177K - views

Possibility Theory and its Applications Where Do we Stand Didier Dubois and Henri Prade IRITCNRS Universite Paul Sabatier Toulouse Cedex France December Abstract This paper provides an overview

Possibility theory lies at the crossroads between fuzzy sets probability and nonmonotonic reasoning Possibility theory can be cast either in an ordinal or in a numerical setting Qualitative possibility theory is closely related to belief revision th

Embed :
Pdf Download Link

Download Pdf - The PPT/PDF document "Possibility Theory and its Applications ..." is the property of its rightful owner. Permission is granted to download and print the materials on this web site for personal, non-commercial use only, and to display it on your personal computer provided you do not modify the materials and that you retain all copyright notices contained in the materials. By downloading content from our website, you accept the terms of this agreement.

Possibility Theory and its Applications Where Do we Stand Didier Dubois and Henri Prade IRITCNRS Universite Paul Sabatier Toulouse Cedex France December Abstract This paper provides an overview






Presentation on theme: "Possibility Theory and its Applications Where Do we Stand Didier Dubois and Henri Prade IRITCNRS Universite Paul Sabatier Toulouse Cedex France December Abstract This paper provides an overview"— Presentation transcript:

2HistoricalBackgroundZadehwasnotthe rstscientisttospeakaboutformalisingnotionsofpossibility.ThemodalitiespossibleandnecessaryhavebeenusedinphilosophyatleastsincetheMiddle-AgesinEurope,basedonAristotle'sandTheophrastus'works[22].MorerecentlytheybecamethebuildingblocksofModalLogicsthatemergedatthebeginningoftheXXthcenturyfromtheworksofC.I.Lewis(seeHughesandCresswell[31]).Inthisapproach,possibilityandnecessityareall-or-nothingnotions,andhandledatthesyntacticlevel.Morerecently,andindependentlyfromZadeh'sview,thenotionofpossibility,asopposedtoprobability,wascentralintheworksofoneeconomist,andinthoseoftwophilosophers.G.L.S.ShackleAgradednotionofpossibilitywasintroducedasafull- edgedap-proachtouncertaintyanddecisioninthe1940-1970'sbytheEnglisheconomistG.L.S.Shackle[127],whocalleddegreeofpotentialsurpriseofaneventitsdegreeofimpossibil-ity,thatis,thedegreeofnecessityoftheoppositeevent.Shackle'snotionofpossibilityisbasicallyepistemic,itisa\characterofthechooser'sparticularstateofknowledgeinhispresent."Impossibilityisunderstoodasdisbelief.Potentialsurpriseisvaluedonadisbeliefscale,namelyapositiveintervaloftheform[0;y],whereydenotestheabsoluterejectionoftheeventtowhichitisassigned.Incaseeverythingispossible,allmutuallyexclusivehypotheseshavezerosurprise.Atleastoneelementaryhypothesismustcarryzeropotentialsurprise.Thedegreeofsurpriseofanevent,asetofelementaryhypotheses,isthedegreeofsurpriseofitsleastsurprisingrealisation.Shacklealsointroducesanotionofconditionalpossibility,wherebythedegreeofsurpriseofaconjunctionoftwoeventsAandBisequaltothemaximumofthedegreeofsurpriseofA,andofthedegreeofsurpriseofB,shouldAprovetrue.ThedisbeliefnotionintroducedlaterbySpohn[130]employsthesametypeofconventionaspotentialsurprise,butusingthesetofnaturalintegersasadisbeliefscale;hisconditioningruleusesthesubtractionofnaturalintegers.D.LewisInhis1973book[109]thephilosopherDavidLewisconsidersagradednotionofpossibilityintheformofarelationbetweenpossibleworldshecallscomparativepossibility.Heequatesthisconceptofpossibilitytoanotionofsimilaritybetweenpossibleworlds.Thisnon-symmetricnotionofsimilarityisalsocomparative,andismeanttoexpressstatementsoftheform:aworldjisatleastassimilartoworldiasworldkis.Comparativesimilarityofjandkwithrespecttoiisinterpretedasthecomparativepossibilityofjwithrespecttokviewedfromworldi.Suchrelationsareassumedtobecompletepre-orderingsandareinstrumentalinde ningthetruthconditionsofcounterfactualstatements.Comparativepossibilityrelationsobeythekeyaxiom:foralleventsA;B;C,ABimpliesC[AC[B:Thisaxiomwaslaterindependentlyproposedbythe