Abstract.Weproposetimed(nite)automatatomodelthebehaviorofrealtimesystemsovertime.Ourdenitionprovidesasimple,andyetpowerful,wayto

Transcription

1 ATheoryofTimedAutomata1 Abstract.Weproposetimed(nite)automatatomodelthebehaviorofrealtimesystemsovertime.Ourdenitionprovidesasimple,andyetpowerful,wayto ComputercienceDepartment,tanfordUniversity RajeevAlur2 tanford,ca DavidL.Dill3 annotatestate-transitiongraphswithtimingconstraintsusingnitelymanyrealvaluedclocks.atimedautomatonacceptstimedwords innitesequencesin intersection,butnotundercomplementation,whereasdeterministictimedmuller whichareal-valuedtimeofoccurrenceisassociatedwitheachsymbol.westudy ditions.weshowthatnondeterministictimedautomataareclosedunderunionand properties,decisionproblems,andsubclasses.weconsiderbothnondeterministic automataareclosedunderallbooleanoperations.themainconstructionofthe timedautomatafromtheperspectiveofformallanguagetheory:weconsiderclosure anddeterministictransitionstructures,andbothbuchiandmulleracceptancecon- Keywords:Real-timesystems,automaticverication,formallanguagesandautomatatheory. (nondeterministic)timedautomaton.wealsoprovethattheuniversalityproblem toautomaticvericationofreal-timerequirementsofnite-statesystems. paperisan(ppace)algorithmforcheckingtheemptinessofthelanguageofa completeinthedeterministiccase.finally,wediscusstheapplicationofthistheory andthelanguageinclusionproblemaresolvableonlyforthedeterministicautomata: bothproblemsareundecidable(1-hard)inthenondeterministiccaseandppace- 1PreliminaryversionsofthispaperappearintheProceedingsofthe17thInternationalColloquiumon necessarilyreectthepositionorthepolicyoftheu..government,andnoocialendorsementofthis workshouldbeinferred. theoryinpractice"(1991) Navy,OceoftheChiefofNavalResearchundergrantN J-1901.Thispublicationdoesnot Automata,Languages,andProgramming(1990),andintheProceedingsoftheREXworkshop\Real-time: 2Currentaddress:AT&TBellLaboratories,600MountainAvenue,Room2D-144,MurrayHill,NJ 3upportedbytheNationalcienceFoundationundergrantMIP ,andbytheUnitedtates

2 1Introduction Modallogicsand!-automataforqualitativetemporalreasoningaboutconcurrentsystems ofsystems.whenthesystemsarenite-state,asmanyare,wecanuseniteautomata, Theseformalismsabstractawayfromtime,retainingonlythesequencingofevents.In thelineartimemodel,itisassumedthatanexecutioncanbecompletelymodeledasa ofthesystemisasetofsuchexecutionsequences.inceasetofsequencesisaformal sequenceofstatesorsystemevents,calledanexecutiontrace(orjusttrace).thebehavior havebeenstudiedingreatdetail(selectedreferences:[36,32,16,28,47,44,37,11]). language,thisleadsnaturallytotheuseofautomataforthespecicationandverication!-regularexpressions,modalformulasof(extended)temporallogic,andsecond-orderformulasofthemonadictheoryofonesuccessor(1) havethesameexpressiveness,aningandanalyzingsystembehavior.theuniversalacceptanceofniteautomataasthe leadingtoeectiveconstructionsanddecisionproceduresforautomaticallymanipulat- nondeterministicbuchiautomata,deterministicandnondeterministicmullerautomata, modelandtheappealofitstheory.inparticular,avarietyofcompetingformalisms canonicalmodelofnite-statecomputationcanbeattributedtotherobustnessofthe vantages,itisultimatelycounterproductivewhenreasoningaboutsystemsthatmust interactwithphysicalprocesses;thecorrectfunctioningofthecontrolsystemofairplanes denetheclassof!-regularlanguages[7,9,33,46,42].consequentlymanyverication andtoastersdependscruciallyuponreal-timeconsiderations.wewouldliketobeableto specifyandverifymodelsofreal-timesystemsaseasilyasqualitativemodels.ourgoal istomodifyniteautomataforthistaskanddevelopatheoryoftimedniteautomata, theoriesarebasedonthetheoryof!-regularlanguages. similarinspirittothetheoryof!-regularlanguages.webelievethatthisshouldbethe Althoughthedecisiontoabstractawayfromquantitativetimehashadmanyad- events,notstates(thetheorywithstate-basedmodelsdiersonlyindetails).withinthis framework,itispossibletoaddtimingtoanexecutiontracebypairingitwithasequence time? i'thevent.atthispoint,however,afundamentalquestionarises:whatisthenatureof rststepinbuildingtheoriesforthereal-timevericationproblem. oftimes,wherethei'thelementofthetimesequencegivesthetimeofoccurrenceofthe Forsimplicity,wediscussmodelsthatconsiderexecutionstobeinnitesequencesof kindsofsynchronousdigitalcircuits,wheresignalchangesareconsideredtohavechanged Modelingtime Onealternative,whichleadstothediscrete-timemodel,requiresthetimesequenceto beamonotonicallyincreasingsequenceofintegers.thismodelisappropriateforcertain behaviorscanbemanipulatedusingordinaryniteautomata.ofcourse,inphysical silenteventasmanytimesasnecessarybetweeneventsintheoriginaltrace.oncethis transformationhasbeenperformed,thetimeofeacheventisthesameasitsposition, intoatracewherethetimesincreasebyexactlyoneateachstep,byinsertingaspecial exactlywhenaclocksignalarrives.oneoftheadvantagesofthismodelisthatitcanbe transformedeasilyintoanordinaryformallanguage.eachtimedtracecanbeexpanded sothetimesequencecanbediscarded,leavinganordinarystring.hence,discretetime 1

3 requiresthatcontinuoustimebeapproximatedbychoosingsomexedquantumapriori, processeseventsdonotalwayshappenatinteger-valuedtimes.thediscrete-timemodel whichlimitstheaccuracywithwhichphysicalsystemscanbemodeled. requiresthesequenceofintegertimestobenon-decreasing.theinterpretationofatimed executiontraceinthismodelisthateventsoccurinthespeciedorderatreal-valued arerecordedinthetrace.thismodelisalsoeasilytransformedintoaconventional formallanguage.first,addtothesetofeventsanewone,calledtick.theuntimed times,butonlythe(integer)readingsoftheactualtimeswithrespecttoadigitalclock tracecorrespondingtoatimedtracewillincludealloftheeventsfromthetimedtrace, Thectitious-clockmodelissimilartothediscretetimemodel,exceptthatitonly tomanipulatethesebehaviorsusingniteautomata,butthecompensatingdisadvantage isthatitrepresentstimeonlyinanapproximatesense. (i+1)'thevents(notethatthisnumbermaybe0).onceagain,itisconceptuallysimple inthesameorder,butwithti+1?tinumberofticksinsertedbetweenthei'thandthe withdensetimeinanite-automataframeworkismoredicultthantheothertwocases, naturalmodelforphysicalprocessesoperatingovercontinuoustime.inthismodel,the timesofeventsarerealnumbers,whichincreasemonotonicallywithoutbound.dealing timedautomatatosupportautomatedreasoningaboutsuchsystems. becauseitisnotobvioushowtotransformasetofdense-timetracesintoanordinary formallanguage.instead,wehavedevelopedatheoryoftimedformallanguagesand Wepreferadense-timemodel,inwhichtimeisadenseset,becauseitisamore elapsedsincethelastreset.thetransitionsoftheautomatonputcertainconstraintson automata.timedautomataaccepttimedwords innitesequencesinwhicharealvaluedtimeofoccurrenceisassociatedwitheachsymbol.atimedautomatonisanite Overview theclockvalues:atransitionmaybetakenonlyifthecurrentvaluesoftheclockssatisfy dentlyofeachother)withthetransitionsoftheautomaton,andkeeptrackofthetime automatonwithanitesetofreal-valuedclocks.theclockscanberesetto0(indepen- Toaugmentnite!-automatawithtimingconstraints,weproposetheformalismoftimed theassociatedconstraints.withthismechanismwecanmodeltimingpropertiessuch as\thechanneldeliverseverymessagewithin3to5timeunitsofitsreceipt".timed periodicity,boundedresponse,andtimingdelays. turessuchasliveness,fairness,andnondeterminism;andquantitativefeaturessuchas automatacancaptureseveralinterestingaspectsofreal-timesystems:qualitativefea- closurepropertiesforthedeterministicclassesaresimilartotheiruntimedcounterparts: bothdeterministicandnondeterministicvarieties,andforacceptancecriteriaweconsider bothbuchiandmullerconditions.weshowthatnondeterministictimedautomataare closedunderunionandintersection,butsurprisingly,notundercomplementation.the deterministictimedmullerautomataareclosedunderallbooleanoperations,whereas deterministictimedbuchiautomataareclosedunderonlythepositivebooleanoperations. Westudytimedautomatafromtheperspectiveofformallanguagetheory.Weconsider Theseresultsimplythat,unliketheuntimedcase,deterministictimedMullerautomata arestrictlylessexpressivethantheirnondeterministiccounterparts. Westudyavarietyofdecisionproblemsforthedierenttypesoftimedautomata.The 2

