A.Madeira,M.A.Martins&L.S.Barbosa85Inthiscontext,thequestforsuitableno - PDF document

A.Madeira,M.A.Martins&L.S.Barbosa85Inthiscontext,thequestforsuitablenotionsofequivalenceandrenementbetweenmodelsofhy-bridisedlogicspecicationsbecomesfundamentaltotheenvisageddesignmethodology.Suchisthepurposeofthepresentpaper.Itscontributionisacharacterisationofbisimilarityandrenementforhybridisedlogicswhichrequiresaformofelementaryequivalence[Hod97]betweenbisimilarstates,asagenericformulationoftheusualinformalrequirementthattruthremainsinvariant.Clearlywhatelementaryequivalentmeansineachcaseboilsdowntothewaythesatisfactionrelationisdenedforthebaselogicusedtospecifythesemanticsoflocalcongurations.Thechoiceofsimilarityandbisimilaritytobaserenementandequivalenceof(modelsof)recon-gurablesystemsseemsquitestandardasanegrainedapproachtoobservationalmethodsforsystemscomparison.Thenotionofbisimulationandtheassociatedconductiveproofmethod,whichisnowper-vasiveinComputerScience,originatedinconcurrencytheoryduetotheseminalworkofDavidPark[Par81]andR.Milnerinthequestforanappropriatedenitionofobservationalequivalenceforcommu-nicatingprocesses.Buttheconceptalsoaroseindependentlyinmodallogicasarenementofnotionsofhomomorphismbetweenalgebraicmodels.Inthesequeltheconceptisrevisitedformodelsofhybridisedlogicsaddinguptothedesignmethodologymentionedabove.Thepaperisorganizedasfollows:Section2recallsinstitutionsasabstractcharacterisationsoflogicsandprovidesabrief,andsimplied,overviewofthehybridizationmethodproposedin[MMDB11,DM].Thisformsthecontextforthepaper'scontribution.Then,Section3introducesageneralnotionofbisimulationforhybridisedlogicsandcharacterizesthepreservationoflogicsatisfactionunderit.Section4followsasimilarpathbutfocussingonrenementaswitnessedbyasimulationrelation.2Background2.1InstitutionsAninstitutionisacategorytheoreticformalisation1ofalogicalsystem,encompassingsyntax,semanticsandsatisfaction.TheconceptwasputforwardbyGoguenandBurstall,intheendoftheseventies,inorderto“formalisetheformalnotionoflogicalsystems”,inresponsetothe“populationexplosionamongthelogicalsystemsusedinComputingScience”[GB92].Theuniversalcharacterofinstitutionsprovedeffectiveandresilientaswitnessedbythewidenumberoflogicsformalisedinthisframework.Examplesrangefromtheusuallogicsinclassicalmathemati-callogic(propositional,equational,rstorder,etc.),totheonesunderlyingspecicationandprogram-minglanguagesorusedfordescribingparticularsystemsfromdifferentdomains.Well-knownexam-plesincludeprobabilisticlogics[BKI02],quantumlogics[CMSS06],hiddenandobservationallogics[BD94,BH06],coalgebraiclogics[Cˆ06],aswellaslogicsforreasoningaboutprocessalgebras[MR07],functional[ST12,SM09]andimperativeprograminglanguages[ST12].Thetheoryofinstitutions(see[Dia08]foraextensiveaccount)wasmotivatedbytheneedtoabstractfromtheparticulardetailsofeachindividuallogicandcharacterisegenericissues,suchassatisfactionandcombinationoflogics,inverygeneralterms.InComputerScience,thisleadtothedevelopmentofasolidinstitution-independentspecicationtheory,onwhich,structuringandparameterisationmecha-nisms,requiredtoscaleupsoftwarespecicationmethods,aredened`onceandforall',irrespectiveoftheconcretelogicusedineachapplicationdomain.Thedenitionisrecalledbelow(e.g.,[GB92,Dia08])andillustratedwithafewexamplestowhichwereturnlaterinthepaper. 