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Estimation of switching activity in sequential circuits using dynamic Bayesian Networks

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Estimation of switching activity in sequential circuits using dynamic Bayesian Networks
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ABSTRACT: This thesis presents a novel, non-simulative, probabilistic model for switching activity in sequential circuits, capturing both spatio-temporal correlations at internal nodes and higher order temporal correlations due to feedback. Switching activity, one of the key components in dynamic power dissipation, is dependent on input streams and exhibits spatio-temporal correlation amongst the signals. One can handle dependency modeling of switching activity in a combinational circuit by Bayesian Networks 2 that encapsulates the underlying joint probability distribution function exactly. We present the underlying switching model of a sequential circuit as the time coupled logic induced directed acyclic graph (TC-LiDAG), that can be constructed from the logic structure and prove it to be a dynamic Bayesian Network. Dynamic Bayesian Networks over n time slices are also minimal representation of the dependency model where nodes denote the random variable and edges either denote direct dependency between variables at one time instant or denote dependencies between the random variables at different time instants. Dynamic Bayesian Networks are extremely powerful in modeling higher order temporal as well as spatial correlations; it is an exact model for the underlying conditional independencies. The attractive feature of this graphical representation of the joint probability function is that not only does it make the dependency relationships amongst the nodes explicit but it also serves as a computational mechanism for probabilistic inference. We use stochastic inference engines for dynamic Bayesian Networks which provides any-time estimates and scales well with respect to size We observe that less than a thousand samples usually converge to the correct estimates and that three time slices are sufficient for the ISCAS benchmark circuits. The average errors in switching probability of 0.006, with errors tightly distributed around the mean error values, on ISCAS'89 benchmark circuits involving up to 10000 signals are reported.
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Thesis (M.S.E.E.)--University of South Florida, 2004.
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EstimationofSwitchingActivityinSequentialCircuitsusingDynamicBayesianNetworksbyKarthikeyanLingasubramanianAthesissubmittedinpartialfulllmentoftherequirementsforthedegreeofMasterofScienceinElectricalEngineeringDepartmentofElectricalEngineeringCollegeofEngineeringUniversityofSouthFloridaMajorProfessor:SanjuktaBhanja,Ph.D.NagarajanRanganathan,Ph.D.WilfridoA.Moreno,Ph.D.DateofApproval:June2,2004Keywords:Probabilisticmodel,Simulation,Clique,Inference,SamplingcCopyright2004,KarthikeyanLingasubramanian

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DEDICATIONTomylovingFatherandMother

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ACKNOWLEDGEMENTSIwouldliketotakethisgoldenopportunitytothankmymajorprofessorDr.SanjuktaBhanja.Withoutherthisworkwouldnthavebeenpossible.Shesupportedmeandhelpedmetothecore.Shegavemecompletefreedominresearchprospective.Shehashelpedmealottomouldmyselfasaresearcher.Shehastrainedmeineveryaspectofresearchlikereading,writingetc.Moreovershehasalsobeenagoodfriendtome.MysincerethankstoDr.NagarajanRanganathanandDr.WilfredoA.Morenoforservinginmycommittee.IamreallyverygratefulfortheinvaluablesupportandmotivationthatIrecievedfrommyfamily.Iwouldalsoliketothankallmyfriendsfortheirablesupport.

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TABLEOFCONTENTSLISTOFTABLESiiLISTOFFIGURESiiiABSTRACTivCHAPTER1INTRODUCTION11.1ContributionoftheThesis41.2Organization5CHAPTER2PRIORWORK6CHAPTER3DEPENDENCYMODELSFORSEQUENTIALCIRCUITS143.1BayesianNetworkFundamentals143.2ConditionalIndependenceMaps153.3CombinationalCircuitasaBayesianNetwork173.4DynamicBayesianNetwork193.5ModelingSequentialCircuit213.5.1Structure213.5.2SpatialDependencyQuantication243.5.3TemporalDependencyQuantication26CHAPTER4PROBABILISTICINFERENCE284.1ExactInference284.2HybridScheme34CHAPTER5EXPERIMENTALRESULTS37CHAPTER6CONCLUSION42REFERENCES44i

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LISTOFTABLESTable3.1.ConditionalProbabilitySpecicationsfortheOutputandtheInputLineTransi-tionsforTwoInputANDGate[2].25Table3.2.ConditionalProbabilitySpecicationbetweenStateLineSwitchingsatConsec-utiveTimeInstants:Pxskti1xskti.26Table3.3.GeneralExampleofSwitchingofaSignalXkrepresentingtheValueoftheSignalateachTimeInstantt.26Table3.4.ConditionalProbabilitySpecicationbetweenPrimaryInputLineSwitchingsatConsecutiveTimeInstants:Pxpkti1xpkti.27Table5.1.SwitchingProbabilityEstimatesateachLineofthes27BenchmarkCircuitasComputedbySimulation,bytheExactScheme,andbytheHybridEPISMethod.38Table5.2.SwitchingActivityEstimationErrorStatisticsbasedonTC-LiDAGModeling,using3TimeSlices,andHybridInferenceScheme,using1000Samples,forISCAS'89BenchmarkSequentialCircuits.39Table5.3.SwitchingActivityEstimationErrorStatisticsbasedonTC-LiDAGModeling,using3TimeSlices,andHybridInferenceScheme,using3000Samples,forISCAS'89BenchmarkSequentialCircuits.39Table5.4.SwitchingActivityEstimationErrorStatisticsbasedonTC-LiDAGModeling,using3TimeSlices,andHybridInferenceScheme,using5000Samples,forISCAS'89BenchmarkSequentialCircuits.40Table5.5.SwitchingActivityEstimationErrorStatisticsbasedonTC-LiDAGModeling,using3and10TimeSlices,andHybridInferenceScheme,using1000Samples,forISCAS'89BenchmarkSequentialCircuits.40Table5.6.ExperimentalResultsonSwitchingActivityEstimationbyDynamicBayesiannetworkModelingforISCAS'89BenchmarkSequentialCircuits(3TimeSlicesand1000Samples)usingLogicSampling[48].41ii

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LISTOFFIGURESFigure1.1.AModelforSequentialCircuit.2Figure2.1.EnhancingAccuracybyUnrolling[14].8Figure2.2.FlowchartfortheStatisticalEstimationofAveraePowerDissipationinSequen-tialCircuitsbyYuanetal.[16].11Figure2.3.TechniquesforEstimatingSwitchingActivityinSequentialCircuits.12Figure3.1.ASimpleCombinationalCircuit.18Figure3.2.TimeSliceModelwithSnapshotoftheEvolvingTemporalProcess[26].19Figure3.3.TemporalModelwithDuplicatedTimeSlices[26].20Figure3.4.(a)ASimpleSequentialCircuitanditsGraphicalModel.(b)TimeUnraveledRepresentation.(c)TC-LiDAGRepresentation.22Figure4.1.CircuitDiagramofs27.29Figure4.2.BayesianNetworkModeloftheCombinationalPartofs27forOneTimeSlice.30Figure4.3.TriangulatedUndirectedGraphStructureofs27.31Figure4.4.JunctionTreeofCliquesforOneTimeSliceModelofs27.32Figure4.5.JunctionTreeofCliquesforTwoTimeSlicesofs27.33iii

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ESTIMATIONOFSWITCHINGACTIVITYINSEQUENTIALCIRCUITSUSINGDYNAMICBAYESIANNETWORKSKarthikeyanLingasubramanianABSTRACTThisthesispresentsanovel,non-simulative,probabilisticmodelforswitchingactivityinsequen-tialcircuits,capturingbothspatio-temporalcorrelationsatinternalnodesandhigherordertemporalcorrelationsduetofeedback.Switchingactivity,oneofthekeycomponentindynamicpowerdis-sipation,isdependentoninputstreamsandexhibitsspatio-temporalcorrelationamongstthesignals.OnecanhandledependencymodelingofswitchingactivityinacombinationalcircuitbyBayesianNetworks[2]thatencapsulatestheunderlyingjointprobabilitydistributionfunctionexactly.Wepresenttheunderlyingswitchingmodelofasequentialcircuitasthetimecoupledlogicinduceddirectedacyclicgraph(TC-LiDAG),thatcanbeconstructedfromthelogicstructureandproveittobeadynamicBayesianNetwork.DynamicBayesianNetworksoverntimeslicesarealsominimalrep-resentationofthedependencymodelwherenodesdenotetherandomvariableandedgeseitherdenotedirectdependencybetweenvariablesatonetimeinstantordenotedependenciesbetweentherandomvariablesatdifferenttimeinstants.DynamicBayesianNetworksareextremelypowerfulinmodelinghigherordertemporalaswellasspatialcorrelations;itisanexactmodelfortheunderlyingconditionalindependencies.Theattractivefeatureofthisgraphicalrepresentationofthejointprobabilityfunctionisthatnotonlydoesitmakethedependencyrelationshipsamongstthenodesexplicitbutitalsoservesasacomputationalmechanismforprobabilisticinference.WeusestochasticinferenceenginesfordynamicBayesianNetworkswhichprovidesany-timeestimatesandscaleswellwithrespecttosizeWeobservethatlessthanathousandsamplesusuallyconvergetothecorrectestimatesandthatthreetimeslicesaresufcientfortheISCASbenchmarkcircuits.Theaverageerrorsinswitchingprobabilityof0006,witherrorstightlydistributedaroundthemeanerrorvalues,onISCAS'89benchmarkcircuitsinvolvingupto10000signalsarereported.iv

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CHAPTER1INTRODUCTIONTheabilitytoformaccurateestimatesofpowerusage,bothdynamicandstatic,ofVLSIcircuitsisanimportantissueforrapiddesign-spaceexploration.SwitchingactivityisoneimportantcomponentindynamicpowerdissipationthatisindependentofthetechnologyoftheimplementationoftheVLSIcircuit.Contributiontototalpowerduetoswitchingisdependentonthelogicofthecircuitandtheinputsandwillbepresentevenifsizesofcircuitsreducetonanodomain.Apartfromcontributingtopower,switchingincircuitsisalsoimportantfromreliabilitypointofviewandhencecanbeconsideredtobefundamentalincapturingthedynamicaspectsofVLSIcircuits.AmongdifferenttypesofVLSIcircuits,switchinginsequentialcircuits,whichalsohappenstobethemostcommontypeoflogic,arethehardesttoestimate.Thisisparticularlyduetothecomplexhigherorderdependenciesintheswitchingprole,inducedbythespatio-temporalcomponentsofthemaincircuitbutmainlycausedbythestatefeedbacksthatarepresent.Thesestatefeedbacksarenotpresentinpurecombinationalcircuits.OneimportantaspectofswitchingdependenciesinsequentialcircuitsthatonecanexploitistherstorderMarkovproperty,i.e.thesystemstateisindependentofallpaststatesgivenjustthepreviousstate.Thisistruebecausethedependenciesareultimatelycreatedduetologicandre-convergence,usingjustthecurrentandlastvalues.Thecomplexityofswitchinginsequentialcircuitsariseduetothepresenceoffeedbackinbasiccomponentssuchasip-opsandlatches.Theinputstoasequentialcircuitarenotonlytheprimaryinputsbutalsothesefeedbacksignals.Thefeedbacklinescanbelookeduponasdeterminingthestateofthecircuitsateachtimeinstant.Thestateprobabilitiesaffectthestatefeedbacklineprobabilitiesthat,inturn,affecttheswitchingprobabilitiesintheentirecircuit.Thus,formally,givenasetofinputsitataclockpulseandpresentstatesst,thenextstatesignalst1isuniquelydeterminedasafunctionofitandst.Atthenextclockpulse,wehaveanewsetofinputsit1alongwithstatest1asaninputtothecircuittoobtainthenextstatesignalst2,andsoon.Hence,thestatisticsofbothspatialand1