4 ofuntimedwordsconsistentwiththetimingconstraintsofatimedautomatonformsan valuedclockvariables,thestatespaceofatimedautomatonisinnite,andtheuntiming mainpositiveresultisanuntimingconstructionfortimedautomata.duetothereal-!-regularset.italsoleadstoappacedecisionprocedurefortestingemptinessofthe algorithmconstructsanitequotientofthisspace.thisisusedtoprovethattheset languageofatimedautomaton.wealsoshowthatthedualproblemoftestingwhether asystemmodeledasaproductoftimedautomatasatisesitsspecicationgivenasa nite-statereal-timesystems.wegiveappacevericationalgorithmtotestwhether thedeterministicversions. languageinclusionproblem.however,boththeseproblemscanbesolvedinppacefor (1-hard)fornondeterministicautomata.Thisalsoimpliestheundecidabilityofthe atimedautomatonacceptsalltimedwords(i.e.,theuniversalityquestion)isundecidable deterministictimedmullerautomaton. Relatedwork Finally,weshowhowtoapplythetheoryoftimedautomatatoprovecorrectnessof havebeenproposedrecently,however,noattempthasbeenmadetodevelopatheory Dierentwaysofincorporatingtimingconstraintsinthequalitativemodelsofasystem Modecharts[25].Inatimedautomaton,unliketheseothermodels,aboundonthetime isbyassociatinglowerandupperboundswithtransitions.examplesoftheseinclude modelhavebeendeveloped. oftimedlanguagesandnoalgorithmsforcheckingreal-timepropertiesinthedense-time bydillthatemploystimers[13].amodelsimilartodill'swasindependentlyproposed transitions,canbedirectlyexpressed.ourmodelisbasedonanearliermodelproposed takentotraverseapathintheautomaton,notjustthetimeintervalbetweenthesuccessive timedpetrinets[38],timedtransitionsystems[35,21],timedi/oautomata[31],and Perhapsthemoststandardwayofintroducingtiminginformationinaprocessmodel transitionscanhappeninatimeintervalofunitlength.ouruntimingconstructiondoes andstudiedbylewis[30].hedenesstate-diagrams,andgivesawayoftranslatinga decidabilityandlowerboundresultspresentedherecarryovertohisformalismalso. notneedthelatterassumption,andhasabetterworst-casecomplexity.wenotethatthe everyedgeisannotatedwithamatrixofintervalsconstrainingvariousdelays.lewisalso developsanalgorithmforcheckingconsistencyofthetiminginformationforaspecial classofstate-diagrams;theonesforwhichthereexistsaconstantksuchthatatmostk circuitdescriptiontoastate-diagram.astate-diagramisanite-statemachinewhere anundecidabilityresult:in[5]itisshownthatthesatisabilityproblemforareal-time model. clocksemantics.inthecaseofthedense-timemodeltheonlypreviouslyknownresultis [6,24,26,35,17,5,20].Mostoftheselogicsemploythediscrete-timeorthectitious- extensionofthelinear-timetemporallogicptlisundecidable(1-hard)inthedense-time Therehavebeenafewattemptstoextendtemporallogicswithquantitativetime 3

5 Figure1:Buchiautomatonaccepting(a+b)a! a,b a given(nite)alphabet(see,forexample,[23]).asopposedtothis,an!-languageconsists 2!-automata a ofallinnitewordsover.!-automataprovideaniterepresentationforcertaintypes Inthissectionwewillbrieyreviewtherelevantaspectsofthetheoryof!-regularlan- ofinnitewords.thusan!-languageoveranitealphabetisasubsetof! theset Themorefamiliardenitionofaformallanguageisasasetofnitewordsoversome 0 1 inputwords.varioustypesof!-automatahavebeenstudiedintheliterature[7,33,9,42]. setofautomatonstates,0isasetofstartstates,andeisasetof edges.theautomatonstartsinaninitialstate,andifhs;s0;ai2ethentheautomaton automaton,butwiththeacceptanceconditionmodiedsuitablysoastohandleinnite of!-languages.an!-automatonisessentiallythesameasanondeterministicnite-state Wewillmainlyconsidertwotypesof!-automata:BuchiautomataandMullerautomata. canchangeitsstatefromstos0readingtheinputsymbola. AtransitiontableAisatupleh;;0;Ei,whereisaninputalphabet,isanite Foraword=12:::overthealphabet,wesaythat withanadditionalsetfofacceptingstates.arunrofaoveraword2!isan isarunofaover,provideds020,andhsi?1;si;ii2eforalli1.forsucharun, denitionofthetransitiontables.abuchiautomatonaisatransitiontableh;;0;ei thesetinf(r)consistsofthestatess2suchthats=siforinnitelymanyi0. Dierenttypesof!-automataaredenedbyaddinganacceptanceconditiontothe r:s01?!s12?!s23?! acceptingruniinf(r)\f6=;.inotherwords,arunrisacceptingisomestatefrom states0isthestartstateands1istheacceptingstate.everyacceptingrunofthe automatonhastheform Example2.1Considerthe2-stateautomatonofFigure1overthealphabetfa;bg.The thesetfrepeatsinnitelyoftenalongr.thelanguagel(a)acceptedbyaconsistsof thewords2!suchthatahasanacceptingrunover. withi2fa;bgfor1inforsomen1.theautomatonacceptsallwordswith onlyanitenumberofb's;thatis,thelanguagel0=(a+b)a!. r:s01?!s02?!n?!s0a 4?!s1a?!s1a?!

6 Figure2:DeterministicMullerautomatonaccepting(a+b)a! b a areknownconstructionsforcomplementingbuchiautomata[41,40]. thelanguagel0ofexample2.1isan!-regularlanguage. intersectionisimplementedbyaproductconstructionforbuchiautomata[9,47].there a vericationproblemreducestothatoflanguageinclusion.theinclusionproblemfor WhenBuchiautomataareusedformodelingnite-stateconcurrentprocesses,the An!-languageiscalled!-regulariitisacceptedbysomeBuchiautomaton.Thus Theclassof!-regularlanguagesisclosedunderalltheBooleanoperations.Language 0 1 containedintheother,wecheckforemptinessoftheintersectionoftherstautomaton!-regularlanguagesisdecidable.totestwhetherthelanguageofoneautomatonis b foracyclethatisreachablefromastartstateandincludesatleastoneacceptingstate. languageofadeterministicautomatoncanbedoneinpolynomialtime[27]. withthecomplementofthesecond.testingforemptinessiseasy;weonlyneedtosearch thatis,j0j=1,and(ii)thenumberofa-labelededgesstartingatsisatmostone [41].However,checkingwhetherthelanguageofoneautomatoniscontainedinthe forallstatess2andforallsymbolsa2.thus,foradeterministictransition Ingeneral,complementingaBuchiautomatoninvolvesanexponentialblow-upinthe numberofstates,andthelanguageinclusionproblemisknowntobeppace-complete table,thecurrentstateandthenextinputsymboldeterminethenextstateuniquely. Consequently,adeterministicautomatonhasatmostonerunoveragivenword.Unlike theautomataonnitewords,theclassoflanguagesacceptedbydeterministicbuchi automataisstrictlysmallerthantheclassof!-regularlanguages.forinstance,thereis AtransitiontableA=h;;0;Eiisdeterministici(i)thereisasinglestartstate, automata(denedbelow)avoidthisproblematthecostofamorepowerfulacceptance condition. nodeterministicbuchiautomatonwhichacceptsthelanguagel0ofexample2.1.muller Buchiautomata,andalsoequalsthatacceptedbydeterministicMullerautomata. F2.ArunrofAoveraword2!isanacceptingruniinf(r)2F.Thatis,a runrisacceptingithesetofstatesrepeatinginnitelyoftenalongrequalssomesetin F.ThelanguageacceptedbyAisdenedasincaseofBuchiautomata. TheclassoflanguagesacceptedbyMullerautomataisthesameasthatacceptedby AMullerautomatonAisatransitiontableh;;0;Eiwithanacceptancefamily consistingofallwordsoverfa;bgwithonlyanitenumberofb's.themulleracceptance Example2.2ThedeterministicMullerautomatonofFigure2acceptsthelanguageL0 familyisffs1gg.thuseveryacceptingruncanvisitthestates0onlynitelyoften. 5

