Abstract.Verifyingpropertiesofhybridsystemscanbehighlycomplex.Toreducetheeortrequiredtoproduceacorrectproof,theuseof
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1 VerifyingHybridSystemsModeledasTimed PresentedatHART'97,Grenoble,France,March26-28,1997 Automata:ACaseStudy? Abstract.Verifyingpropertiesofhybridsystemscanbehighlycomplex.Toreducetheeortrequiredtoproduceacorrectproof,theuseof mechanicalvericationtechniquesispromising.recently,weextended amechanicalvericationsystem,originallydevelopedtoreasonabout Code5546,NavalResearchLaboratory,Washington,DC20375 MylaArcherandConstanceHeitmeyer deterministicreal-timeautomata,toverifypropertiesofhybridsystems. solution,basedonthelynch-vaandragertimedautomatamodel,ofthe Toevaluateourapproach,weappliedourextendedproofsystemtoa perreviewsourmechanicalvericationsystem,whichbuildsonsri's SteamBoilerControllerproblem,ahybridsystemsbenchmark.Thispa- PrototypeVericationSystem(PVS),anddescribesthefeatureswe condencethatreal-timesystemssatisfytheirrequirements,especiallytheircrittectedinapplyingoursystemtothebenchmarkproblem.weconclude addedtohandlehybridsystems.italsodiscussessomeerrorswedeicalrequirements.however,applyingformalmethodstopracticalsystemsrequiresthesolutionofseveralchallengingproblems,e.g.,howtomakeformal andverifyhybridsystems. timesystems[9].suchmethodscangivesystemdevelopersandcustomersgreater 1Introduction Researchershaveproposedmanyinnovativeformalmethodsfordevelopingreal- withasummaryofinsightsweacquiredinusingoursystemtospecify descriptionsandformalproofsunderstandabletodevelopersandhowtodesign approachistobuildformalreasoningtoolsthatarecustomizedforspecifying real-timesystemsthatisdesignedtoaddressthesechallengingproblems.our softwaretoolsinsupportofformalmethodsthatareusablebydevelopers. andverifyingsystemsrepresentedintermsofaspecicmathematicalmodel. In[2],wedescribehowweareusingthemechanicalproofsystemPVS[18,19] Vaandragertimedautomata[15,14].Reference[2]alsopresentstheresultsof tosupportformalspecicationandvericationofsystemsmodeledaslynch- solutiontothegeneralizedrailroadcrossing(grc)problem[7]. acasestudyinwhichweappliedourmethodtoproveinvariantpropertiesofa Wearebuildingamechanizedsystemforspecifyingandreasoningabout orsheusesoursystem,aversionofpvsaugmentedwithasetofspecialized PVSproofstrategies,toverifypropertiesoftheautomaton.Useofoursystemfor timedautomaton,theuserllsinatemplateprovidedbyoursystem.then,he andreasonaboutreal-timesystemsinadirectmanner.tospecifyaparticular?thisworkisfundedbytheoceofnavalresearch.urlsfortheauthorsare Oursystemprovidesmechanicalassistancethatallowshumanstospecify 1
2 specicationandvericationisusuallyquitestraightforwardbecauseoursystem PVSproofswithaverysimilarstructure.Moreover,mostofthePVSproofs ofinvariantpropertiesinthegrcsolutionweretranslatedintocorresponding providesbothastructureandasetofspecializedtheoriesusefulinconstructing timedautomatamodelsandprovingpropertiesaboutthem.byfocusingona notmasterthebaselogicandthecompleteuserinterfaceoftheunderlyingproof system,pvs. particularmathematicalmodel thetimedautomatamodel oursystemallows ausertoreasonwithinaspecializedmathematicalframework.theuserneed wheretheeectsoftimepassageandothereventscanbenondeterministic.to proofsofstateinvariantswasexcessive. demonstratedthatthesameapproachcouldbeappliedtohybridautomata, toenterthespecicationsusingourtemplatenorthetimerequiredtocheckthe couldbedoneusingourspecializedstrategiesalone.neitherthetimerequired Theresultsofourinitialstudyprovedencouraging.Allofthehandproofs