An introduction to DSmT by Florentin Smarandache

AnintroductiontoDSmT

JeanDezertFlorentinSmarandache

TheFrenchAerospaceLab.,ChairofMath.&SciencesDept., ONERA/DTIM/SIF,UniversityofNewMexico, ChemindelaHuni`ere,200CollegeRoad, F-91761PalaiseauCedex,France.Gallup,NM87301,U.S.A. jean.dezert@onera.frsmarand@unm.edu

Abstract– Themanagementandcombinationofuncertain,imprecise,fuzzyandevenparadoxicalorhighlyconflicting sourcesofinformationhasalwaysbeen,andstillremainstoday,ofprimalimportanceforthedevelopmentofreliable moderninformationsystemsinvolvingartificialreasoning.Inthisintroduction,wepresentasurveyofourrecenttheory ofplausibleandparadoxicalreasoning,knownasDezert-SmarandacheTheory(DSmT),developedfordealingwithimprecise,uncertainandconflictingsourcesofinformation. WefocusourpresentationonthefoundationsofDSmTandon itsmostimportantrulesofcombination,ratherthanonbrowsingspecificapplicationsofDSmTavailableinliterature. Severalsimpleexamplesaregiventhroughoutthispresentationtoshowtheefficiencyandthegeneralityofthisnewtheory.

Keywords: Dezert-SmarandacheTheory,DSmT,quantitativeandqualitativereasoning,informationfusion.

MSC2000:68T37,94A15,94A17,68T40.

1Introduction

Themanagementandcombinationofuncertain,imprecise,fuzzyandevenparadoxicalorhighlyconflicting sourcesofinformationhasalwaysbeen,andstillremainstoday,ofprimalimportanceforthedevelopmentof reliablemoderninformationsystemsinvolvingartificialreasoning.Thecombination(fusion)ofinformation arisesinmanyfieldsofapplicationsnowadays(especiallyindefense,medicine,finance,geo-science,economy, etc).Whenseveralsensors,observersorexpertshavetobecombinedtogethertosolveaproblem,orifone wantstoupdateourcurrentestimationofsolutionsforagivenproblemwithsomenewinformationavailable, weneedpowerfulandsolidmathematicaltoolsforthefusion,speciallywhentheinformationonehastodeal withisimpreciseanduncertain.Inthischapter,wepresent asurveyofourrecenttheoryofplausibleandparadoxicalreasoning,knownasDezert-SmarandacheTheory(DSmT)intheliterature,developedfordealingwith imprecise,uncertainandconflictingsourcesofinformation.Recentpublicationshaveshowntheinterestand theabilityofDSmTtosolveproblemswhereotherapproaches fail,especiallywhenconflictbetweensources becomeshigh.WefocusthispresentationratheronthefoundationsofDSmT,andonthemainimportantrules ofcombination,thanonbrowsingspecificapplicationsofDSmTavailableinliterature.Successfulapplications ofDSmTintargettracking,satellitesurveillance,situationanalysis,robotics,medicine,biometrics,etc,canbe foundinPartsIIof[31,35,37]andontheworldwideweb[38]. Severalsimpleexamplesaregiveninthispaper toshowtheefficiencyandthegeneralityofDSmT

2FoundationsofDSmT

ThedevelopmentofDSmT(Dezert-SmarandacheTheoryofplausibleandparadoxicalreasoning[8,31])arises fromthenecessitytoovercometheinherentlimitationsofDST(Dempster-ShaferTheory[24])whichare closelyrelatedwiththeacceptanceofShafer’smodelforthefusionproblemunderconsideration(i.e.theframe of discernment Θ isimplicitlydefinedasafinitesetof exhaustive and exclusive hypotheses θi, i =1,...,n sincethemassesofbeliefaredefinedonlyonthepowersetof Θ -seesection2.1fordetails),thethirdmiddleexcludedprinciple(i.e.theexistenceofthecomplementforanyelements/propositionsbelongingtothepowerset Thispaperisbasedonthefirstchapterof[37].

of Θ),andtheacceptanceofDempster’sruleofcombination(involvingnormalization)astheframeworkforthe combinationofindependentsourcesofevidence.DiscussionsonlimitationsofDSTandpresentationofsome alternativerulestoDempster’sruleofcombinationcanbefoundin[11,15,17–19,21,23,31,40,48,52,53,56–59] andthereforetheywillbenotreportedindetailsinthisintroduction.Wearguethatthesethreefundamental conditionsofDSTcanberemovedandanothernewmathematicalapproachforcombinationofevidenceis possible.ThisisthepurposeofDSmT.

ThebasisofDSmTistherefutationoftheprincipleofthethirdexcludedmiddleandShafer’smodel,since forawideclassoffusionproblemstheintrinsicnatureofhypothesescanbeonlyvagueandimpreciseinsuch awaythatpreciserefinementisjustimpossibletoobtaininrealitysothattheexclusiveelements θi cannot beproperlyidentifiedandpreciselyseparated.Manyproblemsinvolvingfuzzycontinuousandrelativeconceptsdescribedinnaturallanguageandhavingnoabsoluteinterpretationliketallness/smallness,pleasure/pain, cold/hot,Soritesparadoxes,etc,enterinthiscategory.DSmTstartswiththenotionof freeDSmmodel,denoted Mf (Θ),andconsiders Θ onlyasaframeofexhaustiveelements θi, i =1,...,n whichcanpotentiallyoverlap. Thismodelis free becausenootherassumptionisdoneonthehypotheses,buttheweakexhaustivityconstraint whichcanalwaysbesatisfiedaccordingtheclosureprincipleexplainedin[31].Nootherconstraintisinvolved inthefreeDSmmodel.WhenthefreeDSmmodelholds,thecommutativeandassociativeclassicalDSmrule ofcombination,denotedDSmC,correspondingtotheconjunctiveconsensusdefinedonthefreeDedekind’s latticeisperformed.

Dependingontheintrinsicnatureoftheelementsofthefusionproblemunderconsideration,itcanhowever happenthatthefreemodeldoesnotfittherealitybecausesomesubsetsof Θ cancontainelementsknownto betrulyexclusivebutalsotrulynonexistingatallatagiventime(speciallywhenworkingondynamicfusion problemwheretheframe Θ varieswithtimewiththerevisionoftheknowledgeavailable).Theseintegrity constraintsarethenexplicitlyandformallyintroducedintothefreeDSmmodel Mf (Θ) inordertoadaptit properlytofitascloseaspossiblewiththerealityandpermittoconstructa hybridDSmmodel M(Θ) onwhich thecombinationwillbeefficientlyperformed.Shafer’smodel,denoted M0(Θ),correspondstoaveryspecific hybridDSmmodelincludingallpossibleexclusivityconstraints.DSThasbeendevelopedforworkingonly with M0(Θ) whileDSmThasbeendevelopedforworkingwithanykindofhybridmodel(includingShafer’s modelandthefreeDSmmodel),tomanageasefficientlyandpreciselyaspossibleimprecise,uncertainand potentiallyhighlyconflictingsourcesofevidencewhilekeepinginmindthepossibledynamicityoftheinformationfusionproblematic.ThefoundationsofDSmTarethereforetotallydifferentfromthoseofallexisting approachesmanaginguncertainties,imprecisionsandconflicts.DSmTprovidesanewinterestingwaytoattack theinformationfusionproblematicwithageneralframeworkinordertocoverawidevarietyofproblems.