7 languages:theyareasexpressiveastheirnondeterministiccounterpart,andtheycanbe complementedinpolynomialtime.algorithmsforconstructingtheintersectionoftwo Mullerautomataandforcheckinglanguageinclusionareknown[10]. 3Timedautomata ThusdeterministicMullerautomataformastrongcandidateforrepresenting!-regular aword.thenweaugmentthedenitionof!-automatasothattheyaccepttimedwords, Inthissectionwedenetimedwordsbycouplingareal-valuedtimewitheachsymbolin andusethemtodevelopatheoryoftimedregularlanguagesanalogoustothetheoryof!-regularlanguages. Denition3.1Atimesequence=12isaninnitesequenceoftimevaluesi2R Wedenetimedwordssothatabehaviorofareal-timesystemcorrespondstoatimed nonnegativerealnumbers,r,ischosenasthetimedomain.awordiscoupledwitha wordoverthealphabetofevents.asinthecaseofthedense-timemodel,thesetof 3.1Timedlanguages withi>0,satisfyingthefollowingconstraints: timesequenceasdenedbelow: overandisatimesequence.atimedlanguageoverisasetoftimedwordsover. 1.Monotonicity:increasesstrictlymonotonically;thatis,i<i+1foralli1. 2.Progress:Foreveryt2R,thereissomei1suchthati>t. Atimedwordoveranalphabetisapair(;)where=12:::isaninniteword manyconsecutiveeventsinthesequence.toaccommodatethispossibilityonecoulduse correspondingcomponentiisinterpretedasthetimeofoccurrenceofi.undercertain iattimei.ifeachsymboliisinterpretedtodenoteaneventoccurrencethenthe circumstancesitmaybeappropriatetoallowthesametimevaluetobeassociatedwith aslightlydierentdenitionoftimedwordsbyrequiringatimesequencetoincreaseonly monotonically(i.e.,requireii+1foralli1).allourresultscontinuetoholdinthis Ifatimedword(;)isviewedasaninputtoanautomaton,itpresentsthesymbol Example3.2Letthealphabetbefa;bg.DeneatimedlanguageL1toconsistofall alternativemodelalso. timedwords(;)suchthatthereisnobaftertime5:6.thusthelanguagel1isgiven byletusconsidersomeexamplesoftimedlanguages. increasing.thelanguagel2isgivenas nate,andforthesuccessivepairsofaandb,thetimedierencebetweenaandbkeeps AnotherexampleisthelanguageL2consistingoftimedwordsinwhichaandbalter- L2=f((ab)!;)j8i:((2i?2i?1)<(2i+2?2i+1))g: L1=f(;)j8i:((i>5:6)!(i=a))g: 6

8 denedfortimedlanguagesasusual.inadditionwedenetheuntimeoperationwhich Thelanguage-theoreticoperationssuchasintersection,union,complementationare Figure3:Exampleofatimedtransitiontable a, x:=0 0 1 Untime(L2)consistsofasingleword(ab)!. discardsthetimevaluesassociatedwiththesymbols,thatis,itconsiderstheprojection ofatimedtrace(;)ontherstcomponent. of2!suchthat(;)2lforsometimesequence. Denition3.3ForatimedlanguageLover,Untime(L)isthe!-languageconsisting Forinstance,referringtoExample3.2,Untime(L1)isthe!-language(a+b)a!,and b, (x<2)? 3.2Transitiontableswithtimingconstraints Nowweextendtransitiontablestotimedtransitiontablessothattheycanreadtimed upontheinputsymbolread.incaseofatimedtransitiontable,wewantthischoiceto words.whenanautomatonmakesastate-transition,thechoiceofthenextstatedepends dependalsouponthetimeoftheinputsymbolrelativetothetimesofthepreviously Example3.4ConsiderthetimedtransitiontableofFigure3.Thestartstateiss0. thetimedtransitiontablesformally,letusconsidersomeexamples. betakenonlyifthecurrentvaluesoftheclockssatisfythisconstraint.beforewedene Witheachtransitionweassociateaclockconstraint,andrequirethatthetransitionmay instant,thereadingofaclockequalsthetimeelapsedsincethelasttimeitwasreset. transitiontable.aclockcanbesettozerosimultaneouslywithanytransition.atany readsymbols.forthispurpose,weassociateanitesetof(real-valued)clockswitheach fromstates1tos0isenabledonlyifthisvalueislessthan2.thewholecyclerepeats theactionofresettingtheclockxwhentheedgeistraversed.imilarlyanannotationof Thereisasingleclockx.Anannotationoftheformx:=0onanedgecorrespondsto theform(x<2)?onanedgegivestheclockconstraintassociatedwiththeedge. moreformally,thelanguageis clockxshowsthetimeelapsedsincetheoccurrenceofthelastasymbol.thetransition whentheautomatonmovesbacktostates0.thusthetimingconstraintexpressedby thistransitiontableisthatthedelaybetweenaandthefollowingbisalwayslessthan2; Theclockxgetssetto0alongwiththistransition.Whileinstates1,thevalueofthe Theautomatonstartsinstates0,andmovestostates1readingtheinputsymbola. f((ab)!;)j8i:(2i<2i?1+2)g: 7

9 clocktoberesetone1,andassociateanappropriateclockconstraintwithe2.notethat clockscanbesetasynchronouslyofeachother.thismeansthatdierentclockscan d, (y>2)? example. Thustoconstrainthedelaybetweentwotransitionse1ande2,werequireaparticular a b c 0 1 Example3.5ThetimedtransitiontableofFigure4usestwoclocksxandy,andaccepts berestartedatdierenttimes,andthereisnolowerboundonthedierencebetween theirreadings.havingmultipleclocksallowsmultipleconcurrentdelays,asinthenext 2 3 x:=0 y:=0 (x<1)? thelanguage itsvaluewhilereadingd,ensuresthatthedelaybetweenbandthefollowingdisalways 0eachtimeitmovesfroms0tos1readinga.Thecheck(x<1)?associatedwiththe c-transitionfroms2tos3ensuresthatchappenswithintime1oftheprecedinga.a similarmechanismofresettinganotherindependentclockywhilereadingbandchecking Theautomatoncyclesamongthestatess0,s1,s2ands3.Theclockxgetssetto L3=f((abcd)!;)j8j:((4j+3<4j+1+1)^(4j+4>4j+2+2))g: multipleclockswhichcanbesetindependentlyofeachother.theabovelanguagel3is banddtheautomatondoesnotputanyexplicitboundsonthetimedierencebetween aandthefollowingb,orcandthefollowingd.thisisanimportantadvantageofhaving theintersectionofthetwolanguagesl13andl23denedas greaterthan2. Noticethatintheaboveexample,toconstrainthedelaybetweenaandcandbetween clock;howevertoexpresstheirintersectionweneedtwoclocks. ofdierentcomponentsinadistributedsystem.alltheclocksincreaseattheuniform EachofthelanguagesL13andL23canbeexpressedbyanautomatonwhichusesjustone Weremarkthattheclocksoftheautomatondonotcorrespondtothelocalclocks L13=f((abcd)!;)j8j:(4j+3<4j+1+1)g; ratecountingtimewithrespecttoaxedglobaltimeframe.theyarectitiousclocks L23=f((abcd)!;)j8j:(4j+4>4j+2+2)g: andcheckedindependentlyofoneanother,butallstop-watchesrefertothesameclock. theautomatontobeequippedwithanitenumberofstop-watcheswhichcanbestarted inventedtoexpressthetimingpropertiesofthesystem.alternatively,wecanconsider 8 Figure4:Timedtransitiontablewith2clocks