studytheutilityofourprooftechniquesforverifyinghybridsystems,weapplied themtoasolutionofthesteamboilercontrollerproblemdescribedin[12].in verifyingthisapplication,weinvestigatedthefollowingissues: {Isitpracticaltouseanautomatictheoremprovertoreasonaboutnonlinear {Howshouldnondeterminismbemodeled? Whileourinitialstudyonlyinvolveddeterministicautomata,werecently Inverifyingthesolutionin[12],wedetectedseveralerrors.Whilemostofthese andtheircorrectionisnontrivial. errorsareminorandeasilyxed,atleasttwooftheerrorsareerrorsinreasoning, {Canthetimedautomatatemplateandproofstrategiesdevelopedinour {HowsuitableisPVSforverifyingpropertiesofhybridsystems? realarithmetic? earliercasestudybeextendedtohandlethemoregeneralproblemofhybrid model.moreover,liketheseotherapproaches,ourmethodscanbeusedtoprove HyTech[10],andCOSPAN[11],ourapproachisbasedonaformalautomata Likeotherapproachestoverifyingreal-timesystems,suchasSMV[16,5], automata? andstandardlogicsandbyprovidingtemplatesfordevelopingspecications,we severalways.first,thepropertiesweproveareexpressedinastandardlogic propertiesofparticularautomataand,likecospan,toprovesimulationsbetweenautomata.1however,ourapproachisdierentfromotherapproachesitionofproofsinourmethodisnotcompletelyautomatic.rather,ourmethod largelyeliminatetheneedforthespecialnotations,logics,etc.,requiredbyother vericationsystems.second,unlikeotherautomata-basedmethods,thegenera- logic,suchasctlorictl,orintermsofautomata.byusingstandardnotations approaches,wherethepropertiestobeprovedareexpressedeitherinatemporal withuniversalandexistentialquantication.thisisincontrasttomostother usesamechanicalproofsystemtocheckthevalidityofhandproofsthatuse deductivereasoning. 1Wehavedesignedsupportforverifyingsimulationproofs,butimplementationofthis supportrequiressomeimprovementstopvs. 2
3 toolswedevelopedinourearliervericationofthegrc.section3introduces sophisticationfromtheuser.butbysupportingreasoningaboutautomataata highlevelofabstraction,wecanprovemorepowerfulresultsthanispossiblewith toolsrequiringmoreconcretedescriptionsofautomata.moreover,ourapproach andalsoprovidesconsiderablefeedbackwhenanerrorisdetected.suchfeedback isextremelyusefulincorrectingerrors. avoidsthestateexplosionprobleminherentinotherautomata-basedapproaches Interactionwithanautomaticproofsystemdoesdemandahigherlevelof thesteamboilercontrollerproblem,presentsthemainhybridautomatonspeci- edin[12],anddiscussesthetechniquesweusedtoadaptourtemplatetoproving propertiesofhybridautomata.section4presentsourmajorresults theerrors wedetectedinapplyingoursystemtotheboilercontrollerproblemandthe timeandeortweneededtoadaptandapplyourmethods.section5presents anexampleofahandproofandthecorrespondingpvsproofforastateinvariantwhoseproofinvolvesbothnondeterminismandnonlinearrealarithmetic. Section2reviewsthetimedautomatamodel,PVS,andthetemplateand verifyinghybridautomata. 2Background 2.1TheTimedAutomataModel itssolution[12]representsboththecomputersystemcontrolleranditsenvironmentaslynch-vaandragertimedautomata.atimedautomatonisavery automatasuitableformodelingnotonlycomputersystemsbutalsoreal-world quantities,suchaswaterlevelsandsteamrates.thetimedautomatamodel describesasystemasasetoftimedautomata,interactingbymeansofcommon steps(a),tobeafunction.howtomodifythedenitiontohandlethenondeterminisminherentinhybridautomatawasoneoftheissuesaddressedinour oflynch-vaandragertimedautomata,whichrequiresthenext-staterelation, tomatarepresenttheboilerandthecontroller;thecommonactionsaresensors