DSmTrefutesalsotheideathatsourcesofevidenceprovidetheirbeliefswiththesameabsoluteinterpretationofelementsofthesameframe Θ andtheconﬂictbetweensourcesarisesnotonlybecauseofthepossible unreliabilityofsources,butalsobecauseofpossibledifferentandrelativeinterpretationof Θ,e.g.whatis consideredasgoodforsomebodycanbeconsideredasbadforsomebodyelse.Thereissomeunavoidable subjectivityinthebeliefassignmentsprovidedbythesourcesofevidence,otherwiseitwouldmeanthatall bodiesofevidencehaveasameobjectiveanduniversalinterpretation(ormeasure)ofthephenomenaunder consideration,whichunfortunatelyrarelyoccursinreality,butwhenbasicbeliefassignments(bba’s)arebased onsome objectiveprobabilities transformations.Butinthislastcase,probabilitytheory canhandleproperly andefﬁcientlytheinformation,andDST,aswellasDSmT,becomesuseless.Ifwenowgetoutoftheprobabilisticbackgroundargumentationfortheconstructionofbba,weclaimthatinmostofcases,thesourcesof evidenceprovidetheirbeliefsaboutelementsoftheframeofthefusionproblemonlybasedontheirownlimited knowledgeandexperiencewithoutreferencetothe(inaccessible)absolutetruthofthespaceofpossibilities.

2.1Thepowerset,hyper-powersetandsuper-powerset

InDSmT,wetakeverycareaboutthemodelassociatedwiththe set Θ ofhypotheseswherethesolutionofthe problemisassumedtobelongto.Inparticular,thethreemainsets(powerset,hyper-powersetandsuper-power set)canbeuseddependingontheirabilitytoﬁtadequatelywiththenatureofhypotheses.Inthefollowing,we

assumethat Θ= {θ1,...,θn} isaﬁniteset(calledframe)of n exhaustiveelements1.If Θ= {θ1,...,θn} isapriorinotclosed(Θ issaidtobeanopenworld/frame),onecanalwaysincludeinitaclosureelement, say θn+1 insuchawaythatwecanworkwithanewclosedworld/frame {θ1,...,θn,θn+1}.Sowithoutloss ofgenerality,wewillalwaysassumethatweworkinaclosedworldbyconsideringtheframe Θ asaﬁnite setofexhaustiveelements.Beforeintroducingthepowerset,thehyper-powersetandthesuper-powersetitis necessarytorecallthatsubsetsareregardedaspropositionsinDempster-ShaferTheory(seeChapter2of[24]) andweadoptthesameapproachinDSmT.

• Subsetsaspropositions:GlennShaferinpages35–37of[24]considersthesubsetsas propositionsin thecaseweareconcernedwiththetruevalueofsomequantity θ takingitspossiblevaluesin Θ.Then thepropositions Pθ(A) ofinterestarethoseoftheform2:

Pθ(A) Thetruevalueof θ isinasubset A of Θ

Anyproposition Pθ(A) isthusinone-to-onecorrespondencewiththesubset A of Θ.Suchcorrespondenceisveryusefulsinceittranslatesthelogicalnotions ofconjunction ∧,disjunction ∨,implication ⇒ andnegation ¬ intotheset-theoreticnotionsofintersection ∩,union ∪,inclusion ⊂ andcomplementation c(.).Indeed,if Pθ(A) and Pθ(B) aretwopropositionscorrespondingtosubsets A and B of Θ,then theconjunction Pθ(A) ∧Pθ(B) correspondstotheintersection A∩B andthedisjunction Pθ(A) ∨Pθ(B) correspondstotheunion A ∪ B A isasubsetof B ifandonlyif Pθ(A) ⇒Pθ (B) and A isthesettheoreticcomplementof B withrespectto Θ (written A = cΘ(B))ifandonlyif Pθ (A)= ¬Pθ(B).In otherwords,thefollowingequivalencesarethenusedbetweentheoperationsonthesubsetsandonthe propositions:

Operations Subsets Propositions

Intersection/conjunction A ∩ B Pθ(A) ∧Pθ(B)

Union/disjunction A ∪ B Pθ(A) ∨Pθ(B)

Inclusion/implication A ⊂ B Pθ (A) ⇒Pθ(B) Complementation/negation A = cΘ(B) Pθ (A)= ¬Pθ(B)

Table1:Correspondencebetweenoperationsonsubsetsandonpropositions.

• Canonicalformofaproposition:InDSmTweconsiderallpropositions/setsinacanonicalform.We takethedisjunctivenormalform,whichisadisjunctionofconjunctions,anditisuniqueinBoolean algebraandsimplest.Forexample, X = A ∩ B ∩ (A ∪ B ∪ C ) itisnotinacanonicalform,butwe simplifytheformulaand X = A ∩ B isinacanonicalform.

• Thepowerset: 2Θ (Θ, ∪)

AsideDempster’sruleofcombination,thepowersetisoneof thecornerstonesofDempster-ShaferTheory (DST)sincethebasicbeliefassignmentstocombinearedeﬁnedonthepowersetoftheframe Θ.Inmathematics,givenaset Θ,thepowersetof Θ,written 2Θ,isthesetofallsubsetsof Θ.InZermelo–Fraenkelsettheory withtheaxiomofchoice(ZFC),theexistenceofthepowerset ofanysetispostulatedbytheaxiomofpower set.Inotherwords, Θ generatesthepowerset 2Θ withthe ∪ (union)operatoronly.

1Wedonotassumeherethatelements θi arenecessaryexclusive,unlessspeciﬁed.Thereisnorestrictionon θi butthe exhaustivity.

2Weusethesymbol tomean equalsbydeﬁnition;theright-handsideoftheequationisthedeﬁnitionoftheleft-hand side.

Moreprecisely,thepowerset 2Θ isdeﬁnedasthesetofallcompositepropositions/subsetsbuiltfrom elementsof Θ with ∪ operatorsuchthat: 1. ∅,θ1,...,θn ∈ 2Θ 2.If A,B ∈ 2Θ,then A ∪ B ∈ 2Θ . 3.Nootherelementsbelongto 2Θ,exceptthoseobtainedbyusingrules1and2.

Examplesofpowersets: • If Θ= {θ1,θ2},then 2Θ={θ1 ,θ2} = {{∅}, {θ1 }, {θ2 }, {θ1,θ2}} whichiscommonlywrittenas 2Θ = {∅,θ1,θ2,θ1 ∪ θ2}. • Let’sconsidertwoframes Θ1 = {A,B} and Θ2 = {X,Y },thentheirpowersetsarerespectively 2Θ1 ={A,B} = {∅,A,B,A ∪ B} and 2Θ2 ={X,Y } = {∅,X,Y,X ∪ Y }.Let’sconsiderareﬁnedframe Θref = {θ1,θ2,θ3,θ4}.Thegranules θi, i =1,..., 4 arenotnecessarilyexhaustive,norexclusive.If A and B areexpressedmorepreciselyinfunctionofthegranules θi byexampleas A {θ1,θ2,θ3}≡ θ1 ∪ θ2 ∪ θ3 and B {θ2,θ4}≡ θ2 ∪ θ4 thenthepowersetscanbeexpressedfromthegranules θi as follows: 2Θ1={A,B} = {∅,A,B,A ∪ B} = {∅, {θ1 ,θ2,θ3} A , {θ2,θ4} B

, {{θ1,θ2,θ3}, {θ2,θ4}} A∪B } = {∅,θ1 ∪ θ2 ∪ θ3,θ2 ∪ θ4,θ1 ∪ θ2 ∪ θ3 ∪ θ4} If X and Y areexpressedmorepreciselyinfunctionoftheﬁnergranules θi byexampleas X {θ1}≡ θ1 and Y {θ2,θ3,θ4}≡ θ2 ∪ θ3 ∪ θ4 then: 2Θ2 ={X,Y } = {∅,X,Y,X ∪ Y } = {∅, {θ1 } X

