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Real delay graphical probabilistic switching model for VLSI circuits

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Real delay graphical probabilistic switching model for VLSI circuits
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ABSTRACT: Power optimization is a crucial issue at all levels of abstractions in VLSI Design. Power estimation has to be performed repeatedly to explore the design space throughout the design process at all levels. Dynamic Power Dissipation due to Switching Activity has been one of the major concerns in Power Estimation. While many Simulation and Statistical Simulation based methods exist to estimate Switching Activity, these methods are input pattern sensitive, hence would require a large input vector set to accurately estimate Power. Probabilistic estimation of switching activity under Zero-Delay conditions, seriously undermines the accuracy of the estimation process, since it fails to account for the spurious transitions due to difference in input signal arrival times.In this work, we propose a comprehensive probabilistic switching model that characterizes the circuit's underlying switching profile, an essential component for estimating data-dependent dynamic and static power. Probabilistic estimation of Switching under Real Delay conditions has been a traditionally difficult problem, since it involves modeling the higher order temporal, spatio-temporal and spatial dependencies in the circuit. In this work we have proposed a switching model under Real Delay conditions, using Bayesian Networks. This model accurately captures the spurious transitions, due to different signal input arrival times, by explicitly modeling the higher order temporal, spatio-temporal and spatial dependencies. The proposed model, using Bayesian Networks, also serves as a knowledge base, from which information such as cross-talk noise due to simulataneous switching at input nodes can be inferred.
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Thesis (M.S.E.E.)--University of South Florida, 2004.
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by Vivekanandan Srinivasan.
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RealDelayGraphicalProbabilisticSwitchingModelforVLSICircuitsbyVivekanandanSrinivasanAthesissubmittedinpartialfulllmentoftherequirementsforthedegreeofMasterofScienceinElectricalEngineeringDepartmentofElectricalEngineeringCollegeofEngineeringUniversityofSouthFloridaMajorProfessor:SanjuktaBhanja,Ph.D.Yun-LeeiChiou,Ph.D.WilfridoA.Moreno,Ph.D.DateofApproval:November1,2004Keywords:BayesianNetworks,Simulation,Inference,SamplingcCopyright2004,VivekanandanSrinivasan

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DEDICATIONToMyParents.

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ACKNOWLEDGEMENTSIwishtoexpressmysinceregratitudetomyprofessorDr.SanjuktaBhanjaforguidingmeinthisthesis.Dr.Bhanja'svisionandguidance,hasimmenselyhelpedmetostaythecourseandcompletethiswork,andhasalsoexposedmetonewvistasintheresearcharea.IwouldalsoliketoexpressmygratitudetoDr.WilfridoMorenoandDr.Yun-LeeiChiouforagreeingtoserveinmycommittee.Iwishtothankmyparentsforthehardshipstheyhaveenduredtoprovidemewiththiseducation.Iwishtothankmygrandparentsforallthesupporttheyhaveofferedme,especiallymygrandmother,anintelligentandcompassionatewoman,shehasbeensupportiveofallmyventures.Iwouldalsoliketothankmyfriends,intheVLSIDesignAutomationandTestLab,Bheem,Nirmal,Saket,Sathish,ShivaandTara,forcrucialhelpandsuggestionsthroughoutthecourseofmywork.IwouldalsoliketothankPonrajandBodkaforsuggestionsinCodingarea.

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TABLEOFCONTENTSLISTOFTABLESiiiLISTOFFIGURESivABSTRACTvCHAPTER1INTRODUCTION11.1LowPowerVLSIDesign11.2StaticPowerDissipation21.3DynamicPowerDissipation21.4ContributionsofthisThesis41.4.1GraphicalRepresentation41.4.2BayesianNetworks41.4.3GateDelayModeling51.5FlowofthisThesis6CHAPTER2PRIORWORK8CHAPTER3BAYESIANNETWORKS123.1BayesianNetworks123.2FunctionalityofLIDAGBN19CHAPTER4REALDELAYPROBABILISTICMODEL214.1TypesofDelayModel224.1.1ZeroDelayModel224.1.2UnitDelayModel224.1.3VariableDelayModel224.2GateDelayModel234.3ExpandedCircuitRepresentation244.4RealDelayProbabilisticalModel264.4.1DependenciesinPrimaryInputs264.4.2HigherOrderSpatio-TemporalDependencies284.4.3HigherOrderTemporalDependencies29CHAPTER5INFERENCETECHNIQUES315.1ExactInference325.1.1PolytreeAlgorithm325.1.2Clustering32i

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5.2ApproximateSamplingAlgorithms325.2.1StochasticSamplingAlgorithms325.2.2ProbabilisticLogicSampling335.2.3ImportanceSampling345.2.4AdaptiveImportanceSampling355.2.5EvidencePre-propagationImportanceSamplingAlgorithm35CHAPTER6RESULTS37CHAPTER7CONCLUSION407.1FutureWork41REFERENCES42ii

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LISTOFTABLESTable3.1.ConditionalProbabilitySpecicationsfortheOutputandtheInputLineTransitionsforTwoInputNANDGate.20Table4.1.ConditionalProbabilitySpecicationsforPrimaryInputDependency.28Table4.2.ConditionalProbabilitySpecicationsforOutputNodes.30Table6.1.MaximumTimeInstantsforSomeCombinationalBenchmarkCircuits.37Table6.2.SwitchingActivityEstimationErrorStatisticsBasedonDelayDAGModeling,Us-ingPLSInferenceScheme,Using1000Samples,forISCAS'85BenchmarkCom-binationalCircuits.38iii

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LISTOFFIGURESFigure1.1.Moore'sLaw.2Figure3.1.2-InputORGate.13Figure3.2.C17BenchmarkCircuit.14Figure3.3.BayesianNetworkofC17BenchmarkCircuit.15Figure4.1.GlitchGeneration.21Figure4.2.VariableDelay.24Figure4.3.ExpandedCircuit.25Figure4.4.DAGofAReal-DelayModel.27Figure5.1.Flowchart.33iv

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REALDELAYGRAPHICALPROBABILISTICSWITCHINGMODELFORVLSICIRCUITSVivekanandanSrinivasanABSTRACTPoweroptimizationisacrucialissueatalllevelsofabstractionsinVLSIDesign.Powerestimationhastobeperformedrepeatedlytoexplorethedesignspacethroughoutthedesignprocessatalllevels.DynamicPowerDissipationduetoSwitchingActivityhasbeenoneofthemajorconcernsinPowerEstimation.WhilemanySimulationandStatisticalSimulationbasedmethodsexisttoestimateSwitch-ingActivity,thesemethodsareinputpatternsensitive,hencewouldrequirealargeinputvectorsettoaccuratelyestimatePower.ProbabilisticestimationofswitchingactivityunderZero-Delayconditions,seriouslyunderminestheaccuracyoftheestimationprocess,sinceitfailstoaccountforthespurioustransitionsduetodifferenceininputsignalarrivaltimes.Inthiswork,weproposeacomprehensiveprobabilisticswitchingmodelthatcharacterizesthecircuit'sunderlyingswitchingprole,anessentialcomponentforestimatingdata-dependentdynamicandstaticpower.ProbabilisticestimationofSwitch-ingunderRealDelayconditionshasbeenatraditionallydifcultproblem,sinceitinvolvesmodelingthehigherordertemporal,spatio-temporalandspatialdependenciesinthecircuit.Inthisworkwehavepro-posedaswitchingmodelunderRealDelayconditions,usingBayesianNetworks.Thismodelaccuratelycapturesthespurioustransitions,duetodifferentsignalinputarrivaltimes,byexplicitlymodelingthehigherordertemporal,spatio-temporalandspatialdependencies.Theproposedmodel,usingBayesianNetworks,alsoservesasaknowledgebase,fromwhichinformationsuchascross-talknoiseduetosimulataneousswitchingatinputnodescanbeinferred.v

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CHAPTER1INTRODUCTION1.1LowPowerVLSIDesignIn1963,CMOScircuitswereinventedasalowpoweralternativetoTTL,today,CMOSisthepredominanttechnologyindigitalintegratedcircuitsalbeitincreasedfunctionality,powerisstilloneofthemajorconcernsoftheVLSIdesigncommunity.ThequestforsmallerdeviceswithincreasedfunctionalityhasdriventheCMOStechnologyintothedeepsubmicronregion.Theshrinkinggatelengthhasleadtoanexponentialincreaseinthenumberoftransistorsperunitarea.Whilethisincreaseindevicedensityhascreatedsmallerandportableelectronicdevices,whichareeasiertomarket,ithasalsoleadtoanexponentialincreaseinthepowerconsumptionandpowerdissipatedperunitarea.Theincreaseinpowerdissipationrequireshighercoolingcostsandahigherbatteryweight,whichadverselyaffectthemarketabilityofanelectronicdevice.ThisexplainstheemergenceoflowpowerdesignasoneofthekeyresearchareasinVLSIdesign.IntelfounderDr.GordonMoore,statesthatsemiconductordenstiy,andhenceperformance,doublesroughlyevery18months.GordonMoore'sinsightfulobservationondeviceadvancementhashadadirectcorrelationwithpowerdissipation.[Figure1.1.]showsthepowerdissipationofsomerepresentativeIntelmicroprocessorsasafunctionoftime.Thescatterindataclearlysuggeststechnologyadvancementstoreducepower.Buttheoveralltrendistheexponentialincreaseinpowerdissipationwithtime.Powerdissipationisfactoredintothefollowingcomponents:1

