Angelika Mader Veri cation of Modal Properties Using Boolean Equation Systems EDITION VERSAL 8

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1 UsingBooleanEquationSystems VericationofModalProperties AngelikaMader EDITIONVERSAL8

2 Band1:E.Kindler:ModularerEntwurf Herausgeber:WolfgangReisig Lektorat:RolfWalter EDITIONVERSAL Band2:R.Walter:PetrinetzmodelleverteilterAlgorithmen. verteiltersystememitpetrinetzen Band4:K.Schmidt:SymbolischeAnalysemethoden Band3:D.Gomm:ModellierungundAnalyse mitpetrinetzen verzogerungs-unabhangigerschaltungen BeweistechnikundIntuition Band5:M.Kohn:FormaleModellierung Band6:D.Barnard:TemporalLanguageofTransitions furalgebraischepetrinetze asynchronersysteme Band8:A.Mader:VericationofModalProperties Band7:U.Jaeger:EventDetectionin UsingBooleanEquationSystems ActiveDataBases andclient-serversystems

3 UsingBooleanEquationSystems VericationofModalProperties AngelikaMader DieterBertzVerlag

4 Systems/AngelikaMader.{Berlin:Bertz,1997 Mader,Angelika: VericationofModalPropertiesUsingBooleanEquation (EditionVersal;Bd.8) Zugl.:Munchen,Techn.-Univ.,Diss.,1997 DieDeutscheBibliothek{CIPEinheitsaufnahme NE:GT ISBN n-x AlleRechtevorbehalten GorlitzerStr.37, c1996bydieterbertzverlag,berlin

5 Abstract expressionscontainingleastandgreatestxpoints.fixpoint-equation model-checking. Themodal-calculuscontainsxpoint-operatorswhichgivegreatexbraicallyweintroducexpoint-equationsystemsasanextensionopressivepower.Inordertotreatthemodel-checkingproblemalge- systemsexpressedinthemodal-calculus.thisapproachiscalled Thethesisisconcernedwithvericationofpropertiesofconcurrent and presentanewalgorithm,similartogaueliminationforlinearequationsystems. BooleanlatticesarecalledBooleanequationsystems.Modelcheck- solvingnitebooleanequationsystems.wediscussexistingmodelcheckingalgorithmsfromtheperspectiveofbooleanequationsystems systemsinterpretedoverthebooleanlatticeoraninniteproductof Asanapplicationweinvestigatealgorithmssolvingtheproblemof ingforsystemswithnitestatespacesisshowntobeequivalentto mutualexclusion,constructformulaeforlivenesspropertiesandverify lencetoanautomata-theoreticproblembygoingviabooleanequa- tionsystems.thereexistedareductionofmodel-checkingtoagame wepresentanalgorithm,similartothegaueliminationalgorithmfor equivalence. Forthecaseofinnitestatespaceswealsoshowthatmodel-checkingis thenitecase. equivalenttosolvinginnitebooleanequationsystems.additionally, themwithanimplementationofthegaueliminationalgorithm. Model-checkinginthemodal-calculushasalreadybeentreatedin automatatheoryandgametheory.weareabletoshowanewequiva- theoreticproblem.usingbooleanequationsystemswecanprovethe

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7 environmentheprovidedforus,andhisliberalattitudes,whichmade acarefreeandconcentratedwayofworkingpossible. fortheconstantsupportofmyallactivitieshere,thecomfortable IamindebtedtomyproofreadersJulianBradeld,EdBrinksmaand EkkartKindler.Theircommentsandcarefulcriticismwereofgreat Acknowledgement helpformeinndingoutwhatiwasdoing,inimprovingmywork, Intherstplace,IwouldliketothankmysupervisorWilfriedBrauer ereeandformostlyilluminatingdiscussions,julianbradeldforhis commentsonpartsofthethesis. IamverygratefultopeopleinEdinburgh:ColinStirlingforbeingref- and,whatisperhapsevenmorevaluable,theyincreasedthefuni hadwhenwritingup.thanksalsotochristinerocklwhogaveuseful EdBrinksmaandPeterRossmanithsupportedmeinndinganexponentialexampleformyversionoftheGaualgorithmanddelivered scienticatmosphereandthegreatvarietyofsinglemaltscontributed enormouslytomyenjoymentofmyvisitstoedinburgh. Kaivolaforclarifyingautomata-theoreticconcepts.Theimpressive forhertheoreticalandpracticalhelpconcerninggames,androope hisinsightinbooleanequationsystemswithme,perditastevens friendshipandpleasantcooperation,kyriakoskalorkoti,whoshared sharptongueofdominikgomm. liketoacknowledgeallpeopleofthegrouphere,andthosewholeftto Berlin.Ispentagoodtimewiththem.Particularly,Iammissingthe Gaualgorithmwasextremlyhelpfultome.Furthermore,Iwould pleasuretome.hisneverendingengagementinimplementingthe mefromalong-termpassion.infrankwallnerifoundacolleague ManythanksgotoBarbaraRoemerwhogavevaluablehintsconcerninglayout. whowasnotafraidofxpointsanddiscussionswithhimwereagreat WithoutGerhard'ssupportIcouldnothavedonethisworkandmany (Sonderforschungsbereich342)forfundingmypositionattheTU. IthankFa.Siemens,ZFE,andtheDeutscheForschungsgemeinschaft otherthingsatthesametimewhilehavingachild.manythanksalso