, {θ2,θ3,θ4} Y

, {{θ1}, {θ2 ,θ3,θ4}} X∪Y } = {∅,θ1,θ2 ∪ θ3 ∪ θ4,θ1 ∪ θ2 ∪ θ3 ∪ θ4}

Weseethatonehasnaturally: 2Θ1 ={A,B} =2Θ2 ={X,Y } =2Θref ={θ1 ,θ2,θ3,θ4} evenifworkingfrom θi with A ∪ B = X ∪ Y = {θ1,θ2,θ3,θ4} =Θref • Thehyper-powerset: DΘ (Θ, ∪, ∩)

OneofthecornerstonesofDSmTisthefreeDedekind’slattice[4]denotedas hyper-powerset inDSmT framework.Let Θ= {θ1,...,θn} beafiniteset(calledframe)of n exhaustiveelements.Thehyper-powerset DΘ isdefinedasthesetofallcompositepropositions/subsetsbuiltfromelementsof Θ with ∪ and ∩ operators suchthat: 1. ∅,θ1,...,θn ∈ DΘ 2.If A,B ∈ DΘ,then A ∩ B ∈ DΘ and A ∪ B ∈ DΘ . 3.Nootherelementsbelongto DΘ,exceptthoseobtainedbyusingrules1or2. Thereforebyconvention,wewrite DΘ =(Θ, ∪, ∩) whichmeansthat Θ generates DΘ underoperators ∪ and ∩.Thedual(obtainedbyswitching ∪ and ∩ inexpressions)of DΘ isitself.Thereareelementsin DΘ whichareself-dual(dualtothemselves),forexample α8 forthecasewhen n =3 inthefollowingexample. Thecardinalityof DΘ ismajoredby 22n whenthecardinalityof Θ equals n,i.e. |Θ| = n.Thegeneration ofhyper-powerset DΘ iscloselyrelatedwiththefamousDedekind’sproblem[3,4] onenumeratingtheset ofisotoneBooleanfunctions.Thegenerationofthehyper-powersetispresentedin[31].Sinceforanygiven finiteset Θ, |DΘ|≥|2Θ| wecall DΘ the hyper-powerset of Θ

Exampleoftheﬁrsthyper-powersets:

• Forthedegeneratecase(n =0) where Θ= {},onehas DΘ = {α0 ∅} and |DΘ| =1.

• When Θ= {θ1},onehas DΘ = {α0 ∅,α1 θ1} and |DΘ| =2.

• When Θ= {θ1,θ2},onehas DΘ = {α0,α1,...,α4} and |DΘ| =5 with α0 ∅, α1 θ1 ∩ θ2,

α2 θ1, α3 θ2 and α4 θ1 ∪ θ2

• When Θ= {θ1,θ2,θ3},onehas DΘ = {α0,α1,...,α18} and |DΘ| =19 with

α0 ∅

α1 θ1 ∩ θ2 ∩ θ3

α10 θ2

α2 θ1 ∩ θ2 α11 θ3

α3 θ1 ∩ θ3

α4 θ2 ∩ θ3

α12 (θ1 ∩ θ2) ∪ θ3

α13 (θ1 ∩ θ3) ∪ θ2

α5 (θ1 ∪ θ2) ∩ θ3 α14 (θ2 ∩ θ3) ∪ θ1

α6 (θ1 ∪ θ3) ∩ θ2 α15 θ1 ∪ θ2

α7 (θ2 ∪ θ3) ∩ θ1 α16 θ1 ∪ θ3

α8 (θ1 ∩ θ2) ∪ (θ1 ∩ θ3) ∪ (θ2 ∩ θ3) α17 θ2 ∪ θ3 α9 θ1 α18 θ1 ∪ θ2 ∪ θ3

Thecardinalityofhyper-powerset DΘ for n ≥ 1 followsthesequenceofDedekind’snumbers[26],i.e. 1,2,5,19,167,7580,7828353,...andanalyticalexpressionofDedekind’snumbershasbeenobtainedrecently byTombakin[47](see[31]fordetailsongenerationandorderingof DΘ).Interestinginvestigationsonthe programmingofthegenerationofhyper-powersetsforengineeringapplicationshavebeendoneinChapter15 of[35]andin[37].

, {{θ1,θ2,θ3}, {θ2,θ4}} A∪B={θ1 ,θ2,θ , {θ1} X

, {θ1,θ2,θ3} A , {θ2,θ4} B 3,θ4} } = {∅,θ2,θ1 ∪ θ2 ∪ θ3,θ2 ∪ θ4,θ1 ∪ θ2 ∪ θ3 ∪ θ4} =2Θ1 ={A,B} If X and Y areexpressedmorepreciselyinfunctionoftheﬁnergranules θi byexampleas X {θ1} and Y {θ2,θ3,θ4} theninassumingthat θi, i =1,..., 4 areexhaustiveandexclusive,onegets DΘ2={X,Y } = {∅,X ∩ Y,X,Y,X ∪ Y } = {∅, {θ1 }∩{θ2,θ3,θ4} X∩Y =∅ ∅

, {{θ1}, {θ2,θ3,θ4}} X∪Y } = {∅, {θ1 } X

, {θ2,θ3,θ4} Y

, {{θ1 }, {θ2,θ3,θ4}} X∪Y }≡ 2Θ2 ={X,Y } 5

Examplesofhyper-powersets: Let’sconsidertheframes Θ1 = {A,B} and Θ2 = {X,Y },thentheircorrespondinghyper-powersetsare DΘ1 ={A,B} = {∅,A ∩ B,A,B,A ∪ B} and DΘ2={X,Y } = {∅,X ∩ Y,X,Y,X ∪ Y }.Let’sconsiderareﬁned frame Θref = {θ1,θ2,θ3,θ4} wherethegranules θi, i =1,..., 4 arenowconsideredas trulyexhaustiveand exclusive.If A and B areexpressedmorepreciselyinfunctionofthegranules θi byexampleas A {θ1,θ2,θ3} and B {θ2,θ4} then DΘ1={A,B} = {∅,A ∩ B,A,B,A ∪ B} = {∅, {θ1 ,θ2,θ3}∩{θ2,θ4} A∩B={θ2 }

Shafer’smodelofaframe:Moregenerally,whenalltheelementsofagivenframe Θ areknown(orare assumedtobe)trulyexclusive,thenthehyper-powerset DΘ reducestotheclassicalpowerset 2Θ.Therefore, workingonpowerset 2Θ asGlennShaferhasproposedinhisMathematicalTheoryofEvidence[24])is equivalenttoworkonhyper-powerset DΘ withtheassumptionthatallelementsoftheframeareexclusive. Thisiswhatwecall Shafer’smodeloftheframe Θ,written M0(Θ),evenifsuchmodel/assumptionhasnot beenclearlystatedexplicitlybyShaferhimselfinhismilestonebook.