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10101010101010101960196519701975198019851990199520052010200001234567891010MOS ArraysMOS Logic 1975 Actual Data1975 ProjectionTransistors Per DieYear1965 Actual DataIntegrated Circuit Complexity 1010 Figure1.1.Moore'sLaw.1.2StaticPowerDissipationStaticDissipationoccursduetoreversebiasleakagebetweendiffusionregionsandthesubstrate.Subthresholdconductionalsocontributestostaticdissipation.Leakagecurrentisgainingsignicanceaswearespeedingintothenanoregion.1.3DynamicPowerDissipationShort-circuitpowerDissipationoccurswhenthereisashortcurrentpulsefromSupplytoGround.ThedirectpathfromSupplytoGroundisduetothep-typeandn-typetransistorsbeing'on'forabriefperiodoftimeduringtransitionfrom'1'to'0'orfrom'0'to'1'.DynamicDissipationalsooccursduetothecharginganddischargingofparasiticcapacitanceduringswitching.Ourquesttoproducefasterdevices,thusincreasingtheclockfrequency,hasbestowedevengreaterimportanceinthiscomponentofpowerdissipation.2

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AverageDynamicpowerdissipationatagateisgivenby,PdynamiciCiV2ifi(1.1)WhereVddisthesupplyvoltage,fclkistheclockfrequency,Cloadisloadcapacitanceandswxistheaverageswitchingactivityoftheoutputnodex.Thefactorsaffectingpowerdissipationbeingsupplyvoltage,loadcapacitanceandswitchingactiv-ity.Supplyvoltageandclockfrequencyareknowntothedesigners.Supplyvoltage,byvirtueofit'squadraticdependenceonpowerdissipation,hasbeenaneffectivetoolinminimizingpowerdissipation.ICpowermanagementrequirestheinclusionofpoweralongwithtimingandarea,asparametersmanagedthroughoutthedesignprocess.Pre-emptionhasbeentheonlyoptionforlowpowerdesigners.Estimationofpowerhasbeenperformedatvariouslevelsofabstraction,namelybehaviorallevel,RTlevel,gatelevel,circuitleveletc.Optimizationisperformedateachlevelbeforesynthesizingtolowerlevels.Thehigherthelevelofabstraction,fasteristhespeedofestimationbuttheaccuracyislower.Thisworkisfocusedatthegatelevelofpowerestimation.Atthislevel,gatecapacitancesareestimatedfromtheknowledgeofthelogicstructure,thechallengeistoestimateswitchingactivity.Estimationofswitchingactivityisachallengingprocess,inputstatistics,correlationbetweennodes,gatetypeandgatedelaysaresomeofthefactorsthataffectswitchingactivity.Switchingactivityatacircuitnode,canbedenedastheaveragenumberoftransitionsatthatnodeperunittime.Tocalculateswitchingactivitytheknowledgeofthepreviousandpresentstateofanodeisrequired,thishelpsinmodelingfortemporalcorrelation.Reconvergentfanoutsandanydependenciesbetweentheprimaryinputsinthecircuitgivesrisetospatialcorrelation.Forbetteraccuracythespatialandtemporalcorrelationsshouldbetakenintoaccountinthemodel.FirstorderTemporalandhigherorderSpatialcorrelationsareaccuratelycapturedintheworkofbhanjaetal.buttheirworkfailstoaccountforgatedelay.Foraccurateestimationofpower,gatedelayhastobetakenintoaccount,signalsarrivingattheinternalnodesofgate,mightarriveatdifferenttimeinstances,causingspurioustransitionstotakeplace.Spurioustransitions,alsocalledglitches,dissipatepower,thishastobetakenintoaccountinordertoaccuratelyestimatepower.Higherordertemporalcorrelations,alongwithspatiotemporalcorrelations,hastobetakenintoaccountforamodelcapturing3

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spurioustransitionsalongwiththedesiredswitchingactivity.Themainfocusofthisworkistodesignamodelthatwouldtakeintoconsiderationthesehigherordertemporalcorrelations,andtherebyaccountforgatedelaytoaccuratelyestimatePowerdissipation.1.4ContributionsofthisThesisProbabilisticmodelingofGatedelayhasbeentraditionallyadifcultproblem.Thisworksuggestsanovelswitchingmodelthatwouldcaptureallthespurioustransitionsarisingduetogatedelay,byexplicitlymodelingforhigherordertemporalcorrelations.Theswitchingmodelisnovelinthat,itcouldalsobeusedtoestimatetheprobabilityofcross-talknoisebetweenanytwonodes,andasobservedintheworkofRamanietalitcouldalsobeusedtoestimatethenodeswithhighprobabilityofleakage.IthasbeenshownthatBayesiannetworkselegantlycapturetherstordertemporalandspatialcor-relationswithreducedcomputationtime.Thisworkmodelsthetemporalcorrelationbyexplicitlymod-ellingthedependenciesbetweendifferenttimeinstancesofanodeandtherebyincreasingtheaccuracyintheestimationprocess.1.4.1GraphicalRepresentationInferencingjudgementfromProbabilisticmodelsinvolvesdeningajointdistributionfunctiononallpropositionsandtheircombinations.Constructingajointdistributionfunctionfornvariables,wouldrequireatablewith2nentries.Thisleadstoanexponentialgrowthinthecomplexityofcalculationalongwithanincreaseinthesizeofthecircuit,alsothehumanminddoesnotreasonwithsuchheavycompu-tations.Hence,GraphicalRepresentationisused,whichapartfrommakinginferenceofdesiredresultseasier,alsofacilitatesthevericationofexistingdependenciesandinvestigationofnewdependencies.1.4.2BayesianNetworksBayesianNetworkhasbeenusedinthisworktoprobabilisticallymodeltheswitchingactivity.BayesianNetworksareDirectedAcyclicGraphs(DAGs),inwhichthevariablesarerepresentedasnodes,andthecausalinuencebetweenthevariablesarerepresentedbydirectedarcs,theeffectaparentnode4

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hasonitschildrenisrepresentedbyaconditionalprobabilitytable.Byintroducingdirectionalityinthegraphs,BayesianNetworksarecapableofdisplayinginducedandnon-transitivedependencies.LogicInducedDirectedAcyclicGraph(LIDAG)isconstructedbasedonthelogicalstructureofthecircuit.Conditionalprobabilitytables,whichindicatesthestrengthbywhichparentnodesaffectachildnode,representtheswitchingactivityandaremappedone-to-one,ontoaBayesianNetworkthuspreservingthedependencyoftheprobabilityfunction.EachnodeintheLIDAG,representsasignalwhichcanhavefourpossiblestates,00,01,10,11.Directededgesaredrawnfromnodesrepresentingthesignalstotheirrespectivefanouts,theconditionalprobabilitytablereectstheattributeofthenodeanddetermineshowtheswitchinginputfromtheparentnodesaffectthenode.TheworkofBhanjaetalprovesthataLIDAG,thusobtained,isaBayesianNetworkwhichisaminimalrepresentationthatcapturesalltheindependencyrelationshipsinthecircuit.1.4.3GateDelayModelingTherearevarioustypesofgatedelaymodels,zerodelay,unitdelayandvariabledelayareafewamongthem.Inthisworkweconstructavariabledelaymodel,assumingdelaysofthegatetobeproportionaltotheirfanout.Inputsarrivingatthegateatdifferenttimeinstantswouldcauseundesiredorspurioustransitionsattheoutput.Thesespuriousswitchingsareaccountedfor,byexplicitlymodellingforhigherordertemporalandspatiotemporalcorrelations.Weinstantiateagateforeachtimeinstantapossiblespurioustransitioncanoccur.Alogicgateataparticulartimeinstantcouldbeaffectedonlybytheinstancesofinputswithtimeinstantsimmediatelyprecedingthetimeofthelogicgate.InthezerogatedelayofBhanjaetal,bydecouplingdelayfromthemodel,theyassumeinstantaneoustransmissionofsignals,whichisnottherealcase,thereisalwayssomedelayinthearrivalofsignals,thushamperingtheaccuracyofpowerestimationprocess.IntheworkofManichetal[9],theyusethisgatedelaymodeltondthecoupleofvectorsthatwouldmaximizetheweightedswitchingactivity,theirworkusesasimulationmodel.Alargenumberofvectorsshouldbeusedforgainingaccuracy,thusmakingtheprocessatimecomplexone.Ourworkusesaprobabilisticmodel,thatwouldgivehigheraccuracyinasmallerruntime.Assaidearlier,themodeluseshigherordertoaccountforglitcheswe5

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wouldneedtohaveanteriortransitionandpresenttransitionasopposedtoanteriorandpresentstates,Hencethemodelwouldhavetheknowledgeofthegatestatesattimeinstants,t2t1t,wheretrepresentsthepresentstate,andallothertimeinstantstillitreachesastabledesiredvalue.Theprobabilisticmodelexplicitlycapturesthetemporalcorrelationbetweenallthedifferenttimeinstants,thisisdonebydrawingedges,fromtheappropriatetimeinstancesofparentnodes,tothetimeinstanceofchildnode,itwouldaffect.Theswitchingprobabilitiesofanodewoulddenotetheprobabilityofswitching,fromit'sstatecorrespondingtopreviousinputtransitionprobabilitytopresentinputtransitionprobability,thisswitchingprobabilityofthechildnodeisofnoconsequenceinthismodel,wewouldhavetocomparetheswitchingbetweensuccessivetimeinstancesofthechildnodetogettheswitchingprobabilitybetweenthesetimeinstances.Let'ssayanodeAattimeinstant2switchesfrom00t1tandthesamenodeAattimeinstant3switchesfrom01t1t,weshouldcomparethestatesattimeinstanttbetweentheinstances2and3ofnodeA,inthiscasethenodehasswitchedfrom01,thuswewouldhavetocalculatethesumofswitchingprobabilitybetweenallthesuccessivetimeinstancesofnodeAtoobtainthenalswitchingprobabilityofnodeA.Switchingprobabilitythuscalculated,capturestheeffect,thepreviousinputleavesbehindintheformofspurioustransitions.Theswitchingmodeldepictedinthisworkisnovelinthat,itaccountsforgatedelay.Theswitchingmodelapartfromaccountingforgatedelay,canbeusedtoidentifythenodeswithhighprobabilityofleakage,byusingthemethodobservedintheworkofRamani[1].Someoftheearlierworksinthisrelatedeld,havefailedtoaccountfordependenciesinprimaryinput,thiscouldbetakencareofinthismodelbyincorporatingthevectorlessapproach,suggestedbyRamalingam[2].Hencebyincorporatingtheabovefeatures,thismodelapartfrombeingaccurate,wouldbecomprehensive,asitcouldidentifynodeswithhighprobabilityofleakageandcross-talknoisealongwiththeswitchingactivity.1.5FlowofthisThesisTheliteraturereviewispresentedinChapter2.ProbabilisticModelisdiscussedinChapter3.Prob-abilisticmodelunderGateDelayconditionsisdiscussedinChapter4.BayesianInferenceTechniquesis6