8 todavidforconsistentlyrelativizingallupsanddownsconcerningmy workandforallthenightshesleptthrough. >FromEd,myparents,familyandfriendsIreceivedvaluablesupport ofvariouskindsduringallthetime,forwhichiowethemgreatthanks.

9 Contents 2Basics. 1Introduction. 2.2Fixpointsandtheirproperties.::::::::::::::22 1.1Generalintroduction.::::::::::::::::::::11 2.1Ordersandlattices.:::::::::::::::::::::19 1.2Synopsis.:::::::::::::::::::::::::: Simplexpoints.:::::::::::::::::: Themodal-calculus. 3Fixpoint-equationsystems. 3.1Fixpoint-equationsystemsforcompletelattices.::::28 4.1Syntaxandsemantics.:::::::::::::::::::45 3.2Booleanequationsystems.::::::::::::::::: Nestedxpoints.::::::::::::::::::24 4.3Propertiesofthemodal-calculus.::::::::::::51 4.2Basicformulae.::::::::::::::::::::::: SolvingBooleanequationsystems. 5Booleanequationsystemsformodelchecking. 5.3ReductionofBooleanequationsystems.:::::::::62 5.1Reductionofthemodelcheckingproblem.::::::::56 5.2Representationandcomplexity.::::::::::::::59 6.1PlainBooleanequationsystems.:::::::::::::

10 106.4Gauelimination.::::::::::::::::::::::81 6.3Tableaux.::::::::::::::::::::::::::76 6.2Approximation.::::::::::::::::::::::: Complexityforthegeneralcase.::::::::: Complexityforsubclasses.::::::::::::: Globalandlocalalgorithm.::::::::::::82 CONTENTS 7Peterson'smutexalgorithm. 8Equivalenttechniques. 6.5Complexity.:::::::::::::::::::::::::94 7.2FairnessandLiveness.::::::::::::::::::: Graphgames.:::::::::::::::::::::::: Alternatingautomata.::::::::::::::::::: ExperimentalResults.::::::::::::::::::: Modellingthealgorithm.:::::::::::::::::: InniteBooleanequationsystems. 9.6Conclusion.::::::::::::::::::::::::: Examples.:::::::::::::::::::::::::: Eliminationmethod.:::::::::::::::::::: Equivalencetothemodelcheckingproblem.::::::: Denitions.::::::::::::::::::::::::: SetbasedBooleanequationsystems.::::::::::: AAppendix 10Conclusion. A.3ProofsofChapter8.::::::::::::::::::::161 A.2ProofsofChapter5.::::::::::::::::::::158 A.1ProofsofChapter3.:::::::::::::::::::: Innitestatespacemodelchecking:::::::::::: Finitestatespacemodelchecking::::::::::::: Bibliography A.4ProofsofChapter9.::::::::::::::::::::

11 Chapter1 Introduction. 1.1Generalintroduction. be,itispossiblethatitshouldbe... Yet,fromtheproposition`itmaybe' Whenitisnecessarythatathingshould fromthatitfollowsthatitisnotnecessary;itcomesaboutthereforethatthe itfollowsthatitisnotimpossible,and ianstoicsalsodealtwithmodallogics,introducingatimebasedinter- pretation:possibleisjustwhateitherisorwillbe;athingisnecessary onlyifitisnowtrueandalwayswillbetrue. Leibnizgaveasemanticmodelforlogicsincludingthemodalities`nec- Aristotle,Hermeneia1 ThebeginningofmodallogicdatesbacktoAristotlewhowasalready concernedwiththelogicofnecessityandpossibility.later,themegar- be;whichisabsurd... thingwhichmustnecessarilybeneednot essarily'and`possibly':heassumedasetofworldsanddeneda propositionbeingnecessarilytrueifitistrueinallworlds,andbeing possiblytrueifthereexistssomeworldwhereitistrue.inaddition, 1see[Boc70]