• Thesuper-powerset: SΘ (Θ, ∪, ∩,c( ))

Thenotionofsuper-powersethasbeenintroducedbySmarandacheintheChapter8of[35].Itcorresponds actuallytothetheoreticalconstructionofthepowersetof theminimal3 refinedframe Θref of Θ. Θ generates SΘ underoperators ∪, ∩ andcomplementation c( ). SΘ =(Θ, ∪, ∩,c( )) isaBooleanalgebrawithrespectto theunion,intersectionandcomplementation.Thereforeworkingwiththesuper-powersetisequivalenttowork withaminimaltheoreticalrefinedframe Θref satisfyingShafer’smodel.Moreprecisely, SΘ isdefinedasthe setofallcompositepropositions/subsetsbuiltfromelementsof Θ with ∪, ∩ and c(.) operatorssuchthat: 1. ∅,θ1,...,θn ∈ SΘ . 2.If A,B ∈ SΘ,then A ∩ B ∈ SΘ , A ∪ B ∈ SΘ 3.If A ∈ SΘ,then c(A) ∈ SΘ 4.Nootherelementsbelongto SΘ,exceptthoseobtainedbyusingrules1,2and3.

Asreportedin[32],asimilargeneralizationhasbeenpreviouslyusedin1993byGuanandBell[14]forthe Dempster-Shaferruleusingpropositionsinsequentiallogicandreintroducedin1994byParisinhisbook[20], page4.

Exampleofasuper-powerset

Let’sconsidertheframe Θ= {θ1,θ2} andlet’sassume θ1 ∩ θ2 = ∅,i.e. θ1 and θ2 arenotdisjointaccording toFig.1where A p1 denotesthepartof θ1 belongingonlyto θ1 (p standsherefor part), B p2 denotesthe partof θ2 belongingonlyto θ2 and C p12 denotesthepartof θ1 and θ2 belongingtoboth.Inthisexample, SΘ={θ1 ,θ2} isthengivenby SΘ = {∅,θ1 ∩ θ2,θ1,θ2,θ1 ∪ θ2,c(∅),c(θ1 ∩ θ2),c(θ1),c(θ2 ),c(θ1 ∪ θ2)} where c(.) isthecomplementin Θ.Since c(∅)= θ1 ∪ θ2 and c(θ1 ∪ θ2)= ∅,thesuper-powersetisactually givenby SΘ = {∅,θ1 ∩ θ2,θ1,θ2,θ1 ∪ θ2,c(θ1 ∩ θ2),c(θ1),c(θ2 )}

DΘ1={A,B} = {∅,A ∩ B,A,B,A ∪ B} = {∅, (θ1 ∪ θ2 ∪ θ3) ∩ (θ2 ∪ θ4),θ1 ∪ θ2 ∪ θ3,θ2 ∪ θ4,θ1 ∪ θ2 ∪ θ3 ∪ θ4} =2Θ1 =

DΘ2={X,Y } = {∅,X ∩ Y,X,Y,X ∪ Y } = {∅,θ1 ∩ (θ2 ∪ θ3 ∪ θ4),θ1,θ2 ∪ θ3 ∪ θ4,θ1 ∪ θ2 ∪ θ3 ∪ θ4} =2Θ2={X,Y }

Therefore,weseethat DΘ2={X,Y } ≡ 2Θ2 ={X,Y } becausetheexclusivityconstraint X ∩ Y = ∅ holdssinceone hasassumed X {θ1} and Y {θ2,θ3,θ4} withexhaustiveandexclusivegranules θi, i =1,..., 4 Ifthegranules θi, i =1,..., 4 arenotassumedexclusive,thenofcoursetheexpressionsof hyper-power setscannotbesimpliﬁedandonewouldhave:

{A,B}

θ1

A p1

θ2 B p2 C p12

Fig.1:VenndiagramofafreeDSmmodelfora2Dframe. Let’snowconsidertheminimalreﬁnement Θref = {A,B,C} of Θ builtbysplittingthegranules θ1 and θ2 depictedonthepreviousVenndiagramintodisjointparts(i.e. Θref satisﬁestheShafer’smodel)asfollows: θ1 = A ∪ C,θ2 = B ∪ C,θ1 ∩ θ2 = C

Thentheclassicalpowersetof Θref isgivenby 2Θref = {∅,A,B,C,A ∪ B,A ∪ C,B ∪ C,A ∪ B ∪ C} Weseethatwecandeﬁneeasilyaone-to-onecorrespondence, written ∼,betweenalltheelementsofthe super-powerset SΘ andtheelementsofthepowerset 2Θref asfollows: ∅∼∅, (θ1 ∩ θ2) ∼ C,θ1 ∼ (A ∪ C ),θ2 ∼ (B ∪ C ), (θ1 ∪ θ2) ∼ (A ∪ B ∪ C ) c(θ1 ∩ θ2) ∼ (A ∪ B),c(θ1) ∼ B,c(θ2) ∼ A

Suchone-to-onecorrespondencebetweentheelementsof SΘ and 2Θref canbedefinedforanycardinality |Θ|≥ 2 oftheframe Θ andthusonecanconsider SΘ asthemathematicalconstructionofthepowerset 2Θref oftheminimalrefinementoftheframe Θ.Ofcourse,when Θ alreadysatisfiesShafer’smodel,thehyper-power setandthesuper-powersetcoincidewiththeclassicalpowersetof Θ.Itisworthtonotethatevenifwehave amathematicaltooltobuilttheminimalrefinedframesatisfyingShafer’smodel,itdoesn’tmeannecessary thatonemustworkwiththissuper-powersetingeneralinrealapplicationsbecausemostofthetimesthe elements/granulesof SΘ havenoclearphysicalmeaning,nottomentionthedrasticincreaseofthecomplexity sinceonehas 2Θ ⊆ DΘ ⊆ SΘ and

| =

Θ| =2

Θ| |

|2Θ| =2|Θ| < |DΘ| < |SΘ| =2|Θref | =22|Θ| 1 (1) Typically,

SΘ

= |2Θref | =22n 1

4 5 23 =8

8 19 27 =128 4 16 167 215 =32768 5 32 7580 231 =2147483648 Table2:Cardinalitiesof 2Θ , DΘ and SΘ 3Theminimalityrefersheretothecardinalityofthereﬁnedframes.

Insummary,DSmTofferstrulythepossibilitytobuildandto workonrefinedframesandtodealwith thecomplementwhenevernecessary,butinmostofapplicationseithertheframe Θ isalreadybuilt/chosen tosatisfyShafer’smodelortherefinedgranuleshavenoclearphysicalmeaningwhichfinallypreventtobe considered/assessedindividuallysothatworkingonthehyper-powersetisusuallysufficientfordealingwith uncertainimprecise(quantitativeorqualitative)andhighlyconflictingsourcesofevidences.Workingwith SΘ isactuallyverysimilartoworkingwith 2Θ inthesensethatinbothcasesweworkwithclassicalpowersets; theonlydifferenceisthatwhenworkingwith SΘ wehaveimplicitlyswitchedfromtheoriginalframe Θ representationtoaminimalrefinement Θref representation.Therefore,inthesequelwefocusourdiscussionsbased mainlyonhyper-powersetratherthan(super-)powersetwhichhasalreadybeenthebasisforthedevelopment ofDST.Butasalreadymentioned,DSmTcaneasilydealwithbelieffunctionsdefinedon 2Θ or SΘ similarly asthosedefinedon DΘ

Genericnotation:Inthesequel,weusethegenericnotation GΘ fordenotingthesets(powerset,hyper-power setandsuper-powerset)onwhichthebelieffunctionsaredeﬁned.