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presentedinChapter5.TheResultsandfutureworkthatcouldbecarriedonwiththismodelisdiscussedinChapter6.7

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CHAPTER2PRIORWORKShrinkingDevicestructureshavecontinuedtocauseanincreaseinthedevicedensityandhenceanincreasedpowerdensity.Powerestimationhasbeenperformedateverystageindesignprocess,toaiddesignersinoptimizingthedesigntoreducePowerconsumption.Despiteshrinkinggatelengthsandvariouschangesintechnology,DynamicpowerdissipationcontinuestobeoneofthemajorissuesinPowerestimation.SwitchingactivityisanimportantparameterinthecalculationofDynamicpowerestimation.Anovelterm'transitiondensity'wasusedbyNajminhiswork[18],toestimateaverageswitchingrate.Denition:[18]Thetransitiondensityofalogicsignalxt,tn,isdenedasDxlimTrnxT T(2.1)wherenxTrepresentsthenumberoftransitionsinxtinthetimeintervalnT2T2.Theexpressionforcalculatingaveragepowerwouldbe,Pav1 2CV2ddlimTrnxT T(2.2)Transitiondensitiesaregivenattheprimaryinputsandarepropagagedintothecircuittogetthetransitiondensitiesattheoutputandinternalnodes.Thismethodassumestheinputsareindependentandhencefailstoaccountforthecorrelationsintheinputnodes.Bhanjaetal[4]introducedBayesianNetworksforswitchingestimationincombinationalcircuits.Intheirworkswitchingactivitywasrepre-sentedasSwXPX0r1PX1r0(2.3)8

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whereX0r1andX1r0denotesasignaltransitionfrom0to1and1to0atnodeXrespectively.ThesignalprobabilityofanodePX1istheaveragefractionofclockcyclethatthenodeXremainsatlogic1.Switchingactivityisaffectedbythefollowingtypesofcorrelations,1.TemporalCorrelationrepresentsthecorrelationbetweenthepreviousvalueofasignalandpresentvalueofasignal2.SpatialCorrelationrepresentsthecorrelationcausedbyre-convergentfanoutsanddependenciesininput.3.Spatio-TemporalCorrelation,acombinationofspatialandtemporalcorre-lation,representsthedependenceofasignaltothepreviousvalueofaspatiallyconnectedsignal.4.Sequentialrepresentsthespatialcorrelationduetofeedbackstatelines.Powerestimationisperformedatvariouslevelsofdesignprocess.Behaviourallevelpowerestima-tion[76,77,78,80,81]isahighlevelofpowerdesign,thisistheearlieststageindesignprocess,designchangesatthisstageareexibleandsavesdesigntime.RegisterTransferLevel(RTL)[82,83,84,85,86]powerestimationisdonewhenthecircuitisstillexpressedasblocks,witheachblockhavingit'sowncombinationalcircuit.Highlevelpowerestimationthoughexibleandsavesdesigntime,isinaccurate.GatelevelPowerestimationthoughcostlyintermsofdesigntimeprovidesgreateraccuracy.InthisworkweestimatepoweratGatelevel.Simulation[45,71,53,60]methodsarethesimplestmethodtoestimatepower,thesemethodswhilebeingaccurate,consumealotoftime.Thesimulationmethodsarepatterndependent.Inthedesignprocess,thedesignersdonothaveknowledgeofthenatureofinputs,thishampersaccuracy.Statisticalsimulationmethodsaresimilartosimulationmodels,theydifferinthat,theinputvectorsareselectedbasedonuserdenedprobabilities,alsotheaccuracyandcondencecanbespeciedbytheuser.[25,70,47,72,73,74,75]havesuggestedsomemethodsusingStatisticalsimulation.WhilethesemethodsprovideanimprovementintimeoverSimulationmethods,theiraccuracydependsontheirknowledgeofinputs.Probabilisticsimulationinvolvetheconstructionofspecialcircuitmodelsbasedontheprobabilisticmethodused.Thecomputationtimeisfastandtractable.Theprobabilisticmethodsprovideacompactwayofrepresentingthelogicsignals,theseprobabilityvaluesarepropagatedthroughthemodel,basedontheinuenceofthesevaluesonthegates,poweriscalculated.Duetotheprobabilisticnatureofthe9

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inputs,changesininputpatterncanbeeasilymodelled.Thedependenciesinthecircuithastobetakencareofinprobabilisticpropagationtogetanaccuratemodel.ParkerandMcCluskey[21]presentedoneoftheearliestmethodsthatusedsignalprobabilityforprobabilisticpropagation.Theestimateobtainedwasgrosslyinaccurateasthetemporalandspatialcorrelationwereneglected.BinaryDecisionDiagrams(BDD)forsignalproabilitywasrstproposedbyChakravartietal[16],Inthismethod,thesignalprobabilityattheoutputiscalculatedbybuilidingOBDDcorrespondingtothefunctionofthenodeintermsofcircuitinputs,andthenperformingapostordertraversaloftheOBDDusingequationPYPxPfxPxPfx(2.4)Ercolanietal[19]describeaprocedureforpropagatingsignalprobabilitiesfromcircuitinputto-wardsoutputusingonlypairwisecorrelationbetweencircuitlines.Thismethodignoreshigherordercorrelations.Theabovemethodsignoredtotakeintoaccountthegatedelay,ignoringthepowerdissipationduetohazardsandglitches.Ghoshetal[31]estimateaveragepowerdissipatedincombinationalandsequentialcircuits,usingageneraldelayformula.Theyusesymbolicsimulationtoproduceasetofbooleanfunctionthatrepresenttheconditionforswitchingatdifferenttimepointsforeachgate.From,theinputswitchingrate,proba-bilityofeachgateswitchingatanytimepointiscalculated.Thesumofswitchingactivityintheentirecircuitoverallthetimepointsforallthegatescorrespondingtoaclockcycleiscalculated.Themajordisadvantageofthismethodisthatformediumtolargecircuitsthesymbolicformulaebecometoolargetobuild.Burchetal[25]introducedtheconceptofprobabilitywaveform.Thewaveformconsistsoftran-sitionedgesoreventsovertimefromtheinitalsteadystatetime,0tonalsteadystatetime,,whereeacheventisannotatedwithanoccurenceprobability.Theprobabilitywaveformrepresentsallpossiblelogicalwaveformsofthatnode.Giventhesewaveforms,switchingactivityofx,whichincludeshazards10

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andglitchesiscalculated,Exswieventlistxpxt0r1pxt1r0(2.5)Najmetal[24]propagatestransitionwaveformsfromcircuitinputsthroughoutthecircuitandestimatestotalpowerconsumption.Thismethoddoesnottakeintospatialcorrelationsduetoreconvergence.Tsuietal[23]proposedataggedprobabilisticsimulationapproach,inthismethodlogicwaveformsatanodearebrokenintofourgroups,eachgroupbeingcharacterizedbyit'ssteadystatevalues.Eachgroupisthencombinedintoaprobabilitywaveformwithappropriatesteady-statetag.Giventhetaggedprobabilitywaveformsattheinputofnoden,itispossibletocomputetaggedprobabilitywaveformsattheoutput.Thecorrelationbetweenprobabilitywaveformsatinputsisapproximatedbycorrelationbetweenthesteadystatevaluesoftheselines,whichiscalculatedbydescribingthenodefunctionintermsofsomesetofintermediatevariablesinthecircuit.Thismethoddoesnottakeintoaccounttheslewinthewaveforms,thehigherorderspatialcorrelationsarealsonotmodelledinthismethod.Najmetal[17]introducedtheconceptoftransitiondensity,D,thetransitiondensitiesarepropagatedthroughoutthecircuit.Transitiondensityofeachnodeiscalculatedasfollows:Dyni1Py xiDxi(2.6)Thedisadvantageofthismethodisthatitassumesindependenceintheinputfunction.TheworkofBhanjaetal[4]whilecapturinghigherorderspatialandrstordertemporalcorrelationgivesanaccurateswitchingestimationforzero-delaymodels,butfailstoaccountfordelayoflogicgates,thusfailingtoaccountforaccurateswitchingestimationofthecombinationalcircuits.InthisworkweproposeaProbabilisticmethodusingBayesianNetworks,toaccuratelymodelanyorderofTemporalandSpatio-Temporalcorrelation,alongwithhigherorderSpatialcorrelation,intheprocessweinfusedynamictendenciesintheBayesianNetwork.Thisenablesthemodeltocaptureallde-siredandundesiredtransitionsthataGateouptutexpereiences.Thus,enablinganaccurateProbabilisticswitchingestimationmodelunderRealDelayconditions.11

