Acommonvericationproblemforhardwaredesignsistodetermineifevery
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1 OnCombiningFormalandInformalVerication JunYuan? bolicmethodsforthevericationofinvariants.themotivationistwo-fold. conjunctionwithtraditionalsimulationresultsinbetter\coverage"relativetothecomputationalresourcesused.additionally,evenondesigns First,therearedesignswhicharetoocomplextobeformallyveried Abstract.Weproposealgorithmswhichcombinesimulationwithsym- usingsymbolicmethods;howeverbytheuseofsymbolictechniquesin JianShen??JacobAbraham??AdnanAziz?? 1Introduction Inthispaperwewillbeconcernedwiththeproblemofdesignverication;specically,theproblemofinvariantcheckingovergate-leveldesigns.Traditionally, designshavebeenveriedbyextensivesimulation.whileoeringthebenetsof designs,thefractionofthedesignspacewhichcanbecoveredinthismethodo- passedextensivesimulation,butwerestillfoundtocontainbugs[4].thishasled simplicityandscalability,simulationoersnoguaranteesofcorrectness;forlarge simulation. whichcanbesymbolicallyveried,theuseofahybridmethodologyoftendetectsthepresenceofbugsfasterthaneitherformalvericationor totheproposalof\formalmethods"fordesignverication;theadjectiveformal logybecomesvanishinglysmall.indeed,therearemanyexamplesofdesignsthat structures"suchasbinarydecisiondiagrams(bdds)toecientlyrepresentand referstotheunambiguousspecicationofthesystemandthepropertiesbeing manipulatethestatespacesofdesigns[1].theprimarylimitationofbddbased cessfullyverifyalargenumberofcomplexdesignsistheuseof\symbolicdata checked,togetherwiththevalidationstepgeneratingamathematicallyrigorous approachestoinvariantcheckingisthatformanydesigns,thebddsconstructed wellonspecicclassesofdesigns.onemethodwhichhasbeenusedtosuc- proofofcorrectness. inthecourseofvericationcangrowextremelylarge,resultinginspace-outsor tured,andthiscanbeexploitedtodeviseheuristicprocedureswhichperform high;pspace-completetobeprecise.inpractise,manydesignsarewellstruc- severeperformancedegradationduetopaging[12]. ndingbugsinthemasearlyaspossible.ashenzingerhaspointedout,\falsi- Facedwiththetwindilemmasofdiminishedcoveragethroughsimulationand Practicingveriersarelessconcernedwithformallyverifyingdesignsthan
2 asquicklyaspossible. certicateofcorrectnessiftheinvariantpasses,oracounterexampleifitfails. bestressedthatneithermethodiscomplete,i.e.,guaranteedtoprovideaformal askhowbesttocombinesymbolicmethodswithsimulation,soastondbugs theinabilityofsymbolicmethodstoformallyverifylargedesigns,itisnaturalto However,allreportedviolationsoftheinvariantaretruebugs. performsapartialtraversalofthestatespace.ateachstep,symbolictechniques betheprogramcounterandstatusbitsinamicro-processor.theprocedure designatesasubsetofthelatchesasbeing\interesting";thesecould,forexample, Inthispaperweprovidetwoanswerstotheproblemposedabove.Itisto wealsodescribeanextensionthatvisitsallcontrolleredges.heuristically,the areusedtocomputethefullsetofcontrolstatesreachablefromthecurrentset; controlportionofthedesign,whilebeingmuchsmallerthanthedatapath,isthe mainsourceofdesignerrors.saturatedsimulationattemptstoexploreasmuch ofthecontrolstatespace,thusincreasingthelikelihoodofndingbugs. Werstdevelopthemethodofsaturatedsimulation,whereinthedesigner theentirestatespace.additionally,fastbddroutinesexistforgeneratingand coupledwiththefactthatthesetofcontrolstatesistypicallymuchsmallerthan manipulatingrepresentativeelementsofequivalenceclasses[8]. tocomputethesymbolicimageofasinglestateevenforverylargedesigns, Theeciencyofthisapproachcomesfromtheobservationthatitisfeasible