12 tury.nowadaysphilosophers,logicians,linguistsandcomputerscien- tistsshareaninterestinthesubject,andvarioussystemsofmodal Formalmathematicaltreatmentofmodallogicstartedinthiscen- logichavebeendeveloped. 12 Infurtherdevelopment,morestructurewasgiventothemodelof heprovedthatweliveinthebestofallpossibleworlds. Chapter1.Introduction. worlds.whendecidingwhethersomepropositionpisnecessaryin areorderedlinearlyintime. oneworldonlyaspeciedsetofworldsmayberelevant,whichneed Incomputersciencemodalandtemporallogicplayaroleinthevericationofsystems.Here,thetaskistoshowthatasystemmeets itsspecicationwhichmayconsistofsetofpropertiesexpressedas systems.theyconsistofasetofstates(representingtheworlds)and formulaeofalogic. ModelsformodallogicareKripkestructures,alsocalledtransition transitionsbetweenthestates(theaccessibilityrelation).atransition pinballmachine.transitionsmaycarryalabelidentifyinganaction (write1toamemorycell,shootthepinball)ormodellingjustthe systemmodelsthedierentstatesanarbitrarysystemcanenter,and actionsleadingfromonestatetoanother.astatecanrepresente.g. on-goingofasystemastimepasses.thelattercaseprovidesamodel fortemporallogic. Propositionsareaboutstatesorpathsofamodel,e.g.forthepinball thecontentofamemory,thevalueofaprogramcounter,astateofa machineinitiallytheonlypossibleactionistoinsertacoin;thereexists arunofthepinballmachine,whereialwaysgetafreegame,or,ifi rolldown. oneworldmeansthatpbeingtrueinallworldsaccessiblefromthe currentone.temporallogicisthendenedasamodallogic,where accessibilitybetweenworldsrepresentstimepassingby,andtheworlds anaccessibilityrelationbetweenworlds,andpisnecessarilytruein notincludeeveryworldinthemodel.thisfeatureisrepresentedby hitthepinballmachineinnitelyoftenthentheballwilleventually

13 tiveprograms.provingcorrectnessforaprogramwastoshowthat [MP69],Park[Par70]andHoare[Hoa69]wereimportantdevelopments givenaspeciedinputtheprogramwouldterminateandproduce aspeciedoutput.theworksoffloyd[flo67],mannaandpnueli Intherstperiod,objectsofvericationweresequentialandimpera- inthiscontext. 1.1.Generalintroduction. 13 Therstmodallogicsforvericationweredynamiclogicsintroduced bypratt[pra76],andmostlyusedinthepropositionalversion.propositionaldynamiclogic(pdl)isbuiltupfrompropositionallogic extendedbythemodalitieshi,whereaprogramisaregularexpressionoverasetofatomicprograms.theformulahipistrueata state,whereitispossiblefortheprogramtoexecuteandresultin astatesatisfyingp.variousrestrictionsandextensionsofpdlhave acteristicsofprograms:terminationandresultsproducedwerenot longernecessaryfeatures,buton-goingandinteractionwithanenvironmentbecamerelevant.pnuelicalledthem\reactivesystems". andpdl-[str81]whereaninniteloop-operatorisaddedtopro- beeninvestigated.themostfamousonesarepdlwithtestprogams, gramexpressions. Theintroductionofconcurrencycausedchangeconcerningthechar- Clarke,EmersonandSistla[CES86],andothersstartedwithanew approach,calledmodel-checking.here,vericationfornitestatesystemsisperformedautomaticallyand,incontrasttoderivingaproof, Pnueli[MP83]foundthattemporallogicissuitableinthiscontext. Theyappliedaproof-theoreticstyleofverication:foragivenprogramtheyderivedasetoftemporalpropertiesandshowedthatthe Provingcorrectnesshererequiredmoreexpressivelogics.Mannaand specifyingpropertywasaconsequenceofthisset(orwasnot). pinballmachinetheninnitelyoftenitwillbeinthestate\tilt"). relevantpropertiesarenot(e.g.ifinnitelyoftenaplayerhitsthe ifthepinballisshotthenitwilleventuallyrolldownagain),butsome (CTL).Inthislogicanumberofusefulpropertiesisexpressible(e.g. analgorithmreceivingaformulaandamodelasinputgivestheresult trueorfalse.thetemporallogictheyusediscomputationtreelogic