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CHAPTER3BAYESIANNETWORKSGraphicalstructureshavebeenusedinthisworktoestimatetheswitchingprobabilitiesofVLSIcircuits.Graphicalstructuresfacilitatesanintuitiveunderstandingofthedependenciesamongthenodesinamodel,asagainstthetraditionalmethodinwhichwewouldhavetoverifyifPxyPxPy,toascertainthedependencybetweenthevariablesxandy.TheessenceofGraphicalrepresentations,istorepresentprobabilisticinferenceinthecontextofhumanreasoningandtoavoidcomplexnumericalcalculations.Graphicalrepresentations,preserveandhighlight,thedependenciesinthecircuitimpervioustochangesinthenumericalinput.Inourcase,werepresentcombinationalcircuitsasgraphs,itiscomparitivelyaneasierjob,sincewehaveasenseofanodesdirectandindirectneighbours.InthisworkwehaverepresentedthecircuitsasBayesianNetworksandwewoulddiscussindetailaboutthesenetworksinthischapter.ThischapterfollowstheowofPEARL's[40],withprobabilisticmodelsrepresentingcombinationalcircuits,tolucidlyexplainthenatureofBayesianNetworksandtoprovethatourmodelworks.3.1BayesianNetworksBayesiannetworksareDirectedAcyclicGraph(DAG)structuresi.e.,thenodesareconnectedbydirectedarcsandthereisnoclosedconnectionwhiletraversingthedirectedarrows(acyclic).Theim-portanceofDirectednetworksasopposedtoundirectednetworkscanbeexplainedbya2-inputORgate.LetAandBbetheinputsatthisORgateandletCbetheoutput.Theundirectedgraphandthedirectedgraphisshownin3.1..FromtheundirectedgraphweinferthatAandBareindependentgivenC.ButinrealityifthevalueofCisknown,say1,learningthevalueofAsay0,wouldaffectthevalueofB.IfweaddalinkbetweenAandB,itwouldindicatethatAandBarenolongerindependentofeachother.TheBayesianNetworkofthelogicgate,denotesthatAandBareindependentifthevalueofCisnot12

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ABCA BC A 2-Input OR Gate Directed GraphUndirected GraphBC Figure3.1.2-InputORGate.known.OnceCisinstantiated,AandBarenolongerindependent.Thisinduceddependencyiscaptured,intheBayesianNetworkshownin3.1.,byvirtueofdirectedconnectivity,calleddseparation.Directedgraphrepresentationsmakeiteasytoquantifythelinkswithlocal,conceptuallymeaningfulparametersthatturnthenetworkasawholeintogloballyconsistentknowledgebase.Itwouldbeonlyappropriatetostress,theimportanceofindependencies,sincethewholenetworkhasbeenweavedaroundthisconcept.Anodewouldactonlybasedonthebeliefsofit'sneighbours,andwoulddiscounttheeffectofallothernodesitisindependentfrom.Thisisthecoreconceptthatmakesthenetworkappealing.Hence,carehastobetakentocapturealltheindependenciesinthenetwork.Henceforth,inthischapter,wewouldprovethatourmodelsatisesallthetheoremsanddenitionsquantifyingaBayesiannetwork.Markovnetworks,duetolackofdirectionalitystatesthat,iftheremovalofsomesubsetZofnodesfromthenetworksrendernodesXandYdisconnected,thenXandYareindependent.Thiswasexplainedabovewiththe2-inputORgateexample.Thefollowingdenitionfrom[40],givesthecriterionforseperationinaDAG.13

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I3I2 I5 G3G2G1I4I1 G6 G5 G4 Figure3.2.C17BenchmarkCircuit.DEFINITION:1IfXY,andZarethreedisjointsubsetsofnodesinaDAGD,thenZissaidtod-separateXfromY,denotedXZYd,ifalongeverypathbetweenanodeinXandanodeinYthereisanodewsatisfyingoneofthefollowingtwoconditions:1.whasconvergingarrowsandnoneofworitsdescendantsareinZ.If,onthecontrary,woritsdescendentsisinZ,thepathissaidtobeactivatedbyZ.2.wdoesnothaveconvergingarrowsandwisinZ.Inthiscase,thepathissaidtobeblockedbyZ.Thecombinationalbenchmarkcircuit,C17isgivenin,3.2..Thebayesiannetworkofabenchmarkcircuit,C17,3.3.hasbeenusedheretoexplainthisseperationcriterion.Inthebayesianmodelofcombinationalcircuit,C17representedabove,nodesrepresentthelogicgatesandtheprimaryinputsrepresenttheprimaryinputsofthecircuit,thedependencybetweenthenodesarerepresentedbydirectedarrows.Considernodes,G2G3andG6.G3liesinthepathofG2andG6,andG3doesnothaveanyconvergingarrows,thereforeG3accordingtotheabovedenitionblocksthepathfromG2toG6,andtherefored-separatesG2andG6.Thistypeofseparationissimilartothecutsetseparationinundirectedgraphs.ConsiderthenodesG1G5andG3.Thesenodesifdenotedbyundirectedgraphs,wouldleadustobelievethat,G1andG3areindependentgivenG5,butthisisanincorrectassumption,inreality,G5denotestheoutputofanandgate,giventhevalueofG5,knowingthestateofG1wouldaffectthestateofG3.Asintheabovebayesianmodel,whendirectionalityandacleard-separtiondenitionistaken14

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I1I2I3I4I5G6G1G2G4G3G5 Figure3.3.BayesianNetworkofC17BenchmarkCircuit.intoconsideration,weunderstandthepathfromG1toG3isactivatedbyG5,andthatlearningthestateofG5,makesG1andG3dependentoneachother.Themathematicalequivalentofindependencycriterionisgivenbythefollowingdenition.DEFINITION:2LetUABbeanitesetofvariableswithdiscretevalues.LetPbeajointprobabilityfunctionoverthevariablesinU,andletXYandZstandforanythreesubsetsofvariablesinU.XandYaresaidtobeconditionallyindependentgivenZifPXYZPXZwheneverPYZ0(3.1)andthisindependencyisdenotedbyIXZYmeaning,GiventhestateofZ,XandYarerenderedindependent.TheindependencyofthenodeG6fromG2givenG3,intheabovegure,couldbeveriedbytheequation,PG6G2G3PG6G3wheneverPG2G3 0(3.2)15

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TheindependencyofnodesisdenotedbyIG2G3G6TheNumericalrepresentationofprobabilitiesmaynotshedlightoninduceddependencies,Axiom-atizationofprobabilitiesallowsustointuitivelyconjurenewtheorems,thatwouldbetterhelpustomodeldependencies.InaProbabilisticmodel,anindependencerelationshipIXZYmustsatisfythefollowingaxiomsaccordingtoPEARL[40],SYMMETRYIXZY"!$#IYZX(3.3)ThesymmetryaxiomstatesthatforanygivenstateofZ,iflearningthestateofXtellsusnothingnewaboutY,thenlearningthestateofYwouldtellusnothingnewaboutXDECOMPOSITIONIXZYUWIXZY&IXZW(3.4)DecompositionaxiomstatesthatgiventhestateofZ,iflearningthestateofboth,YandWtellsusnothingnewaboutX,thenlearningYandWseparatelywouldalsotellusnothingnewaboutXWEAKUNIONIXZYUWIXZUWY(3.5)WeakunionaxiomstatesthatlearningirrelevantinformationWcannothelpirrelevantinformationYbecomerelevanttoX.CONTRACTIONIXZY&IXZUYWIXZYUW(3.6)ContractionaxiomstatesthatifwejudgeWirrelevanttoXafterlearningsomeirrelevantinformationY,thenWmusthavebeenirrelevantbeforewelearnedY.ADependencemodel,Mofadomainshouldcapturealltheconditionalindependencies,IXZYamongstthevariablesinthatdomain.Denition:3AnundirectedgraphGisadependencymaporDmapofMifthereisaone-to-onecorrespondencebetweentheelementsofUandthenodesVofG,suchthatforalldisjointsubsetsXYZ16

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ofelementswehaveIXZYm%XZYg(3.7)Similarly,GisanindependencymaporImapofMifIXZYmXZYg(3.8)GissaidtobeaperfectmapofMifitisbothaDmapandanImap.DEFINITION:4ADAGDissaidtobeanImapofadependencymodelMifeveryd-separationconditiondisplayedinDcorrespondstoavalidconditionalindependencerelationshipinMi.e.,ifforeverythreedisjointsetsofverticesXYandZwehaveXZYIXZYm(3.9)ADAGisaminimalI-mapofMifnoneofitsarrowscanbedeletedwithoutdestroyingitsI-mapness.ThegurerepresentsaminimalI-map,sinceitrepresentsalltheconditionalindependenciesanddestroyinganyedgewoulddestroyitsI-mapness.DEFINITION:5GivenaprobabilitydistributionPonasetofvariablesU,aDAGD=UEiscalledaBayesiannetworkofPifDisaminimalI-mapofP.SinceourgureisaminimalI-mapitisindeedaBayesianNetwork.DEFINITION:6AMarkovblanketBL1(A)ofanelementAbelongstoUisanysubsetSofelementsforwhichIASUSAandAS(3.10)AsetiscalledaMarkovboundaryofA,denotedB1(A),ifitisaminimalMarkovblanketofA,i.e.,noneofitspropersubsetssatisfytheaboveequation.THEOREM:1EveryelementAbelongstoUinadependencymodelsatisfyingsymmetry,decompo-sition,intersection,andweakunionhasauniqueMarkovboundaryB1(A).Moreover,B1(A)coincideswiththesetofverticesBg0(A)adjacenttoAintheminimalI-mapG0.17

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DEFINITION:7LetMbeadependencymodeldenedonasetUX1X2&XNofelements,andletdbeanorderingX1X2'Xi(oftheelementsofU.TheboundarystrataofMrelativetodisanorderedsetofsubsetsofU,B1B2&Bi(,suchthateachBiisaMarkovboundaryofXiwithrespecttothesetUiX1X2'Xi1,i.e.,BiisaminimalsetsatisfyingBi)UiandIXiBiUiBi.TheDAGcreatedbydesignatingeachBiasparentsofvertexXiiscalledaboundaryDAGofMrelativetod.TheboundaryDAGforthegure[3.3.],representingaC17combinationalcircuit[3.2.]isdonesuchthatanodesmarkovboundaries(inourcaseanode'sparents)arecreatedbeforethecreationofnode,thisisunderstoodintuitively,asitreectsthestructureofthecombinationalcircuit.Theboundarystrata,BMof3.3.isgivenasBM+*fffffI1I2I2I4I5X2X2I3X3X4X1X3,(3.11)THEOREM:2[Verma1986]:LetMbeanysemi-graphoid.IfDisaboundaryDAGofMrelativetoanyorderingd,thenDisaminimalI-mapofM.ThedenitionsandtheoremsgivenabovedenethestructureofBayesianNetwork.Itshouldbestatedherethat,theaboveDenitions,AxiomsandTheorems,religiouslyfollowPEARL,andcommentshavebeenaddedtoexplaintheminourcontext.BayesianNetworkswereintroducedforswitchingactivityestimationincombinationalcircuitsintheworkofBhanjaetal.TheyproposedaLIDAG,LogicInducedDirectedAcyclicGraph,toreectthecombinationalcircuitsasZero-delaymodel.DenitionandTheoremfromBhanjaetal.wouldbeusedtoprovethatthe3.3.isaLIDAG.Thenodes*G1G2G6,inthe3.3.representgateoutputs,andthenodes*I1I2I5,repre-senttheprimaryinputsofthegate.Theoutputofagatedependsontheinputstothegate,whichmaybeprimaryinputsoroutputsfromothergates,andthenatureofit'sconditionalprobabilitytable.Theswitchingoftheinputnodesandthegateoutputnodesaretherandomvariablesofinterest,*I1I2-I5G1G6,.Eachnodetakesfourpossiblestates*x00x01x10x11,representingtheprobabilityoftheswitchingtran-sitions:00,01,10and11.18

