NAND,andNOR).Letlinedenoteaninterconnectionbetweentwogates,)denotetheg - PDF document

NAND,andNOR).Letlinedenoteaninterconnectionbetweentwogates,)denotethegateforwhichlineisaninput,and)denotethesetofinputsofgate.Let)representthestablesignalvalueofavector,whereiseitheragateoraline.Aphysicalpathisasequence(;:::;lwhere(;:::;g)aregates,(;:::;l)arelines,aprimaryinput(PI),andisaprimaryoutput(PO).Lines(;:::;l)arecalledon-pathinputs.Aniscalledaside-input.Therearetwologicalpathsand,associatedwith,correspondingtoarisingandafallingtransitionrespectivelyon.Afaultylogicalpathiscalledapathdelayfault.Inthispaper,weusethetermspathdelayfaultspathfaultsandlogicalpathsinterchangeably.AtestforapathdelayfaultconsistsofavectorpairDenition2.1([11])Avectorpairissaidtofunctionallysensitizeapathfault;:::;gwhere,i:(1) ,and(2)ifanon-pathinputofhasanon-controllingvalueunder,thecorrespondingside-inputshavenon-controllingvaluesunderDenition2.2([14])Avectorpairissaidtobeanon-robusttestforapathdelayfault;:::;gwhere,i:(1) ,and(2)allside-inputsofhavenon-controllingvaluesun-Denition2.3([1,2])Avectorpairsaidtobearobusttestforapathdelayfault;:::;gwhere,ifitguaranteesdetec-tionofthefaultirrespectiveofthedelaysofallothersignalsinthecircuit.Thisconditionissatisedi:(1)isanon-robusttestfor,and(2)ifanon-pathinputofhasacontrollingvalueun-,thecorrespondingside-inputshavesteadynon-controllingvaluesonbothvectors.Remark2.1Theexistenceofavector,thatsatisestherequirementonitgivenbyDenitions2.1and2.2,canbeguaranteedforacircuitwithoutanyrestrictionsonitsinputvalues.Remark2.2Thesetoffunctionallysensitizablepathfaultsisasupersetofthesetofnon-robustlytestablepathfaults,whichinturnisasupersetofthesetofrobustlytestablepathfaults.Remark2.3Onlytherobustnesscriterionimposestherequirementwhichinturnim-Wesaythatapathfaultsensitizedonavectorpairiseitherfunctionallysensitized,robustlytested,ornon-robustlytestedbyLemma2.1Ifapathfault;:::;g,where,issensitizedonavectorpairifthenumberofinversionsbetweenandtheoutputofiseven; otherwiseProof:Wewillprovethelemmabyinduction.InductionBasis:Sinceissensitizedbyfromtheabovedenitions,weknowthatInductionHypothesis:SupposetheclaimistrueforInductionStep:Letandbenon-inverting.Sinceisnon-inverting,thenumberofinversionsbetweenandtheoutputofisequaltothenumberofinversionsbetweenandtheoutputof.Ifisthecontrollingvalueofisthenon-controllingvalueof,bytheabovedenitions,theside-inputscorrespondingtotobenon-controllingfortobesensitizedandhence,.Since)andthenumberofinversionsremainsunchanged,bytheinductionhy-pothesis,theclaimholdsfor.Asimilarargumentcanbemadeforthecasewhenisinverting.3Implication-basedAnalysisWeassumethatlogicimplicationsofbothvalueas-signments(0and1)foreverylineinthecircuitareavailable.Wepresentlemmasthatuselogicimplica-tionsonalinetohelpidentifyasetoflines,suchthateverypathfaultpassingthroughandisuntestablewithrespecttosomecombinationofsig-nalvalues.Let),where;,denotethesetofpathfaultspassingthroughlinesandsuchthatforeverypathfaulttx(V2)=]and[]arenecessaryconditionsfortobesensitizedonavectorpair(refertoLemma2.1).3.1RobustUntestabilityAnalysisWesaythat[]ifallpathfaultsintheset)arerobustlyuntestable.Lemma3.1Forlinesand,ifwhere;andisthecontrollingvalueof,thennSP(x;z)=RU].Proof:Considerapathfault.Lineisaside-inputof.Foravectorpairtobearobusttestfor,thefollowingconditionsarenecessary:(1)since),and(2