actionisscheduled.thedenitionoftimedautomatabelow,whichisbasedon andactuatorscontrollingwhethertostoptheboilerandwhenthenextsensor actions.intheboilercontrollersolutionpresentedin[12],separatetimedau- TheformalmodelusedinthespecicationoftheBoilerControllerproblemand Section6presentssomeconclusionsabouttheusefulnessofourmethodsfor generalautomaton,i.e.,alabeledtransitionsystem.itneednotbenite-state: forexample,thestatecancontainreal-valuedinformation,suchasthecurrent time,theboiler'swaterlevel,andtheboiler'ssteamrate.thismakestimed currentstudy. reportingthewaterlevel,steamrate,andnumberofactivepumpsintheboiler, thedenitionsin[8,7],wasusedinourearliercasestudy[2].itisaspecialcase {start(a)states(a),anonempty(niteorinnite)setofstartstates. {states(a),a(niteorinnite)setofstates. {Amappingnowfromstates(A)toR0,thenon-negativerealnumbers. {acts(a),asetofactions(orevents),whichincludespecialtime-passageactions {steps(a):states(a)acts(a)!states(a),apartialfunctionthatdenesthe AtimedautomatonAconsistsofvecomponents: asinputandoutputactions. (t),wheretisapositiverealnumber,andnon-time-passageactions,classied possiblesteps(i.e.,transitions).3
4 References 1.J.-R.Abrial,E.Boerger,andH.Langmaack.PreliminaryreportfortheDagstuhl- Seminar9523:MethodsforSemanticsandSpecication.Dagstuhl,June M.ArcherandC.Heitmeyer.Mechanicalvericationoftimedautomata:A casestudy.inproc.1996ieeereal-timetechnologyandapplicationssymp. (RTAS'96).IEEEComputerSocietyPress, M.ArcherandC.Heitmeyer.Verifyinghybridsystemsmodeledastimedautomata:Acasestudy.Technicalreport,NRL,Wash.,DC,1997.Inpreparation. 4.R.BoyerandJ.Moore.AComputationalLogic.AcademicPress, S.Campos,E.Clarke,andM.Minea.Analysisofreal-timesystemsusingsymbolic techniques.informalmethodsforreal-timecomputing,chapter9.johnwiley &Sons, M.J.C.GordonandT.Melham,editors.IntroductiontoHOL:ATheoremProvingEnvironmentforHigher-OrderLogic.CambridgeUniversityPress, C.HeitmeyerandN.Lynch.TheGeneralizedRailroadCrossing:Acasestudyin formalvericationofreal-timesystems.inproc.,real-timesystemssymp.,san Juan,PuertoRico,Dec C.HeitmeyerandN.Lynch.TheGeneralizedRailroadCrossing:Acasestudyin formalvericationofreal-timesystems.technicalreportmit/lcs/tm-51,lab. forcomp.sci.,mit,cambridge,ma,1994.alsotr7619,nrl,wash.,dc C.HeitmeyerandD.Mandrioli,editors.FormalMethodsforReal-TimeComputing.Number5inTrendsinSoftware.JohnWiley&Sons, T.HenzingerandP.Ho.Hytech:TheCornellHybridTechnologyTool.Technical report,cornelluniversity, R.P.Kurshan.Computer-AidedVericationofCoordinatingProcesses:the Automata-TheoreticApproach.PrincetonUniversityPress, G.LeebandN.Lynch.ProvingsafetypropertiesoftheSteamBoilerController: Formalmethodsforindustrialapplications:Acasestudy.InJ.-R.Abrial,etal., eds.,formalmethodsforindustrialapplications:specifyingandprogrammingthe SteamBoilerControl,vol.1165ofLect.NotesinComp.Sci.Springer-Verlag, V.Luchangco.Usingsimulationtechniquestoprovetimingproperties.Master's thesis,massachusettsinstituteoftechnology,june N.LynchandF.Vaandrager.Forwardandbackwardsimulations{PartII: Timing-basedsystems.ToappearinInformationandComputation. 15.N.LynchandF.Vaandrager.Forwardandbackwardsimulationsfortiming-based systems.inproc.ofrexworkshop\real-time:theoryinpractice",volume600 oflecturenotesincomputerscience,pages397{446.springer-verlag, K.L.McMillan.SymbolicModelChecking.KluwerAcademicPublishers, M.Merritt,F.Modugno,andM.R.Tuttle.Timeconstrainedautomata.InJ. C.M.BaetenandJ.F.Goote,eds.,CONCUR'91:2ndIntern.Conferenceon ConcurrencyTheory,vol.527ofLect.NotesinComp.Sci.Springer-Verlag, S.Owre,J.Rushby,N.Shankar,andF.vonHenke.Formalvericationforfaulttolerantarchitectures:ProlegomenatothedesignofPVS.IEEETransactionson SoftwareEngineering,21(2):107{125,Feb N.Shankar,S.Owre,andJ.Rushby.ThePVSproofchecker:Areferencemanual. Technicalreport,ComputerScienceLab.,SRIIntl.,MenloPark,CA, J.VittandJ.Hooman.AssertionalSpecicationandVericationUsingPVSof thesteamboilercontrolsystem.inj.