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DEFINITION:8InaLIDAGthenodesrepresenttheswitchingatgateoutputlines,thearcsrepresenttheinputlinesonwhichthenodedepends,thearcsmightoriginatefromprimaryinputnodesorothernodesrepresentinggateoutputline.DAGshownin3.3.correspondstotheC17benchmarkcircuit.THEOREM:3TheLIDAGstructure,correspondingtocombinationalcircuitisaminimalI-mapoftheunderlyingswitchingdependencymodelandhenceisaBayesianNetwork.Proof:FromthedenitionofLIDAGabove,itisunderstoodthatintheLIDAGstructuretheparentsofeachnodeareitsMarkovboundaryelements,TheLIDAGisaboundaryDAGofM,whichcouldbeveriedbyEQUATION,hencefromTHEOREM9itfollowsthatDisaminimalI-mapofM.3.2FunctionalityofLIDAGBNLetustrytoconstructaLIDAGBNforthebenchmarkC17circuit,Eachnodeisassignedfourpossiblestates,x00x01x10x11.xa.b,denotestheprobabilityoftransitionfrompreviousvalueaattime,t1topresentvaluebattime,t.ab*01,.Bydeningthestatesasswitchingprobabilities,wehaveknowledgeofthenode'spreviousandpresentvalue,thuscapturingrstordertemporaldependencies.Thehigherorderspatialcorrelationsarecapturedbythearcsbetweenthedependentnodesandthenatureofdependencyisquantiedbytheconditionalprobabilitytable.TheDAGnatureofthemodelhelpsusinintuitiveidenticationofconditionalindependencies.This,facilitatesthecomputationofthejointprobabilityfunction,whichcanbeexpressedasaproductofconditionalprobabilities.Px1&xN'vPxvxparentv(3.12)ThejointprobabilitydistributionfunctionfortheLIDAGrepresentedingure[3.3.]isexpressedbythefollowingfactoredform.Px1'x6Px6x4x3Px5x1x3Px4I5x2Px3x2I3Px2I2I4Px1I1I2PI1PI2PI3PI4PI5(3.13)19

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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 Theconditionalprobabilitytablereectsthenatureofthegate.AcompleteconditionalprobabilitytableforPx6x4x3isgivenby43entries,sinceeachvariablehas4states.ConditionalpropabilitytablesofaNandislistedinTable[3.1.].InthischapterwehavediscussedthenatureandformationofBayesianNetworksinthenextchapterwewoulddiscusstheconstructionofRealdelaygraphicalprobabilisticmodel.20

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CHAPTER4REALDELAYPROBABILISTICMODELPowerDissipationhasbeenamajorconcernforVLSIdesignerforoveradecade.Dynamicpowerdissipationhasbeenandcontinuestobeoneofthemajorsourcesofpowerdissipation.Thecharginganddischargingofloadcapacitances,duetoswitching,isoneofthemainfactorsofDynamicpowerdissipation.Thedifferenceinarrivaltimesofsignalsatagateinput,leadstospurioustransitions,alsocalledasglitches.Glitchgenerationshownin[Figure4.1.]isduetosignalstravellingthroughgateswithdifferentdelays,beforearrivingattheinputs.Thesespurioustransitionsalsoplayamajorroleinpowerdissipation.PowerestimationusingZero-delaymodelsdonottakethesespurioustransitionsintoconsiderations,thusgrosslyunderestimatingthepowerdissipationforcircuitswithunbalancedpathlengths. d=1t=1t=0 t=0t=0t=1t=2steady 0SPURIOUS TRANSITION AT A NODE [K. ROY]d=1d=1 Figure4.1.GlitchGeneration.21

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4.1TypesofDelayModel4.1.1ZeroDelayModelZeroDelayModelassumesthatthegateshavenodelay,henceaninstantaneoustransmissionofsignalsthroughgatesisassumed.Thisassumptionneglectstheswitchingactivityduetospurioustran-sitions,henceitisinaccurate.Thistypeofmodelcanberepresentedprobabilisticallybytakingintoaccountrstordertemporalandspatio-temporaldependencies.4.1.2UnitDelayModelUnitDelayModelassumesaunitdelayforallit'sgatesi.e.,irrespectiveofit'stypeandloadit'sdrivingallthegatesareassumedtohavethesamedelay.Thismodelwhilebetterthanzerodelaymodelisalsoinaccurate.Thismodelcanberepresentedprobabilisticallybytakingintoaccounthigherordertemporalandspatio-temporaldependencies.4.1.3VariableDelayModelVariableDelayModelareofvarioustypes,thistypeofmodel,assignsdifferentdelaystogatesbasedontheirlogictype,fanout,orthefaninofthegatestheydrive.Inourworkafanoutdependentdelayhasbeenassignedtogates,itisknownthatsignalsfallandriseslowly,ifthenumberoffanoutincreases,owingtoanincreasedloadcapacitance.In[[100]]Gatedelay,dduetopropagationhasbeenrepresentedasdfp(4.1)whereprepresentsparasiticdelay,i.e.,delayofagatewithnoloadcapacitance,frepresentseffortdelay,delayrepresentingthesizeandfanoutofthegate.Effortdelay,fcanberepresentedasfgh(4.2)22

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hereg,logicaleffort,representsthecomplexityofthegate,forexamplethelogicaleffortofaninverterissaidtobe1,whilethelogicaleffortofa2-inputNORgateissaidtobe5/3.fanouteffort,h,isrepresentedbyhCout Cin(4.3)whereCoutrepresentstheexternalloadbeingdrivenandCinrepresentstheinputcapacitanceofthegate.Inthiswork,weconsiderthefanouteffort,fordelayassignmenttoindividualgates.Weneglecttheinputcapacitanceandapproximatetheloadcaapacitanceofthegateasthenumberoffanoutagatehas,thisapproximationisjustiedsincetheloadcapacitanceincreaseslinearlyasthefanoutincreases.4.2GateDelayModelIthasbeenproventhatlargefanoutscausethesignalstofallandriseslowly,thisisbecausegateswithlargefanoutswouldhavetochargeanddischargethroughlargecapacitances.Inthisworkwehaveassignedvariablefanoutdelaytothegates.S.Manichetal[9]haveusedthefanoutdelaymodelintheirworktondthepairofvectorsunderwhichmaximumweightedswitchingactivityoccurs.Considerthe[4.2.],gateshavebeenassigneddelaybasedontheirfanouts.GateX4,hasbeenassignedadelayof2units,GateX5andX6havebeenassignedadelayof1unit.Fromtheassigneddelays,Signalarrivaltimesatinputsofgatehavebeencalculated.Basedonthesignalarrivaltimeandthedelayofgate,timeinstantsatwhichapossiblesignaltransitioncanoccurhavebeencalculatedandassociatedwiththegates.Theinputsignalshavebeenassumedtobetheoutputofaregister.Thisassumptionhasenabledustotakeintoaccountjustthepreviousinputvectorandpresentinputvectorforaccountingundesiredswitchingactivityduetoglitches.Thepreviousinputvectorhasbeenassumedtohavetakenplaceattime0t,andthepresentinputvectorhasbeenassumedtohavetakenplaceattime0t.In[4.2.],theinputvectors*X1,X2,X3,,havebeenassignedtwotimeinstants*-0t,0t,.Gate*X4,,hasbeenassignedtwotimeinstants*-0t,2t,.*X4(-0t),isaffectedbyinputs*X1(-0t),X2(-0t),,ithastobenotedherethat*X4(-0t),,representssteadystateoutputat*X4,,duetopreviousinputvectors.*X4(2t),isaffectedbyinputs*X1(0t),X2(0t),.Sinceonlytwotimeinstantsareassociated23

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X4X2X1(-0t, 0t)(-0t, 0t)(-0t, 2t) (-0t, 1t, 3t)(-0t, 0t)X3(-0t, 2t, 3t, 4t)21X5X61 Figure4.2.VariableDelay.with*X4,,onlyonetransitioncouldoccuratthisnode,duetotransitionfrompreviousinputvectortopresentinputvectorandthetransitionisrepresentedby*X4n0t/X42t',.Gate*X5,,hasbeenassignedthreetimeinstants*0t1t3t,.*X5n0t',isaffectedbyinputs*X4n0tX3n0t',.*X51t',isaffectedbyinputs*X4n0tX30t',.*X53t',isaffectedbyinputs*X42tX30t',.Threetimeinstantsareassociatedwiththenode*X5,.Thetwopossibletransitions,associatedwiththisnodecanberepresentedby*X5n0t/X51t',0*X51tX53t',.Gate*X6,,hasbeenassignedfourtimeinstants*0t2t3t4t,.*X6n0t',isaffectedbyinputs*X4n0tX5n0t',.*X62t',isaffectedbyinputs*X4n0tX51t',.*X63t',isaffectedbyinputs*X42tX51t',.*X64t',isaffectedbyinputs*X42tX53t',.Threepossibletransitionsassociatedwiththisnodecanberepresentedby*X6n0t/X62t',0*X62t1X63t',*X63t2X64t',.Ifthecircuithadbeenrepresentedasazero-delaymodel,thenwewouldhavecapturedatmost1transitioninthegates,andwithunit-delaymodel,wewouldhavecapturedalimitednumberoftransitionsinthegates,whileinVariabledelaymodel,wecouldcaptureallpossiblespurioustransitionsandalsopropagatethesespurioustransitionstothesuccessivenodes.4.3ExpandedCircuitRepresentationTheCircuitin[4.2.]hasbeenexpandedbycreatinginstancesofthegate,forallpossibletimeinstantsatwhichadesiredoranundesired,eventcouldoccurattheouputofthegate.ThegateX4wouldhave2instances,(X4,X4 1),correspondingtotimeinstantsn0t2t,associatedwiththisgate.Gate24