) ,thenon-controllingvalueof(byDenition2.3).However,since(),avectorthatsatisesconditions(1)and(2)cannotexistandhence,isrobustlyuntestable. Lemma3.2Forlinesand,ifwhereisthenon-controllingvalueofandisthecontrollingvalueof,thennSP(a;b)=RU]Proof:Considerapathfault,whereandisthesetofallpathfaultsthatpassthoughandandareside-inputsof.ByDenition2.3,foravectorpairtobearobusttestfor,thefollowingconditionsarenecessary:(1),thenon-controllingvalueof,and(2) ,thenon-controllingvalueof.However,since(avectorthatsatisesconditions(1)and(2)cannotexistandhence,isrobustlyuntestable.Lemma3.3Forlinesand,ifwhere;andisthecontrollingvalueof,then Proof:Considerapathfault .Lineisaside-inputof.Foravectorpairtobearobusttestfor,thefollowingconditionsarenecessary:(1) since )andhence(byRemark2.3),and(2) ,thenon-controllingvalueof(byDenition2.3(2)).However,since(avectorthatsatisesconditions(1)and(2)cannotexistandhence,isrobustlyuntestable.Lemma3.4Forlinesand,if,where;,then and Proof:Bydenition,tosensitizeapathfault onavectorpair and[]arenecessaryconditions.Hence,byRe-mark2.3,fortobearobusttestfor,thefollowingconditionsarenecessary:(1)[],and(2)[ ].However,since(),avectorthatsatisesconditions(1)and(2)cannotexistandhence,isrobustlyuntestable.Similarly,torobustlytestapathfault (1)[],and(2)[ ]arenecessarycondi-tions.Sinceavectorthatsatisesconditions(1)and(2)cannotexist,isrobustlyuntestable.Lemma3.5Ifalineisidentiedashavingaconstantvalueassignment,thennSP(x;x)=RU].Proof:Sinceavectorpairthatsatisescannotexist,byRemark2.3,allpathfaultsthatpassthrougharerobustlyuntestable. n Figure1:PortionofISCAS-85benchmark3.2FunctionalUnsensitizabilityAnalysisFunctionallyunsensitizablepathfaultscanbeig-noredduringdelayfaulttestingandtiminganalysis[11].Wesaythat[]ifallpathfaultsinthe)arefunctionallyunsensitizable.Lemma3.6Forlinesand,ifwhere;,then Proof:SimilartotheproofofLemma3.4.Lemma3.7Ifalineisidentiedashavingaconstantvalueassignment,then  Proof:SimilartotheproofofLemma3.5.OtherlemmassimilartothosepresentedinSec-tion3.1canbederivedforidentifyingfunctionallyun-sensitizablepathfaults.However,inourexperiments,wefoundthattheydonothelpidentifyanyadditionalfunctionallyunsensitizablepathfaults.Figure1illustratesanexampleofidentifyingafunc-tionallyunsensitizablepathfaultusingLemma3.6.Considerthepathfaultc;f;k;m;n;p;q;r;s).Wemaketwoobservations:Foravectorpairtofunctionallysensitize,thefollowingconditionsarenecessary:(1)[0](fromDenition2.1),and(2)[)=1]sincethenumberofinversionsbetweenandisseven(fromLemma2.1).Hence[=0)=0)fromstaticimplicationlearning[10].FromLemma3.6,wecanconcludethatisfunctionallyunsensitizable.Similarlyc;f;k;m;n;o;q;r;s)isfunctionallyunsensitizable.Weexplicitlyenumerateuntestablepathfaultsinthisex-ampleonlyforillustration.Ouralgorithmobtainsthenumberofuntestablepathfaultswithoutenumerating3.3Non-robustUntestabilityAnalysisWesaythat[]ifallpathfaultsinthe)arenon-robustlyuntestable.Toidentifynon-robustlyuntestablepathfaults,Lemmas3.1and3.2canbeusedbyreplacinginthemandLemmas3.6and3.7canbeusedbyreplacinginthem. 