rstauthor[42]inanattempttoderiveapossibilisticcounterparttocomparativeprobabilities.Independently,theconnection2 Possibilitytheoryisdrivenbytheprincipleofminimalspeci city.Itstatesthatanyhypothesisnotknowntobeimpossiblecannotberuledout.Apossibilitydistributionissaidtobeatleastasspeci casanother0ifandonlyifforeachstateofa airss:(s)0(s)(Yager[141]).Then,isatleastasrestrictiveandinformativeas0.Inthepossibilisticframework,extremeformsofpartialknowledgecanbecaptured,namely:Completeknowledge:forsomes0;(s0)=1and(s)=0;8s6=s0(onlys0ispossible)Completeignorance:(s)=1;8s2S(allstatesarepossible).Givenasimplequeryoftheform\doeseventAoccur?"whereAisasubsetofstates,theresponsetothequerycanbeobtainedbycomputingdegreesofpossibilityandnecessity,respectively(ifthepossibilityscaleL=[0;1]):(A)=sups2A(s);N(A)=infs=2A1�(s):(A)evaluatestowhatextentAisconsistentwith,whileN(A)evaluatestowhatextentAiscertainlyimpliedby.Thepossibility-necessitydualityisexpressedbyN(A)=1�(Ac),whereAcisthecomplementofA.Generally,(S)=N(S)=1and(;)=N(;)=0.Possibilitymeasuressatisfythebasic\maxitivity"property(A[B)=max((A);(B)).Necessitymeasuressatisfyanaxiomdualtothatofpossi-bilitymeasures,namelyN(A\B)=min(N(A);N(B)).Onin nitespaces,theseaxiomsmustholdforin nitefamiliesofsets.Humanknowledgeisoftenexpressedinadeclarativewayusingstatementstowhichbeliefdegreesareattached.Itcorrespondstoexpressingconstraintstheworldissupposedtocomplywith.Certainty-quali edpiecesofuncertaininformationoftheform\Aiscertaintodegree "canthenbemodeledbytheconstraintN(A) .Theleastspeci cpossibilitydistributionre ectingthisinformationis[67]:(A; )(s)=1;ifs2A1� otherwise(1)Thispossibilitydistributionisakey-buildinglocktoconstructpossibilitydistributions.Acquiringfurtherpiecesofknowledgeleadstoupdating(A; )intosome(A; ).ApartfromandN,ameasureofguaranteedpossibilitycanbede ned[71,54]:(A)=infs2A(s).ItestimatestowhatextentallstatesinAareactuallypossibleaccordingtoevidence.(A)canbeusedasadegreeofevidentialsupportforA.Uncertainstatementsoftheform\Aispossibletodegree "oftenmeanthatallrealizationsofAarepossibletodegree .Theycanthenbemodeledbytheconstraint(A) .Itcorrespondstotheideaofobservedevidence.Thistypeofinformationisbetterexploitedbyassuminganinformationalprincipleoppositetotheoneofminimalspeci city,namely,4 Animportantexampleofapossibilitydistributionisthefuzzyinterval,whichisafuzzysetofthereallinewhosecutsareintervals[62,67].Thecalculusoffuzzyintervalsisanextensionofintervalarithmeticsbasedonapossibilisticcounterpartofacomputationofrandomvariable.TocomputetheadditionoftwofuzzyintervalsAandBonehastocomputethemembershipfunctionofABasthedegreeofpossibilityAB(z)=(f(x;y):x+y=zg),basedonthepossibilitydistributionmin(A(x);B(y)).Thereisalargeliteratureonpossibilisticintervalanalysis;see[58]forasurveyofXXthcenturyreferences.4QualitativePossibilityTheoryThissectionisrestrictedtothecaseofa nitestatespaceS,supposedtobethesetofin-terpretationsofaformalpropositionallanguage.Inotherwords,SistheuniverseinducedbyBooleanattributes.Aplausibilityorderingisacompletepre-orderofstatesdenotedby,whichinducesawell-orderedpartitionfE1;;EngofS.Itisthecomparativecounterpartofapossibilitydistribution,i.e.,ss0ifandonlyif(s)(s0).Indeeditismorenaturaltoexpectthatanagentwillsupplyordinalratherthannumericalinfor-mationabouthisbeliefs.ByconventionE1containsthemostnormalstatesoffact,Entheleastplausible,ormostsurprisingones.Denotingbymax(A)anymostplausiblestates02A,ordinalcounterpartsofpossibilityandnecessitymeasures[42]arethende nedasfollows:fsg;foralls2SandABifandonlyifmax(A)max(B)ANBifandonlyifmax(Bc)max(Ac):PossibilityrelationsarethoseofLewis[109]andsatisfyhischaracteristicpropertyABimpliesC[AC[Bwhilenecessityrelationscanalsobede nedasANBifandonlyifBcAc,andsatisfyasimilaraxiom:ANBimpliesC\ANC\B:Thelattercoincidewithepistemicentrenchmentrelationsinthesenseofbeliefrevisiontheory[92,69].Conditioningapossibilityrelationbyannon-impossibleeventC�;meansderivingarelationCsuchthatACBifandonlyifA\CB\C:ThenotionofindependenceforcomparativepossibilitytheorywasstudiedinDuboisetal.[46],forindependencebetweenevents,andBenAmoretal.