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Graphical probabilistic switching model inference and characterization for power dissipation in VLSI circuits
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Ramani, Shiva Shankar
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simulation
sampling
entropy
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ABSTRACT: Power dissipation in a VLSI circuit poses a serious challenge in present and future VLSI design. A switching model for the data dependent behavior of the transistors is essential to model dynamic, load-dependent active power and also leakage power in active mode - the two components of power in a VLSI circuit. A probabilistic Bayesian Network based switching model can explicitly model all spatio-temporal dependency relationships in a combinational circuit, resulting in zero-error estimates. However, the space-time requirements of exact estimation schemes, based on this model, increase with circuit complexity 5, 24. This work explores a non-simulative, importance sampling based, probabilistic estimation strategy that scales well with circuit complexity. It has the any-time aspect of simulation and the input pattern independence of probabilistic models.Experimental results with ISCAS'85 benchmark shows a significant savings in time (nearly 3 times) and significant reduction in maximum error (nearly 6 times) especially for large benchmark circuits compared to the existing state of the art technique (Approximate Cascaded Bayesian Network) which is partition based. We also present a novel probabilistic method that is not dependent on the pre-specification of input-statistics or the availability of input-traces, to identify nodes that are likely to be leaky even in the active zone. This work emphasizes on stochastic data dependency and characterization of the input space, targeting data-dependent leakage power. The central theme of this work lies in obtaining the posterior input data distribution, conditioned on the leakage at an individual signal.We propose a minimal, causal, graphical probabilistic model (Bayesian Belief Network) for computing the posterior, based on probabilistic propagation flow against the causal direction, i.e. towards the input. We also provide two entropy-based measures to characterize the amount of uncertainties in the posterior input space as an indicator of the likelihood of the leakage of a signal. Results on ISCAS'85 benchmark shows that conclusive judgments can be made on many nodes without any prior knowledge about the input space.
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Thesis (M.S.E.E.)--University of South Florida, 2004.
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GraphicalProbabilisticSwitchingModel:InferenceandCharacterizationforPowerDissipationinVLSICircuitsbyShivaShankarRamaniAthesissubmittedinpartialfulllmentoftherequirementsforthedegreeofMasterofScienceinElectricalEngineeringDepartmentofElectricalEngineeringCollegeofEngineeringUniversityofSouthFloridaMajorProfessor:SanjuktaBhanja,Ph.D.Yun-LeeiChiou,Ph.D.WilfridoA.Moreno,Ph.D.DateofApproval:September8,2004Keywords:GateLevel,BayesianNetwork,Simulation,Sampling,EntropycCopyright2004,ShivaShankarRamani

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DEDICATIONInmemoryofmyparents

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ACKNOWLEDGEMENTSIwouldliketotakethisopportunitytoexpressmysinceregratitudetomyadvisorDr.San-juktaBhanjaforherconstantencouragement,supportandguidancethroughoutthecourseofthisresearch.Shehashelpedmealottomouldmyselfasaresearcher.IwouldalsoliketoexpressmydeepappreciationtomycommitteemembersDr.Yun-LeeiChiouandDr.WilfridoA.Morenofortheirvaluabletime.MySincerethankstoDr.Yun-LeeiChiouforprovidingmewithresearchprojectsduringtheinitialpartofmygraduatestudies.Iamgratefultomyuncleforstayingbymeduringthehardtimesandfurtherassistingmenancially.IwouldalsoliketotakethisopportunitytothankEssieUtley,BarbaraRoberts,JenevieBrown,Mina,andEvaforsupportingandprovidingmoralsupportthroughoutmystaywiththeUMSA.IamreallygratefulfortheinvaluablesupportandmotivationthatIreceivedfrommyfamilyandfriends(especiallymyroom-mates).

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TABLEOFCONTENTSLISTOFTABLESiiiLISTOFFIGURESivABSTRACTviCHAPTER1INTRODUCTION11.1NeedforLowPowerVLSIDesign11.2ComponentsofPowerDissipation21.2.1DynamicPowerDissipation21.2.2LeakagePowerDissipation41.2.2.1ReverseBiasedPN-junctionI151.2.2.2SubthresholdChannelLeakageI251.2.2.3GateOxideTunnelingI371.2.2.4HotCarrierInjectionfromSubstratetoGateOxideI471.2.2.5GateInducedDrainLeakageI571.2.2.6PunchthroughI671.3PowerAnalysisTechniques101.3.1ProbabilisticBayesianNetworkModel121.4ContributionsofThisThesis151.5Organization17CHAPTER2PRIORWORK182.1ExistingDynamicPowerEstimationTechniques182.2ExistingLeakagePowerEstimationTechniques24CHAPTER3MODELINGUSINGBAYESIANNETWORKS283.1BayesianNetworkFundamentals283.2MathematicalFormalism303.3FormationoftheLIDAG-BN36CHAPTER4BAYESIANINFERENCEFORSWITCHINGACTIVITYESTIMATION394.1ProbabilisticInference394.2StochasticInferenceAlgorithms404.2.1ProbabilisticLogicSampling424.2.2AdaptiveImportanceSampling434.2.3HybridScheme444.2.3.1LoopyBeliefPropagation45i

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4.3ExperimentalResults47CHAPTER5ENTROPYBASEDINPUTCHARACTERIZATIONFORDATA-DEPENDENTLEAKAGEPOWERANALYSIS525.1WhyDoWeNeedInputCharacterization?535.2InputCharacterization565.3ComputingPosteriorInputDistributions585.4ResultsandConclusions60CHAPTER6CONCLUSION66REFERENCES67ii

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LISTOFTABLESTable1.1.ProbabilisticSwitchingActivityEstimationTechniques.11Table3.1.ConditionalProbabilitySpecicationsfortheOutputandtheInputLineTran-sitionsforTwoInputNANDGate.37Table3.2.ConditionalProbabilitySpecicationsfortheOutputandtheInputLineTran-sitionsforTwoInputORGate.38Table4.1.ExperimentalResultsComparingApproximateCascadedBayesianNetworkModelandProbabilisticLogicSampling.48Table4.2.ExperimentalResultsusingAISAlgorithmforvariousSamples.49Table4.3.ExperimentalResultsusingEPISAlgorithmforvariousSamples.50Table5.1.LeakageProleforKnownDataTraces.61Table5.2.EntropyBasedInputCharacterizationforLeakageCondition.62Table5.3.RelativeEntropyBasedInputCharacterizationforLeakySignals.65iii

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LISTOFFIGURESFigure1.1.GraphDepictingMoore'sLaw[1].2Figure1.2.CMOSInverterandaGraphShowingitsShortCircuitCurrent[3].4Figure1.3.LeakageCurrentMechanisminDeep-submicron[14].6Figure1.4.AccuracyvsSpeedofPowerEstimationatVariousLevels[3].8Figure1.5.TwoDifferentSignalshavingIdenticalFrequency[3].10Figure2.1.TechniquesforEstimatingSwitchingActivityinCombinationalCircuits.20Figure2.2.2-InputNandGateandaTableShowingtheDependenceofLeakageonIn-puts.25Figure3.1.BayesianGraphicalModel.29Figure3.2.ACombinationalCircuit.31Figure3.3.BayesianNetworkCorrespondingtotheCircuitinFigure3.2.32Figure3.4.BayesianNetworks:MarriagebetweenGraphicalandProbabilisticModels.33Figure4.1.ProbabilisticInferenceUsingLocalMessagePassing.45Figure4.2.GraphShowingtheTimeAccuracyTrade-offforc432.49Figure4.3.GraphShowingtheTimeAccuracyTrade-offforc1355.50Figure4.4.GraphShowingtheTimeAccuracyTrade-offforc6288.51Figure5.1.GraphShowingRiseinLeakagePowerwithTechnology[1].53Figure5.2.TheCorrelationsamongtheOutputLinesofc432withRandomInputs.TheOutputLinesofOneBlockaretheInputLinesofAnother.54Figure5.3.GraphShowingtheBreak-upofLeakageProleforDifferentSwitchingPro-leforc432.62Figure5.4.GraphShowingtheBreak-upofLeakageProleforDifferentSwitchingPro-lefor1908.63iv

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Figure5.5.GraphShowingtheBreak-upofLeakageProleforDifferentSwitchingPro-leforc3540.64v

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GRAPHICALPROBABILISTICSWITCHINGMODEL:INFERENCEANDCHARACTERIZATIONFORPOWERDISSIPATIONINVLSICIRCUITSShivaShankarRamaniABSTRACTPowerdissipationinaVLSIcircuitposesaseriouschallengeinpresentandfutureVLSIdesign.Aswitchingmodelforthedatadependentbehaviorofthetransistorsisessentialtomodeldynamic,load-dependentactivepowerandalsoleakagepowerinactivemode-thetwocomponentsofpowerinaVLSIcircuit.AprobabilisticBayesianNetworkbasedswitchingmodelcanexplicitlymodelallspatio-temporaldependencyrelationshipsinacombinationalcircuit,resultinginzero-errores-timates.However,thespace-timerequirementsofexactestimationschemes,basedonthismodel,increasewithcircuitcomplexity[5,24].Thisworkexploresanon-simulative,importancesamplingbased,probabilisticestimationstrategythatscaleswellwithcircuitcomplexity.Ithastheany-timeaspectofsimulationandtheinputpatternindependenceofprobabilisticmodels.Experimentalre-sultswithISCAS'85benchmarkshowsasignicantsavingsintime(nearly3times)andsignicantreductioninmaximumerror(nearly6times)especiallyforlargebenchmarkcircuitscomparedtotheexistingstateofthearttechnique(ApproximateCascadedBayesianNetwork)whichispartitionbased.Wealsopresentanovelprobabilisticmethodthatisnotdependentonthepre-specicationofinput-statisticsortheavailabilityofinput-traces,toidentifynodesthatarelikelytobeleakyevenintheactivezone.Thisworkemphasizesonstochasticdatadependencyandcharacterizationoftheinputspace,targetingdata-dependentleakagepower.Thecentralthemeofthisworkliesinob-tainingtheposteriorinputdatadistribution,conditionedontheleakageatanindividualsignal.Weproposeaminimal,causal,graphicalprobabilisticmodel(BayesianBeliefNetwork)forcomputingtheposterior,basedonprobabilisticpropagationowagainstthecausaldirection,i.e.towardstheinput.Wealsoprovidetwoentropy-basedmeasurestocharacterizetheamountofuncertaintiesinvi

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theposteriorinputspaceasanindicatorofthelikelihoodoftheleakageofasignal.ResultsonISCAS'85benchmarkshowsthatconclusivejudgmentscanbemadeonmanynodeswithoutanypriorknowledgeabouttheinputspace.vii

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CHAPTER1INTRODUCTION1.1NeedforLowPowerVLSIDesignGordonMoore(Moore'slaw)predictedanexponentialgrowthinthenumberoftransistorspersquareinch,nearlydoublethenumberevery18monthsandfurtherexpectedthistrendtocontinuefortheforeseeablefutureFigure1.1.Thisdrasticincreaseinchipdensity,togetherwithdecreaseinfeaturesizeshavemadepowerdissipationamajorissueinVLSIcircuits.Theincreaseindevicedensityeveryyearalsodemandsforhighoperatingfrequency.Asaresult,theamountofpowerdisspatedperunitareaorthepowerdensityisboundtoincreasewhichnecessiatestheuseofcostlypackagingandheatsinkstokeepthetemperaturelevelsofthechipwithinitslimits.Astagehasreachedwerewehavetostartanalyzingourdesignsforpowerapartfromareaandspeedconstraintsforbetterimplementation.ThishasmadeLowPowerDesignthefocusofVLSIresearchanddevelopmentoverthelastdecade.Anotherfactorthatdrivestheneedforlowpowerdesignistherapidlyincreasingdemandforportableelectronicsystems,whichimposessevererestrictionsonitssize,weightandpower.Portablesystemsdemandforlowpowerchipstoprolongbatterylife.Batterylifeplaysamajorrolewhenitcomestomakingachoiceofaparticularportableelectronicitem.Thespecicweight,whichisthestoredenergyperunitweight,ofabatteryisnotexpectedtohavearevolutionarychangethatmeetswiththeexpandingapplicationsofportablesystems.Hence,estimationandoptmizationofadesignforpowerapartfromareaandtimeasconstraintsbecomesabsolutelynecessarytomeetwiththedemandsinportablesystemsdesign.Tosummaraize,theneedforlowpowerdesignisduetofollowingreasons:ReducedbatterylifeduetohighenergyconsumedbyVLSIcircuits.Reducedreliabilityandspeedduetoincreaseinpowerdissipation.1

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10101010101010101960196519701975198019851990199520052010200001234567891010MOS ArraysMOS Logic 1975 Actual Data1975 ProjectionTransistors Per DieYear1965 Actual DataIntegrated Circuit Complexity 1010 Figure1.1.GraphDepictingMoore'sLaw[1].Increasedcostsduetoadditionalpackagingandcoolingsystemtoreducetemperature.Environmentalconcernsduetounnecessaryenergyconsumptionandheat.1.2ComponentsofPowerDissipationThissectionisdevotedtogiveanoverviewonvarioussourcesofpowerdissipationinCMOScircuits.AveragepowerconsumptioninCMOScircuitsisdueto2components.Theyaredynamicpowerandstatic(leakage)power.AveragepowercanbeexpressedbythefollowingequationPavgPdynamicPleakage(1.1)1.2.1DynamicPowerDissipationDynamicpowerconsumptionarisesduetofrequentcharginganddischargingoftheparasiticcapacitanceduringswitching.Sofar,dynamicpowerhasbeenthedominantcomponentofpowerandaccountsforabout75%ofthetotalpowerdissipation.Interestedreaderisdirectedto[3]fordetailedunderstandingonthederivationsfordynamicpower.2

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Ifthechargingandthedischargingcycletakesplaceatafrequencyf,thetotalpowerdissipatedinaninverterisgivenasPdynamicEsfCLV2f(1.2)TheaboveEquation1.2isanimportantequationinCMOSVLSIdesignrepresentingthedy-namicpowerdissipiationofasinglegatewithloadcapacitanceCL.Ingeneral,thetotalpowerofacircuitwithngatesisgivenas:PdynamiciCiV2ifi(1.3)VoltageViintheEquation1.3issameforallgatesinacircuitandfiisthefrequencyofswitchingforaparticulargate.FromtheEquation1.3,wecannoticethatthedynamicpowerisdirectlydependentonthefrequencyofswitching.Henceanactivecircuitwilldissipatemorepowerthananidlecircuit.Accurateestimationofdynamicpowerrequiresacarefulanalysisontheswitchingproleforeachgate.Firstpartofthisthesisisfocussedtowardestimatingtheswitchingactivityofacombinationalcircuitatgatelevelunderzerodelayassumption.Anothercomponentofpowerthatiscausedduringinputsignaltransitionistheshort-circuitpowerconsumption.ThiscomponentisduetoadirectpathfromVddtoground,whenbothnmosandpmosconductforashortwhileduringswitching.ConsideraninvertercircuitshownintheFigure1.2.Duringinputtransition,abriefperiodexistsduringwhichbothtransistors,thatis,nmosandpmosconduct.Thereasonis,PMOSturnsoniftheinputsignallevelisbelowVtp(PMOSthresholdvoltage)whereasforNMOStheinputsignallevelhastobeaboveVtn(NMOSthresholdvoltage),asshowninFigure1.2.,thuscausingadirectowofcurrentfromthevoltagesourcetoground.ThepowerdissipatedduringtheinputsignaltransitionphaseisreferredtoastheshortcircuitpowerandisgivenbyEquation1.4[4]PshortKb 12Vdd2VT3ft(1.4)bisthegainfactorofaMOStransistor,VTisthethresholdvoltage,andtistherise/falltimeofthegateinputs.Shortcircuitpoweraccountsfor5%-10%oftheoverallpowerdissipationandis3