4UsingPre-computedImplicationsWeassumethatsomesetofimplicationsofbothvalueassignmentsforeverylineinthecircuitareavailable.4.1MaintainingUntestabilityInformationUsingourfault-independentimplicationanalysisateachline,weconstructfoursetsoflinesassociated),andDenition4.1Asetassociatedwithalinewhere;,consistsoflinesesSP(g;x)=UntestableDependingonthespecictestabilitycriterionused,rulesthathelpconstructthesetscanbederivedfromthelemmaspresentedinSection3.Forconcise-ness,weonlypresentanexampleinvolvingthefunc-tionalsensitizationcriterion.Ifweknowthat(),where;,andisinthefanoutcone,fromLemma3.6,isaddedto 4.2CountingUntestablePathFaultsPomeranzandReddyproposealinear-timecountingalgorithm[13]thatcancomputethenumberofpathsinasinglepassfromPOstoPIs.Basedontheiridea,weproposeacountingalgorithmthatusesthesetsoneverylinetocomputealowerboundonthenumberofuntestablepathfaults.partialpathfaultisapathfaultwithouttherestric-tionthatitsoriginshouldbeaPI.Inthissection,weusethetermspathfaultsandpartialpathfaultsinter-changeably.Consideracircuitwithlines.Letdenotethesetofallbranchesofafanoutstem.Foralinethatisnotastem,letdenotetheoutputof).Weassociateafewvariableswitheachline):numberoftestablepathfaults,asde-terminedbyourprocedure,thatoriginateatandrequireavalueof1(0)ontobesensitized:temporaryvaluesofonaspeciciterationeval:indicateswhetherandhavebeencomputedonaspeciciteration):setofallimmediatepredecessorlinesofred:indicatesifUntestable,onaspeciciterationToavoidreseting agssuchasevalandred,weuseuniqueidentiersoneachiteration.Withnoim-plicationsavailable,allpathfaultsoriginatingatalinefanoutconeoflinecontributetowardsthecomputationof.Suppose.Bydef-inition,cannowbeignoredwhencomputingHowever,itispossiblethatmaycontributetowardsthecomputationofvaluesforotherlinesinthefaninconeof.Assumingthatthesetsassociatedwitheverylineareavailable,Algorithm4.1processeslinesinareversetopologicalorder(POstoPIs)tocomputealowerboundonthenumberofuntestablepathfaults.Algorithm4.1testable-path-count()evalredred1//reversetopologicalorderforeachinsert(redforeachinsert(redlabel(foreachinsert(redforeachinsert(redlabel(#oftestablefaults(iisaPI#ofuntestablefaults=Total#ofpathfaults-label(insert(x)while=dequeue())evalredisconnectedtoaPO)isastem)choose(;b;id=choose(;id=1if)isnon-inverting;0otherwise)//lesserofthetwovaluesredisconnectedtoaPO)isastem)choose(;b;id=choose(;id=0if)isnon-inverting;1otherwiseorifisclosertoaPOthanx)insert(choose(x;y;n;idevalinsert(insertintoaneventlistdequeue()isnon-empty,dequeueandreturnelementthatisclosesttoaPO;return0otherwise4.3CountingUntestableSegmentFaultsThediscussioninSections3and4.1isalsoapplicabletothesegmentdelayfaultmodel.Wemodifytheseg-mentcountingalgorithmpresentedintheliterature[3]touseimplicationstodeterminealowerboundonthenumberofuntestablesegmentfaults. Table1:Alowerboundonthenumberofuntestabledelayfaults Ckt. Segmentfaults Pathfaults Name Robustlyuntestable Robustly Non-robustly Functionally L L=5 untestable untestable unsensitizable # % # % # % cpu(s) # % # % cpu(s) 0 0 0.0 326 1.9 0 163 0.9 163 0.9 0 8.8 7,840 33.9 8,005,696 95.9 2 7,150,240 85.7 6,745,120 80.8 1 8 1,260 6.0 1,070,307 73.4 4 1,067,159 73.2 442,048 30.3 1 2.6 1,307 6.7 1,321,906 97.2 2 1,317,795 96.9 1,314,962 96.7 1 4.7 4,129 12.7 53,610,698 93.5 18 52,488,315 91.5 34,300,319 59.8 6 1.1 1,486 3.6 2,013,498 75.1 14 1,865,548 69.5 1,129,995 42.1 4 8.1 23,577 26.1 1:97810 3 1:97810 1:97810 3 1.5 3,709 4.1 981,720 67.6 24 910,926 62.7 555,050 38.2 7 s5378 284 1.8 815 3.0 6,396 23.6 9 4,869 18.0 3,718 13.7 4 s9234 1,225 4.3 4,275 8.7 442,526 90.4 58 413,785 84.5 282,149 57.6 21 s13207 1,817 5.0 5,091 8.8 2,300,812 85.5 50 1,870,582 