[11]betweenvariables.6 4.2PossibilisticLogicQualitativepossibilityrelationscanberepresentedby(andonlyby)possibilitymeasuresrangingonanytotallyorderedsetL(especiallya niteone)[42].Thisabsoluterepresen-tationonanordinalscaleisslightlymoreexpressivethanthepurelyrelationalone.Whenthe nitesetSislargeandgeneratedbyapropositionallanguage,qualitativepossibilitydistributionscanbeecientlyencodedinpossibilisticlogic[90,59,75].ApossibilisticlogicbaseKisasetofpairs(; ),whereisaBooleanexpressionand isanelementofL.ThispairencodestheconstraintN() whereN()isthedegreeofnecessityofthesetofmodelsof.Eachprioritizedformula(; )hasafuzzysetofmodels(describedinSection3)andthefuzzyintersectionofthefuzzysetsofmodelsofallprioritizedformulasinKyieldstheassociatedplausibilityorderingonS.Syntacticdeductionfromasetofprioritizedclausesisachievedbyrefutationusinganextensionofthestandardresolutionrule,whereby(_ ;min( ; ))canbederivedfrom(_; )and( _:; ).Thisrule,whichevaluatesthevalidityofaninferredpropositionbythevalidityoftheweakestpremiss,goesbacktoTheophrastus,adiscipleofAristotle.Possibilisticlogicisaninconsistency-tolerantextensionofpropositionallogicthatprovidesanaturalsemanticsettingformechanizingnon-monotonicreasoning[17],withacomputationalcomplexityclosetothatofpropositionallogic.Anothercompactrepresentationofqualitativepossibilitydistributionsisthepossibilis-ticdirectedgraph,whichusesthesameconventionsasBayesiannets,butreliesonanordinalnotionofconditionalpossibility[67](BjA)=1;if(B\A)=(A)(B\A)otherwise.Jointpossibilitydistributionscanbedecomposedintoaconjunctionofconditionalpossi-bilitydistributions(usingminimum)inawaysimilartoBayesnets[14].Itisbasedonasymmetricnotionofqualitativeindependence(B\A)=min((A);(B))thatisweakerthanthecausal-likecondition(BjA)=(B)[46].BenAmorandBenferhat[12]investi-gatethepropertiesofqualitativeindependencethatenablelocalinferencestobeperformedinpossibilisticnets.Uncertaintypropagationalgorithmssuitableforpossibilisticgraphicalstructureshavebeenstudied[13].Othertypesofpossibilisticlogiccanalsohandleconstraintsoftheform() ,or() [75].Possibilisticlogiccanbeextendedtologicprogramming[1,10],similarityreasoning[2],andmany-valuedlogicasextensivelystudiedbyGodoandcolleagues[38].4.3Decision-theoreticfoundationsZadeh[142]hintedthat\sinceourintuitionconcerningthebehaviourofpossibilitiesisnotveryreliable",ourunderstandingofthem\wouldbeenhancedbythedevelopmentofanaxiomaticapproachtothede nitionofsubjectivepossibilitiesinthespiritofaxiomatic8 1.(XS;)isacompletepreorder.2.Therearetwoactssuchthatfg.3.8A;8gandhconstant,8f;ghimpliesgAfhAf.4.Iffisconstant,fhandghimplyf^gh.5.Iffisconstant,hfandhgimplyhf_g.thenthereexistsa nitechainL,anL-valuedmonotonicset-function onSandanL-valuedutilityfunctionu,suchthatisrepresentablebyaSugenointegralofu(f)withrespectto .Moreover isanecessity(resp.possibility)measureassoonasproperty(4)(resp.(5))holdsforallacts.ThepreferencefunctionalisthenW�(f)(resp.W+(f)).Axioms(4-5)contradictexpectedutilitytheory.Theybecomereasonableifthevaluescaleis nite,decisionsareone-shot(nocompensation)andprovidedthatthereisabigstepbetweenanylevelinthequalitativevaluescaleandtheadjacentones.Inotherwords,thepreferencepatternfhalwaysmeansthatfissigni cantlypreferredtoh,tothepointofconsideringthevalueofhnegligibleinfrontofthevalueoff.Theaboveresultprovidesdecision-theoreticfoundationsofpossibilitytheory,whoseaxiomscanthusbetestedfromobservingthechoicebehaviorofagents.See[49]foranotherapproachtocomparativepossibilityrelations,morecloselyrelyingonSavageaxioms,butgivingupanycomparabilitybetweenutilityandplausibilitylevels.Thedrawbackoftheseandotherqualitativedecisioncriteriaistheirlackofdiscriminationpower[47].Toovercomeit,re nementsofpossibilisticcriteriawererecentlyproposed,basedonlexicographicschemes[89].Thesenewcriteriaturnouttoberepresentablebyaclassical(butbig-stepped)expectedutilitycriterion.Qualitativepossibilisticcounterpartsofin uencediagramsfordecisiontreeshavebeenrecentlyinvestigated[98].Morerecently,possibilisticqualitativebipolardecisioncriteriahavebeende