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2亿+学生的选择
Julia’s Chinese name is Liu Min .She is in Class 6 and her telephone number is 756－4321. She has a brother, his name is Adam. He is a tall boy, he has(有) some good friends at school. Julia and her brother like apples very much. Julia’s favourite colour is white, but Adam likes black. 黑色是他最爱的颜色. They are in China now, their Chinese teacher is Miss Wang. She is a good teacher.小题1:把第（1）句译成汉语。_____________________________________________________.小题2:把第（黑色是他最爱的颜色）句译成英语。____________________________________________________.回答问题。小题3:Who is their Chinese teacher?&____________________________________________________.小题4:What is Julia’s telephone number? ___________________________________________________.小题5:A& re Julia and Adam in China now?_____________________________________________________
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2亿+学生的选择
小题1:朱丽娅的中文名字叫刘敏。（答案不唯一）小题2:Black is his favourite colour.小题3:Miss Wang .小题4:It is 756—4321.小题5:Yes, they are.
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扫描下载二维码Mode Estimation of PCCAs is based on a Best-First Trajectory08
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Mode Estimation of PCCAs is based on a Best-First Trajectory08
DNNF-basedBeliefStateEst；PaulElliottandBrianWilli；(pelliott,williams)@mit.；Abstract；Asembeddedsystemsgrowinc；ThispaperintroducesanewD；Introduction；Thepurposeofestimationis；?Copyrig
DNNF-basedBeliefStateEstimation?PaulElliottandBrianWilliams(pelliott,williams)@mit.eduMITSSLandCSAILCambridge,MA02139AbstractAsembeddedsystemsgrowincreasinglycomplex,thereisapressingneedfordiagnosingandmonitoringcapabilitiesthatestimatethesystemstaterobustly.Thispaperisbasedonapproachesthataddresstheproblemofrobustnessbyrea-soningoverdeclarativemodelsofthephysicalplant,repre-sentedasavariantoffactoredHiddenMarkovModels,calledProbabilisticConcurrentConstraintAutomata.PriorworkonModeEstimationofPCCAsisbasedonaBest-FirstTrajec-toryEnumeration(BFTE)algorithm.TwoalgorithmshavesincemadeimprovementstotheBFTEalgorithm:1)theBest-FirstBeliefStateUpdate(BFBSU)algorithmhasim-provedtheaccuracyofBFTEand2)theMEXECalgorithmhasintroducedapolynomial-timeboundedalgorithmusingasmoothdeterministicdecomposablenegationnormalform(sd-DNNF)representation.ThispaperintroducesanewDNNF-basedBeliefStateEsti-mation(DBSE)algorithmthatmergesthepolynomialtimeboundoftheMEXECalgorithmwiththeaccuracyoftheBF-BSUalgorithm.ThispaperalsopresentsanencodingofaPCCAasaCNFwithprobabilisticdata,suitableforcompi-lationintoansd-DNNF-basedrepresentation.Thesd-DNNFrepresentationsupportscomputingkbeliefstatesfromkpre-viousbeliefstatesintheDBSEalgorithm.IntroductionThepurposeofestimationistodeterminethecurrent,hid-denstateofthesystem.Anestimatorinfersthecurrentstatebyreasoningoveramodelofthesystemdynamics,thecom-mandsthathavebeenexecutedandtheresultingsensoryob-servations.Inmodel-basedprogramming,themodelswrit-tenbythesystemengineerscanbeusedtodiagnosethesys-tem,usingamodeestimator.Themodeestimatoriscapa-bleofautomaticallydoingsystem-widediagnosticreason-ingandiscompletelyreusablefordifferentapplications.ThispaperintroducesanewDNNF-basedBeliefStateEstimation(DBSE)algorithmthatcombinesfeaturesoftwootheralgorithms,theBest-FirstBeliefStateUpdate(BFBSU)algorithm(Martin,Ingham,&Williams2005;Martin2005)andtheMEXECalgorithm(Barrett2005).TheBFBSUandMEXECalgorithms,inturn,areextensions?CopyrightThiswork??c2006,wasfundedAmericaninpartAssociationundergrantfor#902364.Arti?cialIntelli-gence(www.aaai.org).Allrightsreserved.oftheBest-FirstTrajectoryEnumeration(BFTE)algorithm(Williamsetal.2003).All4algorithms,DBSE,BFBSU,MEXEC,andBFTE,reasonoveravariantoffactoredHiddenMarkovModels(HMMs),calledProbabilisticConcurrentConstraintAu-tomata(PCCA)(Williams&Ingham2002).PCCAsfactortheHMMbybothrequiringthatcomponentshaveindepen-dentprobabilitydistributionsandbyencodingzeroproba-bilitiesasaconstrainttheory.TheBFTEalgorithmcomputesthekmostlikelytrajecto-ries,approximatingthemostlikelybeliefstatesasthemostlikelytrajectories.Trajectoriesarecomputedbytestingcan-didateconsistencyagainstprimeimplicatesinanOptimalConstraintSatisfactionProblem(OCSP)solver(Williamsetal.2003).Theconsistencytestusedhasaworst-casetimecomplexitythatisexponentialinthesizeofthemodel,thoughinpracticeitisoftenpolynomial.ThetrajectoryapproximationmadebyBFTEisapoorapproximationwhentwoormoreoftheleadingbeliefstatesarecloseinprobabilityandcomposition(Martin2005).Inthesecases,thetwobeliefstatesmaytransitionintothesamenextstate,butbydifferenttrajectories,astheyhavediffer-entinitialconditions.TheBFTEapproximationtreatseachtrajectorythatendsinthesamestateasadifferentestimate,whichmeansthesamestatemayappearmorethanonceinthekestimatesthatthealgorithmkeeps.Thealgorithmalsounderestimatestheprobabilityofthatstate,asithasdividedtheprobabilityacrossmultipletrajectories.TheBFBSUalgorithmimprovesupontheaccuracyoftheBFTEalgorithmbyinsteadenumeratingthekbestbeliefstatesusingthesameOCSPsolverinsteadoftrajectories.BFBSUalsoimprovesupontheaccuracyoftheBFTEal-gorithmbyestimatingtheprobabilityofgettinganobserva-tionforaparticularstate.BFBSUusesasparseobservationprobabilitytableforthispurpose.Thetableapproximatesthenumberofpossibleobservations,givenastate,sotheprobabilityofanobservationisoneoverthisvalue,assum-ingauniformdistribution.ThemicroexecutiveMEXECalgorithmreducesthecom-plexityboundoftheBFTEalgorithmbyreplacingtheprimeimplicatesusedintheOCSPsolverwithasmoothdetermin-isticdecomposablenegationnormalform(sd-DNNF)(Dar-wiche2001)representation.Thesd-DNNFrepresentationsupportsthequeriesneededbytheBFTEalgorithmwithatimeboundthatispolynomialinthesizeofthesd-DNNF.TheMEXECalgorithm,like(Kurien&Nayak2000),alsoaddstheabilitytoestimateoveranN-steptimewindow,whereBFTEwasonlyabletohandleasinglestep.ThehardpolynomialupperspaceandtimeboundsoftheMEXECal-gorithmareespeciallyimportantforembeddedapplications,wheresuf?cientmemoryisgenerallyallocatedfortheworstcaseandresponsetimesneedtobeknown.ThenewDBSEalgorithmpresentedinthispapermergesthepolynomialspaceandtimeboundoftheMEXECalgo-rithmwiththeaccuracyofBFBSUalgorithmbyusingansd-DNNFrepresentationtoestimatethebestkbeliefstates.TheDBSEalgorithmimprovesupontheaccuracyoftheBF-BSUalgorithmintwoways:1)thesd-DNNFrepresentationadmitspolynomial-timemodelcounting,sotheobservationprobabilitiesarenolongerapproximateand2)thealgorithmisabletocorrectlyhandletransitionsbetweencomponentsthatarenotfullyindependent,becausetheydependonanunobservedcondition.Thisalgorithmdoesnotincludethen-stepcapabilityoftheMEXECalgorithm.ThispaperalsoprovidesacompleteencodingofthePCCAmodelasaConjunctiveNormalForm(CNF)modelplusasetofOCSPprobabilitiesforthetransitionvariables.ThisCNFencodingisusedwiththeC2D(Darwiche2005)compilertogenerateansd-DNNFsuitableforusebytheDBSEalgorithm.ThispaperstartsoutbypresentingthepartsofthePCCAmodel.ItthenintroducesanencodingofthePCCAmodelasaCNFtheorysuitableforcomputingtheHMMupdateequations.TheCNFencodingcanbecompiledwiththeC2Dcompilerintoasd-DNNFthatissuitableforonlineestimation.ThepaperthenprovidestheDBSEestimationalgorithm,includingtherunningtime.PCCAModelWemodelthephysicalplantbeingdiagnosedasafactoredHMMthatiscompactlyencodedasaPCCA(Williamsetal.2003).APCCArepresentsasetofdiscrete,partially-observable,andconcurrently-operatingcomponentsthatareconnectedtogethertoformasystem.Eachautomatonhasasetofconditionalprobabilistictransitions,whichcapturebothnominalandfaultybehavior.APCCAisacompositionofProbabilisticConstraintAu-tomata(PCA).APCAAaisde?nedbythetripleAa=??Πa,Ma,Ta??:1.Πa=Πma∪Πcwhicha∪Πocompletelya∪Πddescribeaisa?nitesetofdiscretevariables,thecomponent.Allx∈Πahavea?nitedomainD(x).m,c,o,anddcorre-spondtomode,control,observation,anddependentvari-ables,respectively.Themodevariablesaretheestimatedvariables.Thedependentvariablesaretheintermediatevariablesneededtode?nethebehaviorofasinglecom-ponent.Wedenotethenon-modevariablesΠcodthecompletesetoffullassignmentstovariablesa.Σ(Π)isΠ,andC(Π)is2.M??thesetofallpossible??constraintsonvariablesΠ.codmapa:Σ(Πmeachmodea)→CΠ????,themodalconstraints,toaconstraintathatmustholdtruewhenthecomponentiswithinthatmode.GHeNFuel Tank2H4Pipe 1SSolenoid ValvePipe 2HydrazineCatalyst BedThrusterInertial SensorFigure1:MonopropellantPropulsionSystemSchematic3.Ta:??Σ(Πm)×C??Πcd??×Σ(Πm)→??[0,1]??repre-sentprobabilistic,aconditionalaasiderasingletransition(σttransitionfunctions.t+1Con-t,t+1thesourcemodeofm,Ct,t+1,σtthecomponentm).σattimemrep-resentst.C,thetransitionguard,representstheconditionsderwhichthistransitioncanoccur,andσtthecomponentattimet+1m+1un-isthetar-getmodeofafterthetran-sition.P??σtm+1The|σttransitionm,Ct,t+1??functionsTtoatransition.aassigntheprobabilityTransitionsindifferentcomponentsmaybeconditionedonthesamevariables,butsemantically,giventhetruthofthetransitionguards,thetransitiontakenforeachcomponentisdecidedindependentlybasedonitslocaltransitionproba-bilities.Whentwotransitionsdependonavariablewhosevalueisnotknownatthecurrenttime,thetransitionguardsareonlypartiallyknownandsoitisnecessarytoconsideredthepossiblevaluesofthevariableandhowtheyimpactthechoiceoftransitions.BFBSUassumestheguardsarealwaysindependent,anassumptionthatisonlycorrectifalljointlyconditionedvari-ablesareknowneveryestimationcycle.Whenajointlycon-ditionedvariableisnotfullyknown,BFBSUallowsimpos-sibletransitioncombinationstooccur.ThisDBSEimple-mentationrelaxesthisconditionsothatimpossibletransitioncombinationsarenevertakentogether,evenwhennotfullyknown.APCCAmodelPisde?nedbythetripleP=??A,Π,Q??.A={A1,...,AΠn}is=??the?nitesetofPCAs;onePCAforeachcomponent.P.Q∈C(Π)ais=1a..nΠconstraintaisthesetofallvariablesde?nedinoverallthevariablesthatcapturestheinterconnectionsbetweencomponents.ExamplePCCAToillustrateaPCCAmodel,considerasimpli?edmonopro-pellantthruster.AschematicofthepropulsionsubsystemisshowninFigure1.Wemodelthispropulsionsubsystemasasetoftwocomponents:afueltankandasolenoidvalve.Weassumethataproperlyopenedsolenoidvalvealwaysleadtoanominalinertialsensormeasurement.FuelTank:ThefueltankPCAmodelAinFigure2.Atankisshowngraphicallytankisde?nedbythetripleAtank=??Πtank,Mtank,Ttank??.ThevariablesareΠtank=(??xflow????= zero)??0.99??filled??0.01??(??x= positive)??empty??flow????x??p??= nominal??x??p??= zero??Figure2:FuelTankPCAAtank.?(??xvalve????-cmd??= close)??(??x(??xvp-out????= zero)??open??valve????-cmd??= close)??(??xclosed??x??0.99??valve????