14 AnextensionofCTLthatcanexpressthe\tilt"-propertycitedabove thetacklingofthesizeofproblemsandthedenitionofmoreexpressivelogics.ofcourse,theproblemsarenotmutuallyindependentof isvericationandthesmalleristhesizeofsolvableproblems. 14 eachother;roughly,themoreexpressivealogicis,themorecomplex Insubsequentdevelopment,workwascenteredmainlyontwoissues: Chapter1.Introduction. iscalledctl*.forthistemporallogicemersonandlei[el86]presentedamodel-checkingalgorithm. MeanwhilealsovariousextensionsofCTLandCTL*havebeeninves- andthexpointoperatorsand.themodalitiesallowonetoexpresspropertiesforonenext-step,whilebymeansofleast(anddually additiontopropositionallogicitcontainsthemodalities[a]andhai modalandtemporallogicsmentionedabove:themodal-calculus.in tigatedwhicharemoreexpressive,butstillsimpleenoughformodel- checking. greatest)xpointimmediatelypropertiesoverniteandinnitepaths Kozen[Koz83]introducedaverypowerfullogic,subsumingallother canbemodelled.thebeautyofthislogicliesinitsexpressivenessin combinationwithitssimplicity.therstmodel-checkingalgorithm forthemodal-calculuswasdevelopedbyemersonandlei[el86]. However,thecomplexityoftheiralgorithmishigherthanthatforless expressivelogicssuchasctl:itisofexponentialcomplexityinthesize byso-called\symbolicmodel-checking".forearlieralgorithmsthe Concerningthesizeofproblemsconsiderableprogresshasbeenachieved thecomplexityofthisproblemhavenotyetbeendetected. -calculushavebeensuggested,yettherehasnotbeenanyessential algorithmsforctl.sincethenanumberofalgorithmsforthemodal improvementconcerningcomplexitysofar,andthelowerboundsfor oftheformulaincontrasttopolynomialcomplexityofmodel-checking model,atransitionsystem,hadtoberepresentedexplicitly.ina newapproachforctlmodel-checkingburch,clarkeandmcmillan [BCM+92]choseBinaryDecisionDiagrams(BDDs)asdata-structure, sizeofproblemsthatcouldbetreatedgrewenormously. whichallowedaverycompactencodingoftransitionsystems,andthe

15 However,thesizeofthetransitionsystemsisstillthemostlimiting probleminthisarea.especiallyforconcurrentsystemstheso-called \statespaceexplosion"makesvericationdicultorevenimpossi- 1.1.Generalintroduction. ble.reductiontechniquesfortransitionsystemshavebeeninvesti- gatedincludinge.g.abstractionsandsymmetries,whichrelativize thepurelyautomaticapproachandreintroduceelementsofproofto 15 ornoteventhesetofreachablestates,buta(hopefullysmall)subset whetherapropertyholdsofpathsstartingfromtheinitialstateofa system.showingitscorrectnessmaynotrequirethewholestatespace, setofallstatessatisfyingaproperty.usually,weareinterestedin model-checking. Themethodofmodel-checkingdescribedaboveis\global"inthesense thatthealgorithmstraversethewholestatespaceanddeterminethe StirlingandWalker[SW89]informofatableausystem. ofit.algorithmsbasedonthisideaarecalled\local".alocalmodelcheckingalgorithmforthemodal-calculuswasrstintroducedby grammars. sistanceisapossibility.bradeldandstirling[bs90,bra92]developed modelsdenede.g.bysomepetri-netclasses[en94],orcontext-free automaticmethods.however,provingpropertieswithcomputeras- atableaumethodallowingcomputer-aidedvericationforformulaeof themodal-calculus.otherworkhasbeendoneinthisareaforinnite Inthecaseofgeneralinnitestate-spacesthereisnohopeforfully Booleanequationsystems.Infact,wecanshowthatthetwoproblems formedtotheproblemofsolvingaclassofequationsystems,called -calculus.theapproachisanalgebraicone:model-checkingistrans- showtheirrelationstoothertechniques,inautomatatheoryandgame areequivalent,forthecaseofnitesystemsaswellasforinniteones. Basedonthisequivalencewediscussmodel-checkingalgorithmsand Alsointhiswork,weareconcernedwithmodel-checkingforthemodal theory.thefollowingsectiongoesontooutlinethisinmoredetail.

16 16 1.2Synopsis. Inthebeginningwegiveabriefcollectionofrelevantdenitionsand factsfromlatticetheoryandthexpointtheoremswhicharestructures Incomputersciencemainlyleastxpointshavebeenconsidered.Propositionsforexpressionscontainingleastandgreatestxpointoperators donotgobeyonddualityargumentssofar.chapter3containstherst contributionofthiswork:anintroductionofxpoint-equationsystems entailsanextensivecollectionofpropertiesofxpoint-equationsystems.thedierencebetweenmoretraditionalequationsystemsand xpoint-equationsystemsconsistsoftheadditionalstructuregivento thelatter:thereisanorderdenedontheequationsandeachequationisequippedwithaminimalityormaximalitycondition.because ofthisstructureknownresultsforsolutionsofequationsystemsover latticesdonotapplyforthexpoint-equationsystems.inthiswork xpoint-equationsystemswillbeinterpretedoverthebooleanlattice fornitestatespacemodel-checkingaswellasoveraninniteproduct ofbooleanlatticesformodel-checkingofinnitestatespaces.section 3.2containsdenitionsandpropertiesforthenitecase,extending Booleanequationsystems. Chapter4containsanintroductiontothemodal-calculus,including propertiesforxpoint-equationsystemsoverarbitrarylattices.the interpretedinthiswayarecalledbooleanequationsystemsandinnite syntax,semantics,basicnotationsandfacts. Themainpointofchapter5istheequivalenceofthemodel-checking innitecasewillbetreatedinchapter9.fixpoint-equationsystems problemfornitestatespacesandtheproblemofsolvingboolean equationsystems.reductionstobooleanequationsystemsforthecase ofnon-alternating-calculusexpressionshavealreadybeentreatedby applyingdirectlytothegeneralcase.thesizeofabooleanequation otherpeople.theextensiontothegeneralcasecouldbedonebythe andfactsbasicforthewholework. Chapter1.Introduction. asageneralizationofnestedandalternatingxpoint-expressions.it well-knownxpointtheorems.here,insection5,wegiveareduction