somebngrowslargerinsizethana(user-specied)thresholdvalue.cyclesimulationisperformedfromaninitialstate;simulationishaltedifastatewhichlielysis[15].startingfromb,thecomplementoftheinvariant,successivepreimagesb;b1;b2;:::arecomputedsymbolically.thisisdonetillthebddfor Bn,sinceeverystateinBncanreachastateinB.Wedescribegreedysearch strategiesforndingpathstobnfromaninitialstatewhichusehammingdistanceasametrictobeminimized.theprimarybenetofretrogradeanalysisis thattheset[ibiistypicallymuchlarger(inthesenseofcardinality)thanb; Wethendescribeanorthogonalapproachreferredtoasretrogradeana- hence,inaheuristicsense,bnoersamuchlarger\target"forsimulation. givenabove. resultsunderlinetheeectivenessthatissuggestedbytheheuristicarguments rogradeanalysisarenoveltothispaper.wehavebeeninuencedbyanumberof relatedworks.thompson's[15]workonretrogradeanalysisprovidedtheinitial impetus.additionally,wewereinuencedbythedramaticimprovementsmade tocyclesimulationbytheuseofbddsbyasharandmalik[1],andmcgeeret TheseroutineshavebeencodedontopofthetoolVIS[2].Ourexperimental alargenumberofstates.thisisdistinctfromourapproach,whereinasubsetis setsencounteredduringreachabilityanalysiswhichhavesmallbddsbutcontain memoryavailableonthemachine.ravietal.[13]attempttopicksubsetsofstate al.[9],whomadecleartheimportanceofmakingmaximumuseofthephysical Tothebestofourknowledge,theprinciplesofsaturatedsimulationandret- chosenwhichattemptstomaximizethenumberofdistinctcontrollerstates.cho etal.[5]picknetstoabstractintoprimaryinputs,consequentlyobtainingsuper-
3 setsofthesetofreachablestates.theworkofhoetal.[6]andhoskoteetal.[7] oncreatingsimulationvectorswhichexcitealargenumberoftransitionsonthe thanstatestoobtaingoodcoverageofcontrollerbehavior.however,theyused controllerstatesofadesignsuggestedtheusefulnessofusingtransitionsrather designersupplied\translationfunctions",ortest-basedtechniquestogenerate simulationinputsequenceswhichexcitedasmuchofthecontrolaspossible;our 2Background InvariantVerication approachisrootedinsymbolicmethods. associatedbooleanfunction.anitestatemachinecanberepresentedbyan aninterconnectedsetofprimaryinputs,gates,andlatches.eachgatehasan atthebehaviorallevelusingnitestatemachines(fsms).anetlistconsistsof forcomputationalmodelsforhardware. Inordertobeabletoanalyticallyreasonabouthardware,werstneedtodevelop edge-labeleddirectedgraph,wheretheverticescorrespondtostates,andthe mathematicalmodelsfordigitalsystems.singhal[14]givesadetailedexposition labelsareinput-outputpairs. Hardwaredesignscanbemodeledatthestructurallevelusingnetlists,or implementedsymbolicallyusingreducedorderedbinarydecisiondiagram[3]. thesetofallstateswhichcanbereachedfromabyapplyinganinputsequenceof lengthone.similarly,thepre-imageofa(denotedbypreimg(a))isthesetoall stateswhichcanreachainonestep.theimgandpre-imageprocedurescanbe state/outputfunctionsarederivedbycomposingthegatefunctions. chinefromit;statesareevaluationstothesetoflatchvariables,andthenext- GivenadesignDandasetofstatesA,theimageofA(denotedbyImg(A))is Foragivenanetlist,thereisanaturalwayofderivinganitestatema- alllieintheinvariant.analternateapproachtocheckinginvariantsisbasedon icallycomputeallstatesreachablefromtheinitialstatesanddeterminethatthey backwardanalysis,whereinthesymbolicpreimgoperatorisiterativelyapplied todetermineallstateswhichcanreachthecomplementoftheinvariant;the asinvariantverication,orassertionchecking. of\goodstates"(referredtoastheinvariant).thisproblemisvariouslyknown statereachablefromadesignatedsetofinitialstateslieswithinaspeciedset Onestraightforwardsolutiontotheinvariantcheckingproblemistosymbol- Acommonvericationproblemforhardwaredesignsistodetermineifevery thecourseofimagecomputationscangrowverylarge. 3SaturatedSimulation Manydesignscanbeseparatedinto\control"and\datapath"asillustratedin Figure1;furthermore,thedesignerisawareofthisdichotomy.Formostsuch invariantfailsiftheinitialstateliesinthisset. TheprimarylimitationofbothapproachesisthattheBDDsencounteredin