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1t 4t 3t 2t -0t X4X5X4_2X5_1X6_3X6_4X6_2X6X3X2X1X5_3X3_PX1_PX2_P Figure4.3.ExpandedCircuit.X5wouldhave3instances,(X5,X5 1,X5 3),correspondingtotimeinstantsn0t1t3tgateX6wouldhave4instancess,(X6,X6 2,X6 3,X6 4),correspondingtotimeinstantsn0t2t3t4t.Theexpandedgure[4.3.]hasbeencreatedbyforming5slots,correspondingtotimeinstants,n0t1t2t3t4t.Thevariousinstancesofgatesareplacedintheslotscorrespondingtotheirassignedtimeinstant.Thegateshavebeenconnectedtotheirappropriatetimeinstancesofinputsignal.ForexamplegateX5 3,wouldhaveatit'sinput,linesfromgatesX3 1andX4 2.Thusaconnectionisformedbetweenallnodes.25

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4.4RealDelayProbabilisticalModelInthelastchapter,wehadseentheconstructionofaZero-DelayBayesianmodel.IthastobenotedherethattheZero-DelayBayesianmodelcaptureshigherorderSpatialCorrelations,butitcapturesonlyrstordertemporalandspatiotemporalcorrelation,zero-delaymodelwouldcaptureonly1transitionatanodeforagivenclockcycleandthiswouldbeachievedaccurately,ifwehavetheknowlegeofnode'spreviousandpresentstate.Inthiswork,wearerepresentingaReal-DelaymodelofthecircuitasBayesianNetwork,thereforewewouldhavetocaptureallthetransitionsthatanodewouldexperience.Wesuccessfullydothatbymodelingexplicitlythehigherordertemporalcorrelationsandhigherorderspatiotemporalcorrelations.TheReal-DelayDAGstructureismodelledontheexpandedcircuit,[4.4.].ThenodesintheDAGrepresentthegatesatdifferenttimeinstances,thearcsrepresentthedependency.IthastobeemphasizedherethattheReal-DelayDAGmodeldiffersfromtheZero-DelayLIDAG-BN,inthattheactivityofanodeiscapturedatvarioustimeinstancesapossibleeventcouldoccur,owingtodifferentinputsignalarrivaltimes.InthisProbabilisticmodel,allthenodesareassumedtohave4statesrepresenting*00301310311',.ThestatesrepresentthetransitionofnodestateX,Xt1Xt,X*01,,fromprevioustimeinstant,t1topresenttimeinstant,t.Probabilityvaluesareassignedtothesestates,basedontheconditionalprobabilitytableassignedtothenodes,andtheprobabilityvaluesoftheparentsignals.Theconditionalprobabilitytablereectsthenatureofthegate.TheconditionalprobabilitytablesinthisRealDelayProbabilisticalModel,areidenticaltothoseoftheLIDAG-BN,discussedinthelastchapter.TheconstructionoftheRealDelayDAG,andthevariousdependenciesmodelledarediscussedbelow,Thegure[4.4.]isaDAGstructureforaRealDelaymodel.4.4.1DependenciesinPrimaryInputsUnliketheLIDAG-BNmodel,wherewehaveassignedonesetofinputtransitionprobabilitiesforalltheprimaryinputs,inthismodelwehaveassignedtwosetsofinputtransitionprobabilityforpri-maryinputs,onerepresentingtheanteriortransitionprobabilitiesandtheotherrepresentingthepresenttransitionprobabilities.Let'ssayt,representsthetimeofpresentstate,t-1,timeofthepreviousstateandt-2,representthetimeofstatebeforethepreviousstate.In[4.4.],theanteriorinputtran-26

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X4_2X5_1X6X5X1X3 X5_3X1_PX2_PX3_PX6_02 X6_23X6_34X5_13X5_01X4_02X6_4X6_3X6_3X6_2X6_2X6X5_3X5_1X5X5_1X4_2X4X4X6_4X6_3X6_2 X2 Figure4.4.DAGofAReal-DelayModel.27

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Table4.1.ConditionalProbabilitySpecicationsforPrimaryInputDependency. TemporalDependencyofPrimaryInput PX1 PX1 forX1 P X1 *x00x01x10x11, = 0.50.500 x00 000.50.5 x01 0.50.500 x10 000.50.5 x11 sitionprobabilityofprimaryinputvectors*X1,X2,X3,,representstheswitchingprobability,fromtimeinstance,*t24t1',andthepresentinputtransitionprobabilityofinputvectors*X1 P,X2 P,X3 P,,representstheswitchingprobability,fromtimeinstance*t1"5t',.Itshouldbenotedherethatweareexplicitlymodelingthetemporalcorrelationbetweenthetwoswitchingtransitions,bymakingthepresenttransitiondependontheprevioustransition.Sincewearespecifyingtheinputvectorstatesoverthreetimeinstants,*t2't1't',,theinputvectorscantake8statesnamely*000'001'010'100'101110111.Thedependencyofthenode,X1 P,representingaprimaryinputnodewithPresentinputtransitionprob-abilities,onX1,representingtheprimaryinputnodewithPreviousinputtransitionprobabilities,isgivenbythefollowingconditionalproababilitytable.4.4.2HigherOrderSpatio-TemporalDependenciesInourReal-DelayDAG[4.4.],let'sconsiderthenodeX6 3,it'sstates,symbolizetheswitchingprobabilitiesofgate,X6,from*t13i,6*t3i,.Thevariablei,canbedenedasaunitfan-outdependentdelay,suchthat,ifN,isthemaximumtimeinstantofacircuit,then*t,7*tNi,7*t1,,heretandt+1,wouldrepresentthetimeinstancesatwhichpresentprimaryinputvectorandnextprimaryinputvectorarrivefromaregister.Theparentsofthenode,X6 3arethenodesX4 2,andX5 1.ThestatesofnodeX4 2,symbolizetheswitchingprobabilitiesofgate,X4,from*t12i,*t2i,.ThestatesofnodeX5 1,symbolizetheswitchingprobabilitiesofgate,X5,from*t11i,8*t1i,.Theswitchingprobabilityofnode,X6 3,giventheknowledgeofnodes,X4 2andX5 1isrepresentedas,PX6 3X4 2X5 1,the28

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conditionalprobabilityquantifyingthisdependenceisthesameasthatforZero-DelayLIDAGBN,givenin[3.1.].ThisdependencydenotesaHigherOrderSpatio-TemporalDependency,Sincethenodes,*X6 3,X4 2,X5 1,,areremovedspatialyandtemporally,byahigherorder.ThegateX6,isdenotedbythevariables,*X6,X6 2,X6 3,X6 4,,atvarioustimeinstants.Theparentsofthesenodesaregivenby*(X4,X5),(X4,X5 1),(X4 2,X5 1),(X4 2,X5 3),.ThustheSwitchingprobabilityofgateX6,isupdatedatvarioustimeintervals.Thisupdation,bringsinaDynamicnaturetotheBayesianNetwork,enablingustoprobabilistically,captureallthedesiredandundesiredtransitions,seenbythegate.TheSwitchingprobabilitiesofanodecanbecalculated,ifwehavetheknowledgeofit'simmediateparents,irrespectiveoftheprobabilitiesatothernodes.Thishasbeenmadepossiblebytheconditionalindependencies,portrayedbytheDAGstructureofBayesianNetworks.4.4.3HigherOrderTemporalDependenciesIn[4.4.],thevariousinstancesofnodeX4are*X4,X4 2,,hereSwitchingprobabilitiesassociatedwithnodeX4,symbolizetheswitchingofthestatesfrom*X4t2X4t1,,nodeX4 2symbolizesswichingprobabilitiesfrom*X4 2t1:92iX4 2t;92i,.TocalculatetheswitchingprobabilityofnodeX4,from*X4t1X4t92i,,achildnodeX4 02iscreated,withX4andX4 2,asit'sparents.Thenode,X4 02capturesthetotalswitchingprobabilityofgate,X4.Thedesiredswitchingprobabilityisthesumrepresentedby*PX4 02t1rt92i01',"*PX4 02t1rt92i10',.TheconditionalprobabilityofthenodeX4 02,isgivenin[4.2.].SimilarlythetotalswitchingprobabilityofgateX5,isthesumofthe01and10,switch-ingprobabilityofnodes,X5 01andX5 13.Theparentsofnodes*X5 01t1rt91iX5 13t91irt93i,,arethenodes*X5X5 1'X5 1X5 3',.ThetotalswitchingprobabilityofgateX6isthesumof01and10,switchingprobabilityofnodes,*X6 02t1rt92iX6 23t92irt93iX6 34t93irt94i,.Theparentsofthesenodesare*X6X6 2'X6 2X6 3'X6 3X6 4',.Let'sconsideranodeX,whosetimeinstantsaredenedinasetZ<*023==N,,ThetotalswitchingprobabilityatanodeXisgivenbytheequation,PXNi1PX i13i(4.4)29