17 equationsystem,weconstructaformulaofthemodal-calculusanda Section5.2showsthereductionintheotherdirection.GivenaBoolean simpleformforequationshastobedenedfollowingknowntechniques. systemlinearinthesizeoftheoriginalmodel-checkingproblemaa 1.2.Synopsis. systemderivedislinearinthesizeofthemodelandlinearinthesize oftheformula.inordertogetarepresentationofabooleanequation 17 relatingittothe\classical"versionofbooleanequationsystemswithoutorderontheequationsandwithoutsideconditionsforxpoints. thethemodelsatisestheformula.thesizeofthemodelisquadratic inthesizeofthebooleanequationsystem,thesizeoftheformulais linear. Chapter6dealswithmethodsforsolvingBooleanequationsystems, localaswellasglobalones.westartwithadiscussionoftheproblem, model,suchthatthebooleanequationsystemhasthesolutiontruei inationforlinearequationsystems.itleadstoboth,alocalanda BooleanequationsystemsbeinginNP\co-NP,andaccordingtothe techniqueforbooleanequationsystemswhichissimilartogauelim- Theknownmethodssolvingthemodel-checkingproblemaretheapproximationtechniqueandatableaumethod.Weinterpretethem equivalenceresultsalsothemodel-checkingproblemiscontainedin globalalgorithm.thelastsectioncontainsasimpleproofforsolving onbooleanequationsystems.inadditionwepresentanewsolving thisclass,whichisaknownresult. Examplesforapplicationarepresentedinchapter7.Here,wefocus inotherframeworks:thereexistreductionstoproblemsinautomata- algorithmsolvingtheproblemofmutualexclusion.theseproperties providenon-trivialexamplesfor-calculusformulae.theyareveri- edwithanimplementationofgaueliminationforbooleanequation systems. Themodel-checkingproblemforthemodal-calculushasbeentreated oncomposingandprovingdierentlivenesspropertiesforpeterson's andgame-theory.intherstcaseallautomataderivedaretree- automata.insection8.1weshowtheequivalenceofmodel-checking andthenon-emptiness-problemofalternatingautomataoninnite

18 playerhasawinningstrategyforagameandsolvingabooleanequationsystem.thereductionofbooleanequationsystemstomodelcheckinggivesimmediatelyareductionfromamodel-checkinggame games.insection8.2,weshowtheequivalenceofdecidingwhethera Themodel-checkingproblemhasalsobeenreducedtomodel-checking wordsoverasingle-letteralphabetwithaparityacceptancecondition. Chapter1.Introduction. 18 xpoint-equationsystemsinterpretedovera(possiblyinnite)productofbooleanlattices.theequivalenceofinnitebooleanequation systemsandthemodel-checkingproblemforinnitestatespacesis Sofarwehaveonlybeenconsideringnitestatespaces.Inchapter toamodel-checkingproblem,whichhasbeenanopenquestion. case.booleanequationsystemsastheyareusedherearederivedfrom provedbyreductionsinbothdirections.theseresultsareonlyuseful whenhavinganiterepresentationoftheproblemwhichisgivenby 9,thetheoryofBooleanequationsystemsisextendedtotheinnite setbasedequationsystems.wepresentaneliminationmethodusing ideasfromgaueliminationforthenitecaseandfromthetableau examplesdemonstratethetechnique. Thethesisendswithconcludingremarksputtingourresultsinageneralframework. methodofbradeldandstirling.itsolvessetbasedequationsystemsandalsothemodel-checkingproblemfortheinnitecase.small

19 Chapter2 Basics. xpointoperatorsofmodallogichavetobedenedviacontinuous interpretedasanorderpreservingfunctionbetweentwolattices.the functions.therefore,wecollectheretherelevantdenitionsandfacts. iscompletelattice.thesemanticofaformulaofmodallogiccanbe 2.1Ordersandlattices. Thebasicstructureinthisworkarelattices;formulaeofmodallogic withimplicationorderformalattice,thepowersetofastatespace Adetailedintroductionintolatticesandorderscanbefound[DP90]. Asetequippedwithapartialorderiscalledanorderedset. (transitivity)xyandyzimplyxz (antisymmetry)xyandyximplyx=y ifforallx;y;z2p: (reexivity) Denition2.1AbinaryrelationonasetPisapartialorder greatestelementofqisa2qifaxforallx2q.dually,the Denition2.2GivenanorderedsetPandasubsetQofPthe xx leastelementofqisa2qifaxforallx2q.

