Possibility is not expressible in LTL Example Wheneve - PDF document

2!|=!M1and )!E:forsomeinÞniterun!A:forallinÞniteruns91/113CTL#(Emerson&Halpern86)DeÞnition:SyntaxoftheComputationTreeLogicCTL!!::=$|p(p%AP)|Â!|!&!|X!|!U!|E!|A!DeÞnition:Semantics:LetM=(S,T,I,AP,")beaKripkestructureand#aninÞnterunofM.M, !suchthat#!(0)=#(i)M, !suchthat#!(0)=#(i)Example:SomespeciÞcations!EF!:!is!AG!:!isaninvariant!AF!:!is!EG!:!holdsgloballyalongsomepathRemark:A!'ÂEÂ!92/113StateformulaeandpathformulaeDeÞnition:Stateformulae!%CTL"isastateformulaif(M,#,#!,i,jsuchthat#(i)=#!(j)wehaveM,#,i|=!)*M,#!,j|=!If!isastateformulaandM=(AP ]]={s%S|p%"(s)}[[Â!]]=S\[[!]][[!1&!2]]=[[!1]]+[[!2]]DeÞnition:AlternativesyntaxStateformulaeAP)|Â!|!&!|E$|A$Pathformulae$::=!|Â$|$&$|X$|$U$93/113Modelcheckingof .Remark:M|=#!i!I,[[E!]]!=-M|=$!i!I.[[A!]]M|=$!i!M!|=#Â!DeÞnition:ModelcheckingproblemsMC"CTL ")andaformula!%CTL DeÞnition:SyntaxoftheComputationTreeLogicCTL!!::=$|p(p%AP)|Â!|!&!|X!|!U!|E!|A!TheoremThemodelcheckingproblemfor Q%{/,(}and !(t),AP="(t)andpQ!%"!(t)i!MCQLTL(M,t,$)LetM!=(AP!,"!)Let!!=![pQ!/Q$]beobtainedfrom!byreplacingeachQ$bypQ!ReturnMC$CTL Question:ExistenceofamodelMandarun#suchthatM,#,0|=!?TheoremThesatisÞabilityproblemforCTL"is2-EXPTIME-complete !U!|A!U!ThesemanticsisinheritedfromCTL".Remark:AllCTLformulaearestateformulae[[!]]M={s%S|M,s|=!}Examples:Macros!EF!=E1U!andAF!=A1U!!EG!=ÂAFÂ!andAG!=ÂEFÂ!!AG CTL-formulaearestateformulae.Hence,wehaveasimplersemantics.LetM=(S,T,I,AP,")beaKripkestructurewithoutdeadlocksandlets%S.s|=pifp%"(s)s|=EX!if/s"s!withs!|=!s|=AX!if(s"s!wehaves!|=!s|=E!U$if/s=s0"s1"s2"ááá j|=$and k|=!forall02kj99/113CTL(Clarke&Emerson81)Example:12345678qp,qqrp,rp,rp,q[[EXp]]={1,2,3,5,6}[[AXp]]={3,6}[[ !,!Â(!U$)=GÂ$&(Â$U(Â!4Â$))!A!U$=ÂEGÂ$4ÂEÂ$U(Â!4Â$)!AG"Fgrant)=AG"AF M=(S,T,I,AP,")beaKripkestructureand!%CTLM|=#!if %I.Remark:M|=#!i!I,[[ |=$!i!M!|=#Â!DeÞnition:ModelcheckingproblemsMC"CTLMC#CTL ")andaformula!%CTL =(S,T,I,AP,")beaKripkestructureand!%CTLM|=#!isdecidableintimeO(|M|á|!|)Proof:Compute[[!]]={s%S|M,s|=!}byinductionontheformula.Theset[[ %[[!]].Thelabelling"isencodedinL:forp%APL[s][p]= 1);semantics(!2)[[!]]:=[[!1]]+[[!2]]O(|S|)case!