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HIDDEN ALGEBRA ABSTRACT Many interesting constructions in hidden algebra seem particularly natural when viewed as constructions on colagebras: final models, behavioural equivalence, products and concurrent connections, and the cofree extensions used in parameterized hidden specifications. THE DECLARATIVE GROUP |
HiddenAlgebraCurrentResearchContentsofthispageHiddenalgebraa nalmodelof'abstractbehaviours'.Behaviouralequivalencearisesa ecificationofAbstractDataTypes.Springer-VerlagLectureNotesin tory,1994.[Revisedversion,1996.]Anotherpossibility,currently h,InA.W.Roscoe(ed.),AClassicalMind:EssaysinHonourofC.A.R.Hoa iddenAlgebraandCoalgebraTherelationshipbetweenhiddenalgebraa onsystemscanbeviewedascoalgebrasofanendofunctorthatusesthepo mparticularlynaturalwhenviewedasconstructionsoncolagebras:fi inimalrealisation.InMagneHaveraaenandOlafOweandOle-JohanDahl enAlgebra:parameterizedobjectsandinheritance.Draftofapaperpr hsaysthatthecategoryofcoalgebrasofaleftexactcomonadisatopos. ndcoalgebraHiddenalgebrawithsubsortsToposesofhiddenalgebrasH fobjectsthathaveobjectsasattributes).Onepromisingavenueofres esentedatthe12thWorkshoponAlgebraicDevelopmentTechniques,Jun nalmodels,behaviouralequivalence,productsandconcurrentconnec ospecifyconstructorsforcomplexclassesofobjects(i.e.,classeso earchistoextendthealgebraicandcoalgebraicnatureofhiddenalgeb rell,Atoposofhiddenalgebras.TechnicalReportPRG-TR18-97,Prog beingexplored,introducesorderedsortswhilemaintiningthelinkwi tions,andthecofreeextensionsusedinparameterizedhiddenspecifi coalgebraic.Hiddenalgebraimposesaconstraintonoperations:they ratoallowoperationsthattakemorethanonehiddensortasargument,w iththeideathatmodelshaveacoalgebraiccomponentthatdescribesth re,Prentice-HallInternationalSeriesinComputerScience,chapter JosephA.Goguen.Provingcorrectnessofrefinementandimplementati thcoalgebra.ToposesofHiddenModelsArecentdevelopmentisthereal ductsinthetopos,andthetruthobjectconsistsofabstractstatesand roofsintheinternallanguageofthetopos.Furtherreading:JamesWor cations.Theseconstructionsareoftenthedualoftheirclassicalalg e1997.SeealsotheBibliographypage.SubsortsThealgebraicspecifi sievesofcontexts.Oneinterestingpossibilitywouldbetolookatref sthekernelofuniquemorphismstothefinalmodel.Thisgivesafundame ebraiccounterparts.Forexample,hiddentheoriesdefinecovarietie nctor.However,thereisnothinginherentlycoalgebraicinhiddenalg maytakenomorethanoneargumentofhiddensort.Thismakesitawkwardt ndRazvanDiaconescu.HidingandBehaviour:anInstitutionalApproac rordefinitionandhandling,coercion,overwriting,multiplerepres isationthatthemodelsofahiddensignaturewithnoconstantsformsat ust1997.[HiddenAlgebraHome|Background|People|Bibliography]5A eading:GrantMalcolm.Behaviouralequivalence,bisimulation,andm s,andtheadjunctionwithunderlyingcarriersetsgivesacomonadicfu tingsystems,partialrecursivefunctions,multipleinheritance,er ebratoorderedsorts,correspondingtodifferentwaysoftreatingerr inementsasgeometricmorphisms,andperhapsconstructcoinductionp ch.HorstReichelnotedthecorrespondancebetweenobjectclassesand wersetfunctor.Manyinterestingconstructionsinhiddenalgebrasee a.Theextensionofhiddenalgebratohiddenordersortedalgebraisnot sheavesonan'anaemic'(withoutproducts)versionofaLawverecatego eirbehaviour,andanalgebraiccomponentforconstructors.Furtherr (eds.),RecentTrendsinDataTypeSpecifications.11thWorkshoponSp ryforthesignature.Behaviouralequivalencearisesnaturallyaspro isetoanendofunctorwhosecoalgebrasarethehiddenmodelsofthesign reallydifficult,butitcannotbetrivial,sinceitcoversnontermina entation,andmore.Infact,therearemanywaysofextendinghiddenalg 5,pages75-92.Prentice-HallInternational,1994.GrantMalcolmand ComputerScience,1996.CorinaCîrstea,CoalgebraSemanticsforHidd opos.ThisseemstoberelatedtoaresultofMoerdijkandMacLane'swhic Anotherwayofviewingthetoposisasaslicecategoryofthetoposofpre coalgebras,anditwasestablishedthateveryhiddensignaturegivesr ature.Moreover,forsignatureswithnoconstantsymbols,thereisafi ntalrelationshipbetweencoinductionandbisimulation,astransiti ebra;infact,thetreatmentofconstantsymbolsismorealgebraicthan orsincomputations.TwopossibilitiesaredescribedinRodBurstalla cationlanguageOBJhasanotionofsubsortbasedonordersortedalgebr ndcoalgebrahasbeenveryproductiveinsuggestingnewareasofresear rammingResearchGroup,OxfordUniversityComputingLaboratory,Aug on.TechnicalMonographPRG-114,OxfordUniversityComputingLabora ugust1997.............................More_Hidden_Algebra... HiddenAlgebraMoreHiddenAlgebraContentsof enotedifferentstates,thosestatescannotbe lyfortheothermethods.TheequationsinFLAGc .optail:State-]State.endthNowaclassofobj beusedtosolvequeries;ingeneral,coinducti hodsontheattributeup?,thoughingeneralequ dLogicParadigmsTheFLAGexampleaboveshowsh lementationoftheSTREAMtheoryisgivenbythe heflagisup.ModelsofFLAGhaveastateset,Fla agf2iffup?f1=up?f2b1RBoolb2iffb1=b2canbe senderobject:thSENDER||SUMisprSUM[SENDER .[HiddenAlgebraHome|Research|People|Bibl isanimplementationofSTREAM,wecanpassSEND andLogicParadigmsParameterizedprogrammin theleftandrightsidesareequalinallcontext rem.Behaviouralequivalence(equalityinall coinduction:ifanybehaviouralcongruencere eequationisbehaviourallysatisfied.Forexa befoundonthebibliographypage.TheObjectan stream,andthemethodtailsendsthisvaluetot ERasanargumenttotheparameterizedtheorySU ].varS:State.eqinputaddS=inputS.eqsumput nticsofparameterizedalgebraicspecificati ldeductionissoundforbehaviouralsatisfact ion,asthisallowstheobjectandlogicparadig ce,whereabehaviouralcongruenceisafamilyo onaboutclassesofobjectswhosebehaviouriss gAsimpleexampleofaHiddenTheoryisthefollo atisfiedbythefirstmodeldescribedabove,bu ughtheleftandrightsidesoftheequationmayd objectparadigm,weneedahiddenHerbrandtheo e.g.,ofobjectorienteddatabases)toreasoni (N,S)=sumS.eqput(N,addS)=addput(N,S).end methodstoraiseit,lowerit,orreverseit:Thi snotationisbasedonthelanguageOBJ,althoug sup,dnandrevchangethestateofflagobjects, ion.Thismeanswecanusestandardtechniquess uchasrewritingtoprovethingsabouthiddensp mstobecombinedatthesemanticlevel.Wedefin atequationaltechniquessuchasnarrowingcan adigmmayinvolvequeriestoanobjectoriented theadofthestreamtotherunningtotal,whichi thInterestingly,thesemanticsofhiddenpara ingaflagtwicehasnoeffect.Thisequationiss fcongruencerelationsforeachsortsuchthatt lsbehaviourallysatisfyrevrevF=F.Moreonco blecode)astheresultofthequery.Parameteri owerofhigherorderprogramminginasimplefir rate'astreambysendingnumberstoit:Here,th andup?isanattributethatsayswhetherornott eset{true,false},withtheobviousoperation sfromFLAG,wecanshowthatup?revrevF=up?Fwh ngthosevariables.Suchaqueryisbehavioural ariablesthatmakeseachtibehaviourallyequi