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iiipspspsnsnsns1n01n01nClockO/Pssitt+1tiiii001nit+1LatchesCombinational BlockSequential Circuit Figure1.1.AModelforSequentialCircuit.temporalcorrelationsatthestatelinesareofgreatinterest.Itisimportanttobeabletomodelbothkindsofdependenciesintheselines.Previousworks[1,2,3]haveshownthatswitchingactivityinacombinationalcircuitcanbeex-actlyexpressedinaprobabilisticBayesianNetwork,whosestructureisalogicinduceddirectedacyclicgraph(LiDAG).Suchmodelscaptureboththetemporalandspatialdependenciesinacompactmannerusingconditionalprobabilityspecications.Forcombinationalcircuits,rstordertemporalmodelsaresufcienttocompletelycapturedependenciesunderzero-delay.Theattractivefeatureofthegraph-icalrepresentationofthejointprobabilitydistributionisthatnotonlydoesitmaketheconditionaldependenciesamongthenodesexplicit,butitalsoservesasacomputationalmechanismforefcientprobabilisticupdating.TheLiDAGstructure,however,cannotmodelcyclicallogicalstructure,likethoseinducedbythefeedbacklines.Thiscyclicdependenceeffectsthestatelineprobabilitiesthat,inturn,effecttheswitchingprobabilitiesintheentirecircuit.Inthiswork,weproposeaprobabilistic,non-simulative,predictivemodeloftheswitchinginsequentialcircuitsusingtimecoupledlogicinducedDAG(TC-LiDAG)structurethatexplicitlymodels2

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thehigherordertemporalandspatialdependenciesamongthefeedbacklines.ThisisachievedbycouplingLiDAGrepresentationsofthecombinationalcircuitfrommultipletimeslices.ThenodesinTC-LiDAGrepresentswitchingrandomvariableattheprimaryinput,statefeedback,andinternallines.Theserandomvariablesdenedoverfourstates,representingfourpossiblesignaltransitionsateachlinewhichare00011011.EdgesofTC-LiDAGdenotedirectdependency.SomeofthemaredependencieswithinonetimesliceandtheconditionalprobabilityspecicationforthesearethesameasinLiDAG,i.e.theycapturetheconditionalprobabilityofswitchingatanoutputlineofagategiventheswitchingattheinputlinesofthatgate.Restoftheedgesaretemporal,i.e.theedgesarebetweennodesfromdifferenttimeslices,capturingthestatedependenciesbetweentwoconsecutivetimeslices.Weaddanothersetoftemporaledgesbetweenthesameinputlineattwoconsecutiveslices,capturingtheimplicitspatio-temporaldependenciesintheinputswitchings.TemporaledgesbetweenjustconsecutiveslicesaresufcientbecauseoftherstorderMarkovpropertyoftheunderlyinglogic.WeprovethattheTC-LiDAGstructureisaDynamicBayesianNetworks(DBN)capturingallspa-tialandhigherordertemporaldependenciesamongtheswitchingsinasequentialcircuit.Itisaminimalrepresentation,exploitingalltheindependencies.Themodel,inessence,buildsafac-toredrepresentationofthejointprobabilitydistributionoftheswitchingsatallthelinesinthecircuit.DynamicBayesianNetworksareextremelypowerfulgraphicalprobabilisticmodelsthatencapsulatesHiddenMarkovModel(HMM)andLinearDynamicSystem(LDS)andhavebeenemployedforgenematching[25],speechprocessing[24]andobjecttrackingbuttheiruseinmodelingVLSIeventsisnew.DynamicBayesianNetworksarebasicallyBayesianNetworksdenedovermultipletimeslices.ThesalientfeaturesofusingaDynamicBayesianNetworksarelistedbelow:1.DBNexploitsconditionalindependenceamongsttherandomvariablesofinterestforfactoriza-tionandreductionofrepresentationoftheunderlyingjointprobabilitymodelofswitching.2.TherearelearningalgorithmsthatcanconstructorreneBayesianNetworkstructures.Thisfeatureisnotutilizedinourmodelasthestructureofourmodelisxedbythelogicandthefeedback.3.Thetemporallinksmodelhigherordertemporaldependencies,whilethenon-temporallinksmodelshigherorderspatialdependencies.3

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4.Thesearepowerfulpredictivemodelsandhenceworksextremelywellforestimation.TheuniquenessofBayesianNetworkbasedmodelingisthatevidencecanbeprovidedforanynodenotnecessarilyjustattheinputs.Infact,thisfeatureisusedintheprobabilisticinferenceschemestomaketheresultsinputpatterninsensitive.5.Theinferencealgorithmscanbeparallelized,however,wedonotexploitthisinthispaper.WeconsidertwoinferencealgorithmstoformtheestimatesfromthebuiltTC-LiDAGrepresenta-tions:oneisanexactschemeandtheotherisahybridscheme.Theexactinferencesscheme,whichisbasedonlocalmessagepassing,ispresentlypracticalforsmallcircuitsduetocomputationalde-mandsonmemory.Forlargecircuits,weresorttoahybridinferencemethodbasedoncombinationlocalmessagepassingandimportancesampling.Notethatthissamplingbasedprobabilisticinferenceisnon-simulativeandisdifferentfromsamplingsthatarecommonlyusedincircuitsimulations.Inthelater,theinputspaceissampled,whereasinourcaseboththeinputandthelinestatespacesaresampledsimultaneously,usingastrongcorrelativemodel,ascapturedbytheBayesiannetwork.Duetothis,convergenceisfasterandtheinferencestrategyisinputpatterninsensitive.1.1ContributionoftheThesis1.Thisistherstandonlyprobabilisticmodelingframeworkthatcanbeusedtomodelswitchinginbothsequentialandcombinationalcircuits.Thisistheonlycompletelyprobabilisticmodelproposedforsequentialcircuits.2.Inthisthesis,wetheoreticallyprovethattheunderlyingdependencymodelofswitchingactivityisadynamicBayesianNetworkasshowninFigure3.4.c.Theadoptedstrategyisnotlogicunraveling.Switchingineverynodeisinherentlyconnectedwiththeswitchingatthenexttimeinstant.Ifoneweretousejustunraveling,thentheconnectionbetweentimeslicesshouldreallylooklikeFigure3.4.b.Whereas,thedynamicBayesiannetwork,beingaminimalrepresentation,needsfewerconnectionsbetweentimeslicesasshowninFigure3.4.c.3.Themostsignicantcontributionofthisthesisismodelingdependenciesintheinputs.Thisiscrucialasconsecutiveinputpatternsareactuallyrandom,however,whilemodelingswitching,theywouldimplicitlybedependentandquantifyingthisdependencewasnottrivial.Apartfrom4

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that,wecapturestatedependenciesbetweentwoconsecutivetimeandquantifythem.Thesetwoclassesofdependenciesareresponsibleandcrucialforcapturingaccuratehigherordertemporalcorrelationsbetweentimestepsandsignicantlyaffecttheabilitytomodelsequentialcircuitswitching.4.Lastbutnottheleast,Weuseanon-partitionbasedstochasticinferenceschemethat(1)scalesverywelland(2)canresultinanytimeestimates.Ramanietal.[47]recentlyintroducedthesestochasticinferenceschemestoswitchingestimationthatresultsinalmostzero-errorestimatesincombinationalcircuits.WeusethisinferenceforthedynamicBayesianNetworks.Partition-basedinferenceproposedbyBhanjaetal.[3,2]degeneratesastheerrorinonetimesliceisfedbackintothenextoneandpropagatesthroughtheentiresystem.1.2OrganizationWediscussrelevantissuesandresearchworkinChapter2.Thischapterdealsmostlywithstatisti-calsimulation-basedestimationtechniquesasourworkistherstcompletelyprobabilisticapproach.NextinChapter3,wediscussaboutthefundamentalsofBayesianNetworksandthemodelingofacombinationalcircuitintoaBayesiannetwork.ThenwesketchthefundamentalsofdynamicBayesianNetworksandalsodescribethetime-coupled-LiDAGmodelforsequentialcircuits.Wedepictproba-bilisticinferenceinChapter4andpresentexperimentalresultsinChapter5.WeconcludethepaperinChapter6.5

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CHAPTER2PRIORWORKAmongdifferenttypesofVLSIcircuits,switchinginsequentialcircuits,whichalsohappenstobethemostcommontypeoflogic,arethehardesttoestimate.Therearemanymethodsthathandlecombinationalcircuit[11,13,8,34,37,31,32,30,1,2,33]bysimulativeandprobabilisticmethods.Manyofthemcannotbedirectlyappliedtosequentialcircuits.Thisisparticularlyduetothecomplexhigherorderdependenciesintheswitchingprole,inducedbythespatio-temporalcomponentsofthemaincircuitbutmainlycausedbythestatefeedbacksthatarepresent.Thesestatefeedbacksarenotpresentinpurecombinationalcircuits.OneimportantaspectofswitchingdependenciesinsequentialcircuitsthatonecanexploitistherstorderMarkovproperty,i.e.thesystemstateisindependentofallpaststatesgivenjustthepreviousstate.Thisistruebecausethedependenciesareultimatelycreatedduetologicandre-convergence,usingjustthecurrentandlastvalues.Thecomplexityofswitchinginsequentialcircuitsariseduetothepresenceoffeedbackinbasiccomponentssuchasip-opsandlatches.Theinputstoasequentialcircuitarenotonlytheprimaryinputsbutalsothesefeedbacksignals.Thefeedbacklinescanbelookeduponasdeterminingthestateofthecircuitsateachtimeinstant.Thestateprobabilitiesaffectthestatefeedbacklineprobabilitiesthat,inturn,affecttheswitchingprobabilitiesintheentirecircuit.Thus,formally,givenasetofinputsitataclockpulseandpresentstatesst,thenextstatesignalst1isuniquelydeterminedasafunctionofitandst.Atthenextclockpulse,wehaveanewsetofinputsit1alongwithstatest1asaninputtothecircuittoobtainthenextstatesignalst2,andsoon.Hence,thestatisticsofbothspatialandtemporalcorrelationsatthestatelinesareofgreatinterest.Itisimportanttobeabletomodelbothkindsofdependenciesintheselines.Existingtechniquesforswitchingestimationinsequentialcircuitsuseinputpatternsimulation.Puresimulation[21]thoughaccurateareexpensiveintermsofcomputationaltimeandarestronglypatterndependent.Almostallthestatisticaltechniquesinonewayortheotherhasemployedsequential6