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pVVtti /i n ddGroundGroundinoutout short cPull-up(PMOS)(NMOS)Pull-downtptni Short circuit currentIICVV Figure1.2.CMOSInverterandaGraphShowingitsShortCircuitCurrent[3].usuallynottakenintoconsiderationduringlowpowerdesign.Factorsthataffectshortcircuitpoweraregivenbelow:Theslopeanddurationoftheinputsignal.Theoutputloadingcapacitance.Italsodependsonprocesstechnology,temperature,etc.DynamicandShortCircuitPowerDissipationdependontheswitchingactivityandhenceinidlestate(whenswitchingactivityiszero)thecircuitshouldactuallynotconsumeanypower.Inreality,thereisanothercomponentcalledleakagepowerthatcausespowerdissipationinthesleeporstaticmodeandithasbeenrecentlydiscoveredthatleakageexistsevenduringnormaloperationofacircuit.1.2.2LeakagePowerDissipationSecondcomponentofpowerdissipationistheStaticLeakagePowerDissipation.Thereasonforthistypeofdissipationcanbeattributedtoreversebiasdiodeleakage,sub-thresholdleakage,gateoxidetunneling,leakageduetohotcarrierinjection,Gate-InducedDrainLeakage(GIDL),andchannelpunchthrough.Notethatthistypeofpowerdissipationdependsonthelogicstatesofacircuitthanitsswitchingactivites.Currently,powerdissipationduetoleakageisnotalarmingbutisexpectedtoincreasebyyear2005asthetechnologymovestowardsnanodimensions.Thetotal4

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leakagepowerdissipationisgivenbyEquation1.5.PleakageIdiodeIsubthresholdIoxidetunnelingIhotcarrierIGIDLIchannelpunchthroughVdd(1.5)1.2.2.1ReverseBiasedPN-junctionI1ThediodeleakageisduetotheformationofPNjunctionsbetweensourceordrainofthetran-sistorandthebulk(substrate).Leakagecurrentowsfromthejunctiontothesubstratewhenthediodeisreversebiased.Themagnitudeofthecurrentdependsonprocessparameters,areaofthePNjunction,biasvoltage,andtemperature.Equation1.6givesdiodeleakagecurrent.IdiodeIsneVVth1(1.6)Isisthereversesaturationcurrentanditisdependentontemperature.Isdoublesforeverytendegreeincreaseintemperature.Vthisthethermalvoltage,whichisgivenbykTrq.ThereversesaturationcurrentIsisoftheorderof15pArm2.Notethatthediodeleakagecurrentoccursevenduringstand-bymode,thatis,whenthereisnoswitching.Hence,thepowerdisspationduetothismechanismwillhaveasignicantimpactonalargechipcontainingseveralmilliontransistors.Notethatforheavilydopedpandnregions,theBTBT(bandtobandtunneling)dominatesthediodeleakage[15].Higheldacrossthereversebiasedpnjunctioncausesasignicantowofelectrons,bythetunnelingprocess,fromthevalencebandofthep-regiontotheconductionbandofthen-region.1.2.2.2SubthresholdChannelLeakageI2Theprimarycontributortoleakagepoweristhesub-thresholdorweakinversionconductioncurrent.Strictlyspeaking,whenatransistorisinoffstate,thereshouldnotbeanycurrentinthechannel.Butinreality,thereisanon-zerocurrentowingthroughthechannelasshownin5

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II I ++SourceGateDrainoxideSubstratep-well34IIIn n 6512 Figure1.3.LeakageCurrentMechanisminDeep-submicron[14].Figure1.3.Hencethetermsub-thresholdleakageasitoccursatavoltagelevelwellbelowthegatevoltage.Asthedevicedimensionscalesdown,thepowercontributedbysub-thresholdleakagebecomesenormousanditexhibitsanexponentialdependenceonthegatevoltage.SubthresholdcurrentisgivenbyEquation1.7.IsubthresholdI0neVgsVt aVth(1.7)Vtisthethresholdvoltage.I0isthecurrentwhenVgsVt.aisaconstantdependentonthedevicefabricationprocess.Sub-thresholdcurrentdependsonfabricationprocess,temperaturevariations,andgatevoltage.DrainInducedBarrierLowering(DIBL)occurswhenthedraindepletionregioninteractswiththesourcenearthechannelsurface,thusloweringthesourcepotentialbarrier.Forshortchanneldevices,whenthedrainvoltageisincreased,itlowersthebarrierheight,resultingindecreaseofthresholdvoltage.Thesourcetheninjectscarriersintothechannelwithouttheinuenceofthegatebias,thusincreasingthesubthresholdcurrent[14].6

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1.2.2.3GateOxideTunnelingI3Asthedevicesizeshrinks,thereisacorrespondingreductioningateoxidethickness.Thisprocessresultsinasignicantincreaseintheelectriceldacrossthegate.Thehighelectriceldtogetherwithlowgateoxidethicknessresultsintunnelingofelectronsfromthesubstratetogateandalsofromgatetosubstratethroughthegateoxide[14].Thetwoformsoftunnelingacrossthegateoxideare,namely,Fowler-NordheimTunnelingandDirectTunneling.1.2.2.4HotCarrierInjectionfromSubstratetoGateOxideI4Shortchanneldevicesaresusceptibletocarrierinjectionintothegateoxide,duetohighelectriceldnearthesilicon-oxideinterface.Electronsorholesgainsufcientenergytocrosstheinterfacebarrierandenterthegateoxide.Thiseffectiscalledashot-carrierinjection[14].1.2.2.5GateInducedDrainLeakageI5GIDLisduetothehighelectriceldinthegate/drainoverlapregionofatransistor.WhentheMOSisintheaccumulationregion,surfaceunderneaththegatebehaveslikeaheavilydopedregionthanthesubstrate,itcausesthedepletionlayeratthesurfacetobemuchnarrowerthanelsewhere.Thisformofleakageoccursatahighdrainbiasandlowergatebias.Then+drainregionunderthegatebecomesdepletedandsometimesinvertedatalowgatebias.Forminoritycarriersthesubstrateisatalowerpotential,hencethecarrierscaughtinthedepletionregionbeneaththegateareswepttothesubstrate.ThiseffectisknownastheGateInducedDrainLeakage.Lightlydopeddrain,highVDDandthinoxidethicknessenhanceGIDL[14].1.2.2.6PunchthroughI6Inshortchanneldevices,theseperationbetweenthesourceanddraindepletionlayersreduceswithacorrespondingincreaseinthereversebias(Vds)voltage.Atasufcientdrainvoltage,thedepletionlayerstouchormergedeepbelowthesurfacecausingpunchthrough.Sincetheregionnearthesiliconsurfaceisheavilydoped(forathresholdadjustimplant),thereisagreaterexpansion7

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WorstBestAlgorithm Behavior RTL Logic CircuitAbstraction Level Speed Accuracy High LowFigure1.4.AccuracyvsSpeedofPowerEstimationatVariousLevels[3].ofthedepletionlayerdeepbelowthesurface(duetolessdoping)ascomparedtothesurface.Thus,punchthroughoccursdeepbelowthesurface[14].Inthissection,wediscussedaboutthetwocomponentsofpower,namely,dynamicandleakage.Aswemovetowardsthenano-domain,dynamiccomponentaswellastheleakagecomponentofthepowerisexpectedtohaveasteadyincreasemainlyduetotheincreaseinnumberoftransistorsperchip.Thetotalpower(Pt)expendedinacircuitcanbeexpressedasthesumofindividualgatepower(Ptg),whichinturncanbebrokenupintoswitchingandleakagecomponents[61].PtgPtgPdgPsg05afclkV2ddCloadwire1aiPleakibi(1.8)Vddisthesupplyvoltage.fclkistheclockfrequency.Cloadisloadcapacitance.whereadenotestheactivityofthenode,and(1-a)istheprobabilityofremaininginadominantleakagestate(namelysignalat0or1).Notethatbaswellasaisdependentontheswitchingstatesoftheinputsandthephysicalparameters.Thus,totalpowerisafunctionoftheinputswitchingstates,andthedeviceparameters.Asindicatedin[57],dynamicpower(indicatedasPdynamic)willbe90%ofthetotalpowerifswitchingactivity(averagenumberofswitchingsinaclockcycle)ismorethan0.1.Furtherfrom[7],bytheyear2005theleakagecomponentofpower,Pleakage,startsdominatingtheoverall8

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powerdissipation.Reliabilityandperformanceofacircuitdegradeswithexcessiveaveragepowerconsumption,requiringtheuseofcostlypackagingandheatsinkstocontroltemperature.Hence,accurateestimationofpowerduringtheearlydesignphasessuchasattransistor,logic,architecturalorevenbehaviorallevelswillreducecomplicated,expensivedesignchangesatlaterstagesduetopowerdissipationconsiderations.Thus,modelingandestimationofdynamicpoweraswellasleakagepowerremaintobeimportantproblemsinlow-powerdesign.Estimationofpowerhasbeenperformedatvariouslevelsofabstraction,namelybehaviorallevel,RTlevel,gatelevel,circuitleveletc.,sothatoptimizationofdesigncanbeperformedateachlevelbeforesynthesizingtothelowerlevels.AsdepictedinFigure1.4.,higherthelevelofabstraction,fasteristhespeedofestimationbutwithlowaccuracy.Thereasonis,atahigherlevelthedesignerdoesnothaveenoughknowledgeabouttheinternalsofamodule,thatis,theimplementationdetailsofamodule.Thisthesispresentsmethodstoestimatethedynamiccomponentofpoweraswellasleakagecomponentofpoweratthegatelevel.Atthislevelsinceweknowthelogicstructure,wecaneasilyestimatethegatecapacitanceandthemajorchallengeliesinestimatingswitchingactivity(0t11t1t10t)andthedominantleakagestate(0t10t1t11t).Thechallengesindynamicpowerestimationliesinassessingtheloadcapacitanceandswitch-ingactivitysincethesupplyvoltageandclockfrequencyareknowntothedesigners.Switchingactivityisthenumberoftransistionsthatanode(inputoroutput)makesperclockcycle.Theswitchingactivityofanodeisaffectedbyvariousfactorssuchastheconnectivityofthecircuit,theinputstatistics,thecorrelationbetweennodes,thegatetype,andthegatedelays,thusmakingtheestimationprocessacomplexprocedure.Thecorrelationsamongthenodesaffectswitchingactivityandithasbeenobservedthatmodelsthatdidnotaccountfornodecorrelationsyieldlessaccuracy.Thereareplethoraoftechniquesavailabletoestimateswitchingactivitynamely,simu-lation,statisticalsimulation,andprobabilistictechniques.Section1.3givesabriefintroductiontothefundamentalsofsimulativeandprobabilisticpoweranalysistechniques.9

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0102030405001020304050 Figure1.5.TwoDifferentSignalshavingIdenticalFrequency[3].1.3PowerAnalysisTechniquesTheideabehindsimulation-basedapproachesistomimicthecircuitbehaviorovertime.Simu-lationsoftwarerecordstheprecisetimeinstantatwhichasignaleventoccurs.However,simulationtechniquesarestronglyinput-patterndependent.Tocompletelysimulateamoduleforallpossi-bleinputvectorsisimpossibleastheinputvectorsdependonthechipinwhichthemoduleisnallyplaced.Hence,forlargecircuitsestimatingswitchingactivitythroughsimualtioniscom-putationallyexpensive.Notethatsimulationtechniquesareaccurate,andtechnologyindependent.Inprobability-basedapproaches,signalsareconsideredasrandomzero-oneprocess.Wenolongerknowtheexactinstantsatwhichthelogicsignaldoesitstransition.Notethattoestimatethefre-quencyofasignal,thereisnoneedtoknowtheexacttimeofswitching,thatis,itissufcienttorecordthenumberoftransitions.Forexample,inFigure1.5.,althoughthesignalsappeardifferent,thenumberofsignaleventsortransitionsisidentical,whichmeansthefrequencyofthesignalsremainthesame.Hence,forsomepurposes(computingpower),thetwosignalsinFigure1.5.,remainindistinquishable.Thereforeexactcharacterizationofasignalbycapturingallitshistorytostudypowerisinefcientandcumbersome.Theproblemofinput-patterndependencecanbesolvedifwecaptureafewessentialstatisticalparametersofasignal.Thiswaywecanconstructacompactdescriptionofasignalandanalyzeitseffectonacircuit.Bydescribingtheprimaryinputsignalsusingthecharacteristicquantitieslikesignalprobability,transitiondensity,etc,andpropagatingtheeffectsofthesequantitiestothe10

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Table1.1.ProbabilisticSwitchingActivityEstimationTechniques. Canithandle Methods Temporal SpatioInput Speed Accuracy-Time Corr.? TemporalCorr.? Corr.? trade-off order1 (any-timeaspect) CREST[27] Yes No Approx. Fast No DENSIM[26] Yes No No Fast No OBDD[28] Yes Yes No Slow No TPS[35] Yes No Approx. Moderate-Fast No Marculescuetal.[56] Yes No Approx. Slow No Schneideretal.[55] Yes No No Moderate-slow No Marculescuetal.[29] Yes No Approx. Moderate-slow No Bhanjaetal.[5,24] Yes Yes Approx. Fast No Thisthesis Yes Yes Approx. Fast Yes(zero-error) internalnodesandoutputofthecircuit,wecanstudythepowerfromthecollectiveinuenceofallthelogicsignals.Howeverthepropagationofthesignalslargelydependontheprobabilisticmodelused.Asstatedbefore,accurateestimationofswitchingactivityrequirescompletedetailsonsignalcorrelations.Butmostoftheprobabilisticmodels(discussedinChapter2)donotcon-siderthecorrelation(assumetemporaland/orspatialindependence)amongthenodes,asshowninTable1.1.Hence,theyresultininaccurateestimates.Ithasbeenestablishedthatforzero-delaymodelofacombinationalcircuit,onlyrstordertemporalcorrelationisexhibited[54],becausesignalspossessrstorderMarkovproperty.Thus,itissufcienttoconsiderjustrstordertem-poralcorrelation,butallhighorderspatialcorrelationstomodelallspatio-temporaldependenciesinthecombinationalcircuit.Inthisthesis,weuseaprobabilisticmodelusingBayesianNetworks,forestimatingswitchingactivity,thatcapturesboththerstordertemporalaswellasallhighorderspatialcorrelationsinacomprehensivemanner,resultinginaccurateestimates.11

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1.3.1ProbabilisticBayesianNetworkModelRecently,weproposedanovelmodel[5,24],forswitchingactivityestimationincombinationalcircuitsusingProbabilisticBayesianNetworks[2],thatcapturesboththetemporalandspatialde-pendenciesinacomprehensivemanner,resultinginzero-errorestimates.BayesianNetworksareaDirectedAcyclicGraph(DAG)representations,whosenodesrepresentrandomvariablesandthelinksdenotedirectdependencies(capturingthespatialdependencies),quantiedbyconditionalprobabilitiesofanodegiventhestateofitsparents.TheDAGstructuremodelsthejointprobabilityoverasetofrandomvariablesinacompactmanner.TheadvantageofBayesianNetworkisthatitcombinestherepresentationalpowerandalgorithmicpowerofgraphtheorywithprobabilitytheory.Thecoreideaofthisthesisistoexpresstheswitchingactivityofacircuitasajointprobabilityfunctionwhichcanbemapped,one-to-one,ontoaBayesianNetworkthuspreservingthedepen-dencymodeloftheprobabilityfunction.WerstconstructaLogic-Induced-Directed-Acyclic-Graph(LIDAG)basedonthelogicalstructureofthecircuit.In[24],theauthorprovesthattheLIDAGstructure,correspondingtothecombinationalcircuitisaminimalrepresentationoftheun-derlyingswitchingdependencymodelandhenceisaBayesianNetwork.EachsignalinthecircuitisconsideredasarandomvariableintheLIDAGthatcanhavefourpossiblestatesindicatingthetran-sitionsfrom0t10t0t11t1t10t1t11t.Specicationofthesefourstatesenablesustocompletelycapturethetemporalcorrelation.Directededgesbetweentherandomvariablesrepre-sentingswitchingdenotestheprobabilisticdependencyamongthesignals.ItisourobservationthattheBayesianNetworkisapowerfultooltomodelswitchingactivitypreservingthevariousdepen-denciesinacircuit.Further,elegantinferencemechanismsexistforBayesianNetworkcomputationthatmaketheestimationtime-efcientandthus,usableforlargecircuits.Theattractivefeatureofthisgraphicalrepresentationofthejointprobabilityfunctionisthatitcannotonlymodelcomplexconditionalindependenceoverasetofvariables,buttheindepen-denciesserveasacomputationalschemeforsmartprobabilisticupdating.Ingeneral,thebeliefupdatingschemescanbeclassiedintoexactandapproximatetechniques.However,thespace-timecomplexityofexactestimationschemesincreasewithcircuitcomplexity.Forinstance,theinferenceschemethatweusedin[5,24],whichwasaclusterbasedscheme,resultedinexactesti-12