Remarkonthelogicalrefinement:TherefinementinlogictheorypresentedrecentlybyCholvy in[2]was actuallyproposedinninetiesbyaGuanandBell[14]andbyParis[20].Thisrefinementisisomorphictothe refinementinsettheorydonebymanyresearchers.If Θ= {θ1,θ2,θ3} isalanguagewherethepropositional variablesare θ1, θ2, θ3,Cholvyconsidersall8possiblelogicalcombinationsofpropositions θi’sornegations of θi’s(calledinterpretations),anddefinesthe 8=23 disjointparts/propositionsoftheVenndiagraminFig.2 [onealsoconsidersasapartthenegationofthetotalignorance]inthesettheory,sothat: i1 = θ1 ∧ θ2 ∧ θ3 i2 = θ1 ∧ θ2 ∧¬θ3 i3 = θ1 ∧¬θ2 ∧ θ3 i4 = θ1 ∧¬θ2 ∧¬θ3 i5 = ¬θ1 ∧ θ2 ∧∧θ3 i6 = ¬θ1 ∧ θ2 ∧¬θ3 i7 = ¬θ1 ∧¬θ2 ∧ θ3 i8 = ¬θ1 ∧¬θ2 ∧¬θ3 where ¬θi meansthenegationof θi Θ θ1 θ1 ∧¬θ2 ∧¬θ3 p1 θ3 θ2

Fig.2:VenndiagramofthefreeDSmmodelfora3Dframe.

¬θ1 ∧ θ2 ∧¬θ3 p2 ¬θ1 ∧¬θ2 ∧ θ3 p3 ¬θ1 ∧ θ2 ∧ θ3 p23 θ1 ∧ θ2 ∧ θ3 p123 θ1 ∧ θ2 ∧¬θ3 p12 θ1 ∧¬θ2 ∧ θ3 p13 ¬θ1 ∧¬θ2 ∧¬θ3 p0

BecauseofShafer’sequivalenceofsubsetsandpropositions,Cholvy’slogicalrefinementisstrictlyequivalenttotherefinementwedidalreadyin2006indefining SΘ -seeChap.8of[35]-butinthesettheory framework.WediditusingSmarandache’scodification(easy tounderstandandread)inthefollowingway: -eachVenndiagramdisjointpart pij ,or pijk representsrespectivelytheintersectionof pi and pj only,or pi and pj and pk only,etc;whilethecomplementofthetotalignoranceisconsidered p0 [p standsfor part].

Thus,wehaveaneasierandclearerrepresentationinDSmTthaninthelogicalrepresentation.Whilethe reﬁnementinDSTusinglogicalapproachfor n verylargeisveryhard,wecansimplyconsiderintheDSmT thesuper-powerset SΘ =(Θ, ∪, ∩,c(.)).So,inDSmTrepresentationthedisjointpartsarenotedasfollows: p123 = θ1 ∧ θ2 ∧ θ3 = i1 p12 = θ1 ∧ θ2 ∧¬θ3 = i2 p13 = θ1 ∧¬θ2 ∧ θ3 = i3 p1 = θ1 ∧¬θ2 ∧¬θ3 = i4 p23 = ¬θ1 ∧ θ2 ∧ θ3 = i5 p2 = ¬θ1 ∧ θ2 ∧¬θ3 = i6 p3 = ¬θ1 ∧¬θ2 ∧ θ3 = i7 p0 = ¬θ1 ∧¬θ2 ∧¬θ3 = i8

Asseeing,inSmarandache’scodificationadisjointVenndiagrampartisequaltotheintersectionofsingletonswhoseindexesshowupasindexesoftheVennpart;forexamplein p12 caseindexes1and2,intersected withthecomplementofthemissingindexes,inthiscaseindex3ismissing. Smarandache’scodificationcaneasilytransformanysetfrom SΘ intoitscanonicaldisjunctivenormalform. Forexample, θ1 = p1 ∪ p12 ∪ p13 ∪ p123 (i.e.allVenndiagramdisjointpartsthatcontaintheindex “1”intheir indexes;suchindexesfrom SΘ are1,12,13,123)canbeexpressedas θ1 =(θ1 ∩ c(θ2) ∩ c(θ3)) ∪ (θ1 ∩ θ2 ∩ c(θ3))(θ1 ∩ c(θ2) ∩ θ3) ∪ (θ1 ∩ θ2 ∩ θ3) wherethesetvaluesofeachpartweredefinedaspreviously. θ1 ∧ θ2 = p12 ∪ p123 (i.e.allVenndiagramdisjoint partsthatcontaintheindex“12”intheirindexes)equalsto (θ1 ∧ θ2 ∧¬θ3) ∨ (θ1 ∧ θ2 ∧ θ3).

TherefinementbasedonVennDiagram,becomesveryhardandalmostimpossiblewhenthecardinalof Θ, n,islargeandallintersectionsarenon-empty(thefreemodel).Suppose n =20,orevenbigger,andwehave thefreemodel.HowcanweconstructaVennDiagramwheretoshowallpossibleintersectionsof20sets?Its geometricalfigurewouldbeveryhardtodesignandveryhardtoread(youdon’tidentifywelleachdisjoint partofasuchVennDiagramtowhatintersectionofsetsitbelongsto).Thelargeris n,themoredifficultis therefinement.Fortunately,basedonSmarandache’scodification,wecanalgebraicallydesigninaneasyway forallsuchintersections(forexample,if n isverybig,wecanusecomputerprogramstomakecombinations ofindexes {1, 2,...,n} takeningroupsor1,of2,...,orof n elementseach),sotherefinementshouldnot beabigproblemfromtheprogrammingpointofview,butwemustalwayskeepinmindifsuchrefinement isreallynecessaryandifithas(ornot)adeepphysicalinterpretationandjustificationfortheproblemunder consideration.

Theassertionin[2],uponMilanDaniel’s,thathybridDSmruleisequivalenttoDubois-Praderuleisuntrue, sinceindynamicfusiontheygivedifferentresults.Suchexamplehasbeenalreadygivenin[7]andisreported insection2.6.3forthesakeofclarificationforthereaders.Theassertionin[2]that“fromanexpressivitypoint ofviewDSmTisequivalenttoDST”ispartiallytruesincethisideaistruewhentherefinementispossible(not alwaysitispractically/physicallypossible),andevenwhenthespacesweworkon, SΘ =2Θref ,wherethe hypothesesareexclusive,DSmTofferstheadvantagethattherefinementisalreadydone(itisnotnecessary fortheusertodo(orimplicitlypresuppose)itasinDST).Also,DSmTacceptsfromtheverybeginningthe possibilitytodealwithnon-exclusivehypothesesandofcourseitcanafortioridealwithsetsofexclusivehypothesisandworkeitheron 2Θ or 2Θref whenevernecessary,whileDSTfirstrequiresimplicitlytoworkwith exclusivehypothesesonly.

ThemaindistinctionsbetweenDSmTandDSTaresummarizedby thefollowingpoints:

1.Thereﬁnementisnotalways(physically)possible,especiallyforelementsfromtheframeofdiscernment whosefrontiersarenotclear,suchas:colors,vaguesets,unclearhypotheses,etc.intheframeof discernment;DSTdoesnotﬁtwellforworkinginsuchcases,whileDSmTdoes;

2.Eveninthecasewhentheframeofdiscernmentcanbereﬁned (i.e.the atomic elementsoftheframe havealladistinctphysicalmeaning),itisstilleasiertouseDSmTthanDSTsinceinDSmTframework thereﬁnementisdoneautomaticallybythemathematicalconstructionofthesuper-powerset;

3.DSmToffersbetterfusionrules,forexampleProportionalConﬂictredistributionRule#5(PCR5)presentedinthesequel-isbetterthanDempster’srule;hybridDSmrule(DSmH)worksforthedynamic fusion,whileDubois-Pradefusionruledoesnot(DSmHisanextensionofDubois-Praderule);

4.DSmToffersthebestqualitativeoperators(whenworking withlabels)givingthemostaccurateand coherentresults;

5.DSmToffersnewinterestingquantitativeconditioningrules(BCRs)andqualitativeconditioningrules (QBCRs),differentfromShafer’sconditioningrule(SCR). SCRcanbeseensimplyasacombinationof apriormassofbeliefwiththemass m(A)=1 whenever A istheconditioningevent;

6.DSmTproposesanewapproachforworkingwithimprecisequantitativeorqualitativeinformationand notlimitedtointerval-valuedbeliefstructuresasproposedgenerallyintheliterature[5,6,49].