69.5 1,722,492 64.0 24 s15850 3,349 7.1 12,321 15.3 322,581,591 97.9 101 303,523,949 92.1 274,843,560 83.4 40 s35932 11,866 12.5 35,372 27.4 354,324 89.9 519 265,863 67.4 248,567 63.0 241 s38417 3,229 2.9 15,149 7.6 1,675,008 60.2 86 1,377,425 49.5 796,701 28.6 42 s38584 1,608 1.3 6,938 3.6 1,623,570 75.1 26 1,169,090 54.1 1,123,814 52.0 15 Onlyasubsetofdirectimplicationswasavailablefors385845ResultsWeimplementedouralgorithminC++andranexperimentsusingaHP9000/735workstationwith256MBofmemory.WeusetheimplicationsgeneratedbyZhaoet.al.[10]forourwork.Directimplications,determinedbyforwardandbackwardpropagationstart-ingatthenodeunderconsideration,andindirectim-plications,foundbyapplyingthecontrapositivelaw[9],transitivelawandbackwardimplications[10]wereavail-ableformostcircuits.Onlyasubsetofdirectimplica-tionswasavailablefor38584.Informationonconstantvalueassignmentsthatisgeneratedasaby-productoftheirimplicationprocedureisalsoused.ForalltheISCAS-85circuits,theirprogramtooklessthan150sec-ondstogeneratetherelevantinformation[10].TimestakenbytheirprogramfortheISCAS-89circuitswerenotavailable.Table1showsresultsofthealgorithmforasubsetofallvaluesthatsegmentlength,aninputparame-tertotheprogram,cantake.Forpathdelayfaults,weconsidertherobust,non-robustandfunctionalsensiti-zationcriteria.Forsegmentfaultswith(=3)and=5),weonlyconsidertherobusttestabilitycri-teria.Columnswiththeheading#showthenum-berofuntestablefaultsdeterminedbyourprocedure.Columnswiththeheading%indicatethepercentageofuntestablefaultswithrespecttothetotalnumberoffaults.TheruntimesinCPUsecondsareshownonlyforidentifyingrobustlyuntestableandfunctionallyunsen-sitizablepathfaults.Runtimesforidentifyingrobustlyuntestablesegmentfaultsandnon-robustlyuntestablepathfaultsweresimilartothoseforidentifyingrobustlyuntestablepathfaults.However,thereisamarginalin-creaseintheruntimesasisincreased.Inallcases,theruntimeofourmethodisverysmallandisindependentofthenumberoffaults.Whileourmethodidentiesmoreuntestablepathfaultsinsomecircuits,previouslypublishedmethods[11,12]dobetterforsomecircuits.SinceweidentifyonlyasubsetofalluntestableTable2:OurresultsversusATPGresults Ckt. Non-robustlyuntestablepathfaults Name Exact[15] Ourprocedure 19.0% 18.0% s9234 87.8% 84.5% s13207 82.3% 69.5% s15850 96.7% 92.1% s35932 85.2% 67.4% s38417 59.1% 49.5% s38584 84.5% 54.1% faults,rmconclusionscannotbedrawn.However,forcircuitssuchas1908,itappearsthatmanynon-robustlyuntestablepathfaultsdonotbelongtothesetoffunctionallyunsensitizablefaults.Suchpathfaultsaretestableonlyasamultiplefault[16].Dealingwithmultiplepathfaultsmaybecomputationallyintractableforlargecircuits.Insuchcases,usingthesegmentdelayfaultmodel,withasmallvalueof,maybeafeasiblealternative.Forcircuitslike2670,mostofthepathfaultsarefunctionallyunsensitizable.Thesepathfaultscanbeignoredforthepurposesofdelaytestingandthe 10 20 30 40 50 60 70 80 90 100 0 20 40 60 80 100 120 140 % robustly untestable Segment length L c6288 s9234 