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Table4.2.ConditionalProbabilitySpecicationsforOutputNodes. Temporallyconnectedgate PXoutputXinput1Xinput2 forXoutput Xinput1 Xinput2 *x00x01x10x11, = = 1000 x00 x00 0100 x00 x01 1000 x00 x10 0100 x00 x11 0010 x01 x00 0001 x01 x01 0010 x01 x10 0001 x01 x11 1000 x10 x00 0100 x10 x01 1000 x10 x10 0100 x10 x11 0010 x11 x00 0001 x11 x01 0010 x11 x10 0001 x11 x11 given,ii1ZThedependencythusmodelled,representsaHigherOrderTemporalDependency,betweenvarioustimeinstancesofagate.Theconditionalprobabilityofthenodes*X4 02'X5 01'X5 13'X6 02'X6 23'X6 34',isgivenin[4.2.].TheSwitchingprobabilitythuscalculatedaccuratelycapturestheeffectofRealDelayonaGate,byexplicitlymodellingforthehigherordertemporalandhigherorderSpatiotemporalcorrelations,whichthepreviousworkshavefailedtoaccountfor.Inthischapter,wehaveseen,howtoconstructaReal-Delaymodelanddiscussedit'sprobabilis-ticnature.InthenextchapterwewoulddiscusssomeoftheInferenceTechniquesusedinBayesianNetworks.30

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CHAPTER5INFERENCETECHNIQUESInthePreviouschapters,wehaddiscussedtheGraphicalRepresentationoftheVLSIcircuits.ThemainpurposeofthisGraphicalRepresentationhasbeentofacilitatehumanreasoningandtoreducecomputationalcomplexityoftheprobabilisticmodel.Numericalmodelsofprobabilityhavereliedonthejointprobabilitydensityfunctions,tocomputeconditionalprobabilities,PxixjPxixjPxj.Calculationofthejointprobabilityinvolvecomplexcomputation.whileBayesianNetworksconstructedwithconditionalindependencyasitscoreconceptrepresentsxj,asaframeofknowledgeanddenesxiinthecontextofxji.e.,xixj.Thejointdistributionfunctioniscalculatedbyrewritingtheaboveformula.PxixjPxixjPxj.Theaboverepresentationcalledproductrule,canbegeneralizetoformachainruleformula.[[40]]statesthatifwehaveasetofneventsE1E2===En,thenthejointprobabilityoftheseeventscanbewrittenas,PE1E2===EnPEnEn1====E2E1===PE2E1PE1.InversionFormula:Bayesianinversionformulafollowsdirectlyfromtheconditionalprobabilityformularepresentedabove,pxixjePxjexiPxiPxjeTheaboveformulaexplainsthebeliefupdationofvariablexi,oncexjhasbeenassignedanevidence,'e'.Pxirepresentsthe'prior'probabilityi.e.,probabilityofPxi,beforexjwasassignedanevidence.Pxixje,representsthe'posterior'probabilityi.e.,theupdatedbeliefonceanevidencehasbeenassigned.Pxjexirepresentsthe'likelihood'ofobtainingtheevidencegivenxiandPxjePxjeH.PH0PxjeH.PnH.ThecalculationofPxjeistrivialforsmallernetworks,butasthenetworksizeincreases,it'scomputationbecomescomplex.DifferentInferencealgorithmsexisttocalculatePxje.InthischapterwewilldiscussaboutsomeofthealgorithmsusedforBayesianInference.31

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5.1ExactInference5.1.1PolytreeAlgorithmPearlsuggestedanefcientmessagepropagationinferencealgorithmforpolytrees,thealgorithmwasexact,buthadapolynomialcomplexityinthenumberofnodes.Thismethodcouldworkonlyforsinglyconnectednetworks.Loopcutsetconditioning,waslaterintroducedbypearl,toaccountformultipleconnectnetworks.Thismethodinstantiatesaselectedsubsetofnodes,referredtoasaloopcutsetandchangestheconnectivityofanetworkintoasingleconnectednetwork.Thissingleconnectednetwork,issolvedusingthepolytreealgorithm,hehadsuggestedearlier.Thecomplexityofthismethodgrowsexponentiallywiththesizeofloopcutset,makingithardtouseonlargenetworks.5.1.2ClusteringTheclique-treepropagationalgorithmsuggestedbyLauritzenandSpiegelhalter,alsocalledtheclus-teringalgorithm,hasbeenthemostpopularexactBayesianNetworkalgorithm.ThismethodwasusedbyBhanjaetal.intheirworktoestimatezerodelayswitchingactivity.InthismethodthedirectionalityisremovedfromtheDAGstructuretoformanundirectedgraph,calledmoralgraph.Thismoralgraphistriangulatedbyaddinglinkstoformcyclesof3nodes.Acollectionofcompletelyconnectednodesiscalledclique,ajunctiontreeofcliquesisformed,withnodesrepresentingcliques.Betweenanytwocliquescicjinatreethereisauniquepath,andelementsintheintersectionsetCi>CjarepresentinallthecliquesinthepathbetweenCiandCj,thisproperty,calledrunningintersectionpropertyisusedforlocalmessagepassingbasedupdatealgorithm.Stepsinvolvedinthepropagationofevidenceinthejunctiontreeisgivenasaowchart,takenfromtheworkofBhanjaetal.5.2ApproximateSamplingAlgorithms5.2.1StochasticSamplingAlgorithmsStochasticsamplingalgorithmsarewidelyusedapproximateinferencealgorithmsinBayesianNet-works.Samplinginvolvesinferringsomepropertyofalargesetofelementsfromthepropertiesofasmall,randomlyselected,subsetofelements.Thismethodgeneratesrandomsamplesaccordingto32

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propagate towards the selected root cliqueCollect evidence from all the cliques and Propagate probabilities from root node to of the cliques and the separatorsFormation of the distribution function all the nodesFigure5.1.Flowchart.theconditionalpropertytablesinthemodel,andestimateprobabilitiesofqueryvariablesbasedonthefrequencyoftheirappearanceinthesample.[[98]].Accuracyofthismethod,iscontrolledbythenumberofsamplesirrespectiveofthestructureofthenetwork.Stochasticsimulationalgorithmscanbedividedintotwomaincategories:importancesamplingalgorithmsandMarkovChainMonteCarloMethods(MCMC).WehaveusedsomeofthefollowingStochasticsamplingalgorithmsforinference.5.2.2ProbabilisticLogicSamplingMaxHenrion[[99]]suggestedProbabilisticLogicSampling(PLS),aschemeemployingstochasticsimulationforprobabilisticinferenceinlarge,multiplyconnectednetworks.InthisapproachaBayesianNetworkisrepresentedbyanitesampleofdeterministicscenariosgeneratedatrandomfromtheprob-abilisticmodel.Theaccuracyofthismethodcanbecontrolledbythesamplesize.Arandomgeneratorisusedtoproduceasampletruthvalueforeachsourcevariablebasedonitsprobabilities,theevidenceispropagatedthroughthenetworkinthedirectionofdirectedarrows,basedontheconditionalprobabilitytableofthenodesthroughwhichittraverses.Thisprocessisrepeatedforntimes,wherenisthesamplesizespeciedbytheuser.Thepriormarginalprobabilityofanodeisobtainedasafractionoftimes,thevalueofthenodeturnsouttobetrue,withrespecttothesamplesize.33

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Theposteriorprobabilityforanyeventconditionalonanysetofobservedvariablescanbeestimatedasthefractionofsamplescenariosinwhichtheeventoccursoutofthescenariosinwhichtheconditionoccurs[[99]].PLSisaforwardsamplingalgorithm,i.e.,thebeliefispropagatedinonlyonedirection.Thismethodignoressampleswhichdonotmeetwiththeevidencevalues.Indiagnosticinference,ifarareeventisgivenasevidence,mostofthesampleswouldbediscarded,thusseverelyhamperingtheaccuracy.Also,increasingthenumberofevidenceswouldseverelyaffectaccuracy.But,thismethodhasproventoworkwellforpredictiveinferences.5.2.3ImportanceSamplingImportanceFunction:[[95]].Letf(X)beafunctionofnvariablesXx1===xnoverdomainW)Rn.ConsidertheproblemofestimatingthemultipleintegralI?WfXdX(5.1)Weassumethatthedomainofintegrationoff(x)isbounded,i.e.,Iexists.Importancesamplingap-proachesthisproblembyestimatingI?WfX gXgXdX(5.2)gx,calledtheimportancefunction,isaprobabilitydensityfunctionsuchthatgX@0foranyx)W.gX,shouldbeeasytosamplefrom.Toestimatetheintegral,wegeneratesamplesX1X2====XNfromgXandusethegeneratedvaluesinthesample-meanformula.ˆI1 NNi1fXi gXi(5.3)ImportancesamplingassignsmoreweighttoregionswherefX2gXandlessweightswerefX1gXtocorrectlyestimateI.IffX20,theoptimalimportancefunctionisgXfX I(5.4)34

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However,sincendingIisequivalenttosolvingtheintegral,afunctioncloseenoughtotheimportancefunctionisusedtogetgoodconvergencerate.Therearevariousalgorithmsthatusearevisedimpor-tancedistributionforsamplingasanapproximationtotheposteriordistributions.AdaptiveImportanceSamplingandEvidenceprepropogationImportanceSamplingaresuchalgorithms.5.2.4AdaptiveImportanceSamplingAdaptiveImportanceSampling[[97]]adressestheissueofinaccuracyduringdiagnosticinferenceinProbabilisticLogicSampling,bylearningasamplingdistributionthatisclosetooptimalimportancesamplingfunction.TheDAGstructureoftheBayesianNetworks,isbettersuitedforndingtheOptimalimportancefunctionandbeliefupdation.Asnewevidencesappeartheimportanceconditionalpropertytable(ICPT)ofthenodes,similarinstructuretoconditonalpropertytable,areupdated.ICPTofanodeXisatableofposteriorprobabilitiesPXpaXEeconditionalontheevidenceandindexedbyitsimmediatepredecessorspaX.Importancefunctionisconstantlyupdatedasnewevidencearrives,aweightingfunctionisintroducedtothesamplingresultsastheimportancefunctionmovesfurtherfromtheoptimalimportancefunctionatdifferentstages.Chengperformstwoheuristicinitializationsthatmakethismodelbetter,1.InitializingtheICPTtableoftheparentsoftheevidencenodestounin-formdistributionimprovesconvergence.2.Settingathresholdprobabilityvalueandreplacingthepriorprobabilityofnodesthatarelessthanthethresholdvaluewiththelattervalue.Thismethodadoptsanimportantfunctionlearningsteptoapproachtheoptimalimportancefunction,thelearningstepisatimeconsumingprocess.[Changhe]hasintroducedanalgorithmthatwoulddirectlycomputeanapproxima-tionoftheoptimalimportancefunction,ratherthanlearningit.Wewoulddiscussthatalgorithminthenextsection.5.2.5EvidencePre-propagationImportanceSamplingAlgorithmEPISalgorithm[[95]]makesuseoftheloopybeliefpropagation[[40]]tocalculatetheimportancefunction.InLoopybeliefpropagation,ifEdenotesasetofevidence,atanodeX,E9,woulddenotetheevidenceconnectedtoXviait'sparentsandE,woulddenotetheevidenceconnectedtoXviait'children.theposteriorbeliefatXwouldbegivenby,35