-cmd??= open)??vp-out??= ??xvp-in????0.01??0.99???(??xvalve????-cmd??= open)??0.01??(??xstuck??vp-out????= zero)??closed??Figure3:SolenoidValvePCAAvalve.{xtank,x?ow,xxp},wherethefueltank,representedbytank,residesinoneoftwodiscretemodes,D(x,empty}.xdescribeswhetherfuelistank)={?lled?ow?owingfromthetank.xpdescribeswhetherthetankispressurized,whichindicateswhetherornotthefueltankcontainsfuel.ThemodalconstraintMtankandtransitionsTtankarexCMtank?lledtank{(xemptyp=nominal)}{(xp=zero)}Ttankxttankxt?owxt+1p?lledzero?lledtank1?lledpositive?lled0.99?lledpositiveempty0.01empty――empty1SolenoidValve:ThesolenoidvalvePCAmodelAshowngraphicallyinFigure3,andisde?nedinamannervalveissimilartotheFuelTank.CombinedPCCAModelCombiningthesecomponents,thePCCAmodelPisde-?nedbythethreecomponents:1)A={AΠand3)Qlinksxtank,Avalve},2)Π=tank∪Πvalve,ptoxvpinandx.Thecomponentsareconnectedthroughasinglevpouttox?owpres-surevariable.Thereis?owwhenthepressureattheoutputofthevalveisnotzero.sd-DNNFModelWewillusethesd-DNNFcompilerC2D(Darwiche2005),whichconvertsaCNFtheoryintoansd-DNNF.Tousethiscompiler,thePCCAmustbeencodedasaCNFtheoryplustransitionprobabilities.Theresultingsd-DNNFencodingofthemodelpresentedhereisdesignedtoallowfortheeasycomputationoftheHMMGiventheobservationsσtbeliefstatepropagationequations.o+1andthecommandsσtpropagationequations(Baumc,thetwoHMMbeliefstate&Petrie1966)are:f??σt+1P??m??=σto+1|σtm+1????P??σtm+1|σti,σtc,σt,t+1??dg??σti??,σti∈Btk(1)t+1g??σt+1??f??σm??m=??σt+1t+1i∈Bkf??σt(2)i+1??.Equation??1,f??tm+1??,computestheprobabilityPσtm+1|σ0o,t,σ0c,t+1??σusingthedistributiong(σtwhichisapproximatedbythekbestbeliefstatesi)attimet,BtprobabilityisthennormalizedbyEquation2,g??k.Theσtisnecessaryform+1??.NormalizationignoringanormalizationtermP??twoσ0o,treasons:,σ0onlyc,t+1??1)weare,sotheresultmaynotsumto1,and2)wearecomputingthekhighestprobabilitiesforthekbestσtsomesolutions.m+1,sowearelosingInEquation1,theobservationprobabilityP??σtiso+1|σtm+1??#Models(σto+1∧σtm+1∧Bt∧σto∧σtc)#Models(σtm+1k∧Btk∧σto∧σtc),where#Modelsdesignatescountingthenumberofmod-elsofthetheoryforwhichitsargumentistrue.Inotherwords,theprobabilityisthenumberofwaysthatwecouldhavegottenourobservationdividedbythenumberofP??waystogetanyobservation.σtm+1|σti,σtc,σt,t+1??Thetransitionprobabilitydiscomputedforeachofthekpre-viousbeliefstatesσttothethebeliefstatei,wheretheprobabilityoftransitioningσttransitionfunctionm+1isencodedonaper-PCAATabasisinthea.EncodingofaPCCAModelforCompilationThissectionshowsanovelencodingofthePCCAModelasaCNFtheoryplusthetransitionprobabilities,suitableforestimatingthenextbeliefstatesusingkpreviousbeliefstates.ThePCCAmodelhasthreetypesofexplicitcon-straints:1)theglobalconstraintQ,whichisalreadyaCNFconstraint,2)themodalconstraintsMconstraintsTa,and3)thetran-sitionconstraints,theresultinga.Inadditiontoencodingtheexplicitsd-DNNFmustsupportcomputingbeliefstatesfromkpriorbeliefstates,whichwillrequireperformingsomeadditionaloperationsontheCNFpriortousingC2D.ModalConstraintsThemodalconstraintsareencodedas(xxa=v)?Ma((xa=v)),foreachmodeassignment(a=v).TransitionEncodingThetransitionsdesignateasingleprobabilisticchoiceperPCA.Choosingaparticulartransi-tionτameansthatthePCAisinthesourcemodeσtmofthetransitionattimetandisinthetargetmodeσtsitionattimet+1.Choosingthetransitionalsom+1ofthetran-meansthattheguardCt,t+1isconsistentoverthetimeinterval[t,t+1].TheprobabilityofthechoiceisTa(τa).Toencodethetransition,wecreateatransitionvariabletforeachPCAAa.Thedomainofatransitionvariableisthesetofallnon-zeroprobabilitytransitionsτturetheconstraintsimposedbythetransition,a∈Twea.Tocap-constructtheconstraint(t=τa)?