=EX 1)[[!]]:=-O(|S|)forall(s,t)%Tdoift%[[!1]]then[[!]]:=[[!]]+{s}O(|T|)case!=AX 1)[[!]]:=SO(|S|)forall(s,t)%Tdoift/%[[!1]][[!]]:=[[!]] );semantics(!2)L:=[[!2]]//thesetListheÒtodoÓlistO(|S|)Z:=-//thesetZistheÒdoneÓlist S|timesInvariant:[[!2]] !1U!2]]t%L;L:=L\{t};Z:=Z+{t}O(1)foralls%T%1(t)do|T|timesifs%[[!1]]\(Z+L)thenL:=L+{s}[[!]]:=ZZisonlyusedtomaketheinvariantclear.Z+Lcanbereplacedby[[ 1);semantics(!2)L:=[[!2]]//thesetLisimlementedwithalistO(|S|)[[!]]:=[[!2]]O(|S|)whileL!=-do|S|timestaket%L;L:=L\{t}O(1)foralls%T%1(t)do %[[!1]]\[[ );semantics(!2)L:=[[!2]]//thesetListheÒtodoÓlistO(|S|)Z:=-//thesetZistheÒdoneÓlist S|timesInvariant:(s%S,c[s]=|T(s)\Z|and[[!2]]+([[!1]],{s%S|T(s).Z}).Z+L.[[ %L;L:=L\{t};Z:=Z+{t}O(1)foralls%T%1(t)do|T|timesc[s]:=c[s]51O(1)ifc[s]=04s%[[!1]]\(Z+L)thenL:=L+{s}[[!]]:=ZZisonlyusedtomaketheinvariantclear.Z+Lcanbereplacedby[[ !)ReplacingZ+Lby[[!]] 1);semantics(!2)L:=[[!2]]O(|S|)[[!]]:=[[!2]]O(|S|)foralls%Sdoc[s]:=|T(s)|O(|S|)whileL!=-do|S|timestaket%L;L:=L\{t}O(1)foralls%T%1(t)do|T|timesc[s]:=c[s]51O(1)ifc[s]=04s%[[!1]]\[[ %CTLMandastatessuchthatM,s|=!?Theorem:Complexity!ThemodelcheckingproblemforCTL!ThesatisÞabilityproblemforCTL !EachprocessisenabledinÞnitelyoften:!iGFruni!Noprocessstaysultimatelyinthecriticalsection:!iÂFGCSi=!iGFÂCSiDeÞnition:FairKripkestructureM=(S,T,I,AP,",F DeÞnition:Syntaxoffair-CTL!::=$|p(p%AP)|Â!|!&!|EfX!|AfX!|Ef!U!|Af!U!DeÞnition:SemanticsasafragmentofCTL!Let MkdeÞnedby:2k2k512k522k53ááá4321ppppÂpÂpÂpÂp!Mk,2k|=EGFpbutMk,2k52!|=EGFp!If!%CTL |á|!|)Proof:ComputationofFair={s%S|M,s|=Ef1}ComputetheSCCofMwithTarjanÕsalgorithm(intimeO(|M|)).LetS!betheunionofthe(nontrivial)SCCswhichintersecteachFi.Then,FairisthesetofstatesthatcanreachS!.Notethatreachability (Fair4!)andEf!U$=E!U(Fair4$)ItremainstodealwithAf!U$.RecallthatA!U$=ÂEGÂ$4ÂEÂ$U(Â!4Â$)ThisformulaalsoholdsforfairquantiÞcationsAfandEf.Hence,weonlyneedtocomputethesemanticsofEfG!.Proof:ComputationofEfG!LetM"betherestrictionofMto[[!]]f.ComputetheSCCofM"withTarjanÕsalgorithm(inlineartime).LetS!betheunionofthe(nontrivial)SCCsofM"whichintersecteachFi.Then,M,s|=EfG!i!M,s|=E!US!i!M",s|=EFS!.Thisisagainareachability