ned,ax-iomatized[48]andempiricallytested[23].TheyarequalitativecounterpartsofcumulativeprospecttheorycriteriaofKahnemanandTverski[133].5QuantitativePossibilityTheoryThephrase\quantitativepossibility"referstothecasewhenpossibilitydegreesrangeintheunitinterval.Inthatcase,aprecisearticulationbetweenpossibilityandprobabilitytheoriesisusefultoprovideaninterpretationtopossibilityandnecessitydegrees.Severalsuchinterpretationscanbeconsistentlydevised:adegreeofpossibilitycanbeviewedasanupperprobabilitybound[70],andapossibilitydistributioncanbeviewedasalikelihoodfunction[60].ApossibilitymeasureisalsoaspecialcaseofaShaferplausibilityfunction[126].Followingaverydi erentapproach,possibilitytheorycanaccountforprobabilitydistributionswithextremevalues,in nitesimal[130]orhavingbigsteps[16].Thereare10 5.2ConditioningTherearetwokindsofconditioningthatcanbeenvisageduponthearrivalofnewinforma-tionE.The rstmethodpresupposesthatthenewinformationaltersthepossibilitydis-tributionbydeclaringallstatesoutsideEimpossible.Theconditionalmeasure(:jE)issuchthat(BjE)(E)=(B\E).ThisisformallyDempsterruleofconditioningofbelieffunctions,specialisedtopossibilitymeasures.Theconditionalpossibilitydistributionrepresentingtheweightedsetofcon denceintervalsis,(sjE)=((s) (E);ifs2E0otherwise.)DeBaetsetal.[33]provideamathematicaljusti cationofthisnotioninanin nitesetting,asopposedtothemin-basedconditioningofqualitativepossibilitytheory.Indeed,themaxitivityaxiomextendedtothein nitesettingisnotpreservedbythemin-basedconditioning.Theproduct-basedconditioningleadstoanotionofindependenceoftheform(B\E)=(B)(E)whosepropertiesareverysimilartotheonesofprobabilisticindependence[34].Anotherformofconditioning[73,37],moreinlinewiththeBayesiantradition,considersthatthepossibilitydistributionencodesimprecisestatisticalinformation,andeventEonlyre ectsafeatureofthecurrentsituation,notofthestateingeneral.Thenthevalue(BjjE)=supfP(BjE);P(E)�0;PgistheresultofperformingasensitivityanalysisoftheusualconditionalprobabilityoverP()(Walley[135]).Interestingly,theresultingset-functionisagainapossibilitymeasure,withdistribution(sjjE)=(max((s);(s) (s)+N(E));ifs2E0otherwise.)Itisgenerallylessspeci cthanonE,asclearfromtheaboveexpression,andbecomesnon-informativewhenN(E)=0(i.e.ifthereisnoinformationaboutE).Thisisbecause(jjE)isobtainedfromthefocusingofthegenericinformationoverthereferenceclassE.Onthecontrary,(jE)operatesarevisionprocessonduetoadditionalknowledgeassertingthatstatesoutsideEareimpossible.SeeDeCooman[37]foradetailedstudyofthisformofconditioning.5.3Probability-possibilitytransformationsTheproblemoftransformingapossibilitydistributionintoaprobabilitydistributionandconverselyismeaningfulinthescopeofuncertaintycombinationwithheterogeneoussources(somesupplyingstatisticaldata,otherlinguisticdata,forinstance).Itisusefultocastallpiecesofinformationinthesameframework.ThebasicrequirementistorespecttheconsistencyprincipleP.TheproblemistheneithertopickaprobabilitymeasureinP(),ortoconstructapossibilitymeasuredominatingP.12 de nedasP(si)=Pj=i;:::;mpj,withpj=P(fsjg).NotethatPisakindofcumula-tivedistributionofP,alreadyknownasaLorentzcurveinthemathematicalliterature[112].Ifthereareequiprobableelements,theunicityofthetransformationispreservedifequipossibilityofthecorrespondingelementsisenforced.Inthiscaseitisabijectivetransformationaswell.Recently,thistransformationwasusedtoprovearathersurpris-ingagreementbetweenprobabilisticindeterminatenessasmeasuredbyShannonentropy,andpossibilisticnon-speci city.Namelyitispossibletocompareprobabilitymeasureson nitesetsintermsoftheirrelativepeakedness(aconceptadaptedfromBirnbaum[21])bycomparingtherelativespeci cityoftheirpossibilistictransforms.NamelyletPandQbetwoprobabilitymeasuresonSandP,Qthepossibilitydistributionsinducedbyourtransformation.ItcanbeprovedthatifPQ(i.e.PislesspeakedthanQ)thentheShannonentropyofPishigherthantheoneofQ[55].ThisresultgivesomegroundstotheintuitionsdevelopedbyKlir[106],withoutassuminganycommensurabilitybetweenentropyandspeci