20 20Denition2.3LetPbeanorderedset.Thegreatestelementof P,ifitexists,iscalledthetopelementofPandwritten>.Dually, Proposition2.4GivenanorderedsetPanysubsetQPisan Pandwritten?. theleastelementofp,ifitexists,iscalledthebottomelementof Chapter2.Basics. orderedset. Proposition2.5Let(P1;1);:::;(Pn;n)beorderedsets.Their productp1:::pncanbeequippedwithapartialorderbypointwisedenition:(x1;:::;xn)(y1;:::;yn)ixiiyifor1in. Denition2.6LetPandQbeorderedsets.Thesetoffunctions fromptoqisdenotedby(p!q).foreachfunctionf2(p!q) Onthesetoffunctions(P!Q)anorderisinheritedfromthe p1p2itisthecasethatf(p1)f(p2). thedomainispandthecodomainisq. Afunctionf2(P!Q)ismonotone,ifforallp1;p22Pwith ThesetofallmonotonefunctionsisdenotedbyhP!Qi. f(a)g(a)foralla2a. orderontheircodomainq:letf;g2(p!q).thenfgif Denition2.7LetPbeanorderedsetandSbeasubsetofP. Thenx2PisanupperboundofS,ifsxforalls2S.Dually x2pisalowerboundofs,ifxsforalls2s. AllupperboundsofSarecollectedinaset"S,thelowerbounds TinsteadofWandV,and[and\insteadof_and^. inmumvfx;yg.whenspeakingaboutpowersetswewillusesand Notation:ForthesupremumWfx;ygwewritex_y,andx^yforthe VS.TheyarealsocalledthesupremumandinmumofS. upperboundofs,anddenotedbyws.thegreatestelementof #Sifitexists,iscalledgreatestlowerboundofS,anddenotedby inaset#s.theleastelementof"s,ifitexists,iscalledleast

21 2.1.Ordersandlattices. Denition2.8LetPbeanon-emptyorderedset.Pisalattice, ifx_yandx^yexistforallx;y2p.pisacompletelattice,if WSandVSexistforallsubsetsSP. Proposition (5)IfPandQare(complete)latticesthenalsothesetsoffunctions (4)ForanysetXitspowersetP(X)equippedwiththesetinclusion (1)InalatticeWSandVSexistforallnitesubsetsSP. (2)Everynitelatticeiscomplete. (3)Inacompletelatticethebottomelement?andthetopelement inmumareobtainedpointwise. (P!Q)andhP!Qiare(complete)lattices.Supremumand orderisacompletelattice. >exist. fop(k1) theoperationssupremum_andinmum^,andasetofoperators sions.thesearebuiltupbyvariablesxfromasetofvariablesx, theoperatorop(ki) Inmostcaseswethinkoffunctionsasrepresentedbyfunctionexpres- f::=xjf_fjf^fjop(ki) monotonefunction,andspsuchthatwsandvsexistin Proposition2.10LetPandQbeorderedsets,f:P!Qa 1;:::;Op(kn) P,andWf(S),Vf(S)existinQ.Thenf(WS)Wf(S)and ngforsomen2in,wherekidenotesthearityof f(vs)wf(s). i. i(f;:::;f) directed,ifeverynitesubsetfofshasanupperboundins. Proposition2.11Productsofcompletelatticesequippedwitha partialorderasinproposition2.5arecompletelattices. Denition2.12Anon-emptysubsetSofanorderedsetPis

22 22Thenf:P!QiscontinuousifforeverydirectedsetinPitisthe casethatf(wd)=wf(d). Denition2.13LetPandQbecompletelattices. Afunctionthatpreserves?,i.e.f(?)=?iscalledstrict. Chapter2.Basics. Proposition2.14LetPandQbecompletelattices.Thenevery 2.2Fixpointsandtheirproperties. Denition2.15GivenalatticePandafunctionf:P!P.An elementx2pisaxpointoffiff(x)=x. monotonefunctionf:p!qisalsocontinuous. TheverybasictheoremcomesfromTarski[Tar55](seealso[LNS82]). Thissectionisacollectionofvariouspropertiesofxpointswhichcan befoundintheliterature.itstartswithpropertiesofsimplexpoints, bothleastandgreatest.thenwelookatthemoregeneralcasewhere xpointoperatorsofpossiblydierenttypearenested. Itguaranteestheexistenceofaleastandgreatestxpointforamonotonefunctionoveracompletelattice Simplexpoints. Wewillusewhenreferringtoeitheror. Thenextproperties(formonotonef)canbefounde.g.in[Koz83]. notemptyandthesystem(p;)isacompletelattice;inparticular theleastxpointisx:f(x)=vfa2ajf(a)agandthe monotonefunction,andpthesetofallxpointsoff.thenpis Theorem2.16Let(A;)beacompletelattice,f:A!Aa greatestxpointisx:f(x)=wfa2ajf(a)ag.