1Thelanguageofcategorytheory[Lan71]isusedtosetthesceneforinstitutions;categories,however,playnoroleinthepaper'scontribution. A.Madeira,M.A.Martins&L.S.Barbosa87Example2.2(Equationallogic)SignaturesintheinstitutionEQofequationallogicarepairs(S;F)whereSisasetofsortsymbolsandF=fFar !sjar 2S;s2Sgisafamilyofsetsofoperationsymbolsindexedbyaritiesar (forthearguments)andsortss(fortheresults).Signaturemorphismsmapbothcom-ponentsinacompatibleway:theyconsistofpairsj=(jst;jop):(S;F)!(S0;F0),wherejst:S!S0isafunction,andjop=fjopar !s:Far !s!F0jst(ar )!jst(s)jar 2S;s2Sgafamilyoffunctionsmappingoperationssymbolsrespectingarities.AmodelMforasignature(S;F)isanalgebrainterpretingeachsortsymbolsasacarriersetMsandeachoperationsymbols2Far !sasafunctionMs:Mar !Ms,whereMar istheproductofthearguments'carriers.Modelmorphismarehomomorphismsofalgebras,i.e.,S-indexedfamiliesoffunctionsfhs:Ms!M0sjs2Sgsuchthatforanym2Mar ,andforeachs2Far !s,hs(Ms(m))=M0s(har (m)).Foreachsignaturemorphismj,thereductofamodelM0,sayM=ModEQ(j)(M0)isdenedby(M)x=M0j(x)foreachsortandfunctionsymbolxfromthedomainsignatureofj.Themodelsfunctormapssignaturestocategoriesofalgebrasandsignaturemorphismstotherespectivereductfunctors.Sentencesareuniversalquantiedequations(8X)t=t0.Sentencetranslationsalongasignaturemorphismj:(S;F)!(S0;F0),i.e.,SenEQ(j):SenEQ(S;F)!SenEQ(S0;F0),replacesymbolsof(S;F)bytherespectivej-imagesin(S0;F0).Thesentencesfunctormapseachsignaturetothesetofrst-ordersentencesandeachsignaturemorphismtotherespectivesentencestranslation.ThesatisfactionrelationistheusualTarskiansatisfactiondenedrecursivelyonthestructureofthesentencesasfollows:Mj=(S;F)t=t0whenMt=Mt0,whereMtdenotestheinterpretationofthe(S;F)-termtinMdenedrecursivelybyMs(t1;:::;tn)=Ms(Mt1;:::;Mtn).Mj=(S;F)(8X)rwhenM0j=(S;F+X)rforany(S;F+X)-expansionM0ofM.Example2.3(PropositionalFuzzyLogic)Multi-valuedlogics[Got01]generaliseclassiclogicsbyre-placing,asitstruthdomain,the2-elementBooleanalgebra,bylargersetsstructuredascompleteresid-uatelattices.Theywereoriginallyformalisedasinstitutionsin[ACEGG90](butseealso[Dia11]forarecentreference).ResiduatelatticesaretuplesL=(L;;^;_;�;?; ),where(L;^;_;�;?)isalatticeorderedby,withcarrierL,with(binary)inmum(^)andsupremum(_),andbigestandsmallestelements�and?; isanassociativebinaryoperationsuchforanyelementsx;y;z2L:–x �=� x=x;–yzimpliesthat(x y)(x z);–thereexistsanelementx)zsuchthaty(x)z)iffx