10 timeconstant.weallowonlythebooleancombinationsofsuchsimpleconstraints.any valuefromq,thesetofnonnegativerationals,canbeusedasatimeconstant.later,in 3.3Clockconstraintsandclockinterpretations Denition3.6ForasetXofclockvariables,theset(X)ofclockconstraintsis Todenetimedautomataformally,weneedtosaywhattypeofclockconstraintsare denedinductivelyby additionofclockvalues,leadstoundecidability. allowedontheedges.thesimplestformofaconstraintcomparesaclockvaluewitha wherexisaclockinxandcisaconstantinq. ection5.5,wewillshowthatallowingmorecomplexconstraints,suchasthoseinvolving is,itisamappingfromxtor.wesaythataclockinterpretationforxsatisesa tions. Observethatconstraintssuchastrue,(x=c),x2[2;5)canbedenedasabbrevia- AclockinterpretationforasetXofclocksassignsarealvaluetoeachclock;that :=xcjcxj:j1^2; andagreeswithovertherestoftheclocks. 3.4Timedtransitiontables clockconstraintoverxievaluatestotrueusingthevaluesgivenby. (x)+t,andtheclockinterpretationtassignstoeachclockxthevaluet(x).for YX,[Y7!t]denotestheclockinterpretationforXwhichassignsttoeachx2Y, Fort2R,+tdenotestheclockinterpretationwhichmapseveryclockxtothevalue Denition3.7AtimedtransitiontableAisatupleh;;0;C;Ei,where Nowwegivetheprecisedenitionoftimedtransitiontables. E2C(C)givesthesetoftransitions.Anedgehs;s0;a;;i Cisanitesetofclocks,and 0isasetofstartstates, isanitealphabet, isanitesetofstates, attime0withallitsclocksinitializedto0.astimeadvances,thevaluesofallclocks Givenatimedword(;),thetimedtransitiontableAstartsinoneofitsstartstates constraintoverc. Cgivestheclockstoberesetwiththistransition,andisaclock representsatransitionfromstatestostates0oninputsymbola.theset change,reectingtheelapsedtime.attimei,achangesstatefromstos0usingsome transitionoftheformhs;s0;i;;ireadingtheinputi,ifthecurrentvaluesofclocks withrespecttothetimeofoccurrenceofthistransition.thisbehavioriscapturedby satisfy.withthistransitiontheclocksinareresetto0,andthusstartcountingtime clocksatthetransitionpoints.foratimesequence=12:::wedene0=0. deningrunsoftimedtransitiontables.arunrecordsthestateandthevaluesofallthe 9

11 withsi2andi2[c!r],foralli0,satisfyingthefollowingrequirements: Denition3.8Arunr,denotedby(s;),ofatimedtransitiontableh;;0;C;Eiover atimedword(;)isaninnitesequenceoftheform Initiation:s020,and0(x)=0forallx2C. Consecution:foralli1,thereisanedgeinEoftheformhsi?1;si;i;i;iisuch that(i?1+i?i?1)satisesiandiequals[i7!0](i?1+i?i?1). r:hs0;0i1?!1hs1;1i2?!2hs2;2i3?!3 Example3.9ConsiderthetimedtransitiontableofExample3.5.Consideratimed listingthevalues[x;y]. Belowwegivetheinitialsegmentoftherun.Aclockinterpretationisrepresentedby word Thesetinf(r)consistsofthosestatess2suchthats=siforinnitelymanyi0. hs0;[0;0]ia?!2hs1;[0;2]ib (a;2)!(b;2:7)!(c;2:8)!(d;5)! i+1aregivenbytheinterpretation(i+t?i).whenthetransitionfromstatesitosi+1 occurs,weusethevalue(i+i+1?i)tochecktheclockconstraint;however,attime Alongarunr=(s;)over(;),thevaluesoftheclocksattimetbetweeniand?! 2:7hs2;[0:7;0]ic?! 2:8hs3;[0:8;0:1]id?!5hs0;[3;2:3]i i+1,thevalueofaclockthatgetsresetisdenedtobe0. Wecancoupleacceptancecriteriawithtimedtransitiontables,andusethemtodene tablea0.wechoosethesetofclockstobetheemptyset,andreplaceeveryedgehs;s0;ai byhs;s0;a;;;truei.therunsofa0areinanobviouscorrespondencewiththerunsofa. 3.5Timedregularlanguages NotethatatransitiontableA=h;;0;Eicanbeconsideredtobeatimedtransition whereh;;0;c;eiisatimedtransitiontable,andfisasetofacceptingstates. timedlanguages. Denition3.10AtimedBuchiautomaton(inshortTBA)isatupleh;;0;C;E;Fi, oftimedlanguagesacceptedbytbastimedregularlanguages. inf(r)\f6=;. f(;)jahasanacceptingrunover(;)g. InanalogywiththeclassoflanguagesacceptedbyBuchiautomata,wecalltheclass Arunr=(s;)ofaTBAoveratimedword(;)iscalledanacceptingruni ForaTBAA,thelanguageL(A)oftimedwordsitacceptsisdenedtobetheset 10

12 Denition3.11AtimedlanguageLisatimedregularlanguageiL=L(A)forsome Example3.12ThelanguageL3ofExample3.5isatimedregularlanguage.Thetimed transitiontableoffigure4iscoupledwiththeacceptancesetconsistingofallthestates. TBAA. Figure5:TimedBuchiautomatonacceptingLcrt b b,(x<2)? a, x:= requiresthatthetimedierencebetweenthesuccessivepairsofaandbformanincreasing Forevery!-regularlanguageLover,thetimedlanguagef(;)j2Lgisregular. AtypicalexampleofanonregulartimedlanguageisthelanguageL2ofExample3.2.It a a, x:=0 Example3.13TheautomatonofFigure5acceptsthetimedlanguageLcrtoverthe sequence. alphabetfa;bg.lcrt=f((ab)!;)j9i:8ji:(2j<2j?1+2)g: timingconstraintstospecifyaninterestingconvergentresponseproperty: TheautomatonofExample3.13combinestheBuchiacceptanceconditionwiththe Anothernonregularlanguageisf(a!;)j8i:(i=2i)g. automatonstartsinstates0,andcyclesbetweenthestatess0ands1forawhile.then, nondeterministically,itmovestostates2settingitsclockxto0.whileinthecycle theresponsetimeis\eventually"alwayslessthan2timeunits. thatthenextbiswithin2timeunits.interpretingthesymbolbasaresponsetoarequest betweenthestatess2ands3,theautomatonresetsitsclockwhilereadinga,andensures denotedbythesymbola,theautomatonmodelsasystemwithaconvergentresponsetime; Thestartstateiss0,theacceptingstateiss2,andthereisasingleclockx.The phabetfa;bg. Example3.14TheautomatonofFigure6acceptsthefollowinglanguageovertheal- Thenextexampleshowsthattimedautomatacanspecifyperiodicbehavioralso. equals3thereisanasymbol.thusitexpressesthepropertythatahappensatalltime valuesthataremultiplesof3. regularintervalsofperiod3timeunits.theautomatonrequiresthatwhenevertheclock Theautomatonhasasinglestates0,andasingleclockx.Theclockgetsresetat f(;)j8i:9j:(j=3i^j=a)g 11