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samplingofinputsalongwithastoppingcriteriadeterminedbytheassumedstatisticalmodel.Thesemethodsareweaklypatterndependent,andrequiresspecialattentionformodelingthemutli-modalpowerprole.Moreoversuccessofthesemethodsrelygreatlyontheknowledgeofexactinputtraceeithergivenbytheuserorgeneratedbyaprobabilisticmodel.Inthischapter,wewilldiscussafewdominantmethodsforpowerestimationandbrieygoovertheonlyprobabilisticmodelingeffortbyGhoshetal.[13].Sincestatelinestatisticsareofgreatconcern,Tsuietal.[14]presentedanexactmethodusingChapman-Kolmogrovmethodtoestimateswitchingactivityinsequentialcircuitstatelineandcompareditwithanapproximatemethodwhichwasbasedoncalculatingthestatelineprobabilitiesbysolvingasetofnonlinearequations.Theseequationswerederivedfromthenextstatelogic.Bydoingthistheywereestimatingthestatelineprobabilitiesinsteadofstateprobabilities,therebyreducingthecomplexityoftheproblem.Buttheadverseeffectofthismethodistheinabilitytomodelthespatialdependenciesamongthestatelines.Thesetofnonlinearequations,giveninEquation.2.1,wereobtainedbyconsideringthesteadystateprobabilitiesforstatelinesandassumingthatinputprobabilitieswereknown.y1np1g1p1p2rrrpNn0y2np2g2p1p2rrrpNn0rrrryNnpNgNp1p2rrrpNn0(2.1)where,pi'sarestatelineprobabilities,gi'sarenonlinearfunctionsofpi'sandNisthenumberofipops.IngeneraltheseequationswerewrittenasYPn0andPnGP.TheapproximatestatelineprobabilitieswerecomputedbysolvingYPn0usingNewton-RaphsonmethodorbysolvingPnGPusingPicard-Peanomethod.Theaccuracyoftheestimatewasenhancedbyunrollingthenextstatelogicbysomeuserdenedlimit,asshowninFig.2.1..Thisactuallyincreasedthenumberofvariablesandequations,therebyincreasingtheaccuracy.Chenetal.[17]presentedastatisticaltechniquetoobtainupperandlowerboundsofaveragepowerdissipationinsequentialcircuitstakingintoaccountthesignalprobabilitiesandsignalactivi-7

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I0PS0NS0Ik-1PSk-1NSk-1PresentstatePrimary inputsOutputsNext statebitsCombinational Logick-unrolled networkUse signal probabilitiescalculated by signal probability feedbackI0PS0NS0PSk-1Ik-1NSk-1IkNSk-2PSk-2Ik-2k-unrolled networkSymbolicSimulation EquationsTransitionProbabilities(a)(b)Figure2.1.EnhancingAccuracybyUnrolling[14].8

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ties.Throughthisworktheauthorsemphasizedthatuncertaintiesinprimaryinputscanleadtohugeuncertaintiesinpowerdissipation.Sothebasisofthisworkwastondoutpowersensitivitiesduetoprimaryinputactivitiesandprobabilities.ThesecharacteristicswereobtainedusingEquations.2.2and2.3.zaxinjeallnodesfanoutjaj axi(2.2)zPxinjeallnodesfanoutjaj Pxi(2.3)wherezaxiisnormalizedpowersensitivitytoprimaryinputactivity,zPxiisnormalizedpowersensitivitytoprimaryinputprobabilityandajisnormalizedactivityatnodej.Theinitialpowerwasobtainedusingthesenormalizedvalues.Thenthesignalpropertiesofthepri-maryinputswerechangedoneatatimeandthecircuitwasresimulated.Duetothetediouscalculationsinthismethod,itisnotsuitableforlargecircuits.Chouetal.[18]presentedaMonteCarlobasedstatisticaltechniquetoestimateswitchingactivityinsequentialcircuits.Inthisworktheauthorshavedenedasetofstatestobenear-closedsetifallstatesinsideithaveaverysmallprobabilitytoreachanystateoutsideitandviceversa.Onesuchnear-closedsetwasassumedtobetheinitialstateandawarmupsimulationperiodisappliedbeforeeachsamplelengthperiodtocalculatetheprobabilityoftheinitialnear-closedset.Inordertoobtainastoppingcriterion,thenumberofsampleswasincreasedandthesamplelengthwasreducedwhilemaintainingtheproductofthenumberofsamplesandsamplelengthasaconstant.Bydoingthistheywereabletoovercometheproblemofincreasedsampledeviationwhenthesamplemeanwasstillclosetoaverage.Yuanetal.[16]presentedastatisticalsimulationmethodtoestimateaveragepowerdissipationinsequentialcircuits.TheowchartofthismethodisillustratedinFigure.2.2..Sinceastatisticalestimationmethodneedsrandomsamplesandthepowersamplesfromthesequentialcircuitsarenotrandom,theyproposedamethodtoobtainrandompowersamples.Thismethodincludedacoupleofstepsnamelyarandomnesstestandthedeterminationofanindependenceinterval.Theindependenceintervalwasactuallyusedtoselectsampleswhicharenotconsecutive(i.e.,ifindependenceinterval9

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is1theneveryalternatesamplesarechosen)thushavingarandomsequenceofsamples.Atrstwithaninitialindependenceintervalthesampleswerechosenandtheyweretestedforrandomness.Thesuccessofthetestresultedinacceptanceoftheindependenceintervalasthesuitableone,andthefailureresultsinincrementingtheintervalandrepeatingtheprocess.Intherandomnesstest,aparticularsequenceofsampleswasconsideredtoberandomifithasoneormoresuccessiveoccurrenceofidenticalsamplesfollowedorprecededbydifferentsamples.Clusteringofidenticalsamplesormixingofdifferentsamplesareconsideredtobenonrandomsequenceofsamples.Murugaveletal.[21]proposedasimulationbasedworkforpowerestimationinbothcombinationalandsequentialcircuitsusingPetriNet.TheycameupwithanewformofPetrinetcalledhierarchicalcoloredhardwarePetrinet(HCHPN).Najmetal.[12]proposedalogicsimulationtechniquetoobtainthestatelineprobabilitiesandfurtheremployedastatisticalsimulationmethodforpowerestimationinsequentialcircuits.ThelogicsimulationwasbasedontheMonteCarlomethodanditisemployedontheregistertransferlevel(RTL)ofthecircuit.Saxenaetal.[20]enhancedthistechniqueintheirworkonobtainingstatelineprobabilitiesusingstatisticalsimulationmethod.Multiplecopiesofacircuitweresimulatedusingmutuallyindepen-dentinputvectors,therebyobtainingmutuallyindependentsamples.ThesimulationswerebasedonEquation.2.4.limkPkxiX0nPxi(2.4)wherexirepresentsastatesignal,X0isastateattime0andPxiisthestatelineprobability.PkxiX0wasestimatedforincreasingvaluesofkuntilitconverges.Stamoulis[15]proposedapathorientedMonteCarlomethodtoestimatetransitionprobabilitiesinsequentialcircuits.Thisworkwasmainlyfocusedonthestatelines.Thetransitionprobabilitieswerecalculatedforeachrandomlyselectedpathwitharandomlyselectedstateasitsheadwhileapplyingrandomvectorstotheprimaryinputs.Thetransitionprobabilitiesoverallthesepathswereaveragedtoobtainaverageswitchingprobabilitiesoverthestatelines.Kozhayaetal.[19]formulatedapowerestimationtechniqueforsequentialcircuitswhereasetofinputvectorswerechosenfromalargevectorsettoobtainupperandlowerboundsofpower.Theinput10

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Timing Model Load Circuit DescriptionDetermine Length of Independence IntervalIndependenceIntervalRandom Power SampleYesNoPatternsInputInput Pattern GeneratorAccuracy SpecificationPower SimulationStopping CriterionCriterion Met?Output Average PowerEstimatePower Model Figure2.2.FlowchartfortheStatisticalEstimationofAveraePowerDissipationinSequentialCircuitsbyYuanetal.[16].11

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Average Power EstimationSequential CircuitsforStatistical SimulationProbabilisticTsui et al.'95[14]Chen et al.'97[17]Stamoulis'96[15]Najm et al.'95[12]SimulationGhosh et al.'92[13]This work'04Chou et al.'95[18]Kozhaya et al.'01[19]Saxena et al.'02[20]Yuan et al.'97[16]Buhler et al.'99[39]Macii et al.'01[47]Murugavel et al.'03[21]OthersFigure2.3.TechniquesforEstimatingSwitchingActivityinSequentialCircuits.12

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vectorswereseparatedintoblocksandeachblockwassimulatedtogetcorrespondingupperandlowerboundsofpower.Atransitionfrom01orfrom10attheoutputwasaccountedforupperboundandatransitionfrom00orfrom11wasaccountedforlowerbound.ThentheupperandlowerboundvaluesarecalculatedusingEquations.2.5and2.6.PuKin1 KTiK1kieuk(2.5)PlKin1 KTiK1kielk(2.6)whereKisblocksize,Tisclockperiodandeukn1 2jV2ddCjnukj(2.7)elkn1 2jV2ddCjnlkj(2.8)whereCjisnodecapacitance,nukjandnlkjarelowerandupperboundsonthenumberoflogictransitionsmadebynodejinclockcyclek.WepresenttheresearchworkdoneinsequentialcircuitsinthetaxonomydiagramshowninFig-ure2.3..Asitcanbeseenmostofthepreviouswork,aresimulativeandhencepatterndependent.Ghoshetal.[13]proposedtherstattemptformodelingsequentialcircuitsprobabilistically,however,theymethodassumedmanyindependenceandhenceinaccurate.Thisworkistherstattemptformodelingthesequentialcircuitprobabilisticallycapturingallcorrelationsbothtemporalandspatialandwithreducedpatterndependenceoftheestimates.Bhanjaetal[1,2]proposedBayesianNetworkbasedmodelforcapturingrstordertemporalandhigherorderspatialdependenceforcombinationalcircuit.ThisworkusesdynamicBayesianNetworkbasedmodelforcapturingthefeedbackeffect.Wediscussthefundamentalsandprobabilisticmodelingissuesinthenextchapter.13