4 Inthissection,wedescribeanapproachwerefertoas\saturatesimulation";this totalnumberoflatches;however,thecontrolportioniswherebugsusuallyoccur. designs,thenumberoflatchesinthecontrollerisusuallyasmallfractionofthe PC approachattemptsheuristicallytoexploreasmuchofthecontrolportionofthe Mem Out designaspossible. Fig.1.PartitioningadesignintoControlandDatapath. Mem Logic ALU OP canbenaturallydesignatedcontroland21whicharedata.hence,thereareno R ALU Registers verylargedesignstocomputetheimageofasmall(inthesenseofcardinality) morethan512dierentpossiblevaluesforthecontrolstate.itisfeasibleevenfor Asanexample,considerthevipermicroprocessor.Itcontains9latcheswhich ALU out M setofstates.inpart,thisfollowsfromthefactthattheconstructionofthebdd forthenext-statelogiccanberestrictedtothecurrentsetofstates.thissuggests Mem Addr thatitmaybepossibletoperforma\partial"reachabilityanalysis,inwhichall distinctcontrolstatesarepreservedateachstep. givenbyanevaluationtoxc[xd. respecttoaif(8c:8d)[(c;d)2a!(9d)[(c;d)2a]] thevariablesassociatedwiththedatapathbexd.thusthestateofthedesignis Denition1.LetAbeasetofstates.AsubsetAofAiscontrol-saturatedwith LetthevariablesassociatedwiththecontrolportionofthedesignbeXcand ationsina,andisassmallaspossible(inthesenseofcardinality).anexample ofaisagoodrepresentativeset itincludesallthedistinctcontrollercongur- ofacontrol-saturatedsubsetisgiveninfigure2(a). ofa.letfbeabooleanfunctiononvariablesx=fx1;x2;:::;xng.linet curringinaoccursina.thuscontrol-saturatedsubsetsofapreserveallthe controllerstatespresentina.heuristically,aminimalcontrol-saturatedsubset Intuitively,Aisacontrol-saturatedsubsetofAifeverycontrolstateoc- whichtakesabddforfandasubsetxxofthevariables,andreturnsa BDDforafunctionfwhichhasthepropertythat al.[8]presentedanecientalgorithm(referredtoasthecprojectoperator) Wenowaddresstheproblemofcomputingminimalcontrol-saturatedsubsets Status Bits
5 BDDtfunctionCntrlSatSim(A,CntrlVars,G)f /*AinitializedtotheBDDfortheresetstates.*/ /*GistheBDDfortheinvariant.*/ Fig.2.Minimalcontrolsaturatedsubsets. cproject cproject Data value First two bits are control, last three are data 11 A ={(111),(1),(1111), (1),(11),(1)} 1 1 A ={(1),(1),(1)} State Space = 16 (a) Ex. Control saturated gif(bddequal(r,a)) if(bddintersects(a,g))/*invariantfails!!*/ ImgA:=BDDImg(A); R:=BDDOr(A,ImgA); returncntrlsatsim(r,cntrlvars,g); R:=BDDCproject(R,CntrlVars); assertfail; returnr; subset (b) Ex. Control reachability Sincesetscanbethoughtofintermsofcharacteristicfunctions,wewillfreely 1.foranyassignmentvtothevariablesinX,sothatf(v)=1,thereisexactly 2.forallu,f(u)=1)f(u)=1. onevaluationvwhichagreeswiththevaluationvoverthevariablesinx sothatf(v)=1,andfurthermore Fig.3.Control-saturatedsimulation. cprojectoperator.thisinturnisusedasthecurrentreachedstateset.the ReachablestatesareiterativelycomputedusingtheImgoperator;ateachstep, controlsaturatedsubsetofa. rstfewstepsareillustratedinfigure2.theprocedureisincomplete,sinceitis acontrol-saturatedsubsetofthecurrentreachedstateiscomputedusingthe greedy:minimalcontrol-saturatedsubsetsofthesetscomputedbythecproject applythecprojectoperatortosets.observethatcproject(a;xc)isaminimal operatorwillnotnecessarilybesucienttocoverallpossiblecontrollerstates. InFigure3wesketchasimplesymbolicprocedureforinvariantverication. Onesimplewayoffurtherenhancingthecoverageachievedbycontrol-saturated Control value