13 Figure6:Timedautomatonspecifyingperiodicbehavior a,b,(x<3)? intersection. Thenexttheoremconsiderssomeclosurepropertiesoftimedregularlanguages. Theorem3.15Theclassoftimedregularlanguagesisclosedunder(nite)unionand 3.6Propertiesoftimedregularlanguages lossofgeneralitythattheclocksetsciaredisjoint.weconstructtbasacceptingthe unionandintersectionofl(ai). Proof.ConsiderTBAsAi=h;i;i0;Ci;Ei;Fii,i=1;2;:::n.Assumewithout 0 a,(x=3)?,x:=0 i-thcomponentofthetuplekeepstrackofthestateofai,andthelastcomponentisused constructionforbuchiautomata[9].thesetofclocksfortheproductautomatonais thedisjointunionofalltheautomata. [ici.thestatesofaareoftheformhs1;:::sn;ki,whereeachsi2i,and1kn.the asacounterforcyclingthroughtheacceptingconditionsofalltheindividualautomata. incetbasarenondeterministicthecaseofunioniseasy.therequiredtbaissimply Initiallythecountervalueis1,anditisincrementedfromkto(k+1)(modulon)ithe currentstateofthek-thautomatonisanacceptingstate.notethatwechoosethevalue Intersectioncanbeimplementedbyatrivialmodicationofthestandardproduct ofnmodntoben. toberesetwiththistransitionis[ii,andtheassociatedclockconstraintis^ii. pereachautomaton,withthesamelabela.correspondingtothisset,thereisajoint ishs01;:::s0n;jiwithj=(k+1)modnifsk2fk,andj=kotherwise.thesetofclocks havingthesamelabel.letfhsi;s0i;a;i;ii2eiji=1;:::ngbeasetoftransitions,one transitionofaoutofeachstateoftheformhs1;:::sn;kilabeledwitha.thenewstate Ai.AtransitionofAisobtainedbycouplingthetransitionsoftheindividualautomata TheinitialstatesofAareoftheformhs1;:::sn;1iwhereeachsiisastartstateof ceptingconditionsofalltheautomataaremet.consequently,wedenetheacceptingset foratoconsistofstatesoftheformhs1;:::sn;ni,wheresn2fn. isnijij.thenumberofclocksisijcij,andthesizeoftheedgesetisnijeij.note thatjejincludesthelengthoftheclockconstraintsassumingbinaryencodingforthe Thecountervaluecyclesthroughthewholerange1;:::ninnitelyoftenitheac- constants. Intheaboveproductconstruction,thenumberofstatesoftheresultingautomaton 12

14 Example3.16ThelanguageacceptedbytheautomatoninFigure7is other.considerthefollowingexample. inaniteintervaloftime.furthermore,thesymbolscanbearbitrarilyclosetoeach Observethatevenforthetimedregularlanguagesarbitrarilymanysymbolscanoccur Figure7:TimedautomatonacceptingLconverge a,(x=1)?,x:=0 a,x:=0 b (x=1)? y:=0 Everywordacceptedbythisautomatonhasthepropertythatthesequenceoftime Lconverge=f((ab)!;)j8i:(2i?1=i^(2i?2i?1>2i+2?2i+1))g: b,(y<1)?,y:=0 bytheautomatonis dierencesbetweenaandthefollowingbisstrictlydecreasing.asamplewordaccepted timemodel.ifwerequireallthetimevaluesitobemultiplesofsomexedconstant, howeversmall,thelanguageacceptedbytheautomatonoffigure7willbeempty. Thisexampleillustratesthatthemodelofrealsisindeeddierentfromthediscrete- (a;1)!(b;1:5)!(a;2)!(b;2:25)!(a;3)!(b;3:125)! staysunchanged. Theorem3.17LetLbeatimedregularlanguage.Foreveryword,2Untime(L)i thesetofrationalsq.onlythedensenessoftheunderlyingdomainplaysacrucialrole. Inparticular,Theorem3.17showsthatifwerequireallthetimevaluesintimesequences toberationalnumbers,theuntimedlanguageuntime[l(a)]ofatimedautomatona Ontheotherhand,timedautomatadonotdistinguishbetweenthesetofrealsRand thereexistsatimesequencesuchthati2qforalli1,and(;)2l. Otherwisechoose0i2Qsuchthatforall0j<i,foralln2N,(0 withallrationaltimevaluessuchthat(;)2l(a),thenclearly,2untime[l(a)]. everyconstantappearingintheclockconstraintsofaisanintegralmultipleof.let 0=0,and0=0.Ifi=j+nforsome0j<iandn2N,thenchoose0i=0j+n. Proof.ConsideratimedautomatonA,andaword.Ifthereexistsatimesequence Nowsupposeforanarbitrarytimesequence,(;)2L(A).Let2Qbesuchthat 0,ifaclockxisresetatthei-thtransitionpoint,thenitspossiblevaluesatthej-th possible. (i?j)<n.notethatbecauseofthedensenessofqsuchachoiceof0iisalways Consideranacceptingrunr=(s;)ofAover(;).Becauseoftheconstructionof 13 i?0j)<ni

15 0=0,andifthei-thtransitionalongrisaccordingtotheedgehsi?1;si;i;i;ii,then Figure8:TimedMullerautomaton a,(x<5)? a,(x<2)? 3.7TimedMullerautomata transitionpointalongthetwotimesequences,namely,(j?i)and(0j?0i),satisfythe r0=(s;0)over(;0)whichfollowsthesamesequenceofedgesasr.inparticular,choose samesetofclockconstraints.consequentlyitispossibletoconstructanacceptingrun WecandenetimedautomatawithMulleracceptanceconditionsalso. set0i=[i7!0](0i?1+0 i?0 i?1).consequently,aaccepts(;0). b,x:=0 c,x:=0 f(;)jahasanacceptingrunover(;)g. inf(r)2f. Denition3.18AtimedMullerautomaton(TMA)isatupleh;;0;C;E;Fi,where Example3.19ConsidertheautomatonofFigure8overthealphabetfa;b;cg.The h;;0;c;eiisatimedtransitiontable,andf2speciesanacceptancefamily. startstateiss0,andthemulleracceptancefamilyconsistsofasinglesetfs0;s2g.oany ForaTMAA,thelanguageL(A)oftimedwordsitacceptsisdenedtobetheset Arunr=(s;)oftheautomatonoveratimedword(;)isanacceptingruni acceptingrunshouldcyclebetweenstatess0ands1onlynitelymanytimes,andbetween theclassoftimedlanguagesacceptedbytmasisthesameastheclassoftimedregular power.thefollowingtheoremstatesthatthesameholdstruefortbasandtmas.thus than2ifthe(2i)-thsymbolisc,andlessthan5otherwise. statess0ands2innitelymanytimes.everyword(;)acceptedbytheautomaton languages.theproofofthefollowingtheoremcloselyfollowsthestandardargumentthat satises:(1)2(a(b+c))(ac)!,and(2)foralli1,thedierence(2i?1?2i?2)isless an!-regularlanguageisacceptedbyabuchiautomatoniitisacceptedbysomemuller automaton. Theorem3.20AtimedlanguageisacceptedbysometimedBuchiautomatoniitis RecallthatuntimedBuchiautomataandMullerautomatahavethesameexpressive timedtransitiontableasthatofa,andwiththeacceptancefamilyf=f0:0\f6= acceptedbysometimedmullerautomaton. ;g.itiseasytocheckthatl(a)=l(a0).thisprovesthe\onlyif"partoftheclaim. Proof.LetA=h;;0;C;E;FibeaTBA.ConsidertheTMAA0withthesame 14