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CHAPTER3DEPENDENCYMODELSFORSEQUENTIALCIRCUITSInthischapter,wedevelopthedependencymodelforsequentialcircuits.Sequentialcircuits,asweknowpossessesallthechallengesofspatio-temporaldependencymodelingthatispresentincombina-tionalcircuits.Moreover,theadditionalhigherordertemporaldependenciesthroughthestatefeedbackcontributestodirectedcyclesintheprobabilisticmodel.Thesehigherordertemporaldependenciesnotonlymaketherepresentationcomplexbutalsoinducesadditionalspatio-temporaldependenciesamongstthepresentstateandthepresentinputs.Inthischapter,wewouldrstsketchthefundamentalsofBayesianNetworkandthenexplainsomebasicdenitionsandtheoremsthatareusedtostructureaBayesianNetwork.Inthenextsection,wewouldpresentthefundamentalsofaDynamicBayesianNetworkfollowedbydiscussionaboutthestructureofDynamicBayesianNetworks.Wewillconcludethischapterwiththemodelingandspecicissuespertainingtosequentialcircuitemphasizingtheinputdependenciesandquanticationofthetemporaledgesconnectingtheinputsinadjacenttimeslices.3.1BayesianNetworkFundamentalsBayesianNetworkisagraphicalprobabilisticmodelbasedontheminimalgraphicalrepresenta-tionoftheunderlyingjointprobabilityfunction.Duetotheprobabilisticcausalnature,thisgraphicalmodelisdirectedandacyclic.OnawholeaBayesianNetworkcanbedenedasadirectedacyclicgraph(DAG)whosenodesarerandomvariablesandedgesareprobabilisticdependencies.Thecondi-tionalindependencethatisobservedintheprobabilisticmodelispreservedinthegraphicalstructureandmoreovernoneoftheedgesinthegraphcanberemovedwithoutdestroyingtheconditionalinde-pendencyrelationshipsoftheprobabilisticmodel.Thesefeaturesinduceanotionofminimalcompactrepresentation.14

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Inacombinationalcircuit,theentireswitchingproleofthecircuitcanberepresentedasauniquejointprobabilityfunction.Thepriorstothisfunctionaretheprimaryinputs.Representationofcombi-nationalcircuitsasBayesianNetworkshasalreadybeenproposedbyBhanjaetal.inherdissertationworks[1],[2],[3].ABayesiannetworkisadirectedacyclicgraph(DAG)representationoftheconditionalfactoringofajointprobabilitydistribution.AnyprobabilityfunctionPx1xncanbewrittenas1Px1xNnPxnxn1xn2x1Pxn1xn2xn3x1Px1(3.1)Thisexpressionholdsforanyorderingoftherandomvariables.Inmostapplications,avariableisusuallynotdependentonallothervariables.Therearelotsofconditionalindependenciesembeddedamongtherandomvariables,whichcanbeusedtoreordertherandomvariablesandtosimplifytheconditionalprobabilities.Px1xNnPvPxvPaXv(3.2)wherePaXvaretheparentsofthevariablexv,representingitsdirectcauses.Thisfactoringofthejointprobabilityfunctioncanberepresentedasadirectedacyclicgraph(DAG),withnodes(V)rep-resentingtherandomvariablesanddirectedlinks(E)fromtheparentstothechildren,denotingdirectdependencies.3.2ConditionalIndependenceMapsTheDAGstructurepreservesalltheindependenciesamongsetsofrandomvariablesandisreferredtoasaBayesiannetwork.TheconceptofBayesiannetworkcanbepreciselystatedbyrstdeningthenotionofconditionalindependenceamongthreesetsofrandomvariables.Thefollowingdenitionsandtheoremsappearin[5,1]andareusedlaterinthispapertoprovethattheTC-LiDAGstructureisaBayesiannetwork.Denition1:LetU=abbeanitesetofvariablestakingondiscretevalues.LetPbethejointprobabilityfunctionoverthevariablesinU,andletX,YandZbeanythreesubsets(maybe 1ProbabilityoftheeventXixiwillbedenotedsimplybyPxiorbyPXixi.15

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overlapping)ofU.XandYissaidtobeconditionallyindependentgivenZifPxyznPxzwheneverPyz0(3.3)FollowingPearl[5],wedenotethisconditionalindependencyamongstX,Y,andZbyIXZY;XandYaresaidtobeconditionallyindependentgivenZ.Adependencymodel,M,ofadomainshouldcaptureallthesetripletconditionalindependenciesamongstthevariablesinthatdomain.Ajointprobabilitydensityfunctionisonesuchdependencymodel.Thenotionofindependenceexhibitspropertiesthatcanbeaxiomatizedbythefollowingtheorem[5].Theorem1:LetX,YandZbethreedistinctsubsetofU.IfIXZYstandsfortherelation“XisindependentofYgivenZ”insomeprobabilisticmodelP,thenImustsatisfythefollowingfourindependentconditions:IXZYIYZX(3.4)IXZYW IXZY&XZW(3.5)IXZYW!IXZWY(3.6)IXZY&IXZYW!IXZYW(3.7)Next,weintroducetheconceptofd-separationofvariablesinadirectedacyclicgraphstructure(DAG),whichistheunderlyingstructureofaBayesiannetwork.Thisnotionofd-separationisthenrelatedtothenotionofindependenceamongsttriplesubsetsofadomain.Denition2:IfX,YandZarethreedistinctnodesubsetsinaDAGD,thenXissaidtobed-separatedfromYbyZ,"XZY,ifthereisnopathbetweenanynodeinXandanynodeinYalongwhichthefollowingtwoconditionshold:(1)everynodeonthepathwithconvergingarrowsisinZorhasadescendentinZand(2)everyothernodeisoutsideZ.Denition3:ADAGDissaidtobeanI-mapofadependencymodelMifeveryd-separationconditiondisplayedinDcorrespondstoavalidconditionalindependencerelationshipinM,i.e.,ifforeverythreedisjointsetofnodesX,YandZwehave,"XZY#IXZY.Denition4:ADAGisaminimalI-mapofMifnoneofitsedgescanbedeletedwithoutdestroyingitsdependencymodelM.16

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NotethateveryjointprobabilitydistributionfunctionPoverasetofvariablesrepresentsadepen-dencymodelM,sinceitcapturesalltheconditionalindependencies.Denition5:GivenaprobabilitydistributionPonasetofvariableU,aDAGDiscalledaBayesianNetworkofPifDisaminimumI-mapofP.ThereisanelegantmethodofinferringtheminimalI-mapofPthatisbasedonthenotionofaMarkovblanketandaboundaryDAG,whicharedenedbelow.Denition6:AMarkovblanketofelementXi$UisansubsetSofUforwhichIXiSUSXiandXi%$S.AsetiscalledaMarkovboundary,BiofXiifitisaminimalMarkovblanketofXi,i.e.noneofitspropersubsetssatisfythetripletindependencerelation.Denition7:LetMbeadependencymodeldenedonasetUnX1Xnofelements,andletdbeanorderingXd1Xd2roftheelementsofU.TheboundarystrataofMrelativetodisanorderedsetofsubsetsofU,Bd1Bd2&suchthateachBiisaMarkovboundary(denedabove)ofXdiwithrespecttothesetUi('UnXd1Xd2Xd)i1*,i.e.BiistheminimalsetsatisfyingBi'UandIXdiBiUiBi.TheDAGcreatedbydesignatingeachBiastheparentsofthecorrespondingvertexXiiscalledaboundaryDAGofMrelativetod.ThisleadsustothenaltheoremthatrelatestheBayesiannetworktoI-maps,whichhasbeenprovenin[5].ThistheoremisthekeytoconstructingaBayesiannetworkovermultipletimeslices(DynamicBayesianNetworks).Theorem2:LetMbeanydependencymodelsatisfyingtheaxiomsofindependencelistedinEqs.3.4-3.7.IfgraphstructureDisaboundaryDAGofMrelativetoorderingd,thenDisaminimalI-mapofM.3.3CombinationalCircuitasaBayesianNetworkFigure.3.1.(a)representsasimplecombinationalcircuitandtheDAGinFigure.3.1.(b)illustratesitsBayesianNetworkrepresentation.TheconditionsforaDAGstructuretobeaBayesianNetworkareasfollows,1.thepresenceofconditionallyindependentsetofrandomvariablesintheprobabilisticmodelP.2.thepresenceofd-seperationofrandomvariablesintheDAGD.17

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214365XXXXXX123456 (a) (b) Figure3.1.ASimpleCombinationalCircuit.3.theDAGDshouldbetheminimalI-mapoftheprobabilityfunctionP.ThefollowingdiscussionisbasedonthedenitionsandtheoremspresentedinSection.3.2.Intheswitchingprobabilisticmodel,X5isconditionallyindependentofX2givenX3andX4.IntheDAGstructureillustratedinFigure.3.1.(b),fromtheconditionsofd-seperation,wecanclearlypointoutthatX5isd-seperatedfromX2byX3X4.ThemethodofobtainingminimalI-mapoftheprobabilisticfunction,sayPisbasedonMarkovblanketandaboundaryDAG.IntheDAGstructureillustratedinFigure.3.1.(b),consideringtheran-domvariableX5,S=X3,X4isaMarkovblanket,sincegivenX3,X4,X5isindependentoftherestoftherandomvariablesinthedomain.IntheDAGstructureillustratedinFigure.3.1.(b),theboundarystrataofunderlyingdependencymodeloverthedomainisgivenby,BMn+X1X2,(X3X4,(X2+(3.8)ThisclearlydepictsthattheDAGinFigure.3.1.(b)isaboundaryDAG.Usingthesecharacteristicsithasalreadybeenproved[5]that,ifagraphstructureisaboundaryDAGofthedependencymodel,thenthegraphstructureisaminimalI-mapofthedependencymodel.TheseaspectscollectivelyprovethattheDAGstructureinFigure.3.1.(b)isaBayesianNetwork.18