23 2.2.Fixpointsandtheirproperties. Proposition2.17 (1)f(X:f(X))=X:f(X) (2)Iff(a)athenX:f(X)a. (3)Iff(a)athenX:f(X)a. (4)Iff(a)g(a)foralla2AthenX:f(X)X:g(X). 23 Thefollowingpropertyisknownasthereductionlemma,seeforexample[Koz83],[Win89]. Lemma2.18aX:f(X)iaf(X:(f(X)_a)) (6)X:f(X)=X:f(f(X)) (5)Iff(a)=f(b)foralla;b2AthenX:f(X)=f(X). generalversion,usingtransniteiteration(see[lns82]). butnoconstructivemethodtoyieldit.thisisthesubjectofthenext Tarski'stheoremshowstheexistenceofaleastandgreatestxpoint, well-knowntheorembasedonapproximants.itispresentedhereinits Denition2.19Let(A;)beacompletelatticeand or,dually,ax:f(x)iaf(x:(f(x)^a)). term,whereisanordinal.theapproximanttermsaredenedby +1X:f(X)def transniteinduction: f:a!aamonotonefunction.thenx:fisanapproximant X:f(X)def X:f(X)def 0X:f(X)def 0X:f(X)def =^<X:f(X) =_<X:f(X) => =f(x:f(x)) =? whereisalimitordinal.

24 24X:f(X)=^ X:f(X)=_ functionf:a!a Proposition2.20Foracompletelattice(A;)andamonotone 2OrdX:f(X) 2OrdX:f(X) Chapter2.Basics. and,dually, thatofasuchthatfor: X:f(X)=X:f(X) Moreoverthereexistsanordinalofcardinalitylessorequalto whereordistheclassofallordinals. andgaremonotoneinbotharguments.asarststepwewilldene wherexandyarevariablesoverlattices(a;)and(b;),andf 2.2.2Nestedxpoints. Wenowwanttoconsidernestedxpoints,suchasX:f(X;Y:g(X;Y)) X:f(X)=X:f(X): theirdomainsareinterpretedindierentways.fortechnicalreasons weassumefromnowonthattherearenottwodierentvariablesina nestedxpointexpressionhavingthesamenames. abusenotationanddonotintroducenewnamesforfandgwhen theinnerxpointy:g(x;y)asafunctiong0fromatob.wewill andthegreatestxpointis Y:g(X;Y)def Y:g(X;Y)def monotonefunctiononabtob.thentheleastxpointwith respecttobisafunctionfromatob Denition2.21Let(A;)and(B;)becompletelattices,ga =Wfg02(A!B)jg(X;g0(X))g0(X)g. =Vfg02(A!B)jg(X;g0(X))g0(X)g

25 2.2.Fixpointsandtheirproperties. Proof:straightforward g0(a)=y:g(a;y)(g0(a)=y:g(a;y))foreverya2a,where isamonotonefunctiong0:a!banditisthecasethat Proposition2.22Theleast(greatest)xpointofg:AB!B g(a;y):b!aandy:g(b;y)followsdenition forwardly.intheremarkbelowg0mightbeavectoroffunctions Themonotonicityofg0impliesthatf(X;g0(X))isamonotonefunction fromatoaanditsxpointsarewelldenedaccordingtodenition (possiblyempty)productsofcompletelattices. resultingfrominnerxpointsandalldomainscouldbeinterpretedas 2.16.Theapplicationtoarbitrarynestingofxpointsworksstraight- Remark2.23Wewanttopointout,thatthereexisttwobasicallydierentinterpretationsoftheinnerxpointswhichhave g0(a)def morecommonone:g0asafunctiononatobisdenedpointwise, consequencesforalgorithmscalculatingthem.therstoneisthe canexplicitlycalculatethefunctiong0,notinapointwisemanner, functiong(a;y)onbtobandtheapplicationofaxpointoperator Yiswelldened.Thisinterpretationgivesrisetotheapproximationbasedalgorithms.Evaluationofg0ataisdonebyasimple Theotherinterpretationfocusesonthefact,thatinsomecaseswe approximationofy:g(a;y)asinproposition2.20. =Y:g(a;Y).Foreveryargumenta2Awegetthesimple howasimultaneousxpointcanbetransformedtoanestedxpoint expression. Bekic'stheorem[Bek84]foreliminationofsimultaneousxpointsshows butasafunctionexpressionwithafreevariabley.heretheevaluationofg0(a)consistsofasimplefunctionevaluationandnotofan f:ab!aandg:ab!bmonotonefunctions. Theorem2.24Let(A;)and(B;)becompletelattices, approximation. a=x:f(x;y:g(x;y)),andb=y:g(a;y): Then(X;Y):(f(X;Y);g(X;Y))=a;b,where

26 26 Chapter2.Basics.