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mates,however,itwasmemoryintensive.So,forcomplexcircuit,wehadtoresorttopartitioningschemes,resultinginanapproximatemodeloftheswitchingactivitiesintermsofasetoflooselycoupledCascadedBayesianNetworks.Thismodelproducedestimateswithlowmeanerror,butduetocouplinglossesattheboundarynodes,itresultedinlargerstandarddeviationandmaximumerror.Fromadesignpointofview,itissometimesdesirabletohaveanestimationstrategywhereonecaneasilytrade-offbetweentimeandaccuracy,essentiallyanany-timeestimationalgorithm.Thisisnotpossiblewiththecurrentinferencescheme.Forthesereasons,thisworkexploresadifferentsetofstochasticinferencealgorithms(approximateBayesianNetworkinferencescheme)forthreeprimaryreasons:Asnumberoftransistorsincreaseundernano-domaindeviceshrinkage,lossesinthepartitionwoulddegeneratetheestimatesevenfurther,andweneedadifferentnon-partitionbasedinferencemechanism.Withincreasednano-domaincomplexity,weneedanany-timealgorithmthatshouldnotonlygenerateaccurateestimatesgivenenoughtime,butevenundertimeconstraints,shouldyieldroughapproximatelyvalidestimates.ItisnotpossibletouseapartitionedcascadedsetofBayesianNetworkforprobabilisticdiagnosisincasewewanttostudythereverse-causaleffects,namelytheeffectofanevidenceinanobservednodeontheprimaryinputs.Inthisthesis,weexplorethreeStochasticImportanceSamplingschemes:ProbabilisticLogicSampling(PLS)[10],AdaptiveImportanceSampling(AIS)[11]andEvidencePre-propagatedIm-portanceSampling(EPIS)[12]basedonthezero-error,pure,BayesianNetworkswitchingmodelforbeliefupdating.Thesealgorithmscombinetheany-timefeatureofsimulativeapproachesandinput-patternindependenceofprobabilisticapproaches.PLS[10],yieldsexcellentestimateswhenusedunderpredictivesituation(especiallywhentheevidenceispresentintherootintheformofpriorprobability)butindiagnosticreasoning,especiallywhenevidenceisunlikely,theaccuracydegenerates.However,thepredictivemechanismproducesthebestaccuracy-timetrade-off.In13

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AIS[11],andEPIS[12]samplingschemes,eachsampledeterminestheposteriorprobabilityoftheunderlyingmodelfortheremainingsamples.Theprobabilityofrandomvariableisproventoconverge[11]tothecorrectvaluesgivenenoughtime.AISandEPIS,produceaccurateestimatesunderdiagnosticsituations,eventhoughthetimeforpredictivemechanismishigherthanPLS.ExperimentalresultswithISCAS'85benchmarkshowsorderofmagnitudereductioninmaximumerror,standarddeviationspeciallyforlargerbenchmarkswithsignicantlylowtime,especiallyforthePLSscheme.Currently,leakagecurrentdrawnisnotsignicantenoughbutisexpectedtobeonparwiththedynamiccomponentofpowerbytheyear2005asthetechnologymovestowardsnanodimen-sions[7].Asthemagnitudeofleakagecurrentincreasesitbecomesamajorcontributortothepowerconsumption,andhencethedesignershavetofacetheaddedburdenduetotheleakagepowerdis-sipationandoptimizetheirdesignsforlowleakagepoweraswell.Notethat,ithasbeenfoundthatcircuitsnotonlyleakduringthesleeporidlemodebutthereisalsoasignicantcontributiontoleakagepowereveninactivemodes[61],[62],[63].Hence,thedesignersshouldalsoconsiderthegatesthatstaydormantduringmostpartofcircuitoperationwhileoptimizingthecircuitforleak-agepower.Secondpartofthisthesisunveilsanewtechniquefortargetingthegatesthatcontributetowardsleakageevenundertheactiveorrun-timemode.TheissueandtheapproachpresentedinthesecondpartofthethesisforLeakagePoweranalysisarenewtoourknowledge.Ourworkprovidesanotheraresenalforthelowpowerdesignerstofocusonleakagemitigationschemesattargetednodes,ratherthanjustconsideringnodes(basedoncriticalpath)orusingasingle(orahandfulofdataproles)totargetatanoptimizationscheme.Howdoweidentifytheleakynodes?Weexploitthebacktracking(reasoningfromtheevidencetocause)featureofBayesianNetworkstodeterminethelikelyinputspace,givenanobservation.ByusingthisaspectofBayesianNetworkthedesignercandeterminethelikelihoodofanodebeinginaleakystate(0t10tor1t11t)mostofthetime,evenwithoutanypriorknowledgeontheinputspace.Inputsignalsaregenerallyunknownduringthedesignphasebecausetheydependonthesysteminwhichthemoduleorchipwillbeused.Also,itisnotpossibletosimulatethecicuitforallpossibleinputvectors.DesignerscanexploitthebacktrackingattributeofBayesianNetworkstodetermine14

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thetransistorthathasahighpossiblitytoleakevenintherun-timemode,withoutpriorknowledgeoftheinputspace.WeuseEvidencePre-propagatedImportanceSampling[12]stochasticinferencetechniquetoaccuratelypropagatebelieffromevidencetoallothervariables.Methods,suchasdynamicthresholdvoltage[59],[60],canuseourmeasuretoselecttargetnodesforoptimization.Wealsointroducetwoentropybasedmeasurestocharacterizetheamountofuncertainitiesintheposteriorinputspaceasanindicatoroftheleakageofasignal.Thecontributionsofthisthesisaresummarizedbelow.1.4ContributionsofThisThesis1.BayesianNetworkmodelsofswitchingactivityareinherentlyzero-errormodels.Theyareinput-patterninsensitive.2.BayesianNetworksmodelsconditionalindependenciesbutdoesanycausalmodellikeBDD.However,therealmeritofaBNisthatituniesagraphicalmodelandaprobabilisticmodelinvariantintermsoftheconditionalindependenciesandtheDAGstructureisactuallyamin-imalcompactI-mapofprobabilisticmodel.Thisprobabilisticgraphicalmodelusesthead-vantageofbothgraph-basedandprobability-basedmodelforefcientprobabilisticupdating.3.BayesianNetworksareuniqueprobabilisticcausalmodelincapturingtheinduceddepen-dencebetweenindependentparentsofanodegivenanobservedstateforthenode.4.BayesianNetworksallowmulti-directionalbeliefow.Themodelcanacceptevidencefromanynodeandpropagateitinanydirection.5.Thetheoreticalcontributionofourthesisisthatthejointprobabilityfunctionofasetofran-domvariablesisexactlymappedcapturingallhigherordercorrelationsbetweenthesignalsaccuratelyusingBayesianNetworkmodel.Thisimpliesthatwecanmodelspatio-temporalcorrelationsofanyorder(rstordertemporalissufcientforzero-delaymodel)andhenceitisanexactswitchingmodel.Moreover,weuseindependencerelationsnotonlytomodeldependenciesexactly,butalsotouseitinourcomputationaladvantageduringbayesianin-ferencing.15

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6.Inthisthesis,weusenon-partitionbasedstochasticinferenceschemesforswitchingestima-tionthatscaleswell.resultsinanytimeestimate(foraccuracy-timetrade-off).We[9],showedthatthestochasticinferenceschemes,namely,ProbabilisticLogicSampling(PLS)[10],AdaptiveImportanceSampling(AIS)[11]andEvidencePre-propagatedImpor-tanceSampling(EPIS)[12]resultinzero-errorestimatesforswitchingactivityestimationincombinationalcircuits.Thesealgorithmscombinetheany-timefeatureofsimulativeap-proachesandinput-patternindependenceofprobabilisticapproaches.Weachievedanorderofmagnitudeimprovementoverthepaststateoftheart[5,24]intermsofmaximumerrorandstandarddeviation.PLS[10],yieldsexcellentestimateswhenusedunderpredictivesituationwhileAISandEPIS,produceaccurateestimatesunderdiagnosticsituations,eventhoughthetimeforpredictivemechanismishigherthanPLS.7.BayesianNetworksareextremelyeffectivetoperformbacktracking,thefeatureofBayesianNetworkthatisbeingexploitedinthesecondpartofmythesis.Notonlycanitpropagateprobabilitiesfrominputtotheoutputinacausalow,itcanalsopropagateprobabilitiesfromknownevidenceorobservationtoitsunknowncause.Thisisusefulandstraight-forwardforanalysisandtoresolvequerieslike“whatinputscancauseaswitchingprobabilityof08ataparticularnodeofinterest?”,whichwouldrequireasearchintheinputspaceforotherprobabilisticorsimulativeframework.8.Itisimpossibletosimulateacircuitforallpossibleinputcombinations.Itdependsonthechipinwhichthemodulebeingdesignedisnallyplaced.Wepresentanewtechniquethatdeterminesthelikelyinputspaceofanode,sayX,giventhatthenodeXisxedatoneofitsstates(inourcasethepossiblestatesofanodeare00011011).Chapter5exploitsthebacktrackingattributeofBayesianNetworkstotargetnodesthatstayidleevenduringtherun-timemode.Weusetwoentropybasedmeasures(absoluteentropy,relativeentropy)toquantifytheinformationcontentoftheposteriorinputspacetodeterminethe16

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possiblityofanodetoleak.ExperimentalresultswithISCAS'85benchmarksuiteshowthatgreaterpartofthecircuitremaindormantespeciallyinactivemodeandhence,theyarethenodesthatshouldbedenitelyconsideredwhileperformingleakagepoweroptimization.1.5OrganizationTherestofthethesisisorganizedasfollows.Chapter2containsaliteraturesurveyofpowerestimationtechniques.InChapter3,wediscussaboutthefundamentalsofBayesianNetworksandthemodelingofacombinationalcircuitintoaBayesiannetwork.WediscussindetailaboutvariousprobabilisticinferenceschemesforestimatingswitchingactivityinChapter4andconcludethechapterwithexperimentalresultsonISCAS'85benchmarkcircuits.InChapter5,wediscussindetailabouttwoinputcharacterizationschemestoinvestigatethelikelihoodofanodetobeataleakystateeveninactivemode.WeconcludethethesisinChapter6.17

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CHAPTER2PRIORWORK2.1ExistingDynamicPowerEstimationTechniquesTheprimarycomponentoflogiclevelpowerestimationisswitchingactivityestimation.An-otherimportantcomponentiscapacitanceestimationwhichissimpliedduetotheknowledgeofthecircuitstructureandtheexistenceoflibrariesofcapacitancesforstandardcells.Theeffectofreducinggatedelayandgrowinginterconnectdelaywillalsobedominantinfutureestimationtechniques.However,switchingactivitywillremainanimportantparametertoestimateeveninthenanodomainmainlyduetoitsdependenceoninputdataandoncorrelationsexhibitedininputsandintheinternalnodes.Denition1:Switchingactivityatanodecanbedenedastheaveragenumberofsignaltran-sitionsinaclockcycle.Inprobabilisticterms,SwXistheprobabilityofoccurrenceofatransitioninanodeXduringaclockperiod.SwXPX01PX10(2.1)whereX01andX10denotesasignaltransitionfrom0to1and1to0atnodeXrespectively.Itisclearthatswitchingactivityrequiressecondorderstatistics.SwitchingactivitySwXcanalsobedenotedastransitionprobabilityofX.Denition2:ThesignalprobabilityofanodePX1istheaveragefractionofclockcyclethatthenodeXremainsatlogic1.Switchingactivityofanodeisaffectedbythecorrelationsexhibited.Therecanbethree/fourtypesofcorrelation.Theyare18

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1.TemporalCorrelation:Thisarisessincethepreviousvalueofasignalcanbecorrelatedwiththepresentvalueofthesignal.2.SpatialCorrelation:Spatialcorrelationariseswhenthetwospatiallyconnectedsignalsaredependentoneachother.Itiscausedingeneralbyre-convergentfan-outs,byfeedbackandbyalreadycorrelatedprimaryinputs.3.Spatio-temporal:Spatio-temporalcorrelationisdependenceofasignaltothepreviousvalueofaspatiallyconnectedsignal.Hence,itisacombinationofspatialandtemporalcorrelation.4.Sequential:Itispartofspatialcorrelationwhenthedependenceisamongstthefeedbackstatelines.Switchingactivitycanbeestimatedatvariouslevelsofabstractionnamelyatarchitectural,behavioral,RTL,logic(discussedextensivelyinthischapter),ortransistorlevels.Asdiscussedbefore,estimationateachlevelhasitsownadvantagesanddisadvantages(Figure1.4.).Morepowersavingscanbeachievedaspowerisestimatedandoptimizedatthehigherlevels,whereastheestimatesaresignicantlymoreaccurateatthetransistorlevel.Thisworkisconcentratedatthelogicorgatelevelswitchingactivityestimation.Switchingactivityestimationstrategiescanbedividedintothreebroadcategories:estimationbysimulation,estimationbystatisticalsimulationandestimationbyprobabilistictechniques.Estimationbypuresimulation[38,40,44,49]thoughtimeconsuming,isextremelyaccurate.Todecreasethetimecomplexity,severalimprovedsimulationtechniqueshavebeenproposed[33,39,41,42,43,48].Manyofthemusevectorcompactionandmodelingofinputsamplespace[45,46,47]andsequencegenerationtoreducethesamplesneededforsimulation.Thesimulation-basedtechniquesarestronglyinputpatterndependent.Ingeneral,allsimulationtoolshavehigheraccuracycomparedtootherexistingmethodsbutwithhighertimerequirement.Instatisticalsimulation,statisticalmethodsareappliedinconjunctionwithsimulationinordertodeterminethestoppingcriterionforthesimulation.Theearliestworksinthisareacanbefoundin[25]and[50].Thesemethodsareefcientintermsofthetimerequiredandifthestatisticaldistributionoftheinputdataismodeledcorrectlytheycanyieldaccurateestimates.However,one19

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(Logic Level) Average Power Estimation for Combinational CicuitsSimulationStatistical SimulationProbabilistic Parker et al.'75 [36]Cirit et al.'87 [31]Bryant et al.'86 [28]Ercolini et al.'92 [51]Chakravarti et al.'89 [34]Burch et al.'88 [80]Najm et al.'90 [27]Ghosh et al.'92 [30]Stamoulis et al.'93 [37]Najm et al.'93 [26]Kapoor et al.'94 [52]Schneider et al.'94 [54]Marculescu et al.'94 [56]Schneider et al.'96 [55]Ding et al.'98 [35]Marculescu et al.'98 [29]Bhanja et al.'02 [5,24]This work'04 [9]Najm et al.'98 [50]Burch et al.'93 [26]Benini et al.'93 [33]Kang et al.'86 [38]Murugavel et al.'02 [40]Yacoub et al.'89 [49]Krodel '91 [39]Deng et al.'88 [41]Tjarnstrom '89 [48]Figure2.1.TechniquesforEstimatingSwitchingActivityinCombinationalCircuits.20