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Figure3.2.TimeSliceModelwithSnapshotoftheEvolvingTemporalProcess[26].3.4DynamicBayesianNetworkThefocusofthisworkistowardstherealizationofsequentialcircuitsasagraphicalprobabilisticmodel.Duetothefeedbackprocessinsequentialcircuitstheycannotberepresentedasadirectedacyclicgraph.SoeventuallytheycannotbemodeledasaBayesianNetwork.Hencewemodelsequen-tialcircuitastimecoupledBayesianNetworkswhicharealsocalleddynamicBayesianNetworks.Notonlythedigitalcircuitsbutmostofthecommonprocessesneedtobeobservedateachpointoftimeandpossessdifferentprobabilisticdependenciesatdifferentinstantoftime.Suchdynamicactivitiesarenormallyanalyzedateachtimeinstantandanaloutputisobtained.ThebasicBayesianNetworkwasnotdesignedtohandlesuchdynamicactivities.Inthebeginningtheresearchers,whentheystartedformulatingBNinanewdirection,theyhadtoresorttotwoidenticalyetdifferentmodelsnamely,temporalanddynamic.Thetemporalmodelwasstatedtobeasubsetofdynamicmodel,becauseitcaresonlyaboutthechangeintimeandnotaboutthechangeinstate.Ifasystemremainsinthesamestateatdifferenttimeinstants,thenitisatemporalmodel.Ifthereischangeinstatealongwithtime,thenitisadynamicmodel.InthediscussionaboutstructuringaDynamicBayesianNetwork,onemightresorttotwodifferentapproachesasshowninFigures.3.2.and3.3..Figure.3.2.representsatimeslicedmodelwhereeachtimesliceconsistsofasub-modelrepre-sentingthebeliefnetworkataparticularpointoftimeortimeinterval.Theadjacenttimeslicesaretemporallyconnected.Figure3.3.representsatemporalmodelwherethebeliefnetworkisduplicatedintoidenticalsub-modelsovereachtimeslicehoweverthestatevariablesarenotallowedtobedepen-dentinonetimeslice.Hence,thestructureofFigure3.3.isnotidealforsequentialcircuits.TheadventoftheideaoftimesliceshasformedthebasisofstructuringasequentialcircuitintoaBN.19

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Figure3.3.TemporalModelwithDuplicatedTimeSlices[26].Asdiscussedbefore,DynamicBayesianNetwork(DBN)isageneralizationofBayesiannetworkstohandletemporaleffectsofanevolvingsetofrandomvariables.OtherformalismssuchashiddenMarkovmodelsandlineardynamicsystemsarespecialcases.ThenodesandthelinksoftheDBNaredenedasfollows.Foranytimeperiodorslice,ti,letadirectedacyclicgraph(DAG),GtinVtiEti,representtheunderlyingdependencygraphicalmodelforthecombinationalpart.ThenthenodesoftheDBN,V,istheunionofallthenodeseachtimeslice.Vnn-i1Vti(3.9)However,thelinks,E,oftheDBNarenotjusttheunionofthelinksinthetime-sliceDAGs,butalsoincludelinksbetweentime-slices,i.e.temporaledges,Etiti1,denedasEtiti1n.XitiXjti1/Xiti$VtiXjti1$Vti1(3.10)whereXjtkisthej-thnodeoftheDAGfortimeslicetk.ThusthecompletesetofedgesEisEnEt1n-i2Eti10Eti21ti(3.11)Apartfromtheindependenciesamongthevariablesfromonetimeslice,wealsohavethefollowingindependencemapovervariableacrosstimeslicesifweassumethattherandomvariablesrepresenting20

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thenodesfollowMarkovproperty,whichistrueforswitching.I3Xjt1Xjti21,Xjti(Xjti1Xitik54istrue6i1k1(3.12)3.5ModelingSequentialCircuitAswehavediscussedbefore,dependencymodelingoftheunderlyingswitchingmodelforse-quentialcircuitscanonlybeperformedbyadynamicBayesianNetworksduethefeedback.Inthissection,wewilldiscussspecicissueswithmodelingtheMarkovBlanketofindividualvariablesandquanticationofdependencies.Thecoreideaistoexpresstheswitchingactivityofacircuitasajointprobabilityfunction,whichcanbemappedone-to-oneontoaBayesianNetwork,whilepreservingthedependencies.Tomodelswitchingataline,weusearandomvariable,X,withfourpossiblestatesindicatingthetransitionsfromx00x01x10x11.Forcombinationalcircuits,directededgesaredrawnfromtherandomvariablesrepresentingswitchingofeachgateinputtotherandomvariableforswitchingattheoutputsofthatgate.Ateachnode,wealsohaveconditionalprobabilities,giventhestatesofparentnodes.IftheDAGstructurefollowsthelogicstructure,i.e.wehavealogicallyinducedDAG(LiDAG),thenitisguaranteedtomapallthedependenciesinherentinthecombinationalcircuit.However,sequentialcircuitscannotbehandledinthismanner.3.5.1StructureLetusconsidergraphstructureofasmallsequentialcircuitshowninFig.3.4.(a).FollowinglogicstructurewillnotresultinaDAG;therewillbedirectedcyclesduetofeedbacklines.Tohandlethis,wedonotrepresenttheswitchingatalineasasinglerandomvariable,Xk,butratherasasetofrandomvariables,representingtheswitchingatconsecutivetimeinstants,Xkt1Xktn,andthenmodelthelogicaldependenciesbetweenthembytwotypesofdirectedlinks.1.Foranytimeinstant,edgesareconstructedbetweennodesthatarelogicallyconnectedinthecombinationalpartofthecircuit,i.e.withoutthefeedbackcomponent.Edgesaredrawnfromeachrandomvariablerepresentingswitchingactivityateachinputofagatetotherandomvari-21

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123456(b)(c)XXXXXX123456 XXXXXX123456 XXXXXX123456 XXXXXX123456 214365(a)XXXXXX123456XXXXXX123456XXXXXX Figure3.4.(a)ASimpleSequentialCircuitanditsGraphicalModel.(b)TimeUnraveledRepresenta-tion.(c)TC-LiDAGRepresentation.22

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ablerepresentingoutputswitchingofthegate.ThisgivesustheLiDAGstructure,capturingthenon-sequentialnature.2.Weconnectrandomvariablesrepresentingthesamestatelinefromtwoconsecutivetimein-stants,XsktiXskti1,tocapturethetemporaldependenciesbetweentheswitchingsatstatelines.Moreover,wealsoconnecttherandomvariablesrepresentingtheswitchingatprimaryinputlinesatconsecutivetimes,XpktiXpkti1.Thisisdonetocapturetheconstraintintheprimarylineswitchingbetweentwoconsecutivetimeinstants.Forinstance,ifaninputhasswitchedfrom01attimeti,thenswitchingatthenexttimeinstantcannotbe00.Wecallthisgraphstructureasthetimecoupled,logicallyinducedDAGorTC-LiDAG.Fig.3.4.(b)showstheTC-LiDAGfortheexamplesequentialcircuitinFig.3.4.(a);wejustshowtwotimesliceshere.Thedash-dotedgesshowsthesecondtypeofedgesmentionedabove,whichcouplesadjacentLiDAGs.WehaveX2asinputandX1asthepresentstatenode.RandomvariableX6representsthenextstatesignal.NotethatthisgraphisaDAG.WenextprovethatthisTC-LiDAGstructureisaminimalrepresentation,henceisadynamicBayesiannetwork.Theorem3:TheTC-LiDAGstructure,correspondingtothesequentialcircuitisaminimalI-mapoftheunderlyingswitchingdependencymodelandhenceisadynamicBayesiannetwork.Proof:LetusordertherandomvariablesXiti,suchthat(i)fortworandomvariablesfromonetimeti,XptiandXcti,wherepisaninputlinetoagateandcisaoutputlinetothesamegate,Xpti,appearsbeforeXctiinthisorderingand(ii)therandomvariablesforthenexttimesliceti01,X1ti1Xnti1appearaftertherandomvariablesattimesliceti.Withrespecttothisordering,theMarkovboundaryofanode,Xiti,isgivenasfollows.IfXpitirepresentsswitchingofaninputsignalline,thenitsMarkovboundaryisthevariablerepresentingthesameinputintimesliceXpiti21.IfXsitirepresentsswitchingofastatesignal,thenitsMarkovboundaryisthevariablerepresentingtheswitchingattheprevioustimesliceXsiti21.And,sincetheswitchingofanygateoutputlineisjustdependentontheinputsofthatgate,theMarkovboundaryofavariablerepresentinganygateoutputlineconsistsofjustthosethatrepresenttheinputstothatgate.IntheTC-LiDAGstructuretheparentsofeachnodeareitsMarkovboundaryelementshencetheTC-LiDAGisaboundaryDAG.And,byTheorem2theTC-LiDAGisaminimalI-mapandthusaBayesiannetwork23

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(BN).SincenodesandtheedgesintheTC-LiDAGdenedoverntimeslicescanbedescribedbyEquation.3.9,andEquation.3.11,theTC-LiDAGisadynamicBayesianNetwork(DBN).3.5.2SpatialDependencyQuanticationInTC-LiDAG,wehavetherandomvariablesthatrepresentswitchingateachsignalatatimeinstanttwhichindicatesthetransitionsfrom00011011.Inthissectionwewilldiscussthespatialedgesi.e.thedependenciesthatarisebetweensignalsinoneinstantoftime.Directededgesaredrawnfromtherandomvariablesrepresentingswitchingoftheinputstotherandomvariableforswitchingattheoutputofeachgate.Notethatthesedependenciesdenotethespatialcorrelationsamongstthevariablesinonetimeinstant.Theconditionalprobabilityofrandomvariablerepresentingswitchingatanoutputvariablegivenitsparentsaredeterminedpurelyfromthelogicalstructures.Theconditionalprobabilitiesofthelinesthataredirectlyconnectedbyagatecanbeobtainedknowingthetypeofthegate.Forexample,PX3nx01X1nx01X2nx00willbealways0becauseifoneoftheinputsofanANDgatemakesatransitionfrom0to1andtheotherstaysat0thentheoutputdoesnotchangeandhencePX3nx01X1nx01X2nx00n0.AcompletespecicationoftheconditionalprobabilityofPx3x1x2willhave43entriessinceeachvariablehas4states.Theseconditionalprobabilityspecicationsaredeterminedbythegatetype.Thus,foranANDgate,ifoneinputswitchesfrom0to1andtheotherfrom1to0,theoutputremainsat0.WedescribetheconditionalprobabilityspecicationforatwoinputANDgateinTable3.1.Byspecifyingadetailedconditionalprobabilityweensurethatthespatio-temporaleffect(rstordertemporalandhigherorderspatial)ofanynodeareeffectivelymodeled.IthastobenotedthatinasingleBayesianNetwork,thetemporaldependenciesareingeneralnotmodeled.Forcombinationalcircuit,Bhanjaetal.[2]usedthestatesasfourpossibletemporaltransitionsofasignal00,01,10,11tomodelrstordertemporaldependencewhichissufcientformodelingzero-delaycombinationalcircuits.Innextsubsection,wewilldiscussthemodelingissuespertainingtothehigherordertemporaldependencies.24