yz:TheresiduatelatticeLiscompleteifanysubsetSLhasinmumandsupremumdenotedbyVSandWS,respectively.GivenacompleteresiduatelatticeL,theinstitutionMVLLisdenedasfollows.MVLL-signaturearePL-signatures.SentencesofMVLLconsistofpairs(r;p)wherepisanelementofLandrisdenedasaPL-sentenceoverthesetofconnectivesf)_;�;?; g.AMVLL-modelMisafunctionM:FProp!L. A.Madeira,M.A.Martins&L.S.Barbosa89SenHI(j)(r?r0)=SenHI(j)(r)?SenHI(j)(r0),?2f_;^;)g;SenHI(j)(@ir)=@jNom(i)SenHI(r);SenHI(j)([l](r1;:::;rn))=[jMS(l)](SenHI(r1);:::;SenHI(rn));SenHI(j)(hli(r1;:::;rn))=hjMS(l)i(SenHI(r1);:::;SenHI(rn)).HI-modelsfunctor.ModelsofthehybridisedlogicHIcanberegardedas(L-)KripkestructureswhoseworldsareI-models.Formally(S;Nom;L)-modelsarepairs(M;W)whereWisa(Nom;L)-modelinREL;MisafunctionjWj!jModI(S)j.Ineachworld(M;W),fWnjn2NomgprovidesinterpretationsfornominalsinNom,whereasrelationsfWljl2Ln;n2wginterpretemodalitiesL.WedenoteM(w)simplybyMw.Thereductdenitionisliftedfromthebaseinstitution:thereductofaD0-model(M0;W0)alongasignaturemorphismj=(jSig;jNom;jMS):D!D0,denotedbyModHI(j)(M0;W0),istheD-model(M;W)suchthatWisthe(jNom;jMS)-reductofW0;i.e.–jWj=jW0j;–foranyn2Nom;Wn=W0jNom(n);–foranyl2L,Wl=W0jMS(l);foranyw2jWj,Mw=ModI(jSig)(M0w):TheSatisfactionRelation.Let(S;Nom;L)2jSignHIjand(M;W)2jModHI(S;Nom;L)j.Foranyw2jWjwedene:(M;W)j=wriffMwj=Ir;whenr2SenI(S),(M;W)j=wiiffWi=w;wheni2Nom,(M;W)j=wr_r0iff(M;W)j=wror(M;W)j=wr0,(M;W)j=wr^r0iff(M;W)j=wrand(M;W)j=wr0,(M;W)j=wr)r0iff(M;W)j=wrimpliesthat(M;W)j=wr0,(M;W)j=w:riff(M;W)6j=wr,(M;W)j=w[l](x1;:::;xn)iffforany(w;w1;:::;wn)2Wlwehavethat(M;W)j=wixiforsome1in.(M;W)j=whli(x1;:::;xn)iffthereexists(w;w1;:::;wn)2Wlsuchthatand(M;W)j=wixiforany1in.(M;W)j=w@jriff(M;W)j=Wjr,Wewrite(M;W)j=riff(M;W)j=wrforanyw2jWj.AsexpectedHIisitselfaninstitution:Theorem2.1([MMDB11])LetD=(S;Nom;L)andD0=(S0;Nom0;L0)betwoHI-signaturesandj:D!D0amorphismofsignatures.Foranyr2SenHI(D),(M0;W0)2jModC(D0)j,andw2jWj,ModHI(j)(M0;W0)j=wriff(M0;W0)j=wSenHI(j)(r):Letusillustratethemethodbyapplyingittothethreeinstitutionsdescribedabove. 92Bisimilarityandrenementforhybrid(ised)logics(v)Foranyl2Lnif(w0;w01;:::;w0n)2W0jMS(l)andwBjw0,thenforeachk2f1;:::;ngthereisawk2jWj,suchthatwkBjw0kand(w;w1;:::;wn)2Wl.Thefollowingresultestablishesthat,forquantier-freehybridisations,the(local)-hybridsatisfactionj=HIisinvariantunderj;Sen-bisimulations:Theorem3.1LetHIbeaquantier-freehybridizationoftheinstitutionIandj2SignHI(D;D0)asignaturemorphism.LetBjjWjjW0jbeaj;Sen-bisimulation.Then,foranywBjw0andforanyr2SenHI(D),(M;W)j=wriff(M0;W0)j=w0SenHI(j)(r):(4)Proof.Theproofisbyinductiononthestructureofthesentences.1.r=iforsomei2Nom:(M;W)j=wi,fdefn.ofj=wgWi=w,f(i)ofDefn3.2gW0j(i)=w0,fdefn.ofj=w0g(M0;W0)j=w0jNom(i),fdefnofSenHI(j)g(M0;W0)j=w0SenHI(j)(i)2.r2SenI(S):(M;W)j=wr,fdefn.ofj=wgMwj=Ir,fbyhypothesisMwjSigM0w0+Cor3.1gM0w0j=SenI(jSig)(r),fdefn.ofj=w0g(M0;W0)j=w0SenI(jSig)(r),fdefnofSenHI(j)g(M0;W0)j=w0SenHI(j)(r)3.r=x_x0forsomex;x02SenHI(D):(M;W)j=wx_x0,fdefn.ofj=wg(M;W)j=wxor(M;W)j=wx0,fI.H.g 