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hastobecarefulinmodelingthestatisticalpatternsattheinputsandspecialattentionhastobegiventonotgettrappedinalocalminima.Theseestimationstrategiesarebasedontheknowledgeoftheinputs,theroleofinputpatternsbecomeveryimportantintermsofwhetherthesamplesetrepresentstheentirepopulationorasubsetofit.Probabilistictechniquesarefastandmoretractable,buttypicallyinvolveassumptionsaboutjointcorrelations.Theprimaryconceptualdifferenceisthat,theinputstatisticsarerstgatheredintermsofprobabilitiesandthentheseprobabilitiesarepropagated.Hence,theabstractedknowledgeaboutinputsareusedtoestimatetheswitchingactivityofinternalnodes.Hence,thesetechniquescanmoreeasilymodelchangesininputpatternefcientlythanothermethods.Unlikesimulationandstatisticalsimulation,weneedtoknowthedependenciesinthecircuitstructuretopropagateprobabilitiesefciently.Moreover,issuesthatdrasticallyhamperprobabilisticpropagation,suchascorrelationsandfeedback,havetobemodeledaccurately.Probabilistictechniquescanbefurtherclassiedintoprobabilisticsimulation[35,27,37],andpurelyprobabilisticmethods[30,28,36].Theearliesteffortinvolvedprobabilisticpropagationofsignalprobability[31].Undertemporalindependence,switchingactivitycanbemodeledbysignalprobability.However,theestimatesaregrosslyinaccurate.Moreover,spatialindependencewasalsoassumed.MostlaterworksuseBinaryDecisionDiagrams(BDD)[28]tocomputesignalprobabilitiesforalltheinternalnodes.TheuseofBDDforsignalprobabilitywasrstproposedbyChakravartietal.[34].Formostlaterprobabilisticmodels,BDDisusedforprobabilisticpropagation.Insomeofthepioneeringworksbasedonprobabilisticsimulation,Najmetal.[27]estimatedthemeanandvarianceofcurrentusingprobabilitywaveforms.Itstartswithaninputprobabil-itywaveform,whichisthenpropagatedthroughoutthecircuit.Probabilitywaveformconsistsofprobabilityofasignaltobe1foracertaintimeintervalandprobabilitiesoftransitionfromlowtohighandfromhightolowataparticulartimeinstantinthewaveform.Najmetal.inCREST[27]accountedfortemporalcorrelationsduringthepropagationofthesignalandthetransitionprobabili-ties.However,spatialindependencewasassumedwhichresultedininaccurateestimates,especiallyinthepresenceofre-convergentfan-outs.21

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Najmetal.[26]introducedtheconceptoftransitiondensity(denotedbyD,ameasureofswitch-ingactivity),whichisalsoapropagationbasedstrategyusingsocalledBooleandifferencealgo-rithm.Forsimplegates,transitiondensityDcanbecalculateddirectlywhereasonecanutilizeBDDforcomplexlogicgates.Thismodelcansomewhatmodeltheeffectofrealdelaybuthavereportedveryhigherrorsduetounderlyingindependenceassumptionsintheinputsofafunction.Insummary,dependencymodelingofswitchingactivityhasbeenperformedbymanyoftheabovemethods,butonlypartially.Presentformalismsarenotabletoaccountforalltypesofspatialdependencies.Someofthepioneeringworks,thatenhancedtheaccuracyofestimationbyaddressingcorrelationanddependencyissuesarediscussedintheremainingpartofthesection.Kapoor[52]hasmodeledstructuraldependenciesapproximatelybypartitioningthecircuitintolocalBDDsforsignalprobability.Toimprovespeed,localBDDshavebecomeincreasinglycom-mon[29,35].Moreover,apartitioningstrategythatwasfollowedin[52],triedtomaximizethenumberofcorrelatednodesineachpartition.Schneideretal.[54]usedone-lagMarkovmodeltocapturetemporaldependence.Therstordertemporalmodelisvalidonlyunderzerodelaymodel,where,thepresentvalueofanodeisindependentofallthepastvaluesgivenjustthepreviousvalue.Thisisnotthecorrectpictureunderreal-delaymodel.Schneideretal.[55]proposedafastestimationtechniquebasedonROBDD.Anapproximatesolutionbasedonpartitioningapproachisproposedtoattributereconvergentspatialcorrelationwithreducedtimecomplexity.Itisnotclear,howaccuratethismodelingisintermsoftheorderofspatialcorrelation.Modelingspatialcorrelationusingpair-wisecorrelationbetweencircuitlineswasrstproposedbyErcolanietal.[51].Tsuietal.[53]modeledrstorderspatialcorrelationefcientlyusingcorrelationcoefcientsandutilizingtheminprobabilisticpropagation.Marculescuetal.[56],studiedtemporal,spatialdependenciesjointly.Inthiswork,conditionalprobabilitieswereusedforthelag-oneMarkovmodeltocapturetemporalcorrelationandtheideaoftransitioncorrelationcoefcientwasintroduced.22

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Insummary,eventhebestexistingpropagationsalgorithmsdonotaccountforhigherorderspatialcorrelation.Underzero-delaymodel,rstordertemporaleffectaresufcienttocapturethetemporaleffectsexhibitedinthecircuit.Infact,inthisthesis,theconditionalindependencerela-tionshipisutilizedcompletelyandwehavecomputationaladvantageinmodelingaccuratehigherorderspatialcorrelationpropagation.Wetabulatethesalientaspectsoftheworksthataremostcloselyrelatedtoourthesis,inchap-ter1Table1.1.toplaceourthesisinthecontextoftheseearlierefforts.OurproposedBayesiannetworkbasedformalismisabletohandletemporalandspatialcorrelationsindetail.Wedonotrequiretheinputstobeindependentandareabletomodelcorrelationamongthem.We[5]modeledswitchingactivity,capturingallhigherorderdependencies,usingBayesianNetworks.IthastobenotedthatconditionalindependenceisusedinBayesianNetworktomodeldependencies,aswellas,toconstructefcientcomputationalinferenceschemes.TheuniquenessofBayesianNetworkbasedmodelisthatitmakesagraphicalmodelandunderlyingswitchingmodelinvariantintermsofconditionalindependencemap(thesetofallconditionalindependencerelationsbetweenanysubsetofrandomvariables).Thismakeselegantinferenceschemespossi-blebasedonlocalmessagepassing.However,they[5]usedclusteringinferenceschemewhichisanexactalgorithmforupdatingtheprobabilitiesofeachnode.Exactalgorithmswhenappliedtonetworkswithlargenumberofnodesrequiresprohibitiveamountofstorageandarecomputationintensive.Weemployedpartitioningscheme[5]toalleviatetheproblemofcomplexity.However,lossesinthepartitionswerehighandcouldaffectintermediatenodessignicantlyresultinginhighmaximumerrorandstandarddeviation.Thegoalofthisworkistouseapproximatestochasticin-ferenceschemesforBayesianNetworkinferencesuchthatestimatesareuniformlyaccurate.Thesealgorithmsisasmartcombinationoftheany-timefeatureofsimulativeapproachandpatternin-sensitivityofprobabilisticapproach.Moreover,weachieveanorderofmagnitudeimprovementoverthepaststateoftheartintermsofmaximumerrorandstandarddeviation.ThetheoreticalcontributionofourthesisisthatthejointprobabilityfunctionofasetofrandomvariablesisexactlymappedcapturingallhigherordercorrelationsbetweenthesignalsaccuratelyusingBayesianNet-workmodel.Thisimpliesthatwecanmodelspatio-temporalcorrelationsofanyorder(rstorder23

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temporalissufcientforzero-delaymodel)andhenceitisanexactswitchingmodel.Moreover,weuseindependencerelationsnotonlytomodeldependenciesexactly,butalsotouseitinourcomputationaladvantageduringBayesianinferencing.Earliereffortseithertreatedthedistributionasacompositionofpair-wisecorrelatedsignalsbetweenallsignals[29,51]oruseanapproximatesolutionforcapturingspatialcorrelation[55].Moreover,theBayesianNetworkmodelsconditionalindependenceofasubsetofsignalsunlikein[29].Asaresult,complexdependenciesexhibitedbetweensetscanbemodeled.Also,incontrastto[29],thepropagationmechanismdoesnotassumeanysignalisotropyofconditionalindependence.Further,BayesianNetworkbasedmodelingisanuniedapproachformodelingandpropagat-ingprobabilities.Wecanmodelcorrelationsbothinthecircuitsaswellasintheinputs.Oureffortincapturingthejointprobabilitydistributionfunctionofthewholecircuitisuniqueandhasenabledustomodelhigherordercorrelationsratherthanjustpair-wisecorrelations.Thepropagationalgo-rithmsalwaysmaintaintheoverallprobabilityequilibriumofthewholeBNandnotjustbetweeninputsandoutputsofagate.2.2ExistingLeakagePowerEstimationTechniquesAstechnologyscalesdown,supplyvoltagemustbereducedtokeepthedynamicpowerwithinitslimits.Toavoidthenegativeimpactoncircuitdelay,duetothesupplyvoltagereduction,thethresholdvoltageofthetransistorhastobescaledproportionately.However,scalingofthresholdvoltagetomaintainspeedofoperationhasanadverseeffectonleakage,thatis,leakageincreasesasthethresholdvoltagescalesdown.Hencedesignershavetostartanalyzingandoptimizingtheirdesignforleakagepoweraswell.Onecrucial,well-established,observationisthatbothdynamicandstaticpoweraredatadependent.Dynamicpowerisobviouslydatadependent.Stand-byleakageisalsodependentoninputstates2.2.[69].Leakagepoweroptimizationtechniqueshavemostlyconsideredstand-bymode[64,65,66,85],andhenceareblindtothedata-dependence.Theaccuracyofleakagepowerestimationmodelisdependentonthestand-byleakagecurrentmodel[82].Sinceleakagepowerdependsontheprimaryinputcombinations,[82],suggeststhat24

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X 0X1ZVdd 1 X X 0Leakage0023.60 nA147.1551.4282.94nAnAnA00111 Figure2.2.2-InputNandGateandaTableShowingtheDependenceofLeakageonInputs.theleakagepowercouldbeminimizedifweapplytheinputcombinationscorrespondingtothemin-imumleakagepower.Toobtaintheminimumandmaximumvaluesforleakagepowerdissipation,in[82],theauthorsdevelopedanaccurateleakagemodelconsideringtheeffectsoftransistorstacksandimplementeditinthegeneticalgorithmframework.Also,inanefforttoestimatetheproperinputcombinationforminimumleakagevector[83],usesarandomsearchtechniquetodeterminelowleakagestates,withoutconsideringthefunctionalityofthecircuit.Theboundsobtainedarenotsotight.[84],introducesanewapproachforaccurateandefcientcalculationoftheaverageleakagecurrentincircuitsbydeterminingthedominantleakagestatesanduseofstateprobabilites.Theyalsousegraphreductiontechniquesandnonlinearsimulationtospeedupthesimulationtimewhileachievingdesiredaccracy.Theauthorin[86],presentsanon-simulative,graph-basedal-gorithmsforestimatingthemaximumleakagepower.Thealgorithmusedispatternindependent.Theleakageestimationtechniqueshaveonlyconsideredthestand-bymodeleakage.Butthegatesdisspiateleakagepowerevenduringtheactivemodes.Hence,foraccurateestimationofleakagepowerwehavetoconsidertheleakageoccuringintheactiveorrun-timemodeaswell.Runtimeleakagemitigationschemes[67,68,69]havealsobeenproposedtodynamicallychangecircuitconditionsinresponsetolowleakagehighleakagesituations.[69]introducesa25

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methodtoidentifytheMinimumLeakageVector(MLV)forleakagepowerreduction.Sincetheleakagepowerdependsontheinputpattern,thecentralideain[69]istoapplymiminum-leakageproducinginputcombinationtothecircuitwhenitisintheinactivemode,tocontroltheleakagepowerdissipation.Anexcellentreviewonrun-timetechniquescanbeobtainedin[70],whichdiscussestheissueoflimitofleakagereductionandperformancepenaltiesassociatedwiththetechniques.Leakagedoesnotjusthappenduringstand-bymodes,leakageisalsopresentduringactivepe-riodandisacrucialcomponentoftotalpoweroptimization[61,62,63],however,thesearehardertomodelandhandle.Circuitswhichareinactivemodemostofthetime,orswitchesfrequentlybetweenactiveandstand-bymodeswillalsohavenodesthatwouldleaksignicantlyduringtheactivemode.And,thisleakagewouldbedatadependent.ThiscomponentisclearlycapturedbyNguyenetal.[61],wherethestaticcomponentofpowerduringactiveregionisdependenton(1-a)whereaisameasureofactivity.Theyproposedalinearprogramming(LP)basedoptimizationframeworkforsimultaneousassignmentofthresholdvoltageandsizing.TheyalsoproposedadualVddextensionoftheproblembyILPformulation.Aheuristicisusedin[63]fordualVdd,dualVthandsizingwheretheyshowoptimizationforthreedifferentswitchingscenario.In[62],thedatadependenceoftheleakagepowerduringactivezoneisnotconsidered.Theoptimizationcriterionusedforpoweroptimizationisitselfdatadependent,whichmakeithardtomakegeneralizedstate-mentsabouttheoptimalityofanyoperatingpointfoundbyoptimizingitsomesetofinputs.Itisnotcleariftheoptimizedvaluesfoundforonesetofinputstatisticswillholdforanothersetofinputstatistics.Thisisaseriousconcernwhendesigningmodularcircuitsthatwillbeeventuallyusedindifferentcontexts.Inthiswork,wefocusindeterminingthegatesthatremainshibernatingeveninactivemode.Deteminingthesenodesiscrucialforleakagepoweranalysisbecausetheycontributesignicantlytoleakagepower.Moreover,achipormoduledesignedforanapplicationmightbeusedinanyenvironment,wemightnothaveanypriorknowledgeabouttheinputvectorstreams,thatis,duringsimulation,appliedtothismodule.Simulatingthecircuitforpowerestimationwouldrequireaconditionalsearchoftheinputspaceinasimulativeframework,whichiscomputationallyexpen-26

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sive.ProbabilisticinferenceusingBayesianNetworksistheonlyconsistentuncertaintycalculustohandlesuchsituationsinanefcientmanner.WeexploitthebacktrackingaspectofBayesianNetworkstodeterminethelikelyinputspaceofanymodule,givenanevidence.Wedothisbyrstforcinganinternalnodeinthemoduletobeataleakystate(0t10tor1t11t)andusebacktrackingtodeterminetheplausibleinputspace(explainationfortheevidence)fortheinternalnodetobeattheassignedleakystate.InChapter5,wepresenttwoentropybasedmeasurestocharacterizethisposteriorinputspacetodeterminethepossibilityofanodetoleak.27

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CHAPTER3MODELINGUSINGBAYESIANNETWORKSInthischapter,weintroducethefundamentalsofBayesianNetworksandtheconceptofcon-ditionalindependence.WefocusontwoimportantaspectofBayesianNetworks:dependencymodeling,andthenotionofminimalrepresentation.Next,weconvertacombinationalcircuittoaLogicInducedDirectedAcyclicGraph(LIDAG)andwethenprovethatLIDAGisaBayesianNetworkfortheunderlyingswitchingmodelandhenceanexactmodelfortheunderlyingjointprobabilitydistributionofthewholecircuit.Anycorrelationthatispresentinthejointprobabilitydistributioniscapturedinsuchadetailedmodeling.Wediscussourchoiceofvariables,states,edges,andtheassignmentofconditionalprobabilitiesintheLIDAG.3.1BayesianNetworkFundamentalsAnyprobabilityfunctionoverasetofrandomvariables(X1XN)canberepresentedasPX1XNpXnXn1Xn2X1pXn1Xn2Xn3X1pX1(3.1)Theaboveexpressionholdsforanyorderingoftherandomvariables.ThisexactrepresentationoftheprobabilisticknowledgerequiresencodingofallentriesinP(X1XN).Asthenumberofrandomvariablesincrease,representationandinferenceoftheprobabilisticknowledgebasedontheabovementionedprobabilisticmodelbecomesintractable.Theexactrepresentationassumesthateveryvariableisdependentoneveryotherrandomvariablepresentintheset.Itdoesnottakeadvantageoftheconditionalindependenciespresentamongthevariables.Giventhestateoftheparentstheconditionoftherestofthecircuitisirrelavanttotheoutput.Forexample,theoutput28