cityindices.PossibilitydistributionsinducedbypredictionintervalsInthecontinuouscase,movingfromobjectiveprobabilitytopossibilitymeansadoptingarepresentationofuncer-taintyintermsofpredictionintervalsaroundthemodeviewedasthe\mostfrequentvalue".Extractingapredictionintervalfromaprobabilitydistributionordevisingaprobabilisticinequalitycanbeviewedasmovingfromaprobabilistictoapossibilisticrepresentation.Namelysupposeanon-atomicprobabilitymeasurePontherealline,withunimodalden-sityp,andsupposeonewishestorepresentitbyanintervalIwithaprescribedlevelofcon denceP(I)= ofhittingit.Themostnaturalchoiceisthemostpreciseintervalensuringthislevelofcon dence.Itcanbeprovedthatthisintervalisoftheformofacutofthedensity,i.e.I =fs;p(s)gforsomethreshold.Movingthedegreeofcon dencefrom0to1yieldsanestedfamilyofpredictionintervalsthatformapossibilitydistributionconsistentwithP,themostspeci coneactually,havingthesamesupportandthesamemodeasPandde nedby([84]):(infI )=(supI )=1� =1�P(I )Thiskindoftransformationagainyieldsakindofcumulativedistributionaccordingtotheorderinginducedbythedensityp.Similarconstructscanbefoundinthestatisticalliterature(Birnbaum[21]).MorerecentlyMaurisetal.[81]noticedthatstartingfromanyfamilyofnestedsetsaroundsomecharacteristicpoint(themean,themedian,...),theaboveequationyieldsapossibilitymeasuredominatingP.Well-knowninequalitiesofprobabilitytheory,suchasthoseofChebyshevandCamp-Meidel,canalsobeviewedaspossibilisticapproximationsofprobabilityfunctions.Itturnsoutthatforsymmetricunimodaldensities,eachsideoftheoptimalpossibilistictransformisaconvexfunction.Givensuchaprobabilitydensityonaboundedinterval[a;b],thetriangularfuzzynumberwhosecoreisthemodeofpandthesupportis[a;b]isthusapossibilitydistribution14 Possibilitytheoryanddefuzzi cationPossibilisticmeanvaluescanbede nedusingChoquetintegralswithrespecttopossibilityandnecessitymeasures[65,37],andcomeclosetodefuzzi cationmethods[134].Afuzzyintervalisafuzzysetofrealswhosemem-bershipfunctionisunimodalandupper-semicontinuous.Its -cutsareclosedintervals.InterpretingafuzzyintervalM,associatedtoapossibilitydistributionM,asafamilyofprobabilities,upperandlowermeanvaluesE(M)andE(M),canbede nedas[66]:E(M)=Z10infM d ;E(M)=Z10supM d whereM isthe -cutofM.ThenthemeanintervalE(M)=[E(M);E(M)]ofMistheintervalcontainingthemeanvaluesofallrandomvariablesconsistentwithM,thatisE(M)=fE(P)jP2P(M)g;whereE(P)representstheexpectedvalueassociatedtotheprobabilitymeasureP.Thatthe\meanvalue"ofafuzzyintervalisanintervalseemstobein-tuitivelysatisfactory.Particularlythemeanintervalofa(regular)interval[a;b]isthisintervalitself.Theupperandlowermeanvaluesarelinearwithrespecttotheaddi-tionoffuzzynumbers.De netheadditionM+NasthefuzzyintervalwhosecutsareM +N =fs+t;s2M ;t2N gde nedaccordingtotherulesofintervalanalysis.ThenE(M+N)=E(M)+E(N),andsimilarlyforthescalarmultiplicationE(aM)=aE(M),whereaMhasmembershipgradesoftheformM(s=a)fora6=0.Inviewofthisproperty,itseemsthatthemostnaturaldefuzzicationmethodisthemiddlepoint^E(M)ofthemeaninterval(originallyproposedbyYager[140]).Otherdefuzzi cationtechniquesdonotgen-erallypossessthiskindoflinearityproperty.^E(M)hasanaturalinterpretationintermsofsimulationofafuzzyvariable[28],andisthemeanvalueofthepignistictransformationofM.Indeeditisthemeanvalueoftheempiricalprobabilitydistributionobtainedbytherandomprocessde nedbypickinganelement intheunitintervalatrandom,andthenanelementsinthecutM atrandom.6SomeApplicationsPossibilitytheoryhasnotbeenthemainframeworkforengineeringapplicationsoffuzzysetsinthepast.However,onthebasisofitsconnectionstosymbolicarti cialintelligence,todecisiontheoryandtoimprecisestatistics,weconsiderthatithassigni cantpotentialforfurtherapplieddevelopmentsinanumberofareas,includingsomewherefuzzysetsarenotyetalwaysaccepted.Onlysomedirectionsarepointedouthere.1.Possibilitytheoryalsoo ersaframeworkforpreferencemodelinginconstraint-directedreasoning.Bothprioritizedandsoftconstraintscanbecapturedbypos-sibilitydistributionsexpressingdegreesoffeasibilityratherthanplausibility[51].Possibilityo