablestates.) Bytheaboveconstruction,thisgivesrisetothefollowingPDAwithinitial Xa?!XB Xc?!" Bb?!" XBb?!X Xc Xa?!":?!XB:pXa pbb pxc?!p"zxbp"zxba?!p" p"zxbc p"zbbb?!pxzbbpxzbba?!pb pxzbbc pbzbbb?!p"zxbzbb ThisisexpressedmoresimplybythefollowingPDAwithinitialstatep. XBb?!X:p"ZXBb?!pX pxzbbb?!p"zxb?!p"zbb & pa pc?!pb?!q pba pbb pbc?!q?!pbb qb qbb?!q % 12

13 NotethatBPAcoincideswiththeclassofsingle-statePDA.However,weshall seeinsection1.3thatanypdapresentationofthetransitionsystemofexample2musthaveatleast2controlstates:thistransitionsystemisnotrepresented exceptthattheyhaverandomaccesscapabilitytothestack. \parallel"or\random-access"push-downautomata;theyaredenedasabove byanybpa. # MSArepresentstheclassofmultisetautomata,whichcanbeviewedas Example4TheBPAtransitionsystemofExample1isisomorphictothat " givenbythefollowingmsawithinitialstatepx. Notethatwhenthestackalphabethasonlyoneelement,PDAandMSAtriviallycoincide.AlsonotethatBPPcoincideswiththeclassofsingle-stateMSA.! pxa?!pbx pxc?!q qbb?!q WeshallseeinSection1.3thatanyMSApresentationofthetransitionsystem ofexample1musthaveatleast2controlstates:thistransitionsystemisnot representedbyanybpp. nets,asisevidentbythefollowinginterpretationofunrestrictedparallelrewrite systems.thevariablesetvrepresentsthesetofplacesofthepetrinet,and eachrewriterulea PNrepresentstheclassof(nite,labellled,weightedplace/transition)Petri NotethataBPPisacommunication-freePetrinet,oneinwhicheachtransition ontheinputandoutputarcsgivenbytherelevantmultiplicitiesinand. inputandoutputplacesrepresentedbyandrespectively,withtheweights?!representsapetrinettransitionlabelledawiththe hasauniqueinputplace. 13

14 ' Example5Thefollowingunrestrictedparallelrewritesystemwithinitial statexandnalstatey $ describesthepetrinetwhichinitsusualgraphicalrepresentationnetwould?!xa XABc?!X YAa berenderedasfollows.(theweightsonallofthearcsis1.) Xb?!XB Xd?!Y YBb?!Y Aa> = a > -X Y 6}?6- d ~ b} b c -B TheautomatonrepresentedbythisPetrinetrecognisesthelanguageconsistingofallstringsfrom(a+b+c)d(a+b)inwhichthenumberofc's j & b's;andinwhichthenumberofa's(respectivelyb's)beforetheoccurrence inanyprexisboundedabovebyboththenumberofa'sandthenumberof ofthedminusthenumberofc'sequalsthenumberofa's(respectivelyb's) aftertheoccurrenceofthed. holdintheparallelcase.weshallinfactdemonstratethattheautomaton rewritetransitionsystems,weshallseeinsection1.3thatthisresultfailsto Althoughinthesequentialcase,PDAconstitutesanormalformforunrestricted% associatedwiththeabovepnisnottheautomatonofanymsa. GivenalabelledtransitionsystemT=hS;;?!;0;Fi,wecandeneits 1.2LanguagesandBisimilarity ofactionswhichlabelrewritetransitionsleadingfromthegivenstatetoanal languagel(t)tobethelanguagegeneratedbyitsinitialstate0,wherethe languagegeneratedbyastateisdenedintheusualfashionasthesequences state. Denition5L()=fw2:w andarelanguageequivalent,writtenl,itheygeneratethesame language:l()=l().?!forsome2fg,andl(t)=l(0). systemisintheprocessofgeneratingaword,thenthepartialwordshould beextendibletoacompleteword.thatis,fromanyreachablestateofthe transitionsystem,analstateshouldbereachable.ifthetransitionsystem Withrespecttothelanguagesgeneratedbyrewritesystems,ifarewrite satisesthisproperty,itissaidtobenormed;otherwiseitisunnormed. 14