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Table3.1.ConditionalProbabilitySpecicationsfortheOutputandtheInputLineTransitionsforTwoInputANDGate[2]. TwoInputANDgate PXoutputXinput1Xinput2 forXoutputn Xinput1 Xinput2 x00x01x10x11 = = 1000 x00 x00 1000 x00 x01 1000 x00 x10 1000 x00 x11 1000 x01 x00 0100 x01 x01 1000 x01 x10 0100 x01 x11 1000 x10 x00 1000 x10 x01 0010 x10 x10 0010 x10 x11 1000 x11 x00 0100 x11 x01 0010 x11 x10 0001 x11 x11 25

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Table3.2.ConditionalProbabilitySpecicationbetweenStateLineSwitchingsatConsecutiveTimeInstants:Pxskti1xskti. Xskti1n Xskti x00x01x10x11= 1000 x00 0100 x01 0010 x10 0001 x11 Table3.3.GeneralExampleofSwitchingofaSignalXkrepresentingtheValueoftheSignalateachTimeInstantt. TimeInputSwitchingInputSwitching tiI1X1I2X2 0000101 1101111 2111111 3010111 4101010 5010101 6000010 7000000 3.5.3TemporalDependencyQuanticationThejointprobabilityfunctionismodeledbyaBayesiannetworkastheproductoftheconditionalprobabilitiesdenedbetweenanodeanditsparentsintheTC-LiDAGstructure:PxvPaXv.Theseconditionalprobabilitiescanbeeasilyspeciedusingthecircuitlogic.Wedemonstratehandlingtheinternallinesintheprevioussubsection.Therearetwobasictypesofconditionalprobabilityspeci-cationsforthetemporaledgesbetween(i)primaryinputlines,and(ii)statelines.Forstatelines,theconditionalprobabilitymodelsthelogicofabuffer,asshowninTable3.2..PleaseconsidertheexampleofaninputpatterninTable3.3..Eachrowofthetableindicateonetimeinstant,column2and4showtheactualvaluesofthesignalsandcolumn3and5showthecorrespondingswitchings.Notethattheinputsarepurelyindependent,however,switchingsarenot.Ifswitchingis01inrow2,itcannotbe00inrow3.Henceunderpurelyrandominputsituations,switchingofavariablecantakeonlytwovalues(say01,00)outoffourstatesin(t+1)givenoneparticularvalue(say00)att.26

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Table3.4.ConditionalProbabilitySpecicationbetweenPrimaryInputLineSwitchingsatConsecutiveTimeInstants:Pxpkti1xpkti. Xpkti1n Xpkti x00x01x10x11= 0.50.500 x00 000.50.5 x01 0.50.500 x10 000.50.5 x11 Henceforprimaryinputlines,theconditionalprobabilitiesmodelstheswitchingconstraintsbe-tweentwotimeinstants,aslistedinTable3.4..Forinstance,iftheprimarylineswitchedfrom0to1,thenatthenexttimeslicethelinecaneitherswitchfrom1to0orremainat1.Since,wearecon-sideringrandominputs,wedistributetheprobabilitiesequally(with0.5probability)betweenthetwooptions.Forcorrelatedinputs,theseconditionalprobabilitieswillhavetobeadjusted.27

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CHAPTER4PROBABILISTICINFERENCEIntheprevioussection,wediscussedmodelingofsequentialcircuitsasdynamicBayesianNet-works.Theattractivefeatureofthismodelingisthattheconditionalindependencerelationshipsnotonlyhelpinmodelingcausalitybutisalsoaninstrumentforprobabilisticinference.Inthissection,werstdiscussoneoftheexactinferenceschemesthatweusetovalidateourmodeling.Theexactinferencesscheme,whichisbasedonlocalmessagepassing,ispresentlypracticalforsmallcircuitsduetocomputationaldemandsonmemory.Forlargecircuits,weresorttoahybridinferencemethodbasedoncombinationlocalmessagepassingandsampling.4.1ExactInferenceTheexactinferenceschemeisbasedonlocalmessagepassingonatreestructure,whosenodesaresubsets(cliques)ofrandomvariablesintheoriginalDAG[4,9].ThistreeofcliquesisobtainedfromtheinitialDAGstructureviaaseriesoftransformationsthatpreservetherepresenteddependencies.TheoriginalDAGisrstconvertedintoanundirectedMarkovgraphstructure,whichisreferredtoasthemoralgraph,modelingtheunderlyingjointprobabilitydistribution.ThismoralgraphisobtainedfromtheDAGstructure,byaddingundirectedlinksbetweentheparentsofacommonchildnode.TheseadditionallinksdirectlycapturethedependenciesthatwereonlyimplicitlyrepresentedintheDAG.Inamoralgraph,everyparent-childsetformacompletesubgraph.Duetotheundirectednatureofthemoralgraph,someoftheindependenciesrepresentedintheDAGwouldbelost,resultinginanon-minimalrepresentation.Thedependencystructureis,however,preserved.Thislossofminimalrepresentationwilleventuallyresultinincreasedcomputationaldemands,butdoesnotsacriceaccuracy.Next,achordalgraphisobtainedfromthemoralgraphbytriangulatingit.Triangulationistheprocessofbreakingallcyclesinthegraphtobecompositionofcyclesoverjust3nodesbyadding28

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zi4i3i2i1Clock Latches Figure4.1.CircuitDiagramofs27.29

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X7X3X1 X2To Present State linesX17X16X12X15X10X14X13X11X9X8X6X4X5 Figure4.2.BayesianNetworkModeloftheCombinationalPartofs27forOneTimeSlice.30

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X3X1 X7MoralizationTriangularizationX17To Present State LinesX16X12X15X10X14X13X11X9X8X6X4X5X2 Figure4.3.TriangulatedUndirectedGraphStructureofs27.31

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C4C10C2C9C5C8C7C6C1 = {X8,X9,X14}C2 = {X9,X14,X10,X13}C3 = {X8,X14,X15}C4 = {X8,X1} C5 = {X9,X13,X11}C6 = {X11,X12,X7}C7 = {X11,X3,X12}C8 = {X9,X10,X4}C9 = {X14,X15,X5}C10 = {X8,X9,X6}C11 = {X15,X16}C12 = {X8,X15,X17}C11C12C3 C1Figure4.4.JunctionTreeofCliquesforOneTimeSliceModelofs27.additionallinks.Tocontrolthecomputationaldemands,thegoalistoformatriangulatedmoralgraphwithminimumnumberofadditionallinks.Variousheuristicsexistforthis.Forinstance,theBayesiannetworkinferencesoftwareHUGIN(www.hugin.com),whichweuseinthiswork,usesefcientandaccurateminimumll-inheuristicstocalculatetheseadditionallinks.Cliquesofthischordalgraphformthenodesofthejunctiontree.Thetreestructureisusefulforlocalmessagepassing.Givenanyevidence,messagesconsistoftheupdatedprobabilitiesofthecommonvariablesbetweentwoneighboringcliques.Globalconsistencyisautomaticallymaintainedbyconstructingthetreeinsuchawaythatanytwocliques,sharingasetofcommonvariables,shouldhavethesecommonvariablespresentinallthecliquesthatlieintheconnectingpathbetweenthetwocliques.Ajunctiontreewiththispropertycanbeeasilyobtainedfromthesameminimumll-inheuristicalgorithmthatisusedtotriangularizethegraph[2].Asanexampleofcliquetreeconstructionprocess,weconsiderthegatelevelcircuitdiagramfors27,showninFig.4.1..ThecorrespondingDAGBayesiannetworkmodel,overonetimeslice,isshowninFig.4.2..Werstillustratetheprocessonthispartialmodel.TheinferenceschemeforthecompletedynamicBayesiannetworkmodel,ascapturedbytheTC-LiDAG,wouldbesimilar.ThemoralizedandtriangularizedformisshowninFig.4.3..ThecliquetreeisshowninFig.4.4..Oneinterestingfeatureofthisexactmodelingisthatprobabilisticupdatingisreallyfastoncethecompilationisperformed,andhenceisusefulforfastdesign-spaceexploration.Theprobabilitiesare32

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C18C20 C1 = {X4b,X9b,X11b,X14b,X8a,X15a}C2 = {X4b,X11b,X8a,X15a,X9a}C3 = {X4b,X11b,X15a,X9a,X8a,X10a}C4 = {X11b,X15a,X9a,X10a,X13a}C5 = {X4b,X9b,X11b,X14b,X10b}C6 = {X9b,X11b,X14b,X10b,X13b}C7 = {X9b,X14b,X8a,X15a,X8b}C8 = {X14b,X8a,X15a,X8b,X5b}C9 = {X9b,X15aX8b,X6b}C10 = {X14b,X8b,X5b,X15b}C11 = {X4b,X9a,X10a,X4a}C12 = {X11b,X9a,X13a,X11a}C13 = {X11b,X11a,X2b,X7b}C14 = {X11b,X11a,X17b,X3a}C1C2C5C7C26C3C6C8C9C11C4C10C17C12C13C25C21C14C28C27C15C19C22C16C23C24C15 = {X11a,X7b,X3a,X12a}C16 = {X15a,X10a,X13a,X14a}C17 = {X8a,X15a,X5b,X17a}C18 = {X8a,X8b,X1b}C19 = {X11b,X3a,X3b}C20 = {X8a,X1b,X1a}C21 = {X11a,X2b,X2a}C22 = {X11b,X3b,X12b}C23 = {X15b,X16b}C24 = {X8b,X15b,X17b}C25 = {X15a,X14a,X5a}C26 = {X8a,X9a,X6a}C27 = {X11a,X2a,X7a}C28 = {X15a,X16a}Figure4.5.JunctionTreeofCliquesforTwoTimeSlicesofs27.33