2.2NotionoffreeandhybridDSmmodels

FreeDSmmodel:Theelements θi, i =1,...,n of Θ constitutethefinitesetofhypotheses/conceptscharacterizingthefusionproblemunderconsideration.Whenthereisnoconstraintontheelementsoftheframe,we callthismodelthe freeDSmmodel,written Mf (Θ).ThisfreeDSmmodelallowstodealdirectlywithfuzzy conceptswhichdepictacontinuousandrelativeintrinsicnatureandwhichcannotbepreciselyrefinedintofiner disjointinformationgranuleshavinganabsoluteinterpretationbecauseoftheunreachableuniversaltruth.In suchcase,theuseofthehyper-powerset DΘ (withoutintegrityconstraints)isparticularlywelladaptedfor definingthebelieffunctionsonewantstocombine.

Shafer’smodel:Insomefusionproblemsinvolvingdiscreteconcepts,alltheelements θi, i =1,...,n of Θ canbetrulyexclusive.Insuchcase,alltheexclusivityconstraintson θi, i =1,...,n havetobeincludedinthe previousmodeltocharacterizeproperlythetruenatureofthefusionproblemandtoﬁtitwiththereality.By doingthis,thehyper-powerset DΘ aswellasthesuper-powerset SΘ reducenaturallytotheclassicalpower set 2Θ andthisconstituteswhatwehavecalled Shafer’smodel,denoted M0(Θ).Shafer’smodelcorresponds actuallytothemostrestrictedhybridDSmmodel.

HybridDSmmodels:BetweentheclassoffusionproblemscorrespondingtothefreeDSmmodel Mf (Θ) andtheclassoffusionproblemscorrespondingtoShafer’smodel M0(Θ),thereexistsanotherwideclass ofhybridfusionproblemsinvolvingin Θ bothfuzzycontinuousconceptsanddiscretehypotheses.In such (hybrid)class,someexclusivityconstraintsandpossibly somenon-existentialconstraints(especiallywhen workingondynamic4 fusion)havetobetakenintoaccount.Eachhybridfusionproblemofthisclasswillthen becharacterizedbyaproperhybridDSmmodeldenoted M(Θ) with M(Θ) = Mf (Θ) and M(Θ) = M0(Θ).

Inanyfusionproblems,weconsiderasprimordialatthevery beginningandbeforecombininginformation expressedasbelieffunctionstodefineclearlytheproperframe Θ ofthegivenproblemandtochooseexplicitly itscorrespondingmodelonewantstoworkwith.Oncethisisdone,thesecondimportantpointistoselectthe properset 2Θ , DΘ or SΘ onwhichthebelieffunctionswillbedefined.Thethirdpoint concernsthechoiceof anefficientruleofcombinationofbelieffunctionsandfinallythecriteriaadoptedfordecision-making.

4i.e.whentheframe Θ and/orthemodel M ischangingwithtime.

Inthesequel,wefocusourpresentationmainlyonhyper-powerset DΘ (unlessspeciﬁed)sinceitisthe mostinterestingnewaspectofDSmTforreadersalreadyfamiliarwithDSTframework,butafortioriwecan worksimilarlyonclassicalpowerset 2Θ ifShafer’smodelholds,andevenon 2Θref (thepowersetoftheminimalreﬁnedframe)wheneveronewantstouseitandifpossible.

Examplesofmodelsforaframe

Θ: • Let’sconsiderthe2Dproblemwhere Θ= {θ1,θ2} with DΘ = {∅,θ1 ∩ θ2,θ1,θ2,θ1 ∪ θ2} andassume nowthat θ1 and θ2 aretrulyexclusive(i.e.Shafer’smodel M0 holds),thenbecause θ1 ∩ θ2 M0 = ∅,onegets DΘ = {∅,θ1 ∩ θ2 M0 = ∅,θ1,θ2,θ1 ∪ θ2} = {∅,θ1,θ2,θ1 ∪ θ2}≡ 2Θ • AsanothersimpleexampleofhybridDSmmodel,let’sconsiderthe3Dcasewiththeframe Θ= {θ1,θ2,θ3} withthemodel M = Mf inwhichweforceallpossibleconjunctionstobeempty,but θ1 ∩ θ2.Thishybrid DSmmodelisthenrepresentedwiththeVenndiagramonFig.3(whereboundariesofintersectionof θ1 and θ2 arenotpreciselydeﬁnedif θ1 and θ2 representonlyfuzzyconceptslike smallness and tallness byexample).

Fig.3:VenndiagramofaDSmhybridmodelfora3Dframe.

2.3Generalizedbelieffunctions

Fromageneralframe Θ,wedeﬁneamap m( ): GΘ → [0, 1] associatedtoagivenbodyofevidence B as m(∅)=0 and A∈GΘ

m(A)=1 (2)

Thequantity m(A) iscalledthe generalizedbasicbeliefassignment/mass (gbba)of A The generalizedbeliefandplausibilityfunctions aredeﬁnedinalmostthesamemanneraswithinDST,i.e. Bel(A)= B⊆A B∈GΘ

m(B) Pl(A)= B∩A=∅ B∈GΘ

m(B) (3)

Werecallthat GΘ isthegenericnotationforthesetonwhichthegbbaisdefined (GΘ canbe 2Θ , DΘ oreven SΘ dependingonthemodelchosenfor Θ).Thesedefinitionsarecompatiblewiththedefinitionsof theclassicalbelieffunctionsinDSTframeworkwhen GΘ =2Θ forfusionproblemswhereShafer’smodel M0(Θ) holds.Westillhave ∀A ∈ GΘ , Bel(A) ≤ Pl(A).NotethatwhenworkingwiththefreeDSmmodel Mf (Θ),onehasalwaysPl(A)=1 ∀A = ∅∈ (GΘ = DΘ) whichisnormal.

Example:Let’sconsiderthesimpleframe Θ= {A,B},thendependingonthemodelwechoosefor GΘ,one willconsidereither: • GΘ asthepowerset 2Θ andtherefore: m(A)+ m(B)+ m(A ∪ B)=1

✫✪ ✬✩ ✫✪ ✬✩ ✫✪ ✬✩ ❅ ❘ θ1 ✠ θ

✛ θ3 12 3 2 1

• GΘ asthehyper-powerset DΘ andtherefore: m(A)+ m(B)+ m(A ∪ B)+ m(A ∩ B)=1

• GΘ asthesuper-powerset SΘ andtherefore: m(A)+ m(B)+ m(A ∪ B)+ m(A ∩ B)+ m(c(A))+ m(c(B))+ m(c(A) ∪ c(B))=1

2.4TheclassicDSmruleofcombination

WhenthefreeDSmmodel Mf (Θ) holdsforthefusionproblemunderconsideration,theclassicDSmruleof combination mMf (Θ) ≡ m(.) [m1 ⊕ m2](.) oftwoindependent5 sourcesofevidences B1 and B2 overthe sameframe Θ withbelieffunctionsBel1(.) andBel2(.) associatedwithgbba m1(.) and m2(.) correspondsto theconjunctiveconsensusofthesources.Itisgivenby[31]: ∀C ∈ DΘ ,mMf (Θ)(C ) ≡ m(C )= A,B∈DΘ A∩B=C

m1(A)m2(B) (4)

Since DΘ isclosedunder ∪ and ∩ setoperators,thisnewruleofcombinationguaranteesthat m( ) isa propergeneralizedbeliefassignment,i.e. m( ): DΘ → [0, 1].Thisruleofcombinationiscommutativeand associativeandcanalwaysbeusedforthefusionofsourcesinvolvingfuzzyconceptswhenfreeDSmmodel holdsfortheproblemunderconsideration.Thisrulehasbeenextendedfor s> 2 sourcesin[31].