16 AF=h;;0;C;E;fFgi,soitsucestoconstruct,foreachacceptancesetF,aTBA languageusingthesimulationofmulleracceptanceconditionbybuchiautomata.let A0FwhichacceptsthelanguageL(AF).AssumeF=fs1;:::skg.TheautomatonA0F tomakesurethateverystateinfisvisitedinnitelyoften.tatesofa0fareofthe AbeaTMAgivenash;;0;C;E;Fi.FirstnotethatL(A)=[F2FL(AF)where usesnondeterminismtoguesswhenthesetfisenteredforever,andthenusesacounter Intheotherdirection,givenaTMA,wecanconstructaTBAacceptingthesame formhs;ii,wheres2andi2f0;1;:::kg.thesetofinitialstatesis0f0g.the automatonsimulatesthetransitionsofa,andatsomepointnondeterministicallysets hasatransitionhhs;0i;hs0;0i;a;;i,and,inaddition,ifs02fitalsohasatransition thesecondcomponentto1.foreverytransitionhs;s0;a;;iofa,theautomatona0f hhs;0i;hs0;1i;a;;i. 4Checkingemptiness j=i.theonlyacceptingstateishsk;ki. setf.foreverya-transitionhs;s0;a;;iwithbothsands0inf,foreach1ik, thereisana0f-transitionhhs;ii;hs0;ji;a;;iwherej=(i+1)modk,ifsequalssi,else Whilethesecondcomponentisnonzero,theautomatonisrequiredtostaywithinthe timedautomaton.theexistenceofaninniteacceptingpathintheunderlyingtransition tableisclearlyanecessaryconditionforthelanguageofanautomatontobenonempty. However,thetimingconstraintsoftheautomatonruleoutcertainadditionalbehaviors. Inthissectionwedevelopanalgorithmforcheckingtheemptinessofthelanguageofa parisonswithrationalconstants.thefollowinglemmashowsthat,forcheckingemptiness, Recallthatourdenitionoftimedautomataallowsclockconstraintswhichinvolvecom- 4.1Restrictiontointegerconstants untimedwordsthatareconsistentwiththetimedwordsacceptedbyatimedautomaton. WewillshowthataBuchiautomatoncanbeconstructedthatacceptsexactlythesetof bymultiplyingallibyt. Lemma4.1ConsideratimedtransitiontableA,atimedword(;),andt2Q.(s;) constants.foratimedsequenceandt2q,lettdenotethetimedsequenceobtained wecanrestrictourselvestotimedautomatawhoseclockconstraintsinvolveonlyinteger transitiontableobtainedbyreplacingeachconstantdineachclockconstraintlabeling ttobetheleastcommonmultipleofdenominatorsofalltheconstantsappearinginthe theedgesofabytd. isarunofaover(;)i(s;t)isarunofatover(;t),whereatisthetimed clockconstraintsofa,thentheclockconstraintsforatuseonlyintegerconstants.inthis denominatorsofalltheoriginalconstants.weassumebinaryencodingfortheconstants. translation,thevaluesoftheindividualconstantsgrowatmostwiththeproductofthe ThusthereisanisomorphismbetweentherunsofAandtherunsofAt.Ifwechoose Proof.Thelemmacanbeprovedeasilyfromthedenitionsusinginduction. 15

17 weencodeconstantsinbinarynotation;ifweuseunaryencodingthenj(at)jcanbe j(at)jisboundedbyj(a)j2.observethatthisresultdependscruciallyonthefactthat exponentialinj(a)j. LetusdenotethelengthoftheclockconstraintsofAbyj(A)j.Itiseasytoprovethat L(A)weconsiderAt.AlsoUntime[L(A)]equalsUntime[L(At)].Intheremainderofthe sectionweassumethattheclockconstraintsuseonlyintegerconstants. 4.2Clockregions Ateverypointintimethefuturebehaviorofatimedtransitiontableisdeterminedby ObservethatL(A)isemptyiL[At]isempty.Hence,todecidetheemptinessof Denition4.2Foratimedtransitiontableh;;0;C;Ei,anextendedstateisapair itsstateandthevaluesofallitsclocks.thismotivatesthefollowingdenition: alsoontheorderingofthefractionalpartsofallclockvalues,thentherunsstartingfrom possiblybuildanautomatonwhosestatesaretheextendedstatesofa.butiftwo extendedstateswiththesamea-stateagreeontheintegralpartsofallclockvalues,and hs;iwheres2andisaclockinterpretationforc. thefractionalpartsisneededtodecidewhichclockwillchangeitsintegralpartrst.for thetwoextendedstatesareverysimilar.theintegralpartsoftheclockvaluesareneeded todeterminewhetherornotaparticularclockconstraintismet,whereastheorderingof incethenumberofsuchextendedstatesisinnite(infact,uncountable),wecannot withclockconstraint(x=1)canbefollowedbyatransitionwithclockconstraint(y=1), example,iftwoclocksxandyarebetween0and1inanextendedstate,thenatransition consequenceindecidingtheallowedpaths. comparedwithaconstantgreaterthanc,thenitsactualvalue,onceitexceedsc,isofno dependingonwhetherornotthecurrentclockvaluessatisfy(x<y). Theintegralpartsofclockvaluescangetarbitrarilylarge.Butifaclockxisnever cxbethelargestintegercsuchthat(xc)or(cx)isasubformulaofsomeclock Denition4.3LetA=h;;0;C;Eibeatimedtransitiontable.Foreachx2C,let clockincappearsinsomeclockconstraint. constraintappearingine. andbtcdenotestheintegralpartoft;thatis,t=btc+fract(t).weassumethatevery Nowweformalizethisnotion.Foranyt2R,fract(t)denotesthefractionalpartoft, 0iallthefollowingconditionshold: 1.Forallx2C,eitherb(x)candb0(x)carethesame,orboth(x)and0(x)are TheequivalencerelationisdenedoverthesetofallclockinterpretationsforC; AclockregionforAisanequivalenceclassofclockinterpretationsinducedby. 2.Forallx;y2Cwith(x)cxand(y)cy,fract((x))fract((y))i 3.Forallx2Cwith(x)cx,fract((x))=0ifract(0(x))=0. fract(0(x))fract(0(y)). greaterthancx. 16

18 012 1y6???? -x6cornerpoints:e.g.[(0,1)] 14Openlinesegments:e.g.[0<x=y<1] uniquelycharacterizedbya(nite)setofclockconstraintsitsatises.forexample, Wewilluse[]todenotetheclockregiontowhichbelongs.Eachregioncanbe 8Openregions:e.g.[0<x<y<1] consideraclockinterpretationovertwoclockswith(x)=0:3and(y)=0:7.every clockinterpretationin[]satisestheconstraint(0<x<y<1),andwewillrepresent thisregionby[0<x<y<1].thenatureoftheequivalenceclassescanbebest Figure9:Clockregions cy=1.theclockregionsareshowninfigure9. Example4.4Consideratimedtransitiontablewithtwoclocksxandywithcx=2and understoodthroughanexample. clockconstraintievery2satises.eachregioncanberepresentedbyspecifying ofa,if0thensatisesi0satises.wesaythataclockregionsatisesa Notethatthereareonlyanitenumberofregions.Alsonotethatforaclockconstraint (1)foreveryclockx,oneclockconstraintfromtheset theupperboundinthefollowinglemma. Bycountingthenumberofpossiblecombinationsofequationsoftheaboveform,weget (2)foreverypairofclocksxandysuchthatc?1<x<candd?1<y<d appearin(1)forsomec;d,whetherfract(x)islessthan,equalto,or greaterthanfract(y). fx=cjc=0;1;:::cxg[fc?1<x<cjc=1;:::cxg[fx>cxg; sizeofthelargestconstantstheclocksarecomparedwith,thenthenumberofregions binaryencoding,andhencetheproductx2c(2cx+2)iso[2j(a)j].incethenumber O[2j(A)j].Notethatifweincrease(A)withoutincreasingthenumberofclocksorthe ofclocksjcjisboundedbyj(a)j,henceforth,weassumethatthenumberofregionsis Lemma4.5Thenumberofclockregionsisboundedby[jCj!2jCjx2C(2cx+2)]. doesnotgrowwithj(a)j.alsoobservethataregioncanberepresentedinspacelinear inj(a)j. Rememberthatj(A)jstandsforthelengthoftheclockconstraintsofAassuming 17