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X2X3X4X5X1Figure3.1.BayesianGraphicalModel.ofadigitalgatedoesnotdependentontheconditionoftherestofthecircuitgiventhestateofitsparents.Thispropertyiscalledasconditionalindependence.Thisleadstotheideatoencodetheprobabilisticknowledge,thatis,thejointprobabilitydistributionofanitesetofvariables,conciselyusinggraphicalmodels,whichcapturesconditionalindependenciesembeddedamongtherandomvariablesandarrivesattheminimalfactoredrepresentationinEquation3.2,whichisaprobabilisticmodelofaBayesianNetwork.PX1XNn'k1PXkPaXk(3.2)Thisformofminimalrepresentationofthejointprobabilityfunctioncanberepresentedasadirectedacyclicgraph(DAG),withnodesrepresentingtherandomvariablesandthelinksbetweenrandomvariablesrepresentingdirectprobabilisticdependencies.Letusconsideranexamplewhichillustratestheconceptofconditionalindependence.IntheFigure3.1.,X4isdependentonX2andX3(andX1byinheritance).GivenX3,X4isconditionallyindependentofX5andviceversa.LetusconsiderevaluatingthejointprobabilitydistributionoftherandomvariablesX1X2X3X4X5.Usingtheexactrepresentation,PX1X2X3X4X5 PX5X4X3X2X1PX4X3X2X1PX3X2X1PX2X1PX1(3.3)29

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Theconservativeassumptionthateveryrandomvariableisdependentoneveryotherrandomvari-ablemakestherepresentationandupdatingoftheprobabilisticknowledgeusingtheexactmodelin-efcient.BayesianNetworksresolvesthisissuebyexploitingtheconditionalindependenceamongrandomvariables.HencebyFigure3.1.,thejointprobabilityfunctionfactorizestoPX1X2X3X4X5PX5X3PX4X3X2PX3PX2X1PX1(3.4)Theattractivefeatureofthisgraphicalrepresentationofthejointprobabilityfunctionisthatitcannotonlymodelcomplexconditionalindependenceoverasetofvariables,buttheindependenciesserveasacomputationalschemeforsmartandefcientprobabilisticupdating.3.2MathematicalFormalismInthissection,wediscussthefundamentalmodelingissuesrelevanttoBayesianNetwork.In-terestedreaderisrecommendedtoread[2]fordetailedunderstanding.Aswementionedbefore,BayesianNetworksarecompactgraphicalprobabilisticmodelfortheunderlyingjointprobabilitydistributionfunction.EachnodeintheDAGstructureisarandomvariablerepresentingswitchingandcanhavefourstates(00011011)forcompletecaptureoftemporaldepen-denceunderzero-delayscenario.EdgesintheDAGdenotescauseandeffectrelationshipintheprobabilisticmodelandisquantiedbytheconditionalprobabilityofachildnodegivenitsparents.Toformalizetheconceptofdependencies,werstpresenttheconceptofconditionalinde-pendence.Webeginwiththedenitionofconditionalindependenceamongthreesetsofrandomvariables.Denition1:LetU=!U1U2Un"beanitesetofvariablesthatcanassumediscreteval-ues.LetP#bethejointprobabilityfunctionoverthevariablesinU,andletX,YandZbeanythreesubsetsofU.X,YandZmayormaynotbedisjoint.XandYaresaidtobeconditionallyindependentgivenZifPxyzPxzwheneverPyz%$0(3.5)30

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87643521 Figure3.2.ACombinationalCircuit.FollowingPearl[2],thisconditionalindependenceamongstX,Y,andZisdenotedasIXZYinwhichXandYaresaidtobeconditionallyindependentgivenZ.ConditionalindependenceimpliesthatknowledgeofZmakesXandYindependentofeachother.InFigure3.2.forexample,letusdenotetheswitchingactivityatlineibyrandomvariableXi.Uisdenedasaset=!X1X8".SwitchinginacombinationalcircuitfollowsdirectedMarkovproperty,thatis,theoutputofagateisdependentonlyonitsinputs.ThustherandomvariableX7iscompletelyindependentofX4given!X5X6".Hence,I(X7!X5X6"&X4)isoneofthemanyindependenciesthatispresentinthecircuit.Adependencymodel,M,ofadomainshouldcaptureallthesetripletsnamelyXZYcondi-tionalindependenciesamongstthevariablesinthatdomain.Thejointprobabilitydensityfunctionisonesuchdependencymodel.Thepropertiesinvolvingthenotionofindependenceareaxiomatizedbythefollowingtheorem.Theorem1:LetX,Y,andZbethreedistinctsubsetsofU.IfIXZYstandsfortherelation“XisindependentofYgivenZ”insomeprobabilisticmodelP,thenImustsatisfythefollowingfourindependentconditions:IXZY'IYZX(symmetry)(3.6)IXZY(W'IXZY&XZW(decomposition)(3.7)31

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X1XXXXXXX2345678Figure3.3.BayesianNetworkCorrespondingtotheCircuitinFigure3.2.IXZY(W'IXZ(WY(weakunion)(3.8)IXZY&IXZ(YW'IXZY(W(contraction)(3.9)Proof:Forproof,see[13].Next,weintroducetheconceptofd-separationofvariablesinaDirectedAcyclicGraphstruc-ture(DAG),whichistheunderlyingstructureofaBayesiannetwork.Thisnotionofd-separationisthenrelatedtothenotionofindependenceamongsttriplesubsetsofadomain.Denition2:IfX,Y,andZarethreedistinctnodesubsetsinaDAGD,thenXissaidtobed-separatedfromYbyZ,)XZY$,ifthereisnopathbetweenanynodeinXandanynodeinYalongwhichthefollowingtwoconditionshold:(1)everynodeonthepathwithconvergingarrowsisinZorhasadescendentinZand(2)everyothernodeisoutsideZ.Ifthereexistsuchapathwheretheabovetwoconditionshold,thepathiscalledanactivepath.ConsidertheexampleDAGinFigure3.3.,letX!X5",Y!X6"andZ!X7".PathX5X7*X6isactivesincegiveninformationonnodeX7,X5andX6arenotd-seperated.Itisworthmentioningthatifanyonepathisactive,eventhoughtheotherpathsareblocked,thenodesarenotd-separated.Inthesameexample,X7isd-separatedfromX2byX5sincetheonlypathX2X5X7isblocked.Denition3:ADAGDissaidtobeanI-mapofadependencymodelMifeveryd-separationconditiondisplayedinDcorrespondstoavalidconditionalindependencerelationshipinM,i.e.,ifforeverythreedisjointsetsofverticesX,Y,andZ,wehave,)XZY$+'IXZY.InFigure3.3.,forexample)X7X5X1$impliestheindependencerelationIX7X5X1inthedependencymodel32

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X3X1X2X4X3X1X2X4X1X2X3X4X3X4X2X1CDB A Figure3.4.BayesianNetworks:MarriagebetweenGraphicalandProbabilisticModels.MformedbytherandomvariablesdepictingswitchingactivityatalineinthecombinationalcircuitinFigure3.2.NotethattheDenition3holdstheunifyingfeatureofthegraphbasedprobabilitymodelinawaythatconnectstheDAGDtotheprobabilisticmodelP.InBayesianNetworks,wenotonlysuggestthatDAGDisadependencymodelforP(becauseallthed-separationsinDimplyacon-ditionalindependenceinP),butalsothenotionofacompactminimalrepresentationisbuiltin.LetusconsidertheexampleofaprobabilisticmodelPoverfourrandomvariables!X1X2X3andX4"asshowninFigure3.4.Notethat,theDAGinFigure3.4.A,allthenodesareconsideredindepen-dentandhenceI-mapofDisgreaterthanthatofPwhichindicatesthatDunder-representsP.InFigure3.4.D,theI-mapofDislessthanthatofPasDisacompleteDAGexhibitingmaximumdependencies.Thismodelwouldgenerateaccurateresultsbutareover-representationandhencethecomputationeffortswouldbelarge.ABayesianNetworkhastobetheDAGwheretheI-mapforDAGmatchestheI-mapofthePandhenceitistheexactrepresentationthatisminimalinstructure.Equation3.1denotestheexactprobabilisticmodeloverrandomvariablesandusingconditionalindependencies(inEquation3.10),wecanarriveattheminimalfactoredrepresentationshownin33

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Equation3.2whichistheprobabilisticmodelofBayesianNetwork.pxixi1xi2x1 pxiPaxi(3.10)Denition4:ADAGisaminimalI-mapofMifnoneofitsedgescanbedeletedwithoutdestroyingitsdependencymodelM.Denition5:GivenaprobabilityfunctionPonasetofvariablesU,aDAGDiscalledaBayesianNetworkofPifDisaminimumI-mapofP.Ingeneral,itishardtondalltheI-mapsgivenaprobabilitydistributionfunctionoragraphicalrepresentation.ThereisanelegantmethodofinferringtheminimalI-mapofPthatisbasedonthenotionofaMarkovblanketandaboundaryDAG,whicharedenednext.Denition6:AMarkovblanketofelementXi,UisasubsetSofUforwhichIXiSUSXiandXi-,S.AsetiscalledaMarkovboundary,BiofXiifitisaminimalMarkovblanketofXi,i.e.,noneofitspropersubsetssatisfythetripletindependencerelation.Denition7:LetMbeadependencymodeldenedonasetU!X1Xn"ofelements,andletdbeanordering!Xd1Xd2."oftheelementsofU.TheboundarystrataofMtermedasBMrelativetodisanorderedsetofsubsetsofU,!Bd1Bd2."suchthateachBdiisaMarkovboundary(denedabove)ofXdiwithrespecttothesetUdi0/U !Xd1Xd2Xd1i12",i.e.BdiistheminimalsetsatisfyingBdi/UandIXdiBdiUdiBdi.TheDAGcreatedbydesignatingeachBdiastheparentsofthecorrespondingvertexXdiiscalledaboundaryDAGofMrelativetod.ItshouldbenotedherethattheonlyorderingrestrictionisthatthevariablesintheMarkovBoundaryset(ofaparticularvariable)havetobeorderedbeforetherandomvariable.ThisleadsustothenaltheoremthatrelatestheBayesiannetworktoI-maps,whichhasbeenprovenin[2].ThistheoremisthekeytoconstructingaBayesiannetwork.Theorem2:LetMbeanydependencymodelsatisfyingtheaxiomsofindependencelistedinEquations3.6-3.9.IfthegraphstructureDisaboundaryDAGofMrelativetoorderingd,thenDisaminimalI-mapofM.Proof:Forproof,see[13].34

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Thistheoremalongwithdenitions2,3,and4above,speciesthestructureoftheBayesiannetwork.WeusethesetoproveourtheoremregardingthestructureofBayesiannetworktocapturetheswitchingactivityofacombinationalcircuit.Letacombinationalcircuitconsistofgates!G1GN"withnprimaryinputsignalsdenotedbytheset!I1In".LettheoutputofgateGibedenotedbyOi.Theinputstoagateareeitheraprimaryinputsignaloroutputofanothergate.Theswitchingoftheseinputsignalandoutputlines,!I1InO1ON",aretherandomvariablesofinterest.Notethatthesetofoutputlinesincludebothintermediatelinesandthenaloutputlines.LetXibetheswitchingatthei-thline,whichiseitheraninputoranoutputline,takingonfourpossiblevalues,!x00x01x10x11",correspondingtothepossibletransitions:000110and11.Denition8:ALogicInducedDirectedAcyclicGraph(LIDAG)structure,LD,correspondingtoacombinationalcircuitconsistsofnodes,Xis,representingtheswitchingateachlineandlinksbetweenthemisconstructedasfollows:Theparentsofarandomvariablerepresentingtheswitchingatanoutputline,Oi,ofagateGiarethenodesrepresentingswitchingsattheinputlinesofthatgate.Eachinputlineiseitheroneof!I1In"oranoutputofanothergate.TheDAGshowninFigure3.3.,isaLIDAGcorrespondingtothecombinationalcircuitshowninFigure3.2.Theorem3:TheLIDAGstructure,LD,correspondingtothecombinationalcircuitisaminimalI-mapoftheunderlyingswitchingdependencymodelandhenceisaBayesiannetwork.ItisinterestingtonotethattheLIDAGstructurecorrespondsexactlytotheDAGstructureonewouldarrivebyconsideringtheprincipleofcausality,whichstatesthatonecanarriveattheappropriateBayesiannetworkstructurebydirectinglinksfromnodesthatrepresentcausestonodesthatrepresentimmediateeffects[2].Thus,directedlinksinthegraphdenoteimmediatecauseandeffectrelationship.Inacombinationalcircuittheimmediatecausesofswitchingatalinearetheswitchingsattheinputlinesofthecorrespondinggate.35

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3.3FormationoftheLIDAG-BNWerstillustratewithanexamplehowswitchinginacombinationalcircuitatcircuitlevelcanberepresentedbyaLIDAGstructuredBayesiannetwork(LIDAG-BN).ThenweshowhowtheconditionalprobabilitiesthatquantifythelinksofLIDAG-BNarespecied.LetusconsiderthecircuitwithvesgatesshowninFigure3.2.Weareinterestedintheswitch-ingateachofthe8numberedlinesinthecircuit.Eachlinecantakefourvaluescorrespondingtothefourpossibletransitions:!x00x01x10x11".Notethatthiswayofformulatingtherandomvariableeffectivelymodelstemporalcorrelationsinceonlyrstordertemporalcorrelationisex-hibitedincombinationalcircuitunderzero-delayscenario[54].Tocaptureallhigherorderspatialcorrelations,weformtheinterconnection(throughedges)andquantifythembytheconditionalprobabilitiesforthechild-parentgroupintheLIDAG.TheprobabilityofswitchingatalinewouldbegivenbyPXix01&PXix101.TheLIDAGstructureforthecircuitisshowninFigure3.3.Dependenceamongthenodesthatarenotconnecteddirectlyisimplicitinthenetworkstructures.Forexample,nodesX1andX2areindependentofeachother,however,theyareconditionallydepen-dentgiventhevalueofsaynodeX5.Orthetransitionatline5,X5,isdependentonthetransitionsatlines1and2,representedbytherandomvariables,X1andX2,respectively.Thus,thetransitionsofline5areconditionallyindependentofalltransitionsatotherlinesgiventhetransitionstatesoflines1and2.FortheBayesiannetworkstructureinFigure3.3.,thecorrespondingjointprobabilitydensityisgivenbythefollowingfactoredform.Ithastobenotedthatthisfactoredformcanonlybeobtainedforcircuitswithoutafeedback.Px1x8 Px8x4Px7x5x6Px6x3x4Px5x1x2Px4Px3Px2Px1(3.11)Theconditionalprobabilitiesofthelinesthataredirectlyconnectedbyagatecanbeobtainedknowingthetypeofthegate.Forexample,PX5x01X1x01X2x00willbealways1becauseifoneoftheinputsofanORgatemakesatransitionfrom0to1andtheotherstaysat0thenthe 1ProbabilityoftheeventXi3xiwillbedenotedsimplybyP4xi5orbyP4Xi3xi5.36

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Table3.1.ConditionalProbabilitySpecicationsfortheOutputandtheInputLineTransitionsforTwoInputNANDGate. TwoInputNANDgate PXoutputXinput1Xinput2 forXoutput Xinput1 Xinput2 !x00x01x10x11" = = 0001 x00 x00 0001 x00 x01 0001 x00 x10 0001 x00 x11 0001 x01 x00 0010 x01 x01 0001 x01 x10 0010 x01 x11 0001 x10 x00 0001 x10 x01 0100 x10 x10 0100 x10 x11 0001 x11 x00 0010 x11 x01 0100 x11 x10 1000 x11 x11 outputalwaysmakesatransitionfrom0to1.Acompletespecicationoftheconditionalprobabil-ityofPx5x1x2willhave43entriessinceeachvariablehas4states.Theseconditionalprobabilityspecicationsaredeterminedbythegatetype.Thus,foraNANDgate,ifoneinputswitchesfrom0to1andtheotherfrom1to0,theoutputremainsat1.WedescribetheconditionalprobabilityspecicationforatwoinputNANDandatwoinputORgateinTable3.1.andinTable3.2.respec-tively.Byspecifyingadetailedconditionalprobabilityweensurethatthespatio-temporaleffect(rstordertemporalandhigherorderspatial)ofanynodeareeffectivelymodeled.ThelastfourtermsintherighthandsideofEq.3.11representthestatisticsoftheinputlines.Giventhestatisticsoftheinputlines,wewouldliketoinfertheprobabilitiesofalltheothernodes.Abruteforcewayofachievingthiswouldbetocomputethemarginalprobabilitiesbysummingoverpossiblestates,thus,Px8x16x27778x7Px1x9.This,obviously,iscomputationallyvery37