94Bisimilarityandrenementforhybrid(ised)logics2Asdirectconsequenceoftheprevioustheoremwegetthefollowingcharacterisationofthepreser-vationof(global)satisfaction,j=HI,underj-bisimilarity:Corollary3.2OntheconditionsofTheorem3.1,let(M;W) j(M0;W0)witnessedbyatotalandsurjectivebisimulation.Then,(M;W)j=HIriff(M0;W0)j=HISenHI(j)(r):(5)Example3.1(BisimulationinHPL)LetusinstantiateDefn.3.2fortheHPLcase(cf.Ex.2.1),consideringj=idandSen0=SenI.AbisimulationBissuchthat(M;W)B(M0;W0),foranytwomodels(M;W);(M0;W0)2jModHPL(P;Nom;flg)j,if(i)foranyi2Nom,wBw0,w=Wiiffw0=W0i;(ii)MwM0w0,i.e.,bisimilarstatessatisfythesamesentences;(iii)foranyi2Nom,WiBW0i;(vi)forany(w;w1)2WlwithwBw0,thereisaw012jW0jsuchthatw1Bw01and(w1;w01)2W0l;(v)forany(w0;w01)2W0lwithwBw0,thereisaw12jWjsuchthatw1Bw01and(w1;w01)2W0l;.Notethatcondition(ii)isequivalenttosaythatbisimilarstateshaveassignedthesamesetofpropositions(foranyp2P,Mw(p)=�iffM0w0(p)=�).Asexpected,thisdenitioncorrespondsexactlytostandardbisimulationforpropositionalhybridlogic(see,e.g.[Cat05,Defn4.1.1]).Thedenitionofbisimulationcomputedinthepreviousexample,canalsocapturethecaseofpropo-sitionalmodallogic:justconsiderpuremodalsignatures(i.e.,withanemptysetofnominals),ascon-dition(i)istriviallysatised.Moreover,instantiatingTheorem3.1wegettheclassicalresultaboutpreservationofmodaltruthbybisimulation.Example3.2(BisimulationforHEQ)Considernowtheinstantiationof3.2forHEQ(cf.Ex2.6).Allonehastodoistoreplacecondition(iv)inDefn3.2byitsinstantiationforalgebras:twoalgebrasareelementarilyequivalentiftherespectivegeneratedvarietiescoincides[Gr¨a79].4RenementsforgenerichybridisedlogicsLetuscomebacktothegeneralcaseofarecongurablesystemdescribedbyasetofcongurationsandatransitionstructureentailingchangesfromonetoanother.Ifequivalenceofspecicationsofsuchsystemscorrespondstoanotionofbisimilarityinwhichbisimilarcongurationsareenforcedtobeelementaryequivalent,arenementrelationcorrespondstosimilarity.Thisentails,ontheonehand,preservation(butnotreection)oftransitions,i.e.,ofrecongurationsteps,fromtheabstracttothecon-cretesystem.And,ontheotherhand,ateachlocalconguration,preservationoftheoriginalpropertiesalonglocalrenement.Formally,Denition4.1LetHIbethehybridisationofaninstitutionI,j2SignHI(D;D0)asignaturemor-phismandSen0asubfunctorofSenI.Aj;Sen0-renementof(M;W)2ModHI(D)by(M0W0)2ModHI(D0)consistsofanon-emtpyrelationRSen0jjWjjW0jsuchthat,foranywRSen0jw0,(f.i)foranyi2Nom,ifWi=wthenW0jNom(i)=w0.(f.ii)MwSen0jM0w0. 96Bisimilarityandrenementforhybrid(ised)logicsthereexists(w;w1;:::;wn)2Wlsuchthat(M;W)j=wkxkforanyk2f1;:::;ng)fBy(f.iii),wehavewkRjw0kforanyk2f1;:::;ng+I.H.gthereexists(w0;w01;:::;w0n)2W0jMS(l)suchthat(M0;W0)j=w0kxkforanyk2f1;:::;ng,fdefn.ofj=w0g(M0;W0)j=w0hjMS(l)i(SenHI(j)(x1);:::;SenHI(j)(xn)),fdefn.ofSenHI(j)g(M0;W0)j=w0SenHI(j)(hli(x1;:::;xn))2Corollary4.1IntheconditionsofTh4.1,foranyr2SenHI+(D),ifRjissurjective,then(M;W)j=rimpliesthat(M0;W0)j=SenHI(j)(r):Thefollowingexamplesillustraterenementsituationsinthissetting.Example4.1(RenementinHMVLL)Figure4.1illustratesanexampleofaSen0-renementinHMVLL4,forL4representedinFigure4.1.ConsiderSen0SenIrestrictingthebasesentencestopropositions,i.e.,Sen0(LProp)=f(p;l)jp2LPropandl2L4g:Conditions(f.i)and(f.iii)areobviouslysatised.In