6 simulationistogenerateseveral\representative"controlstates.therearesimple modicationstothecprojectoperatorwhichcanachievethiseect.another iteration. Afundamentalextensiontoobtainenhancedcoverageistoperformapartial 3.1Control-edgeSaturatedSimulation approachistoapplycprojectonlytothefrontierofthereachedstatesateach reachabilityanalysisandateachsteppickasubsetoftheimagewhichpreserves all\controllertransitions"totheimagefromthecurrentset.hoetal.[6]and Abrahametal.[7]createdsimulationvectorswhichexcitealargenumberof bugswiththesevectorsunderlinestheusefulnessofusingtransitionsratherthan statestoobtaingoodcoverage.asanexample,consideramicroprocessorwhere controltransitionsindesigns;thehighqualityoftheirresultsintermsofnding thecontrolstateisthevalueoftheprogramcounter.twostateswhichcorrespond withdierentdatavalues;inthiscase,itisnaturaltokeeptheresultingstates todierentlinesintheprogrammaybothtransitionthesameprogramline Denition2.LetAbeasetofstates.AsubsetBofImg(A)issaidtobe dierent. control-edgesaturatedwithrespecttoaif (8c:8d:8c:8d)[(c;d)2A^(c;d)2Img(f(c;d)g)]! (9d:9d)[(c;d)2B^(c;d)2A^(c;d)2Img(f(c;d)g)] Wenowdescribehowtoexploreedgesinthecontrolstatespace. inbandastate(c;d)inasothat(c;d)!(c;d). everytransition(c;d)!(c;d)fromatoimga,thereisastate(c;d) InEnglish,theabovedenitionsaysthatBiscontrol-edgesaturatedwhenfor foreverycontrollatchxc,addanewlatchxswhich\shadows"xc,thatis, subsetisgiveninfigure4. thenextstateofxsisthepresentstateofxc.denotethesetofshadowstate andisassmallaspossible.anexampleofaminimalcontrol-edgesaturated distinctcontrollercongurationsresultinginimg(a)fromtransitionsfroma, thecontrollertransitionsoriginatingata.heuristically,aminimalcontrol-edge saturatedsubsetofimg(a)isagoodrepresentativeset itincludesallthe Minimalcontrol-edgesaturatedsetscanbecomputedaugmentingthedesign: Thusinsomesense,control-edgesaturatedsubsetsofImg(A)preserveall variablesthusintroducedbyxs.clearlythenext-stateofthelatchesindexed byxc[xdisindependentofthatoftheshadowlatches.thefollowinglemma demonstratesthatminimalcontrol-edgesaturatedsetscanbecomputedfrom theaugmenteddesign. Lemma3.LetAbeAliftedfromXc[XdtoXc[Xd[Xs.DeneBtobe minimalcontrol-edgesaturatedwithrespecttoa. theexistentialquanticationofcproject(img(a);xc[xs)byxs.thenbis
7 Proof.Observingthatcproject(?;)isalwayssubsetof?,itfollowsthat non-shadowlatchesdoesnotdependontheshadowlatches,itfollowsthatthe subsetofimg(a). existentialquanticationofimg(a)byxsisequaltoimg(a),andsobisa cproject(img(a);xc[xs)isasubsetofimg(a).sincethenextstateof (namelyb)willcontain(c;d).since(c;d)liesintheimageof(c;d), tifyingthexsvariablesfromcproject(img(a);xc[xs),theresultingset Img(A);letitlieintheimageof((C;S);D).Hence,onexistentiallyquan- Xs)containsastateoftheform((C;C);D).Note((C;C);D)liesin foranarbitraryassignmentstotheshadowlatches.hencecproject(img(a);xc[ Fromtheconstructionoftheaugmenteddesign,((C;C);D)isinImg((C;S);D) atransitionfrom(c;d)2ato(c;d),i.e.,(c;d)2img(f(c;d)g). (C;D)satisfythe\if"portionoftheimplicationinDenition2.Thenthereis WenowshowBiscontrol-edgesaturatedwithrespecttoA.Let(C;D)and previoussection. indenition2. DandDareexistentialwitnessesforthe\then"portionoftheimplication WecodedtheroutinesdescribedintheprevioussectionaspartoftheVISprogram[2].Resultsareprovidedontwobenchmarks{the885,andvipermicroprocessors.The885isapproximately4gateequivalents,andcontains242 MinimalityofBfollowsfromthepropertiesofcprojectdescribedinthe 3.2ExperimentalResults{SaturatedSimulation latches,ofwhich33wereidentiedasbeingcontrol.theviperisalso4gate 128MBytesofmainmemory.Atimeoutof2secondswasusedforallviperexperiments,and1secondsfor885experiments.Sifting-baseddynamic reorderingwasenabledthroughouttheexperiments. perimentswereconductedonanultrasparc1,witha17mhzprocessor,and equivalents,andcontains218latchesofwhich9werefromthecontrol.allex- theuseofacontrol-statesaturatedsimulation(asgiveninfigure3).for885,we fourreachabiltysteps,whichwereeasilyperformed.)table2presentsresultson forviperintable1stemsfromthefactthattheprogramtimedoutaftertherst anorderofmagnitudemore.table3presentsresultsontheuseofcontrol-edge BDDencounteredduringreachabilityanalysis.