ersanaturalsettingforfuzzyoptimizationwhoseaimistobalancethelevelsofsatisfactionofmultiplefuzzyconstraints(insteadofminimizinganoverall16 [79].Thisapproachcanbesystematizedtofuzzyoruncertainversionsofformalconceptanalysis.GeneralisedpossibilisticlogicPossibilisticlogic,initsbasicversion,attachesdegreesofnecessitytoformulas,whichturnthemintogradedmodalformulasofthenecessitykind.Howeveronlyconjunctionofweightedformulasareallowed.Yetveryearlywenoticedthatitmakessensetoextendthelanguagetowardshandingconstraintsonthedegreeofpossibilityofaformula.Thisrequiresallowingfornegationanddisjunctionsofnecessity-quali edproposition.Thisextension,stillunderstudy[78],putstogethertheKDmodallogicandbasicpossibilisticlogic.Recentlyithasbeenshownthatnon-monotoniclogicprograminglanguagescanbetranslatedintogeneralizedpossibilisticlogic,makingthemeaningofnegationbydefaultinrulemuchmoretransparent[85].Thismovefrombasictogeneralizedpossibilisticlogicalsoenablesfurtherextensionstothemulti-agentandthemulti-sourcecase[76]tobeconsidered.Besides,ithasbeenrecentlyshownthataSugenointegralcanbealsorepresentedintermsofpossibilisticlogic,whichenablesustolaybarethelogicaldescriptionofanaggregationprocess[80].Qualitativecapacitiesandpossibilitymeasures.Whileanumericalpossibilitymea-sureisequivalenttoaconvexsetofprobabilitymeasures,itturnsoutthatinthequalitativesetting,amonotoneset-functioncanberepresentedbymeansofafamilyofpossibilitymeasures[5,43].ThislineofresearchenablesqualitativecounterpartsofresultsinthestudyofChoquetcapacitiesinthenumericalsettingstobeestab-lished.Especially,amonotoneset-functioncanbeseenasthecounterpartofabelieffunction,andvariousconceptsofevidencetheorycanbeadaptedtothissetting[119].Sugenointegralcanbeviewedasalowerpossibilisticexpectationinthesenseofsection4.3[43].Theseresultsenablethestructureofqualitativemonotonicset-functionstobelaidbare,withpossibleconnectionwithneighborhoodsemanticsofnon-regularmodallogics.RegressionandkrigingFuzzyregressionanalysisisseldomenvisagedfromthepointofviewofpossibilitytheory.OneexceptionisthepossibilisticregressioninitiatedbyTanakaandGuo[132],wheretheideaistoapproximatepreciseorset-valueddatainthesenseofinclusionbymeansofaset-valuedorfuzzyset-valuedlinearfunctionobtainedbymakingthelinearcoecientsofalinearfunctionfuzzy.ThealternativeapproachisthefuzzyleastsquaresofDiamond[40]wherefuzzydataareinterpretedasfunctionsandacrispdistancebetweenfuzzysetsisoftenused.However,fuzzydataarequestionablyseenasobjectiveentities[110].Theintroductionofpossibilitytheoryinregressionanalysisoffuzzydatacomesdowntoanepistemicviewoffuzzydatawherebyonetriestoconstructtheenvelopeofalllinearregressionresultsthatcouldhavebeenobtained,hadthedatabeenprecise[44].Thisviewhasbeenappliedtothekrigingproblemingeostatistics[111].Anotheruseofpossibilitytheoryconsistsin18 [12]N.BenAmor,S.Benferhat,Graphoidpropertiesofqualitativepossibilisticindepen-dencerelations.Int.J.Uncert.Fuzz.&Knowl.-B.Syst.13,59-97,2005.[13]N.BenAmor,S.Benferhat,K.Mellouli,Anytimepropagationalgorithmformin-basedpossibilisticgraphs.SoftComput.,8,50-161,2003.[14]S.Benferhat,D.Dubois,L.GarciaandH.Prade,Onthetransformationbetweenpos-sibilisticlogicbasesandpossibilisticcausalnetworks.Int.J.ApproximateReasoning,29:135-173,2002.[15]S.Benferhat,D.DuboisandH.Prade,Nonmonotonicreasoning,conditionalobjectsandpossibilitytheory,Arti cialIntelligence,92:259-276,1997.[16]S.Benferhat,D.DuboisandH.PradePossibilisticandstandardprobabilisticseman-ticsofconditionalknowledgebases,J.LogicComput.,9,873-895,1999.[17]S.Benferhat,D.Dubois,andH.Prade,PracticalHandlingofexception-taintedrulesandindependenceinformationinpossibilisticlogicAppliedIntelligence,9,101-127,1998.[18]S.Benferhat,D.Dubois,H.PradeandM.-A.Williams.Apracticalapproachtore-visingprioritizedknowledgebases,StudiaLogica,70,105-130,2002.[19]S.Benferhat,D.Dubois,S.Kaci,H.Prade,Bipolarpossibilitytheoryinpreferencemodeling:Representation,fusionandoptimalsolutions.InformationFusion,7,135-150,2006.[20]S.Benferhat,D.Dubois,S.Kaci,H.Prade,Modelingpositiveandnegativeinforma-tioninpossibilitytheory.Int.J.Intell.Syst.,23,1094-1118,2008.