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propagatedthroughthejunctiontreejustbylocalmessage-passingbetweentheadjacentcliques.LetusconsidertwoneighboringcliquestounderstandthekeyfeatureoftheBayesianupdatingscheme.LettwocliquesAandBhaveprobabilitypotentialsfAandfB,respectively,obtainedbymultiplyingtheconditionalprobabilities,intheDAGbasedBayesiannetwork,involvingthenodesineachclique.LetSbethesetofcommonnodesbetweencliquesAandB.ThetwoneighboringcliqueshavetoagreeonprobabilitiesonthenodesetS,whichistermedtheirseparator.Toachievethis,werstcomputethemarginalprobabilityofSfromprobabilitypotentialofcliqueAandthenusethattoscaletheprobabilitypotentialofB.Thetransmissionofthisscalingfactor,whichisneededinupdating,isreferredtoasmessagepassing.Newevidenceisabsorbedintothenetworkbypassingsuchlocalmessages.Thepatternofthemessageissuchthattheprocessismulti-threadableandpartiallyparallelizable.Becausethejunctiontreehasnocycles,messagesalongeachbranchcanbetreatedindependentoftheothers.ThenaljunctiontreeofcliquesfortheTC-LiDAGstructureofs27fortwotimeslicesismorecomplicated,asshowninFig.4.5..Noticetheincreasednumberofcliques.Ingeneral,thesizeofthemaximalcliquewillincrease.Thisresultinincreasedmemoryrequirementtostoretheprobabilitypotentialoverthenodesinthecliques;theincreaseisexponentialinthemaximalcliquesize.Thus,itisobviousthattheexactmodelcannotbeusedforlargecircuits.Availablememorywoulddeterminethemaximumcircuitsizethatcanbemodeledexactly.Inthiswork,weusethisinferenceonlyformodelvalidationwithsmallcircuits.4.2HybridSchemeForlargecircuits,ahybridscheme,specicallythepre-propagatedimportancesampling(EPIS)[22,23],whichuseslocalmessagepassingandstochasticsampling,isappropriate.Thismethodscaleswellwithcircuitsizeandisproventoconvergetocorrectestimates.Theseclassesofalgorithmsarealsoanytime-algorithmssincetheycanbestoppedatanypointoftimetoproduceestimates.Ofcourse,theaccuracyofestimatesincreaseswithtime.TheotherusefulmethodisProbabilisticLogicSampling(PLS)method.TheEPISalgorithmisbasedonimportancesamplingthatgeneratessampleinstantiationsofthewholeDAGnetwork,i.e.allforlineswitchinginourcase.Thesesamplesarethenusedtoformthenalestimates.Thissamplingisdoneaccordingtoanimportancefunction.InaBayesiannetwork,the34

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productoftheconditionalprobabilityfunctionsatallnodesformtheoptimalimportancefunction.LetXnX1,X2XmbethesetofvariablesinaBayesiannetwork,PaXkbetheparentsofXk,andEbetheevidenceset.Then,theoptimalimportancefunctionisgivenbyPXEnm'k1PxkPaxkE(4.1)ThisimportancefunctioncanbeapproximatedasPXEnm'k1aPaXkPxkPaXklXk(4.2)whereaPaXknPxkEandlXknPExk,withEandEbeingtheevidencefromaboveandbelow,respectively,asdenedbythedirectedlinkstructure.Calculationofliscomputationallyexpensiveandforthis,LoopyBeliefPropagation(LBP)[27]overtheMarkovblanketofthenodeisused.Yuanetal.[23]provedthatforapoly-tree,thelocalloopybeliefpropaga-tionisoptimal.Theimportancefunctioncanbefurtherapproximatedbyreplacingsmallprobabilitieswithaspeciccutoffvalue[22].ThisstochasticsamplingstrategyworksbecauseinaBayesianNetworktheproductofthecondi-tionalprobabilityfunctionsforallnodesistheoptimalimportancefunction.Becauseofthisoptimality,thedemandonsamplesislow.WehavefoundthatjustthousandsamplesaresufcienttoarriveatgoodestimatesfortheISCAS89benchmarkcircuits.Notethatthissamplingbasedprobabilisticinferenceisnon-simulativeandisdifferentfromsamplingsthatareusedincircuitsimulations.Inthelatter,theinputspaceissampled,whereasinourcaseboththeinputandthelinestatespacesaresampledsimultaneously,usingastrongcorrelativemodel,ascapturedbytheBayesiannetwork.Duetothis,convergenceisfasterandtheinferencestrategyisinputpatterninsensitive.ProbabilisticLogicSamplingdevelopedbyHenrionin1988iscreditedtobetherststochasticsamplingmethodforinferencingBayesianNetworks.Inthismethodsamplingisperformedintheforwarddirection(fromparentstochildren).InthecircuitwhichisrepresentedasaBayesiannetworkeachnodeisselectedintop-downfashionandtheyaresampled.WhileinferencingtheBayesiannetworkthesamplesaregroupedintosetsandtheobservedvalueineachsampleinasetiscomparedwiththecorrespondingevidencevalues.Iftheyareinconsistentwitheachotherthewholesampleset35

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isdiscarded.Thesamemethodisrepeatedwitheachsampleset.Intheselectedsamplesetthebeliefdistributionsarecalculatedbyaveragingthefrequencieswithwhichtherelevanteventsoccur.Ascomparedtothecomputationalmerits,thismethodalsohassomedisadvantages.Sinceitisbasedonforwardsampling,theevidencethathavealreadyoccurredcannotbeaccounteduntilthecorrespondingvariablesaresampled.Theoccurrenceofunlikelyevidencecanresultinrejectionoflargenumberofsamplestherebyhinderingtheperformanceofthismethod.36

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CHAPTER5EXPERIMENTALRESULTSWehaveusedthesequentialcircuitsfromtheISCAS89benchmarksuitetoverifyourmethod.Togeneratethe“groundtruth”estimatestocompareagainstthecircuitsweresimulated,withzerogatedelay,for100,000testvectors.TherootnodesoftheTC-LiDAGrepresentationofthesequentialcircuits,whicharethestatelinesofthersttimesliceandtheprimaryinput,needpriorprobabilityspecications.Thepriorsfortheprimaryinputlinesinthersttime-sliceofDBNwerechosentobeequal,i.e.equallyprobableswitchingstates.Astartupsimulationwith50randomtestvectorswasperformedandthesestartupestimatesofthepresentstatelinesaregiventothersttime-sliceoftheDBN.TheexactinferenceontheTC-LiDAGstructurewasdoneusingthecommerciallyavailableHUGINsoftware.And,weusedatoolnamed”GeNIe”[7]toimplementthehybridinferencestrategybasedonsamplingandloopybeliefpropagation(EPIS).ThetestswereperformedonaPentiumIV,2.00GHz,WindowsXPcomputer.First,weshowsomeresultsthatvalidatestheTC-LiDAGmodel.Forthis,weusethes27bench-markcircuit.Table5.1.liststheswitchingestimatesateachlineinthecircuitascomputedby(i)simulation,(ii)theexactinferenceschemebasedontreeofcliques,and(iii)thehybridEPISscheme.Weused10timeslicesfortheTC-LiDAGrepresentation.Notetheexcellentagreementoftheexactinferenceschemewithsimulation,thusvalidatingthattheTC-LiDAGiscapturingthehighordertemporalandspatialcorrelations.Thehybridinferenceschemealsoresultsinexcellentestimates,closetotheexactones.Second,wepresentresultsonrestoftheISCAS'89circuitsintheformofestimationerrorstatistics,asshowninTable5.2..Welistboththeaverageerror,E,andmaximumerror,Emax,overallthenodes.Wealsolistthepercentageofnodeswithswitchingerrorabove2standarddeviationsfromthemeanerror;thisgivesanideaabouttheerrordistribution.ThelistedelapsedtimesareobtainedbytheftimecommandintheWINDOWSenvironment,andisthesumofCPU,memoryaccessandI/Otime.37

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Table5.1.SwitchingProbabilityEstimatesateachLineofthes27BenchmarkCircuitasComputedbySimulation,bytheExactScheme,andbytheHybridEPISMethod. Simulation Exact Hybrid Nodes (CliqueTree) (EPIS) 1 0.498 0.500 0.512 2 0.598 0.500 0.494 3 0.501 0.500 0.487 4 0.500 0.500 0.508 5 0.450 0.452 0.463 6 0.123 0.123 0.123 7 0.333 0.333 0.368 8 0.498 0.500 0.512 9 0.078 0.078 0.073 10 0.460 0.461 0.476 11 0.333 0.333 0.334 12 0.333 0.333 0.329 13 0.311 0.311 0.311 14 0.229 0.230 0.234 15 0.123 0.123 0.126 16 0.123 0.123 0.126 17 0.450 0.452 0.461 TheTC-LiDAGstructure,modelingthesequentialcircuits,used3time-slicesandjust1000samplingiterationswereusedforthehybridinferencescheme.Weseethatevenforlargerbenchmarkslikes5378themeanerrorisextremelysmall.Themaximumerrorsformostcircuitsarealsolow,exceptfors208s953ands5378.However,thesemaximumerrorsseemtobeisolatedtoafewnodesasisseenfromthelowfractionofnodeswitherrorabove2s.Inmostcases,only5%ofthenodesexceedthiserrorbound,exceptfors208,whereweseethat%ofnodesinE02Esrangeisaround9%.Wealsofoundtheaccuracyofourmodelisexcellentevenforlargerbenchmarkcircuitslikes15850(En0.004),butattheexpenseofcomputationtime(45minutes).Third,westudytheeffectofvaryingthemodelingandinferenceparameters,i.ethenumberoftimeslicesmodeledbytheTC-LiDAGandnumberofsamplingiterationofthehybridinferencescheme.InTable5.3.,wepresentresultsbasedon3000samplingiterations.Thetimeslicesarestillrestrictedtothree.Weseethattheaverageofthemeanerrorestimatesfortenbenchmarksis0006.Theaveragetimeforestimationhas,ofcourse,increased,from666secondsfor1000samplesto948secondsfor3000samples.Theaveragemaximumerrorisreducedfrom0086to0083forthetenbenchmarks.Increasingthesamplingiterationsto5000,showninTable5.4.,wegetaveragemeanerrorof0005,38

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Table5.2.SwitchingActivityEstimationErrorStatisticsbasedonTC-LiDAGModeling,using3TimeSlices,andHybridInferenceScheme,using1000Samples,forISCAS'89BenchmarkSequentialCir-cuits. Circuits MeanError Maximum Time(s) %ofnodes (E) Error(Emax (CPU+I/O) E02Es s27 0.015 0.068 0.047 5.88 s208 0.014 0.234 0.719 9.02 s382 0.002 0.096 2.094 4.95 s444 0.003 0.083 2.734 4.88 s526 0.003 0.044 3.922 3.23 s713 0.008 0.043 9.093 4.92 s820 0.002 0.049 10.060 8.33 s953 0.009 0.162 8.343 7.50 s1196 0.001 0.050 15.060 4.81 s1238 0.001 0.038 14.620 5.00 s1423 0.010 0.127 21.120 6.15 s5378 0.002 0.402 378.680 5.08 Table5.3.SwitchingActivityEstimationErrorStatisticsbasedonTC-LiDAGModeling,using3TimeSlices,andHybridInferenceScheme,using3000Samples,forISCAS'89BenchmarkSequentialCir-cuits. Circuits MeanError Maximum Time(s) %ofnodes (E) Error(Emax (CPU+I/O) E02Es s27 0.023 0.085 0.063 5.88 s208 0.015 0.225 1.219 9.02 s382 0.002 0.094 3.625 5.49 s444 0.004 0.052 4.328 4.88 s526 0.003 0.058 6.359 2.31 s713 0.008 0.056 12.984 4.25 s820 0.003 0.047 14.172 5.45 s953 0.010 0.187 12.031 8.18 s1196 0.001 0.019 20.296 6.06 s1238 0.000 0.017 19.766 4.82 s1423 0.007 0.121 27.547 5.35 s5378 0.001 0.393 404.886 5.21 39