Figure1:RenementinHMVLLwhatconcernsthevericationofcondition(f.ii)forwhich(p;l)2Sen0(LProp),Mwj=MVLL4LProp(p;l))M0w0j=MVLL4LProp(p;l),itissufcienttobethat,(Mwj=p)(M0w0j=p),p2LProp.Example4.2(RenementinHEQ)ConsiderastoresystemabstractlymodelledastheinitialalgebraAofthe((S;F);G)whereS=fmem;elemg,Fmemelem!mem=fwriteg,Fmem!mem=fdelgandFar !s=/0otherwiseandG=fdel(write(m;e))=mg.Supposeoneintendstorenethisstructureintoareadfunctioncongurableintwodifferentmodes:inoneofthemitreadstherstelementinthestore,intheotherthelast.Recongurationbetweenthetwoexecutionmodesisenforcedbyanexternaleventshift.Notethattheabstractmodelcanbeseenasthe�(S;F);/0;fshiftg
-hybridmodelM=(M;W),takingjWj=f?g,Wshift=/0andM?=A.Then,wetaketheinclusionmorphismjSig:(S;F),!(S;F0)whereF0 A.Madeira,M.A.Martins&L.S.Barbosa97 Figure2:RenementinHEQextendsFwithFmem!elem=readandFmem=femptyg.FortheenvisagedrenementletusconsiderthemodelM0=(M0;W0)whereW0=fs1;s2gandW0shift=f(s1;s2);(s2;s1)gandwhereMs1andMs2aretheinitialalgebrasoftheequationspresentedinFigure4.2.ItisnotdifculttoseethatR=f(?;s1);(?;s2)gisaj-renementrelation:conditions(f.i)and(f.iii)aretriviallyfullledand,condition(f.ii)isadirectconsequenceofpropertiesrepresentabilityoftheinitialmodels.5ConclusionsThepaperintroducednotionsofequivalenceandrenementbetweenmodelsofhybridisedlogicspeci-cations,i.e.specicationsformalisedinhybridisedversionsofbaselogicsusedtodescribeasystems'possiblecongurations.Thedenitionisparametriconpreciselythebaselogicrelevantforeachapplica-tion.Currentworkonthistopicincludesresearchonafullequivalencetheorem,showing,inparticular,inwhichcasesHIlogicalequivalenceentailsbisimilarity.AnothertopicconcernsthestudyoftypicalconstructionsonKripkestructures(e.g.boundedmorphismimages,substructuresanddisjointunions)andtheircharacterisationunderbisimilarityandrenement.AcknowledgementsWorkfundedbytheERDFthroughtheProgrammeCOMPETEandthePortugueseGovernmentthroughFCT-FoundationforScienceandTechnology,undercontractFCOMP-01-0124-FEDER-028923,CentrodeInvestigac¸˜aoeDesenvolvimentoemMatem´aticaeAplicac¸˜oesofUniversidadedeAveiro,anddoctoralgrantSFRH/BDE/33650/2009supportedbyFCTandCriticalSoftwareS.A.,Portugal.References[ACEGG90]JaumeAgusto-Cullell,FrancescEsteva,PereGarcia&LluisGodo(1990):FormalizingMultiple-ValuedLogicsasInstitutions.InB.Bouchon-Meunier,R.Yager&L.A.Zadeh,editors:UncertaintyinKnowledgeBases,IPMU90,LectNotesinComputerScience(512),Springer,pp.269–278,doi:10.1007/BFb0028112.[BD94]RodBurstall&RazvanDiaconescu(1994):Hidingandbehaviour:aninstitutionalapproach.InW.Roscoe,editor:AClassicalMind:EssaysinHonourofC.A.R.Hoare,Prentice-Hall,pp.75–92.[BH06]MichelBidoit&RolfHennicker(2006):Constructor-basedobservationallogic.J.Log.Algebr.Program.67(1-2),pp.3–51,doi:10.1016/j.jlap.2005.09.002.