Figure2:%ofuntestablefaultsversussegmentlengthpathfaultmodelmaybepracticalinsuchcases.Table2comparesourresultswiththeexactresultsobtainedfromatestgenerator[15]fortheISCAS-89circuits.Theexactnumberofuntestablefaultsisun-knownformanyoftheISCAS-85circuits.Ourmethodgivesonlyalowerboundonthenumberofuntestablefaultssinceitisbasedonanincompletesetofimpli-cations.DuringanATPGrun,alowerboundonthenumberofuntestablefaultsmaybeusefulasastoppingcriterioniftherequiredfaulteciencyisreached.Inourexperiments,thepercentageofuntestablefaultsincreasesmonotonicallywithandthende-creasesasapproachesthemaximumlogicdepth.Fig-ure2showsthistrendforrobustlyuntestablefaultsin6288(maximumlogicdepth:125)and9234(max-imumlogicdepth:59).For6288,weidentiedrobustlyuntestablepathfaults,ofwhichwerealsofunctionallyunsensitizable.6ConcludingRemarksOuralgorithmusesstaticlogicimplicationsandrapidlycomputesalowerboundonthenumberofro-bustlyuntestable,non-robustlyuntestable,andfunc-tionallyunsensitizabledelayfaultsbyusinganon-enumerativecountingprocedure.Untestablefaultscanalsobelistedifdesiredandtargetingthemfortestgen-erationbyanATPGtoolcanbeavoided.Oneofthemainfeaturesofthealgorithmisthatitscomplexityofcomputationdoesnotgrowwiththenumberofdelayfaults.Thisisespeciallyimportantwhenconsideringthepathdelayfaultmodelsincecircuitstypicallyhavealargenumberofpathfaults.Theimplicationanaly-sispresentedinthispaperconsidersoneimplicationatatime.Itmaybepossibletoobtainbetterresultsbyconsideringmultipleimplicationssimultaneously.References[1]G.L.Smith,\ModelforDelayFaultsBasedUponPaths,"inProc.InternationalTestConf.,pp.342{349,Nov.1985.[2]C.J.LinandS.M.Reddy,\OnDelayFaultTestinginLogicCircuits,"IEEETrans.onCAD,vol.6,pp.694{703,Sept.1987.[3]K.Heragu,J.H.Patel,andV.D.Agrawal,\SegmentDelayFaults:ANewFaultModel,"inProc.VLSITest,pp.32{39,Apr.1996.[4]K.Heragu,J.H.Patel,andV.D.Agrawal,\SIGMA:ASimulatorforSegmentDelayFaults,"inProc.Inter-nationalConf.CAD,pp.502{508,Nov.1996.[5]I.PomeranzandS.M.Reddy,\OnAchievingCom-pleteTestabilityofSynchronousSequentialCircuitswithSynchronizingSequences,"inProc.InternationalTestConf.,pp.1007{1016,Oct.1994.[6]V.D.AgrawalandS.T.Chakradhar,\CombinationalATPGTheoremsforIdentifyingUntestableFaultsinSequentialCircuits,"IEEETrans.onCAD,vol.14,pp.1155{1160,Sep.1995.[7]M.A.IyerandM.Abramovici,\FIRE:AFault-IndependentCombinationalRedundancyIdenticationAlgorithm,"IEEETrans.onVLSISystems,vol.4,pp.295{301,Jun.1996.[8]M.A.Iyer,D.E.Long,andM.Abramovici,\Identify-ingSequentialRedundanciesWithoutSearch,"inProc.33rdDesignAutomationConf.,Jun.1996.[9]W.KunzandD.K.Pradhan,\AcceleratedDynamicLearningforTestPatternGenerationinCombinationalCircuits,"IEEETrans.onCAD,vol.12,pp.684{694,May1993.[10]J.Zhao,E.M.Rudnick,andJ.H.Patel,\StaticLogicImplicationwithApplicationtoRedundancyIdenti-cation,"inProc.VLSITestSymp.,pp.288{293,Apr.[11]K.T.ChengandH.C.Chen,\DelayTestingforNon-RobustUntestableCircuits,"inProc.InternationalTestConf.,pp.954{961,Oct.1993.[12]S.Kajihara,K.Kinoshita,I.Pomeranz,andS.Reddy,\AMethodforIdentifyingRobustDependentandFunctionallyUnsensitizablePaths,"inProc.10thInter-nationalConf.onVLSIDesign,pp.82{87,Jan.1996.[13]I.PomeranzandS.M.Reddy,\AnEcientNon-EnumerativeMethodtoEstimatethePathDelayFaultCoverageinCombinationalCircuits,"IEEETrans.,vol.13,pp.240{250,Feb.1994.[14]E.S.ParkandM.R.Mercer,\RobustandNonrobustTestsforPathDelayFaultsinaCombinationalCir-cuit,"inProc.InternationalTestConf.,pp.1027{1034,Sept.1987.[15]K.Fuchs,M.Pabst,andT.Rossel,\RESIST:ARecur-siveTestPatternGenerationAlgorithmforPathDelayFaultsConsideringVariousTestClasses,"IEEETrans.onCAD,vol.13,pp.1550{1561,Dec.1994.[16]W.KeandP.R.Menon,\SynthesisofDelay-veriableCombinationalCircuits,"IEEETrans.onCAD,vol.44,pp.213{222,Feb.1995.