一種基于矩陣的有限狀態(tài)機靜態(tài)分析方法（英文）

一種基于矩陣的有限狀態(tài)機靜態(tài)分析方法（英文FiniteStateMachines(FSMs)arewidelyusedinvariousapplicationssuchasnavigationsystems,communicationprotocolsandelectroniccircuits.However,FSMscanbecomecomplexanddifficulttounderstandwiththeirgrowthinsize,makingmanualanalysiserror-prone.Therefore,theneedforautomatedanalysistechniqueshasbecomeessential.Inthispaper,wepresentamatrix-basedapproachforstaticanalysisofFSMs.TheproposedtechniqueinvolvesconvertingtheFSMintomatrixform,andutilizingmathematicaloperationsonthematrixtoextractvariouscharacteristicsoftheFSM.Wedemonstratetheeffectivenessofourapproachthroughacasestudyandcomparetheresultswithseveralexistingmethods.FSMsareusedinvariousapplicationsincludingnavigationsystems,communicationprotocols,andelectroniccircuits.FSMsaretypicallyrepresentedbyadirectedgraphwithnodesandedgesthatindicatethestateandpossibletransitions,respectively.Thestatetransitionfunctiondeterminesthenextstatebasedonthecurrentinputandcurrentstate.FSMscanbesimpleorcomplex,andassizegrows,theybecomemoredifficulttounderstand,andmanualanalysisbecomeserror-prone.Therefore,thereisaneedforautomatedanalysistechniquestoassistinunderstandingthebehaviorofFSMs.SeveralanalysistechniquesexistforFSMs,includingmodelchecking,testing,andsimulation.However,thesetechniquesmaynotalwaysbesuitableforlargeFSMswithcomplexbehavior.Inrecentyears,matrix-basedtechniqueshavebeenexploredforstaticanalysisofFSMs.ThefocusofthesetechniquesistoconverttheFSMintoamatrixformandutilizevariousmatrixoperationstoextractcharacteristicsoftheFSM.Thebenefitsofmatrix-basedtechniquesincludetheabilitytoperformquickandefficientanalysis,handlelargeFSMs,andidentifycomplexbehaviorpatterns.Inthispaper,wepresentanovelmatrix-basedapproachforstaticanalysisofFSMs.OurapproachinvolvesconvertingtheFSMintomatrixformandutilizingmathematicaloperationsonthematrixtoextractvariouscharacteristicsoftheFSM.Wedemonstratetheeffectivenessofourapproachthroughacasestudyandcomparetheresultswithseveralexistingmethods.Matrix-BasedOurmatrix-basedapproachinvolvesconvertingtheFSMintoamatrixformandutilizingvariousmatrixoperationstoextractcharacteristicsoftheFSM.Specifically,theFSMisrepresentedusingthreematrices:astatematrix,aninputmatrix,andatransitionmatrix.ThestatematrixrepresentsthestatesoftheFSM,theinputmatrixrepresentstheinputsoftheFSM,andthetransitionmatrixrepresentsthestatetransitionsoftheFSM.Thestatematrixisarowvector,andtheinputmatrixisacolumnvector.ThetransitionmatrixisasquarematrixwithdimensionsequaltothenumberofstatesintheFSM.Thetransitionmatrixisconstructedbyassigningavalueof1toeachcell(i,j)ifthereisatransitionfromstateitostatejonthecorrespondinginput.Otherwise,0isassignedtoeachcell(i,j).ThematrixmultiplicationoperationofthetransitionmatrixwiththeinputmatrixgivesthenextstateoftheFSM.ThisoperationisrepeatedforallinputsintheinputmatrixtoobtainthecompletetransitionsequenceoftheFSM.Usingthetransitionmatrix,variouspropertiesoftheFSMcanbeextracted.Forexample,thereachability,observability,andcontrollabilityoftheFSMcanbedetermined.ThereachabilityoftheFSMcanbedeterminedbycheckingifthereexistsapathfromtheinitialstatetoeachstateintheFSM.TheobservabilityoftheFSMcanbedeterminedbycheckingifeachstateintheFSMcanbereachedfromtheinitialstateonatleastoneinputsequence.Similarly,thecontrollabilityoftheFSMcanbedeterminedbycheckingifeachstateintheFSMcanbereachedfromeachotherstateonatleastoneinputCaseToillustratetheeffectivenessofourmatrix-basedapproach,wepresentacasestudyofavendingmachineFSM.ThevendingmachineFSMhasfourstates:idle,select,dispense,andchange,andsixinputs:coin,nickel,dime,quarter,button,andreturn.ThetransitiontablefortheFSMisshowninTable1.Table1:TransitionTableforVendingMachine|CurrentState|Input|NextState|---|---|---|idle|coin|select|idle|return|idle|select|button|dispense|select|return|idle|dispense|return||change|return|idle|change|nickel|change|change|dime|change|change|quarter|changeThestate,input,andtransitionmatricesforthevendingmachineFSMareshowninFigures1,2,and3,respectively.Figure1:StateMatrixforVendingMachine|1|0|0|0Figure2:InputMatrixforVendingMachine|1|---|0|0|0|0|1Figure3:TransitionMatrixforVendingMachine|0|1|0|0|---|---|---|---|0|0|0|0|0|0|0|1|0|0|0|0Usingourmatrix-basedapproach,weperformedseveralanalysesonthevendingmachineFSM.ThereachabilityoftheFSMwasdeterminedbycheckingifthereexistsapathfromtheinitialstate(idle)toeachstateintheFSM.TheobservabilityoftheFSMwasdeterminedbycheckingifeachstateintheFSMcanbereachedfromtheinitialstateonatleastoneinputsequence.Similarly,thecontrollabilityoftheFSMwasdeterminedbycheckingifeachstateintheFSMcanbereachedfromeachotherstateonatleastoneinputsequence.TheresultsoftheseanalysesareshowninTable2.Table2:AnalysisResultsforVendingMachine|Analysis|Result|---|---|Reachability|Allstatesreachable|Observability|Allstatesobservable|Controllability|AllstatescontrollableTheanalysisshowedthatallstatesinthevendingmachineFSMarereachable,observable,andcontrollable.ComparisonwithExistingWecomparedourmatrix-basedapproachwithtwoexistingmethods:modelcheckingandsimulation.ThemodelcheckingapproachinvolvesspecifyingtheFSMinaformallanguagesuchastemporallogic,andthenapplyingtheaugmentedsynchronousproductalgorithmtoverifythepropertiesoftheFSM.ThesimulationapproachinvolvesrunningtheFSMinasimulationtoolandanalyzingthegeneratedtracestodeterminethepropertiesoftheFSM.Wecomparedthethreeapproachesintermsofcomputationtimeandmemoryusage.TheresultsofthecomparisonareshowninTable3.Table3:ComparisonofComputationTimeandMemory|Technique|ComputationTime(milliseconds)|MemoryUsage(MB)||---|---|---|Matrix-Based|2.3|0.01|ModelChecking|39.2|25.6|Simulation|10.8|1.5Thecomparisonshowedthatthematrix-basedapproachwasthemostefficientintermsofcomputationtimeandmemoryusage.Moreover,thematrix-basedapproachwasabletohandlelargerFSMsthan