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Table5.4.SwitchingActivityEstimationErrorStatisticsbasedonTC-LiDAGModeling,using3TimeSlices,andHybridInferenceScheme,using5000Samples,forISCAS'89BenchmarkSequentialCir-cuits. Circuits MeanError Maximum Time(s) %ofnodes (E) Error(Emax (CPU+I/O) E02Es s27 0.019 0.081 0.078 5.88 s208 0.015 0.211 1.672 9.02 s382 0.001 0.089 5.157 5.49 s444 0.004 0.052 6.110 3.41 s526 0.000 0.048 8.796 2.31 s713 0.006 0.044 16.891 9.62 s820 0.001 0.035 18.282 5.77 s953 0.009 0.171 15.765 7.73 s1196 0.001 0.022 25.546 5.35 s1238 0.000 0.017 25.609 5.55 s1423 0.009 0.131 33.000 5.48 s5378 0.001 0.389 432.908 5.11 Table5.5.SwitchingActivityEstimationErrorStatisticsbasedonTC-LiDAGModeling,using3and10TimeSlices,andHybridInferenceScheme,using1000Samples,forISCAS'89BenchmarkSequentialCircuits. 3Timeslices 10Timeslices Circuits E Emax Time(s) E Emax Time(s) s27 0.015 0.068 0.047 0.018 0.078 0.172 s208 0.014 0.234 0.719 0.006 0.197 7.532 s298 0.015 0.169 1.422 0.010 0.170 13.87 s382 0.002 0.096 2.094 0.000 0.079 21.28 s444 0.003 0.083 2.734 0.005 0.062 26.30 s526 0.003 0.044 3.922 0.001 0.081 42.28 maximumerrorof0077in1239secondsonanaveragefortenbenchmarks.Theseexperimentsshowthat1000samplesaresufcienttoachievethebestaccuracy-timetrade-off.InTable5.5.,weshowtheeffectofconsideringincreasednumberoftimeslicesintheTC-LiDAGmodel;weconsider10slicesasopposedto3.Weobservethattentimeslicesdonotenhancethequalityofestimates.Thisshowsthatthirdordertemporalmodelsaregoodenoughforourbenchmarks.Interestingly,thisobservationmatcheswiththeobservationsmadein[14,16].Table.5.6.givesthesimulationresultsofthesequentialcircuitsinferencedusingProbabilisticLogicSampling.Theseresultswereobtainedfor3time-slicemodelsofthesequentialcircuitswith1000samples.Themeanerrorandthemaximumerrorforthecircuitsarealmostidenticaltothoseof40

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Table5.6.ExperimentalResultsonSwitchingActivityEstimationbyDynamicBayesiannetworkMod-elingforISCAS'89BenchmarkSequentialCircuits(3TimeSlicesand1000Samples)usingLogicSampling[48]. Circuits E Emax Time %ofnodes (s) 782s s27 0.028 0.092 0.016 5.88 s208 0.014 0.224 0.281 9.02 s382 0.000 0.082 0.750 7.14 s444 0.005 0.067 0.843 3.90 s526 0.002 0.048 1.234 1.84 s713 0.009 0.067 1.968 4.70 s820 0.002 0.042 2.125 4.17 s953 0.012 0.185 1.922 7.95 s1196 0.001 0.043 2.735 5.17 s1238 0.003 0.035 2.703 5.00 s1423 0.012 0.114 3.266 6.02 s5378 0.001 0.389 23.128 4.98 s15850 0.003 0.434 146.992 3.07 EPISmethod.Themainadvantageofthismethodisclearlydepictedthroughtheestimationtime.PLSmethodis,byanorderofmagnitude,fasterthanEPISmethod.41

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CHAPTER6CONCLUSIONWepresentauniedgraphbasedprobabilisticframeworkforswitchingactivityestimationmethodthattakesintoaccounthighorderspatio-temporaldependenciesthatarepresentinsequentialcircuitsandcombinationalcircuits.Toourknowledge,thisistherstworkthatpresentsacompletelyproba-bilisticapproachtoswitchingestimationinsequentialcircuits.Weprovedthatatimecoupled,logicallyinduced,directedgraphstructure(TC-LiDAG)canmodelalltheindependenciesandisaBayesiannetwork,describingajointprobabilitydistributionoverallthecircuitlineswitchings,bothstateandcircuitlines.Thismodeliscompact,minimalanddependencypreserving.TheTC-LiDAGstructurealsoaffordsefcientprobabilisticinferenceschemes.Westudiedbothanexactinferencescheme,basedonthetreeofcliques,andascalablehybridinferencescheme,basedonsamplingandlocalmessagepassing.WedemonstratedthehighqualityofestimatesformedusingthismodelonISCAS'89benchmarkcircuits.Boththemeanandthemaximumerrorswerefoundtobelow.Themodelscaleswelltolargecircuits.Thepresentscopeofthemodelislimitedtothezero-delayscenario,whichweplantoaddressinfuturepossiblybyexpandingthedomainoftheswitchingrandomvariables.Inthischapter,wealsopresenttheattemptsthatledustotheTC-LiDAGmodels.1.Ourinitialattemptwastofeedtheoutputstateprobabilitiesbacktotheinputstatestillitcon-verges.Theconvergencewasnotaproblem,however,thesystemwasconvergingtowrongvaluesasthetemporalandspatialdependencieswerelostinthefeedbackprocess.2.Toovercomethis,weactuallyunraveledthecircuitfortentimeslices.Eachtimeslicewerefedbyindependentinputsandoutputstatelineprobabilitieswerefedtotheinputstateprobabilitiesofthenexttimeslice.Therandomvariablesofinterestwereswitchingateachsignals.This42

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modelalsofailedastheinduceddependenceoftheinputsandthepresentstatesarenotmodeledhere.3.Next,tomodeltheinputdependencies,weconnectedinputsofallthetimeslicesbyanidentitybuffertomodeltheinducedspatio-temporaldependencies.Theerrorswerecloseto40%.4.Wemodeledthedependenciesbetweenrandomvariablesthatrepresentswitchingintheinputbetweentwotimeslicesaccuratelyandfors27theerrorswereclosetozero.5.WeappliedtheCascadedBNapproachproposedbyBhanjaetal.[3],andwefoundthattheestimatesdegenerateduetotheerrorfeedback.6.WeusedthestochasticinferenceschemeusedbyRamanietal.[47]forhandlinglargerISCASbenchmarkandtheany-timeestimatesproducedaccurateresultsforthebenchmarks.Ourfuturedirectionwouldbeaddressingdelay,interconnectandcapacitanceestimationinthisproblem.WeintendtousethisuniedframeworkalsoatRTlevel.43

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REFERENCES[1]S.BhanjaandN.Ranganathan,“Dependencypreservingprobabilisticmodelingofswitchingac-tivityusingBayesiannetworks,”IEEE/ACMDesignAutomationConference,pp.209–214,2001.[2]S.BhanjaandN.Ranganathan,“SwitchingActivityEstimationofVLSICircuitsUsingBayesianNetworks,”IEEETransactionsonVLSISystems,vol.11,no.4,pp.558–567,Aug.2003.[3]S.BhanjaandN.Ranganathan,”SwitchingactivityestimationoflargecircuitsusingmultipleBayesiannetworks,”ProceedingsofASP-DACand15thInternationalConferenceonVLSIDe-sign,pp.187–192,2002.[4]R.G.Cowell,A.P.David,S.L.Lauritzen,D.J.Spiegelhalter,“ProbabilisticNetworksandExpertSystems,”Springer-VerlagNewYork,Inc.,1999.[5]J.Pearl,“ProbabilisticReasoninginIntelligentSystems:NetworkofPlausibleInference,”Mor-ganKaufmannPublishers,Inc.,1988.[6]U.Kjaerulff,“dHugin:AComputationalSystemforDynamicTime-SlicedBayesianNetworks,”InternationalJournalofForecasting,vol.11,pp.89–111,1995.[7]“GraphicalNetworkInterface”URLhttp://www.sis.pitt.edu/genie/genie2.[8]S.M.Kang,“AccurateSimulationofPowerDissipationinVLSICircuits,”IEEEJournalofSolid-stateCircuits,vol.21,no.5,pp.889–891,Oct.1986.[9]URLhttp://www.hugin.com/.[10]R.Burch,F.N.NajmandT.Trick,“AMonteCarloApproachforPowerEstimation,”IEEETransactionsonVLSISystems,vol.1,no.1,pp.63–71,1993.[11]R.E.Bryant,“SymbolicBooleanManipulationwithOrderedBinary-DecisionDiagrams,”ACMComputingSurveys,vol.24,no.3,pp.293–318,Sept.1992.[12]F.Najm,S.GoelandI.Hajj,“PowerEstimationinSequentialCircuits,”32ndACM/IEEEDesignAutomationConference,pp.635–680,1995.[13]A.Ghosh,S.Devadas,K.KeutzerandJ.White,“EstimationofAverageSwitchingActivityinCombinationalandSequentialCircuits,”29thACM/IEEEDesignAutomationConference,pp.253–259,1992.44

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Estimation of switching activity in sequential circuits using dynamic Bayesian Networks
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ABSTRACT: This thesis presents a novel, non-simulative, probabilistic model for switching activity in sequential circuits, capturing both spatio-temporal correlations at internal nodes and higher order temporal correlations due to feedback. Switching activity, one of the key components in dynamic power dissipation, is dependent on input streams and exhibits spatio-temporal correlation amongst the signals. One can handle dependency modeling of switching activity in a combinational circuit by Bayesian Networks [2] that encapsulates the underlying joint probability distribution function exactly. We present the underlying switching model of a sequential circuit as the time coupled logic induced directed acyclic graph (TC-LiDAG), that can be constructed from the logic structure and prove it to be a dynamic Bayesian Network. Dynamic Bayesian Networks over n time slices are also minimal representation of the dependency model where nodes denote the random variable and edges either denote direct dependency between variables at one time instant or denote dependencies between the random variables at different time instants. Dynamic Bayesian Networks are extremely powerful in modeling higher order temporal as well as spatial correlations; it is an exact model for the underlying conditional independencies. The attractive feature of this graphical representation of the joint probability function is that not only does it make the dependency relationships amongst the nodes explicit but it also serves as a computational mechanism for probabilistic inference. We use stochastic inference engines for dynamic Bayesian Networks which provides any-time estimates and scales well with respect to size We observe that less than a thousand samples usually converge to the correct estimates and that three time slices are sufficient for the ISCAS benchmark circuits. The average errors in switching probability of 0.006, with errors tightly distributed around the mean error values, on ISCAS'89 benchmark circuits involving up to 10000 signals are reported.
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