19 Therststepinthedecisionprocedureforcheckingemptinessistoconstructatransition tablewhosepathsmimictherunsofainacertainway.wewilldenotethedesired 4.3Theregionautomaton ofthetimedtransitiontablea,andtheequivalenceclassofthecurrentvaluesofthe transitiontablebyr(a),theregionautomatonofa.astateofr(a)recordsthestate clocks.itisoftheformhs;iwiths2andbeingaclockregion.theintended interpretationisthatwhenevertheextendedstateofaishs;i,thestateofr(a)is labeledwithaiainstateswiththeclockvalues2canmakeatransitiononato andtheclockinterpretation0assigns0toeveryclock.thetransitionrelationofr(a) hs;[]i.theregionautomatonstartsinsomestatehs0;[0]iwheres0isastartstateofa, theextendedstatehs0;0iforsome020. isdenedsothattheintendedsimulationisobeyed.ithasanedgefromhs;itohs0;0i Denition4.6Aclockregion0isatime-successorofaclockregioniforeach2, thereexistsapositivet2rsuchthat+t20. bevisitedbyaclockinterpretation2astimeprogresses. clockregions.thetime-successorsofaclockregionarealltheclockregionsthatwill Theedgerelationcanbeconvenientlydenedusingatime-successorrelationoverthe ofaregionaretheregionsthatcanbereachedbymovingalongalinedrawnfromsome pointininthediagonallyupwardsdirection(paralleltothelinex=y).forexample, theregion[(1<x<2);(0<y<x?1)]has,otherthanitself,thefollowingregionsas Example4.7ConsidertheclockregionsshowninFigure9again.Thetime-successors [(x>2);(y>1)]. time-successors:[(x=2);(0<y<1)],[(x>2);(0<y<1)],[(x>2);(y=1)]and Tocomputeallthetime-successorsofweproceedasfollows.Firstobservethatthe (d?1<y<d)appearin(1),theorderingrelationshipbetweenfract(x)andfract(y). or(c?1<x<c)or(x>cx),and(2)foreverypairxandysuchthat(c?1<x<c)and time-successorrelationisatransitiverelation.weconsiderdierentcases. clockregionisspeciedbygiving(1)foreveryclockx,aconstraintoftheform(x=c) Firstsupposethatsatisestheconstraint(x>cx)foreveryclockx.Theonly Nowletusseehowtoconstructallthetime-successorsofaclockregion.Recallthata time-successorofisitself.thisisthecasefortheregion[(x>2);(y>1)]infigure9. below:(1)forx2c0,ifsatises(x=cx)thensatises(x>cx),otherwiseif (x=c)forsomeccx,isnonempty.inthiscase,astimeprogressesthefractional time-successorsofaresameasthetime-successorsoftheclockregionspeciedas partsoftheclocksinc0becomenonzero,andtheclockregionchangesimmediately.the NowsupposethatthesetC0consistingofclocksxsuchthatsatisestheconstraint (2)Forclocksxandysuchthatx<cxandy<cyholdsin,theordering relationshipinbetweentheirfractionalpartsisthesameasin. satises(x=c)thensatises(c<x<c+1).forx62c0,theconstraint inisthesameasthatin. 18

20 Forinstance,inFigure9,thetime-successorsof[(x=0);(0<y<1)]aresameasthe time-successorsof[0<x<y<1]. thiscase,astimeprogresses,theclocksinc0assumeintegervalues.letbetheclock regionspeciedby clocksyforwhichdoesnotsatisfy(y>cy),fract(y)fract(x)isaconstraintof.in doesnotsatisfy(x>cx)andwhichhavethemaximalfractionalpart;thatis,forall Ifboththeabovecasesdonotapply,thenletC0bethesetofclocksxforwhich Inthiscase,thetime-successorsofinclude,,andallthetime-successorsof.For (1)Forx2C0,ifsatises(c?1<x<c)thensatises(x=c).For instance,infigure9,time-successorsof[0<x<y<1]includeitself,[(0<x<1);(y= (2)Forclocksxandysuchthat(c?1<x<c)and(d?1<y<d)appear asin. x62c0,theconstraintinissameasthatin. 1)],andallthetime-successorsof[(0<x<1);(y=1)]. Nowwearereadytodenetheregionautomaton. in(1),theorderingrelationshipinbetweentheirfractionalpartsissame Denition4.8ForatimedtransitiontableA=h;;0;C;Ei,thecorrespondingregionautomatonR(A)isatransitiontableoverthealphabet. ThestatesofR(A)areoftheformhs;iwheres2andisaclockregion. Theinitialstatesareoftheformhs0;[0]iwheres020and0(x)=0forallx2C. isfa;b;c;dg.everystateoftheautomatonisanacceptingstate.thecorresponding Example4.9ConsiderthetimedautomatonA0showninFigure10.Thealphabet R(A)hasanedgehhs;i;hs0;0i;aiithereisanedgehs;s0;a;;i2Eandaregion regionautomatonr(a0)isalsoshown.onlytheregionsreachablefromtheinitialregion 00suchthat(1)00isatime-successorof,(2)00satises,and(3)0=[7!0]00. automatonensurethatthetransitionfroms2tos3isnevertaken.theonlyreachable hs0;[x=y=0]iareshown.notethatcx=1andcy=1.thetimingconstraintsofthe regionwithstatecomponents2satisestheconstraints[y=1;x>1],andthisregionhas mostoneedgeoutofhs;iforeveryedgeoutofsandeverytime-successorof.itfollows iso[jj2j(a)j].aninspectionofthedenitionofthetime-successorrelationshowsthat everyregionhasatmostx2c[2cx+2]successorregions.theregionautomatonhasat canfollowab-transition. nooutgoingedges.thustheregionautomatonhelpsusinconcludingthatnotransitions thatthenumberofedgesinr(a)iso[jej2j(a)j].notethatcomputingthetime-successor relationiseasy,andcanbedoneintimelinearinthelengthoftherepresentationofthe region.constructingtheedgerelationfortheregionautomatonisalsorelativelyeasy;in Fromtheboundonthenumberofregions,itfollowsthatthenumberofstatesinR(A) additiontocomputingthetime-successors,wealsoneedtodeterminewhethertheclock 19

21 a 0 1 y:=0 2 b,(y=1)? c,(x<1)? c,(x<1)? a,(y<1)?,y:=0 3 d,(x>1)? 0 x=y=0 a a Figure10:AutomatonA0anditsregionautomaton a b b 1 0=y<x<1 1 y=0,x=1 1 y=0,x>1 b 2 1=y<x a c a a graphcanbeconstructedintimeo[(jj+jej)2j(a)j]. a d 3 d 3 3 d 3 d 0<y<x<1 0<y<1<x d 1=y<x x>1,y>1 Denition4.10Forarunr=(s;)ofAoftheform R(A). constraintlabelingaparticulara-transitionissatisedbyaclockregion.theregion NowweproceedtoestablishacorrespondencebetweentherunsofAandtherunsof d d r:hs0;0i1?!1hs1;1i2?!2hs2;2i3 d deneitsprojection[r]=(s;[])tobethesequence [r]:hs0;[0]i1?!hs1;[1]i2?!hs2;[2]i3?!3 20?!