[21]Z.W.BirnbaumOnrandomvariableswithcomparablepeakedness.Ann.Math.Stat.19,76-81,1948.[22]I.M.Bochenski,LaLogiquedeTheophraste.Librairiedel'UniversitedeFribourgenSuisse.1947.[23]J.-F.Bonnefon,D.Dubois,H.Fargier,S.Leblois,Qualitativeheuristicsforbalancingtheprosandthecons,TheoryandDecision,65,71-95,2008.[24]C.Borgelt,J.GebhardtandR.Kruse,Possibilisticgraphicalmodels.InG.DellaRicciaetal.editors,ComputationalIntelligenceinDataMining,Springer,Wien,pages51-68,2000.[25]P.BoscandH.Prade,Anintroductiontothefuzzysetandpossibilitytheory-basedtreatmentofsoftqueriesanduncertainofimprecisedatabases.In:P.Smets,A.Motro,eds.UncertaintyManagementinInformationSystems,Dordrecht:Kluwer,285-324,1997.20 [40]P.Diamond(1988)Fuzzyleastsquares.InformationSciences,46,141-157[41]Y.Djouadi,H.Prade:Possibility-theoreticextensionofderivationoperatorsinformalconceptanalysisoverfuzzylattices.FuzzyOptimization&DecisionMaking10,287-309,2011.[42]D.Dubois,Beliefstructures,possibilitytheoryanddecomposablemeasureson nitesets.ComputersandAI,5,403-416,1986.[43]D.DuboisFuzzymeasureson nitescalesasfamiliesofpossibilitymeasures,Proc.7thConf.oftheEurop.Soc.forFuzzyLogicandTechnology(EUSFLAT'11),Annecy,AtlantisPress,822-829,2011.[44]D.DuboisOnticvs.epistemicfuzzysetsinmodelinganddataprocessingtasks,Proc.Inter.JointConf.onComputationalIntelligence,Paris,pp.IS13-IS19.[45]D.Dubois,F.DupindeSaint-Cyr,H.Prade:Apossibility-theoreticviewofformalconceptanalysis.Fundam.Inform.,75,195-213,2007.[46]D.Dubois,L.FarinasdelCerro,A.HerzigandH.Prade,Qualitativerelevanceandin-dependence:Aroadmap,Proc.ofthe15hInter.JointConf.onArtif.Intell.,Nagoya,62-67,1997.[47]D.Dubois,H.Fargier.Qualitativedecisionrulesunderuncertainty.InG.DellaRiccia,etal.editors,PlanningBasedonDecisionTheory,CISMcoursesandLectures472,SpringerWien,3-26,2003.[48]D.Dubois,H.Fargier,J.-F.Bonnefon.Onthequalitativecomparisonofdecisionshavingpositiveandnegativefeatures.J.ofArti cialIntelligenceResearch,AAAIPress,32,385-417,2008.[49]D.Dubois,H.Fargier,andP.PernyH.Prade,Qualitativedecisiontheorywithpref-erencerelationsandcomparativeuncertainty:Anaxiomaticapproach.Arti cialIn-telligence,148,219-260,2003.[50]D.Dubois,H.FargierandH.Prade:Fuzzyconstraintsinjob-shopscheduling.J.ofIntelligentManufacturing,6:215-234,1995.[51]D.Dubois,H.Fargier,andH.Prade,Possibilitytheoryinconstraintsatisfactionproblems:Handlingpriority,preferenceanduncertainty.AppliedIntelligence,6:287-309,1996.[52]D.Dubois,H.Fargier,andH.Prade,Ordinalandprobabilisticrepresentationsofacceptance.J.Arti cialIntelligenceResearch,22,23-56,200422 [67]D.DuboisandH.Prade,PossibilityTheory,NewYork:Plenum,1988.[68]D.DuboisandH.Prade,Randomsetsandfuzzyintervalanalysis.FuzzySetsandSystems,42:87-101,1991.[69]D.DuboisandH.Prade,Epistemicentrenchmentandpossibilisticlogic,Arti cialIntelligence,1991,50:223-239.[70]D.DuboisandH.Prade,Whenupperprobabilitiesarepossibilitymeasures,FuzzySetsandSystems,49:s65-74,1992.[71]D.Dubois,H.Prade:Possibilitytheoryasabasisforpreferencepropagationinau-tomatedreasoning.Proc.ofthe1stIEEEInter.Conf.onFuzzySystems(FUZZ-IEEE'92),SanDiego,Ca.,March8-12,1992,821-832.[72]D.DuboisandH.Prade,Whatarefuzzyrulesandhowtousethem.FuzzySetsandSystems,84:169-185,1996.[73]D.DuboisandH.Prade,Bayesianconditioninginpossibilitytheory,FuzzySetsandSystems,92:223-240,1997.[74]D.DuboisandH.Prade,Possibilitytheory:Qualitativeandquantitativeaspects.In:D.M.GabbayandP.SmetsP.,editorsHandbookofDefeasibleReasoningandUncertaintyManagementSystems,Vol.1.,Dordrecht:KluwerAcademic,169-226,1998.[75]D.Dubois,H.Prade.Possibilisticlogic:Aretrospectiveandprospectiveview.FuzzySetsandSystems,144,3-23,2004.[76]D.Dubois,H.Prade.Towardmultiple-agentextensionsofpossibilisticlogic.Proc.IEEEInter.Conf.onFuzzySystems(FUZZ-IEEE'07),London,July23-26,187-192,2007.[77]D.Dubois,H.Prade.Anoverviewoftheasymmetricbipolarrepresentationofpositiveandnegativeinformationinpossibilitytheory.FuzzySetsandSystems,160,1355-1366,2009.