(TheabnormallylowpeakBDD computealmosttwiceasmanyreachablecontrolstatesandtransitions;forviper, saturatedsimulation.inthesametime,moreedgesarevisited;thiscomesatthe analysisonthetwobenchmarks.peakbddisthenumberofnodesinthelargest Table1presentsresultsontheuseofacompleteBDD-basedreachability expenseofhighermemoryconsumptionwithrespecttocontrol-statesaturated simulation.interestingly,fewercontrolstatesarevisited;weascribethistothe factthatthecontrol-statesaturatedsimulationisfaster,andsomanagestogo deeperintothestatespaceinthesameamountoftime;thisisseeninthedepth column. 9]inTable4.Forviper,weperformed1setsofsimulations,eachcomprising Wecomparesaturatedsimulationwithfastlookupbasedcyclesimulation[1,
8 ExamplejRchd.StatesjPeakBDDjControlStatesjjControlEdgesjdepth viper1: : ExamplePeakBDDjControlStatesjjControlEdgesjdepth viper1618 Table1.CompleteBDDbasedreachabilityanalysis worsethansaturatedsimulation. of2vectors;for885weperformed4setsoflength2.eventhoughwe gavecyclesimulationtwoordersofmagnitudemoretime,itstillperformedfar Table2.Partialreachabilityanalysisusingcontrol-statesaturatedsubsets RetrogradeAnalysis RetrogradeAnalysis(RA)isanimportantsearchtechniquedevelopedwithin theeldofarticialintelligence.initssimplestform,rarstmarksallend determiningthegametheoreticvaluesofallpositionsinthesearchspace. positions(e.g.,checkmate),andthenbymakingunmovesfromtheendpositions worksitswaybacktothepositionsfarthestfromtheendposition,ontheway B;B1;:::whereBisthecomplementoftheinvariantandBi+1=PreImg(Bi). Bl,searchforaninputsequencewhichtakesaninitialstatetoastateinBl. AnalogouslytotheWi'sabove,theBi'sareeectivelybadstates.TheB'scan themcompactly.finally,whenmainmemoryisnearlyexhausted,sayattheset growverylargeintermsofcardinality;itisnaturaltousebddstorepresent RAcannaturallybeappliedtoinvariantchecking:constructthesetsofstates tothenumberofbitsinthestate,andisindependentofthesizeofthebdd. ifastateliesinthesetdenedbyabddisveryfast ittaketimeproportional fashionliesinbl.thisapproachisillustratedinfigure5(a).notethatchecking ingfromarandominitialstate;thesearchhaltsifsomestatereachedinthis Thesimplestsearchstrategyisthesimulationofrandominputvectorsstart- Table3.Partialreachabilityanalysisusingcontrol-edgesaturatedsubsets. ExamplePeakBDDjControlStatesjjControlEdgesjdepth viper
9 ExampleSaturatedSimulation viper Time(sec)jCtlStatesjjCtlEdgesjTimeSizejCtlStatesjjCtlEdgesj Table4.Comparingsaturatedsimulationwithcyclesimulation CycleSimulation tothetargetstates,i.e.,tobl.weproposetheuseofhammingdistanceasa measureofcloseness. isthenumberofpositionsinwhichtheandvectorsdier.considertherelationsh;h1;h2;:::;hnf;1g2nwhere(;)2hki,(;)k.the RecallthattheHammingdistancebetween;2f;1gn(denotedby(;)) Amoresophisticatedsearchstrategyistopickaninitialstatewhichis\close" Fig.4.Aminimalcontrol-edgesaturatedsubset. thefollowingidentity: relationh1canbeconstructeddirectlyusingbdds.therelationhi+1satises thresholdsize.fromtheoutermostring,pickastate(say)whichisclosest reachabilityfromtheinitialstatestillthebddforreachedstatesreachesa Hence,theBDDsforH;H1;H2;:::;Hnf;1g2ncanbeeasilyconstructed; smallfortheinterleavedvariableordering. furthermoreasimpleargumentbasedoncountingcofactorsshowsthattheyare