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Table3.2.ConditionalProbabilitySpecicationsfortheOutputandtheInputLineTransitionsforTwoInputORGate. TwoInputORgate PXoutputXinput1Xinput2 forXoutput Xinput1 Xinput2 !x00x01x10x11" = = 1000 x00 x00 0100 x00 x01 0010 x00 x10 0001 x00 x11 0100 x01 x00 0100 x01 x01 0001 x01 x10 0001 x01 x11 0010 x10 x00 0001 x10 x01 0010 x10 x10 0001 x10 x11 0001 x11 x00 0001 x11 x01 0001 x11 x10 0001 x11 x11 expensiveand,inaddition,doesnotscalewell.Inthenextchapter,weshowhowthestructureoftheBayesiannetworkcanbeusedtoefcientlycomputetherequiredprobabilities.38

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CHAPTER4BAYESIANINFERENCEFORSWITCHINGACTIVITYESTIMATIONInthepreviouschapter,weprovedthattheBayesianNetworkmodelstheexponentiallysizedjointprobabilitydistributioninacompactmannerbyexploitingtheconditionalindependencerela-tionshipspresentamongtherandomvariables.Theattractivefeatureofthisgraphicalrepresentationofthejointprobabilityfunctionisthatitcannotonlymodelcomplexconditionalindependenceoverasetofvariables,buttheindependenciesserveasacomputationalschemeforsmartprobabilisticupdating.Thischapterstartswithabriefintroductionto”whatisprobabilisticinference?”,andthetwoimportantinferenceschemesusedtoestimatethebeliefsorprobabilitiesinaBayesianNetwork.Majorpartofthischapterisdevotedtowardsexploringdifferentsetofstochasticim-portancesampling(approximatebayesianinferencescheme)schemesnamely,ProbabilisticLogicSampling[10],AdaptiveImportanceSampling(AIS)[11]andEvidencePre-propagatedImpor-tanceSampling[12]forbayesianinferencing.WeconcludethischapterwithexperimentalresultsonISCAS'85benchmarkcircuits.4.1ProbabilisticInferenceProbabilisticinferenceorcommonlyreferredasbeliefupdatingamountstocalculatingtheprob-abilitydistributionofaquerynodegivenanobservationorevidence.ThisamountstocomputingPXE(bayestheorem).PXEePXrEEe PEe(4.1)39

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ComputationoftheprobabilityofevidenceisP(E=e)requiressummationoverallthevariablesinthesetexcepttheevidencevariablesandthisisexpressedinEquation4.2.PEeXEPXrEEe(4.2)Forsmallnetworks,computationoftheprobabilityofevidenceusingtheexactrepresentationthatis,Equation4.2issimple.However,asnetworksizeincreasescomputationofPEeefcientlybecomesacomputationalcomplexproblem.DifferentschemestoprocesstheEquation4.2givedifferentinferencealgorithms.TwoimportantBayesianNetworkinferencealgorithmsare1)exactinference,2)approximateinference.Exactinferencealgorithmslikeclustering,pearl'spolytreealgorithmetc.,provideexactestimate[13,23],howeverforverylargenetworkstheystumbleduetoNP-hardnessofinference[16].Exactinferenceappliedonlargenetworksareeitherstorageintensiveorcomputationallyextensive.ToresolvethisissueapproximateinferencemethodslikeModelSimplication,Searchbased,LoopyBeliefPropagation,StochasticImportanceSamplingweredeveloped.WhileapproximateinferenceisprovedtobeNP-hardaswell[17],itistheonlyalternativewaytoarriveatanestimateforlargeandcomplexcircuits.Aprominentsubclassofapproximateinferencealgorithmsarestochasticsamplingalgorithms.SomeinstancesoftheseareProbabilisticLogicSampling[10],Likelihoodweighting[18,19],backwardsampling[20],andimportancesampling[19].Inthischapter,weexplorethreeimportantstochasticimportancesam-plingschemes:ProbabilisticLogicSampling[10],AdaptiveImportanceSampling(AIS)[11]andEvidencePre-propagatedImportanceSampling[12]forBayesianinferencing.Thesealgorithmscombinetheany-timefeatureofsimulativeapproachesandinput-patternindependenceofproba-bilisticapproaches.4.2StochasticInferenceAlgorithmsStochasticsamplingalgorithmsareapproximateBayesianNetworkinferenceschemes.Proba-bilitiesareinferredbyacompletesetofsamplesorinstantiationsthataregeneratedforeachnodeinthenetworkaccordingtotheimportanceconditionalprobabilitydistributionofthisnodegiventhe40

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valuesoftheparents.Inthesesamplingschemes,eachsampledeterminestheposteriorprobabilityoftheunderlyingmodelfortheremainingsamples.Theprobabilityofarandomvariableisproventoconverge[11]toitscorrectvaluegivenenoughtime.Thesalientfeaturesofthesealgorithmsare:Theyscaleextremelywellforlargersystemsmakingthematargetinferencefornano-domainbilliontransistorscenario.Theyareany-timealgorithm,providingadequateaccuracy-timetrade-off.Thesamplesarenotbasedoninputsandtheapproachisinput-patterninsensitive.TheclassesofalgorithmsselectedhereareknownasImportanceSamplingalgorithms[10,11,12]whicharenotonlygoodpredictorsorestimators,predictingthebehaviorsofdescendentnodes(intermediateones)givensomepropertiesoftheprimaryinputs,butalsoaccuratediagnostictoolthatwouldprovidepossiblepatternoftheinputsgivenaparticularsetofbehavior(evidence)onanyinternalnodes.Beforewereviewvariousstochasticinferencemethods,itisextremelynecessarytounderstandthetheorybehindimportancesampling,whichactsasthebackboneforthesestochasticinferencemethods.ReadersinterestedinmoredetailsaredirectedtotheliteratureonMontecarlomethodsinniteintegralscomputation[58].LetusconsidertheapproximatecomputationoftheintegralJ,J9QbXdX(4.3)LetbXbeafunctionofkvariablesXX1X2...XkoveradomainQ/Rk.TheintegralinEquation4.3canbesolvedbynumericalintegartiontechniqueslikeTrapezoidalrule,Simpson'srule,Montecarlomethod.TheintegralinEquation4.3isusuallycomputedbymeansofeitherTrapezoidalruleorSimpson'sruleastheyarriveatpreciseestimates.Butformultivaluedinte-grals,useoftheabovemethodsiscomputationallyintensiveandhenceweresorttoMonteCarlomethod.Inthissection,wewilluseMonteCarlobasedtechniquefortheapproximateevaluationoftheintegralJ.MonteCarlomethodsuserandomnumberstoperformnumericalintegration.For41

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accuarcy,samplingofrandomnumbersisperformedonadistributionthatisareasonableapprox-imationtotheactualfunctionbX.Hence,weintroduceanarbitrarydensityfunctionalsocalledastheimportancefunctioniXinthisintegral,J;:Qb1X2 i1X2iXdX.TheimportancefunctioniXisaprobabilitydensityfunctionsuchthatiX<$0foranyX/Q.AftersamplingtheimportancefunctionoverMinstantiationsX1X2...XM,theapproximatevalueoftheintegraliscalculatedasfollows,ˆJ1 MMi1bXi iXi(4.4)ForlargevaluesofM,thedistributionofiXapproachesthedistributionofbXandhencetheaccuracyofˆJincreases.ThevariancebetweenthetwodistributionsisminimizedwheniXisproportionaltobX.ThemaingoaloftheImportanceSamplingalgorithmisachievingtheimportancefunction.WhileAISarrivesatanimportancefunctionbylearningfromthesamplesgeneratedduringeachiteration,EPISarrivesatanimportancefunctionbyusingyetanotherap-proximateinferenceschemecalledtheLoopyBeliefPropagation.Notethat,itispossibletousebXasaguideinchoosingiX.ThereasonwhyimportancesamplingtechniqueisusedbecomesevidentwhenwecomparetheEquation4.2andEquation4.3andconcludethattheyarealmostidenticalexceptfortheintegrationwhichisreplacedbysummationandthedomainQisreplacedbyXrE.ThestochasticsamplingstrategyworksbecauseinaBayesianNetworktheproductoftheconditionalprobabilityfunctionsforallnodesistheoptimalimportancefunction.4.2.1ProbabilisticLogicSamplingProbabilisticLogicSampling(PLS)istherstandthesimplestsamplingalgorithmsproposedforBayesianNetworks[10].Theowofthealgorithmisasfollows:1.CompletesetofsamplesaregeneratedfortheBayesianNetworkusingtheimportancefunc-tion,whichisinitializedtojointprobabilityfunctionPX.Theimportancefunctionisneverupdatedonceitsinitialized.Withoutevidence,PXistheoptimalimportancefunctionfortheevidenceset.2.Samplesthatareincompatiblewiththeevidencesetarediscarded.42

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3.Theprobabilityofallthequerynodesareestimatedbasedoncountingthefrequencywithwhichtherelevanteventsoccurinthesample.Inpredictiveinference,logicsamplinggen-eratesprecisevaluesforallthequerynodesbasedontheirfrequencyofoccurrencebutwithdiagnosticreasoning,thissystemfailstoprovideaccurateestimatesbecauseoflargevari-ancebetweentheoptimalimportancefunctionandtheactualimportancefunctionused.Thedisadvantageofthisapproachisthatincaseofunlikelyevidence,wehavetodiscardmostsamplesandthustheperformanceofthePLSapproachdeteriorates.4.2.2AdaptiveImportanceSamplingOurobjectiveisestimatingtheprobabilityofevidencePEe.Theposteriorprobabilityisgivenbyequation4.1.TheoptimalimportancefunctionforcalculatingPEeisPXEe[11].Althoughweknowthemathematicalexpressionfortheoptimalimportancefunction,itiscomputationallyexpensivetoobtainthisfunctionexactly.ByexploitingthestructuraladvantageofBayesianNetwork(thejointprobabilityfunctionthatismodeledbyaBNcanbeexpressedastheproductoftheconditionalprobabilityofthenodesgivenitsparentnodes)wecanarriveatanapproximateimportancefunction.Theapproximateimportancefunctionisgivenas,rXrE=m'k1PXkPaXkE(4.5)Thisfunctionconsiderstheeffectofevidenceonrestofthecircuit.PEe
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1.Thenodesarearrangedintopologicalorder.Eachevidencenodeisinstantiatedtoitsobservedstateandisomittedfromfurthersamplegenerations.Eachrootnodeisrandomlyinstantiatedtooneofitspossiblestatesaccordingtotheimportancepriorprobabilityofthenode.2.Eachnodewhoseparentswerealreadyinstantiatedwillbeinstantiatedtooneofitspossibleoutcomes,accordingtoitsimportanceconditionalprobabilitytable,whichcanbederivedfromtheimportancefunction.3.ConditionalprobabilityoftheevidencesetgiventhesampleinstantiationiscalculatedandstoredandusedtoupdatetheimportancefunctionafterafewrunbyapplyingBayesianNetworklearningalgorithms.Thisfunctionwillthenbeusedforthenextstageofsampling.Theposteriorprobabilitiesarethencalculatedfromthesamples.4.2.3HybridSchemeForlargecircuits,ahybridscheme,specicallytheEvidencePre-propagatedImportanceSam-pling(EPIS)[12],whichuseslocalmessagepassingandstochasticsampling,isappropriate.Thismethodscaleswellwithcircuitsizeandisproventoconvergetocorrectestimates.Theseclassesofalgorithmsarealsoanytime-algorithmssincetheycanbestoppedatanypointoftimetoproduceestimates.Ofcourse,theaccuracyofestimatesincreaseswithtime.TheEPISalgorithmisbasedonImportanceSamplingthatgeneratessampleinstantiationsofthewholeDAGnetwork,i.e.allforlineswitchinginourcase.Thesesamplesarethenusedtoformthenalestimates.Thissamplingisdoneaccordingtoanimportancefunction.InaBayesianNet-work,theproductoftheconditionalprobabilityfunctionsatallnodesformtheoptimalimportancefunction.LetX!X1,X2Xm"bethesetofvariablesinaBayesianNetwork,PaXkbetheparentsofXk,andEbetheevidenceset.Then,theoptimalimportancefunctionisgivenbyPXE=m'k1PXkPaXkE(4.6)44

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Z1l(x)YmlY1(x)lX(z1)pX(z1)lX(zn)Y1(x)p(x)YmpX(zn)p Z1nYYmXFigure4.1.ProbabilisticInferenceUsingLocalMessagePassing.ThisimportancefunctioncanbeapproximatedasPXE=m'k1aPaXkPxkPaXklXk(4.7)whereaPaXkisanormalizingconstantdependentonPaXkandlXk>PExk,withEandEbeingtheevidencefromparentsetandchildset,respectively,asdenedbythedirectedlinkstructure.Calculationofliscomputationallyexpensiveandforthis,LoopyBeliefPropagation(LBP)[21]overtheMarkovblanketofthenodeisused.Yuanetal.[12]provedthatforapoly-tree,thelocalloopybeliefpropagationisoptimal.Theimportancefunctioncanbefurtherapproximatedbyreplacingsmallprobabilitieswithaspeciccutoffvalue.4.2.3.1LoopyBeliefPropagationInthispart,wewouldoutlinePearl's[2],[81]distributedlocalmessagepassingschemethatallowsefcientbacktrackinginpoly-treeandshowanapproximationofthepoly-treecalledloopybeliefpropagation(LBP)[21]whichextendstonetworkwithloop(re-convergence)formanyap-plications.45

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Here,webrieysummarizePearl'sbeliefpropagationalgorithmontheFigure4.1.thatisapoly-tree.EachnodeXcomputesitsposteriorprobabilitybasedontheinformationobtainedfromitsneighbors.i.e.,Belx>PXxE,whereErepresentstheevidenceset.Inapoly-tree,anynodeXd-separatesEinto2subsets,ExwhichistheevidenceconnectedtonodeXthroughitsparentsZandExistheevidenceconnectedtonodeXthroughitschildrenY.Now,thenodeXcancomputeitsbeliefbyseparatelycombiningthemessagesobtainedfromitsparentsandchildren.Belx alxpx(4.8)wherelxandpxaregivenbylx 'UlUx(4.9)UisasetcontainingallchildrenofX.px z1z2777znPxz1z2znn'i1pXzi(4.10)whereZ1Z2ZnareparentsofnodeX.Oncethenodecomputesitsbeliefitpropagatestheupdatedmessagestoitsneighborsandthisiterationiscarrieduntiltheconvergenceoftheposteriors.ThemessagetotheparentZlofnodeXisgivenby:lXzlxz1z2777zkPxzlz1z2zkk'i1i?lpXzilx(4.11)whereZ1Z2...ZlZkareotherparentsofX.ThemessagefromnodeXtoitschildisgivenby:pYxpx'C@CHXYlCx(4.12)Pearl'sbeliefpropagationalgorithmcanbeappliedtonetworkswithloopswherethebeliefofanodeiscontinuouslyupdatedinalooptillbeliefhasconverged.ManyapplicationshaveshownenormoussuccessandcorrectconvergenceusingLBP.In[21]itisshowntobeconnectedwiththeKikuchiapproximationofvariationalBethefreeenergyinstatisticalphysics.Notethat,wearenotusingLBPdirectly,weuseLBPtoarriveatanimportancefunctionforstochasticinferencedis-46