[78]D.Dubois,H.Prade:Generalizedpossibilisticlogic.Proc.5thInter.Conf.onScalableUncertaintyManagement(SUM'11),Dayton,Springer,LNCS6929,428-432,2011.[79]D.Dubois,H.Prade.Possibilitytheoryandformalconceptanalysis:Characterizingindependentsub-contextstoappearinFuzzySetsandSystems.[80]D.Dubois,H.Prade,A.Rico.ApossibilisticlogicviewofSugenointegrals.Proc.EurofuseWorkshoponFuzzyMethodsforKnowledge-BasedSystems(EUROFUSE24 [94]J.GebhardtandR.Kruse,Thecontextmodel.IInt.J.ApproximateReasoning,9,283-314,1993.[95]M.Gil,ed.,FuzzyRandomVariables,SpecialIssue,InformationSciences,133,2001.[96]J.F.GeerandG.J.Klir,Amathematicalanalysisofinformation-preservingtransfor-mationsbetweenprobabilisticandpossibilisticformulationsofuncertainty,Int.J.ofGeneralSystems,20,143-176,1992.[97]M.Grabisch,T.MurofushiandM.Sugeno,editors,FuzzymeasuresandIntegralsTheoryandApplications,Heidelberg:Physica-Verlag,2000.[98]W.Guezguez,N.BenAmor,K.Mellouli,Qualitativepossibilisticin uencediagramsbasedonqualitativepossibilisticutilities.Europ.J.ofOperationalResearch,195,223-238,2009.[99]D.Guyonnet,B.Bourgine,D.Dubois,H.Fargier,B.C^ome,J.-P.Chiles,Hybridapproachforaddressinguncertaintyinriskassessments.J.ofEnviron.Eng.,129,68-78,2003.[100]J.C.Helton,W.L.Oberkampf,editors(2004)AlternativeRepresentationsofUncer-tainty.ReliabilityEngineeringandSystemsSafety,85,369p.[101]E.Huellermeier,D.DuboisandH.Prade.Modeladaptationinpossibilisticinstance-basedreasoning.IEEETrans.onFuzzySystems,10,333-339,2002.[102]M.Inuiguchi,H.Ichihashi,Y.KumeModalityconstrainedprogrammingproblems:Auni edapproachtofuzzymathematicalprogrammingproblemsinthesettingofpossibilitytheory,InformationSciences,67,93-126,1993.[103]C.Joslyn,Measurementofpossibilistichistogramsfromintervaldata,Int.J.ofGeneralSystems,26:9-33,1997.[104]S.Kaci,H.Prade,Masteringtheprocessingofpreferencesbyusingsymbolicprioritiesinpossibilisticlogic.Proc.18thEurop.Conf.onArti cialIntelligence(ECAI'08),Patras,July21-25,376-380,2008.[105]G.JKlirandT.FolgerFuzzySets,UncertaintyandInformation.EnglewoodCli s,NJ:PrenticeHall,1988.[106]G.J.Klir,Aprincipleofuncertaintyandinformationinvariance,Int.J.ofGeneralSystems,17:249-275,1990.[107]G.J.KlirandB.ParvizB.Probability-possibilitytransformations:Acomparison,Int.J.ofGeneralSystems,21:291-310,1992.26 [122]H.Prade,M.Serrurier.Maximum-likelihoodprincipleforpossibilitydistributionsviewedasfamiliesofprobabilities.Proc.IEEEInter.Conf.onFuzzySystems(FUZZ-IEEE'11),Taipei,2987-2993,2011.[123]A.Puhalskii,LargeDeviationsandIdempotentProbability,ChapmanandHall,2001[124]E.Raufaste,R.DaSilvaNeves,C.MarineTestingthedescriptivevalidityofpossi-bilitytheoryinhumanjudgementsofuncertainty.Arti cialIntelligence,148:197-218,2003.[125]L.J.SavageTheFoundationsofStatistics,NewYork:Dover,1972.[126]G.Shafer,Belieffunctionsandpossibilitymeasures.InJ.C.Bezdek,editor,AnalysisofFuzzyInformationVol.I:MathematicsandLogic,BocaRaton,FL:CRCPress,pages51-84,1987.[127]G.L.S.ShackleDecision,OrderandTimeinHumanA airs,2ndedition,Cam-bridgeUniversityPress,UK,1961.[128]R.SlowinskiandM.Hapke,editors(2000)SchedulingunderFuzziness,Heidelberg:Physica-Verlag,2000.[129]P.Smets(1990).Constructingthepignisticprobabilityfunctioninacontextofun-certainty.In:M.Henrionetal.,editors,UncertaintyinArti cialIntelligence,vol.5,Amsterdam:North-Holland,pages29-39,1990.[130]W.Spohn,Ageneral,nonprobabilistictheoryofinductivereasoning.InR.D.Shachter,etal.,editors,UncertaintyinArti cialIntelligence,Vol.4.Amsterdam:NorthHolland.pages149-158,1990.[131]T.Sudkamp,Similarityandthemeasurementofpossibility.ActesRencontresFran-cophonessurlaLogiqueFloueetsesApplications(Montpellier,France),Toulouse:CepaduesEditions,pages13-26,2002.[132]H.Tanaka,P.J.Guo,PossibilisticDataAnalysisforOperationsResearch,Heidelberg:Physica-Verlag,1999.[133]A.TverskyandD.Kahneman.Advancesinprospecttheory:cumulativerepresenta-tionofuncertainty.J.ofRiskandUncertainty,5,297-323,1992.[134]W.VanLeekwijckandE.E.Kerre,Defuzzi cation:criteriaandclassi cation,FuzzySetsandSystems,108:303-314,2001.[135]P.Walley,StatisticalReasoningwithImpreciseProbabilities,ChapmanandHall,1991.28