ThesearchforstatesinBlcanbeenhancedbybyrstperformingforward Hi+1=Hi[(9)[(;)2Hi^(;)2H1] tobl,andthenperformrandomcyclesimulationfrom.thisisillustrated recursivelyapplied.thisillustratedinfigure5(c). infigure5(b).insteadofcyclesimulationfrom,acombinationofsymbolic forwardreachabilityanalysiscoupledwiththethehammingheuristiccanbe A (1) (1111) (11) Input Img(A) (1) (111) (1111) (11) (111) (1) (1) (111) (1111) (11) (111) First two bits are control, last three are data Same control values, but correspond to different controller transitions => keep both
10 4.1ExperimentalResults{RetrogradeAnalysis areprovidedontwobenchmarks{mesh4isaroutingalgorithmona4by4mesh WecodedtheroutinesdescribedintheprevioussectionaspartoftheVISprogram[2],andexperimentedwithanumberofexamples.Representativeresults ofnodes,andcube4ishypercubebasedroutingprotocol.forbothexamples,we Fig.5.RetrogradesearchforInvariantchecking A init A1 A2 Ai A1 A2 A Ai A1 A2 A Bl Ai B4 choseaninvariantwhichfails. B3 Bl B2 B1 B B2 B1 B B2 B1 B State Space State Space State Space plotthenumberofsimulationtrialsneededtoreachapre-image,startingfrom aftersuccessivepre-imagesinfigure6(a);bothgrowquickly.infigure6(b)we theinitialstateagainstthenumberofpreimagestepstaken;eachtrialconsists ResultsonMesh4arereportedinFigure6.WeplotBDDsizeandcardinality Vanilla RA (b) RA Hamming; closest states hatched (c) Enhanced RA Hamming decreasesrapidly. Figures7(a)and7(b)areasbefore.InFigure7(c),weshowtheeectoftaking ofapplying1randomvectors.itisclearfromthepicturethatthisnumber oneforwardstep,andthenpickingastateintheimagewhichisclosetothetarget asopposedtoarandomstateintheimage;infigure7(d)wetaketwoforward distanceisused.interestingly,whenastateintheimageispickedatrandom, appreciabledecreaseinthenumberofsimulationtrialsneededwhenhamming steps,andthenpickastatewhichisclosetothetarget.inbothcases,thereisan theperformanceisactuallyworsethatsimplystartingattheinitialstate. TheeectofHammingdistanceisgiveninFigure7fortheCube4example. 5Conclusion Weinvestigatedwaysinwhichtocombinesymbolicvericationwithsimulation. Specically,wegaveheuristicjusticationforsaturatedsimulationandretrogradeanalysis.Experimentalevidencecorroboratesthattheseapproachesyield enhancedcoverageandrobustness.thusthecombinationofformalandinformal vericationoersbenetsnotavailableineachindependently. controllers. formalmethods,particularlytheproblemofvalidatingsoftwareforembedded Inthefutureweintendtobuilduponthethemeofrelatingformalandin-
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12 1 Target state number Simulation steps without Hamming 8 BDD size 4 BDD size Fig.7.EectofHammingDistanceonCube No. of backward steps No. of backward steps (a) (b) BDD size Simulation steps Simulation steps 4 BDD size without Hamming 4 without Hamming Simulation steps Simulation steps with Hamming with Hamming R.Ranjan,J.Sanghavi,R.K.Brayton,andA.L.Sangiovanni-Vincentelli.High 11.R.MotwaniandP.Raghavan.RandomizedAlgorithms.CambridgeUniversity K.RaviandF.Somenzi.HighDensityReachabilityAnalysis.InProc.Intl.Conf VigyanSinghal.DesignReplacementsforSequentialCircuits.PhDthesis,Uni- Press, K.Thompson.Retrogradeanalysisofcertainendgames.ICCAJournal,9(3):131{ PerformanceBDDPackageBasedonExploitingMemoryHierarchy.InProc.of thedesignautomationconf.,lasvegas,nv,june1996. oncomputer-aideddesign,santaclara,ca,november1995. eering,universityofcalifornia,berkeley,ca9472,1996. versityofcaliforniaberkeley,electronicsresearchlaboratory,collegeofengin ,1986. No. backward steps No. backward steps (c) (d) ThisarticlewasprocessedusingtheLATEXmacropackagewithLLNCSstyle
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