22 over.incetimeprogresseswithoutboundalongr,everyclockx2ciseitherreset followingdenition: Denition4.11Arunr=(s;)oftheregionautomatonR(A)oftheform innitelyoften,orfromacertaintimeonwardsitincreaseswithoutbound.hence,for allx2c,forinnitelymanyi0,[i]satises[(x=0)_(x>cx)].thispromptsthe FromthedenitionoftheedgerelationforR(A),itfollowsthat[r]isarunofR(A) Lemma4.13impliesthatprogressiverunsofR(A)preciselycorrespondtotheprojected [(x=0)_(x>cx)]. isprogressiveiforeachclockx2c,thereareinnitelymanyi0suchthatisatises ThusforarunrofAover(;),[r]isaprogressiverunofR(A)over.Thefollowing r:hs0;0i1?!hs1;1i2?!hs2;2i3?! again. Example4.12ConsidertheregionautomatonR(A0)ofFigure10.Everyrunrof runsofa.beforeweprovethelemmaletusconsidertheregionautomatonofexample4.9 (ii),eventhoughthevalueofxisnotbounded,theclockyisresetonlynitelyoften, thoughygetsresetinnitelyoften,thevalueofxisalwayslessthan1.forrunsoftype theregionshs1;[y=0<x<1]iandhs3;[0<y<x<1]i,(ii)theautomatonstaysinthe hs3;[x>1;y>1]i. regionhs3;[0<y<1<x]iusingtheself-loop,or(iii)theautomatonstaysintheregion R(A0)hasasuxofoneofthefollowingthreeforms:(i)theautomatoncyclesbetween R(A0)oftype(iii). andyet,itsvalueisbounded.thuseveryprogressiverunofa0correspondstoarunof Lemma4.13IfrisaprogressiverunofR(A)overthenthereexistsatimesequence Onlythecase(iii)correspondstotheprogressiveruns.Forrunsoftype(i),even thattheextendedstateofaishsi;iiattimeiwithi2i.thereisanedgeinr(a) fromhsi;iitohsi+1;i+1ilabeledwithi+1.fromthedenitionoftheregionautomaton itfollowsthatthereisanedgehsi;si+1;i+1;i+1;i+1i2eandatime-successor0i+1of r0andthetimesequencestepbystep.asusual,r0startswithhs0;0i.nowsuppose andarunr0ofaover(;)suchthatrequals[r0]. isuchthat0i+1satisesi+1andi+1=[i+17!0]0i+1.fromthedenitionoftimesuccessor,thereexistsatimei+1suchthat(i+i+1?i)20i+1.nowitisclearthe Proof.Consideraprogressiverunr=(s;)ofR(A)over.Weconstructtherun Usingthisconstructionrepeatedlywegetarunr0=(s;)over(;)with[r0]=r. nexttransitionofacanbeattimei+1toanextendedstatehsi+1;i+1iwithi+12i+1. thattheautomatoncanfollowthesamesequenceoftransitionsasr0butattimes0i. runtoconstructanothertimesequence0satisfyingtheprogressrequirementandshow condition.upposethatisaconvergingsequence.weusethefactthatrisaprogressive sequence,afteracertainpositiononwards,everyclockinc0getsresetbeforeitreaches thevalue1.incerisprogressive,everyclockxnotinc0,afteracertainposition Theonlyproblemwiththeaboveconstructionisthatmaynotsatisfytheprogress LetC0bethesetofclocksresetinnitelyoftenalongr.inceisaconverging 21

23 onwards,nevergetsreset,andcontinuouslysatisesx>cx.thisensuresthatthere existsj0suchthat(1)afterthej-thtransitionpointeachclockx62c0continuously satises(x>cx),andeachclockx2c0continuouslysatises(x<1),and(2)foreach constructanothersequencer00=(s;0)withthesequenceoftransitiontimes0asfollows. Thesequenceoftransitionsalongr00issameasthatalongr0.Ifi62fk1;k2:::gthen isresetatleastoncebetweentheki-thandki+1-thtransitionpointsalongr.nowwe k>j,(k?j)islessthan0:5. werequirethe(i+1)-thtransitiontohappenafteradelayof(i+1?i),otherwisewe transitionpointsislessthan1.consequently,inspiteoftheadditionaldelays,thevalue requirethedelaytobe0:5.observethatalongr00thedelaybetweentheki-thandki+1-th Letj<k1<k2;:::beaninnitesequenceofintegerssuchthateachclockxinC0 andisarunofa.furthermore,[r00]=[r0]=r. (ascomparedtor0).fromthisweconcludethatr00satisestheconsecutionrequirement, alltheclockconstraintsandtheclockregionsatthetransitionpointsremainunchanged ofeveryclockinc0remainslessthan1afterthej-thtransitionpoint.othetruthof ForatimedautomatonA,itsregionautomatoncanbeusedtorecognizeUntime[L(A)]. ThefollowingtheoremisstatedforTBAs,butitalsoholdsforTMAs. 4.4Theuntimingconstruction quirement.hencer00istherunrequiredbythelemma. ince0hasinnitelymanyjumpseachofduration0:5,itsatisestheprogressre- theregionautomatoncorrespondingtothetimedtransitiontableh;;0;c;ei.the Theorem4.14GivenaTBAA=h;;0;C;E;Fi,thereexistsaBuchiautomaton acceptingsetofa0isf0=fhs;ijs2fg.,thelemmagivesatimesequenceandarunr0ofaover(;)suchthatrequals[r0]. overwhichacceptsuntime[l(a)]. A0over.TheconversefollowsfromLemma4.13.GivenaprogressiverunrofA0over Ifrisanacceptingrun,soisr0.Itfollowsthat2Untime[L(A)]iA0hasaprogressive, IfrisanacceptingrunofAover(;),then[r]isaprogressiveandacceptingrunof Proof.WeconstructaBuchiautomatonA0asfollows.ItstransitiontableisR(A), acceptingrunoverit. Example4.15LetusconsidertheregionautomatonR(A0)ofExample4.9again.ince progressiveisomestatefromeachfxrepeatsinnitelyoften.itisstraightforwardto overia00hasanacceptingrunover. constructanotherbuchiautomatona00suchthata0hasaprogressiveandacceptingrun Forx2C,letFx=fhs;ijj=[(x=0)_(x>cx)]g.RecallthatarunofA0is itfollowsthatthetransitiontabler(a0)canbechangedtoabuchiautomatonbychoosingtheacceptingsettoconsistofasingleregionhs3;[x>1;y>1]i.consequently allstatesofa0areaccepting,fromthedescriptionoftheprogressiverunsinexample4.12 TheautomatonA00isthedesiredautomaton;L(A00)equalsUntime[L(A)]. Untime[L(A0)]=L[R(A0)]=ac(ac)d!: 22

24 incharacter;itsconsistencycanbecheckedbyanite-stateautomaton.anequivalent formulationofthetheoremis Theorem4.14saysthatthetiminginformationinatimedautomatonis\regular" theproofoftheorem4.14.thenexttheoremfollows. fortheemptinessofthelanguageofthecorrespondingbuchiautomatonconstructedby Theorem4.16GivenatimedBuchiautomatonA=h;;0;C;E;Fitheemptinessof Furthermore,tocheckwhetherthelanguageofagivenTBAisempty,wecancheck IfatimedlanguageListimedregularthenUntime(L)is!-regular. L(A)canbecheckedintimeO[(jj+jEj)2j(A)j]. boundofthetheoremfollows. ofthesetsfx.thiscanbecheckedintimelinearinthesizeofa0[41].thecomplexity fromsomestartstateofa0andccontainsatleastonestateeachfromthesetf0andeach orem4.14.recallthatinection4.3wehadshownthatthenumberofstatesina0is O[jj2j(A)j],thenumberofedgesisO[jEj2j(A)j]. Proof.LetA0betheBuchiautomatonconstructedasoutlinedintheproofofThe- constants,weneedtoapplytheabovedecisionprocedureonatfortheleastcommon RecallthatifwestartwithanautomatonAwhoseclockconstraintsinvolverational ThelanguageL(A)isnonemptyithereisacycleCinA0suchthatCisaccessible thesizeoftheclockconstraints;wehave[at]=o[(a)2]. emptinessofl(a). denominatortofalltherationalconstants(seeection4.1).thisinvolvesablow-upin 4.5Complexityofcheckingemptiness amuller(or,buchi)automatonwhichacceptsuntime[l(a)],anduseittocheckforthe automata.inparticular,givenatimedmullerautomatonawecaneectivelyconstruct Theabovemethodcanbeusedevenifwechangetheacceptanceconditionfortimed ThecomplexityofthealgorithmfordecidingemptinessofaTBAisexponentialinthe numberofclocksandthelengthoftheconstantsinthetimingconstraints.thisblow-up incomplexityseemsunavoidable;wereducetheacceptanceproblemforlinearbounded tobeppace-completebyarguingthatthealgorithmofection4.4canbeimplemented automatona,isppace-complete. inpolynomialspace. Theorem4.17Theproblemofdecidingtheemptinessofthelanguageofagiventimed automata,aknownppace-completeproblem[23],totheemptinessquestionfortbas toprovetheppacelowerboundfortheemptinessproblem.wealsoshowtheproblem automatonbyguessingapathofthedesiredformusingonlypolynomialspace.thisisa fairlystandardtrick,andhenceweomitthedetails. table.butitispossibleto(nondeterministically)checkfornonemptinessoftheregion isexponentialinthenumberofclocksofa,wecannotconstructtheentiretransition Proof.[PPACE-membership]incethenumberofstatesoftheregionautomaton 23