σtm∧σtm+1∧Ct,t+1,foreachtransitionassignment(t=τa).ThetransitionprobabilityP??isneededtocomputeσtm+1|σti,σtc,σt,t+1??d.ThisprobabilitycannotbeencodedintheCNFandsoitisstoredexternally.Thereisaone-to-onecorrespondencebetweenassignmentsσeasytoassociatetheprobabilitiestandprobabilities,henceitiswiththesd-DNNFaftercompilation.BeliefStatesEncodingDBSEsupportstrackingthekbestbeliefstates,wherekisaninputparametertothecom-pilationprocess.Abeliefstaterepresentsapossiblestateofthesystem,andfromeachofthekpossiblepreviousbeliefstates,theDBSEalgorithmcomputesthekbestnextbe-liefstates.Tosupportkindependentpreviousbeliefstatesalltransitioningtothesamenextbeliefstate,weencodeksetsofthemodelvariablesattimet,alongwithksetsoftheconstraintsC1,...,Ckcontainingthosemodelvari-ables.Thepreviousbeliefstatesmuststillshareasingleco-herentnextstate,sotheremainingvariablesandconstraintsareunchanged.Thechoiceofwhichpreviousbeliefstateledtoaparticu-larstateisencodedbycreatingavariablebwiththedomain[1,k].Choosingabeliefstateimpliesthatthepreviousbe-liefstateassignmenthasbeenchosenwiththeprobabilityofthepreviousbeliefstate.Usingthisvariableb,thechoiceisencodedas(b=i)?Cduplicatei,foreachbeliefstatevariableas-signment(b=i)andithconstraintCresponsibleforassociatingtheithi.TheonlinealgorithmisbeliefstatewiththevariablesoftheconstraintCi.Theonlinealgo-rithmcandirectlyassociatethebeliefstateprobabilitywith(b=i).sd-DNNFRepresentationThecompiledPCCArepresentationconsistsofthetwoparts:1)ansd-DNNFrepresentationDoftheCNFencod-ing,asoutputfromC2D,and2)amappingoftransitionvariablestotheircorrespondingmodeledprobabilityP(τmodel.a),fromthePCCAEstimationAlgorithmTheonlineestimationalgorithmneedstocomputethetran-sitionandobservationprobabilitiesforeachcandidate.Thetrueprobabilityofthecandidateistheproductofthetwo,perEquation1.Traditionally,onecomputesthe?rstandthenthesecond,asoneneedstoknowwhattheexpectedstateisinordertocomputetheobservationprobability.Akeybene?tofusingsd-DNNFsinthisestimationprocessisthattheobservationandtransitionprobabilitiescanbecom-putedinparallel,inafactoredfashion,foreverycandidate,allowingtheestimationalgorithmtoselectthebestcandi-datesef?cientlyusinganexactmetricoftheirfactoredprod-uct.Speci?cally,thesd-DNNFtheoryisfactoredintohier-archicallyindependentsetsofdeterministicor-nodes,whichcanbechosenbetweenarbitrarily.Wecancomputetheob-servationprobabilityofeachchoiceatthesametimeasthetransitionprobabilityofeachchoice,andthenef?cientlyde-cidebetweenthechoiceslocallybasedontheproductofthetwoprobabilities.Webrie?yexplainwhythedeterministicnodescanbede-cidedindependently.Wedistinguishbetweentwotypesofor-nodes:deterministicandindistinguishable.Adetermin-isticor-nodeisoneinwhichdifferentpathsensuredifferentmodelsarechosenovertheremainingvariables,inourtheestimatedvariablesσtm+1case,incompliancewiththedeter-ministicrequirementofthesd-DNNF.Anindistinguishablenode,bycontrast,isadisjunctionoverindistinguishablemodels,whichhavebecomeindistinguishablethroughpro-jection,andarenotallowedtobepartofapropersd-DNNF.Whenprojectingansd-DNNF,thesd-DNNFisguaranteedtoonlyhaveindistinguishablenodesattheleavesifthepro-jectedvariablesareall?rstinthedecomposition.IntheDBSEalgorithm,allvariablesarevaluesexcepttheσt+1summed-outorassignedinthedecomposition.mvariables,sothesevariablesmustappear?rstTomakethesd-DNNFcompliantagain,weneedtomakeasinglewalkthroughtheDNNFandsum-outtheindistinguishablenodes.Tosum-outanindistinguishablenode,weaddbothmodelcountsandthetransitionprobabilitiesofallthenode’schil-drentogetherintoasinglepairofmodelcountsandasingletransitionprobability.Theindistinguishablenodecanthenbereplacedbyanewnodecontainingthesethreevalues.Theand-nodesthataredescendantsalsoneedtobeelimi-natedbymultiplyingtheirchildren,ratherthanadding.Theand-node,too,canbereplacedbyanewnodecontainingtheproductofitschildren.Oncewehavechosenapathforeverydeterministicnodeandeliminatedeveryindistinguishablenode,wecancol-lapsetheentiresd-DNNFintoasingleand-nodewithalistofliteralsasitsleaves.Sinceand-nodesmultiplytheirchil-drenandmultiplicationisassociative,thetransitionandob-servationprobabilitiesofeachor-nodewillbemultipliedintothe?nalanswer.Thus,theprobabilitiescanbecom-putedindependently,perdeterministicnode,andthedeci-sionforeachdeterministicnodecanbemadebasedonthelocalmultipliedpairofprobabilities.OnlineTheonlinealgorithm,showninFigure4,isresponsibletakingansd-DNNFD,uptokbeliefstatesBtforσtk,thepreviousthenoandnextσtcomputingo+1observations,andthecommandsσttheknextbeliefstatesBtcincrementalandonlyrequiresk+1and.Thealgorithm,therefore,iskeepingtrackoftheprevioustimestep.The?rststepofthealgorithmassignsthekbestbeliefstatesfromthepreviousstatetothekbeliefstatevariables.Btk+1DBSE??D,Btk,σto,σtc,σto+1??1.D←D|(σtm∈Btk)∧σto∧σtc2.AssignP(Btk(i))to(b=pute#Models(D),#Models??D|σto+1??,andP(D)4.D←D?Πmt+15.Btk+1←EnumeratethekbestmodelsofD6.NormalizeandreturnBtk+1Figure4:Analgorithmthatcomputeskbeliefstatesusingasd-DNNFmodel.Italsoassignstheobservationsgatheredatthepreviouspointintimeandthecommandsthathavesincebeenissued.As-signmentisperformedbyconditioningthesd-DNNFontheconjunctionoftheassignments.Thesecondstepassignsthebeliefstateprobabilitiestothebeliefstatevariables.Inthethirdstep,wewalkthroughthesd-DNNFDandcomputethethreepartsoftheupdateequation,Equation1:1)Thedenominatoroftheobservationprobability,2)thenu-meratoroftheobservationprobability,and3)thetransitionprobability.Asdiscussedearlier,thesecanonlybecom-binedintoasingleprobabilityatdeterministicnodes,wherewecanguaranteetheonlyoperationismultiplication.ThevaluesarecomputedforeachnodeintheDNNF.ThefourthstepprojectstheDNNFontothemodevari-ablesthatweareestimating.TheprojectionusestheDNNFprojectalgorithm,exceptwekeeptrackofthenumberofmodelsthatweareprojecting,ascomputedinstep3.Were-fertothisassumming-outthevariables.Inthisprocess,allindistinguishablenodesareeliminated,astheyarenolongerneeded.Thisensuresthateveryor-nodeisadeterministicnode,choosingbetweendifferentmode-variableinstantia-tions,soeachmodelofthesd-DNNFappearsexactlyonce.The?fthstepinvolvesenumeratingthekbestmodelsofD,usingthevalueofEquation1computedforeachdeter-ministicnode.Alinearprogrammingalgorithmisusedtocomputethekmostprobablechoicesforeachnodebasedonthecombinationofthekbestcandidatesofitschildren.Nodeswillonlyhaveanestimateiftheyhaveadeterministicnodeasachildoriftheyarethemselvesdeterministicnodes.Whenthealgorithmcompletes,therootnodewillhavetheprobabilitiesofkmostlikelybeliefstatesandwhichchil-drenledtothoseprobabilities.ItthencanwalkbackthroughtheDNNFandbuildasetofkbeliefstatesbasedonwhichchoiceswerepartofeachestimate.The?nalstepre-normalizesthebeliefstatessotheproba-bilitiessumto1.ThisstepperformsthecalculationofEqua-tion2.Wethenreturnthenewkbestbeliefstates.RunningTimeThissectionaddressestherunningtimeofthealgorithm.The?rststepofthealgorithminFigure4requiresasin-gleiterationthroughD,assigningeachleafliteralavalueoftrueorfalse.ThesecondstepsassignskprobabilitiestokFigure5:Thenumberofnodesandedgesneededtorep-resenttheMarsEDLmodelasafunctionofthenumberofbeliefstateskencodedintheCNFformula.Thenumberofnodesis71×103×kandthenumberofedgesis244×103×kfork≥4,withanRMSerrorof2.5×103and4.7×103,re-spectively.Fork=1,2,thenumberofnodesis4.4×103×kandthenumberofedgesis14.8×103×k,withanRMSerrorof370and340,respectively.leafliterals,andthusthetimetakenisO(k).ThethirdsteprequiresoneiterationthroughD,visitingeveryedgeonce.Thefourthstep,likethethird,makesonepassthroughD,visitingeachedgeandleafonce.The?fthsteprequirestwoiterationsoverD.The?rstiterationidenti?esthebestbe-liefstatesandthenthesecondextractsthebestbeliefstates.The?rstiterationrequiresO(k)spaceperedgeandnodeofthesd-DNNF.Themergingofthechild’sestimateswiththeparent’sestimatesrequiresO(k)timeperedgeforor-nodeparentsandO(k2)timeperedgeforand-nodeparents.Theseconditerationpropagatestheselectionattherootdowneachedgeforeachofthekestimates.The?nalsteprequiresaveragingthekprobabilities.ForaDNNFwithnnodes,eedgesandsupportingkbeliefstates,intotaltheDBSEalgorithmhasaspacecomplexityofO(kn+ke)andatimecomplexityofO(kn+k2e).Sincen≤eforallsd-DNNFs,wecansimplifytheseboundstoO(ke)andO(k2e)forspaceandtime,respectively.PreliminaryResultsTheencodingpresentedinthispaperhasbeenusedtogen-erateasetofsd-DNNFsusingaMarsEntry,Descent,andLanding(EDL)model(Ingham2003),whichisroughlythesizeofaspacecraftsubsystem.Themodelhasforty-twovariables,ofwhichtenaredependentvariablesandtenarestatevariables,withanaveragedomainsizeof4.4.Fortheset,wevariedkfrom1to30.TheC2DalgorithmemploysrandomizationingeneratingadecompositionoftheCNFtheory,andwasgivennoguidanceastowhichdecompo-sitionstoconsider.Thus,tomakethedatamoreregular,theC2Dcompilerwasrun10timesoneachCNFformula.ThenumberofnodesandedgesofthesmallestmodelwaskeptandplottedinFigure5.ItcanbeseeninFigure5thatthesd-DNNFislinearink.Fork≥4,thenumberofnodesis71×103×kandthenumberofedgesis244×103×k.TheRMSerrorfortheslopeofthelineis2.5×103and4.7×103forthenumberofnodesandedges,respectively.The?rsttwodatapoints,however,tellasomewhatdifferentpicture.Forthe?rsttwodatapoints,whichhaveonly2or3copiesofthevariablesandconstraints,respectively,theC2Dcompilerisableto?ndamuchsmallerencodingthanfortheremainingdatapoints.Forthese?rsttwopointsthenumberofnodesis4.4×103×kandthenumberofedgesis14.8×103×k,withanRMSerrorof370and340,respectively.Forthese?rsttwovaluesofk,theC2D’sbinaryvariablerepresentationisabletoencodeexactlythepossibleassignmentstothebeliefstatevariable.Thus,theC2Dcompilerisforcedtoidentifyasingledecisionpointatwhichthemodelbreaksintokinde-pendentpieces.Forlargerk,thedecisionastowhichbeliefstateisactiveisspreadacrossmanydecisionpoints,whichwebelieveiswhatleadstoalessef?cientdecomposition,asrepresentedbythelargerslope.Webelievethatgeneratingconsistentlysmallsd-DNNFs,atasizeproportionaltothe?rsttwodatapoints,ispossiblebyguidingthedecomposition.TheCNFformulacontainsk+1copiesofthesamevariablesandk+1copiesofthestateconstraintsandglobalconstraints.Italsohaskcopiesofthetransitionconstraints.TheC2Dcompileriscurrentlyexpectedto?ndthelargenumberofsymmetriesandthenorderthedecompositiontotakeadvantageofthem.Thelinearityofthesd-DNNFwithrespecttokallowsustoexpressthespaceandtimeboundsgivenintheprevioussectionmorepreciselyintermsofthemoreprecisemodelsize.ThenewspaceboundisO(k2e(k3ethesizen)andthenewtimeboundsisOn),whereenisofthemodelfork=1.FortheMarsEDL3model,enwasshowntobeap-proximately244×10fork≥4.However,webelievethatbyprovidingguidancetotheC2Dcompiler,3ereducedtothek=1,2modelsizeof14.8×10edges.nmaybeWenowcomparetheseresultstothetwopreviousalgo-rithms.TheBFBSUalgorithmdisplayedalinearspaceandtimeboundasafunctionofkin(Martin2005).FortheMEXECalgorithm,alinearspaceandtimeboundasafunc-tionofthesizeofthesd-DNNFwaspublishedin(Barrett2005).Withinthesetimeandspacebounds,theMEXECal-gorithmcanextractanyoneofthebest,equally-likelylead-ingcandidates.TheMEXECalgorithmisnotabletoex-tractthenextmostlikelycandidate(s).Thesd-DNNFoftheMEXECalgorithmisassumedtobelinearasafunctionofn,thesizeofthetime-stepwindowthattheestimationalgo-rithmuses.ForcomparisonwiththeMEXECalgorithm,letk=n.TheDBSEalgorithmrequiresktimesmorespaceandk2moretimethantheMEXECalgorithm.Thisextracomputa-tionaloverheadisnecessarytostoretheinformationneededtoextractthekbestestimates,ratherthanjustthesetofestimatesthatallsharethesamelikelihood.Thenumberofestimatesthatsharethesamelikelihoodvariessigni?-cantlyfromestimationcycletoestimationcycle,depend-ingonhowmuchuncertaintythereisastothecurrentstate.WithrespecttoBFBSU,theDBSEalgorithmrequiresktimesmorespacethanBFBSU,whichispresentlyneces-sarytostorekcopiesofthecompiledconstraints.However,unlikeBFBSU,whichhasexperimentallyshownanaveragecasebound,thisspaceboundisaknownupperbound.Sim-ilarly,thek2factormorethantheBFBSUalgorithmforthetimeboundisanknownupper-bound,asopposedtoBF-BSU’stimebound.ThealgorithmalsocomputesamoreaccurateestimatethanBFBSU.ConclusionThispaperhaspresentedtheDBSEalgorithmforestimatingthekbestbeliefstatesusingansd-DNNFrepresentation.Thealgorithmhasaspacerequirementthatisquadraticinktimesthesizeofthe(k=1)sd-DNNF.Ithasarunningtimethatiscubicinktimesthesizeofthe(k=1)sd-DNNF.WhilekeepingthepolynomialtimeboundsoftheMEXECalgorithm,thisalgorithmimprovestheaccuracyofboththeBFBSUandtheMEXECalgorithms.ReferencesBarrett,A.2005.Modelcompilationforreal-timeplan-ninganddiagnosiswithfeedback.InKaelbling,L.P.,andSaf?otti,A.,eds.,IJCAI,.ProfessionalBookCenter.Baum,L.,andPetrie,T.1966.Statisticalinferenceforprobabilisticfunctionsof?nite-stateMarkovchains.An-nalsofMathematicalStatistics37:.Darwiche,A.2001.Decomposablenegationnormalform.J.ACM48(4):608C647.Darwiche,A..20.http://reasoning.cs.ucla.edu/c2d.Ingham,M.2003.TimedModel-basedProgramming:ExecutableSpeci?cationsforRobustMission-CriticalSe-quences.PhDthesis,MassachusettsInstituteofTechnol-ogy,DepartmentofAeronauticsandAstronautics.Kurien,J.,andNayak,P.P.2000.Backtothefutureforconsistency-basedtrajectorytracking.InAAAI/IAAI,370C377.Martin,O.;Ingham,M.;andWilliams,B.2005.DiagnosisasApproximateBeliefStateEnumerationforProbabilisticConcurrentConstraintAutomata.InProceedingsoftheAAAI.Martin,O.2005.Accuratebeliefstateupdateforproba-bilisticconstraintautomata.Master’sthesis,MassachusettsInstituteofTechnology,MITMERS.Williams,B.C.,andIngham,M.D.2002.Model-basedProgramming:ControllingEmbeddedSystemsbyReason-ingAboutHiddenState.InEighthInt.Conf.onPrinciplesandPracticeofConstraintProgramming.Williams,B.C.;Ingham,M.;Chung,S.H.;andElliott,P.H.2003.Model-basedProgrammingofIntelligentEm-beddedSystemsandRoboticSpaceExplorers.InProceed-ingsoftheIEEE,volume9,212C237.三亿文库包含各类专业文献、各类资格考试、应用写作文书、中学教育、幼儿教育、小学教育、文学作品欣赏、Mode Estimation of PCCAs is based on a Best-First Trajectory08等内容。