bra(seetheBibliographypage),whichmeansth rieswithotherhiddentheories,weobtainthep ons;seetheBibliographypageformoredetails ly,aflagobjectcanbeeitherupordown,andhas anbethoughtofasdefiningtheeffectofthemet ails.CoinductionandBisimulationEquationa sallowsustoshowbehaviouralsatisfactionby inductionanditsrelationtobisimulationcan pecifiedbyequationswithuniversallyquanti bedefinedforotherlogicalconnectives.Ofpa rticularinterestisexistentialquantificat valenttoti'.Toestablishthecombinedlogic- onemightdescribeaholidaypackage(orasoftw wingspecificationofflagobjects.Intuitive meofthehiddensortofflagobjects.Themethod entheyarebehaviourallyequivalent,andsoth showntobeabehaviouralcongruence,andsince up?revrevF=up?F,itfollowsthatallFLAGmode databasewheretheresultingobjectisnotjust reFLAGisthenameofthemoduleandFlagisthena andattributes.Forexample,onemodelhasstat ichisvalidforallFLAGmodels.Morepowerfult hiddenHerbrandtheoremexistsforhiddenalge zedProgrammingByparameterizinghiddentheo thispageCoinductionandbisimulationObject eahiddenquerytobeasentenceoftheform(Exis tsX)t1=t1'and...andtn=tn'whereXisasetofv ngoverasingle'Herbranduniverse'termalgeb arepackage)thatonewants,andthenactuallyg ecification,sobehaviouralsatisfactionall fiedvariables.Behaviouralsatisfactioncan remtoreducereasoningoverarbitrarymodels( followingspecificationofobjectsthat'gene hestream.ThetailmethodisimportedfromtheS themethodupappen th FLAG is satisfytheequati sort Flag . ecifications.For op up?_ : Flag -> Bool . ra,asinordinaryl ops (up_) (dn_) (rev_) : Flag -> Flag . onwillalsobeneed var F : Flag . inagivenstreamca eq up? up F = true . TREAMtheoryinthe eq up? dn F = false . s,whereacontexti eq up? rev F = not up? F . sforup,dn,etc.An endth ationscanspecifyabstractbehaviours.Forex ample,theequationrevrevF=Fsaysthatrevers owsustoabstractawayfromimplementationdet echniquesfortheo th SUM[S :: STREAM] is wingtheory,which op sum : State -> Nat . htheintendedsema op add : State -> State . g,andoperationsc var N : Nat . ,dn,rev}*,theset var S : State . herelationsonvis eq sum(add(S)) = head(S) + sum(S) . latestheleftandr eq sum(tail(S)) = sum(S) . lysatisfiedifthe eq head(add(S)) = head(S) . Here,themethodad endth sgivenbytheattributesum.Amoreconcreteimp M,tobuildasystemthatsumsthevaluessentbya wedbyanattribute.Inotherwords,bothmodels distinguishedbya th SENDER is contexts)isthela pr STREAM . mple,thefamilyof op input : State -> Nat . ariablesandeacht op put : Nat State -> State . implehiddentheor var N : Nat . rs:thSTREAMissor var S : State . ectsthatkeeparun eq head(put(N,S)) = head(S) . emethodputsetsth eq input(put(N,S)) = N . ourallysatisfyth eq head(tail(S)) = input(S) . owhiddenalgebrac eq input(tail(S)) = input(S) . meterizedtheorie endth tnotbythesecond.However,bothmodelsbehavi evrevF=up?upFup?dnrevrevF=up?dnF...Altho ettheticketsandreservations(ortheexecuta stordersetting.Asanexample,webeginwithas retrieved,butactuallycreated.Forexample, iography]5August1997.................PbN |
ALGEBRA HIDING MACHINE |
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REFERENCES |
OXFORD UNIVERSITY Computing Laboratory The Declarative Group HIDDEN ALGEBRA
www.comlab.ox.ac.uk/oucl/groups/declarative/HSA/ha.html
MORE HIDDEN ALGEBRA
www.comlab.ox.ac.uk/oucl/groups/declarative/HSA/backgrnd.html
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