AccordingtoTable2,thisclassicDSmruleofcombinationlooksveryexpensiveintermsofcomputations andmemorysizeduetothehugenumberofelementsin DΘ whenthecardinalityof Θ increases.Thisremark ishowevervalidonlyifthecores(thesetoffocalelementsofgbba) K1(m1) and K2(m2) coincidewith DΘ , i.e.when m1(A) > 0 and m2(A) > 0 forall A = ∅∈ DΘ.Fortunately,itisimportanttonoteherethatin mostofthepracticalapplicationsthesizesof K1(m1) and K2(m2) aremuchsmallerthan |DΘ| becausebodies ofevidencegenerallyallocatetheirbasicbeliefassignmentsonlyoverasubsetofthehyper-powerset.This makesthingseasierfortheimplementationoftheclassicDSmrule(4).TheDSmruleisactuallyveryeasy toimplement.Itsufﬁcesforeachfocalelementof K1(m1) tomultiplyitwiththefocalelementsof K2(m2) andthentopoolallcombinationswhichareequivalentunder thealgebraofsets.Whileverycostlyintermon memorystorageintheworstcase(i.e.whenall m(A) > 0, A ∈ DΘ or A ∈ 2Θref ),theDSmrulehowever requiresmuchsmallermemorystoragethanwhenworkingwith SΘ,i.e.workingwithaminimalreﬁnedframe satisfyingShafer’smodel.

Inmostfusionapplicationsonlyasmallsubsetofelementsof DΘ haveanonnullbasicbeliefmassbecauseallthecommitmentsarejustusuallyimpossibletoobtainpreciselywhenthedimensionoftheproblem increases.Thus,itisnotnecessarytogenerateandkeepinmemoryallelementsof DΘ (oreventually SΘ)but onlythosewhichhaveapositivebeliefmass.Howeverthereisarealtechnicalchallengeonhowtomanage efficientlyallelementsofthehyper-powerset.Thisproblemisobviouslymuchmoredifficultwhentryingto workonarefinedframeofdiscernment Θref ifonereallypreferstouseDempster-Shafertheoryandapply Dempster’sruleofcombination.Itisimportanttokeepinmindthattheultimateandminimalrefinedframe consistinginexhaustiveandexclusivefinitesetofrefinedexclusivehypothesesisjustimpossibletojustifyand todefinepreciselyforallproblemsdealingwithfuzzyandill-definedcontinuousconcepts.Adiscussionon refinementwithanexamplehasbeincludedin[31].

5Whileindependenceisadifﬁcultconcepttodeﬁneinalltheoriesmanagingepistemicuncertainty,wefollowherethe interpretationofSmetsin[39]and[40],p.285andconsider thattwosourcesofevidenceareindependent(i.edistinctand noninteracting)ifeachleavesonetotallyignorantabouttheparticularvaluetheotherwilltake.

WhenthefreeDSmmodel Mf (Θ) doesnotholdduetothetruenatureofthefusionproblemunderconsiderationwhichrequirestotakeintoaccountsomeknownintegrityconstraints,onehastoworkwithaproper hybridDSmmodel M(Θ) = Mf (Θ).Insuchcase,thehybridDSmrule(DSmH)ofcombinationbasedon thechosenhybridDSmmodel M(Θ) for k ≥ 2 independentsourcesofinformationisdeﬁnedforall A ∈ DΘ as[31]: mDSmH (A)= mM(Θ)(A) φ(A) S1(A)+ S2(A)+ S3(A) (5)

whereallsetsinvolvedinformulasareinthecanonicalform and φ(A) isthe characteristicnon-emptiness function ofaset A,i.e. φ(A)=1 if A/ ∈ ∅ and φ(A)=0 otherwise,where ∅ {∅M, ∅} ∅M isthesetof allelementsof DΘ whichhavebeenforcedtobeemptythroughtheconstraintsof themodel M and ∅ isthe classical/universalemptyset. S1(A) ≡ mMf (θ)(A), S2(A), S3(A) aredeﬁnedby S1(A) X1 ,X2,...,Xk∈DΘ X1 ∩X2∩ ∩Xk =A

k i=1 mi(Xi) (6) S2(A) X1 ,X2,...,Xk∈∅ [U =A]∨[(U∈∅)∧(A=It)]

k i=1 mi(Xi) (7) S3(A) X1 ,X2,...,Xk∈DΘ X1 ∪X2∪...∪Xk =A X1 ∩X2∩ ∩Xk ∈∅

k i=1 mi(Xi) (8) with U u(X1)∪u(X2 )∪ ∪u(Xk) where u(X ) istheunionofall θi thatcompose X, It θ1 ∪θ2 ∪ ∪θn isthetotalignorance. S1(A) correspondstotheclassicDSmrulefor k independentsourcesbasedonthefree DSmmodel Mf (Θ); S2(A) representsthemassofallrelativelyandabsolutelyemptysetswhichistransferred tothetotalorrelativeignorancesassociatedwithnonexistentialconstraints(ifany,likeinsomedynamic problems); S3(A) transfersthesumofrelativelyemptysetsdirectlyontothe canonicaldisjunctiveformof non-emptysets.

ThehybridDSmruleofcombinationgeneralizestheclassicDSmruleofcombinationandisnotequivalent toDempter’srule.Itworksforanymodels(thefreeDSmmodel,Shafer’smodeloranyotherhybridmodels) whenmanipulating precise generalized(oreventuallyclassical)basicbelieffunctions.Anextensionofthis ruleforthecombinationof imprecise generalized(oreventuallyclassical)basicbelieffunctionsispresented innextsection.Asalreadystated,inDSmTframeworkitisalsopossibletodealdirectlywithcomplements ifnecessarydependingontheproblemunderconsiderationandtheinformationprovidedbythesourcesof evidencethemselves.

Theﬁrstandsimplestwayistoworkwith SΘ onShafer’smodelwhenaminimalreﬁnementispossibleand makessense.Thesecondwayistodealwithpartiallyknownframeandintroducedirectlythecomplementary hypothesesintotheframeitself.Byexample,ifoneknowsonlytwohypotheses θ1, θ2 andtheircomplements

itisnotnecessarythattheframeisbuiltonpure/simple(possiblyvague)hypotheses θi asusuallydonein alltheoriesmanaginguncertainty.Theframe Θ canalsocontaindirectlyaselementsconjunctionsand/or disjunctions(ormixedpropositions)andnegations/complementsofpurehypothesesaswell.TheDSmrules alsoworkinsuchnon-classicframesbecauseDSmTworksonanydistributivelatticebuiltfrom Θ anywhere Θ isdeﬁned.