-r.abrialetal.,editors,formalmethods forindustrialapplications:specifyingandprogrammingthesteamboilercontrol, volume1165oflect.notesincomp.sci.springer-verlag,
5 AAppendix.ConstantandVariableDenitions Table1.ConstantsfortheBoilerandControllerModels Table2.VariablesfortheBoilerandControllerModels ThisarticlewasprocessedusingtheLATEXmacropackagewithLLNCSstyle Table3.AdditionalDenitionsfortheControllerModel 15 Name Type Unit Description I real, >0 s time between the periodical sensor reading S real, >0 s delay to activate pumps after the last sensor reading U 1 real, >0 l /s 2 maximum gradient of the increase of the steam rate U 2 real, >0 l /s 2 maximum gradient of the decrease of the steam rate M 1 real, >0 l minimum amount of water before boiler becomes critical M 2 real, >0 l maximum amount of water before boiler becomes critical W real, >0 l /s maximum steam rate before boiler becomes critical P real, >0 l /s exact rate at which one active pump supplies water to the boiler #pumps int, >0 Number of pumps that can supply water to the boiler in parallel C real, >0 l Capacity of the boiler Name Initial Type Values Unit Description Value Range now 0 real [0, ) s current time pr 0 integer {0,..., #pumps} number of pumps actively supplying water to the boiler q >>M 1, real [0, C] l actual water level in the boiler <<M 2 v 0 real [0, ) l /s steam rate of the steam currently leaving the boiler pr_new 0 integer {0,..., #pumps} number of pumps that are supposed to supply water after the activation delay error 0 integer {0,..., pr_new } number of pumps that fail to supply water to the boiler after activation do_sensor true boolean {true, false} enable a single sensor reading set S real [0, ) s next time the pumps change to the new settings read 0 real [0, ) s next time the sensors will be read stop false boolean {true, false} flag whether emergency shut down is activated do_output false boolean {true, false} flag that enables the output. This represents a kind of program counter. stopmode false boolean {true, false} flag to activate the shut down wl q real [0, C] l current water level reading sr 0 real [0, W] l /s current steam rate reading now 0 real [0, ) s current time pumps 0 integer {0,..., #pumps} number of currently active pumps suppling water to the boiler px 0 integer {0,..., #pumps} number of pumps that shall supply water next Name Type Unit Value Description max_pumps_af ter_set integer #pumps maximum number of pumps that can supply water to the boiler after the delay considering the pump failure model. min_pumps_af ter_set integer 0 minimum number of pumps that can supply water to the boiler after the delay considering the pump failure model. max_steam_water (sr ) real l max(0, (sr U 2 *I/2)*I ) minimum amount of water that can evaporate into steam until the next sensor reading min_steam_water (sr ) real l (sr + U 1 *I/2)*I maximum amount of water that can evaporate into steam until the next sensor reading
Myla Archer and Constance Heitmeyer. farcher, heitmeyerg@itd.nrl.navy.mil. mechanical verication techniques is promising. Recently, we extended
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