AFuntorialApproahtoRenementYoshikiKinoshitaandHiroshiWatanabeSemantisGroupEletrotehnialLaboratoryAmagasaki661-0974,Japan.AbstratWeintrodue-algebrasand-algebrasassemantidomainsfordatarenementofimperativeprogramminglanguages.Thefuntorialsemantisof-alulusisgivenbyusingtheadjuntionbetweentheategoryof-algebrasandtheategoryofsmallloallyorderedategories.Wedenethenotionofupwardanddownwardsimula-tionbetweentheinterpretationsofatomiommands,andrenementsbetweentheinterpretationsofommands,byusinglaxoroplaxtransformations.Weshowthattheadjuntionextendstoanenrihedadjuntioninthesenseoflax,andwhihprovidethefundamentalmahineryintheproofofsoundnessandompletenessofthedownwardsimulationwithrespettotherenement.1IntrodutionHoare,HeandSanders[3,1℄givesatheoryofdatarenementforimpera-tivelanguageswithnondeterminismandthereursionsthataresupposedtobeinterpretedasgreatestxpoints.Onthesettingofarelationalsemantis,theygaveadenitionofupwardanddownwardsimulationbetweentheinter-pretationsofatomisymbols,andrenementbetweentheinterpretationsofommands.TheyshowedSoundnessIfthereisadownwardoranupwardsimulationbetweentheinterpretationsofatomisymbols,thenthereisarenementbetweentheinterpretationsofommands,whiharenaturallyextendedfromthoseofatomisymbols.JointompletenessIfthereisarenementbetweentheinterpretationsofommands,whihareextensionsofthoseofatomisymbols,thenthereare1TheauthorsaknowledgethesupportofSTAthroughCOEbudgetforGlobalInformationProessingProjet.PreprintsubmittedtoElsevierPreprintanupwardsimulationandadownwardsimulationbetweentheinterpreta-tionsofatomisymbols.Thesoundnessassuresthevalidityofsimulationmethodstodatarenement.Later,\Singleompletenesstheoremratherthan\Jointompletenesstheo-remisarguedin[2,4℄.Theaimofthispaperistogiveafoundationofthefuntionalsemantisapproah,asin[5,7℄,tothealulus,whihweall-alulus,introduedin[3℄.FuntorialsemantisgivenbyadjuntionInterpretationsofbothatomisymbolsandommandsaregivenbyfuntors.Anexisteneofadjuntionenablestheextensionofaninterpretationofatomisymbolstothatofom-mands.SimulationsandrenementsbylaxoroplaxtransformationsAllthenotions,upwardanddownwardsimulationsandrenementsarelaxoroplaxtransformationsbetweenfuntors.SoundnessandCompletenessassuredbyenrihedadjuntionTheva-lidityofsoundnessandompletenessdependsonwhethertheadjuntionextendstoanenrihedadjuntioninthesenseoflaxoroplax.Wegive-algebrasasthesemantidomainforthe-alulus,andshowthattheforgetfulfuntorfromtheategory-Algof-algebrastotheategoryLoOrdofsmallloallyorderedategorieshasaleftadjoint.Thefuntorialsemantisof-alulusisgivenbyusingtheadjuntion.Wedeneupwardsimulationbyusingoplaxtransformationbetweenfuntors,anddownwardbylax.Wealsogivedenitionsofrenements.Weshowthattheadjuntionextendstoenrihedadjuntioninthesenseoflax,henewehavesoundnessandompletenessofdownwardsimulationwithrespettorenement.Theexisteneoftheenrihedadjuntionbetween-AlgandLoOrdliesintheheartofthispaper.Theenrihmentofadjuntionforlaxisalmosttrivialfromabstratategorytheorysine-algebraisanalgebraistrutureontheategoryofLoOrdl(see[4,5℄andreferenesinthere).Inordertoenrihtheadjuntioninthesenseofoplax,itisneededtomodifythedenitionof-algebra.Wealsogivethedenitionof-algebraasthesemantidomainforthe-aluluswhihhasnondeterminismandreursionsthataresupposedtobeinterpretedbyleastxpoints.Theenrihmentoftheadjuntionfor-algebrahasessentiallyarguedin[4℄,whihhasappliedtoshowthesafetypropertyofgarbageolletionin[6℄.Weproeedasfollows.InSetion2,wegivethedenitionsof-algebraand-algebra.Wealsostatethatthereisanadjuntionbetweentheategory-AlgandLoOrd,whihprovidesfuntorialsemantisof-alulusinSetion4.We2givedenitionsofrenements,upwardanddownwardsimulationsinSetion5.Theadjuntionbetween-AlgandLoOrdextendstoanenrihedadjuntioninthesenseoflax,whihassuresthesoundnessandompletenessofdownwardsimulationwithrespettorenementinSetion6.2-alulusand-alulusThe-alulusandthe-alulusarethealuliofimperativeprogramminglanguagewithnondeterminism.Theoriginalformofthe-alulushastheleastxpointoperator,butweintroduea\variablefreeformofthe-aluluswheretheleastupperboundoperatorWofountablearityreplaesthevariablesand.Similarly,weintrodueavariablefreeformofthe-aluluswherethegreatestlowerboundoperatorVofountablearityreplaesthevariablesandthegreatestxpointoperator.For-alulus,wewishVtobeappliedonlytodereasingsequenes,andthatompliatesthedenitionofsyntax,asweshallseebelow.2.1-alulusGivenasetAtomofatomisymbols,thesetCommoftermsofthe-alulusisdenedindutivelyasfollows.Ifa2Atomthena2Comm.skip2Comm.Ifand0areterms,then;0isaterm.?2CommIf=[!3i7!i2Comm℄isaountablesequeneofterms,Wisaterm.Termsarethoseobtainedbyatmostountablymanyappliationsoftheaboverules.Notethataountablesequeneoftermsisnothingbutamapfromtheset!ofnaturalnumberstoComm.Henethenotation[!3i7!i2Comm℄forountablesequenes.WedenetobetheleastpreorderonCommwhihsatisesthefollowing.Leastelement?.Unitof;skip;,skip;.;skip,;skip.3Assoiativity(;0);00;(0;00),;(0;00)(;0);00.LeastupperboundForeah=[!3i7!i2Comm℄,jWforeahj.((8j2!)j)impliesW.;w
本文标题:A Functorial Approach to Renement Yoshiki Kinoshit
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