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cussedintheprevioussection.Importancesamplingalgorithmshavebeenshown[11]toconvergecorrectlyevenwhentheimportancefunctionvariesslightlyfromtheoptimumone.Theabovesetofstochasticsamplingstrategiesdiscussedinsubsection4.2.1,4.2.2,and4.2.3workbecauseinaBayesianNetworktheproductoftheconditionalprobabilityfunctionsforallnodesistheoptimalimportancefunction.Becauseofthisoptimality,thedemandonsamplesislow.WehavefoundthatjustthousandsamplesaresufcienttoarriveatgoodestimatesfortheISCAS85benchmarkcircuits.Notethatthissamplingbasedprobabilisticinferenceisnon-simulativeandisdifferentfromsamplingsthatareusedincircuitsimulations.Inthelatter,theinputspaceissampled,whereasinourcaseboththeinputandthelinestatespacesaresampledsimultaneously,usingastrongcorrelativemodel,ascapturedbytheBayesianNetwork.Duetothis,convergenceisfasterandtheinferencestrategyisinputpatterninsensitive.4.3ExperimentalResultsWeexperimentedwiththecombinationalcircuitsfromtheISCAS85benchmarksuite.WerstmappedtheISCAScircuitstotheircorrespondingDAGstructuredBayesianNetworks.EachnodeintheBayesianNetworktakesfourpossibleoutcomes(x00x01x10x11).Theconditionalprobabilityofeachnodeisformedbasedontheknowledgeoftypeofgateconnectingtheparentandthechild.Theexperimentalset-upof”GeNIe”[22],agraphicalnetworkinterfaceisusedforourexperimentation.ThetestswereperformedonaPentiumIV,2.00GHz,WindowsXPcomputer.Forcomparison,weperformedzero-delaylogicsimulationontheISCAS85benchmarkcircuits,whichprovidesaccurateestimatesofswitching.Table4.1.showsthemean,standarddeviation,maximumerrorandthetimeelapsedfortheISCAScircuitswithPLSandwecomparetheresultswiththatobtainedusingthepriorapproximateCascadedBayesianNetworks(CBN)[5].Columns2,3,4and5inthistablerepresentsthemeanerror(E),standarddeviationoftheerror(sE),maximumerror(MxE),andtheelapsedtime(T)forswitchingactivity.ItcanbeeasilyseenthateventhoughgoodmeanerrorsareobtainedbyapproximateCBNmethods,thestochasticPLSprovidesbetterestimatesintermsofstandarddevi-ationandshowssignicantimprovementoverthemaximumerror.Thetotalelapsedtime,whichis47

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Table4.1.ExperimentalResultsComparingApproximateCascadedBayesianNetworkModelandProbabilisticLogicSampling. Approx.CBNmodel[5] PLS:1000samples E sE MxE T(s) E sE MxE T(s) c432 0.00 0.02 0.28 3 0.00 0.00 0.04 0.40 c499 0.00 0.00 0.00 9.03 0.00 0.01 0.04 0.45 c880 0.00 0.00 0.04 2.52 0.00 0.01 0.05 1.05 c1355 0.00 0.00 0.09 1.81 0.00 0.01 0.06 1.75 c1908 0.00 0.01 0.15 10.70 0.00 0.01 0.05 2.7 c3540 0.00 0.04 0.26 18.86 0.00 0.00 0.04 5.96 c6288 0.01 0.04 0.37 38.75 0.00 0.01 0.06 11 thesumofCPU,memoryaccessandI/Otime(computedusingtheftimecommandinWINDOWSenvironment)isalsosignicantlylowforPLS.HighmaximumerrorintheapproximateCascadedBayesianNetwork(CBN)modelisattributedtopartitioningofthenetwork,whichresultsinlossofinformationattheboundarynodes.ThePLSschemeconvergestoaccurateestimateswhenpropagatingevidencealongthecausallinks,butfordiagnosticreasoningtheestimatesobtainedthroughthisapproachdeteriorates.Ta-bles4.2.and4.3.showtheerrorstatisticsforpredictiveaswellasdiagnosticinferenceusingAdap-tiveImportanceSamplingandEvidencePre-propagationImportanceSamplingfor500,1000sam-ples.Comparisonofboththetablesshowthetwoalgorithmsconvergeclosetoaccurateestimateswithin500samples.Themeanandstandarddeviationoftheerrorandthemaximumerrorareex-tremelylowforboththemodelsevenforlargerbenchmarkcircuitslikec3540,c6288.Thiscanbeattributedtotheformationofagoodimportancefunctionthatisclosetotheoptimalimpor-tancefunction.However,EPISshowsfasterconvergencethanAISasitavoidsthecostlylearningprocessinAISalgorithm.NotethatthediagnosticfeaturethatbothAISandEPISmethodsofferovertheApproximateCascadedBayesianNetworkmethodsisoneofourkeymotivationsforusingstochasticinference.Figures4.2.,4.3.,and4.4.,correspondingtoc432,c1355,c6288benchmarkcircuits,respec-tively,showthevariationoferrors,obtainedusingAIS,EPISandPLS.Analysisofthegraphshowsthattheestimatesconvergefasterwithinasmallsamplespaceandestimatescanalwaysbeformedevenwhenthesamplespaceissmallorinsufcient(any-time).48

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10020030040050060070080090010000.010.020.030.040.050.06EPISSAMPLESc432PLSAISERROR Figure4.2.GraphShowingtheTimeAccuracyTrade-offforc432.Table4.2.ExperimentalResultsusingAISAlgorithmforvariousSamples. AIS:500samples AIS:1000samples E sE MxE T(s) E sE MxE T(s) c432 0.004 0.014 0.056 16.42 0.001 0.009 0.041 16.68 c499 0.001 0.012 0.097 19.42 0.000 0.009 0.041 19.67 c880 0.000 0.013 0.057 40.29 0.000 0.010 0.043 40.82 c1355 0.001 0.013 0.064 62.48 0.000 0.009 0.052 63.68 c1908 0.002 0.015 0.069 97.75 0.000 0.010 0.044 99.62 c3540 0.001 0.012 0.065 205.7 0.001 0.009 0.048 212.8 c6288 0.001 0.014 0.085 389.33 0.002 0.010 0.056 394.78 49

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10020030040050060070080090010000.010.020.030.040.050.06EPISSAMPLESc1355PLSERRORAIS Figure4.3.GraphShowingtheTimeAccuracyTrade-offforc1355.Table4.3.ExperimentalResultsusingEPISAlgorithmforvariousSamples. EPIS:500samples EPIS:1000samples E sE MxE T(s) E sE MxE T(s) c432 0.004 0.011 0.049 0.72 0.002 0.009 0.048 1.04 c499 0.001 0.012 0.055 0.94 0.001 0.008 0.039 1.03 c880 0.000 0.014 0.078 2.82 0.002 0.010 0.056 3.36 c1355 0.002 0.019 0.056 6.95 0.001 0.009 0.051 7.82 c1908 0.004 0.015 0.067 15.42 0.001 0.009 0.044 16.64 c3540 0.002 0.013 0.070 52.34 0.001 0.009 0.042 54.76 c6288 0.002 0.012 0.069 143.23 0.001 0.009 0.052 144.33 50

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10020030040050060070080090010000.010.020.030.040.050.06EPISAISSAMPLESERRORPLSc6288 Figure4.4.GraphShowingtheTimeAccuracyTrade-offforc6288.51

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CHAPTER5ENTROPYBASEDINPUTCHARACTERIZATIONFORDATA-DEPENDENTLEAKAGEPOWERANALYSISTheaveragepowerdissipationinCMOSissumoftwofactors,namely,dynamicpowerandleakagepower.Tradiationally,themostsignicantcontributortopowerdissipationinCMOScir-cuitshasbeenthedynamicpowerdissipation.Toattenuatethisproblemdesignersreliedonscalingdownthesupplyvoltageduetothequadraticdependenceofdynamicpoweronsupplyvoltage.Amajordisadvantageofthistechniqueisthatitaffectstheswitchingspeedofthecircuit.Sustenanceofthecircuitspeedrequiresaproportionatedecreaseinthethresholdvoltage.Downscalingofthethresholdvoltageaggravatesleakagepowerdissipation(duetoanexponentialincreaseinsub-thresholdleakagecurrent).Also,asthedevicedensityincreasesleakagepowerstartsdominatingthetotalpowerdissipation.Leakagepowerdissipationcanaccountformorethan50%oftotalpowerdissipationin65nmIC's.Aneffectivestrategyformitigatingleakagepoweristousedualthresholdvoltagecells(placinglowthresholdvoltagecellsinthecriticalpathtomaintainperformanceandhavinghighthresholdvoltagecellsinthenon-criticalpathstoreduceleakage).Itshouldbenotedthat,leakagedoesnotonlyhappenduringstand-bymodes,itisalsopresentduringactiveperiodandisacrucialcomponentoftotalpowerdissipation,asinFigure[5.1.],[61],[62].ThischapterintroducesanoveltechniqueusingBayesianNetworksthatidentiesgateswhicharedormantevenintherun-timemode.Designerscantargetthesegatesascandidategatesforleakagepoweropti-mization.Theintensionofthischapterisnottofocusonthemethodsofleakageoptimizationbuttoprovidedesignerswithanothertechniquetotargetgatesforleakagepoweroptimization.52

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Active PowerActive Leakage PowerPower (Watts)050150200250100Technology0.25m 0.18m 0.13m 0.1m 0.07m Figure5.1.GraphShowingRiseinLeakagePowerwithTechnology[1].5.1WhyDoWeNeedInputCharacterization?Thetotalpower(Pt)expendedinacircuitcanbeexpressedasthesumofindividualgatepower(Ptg),whichinturncanbebrokenupintoswitchingandleakagecomponents[61].PtgPtgPdgPsg05afV2ddCloadwire1aiPleakibi(5.1)whereadenotestheactivityofthenode,andbistheprobabilityofremaininginadominantleakagestate(namelysignalat0or1).Notethatbaswellasaisdependenton!yi"A,!0t10t0t11t1t10t1t11t"theswitchingstatesoftheinputsandthephysicalparametersofthegate,q.Thus,totalpowerisafunctionoftheinputswitchingstates,y1yN,andthedeviceparameters,q,i.e.Pty1yNq.Givenaninputtrace,oneusuallycomputes1y17778yN2Pty1yNq,whichisakintocomputingtheexpectedvalueofthetotalpower,EPt,withtheparticularinputtrace.Theoptimizationprob-lemcanthenbeexpressedas:minqEPtBminqy17778yNPty1yNqpy1pyN(5.2)53

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F2F3F4F5F6F7 F8F9F10F1Figure5.2.TheCorrelationsamongtheOutputLinesofc432withRandomInputs.TheOutputLinesofOneBlockaretheInputLinesofAnother.Theoptimalpoint,qopt,thusfoundwillbeafunctionoftheassumedinputstatistics,py1pyN1.Notethat,poweroptimizedoperatingpointarrivedatforasingleinputswitchingpoint,orafewsetofinputstatistics,isnotsufcient.Thespaceofallpossibleinputswitchingstatisticsisjusttoolarge,infactitismuchlargerthanthesizeoftheinputspaceitself;itisthesetofallpossibleprobabilitydistributionsovernottheinputstates,buttheinputswitchingstates.Moreover,sim-plisticrandominputassumptionsontheinputspaceisaseriousproblem.Inputsofonecircuitsaretheoutputsofanother,andhencecanexhibitstrongcorrelations.Forinstance,Figure.5.2.showstheprobabilisticdependenciesamongsttenoftheoutputsofc432(4bitALU)benchmark.Weproviderandominputstotheprimaryinputsofc432andlearnacausalprobabilisticgraphicalmodelamongsttheoutputs(thelearningprocessusedisnotinthescopeofthisthesis).Theprob-abilisticdependencybetweenthesenodesdenitelyarefarfromrandomandevenmodelingusingjustbiasedinputs(low/highswitching)isnotsufcient.Clearly,thereisneedforastatisticalinputcharacterizationmeasureforleakageoptimizationthatisnotbasedonpriorknowledgeofinputtrace.Ofcourse,ifcompletedatatracespecicationsforanapplicationdomainareknown,anyestimationalgorithmcanpredictthelikelihoodofasignaltoleak.However,inpracticethisishardtoaccomplish,especiallyinthenano-domain,whereduetothemulti-objectiveoptimizationneeds,thesizeofthedatatracerequirementsincrease,soastobeabletoexerciseallthe“modes”oftheobjectivefunctions. 1Inthechapter,weusecapitals,e.g.Y,todenoterandomvariables,correspondingsmallletters,e.g.ytodenotevalues.Wealsousep4y5todenotep4Y3y5,i.e.theprobabilityoftheeventY3y54

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Insteadofconsideringtotalpowerbasedonapriorovertheinputspace,whichhasbeentheusualpractice,weconsidertheposteriorovertheinputspace,conditionedonstatesofinternalnodes.Usingtheseposteriordistributionsovertheinputs,weidentifynodesthatarelikelytobeleakyintheactivezone.Itdoesnotrequirethepre-specicationofinput-statisticsorinput-traces.Mathematically,weconsiderpC!yi"Xj0t10t,whereXjrepresentsarandomvariableinternaltothecircuitandpC!yi"istheproleofinputsetthatgeneratestheobservationXj0t10t.Weusetheconceptofentropyofthisposteriordistributiontocharacterizetheinputspace.Theconceptofentropyinitselfisnotnewandisusedinalmostallresearchdisciplines.Excellentintroductiontoaxiomaticbasisofentropyconceptscanbefoundin[73,74,75].Entropyisameasureofuncertaintyinanitesystem.Highentropyindicateshighuncertainty.Completelyrandominputswouldbeassociatedwithhighentropy.Inthiswork,wegenerateanupperboundoftheentropyoftheposteriorinputspacedistributionbyconsideringindependentinputs.ifnopriorinformationabouttheinputisknown,theconceptofabsoluteentropycanbeused.However,ifsomeknowledgeoftheinputisavailable,suchaswhenoneknowsthattheinputsaretheoutputsofanotherlogicblock,theninsuchcase,wecanuserelativeentropyasadistancemeasurebetweentheknowninputspaceandtheposteriorinputspaceundertheevidence/observation.WeuseBayesianBeliefNetworkbasedmodelingtocomputetheinputposteriors.WhyBayesianNetworks?Simplybecausethesecausal,graphicalmodelsareminimalrepresentationscompletelycapturingtheunderlyingjointprobabilitydensityfunction(pdf);itinducesanoptimalfactoriza-tionofthejointpdf[2].Theminimalityoftherepresentationmanifestsinthereducednumberofthelinksinthegraphrepresentation,whichinturnfacilitatesfastbeliefpropagationschemes.Theseprobabilisticbeliefpropagationschemesarenotdirection-sensitive,eventhoughtheunderly-inggraphrepresentationhasdirectedlinksbetweencauseandeffect;duringupdatingmessagesarepassedinbothdirections.Infact,thisprobabilisticmodelispattern-insensitiveforpredictivepur-poses,whereweknowthepossiblecausesandwewanttoknowtheeffects.But,therealpowerofprobabilisticmodelsisthatgivenanobservation,wecancharacterizetheplausiblecausesthatcanproducethegivenobservation,thatis,theprobabilisticmodelsprovidetheMostProbableExpla-55

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nationthatcancausetheobservation.WeuseEvidencePre-propagatedImportanceSampling[12]stochasticinferencetechniquetoaccuratelypropagatebelieffromevidencetoallothervariables.Theissueandtheapproachpresentedinthischapterarenewtoourknowledge.Foradifferentcontextofdynamicpower,dataspecication,orratherlackofspecication,issueshasbeenstudiedusingpowersensitivity[76],whereupper/lowerboundsonaveragedynamicpowerwascomputed.Notethatthistypeoftreatmentdependsonafewsimulation-based(pattern-sensitive)points,aroundwhichsensitivityiscomputedandtreatsonlythedynamiccomponentofpower.Withregardtotheconceptofentropy,ithasbeenusedinRTpowerasanupperboundofswitchingactivitybyNemanietal.[77]andnotforinputspacecharacterization.WithregardstoBayesiannetworks,itwasrstproposedin[8]andthenin[78].However,thecontributionshavebeenlimitedtoswitchingesti-mationandtimingpredictionthatdonotexploitthebacktrackingaspectofprobabilisticreasoning.Ourworkprovidesanotherarsenalforthelowpowerdesignerstofocusleakagemitigationschemesattargetednodes,ratherthanjustconsideringnodes(basedoncriticalpath)orusingasingle(orahandfulofdataproles)totargettheoptimizationschemes.Methodsthatusemeasuresoneverytransistors,suchasdynamicthresholdvoltage[67,68],canuseourmeasuretoselectthetargetnodes.5.2InputCharacterizationWeapproachtheinputspacecharacterizationproblemasaprocesstondouttheplausiblecausesofanobservation.Inourcase,supposewewanttogetameasureofleakiness(Xi0t10tXi1t11t)oramountofswitching(Xi0t11tXi1t10t)ofaninternalsignalXi,predic-tiveinferencestrategieswouldactuallyconsidertheconditionalprobabilitypXi0t10ty1yNwhereknowledgeofinputswitchingstatesy1yNareassumedtobeknownapriori.Ontheotherhand,diagnosticinferencewouldanalyzepy1yNXi0t10tandthenusetheposte-riorinputdistributiontocharacterizetheinputspacefortheobservationXi0t10t.Beforewediscuss,inSection5.3,howtocomputethisinputposterior,wepresenttheentropybasedcharacter-izationthatweadvocate.Notethat,wewillbeusinguppercasealphabets(namelyXY)todenote56