15 writtenn(),tobethelengthofashortestrewritetransitionsequencewhich Denition6Wedenethenormofanystateofalabelledtransitionsystem, takestoanalstate,thatis,thelengthofashortestwordinl().by convention,wedenen()=1ifthereisnosequenceoftransitionsfrom Notethat,duetotheassumptionfollowingDenition2ontheaccessibilityof toanalstate,thatis,l()=;.thetransitionsystemisnormedievery allthevariables,ifatype2rewritetransitionsystemisnormed,thenallofits reachablestatehasanitenorm;otherwiseitisunnormed. variablesmusthavenitenorm.thefollowingthenisabasicfactaboutthe normsoftype2(bpaandbpp)states. BPP),n()=n()+n(). Lemma7Givenanystateofatype2rewritetransitionsystems(BPAor u1u2un2l()andv1v2vn2l()g.theresultfollowseasilyfromthis. ProofForthesequentialcase,L()=L()L()=fuv:u2L()and v2l()g.fortheparallelcase,l()=l()kl()=fu1v1u2v2unvn: Denition8Tisdeterministiciforeveryreachablestateandeverylabel Afurthercommonpropertyoftransitionsystemsisthatofdeterminacy. 2 Forexample,thetwonitestateautomatapresentedinExample0areboth athereisatmostonestatesuchthata normedtransitionsystems,whileonlytherstisdeterministic.allotherexampleswhichwehavepresentedhavebeenbothnormedanddeterministic.?!. stratedierentdeadlockingcapabilitiesduetothenondeterministicbehaviour tobetoocoarseanequivalence.forexample,itequatesthetwotransition systemsofexample0whichgeneratethesamelanguagefab;acgyetdemon- Intherealmofconcurrencytheory,languageequivalenceisgenerallytaken exhibitedbythesecondtransitionsystem.manynerequivalenceshavebeen proposed,withbisimulationequivalencebeingperhapsthenestbehavioural concurrency'equivalencessuchasthosebasedonpartialorders.)bisimulationequivalencewasdenedbypark[125]andusedtogreateectbymilner[113,115].itsdenition,inthepresenceofnalstates,isasfollows. Denition9AbinaryrelationRonstatesofatransitionsystemisabisimulationiwhenever(;)2Rwehavethat ifa ifa?!0thena?!0thena?!0forsome0with(0;0)2r; equivalencestudied.(notethatwedonotconsiderhereanyso-called`true andarebisimulationequivalentorbisimilar,written,i(;)2r 2Fi2F.?!0forsome0with(0;0)2R; forsomebisimulationr.thatis,=sr:risabisimulationrelation. 15

16 Lemma10isthelargestbisimulationrelation. ProofAnarbitraryunionofbisimulationsisitselfabisimulation. Lemma11isanequivalencerelation. 2 sincethecompositionoftwobisimulationsisabisimulation. holdssincetheinverseofabisimulationisabisimulation;andtransitivityholds ProofReexivityholdssincetheidentityrelationisabisimulation;symmetry two-playergames[136,91].startingwithapairofstatesh;i,thetwoplayers Bisimulationequivalencehasanelegantcharacterisationintermsofcertain 2 alternatemovesaccordingtothefollowingrules. 1.Ifexactlyoneofthepairofstatesisanalstate,thenplayerIisdeemed tobethewinner.otherwise,playerichoosesoneofthestatesandmakes sometransitionfromthatstate(eithera impossible,duetobothstatesbeingterminal,thenplayeriiisdeemedto bethewinner.?!0ora?!0).ifthisproves 2.PlayerIImustrespondtothemovemadebyplayerIbymakingan a identically-labelledtransitionfromtheotherstate(eithera?!0).ifthisprovesimpossible,thenplayeriisdeemedtobethe?!0or Thefollowingresultisthenimmediatelyevident. 3.Theplaythenrepeatsitselffromthenewpairh0;0i.Ifthegame continuesforever,thenplayeriiisdeemedtobethewinner. Fact12iPlayerIIhasawinningstrategyinthebisimulationgame ProofAnybisimulationrelationdenesawinningstrategyforplayerIIfor gamestartingwiththepairh;i. Conversely,6iPlayerIhasawinningstrategyinthebisimulation pairiscontainedinthebisimulation. merelyhastorespondtomovesbytherstinsuchawaythattheresulting thebisimulationgamestartingfromapairintherelation:thesecondplayer thatpair,namelythecollectionofallpairswhichappearaftereveryexchange ingfromaparticularpairofstatesdenesabisimulationrelationcontaining ofmovesduringanyandallgamesinwhichplayeriiusesthisstrategy. Conversely,awinningstrategyforplayerIIforthebisimulationgamestart- 2 16