27 Chapter3 Fixpoint-equation systems. pretedoverarbitrarycompletelattices.fortheissueofthisworkthe caseofxpoint-equationsystemsisinvestigated,wheretheyareinter- propertiesofxpoint-equationsystems.intherstsectionthegeneral nitionsofsyntaxandsemanticsitcontainsanextensivecollectionof technicalbasisfortherestofthework.therefore,apartfromde- Weintroducexpoint-equationsystemsextendingthenotionofnested requireddomainsarethebooleanlatticeandapossiblyinniteprod- xpointexpressions.theintentionofthischapteristoprovidethe uctofbooleanlattices.thesecondsectionfocusesonthexpoint- equationsystemsoverthebooleanlattice,booleanequationsystems. Forthiscasesomedenitionssimplifyandwegetanumberoffurther properties.proofsofthischapterareshiftedtotheappendix.

28 3.1Fixpoint-equationsystemsfor 28 fromxpointexpressionstoxpointequationsystems.themainpart Firstsyntaxandsemantics1aredened,thenwegiveatranslation completelattices. Chapter3.Fixpoint-equationsystems. ineachfunction.insteadofperformingexplicitlythesubstitutionin environment.;1;:::willrangeoverenvironments,whereeachis equationsystems. Inthefollowingweconsidersequencesoffunctionsf1;f2;:::overalattice(A;).Often,freevariableswillbesubstitutedbythesamevalues eachfunctionwecollectthevaluesofthevariablesinavaluation,called ofthissectioncontainsanextensivecollectionofpropertiesofxpoint- fby(x).by[x=a]wedenotetheenvironmentthatcoincideswith afunction:x!a. Afunctionfcanbeappliedtoanenvironment,andtheresultf() isthevalueofthefunctionfaftersubstitutingeachfreevariablexof forallvariablesexceptx,i.e.(y)=([x=a])(y)fory6x,and Theorderonalattice(A;)extendsnaturallytoanorderonenvironmentsoverA(seeDeniton2.6).Wehave12iforallvariables latticeoperations_and^canbeappliedalsotoenvironmentswhen ments(foraxedsetofvariablesx)formsalattice.obviously,the ([X=a])(X)=a.Intheremainder[X=a]haspriorityoverallother interpretingthempointwise. operations,and[x=a]alwaysstandsfor([x=a]). X2Xitisthecasethat1(X)2(X).Thusthesetofenviron- pointedmetoitforthespecialcaseofxpoint-equationsystemsovertheboolean Axpoint-equationsystemoverAisanitesequenceofequations oftheform(x=f),wheref:an!aforsomen2inisa Denition3.1Let(A;)beacompletelattice. lattice.itturnedouttobemorecompactthanearlierversions. monotonefunction. Theemptysequenceisdenotedby. 1TheversionofnotationusedherewasinspiredfromVergauwen[Ver95]who

29 rightsideofanequationofearecollectedinthesetrhs(e).variables whichappearonthelefthandsideofanequationofearecollectedin thesetlhs(e),i.e.lhs((x=f)e)def equationsystemehavethesamelefthandsidevariable.variables Fortechnicalreasonsweassumethatnotwoequationsofaxpoint- 3.1.Fixpoint-equationsystemsforcompletelattices. InthefollowingE;E0;E1;:::willrangeoverxpoint-equationsystems. =fxg[lhs(e).variablesonthe 29 ofrhs(e)whicharecontainedinlhs(e)arecalledbound.variables whicharenotboundarefree,free(e)def axpoint-equationsystemeisasetofconsecutiveequationsofeall havingthesamexpointoperatorinfront. Theorderdenedbelowreectsthelinearorderofequationsina xpoint-equationsystem.itwillbeappliedtobothequationsand variables. Denition3.2Let(X=f)Ebeaxpoint-equationsystemand =rhs(e)nlhs(e).ablockin respecttoe,iffree(e0)free(e). systeme,ifforeachpairofequationswith(xx=fx)c(yy=fy) Axpoint-equationsystemE0isasubsystemofaxpoint-equation AsubsystemE0ofaxpoint-equationsystemEiscalledclosedwith ine0bothequationsarecontainedineandorderedinthesameway. AsusualXEYabbreviates(XCYorX=Y). 0Y=ganequationofE.ThenX=fC0Y=gandalsoXCY. Denition3.3Let(A;)beacompletelattice,(X=f)Ea Thesolutionofaxpoint-equationsystemrelativetoisan environmentdenedbystructuralinduction: xpoint-equationsystemovera,and:x!aanenvironment. [(X=f)E]def [(X=f)E]def []def X:f([E])=Wfajaf([E][X=a])g X:f([E])=Vfajaf([E][X=a])g where=[e][x=x:f([e])] = =[E][X=X:f([E])]

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