Possibility Theory and its Applications Where Do we Stand Didier Dubois and Henri Prade - PDF document

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casanother0ifandonlyifforeachstateofaairss:(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(;),whereisaBooleanexpressionandisanelementofL.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].Followingaverydierentapproach,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)=Z10infMd;E(M)=Z10supMdwhereMisthe-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;t2Ngde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nedbypickinganelementintheunitintervalatrandom,andthenanelementsinthecutMatrandom.6SomeApplicationsPossibilitytheoryhasnotbeenthemainframeworkforengineeringapplicationsoffuzzysetsinthepast.However,onthebasisofitsconnectionstosymbolicarticialintelligence,todecisiontheoryandtoimprecisestatistics,weconsiderthatithassignicantpotentialforfurtherapplieddevelopmentsinanumberofareas,includingsomewherefuzzysetsarenotyetalwaysaccepted.Onlysomedirectionsarepointedouthere.1.Possibilitytheoryalsooersaframeworkforpreferencemodelinginconstraint-directedreasoning.Bothprioritizedandsoftconstraintscanbecapturedbypos-sibilitydistributionsexpressingdegreesoffeasibilityratherthanplausibility[51].Possibilityoersanaturalsettingforfuzzyoptimizationwhoseaimistobalancethelevelsofsatisfactionofmultiplefuzzyconstraints(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.EnglewoodClis,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,OrderandTimeinHumanAairs,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