17 ' Example6ThestatesXandAfromExample0arelanguageequivalent, astheydenethesamelanguagefab;acg.however,theyarenotbisimu-lationequivalent:thereisnobisimulationrelationwhichrelatesthem.to seethis,wecandemonstrateanobviouswinningstrategyforplayeriin & IcannotrespondfromB,andinthelattercaseplayerImaymakethe exchangeofmoves,thenewpairofstatesmustbeeitherhy;biorhy;ci;in thebisimulationgamestartingwiththepairofstateshx;ai.aftertherst theformercase,playerimaymakethetransitionyc transitionyb?!"towhichplayer transitionsfromanygivenreachablestate;anditisimage-niteifthereare Atransitionsystemisnite-branchingifthereareonlyanitenumberof?!"towhichplayerIcannotrespondfromC. onlyanitenumberoftransitionswithagivenlabelfromanygivenreach-ablestate.forimage-nitetransitionsystems,wehavethefollowingstratied characterisationofbisimulationequivalence[115]. Denition13Thestratiedbisimulationrelationsnaredenedasfollows. k+1i 0forallstates. {ifa {ifa?!0thena?!0thena?!0forsome0with0k0; Intermsofthegamecharacterisationofbisimilarity,niPlayerIcannot {2Fi2F.?!0forsome0with0k0; forceawinwithintherstnexchangesofmoves. Lemma14Ifandareimage-nite,theninforalln0. isabisimulation. ProofIfthenaninductionproofgivesthatnforeachn0. Conversely,f(;):andareimage-niteandnforalln0g applies.itisequallyclearthateachoftherelationsnisdecidable,and Itisclearthatrewritetransitionsystemsareimage-nite,andthatthislemma 2 tems.hencethedecidabilityresultswouldfollowfromdemonstratingthesemi- decidabilityofbisimilarity. thatthereforenon-bisimilarityissemi-decidableoverrewritetransitionsys- 17

18 ' Example7Considerthefollowinginnite-branchingtransitionsystem. X $ A0Y A1 A2?6 RA3 Z I - a a Y a a a a & ThestatesXandYareclearlynotbisimilar,asthestateZcannotbebisimilartoAnforanyn0.However,XnYforeveryn0asZnAnfor itsaccompanyingcorollaryrelatingbisimulationequivalencetolanguageequivalence. Animmediatelyevidentyetimportantpropertyisthefollowinglemmawith% Lemma15Ifandw ProofGivenabisimulationRcontaining(;),aninductiononthelength that00.?!0withw2,thenw eachn0. ofw2demonstratesthatifw?!0thenw?!0forsome0such Corollary16IfthenL.?!0with(0;0)2R. 2 ProofIfthenw2L()iw thepreviouslemma holdsiw holdsiw2l().?!0where0isanalstate,whichinturn?!0where0isanalstate,which by algebraicformalisms,bisimilarityhasseveralpleasingmathematicalproperties Apartfrombeingthefundamentalnotionofequivalenceforseveralprocess 2 notsharedbyotherequivalencessuchaslanguageequivalence.furthermoreas overtheclassofnormeddeterministicprocesses. Lemma17Forstatesandofanormeddeterministictransitionsystem, givenbythefollowinglemma,languageequivalenceandbisimilaritycoincide iflthen.thus,takenalongwithcorollary16,landcoincide. statesofanormeddeterministictransitionsystemgisabisimulationrelation. ProofItsucestodemonstratethattherelationf(;):Land;are 18 2