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BELxaPexxPxe9x(5.5)BELxalxpx(5.6)wherepxdenotesthemessagecommunicatedbyx'sparent,andlxdenotesthemessagecommuni-catedbyx'schild.ThefollowingTheoremshowsthatImportancefunctioncanbecalculateddirectlyfrompolytrees.THEOREM:1[[95]]LetXibeavariableinapolytree,andEbethesetofevidence.TheexactICPTPXipaXiEforXiisapaXiPXipaXilXi(5.7)whereapaXiisanormalizingconstantdependentonpaXi.Proof:Referto[[95]]Corollary[[95]]:Forapolytree,theoptimalimportancefunctionisgivenby,rXEn'i1apaXiPXipaXilXi(5.8)usingloopybeliefpropataion,thusanoptimalimportancefunctionisobtained.SimilartoAIS,thismethodalsousessomethreshold,e,forreplacingsmallerprobabilityinthenetworkbye.Theposteriorprobabilitiesarecalculatedfromthesamplesthataregeneratedbasedontheimportancefunction.36

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CHAPTER6RESULTSTheISCAS-85Benchmarksuite,introducedinnetlistformattheInternationSymposiumofCircuitsandSystemsin1985,hasbeenusedinthiswork.TheISCAS-85circuitshavewell-dened,highlevelstructuresandfunctionsbasedonbuildingblockssuchasmultiplexers,ALU's,anddecoders.Inthiswork,wehaveproposedaprobabilisticswitchingmodel,takingintoaccounttheeffectsofGatedelay,thishasbeendonebycreatingdifferentinstancesofanodeatdifferenttimeinstants,asexplainedinChapter5.TheISCAS-85netlisthasbeenmodiedtocapturethetemporaldependenciesandspatio-temporaldependenciesbetweenthenodesatdifferenttimeinstants.ThismodiednetlistisusedforconstructingtheBayesianNetwork,andaSimulator.Thesimulatorresultsserveasagroundtruth,theresultsfromBayesianNetworkarecomparedwiththeresultfromthesimulator,andtheerrorshavebeentabulated.TheConversionofISCAS-85netlistintoaDelaynetlistwasdoneusingCprogrammingLanguage,themaximumtimeinstanceofeachcircuithasbeentabulated[6.1.].FromthisDelaynelistle,theSimulator,forvericationpurposes,andtheProbabilisticDependencymodelwerecreated.TheProbabilisticBayesianNetworkmodelhasbeenimplantedwithDynamictendencies,byex-plicitmodellingforhigherorderspatio-temporalandhigherordertemporalcorrelations.TheBayesianTable6.1.MaximumTimeInstantsforSomeCombinationalBenchmarkCircuits. Circuits MaximumTimeInstance C432 55 C499 30 C880 51 C1355 55 C2670 81 C5315 90 37

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Table6.2.SwitchingActivityEstimationErrorStatisticsBasedonDelayDAGModeling,UsingPLSInferenceScheme,Using1000Samples,forISCAS'85BenchmarkCombinationalCircuits. Circuits AverageError Time(s)s (E) CPUIO C432 0.023689 4.501 C499 0.013762 2.734 C880 0.018867 17.422 C1355 0.018578 43.643 C2670 0.021974 93.59 C5315 0.034694 641.51 NetworkwascreatedusingGeNIe,atoolfromDecisionSystemLaboratories,UniversityofPittsburgh,theinferencingwasdoneusingProbabilisticLogicSampling,anefcientstochasticinferencemethodavailableinthesametool.ThetestswereperformedusingPentiumIV,2.00GHz,WindowsXPcomputer.TheSimulatorrequirestwoinputvectors,thepreviousinputvectorandthepresentinputvector,atanygiventime,toaccountforspurioustransitionduetodelay.Thevectorsweremadedependenttemporally.TheSimulationwasrunfor100000vectors,thevectorsweregeneratedusingapseudo-randomgeneratorandtheswitchingvalueofeachnodewascalculated.TheBayesianNetworksrequirestheprobabilityoftwoSwitchinginputtransitions,theanteriorin-puttransitionandpresentinputtransition.Theresultswereobtainedbybasingtheinputswitchingprobabilities,forallthefourstates,*00,01,10,11,ofallprimaryinputnodesas0.25.Thetemporalde-pendenciesbetweentheprimaryinputswasmodelledexplicitlyintheBayesianNetwork,aswerethehigherordertemporalandhigherorderspatio-temporalcorrelationsamongthenodes.TheInferencingwasdoneusingProbabilisticLogicSampling,forasamplesizeof1000.TheresultswhencomparedtotheSimulationresultshaveshowedverylowmeanerror,evidentinthegiventable[6.2.].ThetablelistedshowstheaverageerrorE,betweentheresultsfromtheprobabilisticmodelandthesimulationmodel.Thetablealsohighlightsthetimeefciencyofourprobabilisticmodel.TheelapsedtimewasobtainedfromtheWINDOWSenvironmentanditisthesumoftheCPU,memoryaccessandI/0time.38

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ThisProbabilisticmodelasevidentfromtheresultshasalowcomputationtimeandahighaccuracy.Thishighaccuracyisattributedtothecapturingoftheinputdependencies,andhigherorderspatial,higherordertemporalandhigherorderspatio-temporalcorrelationsinthecircuit.39

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CHAPTER7CONCLUSIONPowerDissipationhasbeenamajorconcernforresearchersintheVLSIDesigneld.ItisknownthatSwitchingActivityatgates,hasbeenamajorcontributortoPowerDissipation.SwitchingActivityEstimationhasbeencarriedoutatalllevelsofdesignabstraction,andithasbeenproventhatGateLevelswitchingestimationhasabetteraccuracycomparedtotheotherlevels.TheLiteratureReviewshowsthatswitchingestimationhasbeencarriedoutusingSimulation,Sta-tisticalsimulationandProbabilisticmethodologies.TheSimulationandStatisticalsimulationbasedmethodologieshaveproventobehighlyinputpatternsensitive.Toacquireanaccurateresultusingthesemethodologies,alargeinputsetwouldberequired,increasingthetimecomplexityoftheprocess.TheexistingProbabilisticmethodsdonothaveauniedframeworkthataccountsforthedifferentissues,likeinputpatterndependency,spatialcorrelations,temporalcorrelationsandspatio-temporalcorrelations,intheSwitchingEstimationprocess,ThusfailingtoaccuratelycapturetheSwtichingActivity.TheworkofBhanjaetal,whileaccountingforhigherorderspatialcorrelationsandrstordertem-poralcorrelation,failtoaccountforGateDelays,ignoringthespurioustransitionsthatoccurduetodifferentarrivaltimesoftheinputsignals.TheotherProbabilisticmodelsthataccountfordelay,donotmodelhigherordertemporalandhigherorderspatio-temporalcorrelations.TheGateDelaymodelinthiswork,usesBayesianNetworkstoaccuratelycapturethespurioustransitions,byexplicitlymodellingforhigherorderspatio-temporalandhigherordertemporalcorrelations.TheinferencingofresultshasbeendoneusingProbabilisticLogicsampling,aprovenefcientinferencingmechanismforpredictiveProbabilisticinference.Theresultswhencomparedtothesimulationresultshaveminimumerrorandthecomputationtimeisverylow,comparedtoothermethods.40

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7.1FutureWork1.IntheNanodomain,InterconnectDelayhasbeengainingdominance.Thisworkcanbeextendedtoaccountforinterconnectdelay,byalteringthedelayassignmentmechanism.2.InthisworkwehaveeffectivelydevelopedaProbabilisticSwitchingmodelforCombinationalcircuits,underRealDelayconditions.Wecouldextendthisworkforswitchingestimationinsequentialcircuits.41

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ABSTRACT: Power optimization is a crucial issue at all levels of abstractions in VLSI Design. Power estimation has to be performed repeatedly to explore the design space throughout the design process at all levels. Dynamic Power Dissipation due to Switching Activity has been one of the major concerns in Power Estimation. While many Simulation and Statistical Simulation based methods exist to estimate Switching Activity, these methods are input pattern sensitive, hence would require a large input vector set to accurately estimate Power. Probabilistic estimation of switching activity under Zero-Delay conditions, seriously undermines the accuracy of the estimation process, since it fails to account for the spurious transitions due to difference in input signal arrival times.In this work, we propose a comprehensive probabilistic switching model that characterizes the circuit's underlying switching profile, an essential component for estimating data-dependent dynamic and static power. Probabilistic estimation of Switching under Real Delay conditions has been a traditionally difficult problem, since it involves modeling the higher order temporal, spatio-temporal and spatial dependencies in the circuit. In this work we have proposed a switching model under Real Delay conditions, using Bayesian Networks. This model accurately captures the spurious transitions, due to different signal input arrival times, by explicitly modeling the higher order temporal, spatio-temporal and spatial dependencies. The proposed model, using Bayesian Networks, also serves as a knowledge base, from which information such as cross-talk noise due to simulataneous switching at input nodes can be inferred.
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