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randomvariablesandlowercasealphabets(namelyxyyiyj)todenotethediscretestatesthattherandomvariablecantake.Beforeprovidingthedenitions,letusintuitivelyseewhyweuseentropy.Entropyofasys-temmeasurestheamountofuncertaintyinthesystem.Amorelikelyeventisboundtohavehigherentropythanalesslikelyevent[74].ConsiderarandomvariableYi,representingthei-thinput.Forourcase,eachYicantakenononeoffourpossibleswitchingstatesvalues,yi,!0t10t0t11t1t10t1t11t".EntropyofthisrandomvariableisgivenbyHYi yipyilogpyi(5.3)Ifrandomswitching,pyi1r4andHYiislog4,themaximumpossible.Ifthestateofvariableisknownforsure,HYiiszero.Entropycanalsobecomputedforasetarandomvariablebyconsideringtheirjointprobabilityfunctionanditcanbeshownthat[79].HY1YNNi1HYiY1Yi1Yi1YN(5.4)WhentherandomvariablesareindependentofeachotherjointentropyissuchthesumoftheindividualentropiesHuY1YNNi1HYi(5.5)ItcanbeshownthatHuDHwhereyi'sarenotmutuallyindependent,orisanupperboundonthejointentropy.Weusethisupperboundentropymeasureontheposteriorinputspaceasaleakagepossibilitycharacterizationmeasurefornode,Xj.HuxjY1YNNi1pyixjlogpyixj(5.6)Wecanuseentropicmeasures,evenforsituationswherewemightknowsomethingabouttheinputstatistics.Insuchcasesweconsidertherelativeentropy,whichisalsoknownastheKullback-Lieblerdistance,crossentropy,discriminationinformation,directeddivergence,orI-divergence.57

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Letqy1qyN,denotetheknowninputstatistics.Thentherelativeentropybetweenthepos-teriordistributionandtheknownoneisgivenbyHrxjpy1yNxjlogpy1yNxj qy1qyN(5.7)Thisservesasameasureofleakagepossibility.Iftherelativeentropyishighbetweentheposteriorinputspace,conditionedonaleakystate,xj,ofanode,Xj,andthegiveninputspace,thenitisunlikelythatparticularnode,Xj,willleak.Andviceversa,ifthedistanceisless.Likeforsimpleentropy,itcanbeshownthattherelativeentropymeasureunderindependenceassumptioncanprovideuswithanupperbound.TheupperboundrelativeentropycanexpressedsimplyasHrxjpy1y2yNlogpy1y2yN qy1y2yN(5.8)py1py2pyNilogpyi qyi(5.9)ispyilogpyi qyi'Ej?ispyj(5.10)ispyilogpyi qyi(5.11)Equation.5.9exploitsthefactorizationthatispossiblebasedonindependence.InEq.5.10,spyjF1.Notethatundertheindependenceassumption,therelativeentropycomputationisinexpensive.5.3ComputingPosteriorInputDistributionsInthissection,wediscusstheprobabilisticmodelandthebackwardpropagationschemesthatareusedtocomputetheposteriordistributionsovertheinputspace.Anyprobabilityfunctionoverasetofrandomvariables(X1XMy1yN),whereXi,aretherandomvariablesrepresenting58

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i-thinternalsignals,Yjisthej-thprimaryinput,canberepresentedasPx1xMy1yNPxMxM1xM2x1y1yNPxM1xM2xM3x1y1yNPy1PyN(5.12)Theaboveexpressionholdsforanyorderingoftherandomvariables.Thisbruteforcerepresen-tationofprobabilisticknowledgerequirestabulationofalltheentriesinPx1xMy1yN,whichareexponentialinnumber.Theexactrepresentationassumesthateveryvariableisdepen-dentoneveryotherrandomvariablepresentintheset.Itdoesnottakeadvantageoftheconditionalindependenciespresentamongthevariables.Untilveryrecently,thecommonstrategyhasbeentomakethenaiveassumptionofcompleteindependenceofthevariables,i.e.Px1xMGPx1PxM.However,thisfailstoexploitthefullpowerofprobabilisticmodeling.Power-fuldirectedacyclicgraph(DAG)basedmodelshavebeenrecentlyproposedthatexploitsalltheindependenciesamongtherandomvariablestoarriveataminimalmodelofthejointprobabilitygraph.ThesearetermedBayesiannetworks,causalnetworks,orbeliefnetworks[2].Thisgraphicalprobabilisticmodelinducesafactoredrepresentationofthejointprobabilityfunctionintermoftheconditionalprobabilitiesofeachrandomvariables,giventhestatesoftheircorrespondingparentsinthegraphstructure.Px1xMy1yN M'k1PxkPaxkN'j1Pyj(5.13)ThepowerofBayesianNetworksisinthefactthatnotonlycanitdopredictiveinference,i.e.computeprobabilitiesoftheoutputs(effects)basedonevidenceabouttheinputs(causes),butitcanalsododiagnosticinference,i.e.computeprobabilitiesabouttheinputs(causes)conditionedontheoutputs(effects).Whilesimulativeapproaches,suchasthoseusedinVLSIestimation,mightsufceforpredictiveinference,theyarenotatallefcientfordiagnosticinference.Oneexampleofdiagnosticqueryis“Whatinputscancauseaswitchingprobabilityof08ataparticularnodeofinterest?”Thiswouldrequireaconditionalsearchoftheinputspaceinasimulativeframework,whichiscomputationallyexpensive.Probabilisticinferenceistheonlyconsistentuncertaintycal-59

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culustohandlesuchsituationsinanefcientmanner.WeuseEPIS(astochasticsamplingtechniquediscussedinchapter4,section4.2.3)techniquetoestimatetheposteriorinputdistributionsgivenanobservation.Forourexperiments,weusedtheBayesiannetworkcomputationalpackageandlibraryknownas”GeNIe”[22]andSMILE.Notethatthissamplingbasedprobabilisticinferenceisnon-simulativeandisdifferentfromsamplingsthatareusedincircuitsimulations.Inthelatter,theinputspaceissampled,whereasinourcaseboththeinputandthelinestatespacesaresampledsimultaneously,usingastrongcorrelativemodel,ascapturedbytheBayesianNetwork.Duetothis,convergenceisfasterandtheinferencestrategyisinputpatterninsensitive.5.4ResultsandConclusionsWeillustratetheideaspresentedinthischapterusingtheISCAS'85benchmarkcircuits.AllcomputationsarerunonaPentiumIV,2.00GHz,WindowsXPcomputer.Table5.1.showstheprolesofnodesthataretestedforknowndatatraces.Wetestedwithtwosets.Therstsetiswheretheinputswitchingismoderate(switchingactivity0.5)andthesecondsetisforabiasedlowswitchinginputs(switchingactivity0.3).Column2and3denotesthenumberofnodesthatwouldbeleaking60-80%andabove80%ofthetimeduringtheactivemodeofoperation(inputswitchingisassumedto0.5).Foranexamplec3540,wouldhave841nodesleakingmorethan80%timeandc1908has362nodesleakingformorethan80%oftime.Notethattheabovedataisreportedonindividualsignals.However,foranNMOSstack,oneoftheinputat1mightnotmakeitaleakystate.Forthetruecomputation,weshouldconsiderthejointinstantiationoftheinputstoeither0t10torto1t11tandthencalculateentropy.Notethatthecalculationeffortofsuchacombinationwouldnotbedifferentfromsingleinstantiation.Figure5.3.,5.4.,5.5.showsthedetailedproleofthenodebreakupofthepercentageoftimeleakingatstatezero,forthreecircuitsandwithtwodifferentinputstatistics;foronecase,primaryinputsareswitching50%ofthetimeandintheothercase,primaryinputsareswitching30%oftime.Noticethatalargernumberofnodeswouldbeleakingsignicantlyduringactivezoneforactivity0.3.DatashowninTable5.1.,aswellasinFigure.5.3.,5.4.,5.5.,aregeneratedbythe60

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Table5.1.LeakageProleforKnownDataTraces. Circuits Numberofnodesleakinginactivezone Inputactivity0.5 Inputactivity0.3 (60-80)%oftime H80%oftime (60-80)%oftime H80%oftime c432 178 37 363 102 c499 54 88 235 88 c880 252 123 610 224 c1355 402 280 743 306 c1908 372 362 1224 395 c3540 896 841 2068 1040 c6288 1997 292 2856 555 sameprobabilisticframeworkandinferencethatisusedfortheposteriorcomputation.Allthedataarenon-simulative,hencepatterninsensitiveandmodelsalldependenciesinthecircuits.InTable5.2.,weshowtheresultsoninputspacecharacterizationbyabsoluteentropy.Notethatthistableisfortheentropycomputationwhenposteriorinputdistributionwerecomputedforcingindividualsignalstoleakatlogiczero.Signalsthatproducedanentropymeasurethatlieswithin30%ofthemaximumentropyarereportedatcolumn3andthesenodesaretargetsforleakagemitigationschemes.Minimumandmaximumentropyisreportedincolumn4and5respectively.Column2reportsnodesarehighlyunlikelytoleakatzero:thesesignalsproducedentropythatwithin30%ofthelowestentropyseenforthosecircuits.Notethatforc880andc1908,therearenodesthatdonotleakatanytimeatzeroforanyinputcombinations.Somecircuitsshowedlargerrangeinentropythanothers.Forexample,inc3540thedifferencebetweenmaximumandminimumentropyis10whereasc499andc1355entropyrangeissmall.Thisindicatesthedatatracesaremoreimportanttoobtainforthelattergroups(c499).Also,wefoundoutafewnodesthatcannotbeleakingatzeroirrespectiveoftheinputspace.Thishappenswhenposteriordistributionisimpossible.InTable5.3.,wepresentresultsontherelativeentropy.Notethat,relativeentropymeasuresdistancefromareferencedistributionandhencesignalsthatgenerateslowHr,aremorelikelytobeleakyatzerogiventheexpectedreferencedistribution.WereportnumberofnodesthatgeneratedhighinputKLdistancemeasure(within30%ofthemaximumHr)incolumn2andnumberof61

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I J K L M N M O M M O N M P M M P N M Q M R M S M T M O M M U V W X Y Z [ \ [ ] ^ Y ` ] X a [ b V c d e f g h i j c i k g l m n o p q r s t u m n o p q r s t vFigure5.3.GraphShowingtheBreak-upofLeakageProleforDifferentSwitchingProleforc432.Table5.2.EntropyBasedInputCharacterizationforLeakageCondition. Circuits Numberofnodesgenerating RangeofEntropy Lowinput Highinput entropy entropy min max c17 4 2 4.14 6.92 c432 2 321 44.66 49.86 c499 31 321 53.95 56.78 c1355 128 793 53.95 56.79 c1908 4 1871 0 45.74 c3540 5 3220 58.91 69.25 c6288 6 4172 37.81 44.33 62

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w x y z { | } | | ~ | | | | € | |  | | ‚ | | ƒ | | „ | | … | | } | | | ‚ | ƒ | „ | … | } | | † ‡ ˆ ‰ Š ‹ Œ  Œ Ž  Š  ‘ Ž ‰ ’ Œ “ ‡ ” • – — ™ š › ” š œ  Ÿ   ¡ ¢ £ ¤ ¦ Ÿ   ¡ ¢ £ ¤ §Figure5.4.GraphShowingtheBreak-upofLeakageProleforDifferentSwitchingProlefor1908.63

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¨ " " Figure5.5.GraphShowingtheBreak-upofLeakageProleforDifferentSwitchingProleforc3540.64

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Table5.3.RelativeEntropyBasedInputCharacterizationforLeakySignals. Circuits Numberofnodesgenerating RangeofHr HighHr LowHr MinHr MaxHr c17 4 2 0.01 2.78 c432 1 303 0.04 4.77 c499 30 322 0.05 2.88 c1355 127 794 0.05 2.88 c1908 13 1760 0.04 5.6 c3540 4 3221 0.06 10.39 c6288 5 4175 0.04 6.55 nodesthatgeneratedlow(bottom30%)inputHrmeasureincolumn3.Columns4and5reports,minimumandmaximumrelativeentropy,respectively,thatweobserveforthecircuits.65

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CHAPTER6CONCLUSIONSwitchingactivityestimationisacomplexproblemthathavebeenresearchedformorethanadecade.Thisthesisintroducesaswitchingactivityestimationtoolforcombinationalcircuitsthatmodelsallthetemporalandspatialdependenciesinacircuitwithhighaccuracy.Wehavedemon-stratedtheresultsoftheestimatedswitchingactivityusingvariousany-timestochasticsamplinginferencealgorithmsnamelyEPIS,AIS,andPLS.WendthatPLSyieldsthebestaccuracy-timetradeoffifusedunderpredictivesituation.Indiagnosticsituation,caseswhenevidenceisunlikely,EPISandAISalgorithmswouldyieldaccurateestimates.WethusconcludethattheBayesianNet-workbasedmodelingofswitchingactivityandinferenceyieldshigheraccuracyinsignicantlylowertime.Thepresentscopeofthismodelislimitedtozero-delayscenario,whichweplantoaddressinfuture.Wealsopresentedanovelprobabilisticframeworkformeasuringleakinesspotentialofasignalduringactiveswitchingmode,withoutanypriorassumptionabouttheinputspace.WeusetheprobabilisticBayesianNetworktopropagatebelieffromobservationtotheplausiblecausesandusetheattributeslikeentropyoftheposteriortodeterminethelikelinessofthesignalstobeatzero.Ourworkprovidesanotherarsenalforthelowpowerdesignerstofocusleakagemitigationschemesattargetednodes,ratherthanjustconsideringnodes(basedoncriticalpath)orusingasingle(orahandfulof)dataprolestotargetoptimizationscheme.Methods,suchasdynamicthresholdvoltage,canuseourmeasuretoselectthetargetnodesforleakageoptimization.66

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Document formatted into pages; contains 83 pages.
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ABSTRACT: Power dissipation in a VLSI circuit poses a serious challenge in present and future VLSI design. A switching model for the data dependent behavior of the transistors is essential to model dynamic, load-dependent active power and also leakage power in active mode the two components of power in a VLSI circuit. A probabilistic Bayesian Network based switching model can explicitly model all spatio-temporal dependency relationships in a combinational circuit, resulting in zero-error estimates. However, the space-time requirements of exact estimation schemes, based on this model, increase with circuit complexity [5, 24]. This work explores a non-simulative, importance sampling based, probabilistic estimation strategy that scales well with circuit complexity. It has the any-time aspect of simulation and the input pattern independence of probabilistic models.Experimental results with ISCAS'85 benchmark shows a significant savings in time (nearly 3 times) and significant reduction in maximum error (nearly 6 times) especially for large benchmark circuits compared to the existing state of the art technique (Approximate Cascaded Bayesian Network) which is partition based. We also present a novel probabilistic method that is not dependent on the pre-specification of input-statistics or the availability of input-traces, to identify nodes that are likely to be leaky even in the active zone. This work emphasizes on stochastic data dependency and characterization of the input space, targeting data-dependent leakage power. The central theme of this work lies in obtaining the posterior input data distribution, conditioned on the leakage at an individual signal.We propose a minimal, causal, graphical probabilistic model (Bayesian Belief Network) for computing the posterior, based on probabilistic propagation flow against the causal direction, i.e. towards the input. We also provide two entropy-based measures to characterize the amount of uncertainties in the posterior input space as an indicator of the likelihood of the leakage of a signal. Results on ISCAS'85 benchmark shows that conclusive judgments can be made on many nodes without any prior knowledge about the input space.
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Adviser: Bhanja, Sanjukta.
653
gate level.
Bayesian network.
simulation.
sampling.
entropy.
690
Dissertations, Academic
z USF
x Electrical Engineering
Masters.
773
t USF Electronic Theses and Dissertations.
4 856
u http://digital.lib.usf.edu/?e14.497