19 ulationequivalencewheninvestigatingdecidabilityresultsforlanguageequiv- alencefordeterministiclanguagegenerators.weshallseeexamplesofthisin Section2. Henceitissensibletoconcentrateonthemoremathematicallytractablebisim- Thehierarchyfromabovegivesusthefollowingclassicationofprocesses. 1.3ExpressivityResults BPA (h) PDA (?) FSA (e) (?) (i) (f) BPP (g) (j) MSA PN Inthissectionwedemonstratethestrictnessofthishierarchybyprovidingexampletransitionsystemswhichliepreciselyinthegaps(a)-(j)indicatedinthe (a) (b) (c) (d) alsolanguage)equivalence.theseresultscomplementthosepresentedforthe infactdomorethanthisbygivingexamplesofnormeddeterministictransitionsystemswhichseparatealloftheseclassesuptobisimulation(andhence classication.(weleaveopenthequestionregardingthenaltwogaps.)we taxonomydescribedbyburkart,caucalandsteen[24]. (b)thetype2rewritesystemwiththetworulesaa (a)therstautomatoninexample0providesanormeddeterministicfsa. risetothesametransitionsystemregardlessofwhetherthesystemis sequentialorparallel;thisisanimmediateconsequenceofthefactthat itinvolvesonlyasinglevariablea.thistransitionsystemisdepictedas?!aaandab?!"gives follows. " braab-aaab -AAAab - Thisisanexampleofanormeddeterministictransitionsystemwhichis (c)examples2and3provideatransitionsystemwhichcanbedescribedby bothabpp(example2)andapda(example3).however,itcannot bothabpaandabppbutnotanfsa. bedescribeduptobisimilaritybyanybpa.toseethis,supposethat 19

20 andletmbeatleastaslargeasthenormofanyofitsvariables.thenthe wehaveabpawhichrepresentsthistransitionsystemuptobisimilarity, transitionsmustleadtothebpastate,whilethetransitionsystemin BPAstatecorrespondingtoXBminExample2mustbeoftheformA wherea2vand2v+.butthenanysequenceofn(a)norm-reducing (d)thefollowingbppwithinitialstatex Bkwherek=n(). Example2hastwosuchnon-bisimilarderivedstates,namelyXBk-1and isnotlanguageequivalenttoanypda,asitslanguageiseasilyconrmed nottobecontextfree.(thewordsinthislanguageoftheformacbde Xa?!XB Xc?!XD Xe?!" Bb?!" Dd?!" (e)examples1and4provideatransitionsystemwhichcanbedescribedby areexactlythoseoftheformakcnbkdne,whichisclearlynotacontextfreelanguage.) bothabpa(example1)andamsa(example4).however,thecontextfreelanguagewhichitgenerates,fancbn:n0g,cannotbegenerated byanybpp,sothistransitionsystemisnotevenlanguageequivalentto BPPstateX.(Astheprocesshasnorm1,thestatemustconsistofa anybpp.toseethis,supposethatl(x)=fancbn:n0gforsome singlevariablex.)letkbeatleastaslargeasthenormofanyofthe nite-normedvariablesofthisbpp,andconsideratransitionsequence acceptingthewordakcbk: wherethec-transitionisgeneratedbythetransitionruleyc haven(y)=k+1>n(y),so6=";hencebi X?!Yc ak?!bk?!" i>0.thuswehavexak?!"andbk-i?!.wemust?!ybi?!yc?!bk-i?!"?!"forsome (f)thefollowingpdawithinitialstatepx fromwhichwegetthedesiredcontradiction:akbicbk-i2l(x)forsome i>0. coincideswiththemsawhichitdenes,sincethereisonlyonestack symbol.thistransitionsystemisdepictedasfollows. pxa?!pxx pxb?!q pxb?!r qxc?!q rxd?!r 20

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