2.5ThehybridDSmruleofcombination

θ1, θ2,thenwecanchoosetoswitchfromoriginalframe Θ= {

1,θ2} tothenewframe Θ= {θ1,θ2, θ1, θ2}.In suchcase,wedon’tnecessarilyassumethat θ1 = θ2 and θ2 = θ1 because θ1 and θ2 mayincludeotherunknown hypotheseswehavenoinformationabout(caseofpartialknownframe).Moregenerally,inDSmTframework,

2.6Examplesofcombinationrules

HerearesomenumericalexamplesonresultsobtainedbyDSmrulesofcombination.Moreexamplescanbe foundin[31].

2.6.1Examplewith Θ= {θ1,θ2,θ3,θ4}

Let’sconsidertheframeofdiscernment Θ= {θ1,θ2,θ3,θ4},twoindependentexperts,andthetwofollowing bbas m1(θ1)=0.6 m1(θ3)=0.4 m2(θ2)=0.2 m2(θ4)=0.8 representedintermsofmassmatrix M = 0 600 40 00 200 8

• Dempster’srulecannotbeappliedbecause: ∀1 ≤ j ≤ 4,onegets m(θj )=0/0 (undeﬁned!).

• ButtheclassicDSmruleworksbecauseoneobtains: m(θ1)= m(θ2)= m(θ3)= m(θ4)=0,and m(θ1 ∩ θ2)=0 12, m(θ1 ∩ θ4)=0 48, m(θ2 ∩ θ3)=0 08, m(θ3 ∩ θ4)=0 32 (partialparadoxes/conﬂicts).

• Supposenowoneﬁndsoutthatallintersectionsareempty(Shafer’smodel),thenoneappliesthehybrid DSmruleandonegets(index h standsherefor hybrid rule): mh(θ1 ∪ θ2)=0.12, mh(θ1 ∪ θ4)=0.48, mh(θ2 ∪ θ3)=0.08 and mh(θ3 ∪ θ4)=0.32

2.6.2GeneralizationofZadeh’sexamplewith Θ= {θ1,θ2,θ3} Let’sconsider 0 <ǫ1,ǫ2 < 1 betwoverytinypositivenumbers(closetozero),theframeofdiscernmentbe Θ= {θ1,θ2,θ3},havetwoexperts(independentsourcesofevidence s1 and s2)givingthebeliefmasses m1(θ1)=1 ǫ1 m1(θ2)=0 m1(θ3)= ǫ1 m2(θ1)=0 m2(θ2)=1 ǫ2 m2(θ3)= ǫ2

Fromnowon,weprefertousematricestodescribethemasses, i.e. 1 ǫ1 0 ǫ1 01 ǫ2 ǫ2

• UsingDempster’sruleofcombination,onegets m(θ3)= (ǫ1ǫ2) (1 ǫ1) 0+0 (1 ǫ2)+ ǫ1ǫ2 =1 whichisabsurd(oratleastcounter-intuitive).Notethatwhateverpositivevaluesfor ǫ1, ǫ2 are,Dempster’sruleofcombinationprovidesalwaysthesameresult(one)whichisabnormal.Theonlyacceptable andcorrectresultobtainedbyDempster’sruleisreallyobtainedonlyinthetrivialcasewhen ǫ1 = ǫ2 =1, i.e.whenbothsourcesagreein θ3 withcertaintywhichisobvious.

• UsingtheDSmruleofcombinationbasedonfree-DSmmodel,onegets m(θ3)= ǫ1ǫ2, m(θ1 ∩ θ2)= (1 ǫ1)(1 ǫ2), m(θ1 ∩ θ3)=(1 ǫ1)ǫ2, m(θ2 ∩ θ3)=(1 ǫ2

)ǫ1 andtheothersarezerowhichappears morereliable/trustable. • GoingbacktoShafer’smodelandusingthehybridDSmruleofcombination,onegets m(θ3)= ǫ1ǫ2, m(θ1 ∪ θ2)=(1 ǫ1)(1 ǫ2), m(θ1 ∪ θ3)=(1 ǫ1)ǫ2, m(θ2 ∪ θ3)=(1 ǫ2)ǫ1 andtheothersare zero. Notethatinthespecialcasewhen ǫ1 = ǫ2 =1/2,onehas m1(θ1)=1/2 m1(θ2)=0 m1(θ3)=1/2 m2(θ1)=0 m2(θ2)=1/2 m2(θ3)=1/2 Dempster’sruleofcombinationsstillyields m(θ3)=1 whilethehybridDSmrulebasedonthesameShafer’s modelyieldsnow m(θ3)=1/4, m(θ1 ∪ θ2)=1/4, m(θ1 ∪ θ3)=1/4, m(θ2 ∪ θ3)=1/4 whichisnormal. 14

WecomparetheresultsprovidedbyDSmTrulesandthemaincommonrulesofcombinationonthefollowingverysimplenumericalexamplewhereonly2independentsources(aprioriassumedequallyreliable)are involvedandprovidingtheirbeliefinitiallyonthe3Dframe Θ= {θ1,θ2,θ3}.Itisassumedinthisexample thatShafer’smodelholdsandthusthebeliefassignments m1( ) and m2( ) donotcommitbelieftointernal conﬂictinginformation. m1( ) and m2( ) arechosenasfollows: m1(θ1)=0.1 m1(θ2)=0.4 m1(θ3)=0.2 m1(θ1 ∪ θ2)=0.3 m2(θ1)=0 5 m2(θ2)=0 1 m2(θ3)=0 3 m2(θ1 ∪ θ2)=0 1

Thesebeliefmassesareusuallyrepresentedintheformofabeliefmassmatrix M givenby M = 0 10 40 20 3 0 50 10 30 1 (9)

whereindex i fortherowscorrespondstotheindexofthesourceno. i andtheindexes j forcolumnsof M correspondtoagivenchoiceforenumeratingthefocalelementsofallsources.Inthisparticularexample,index j =1 correspondsto θ1, j =2 correspondsto θ2, j =3 correspondsto θ3 and j =4 correspondsto θ1 ∪θ2.

Nowlet’simaginethatoneﬁndsoutthat θ3 isactuallytrulyemptybecausesomeextraandcertainknowl edgeon θ3 isreceivedbythefusioncenter.Asexample, θ1, θ2 and θ3 maycorrespondtothreesuspects (potentialmurders)inapoliceinvestigation, m1( ) and m2( ) correspondstotworeportsofindependentwitnesses,butitturnsoutthatﬁnally θ3 hasprovidedastrongalibitothecriminalpoliceinvestigatoroncearrested bythepolicemen.Thissituationcorrespondstosetupahybridmodel M withtheconstraint θ3 M = ∅

Let’sexaminetheresultofthefusioninsuchsituationobtainedbytheSmets’,Yager’s,Dubois&Prade’s andhybridDSmrulesofcombinations.Firstnotethat,based onthefreeDSmmodel,onewouldgetby applyingtheclassicDSmrule(denotedherebyindex DSmC)thefollowingfusionresult

mDSmC (θ1)=0 21 mDSmC (θ2)=0 11

mDSmC (θ3)=0 06 mDSmC (θ1 ∪ θ2)=0 03

mDSmC (θ1 ∩ θ2)=0.21 mDSmC (θ1 ∩ θ3)=0.13

mDSmC (θ2 ∩ θ3)=0 14 mDSmC (θ3 ∩ (θ1 ∪ θ2))=0 11

Butbecauseoftheexclusivityconstraints(imposedhereby theuseofShafer’smodelandbythenonexistentialconstraint θ3 M = ∅),thetotalconﬂictingmassisactuallygivenby k12 =0.06+0.21+0.13+0.14+ 0.11=0.65.

• Ifoneapplies Dempster’srule [24](denotedherebyindex DS),onegets:

mDS (∅)=0

mDS (θ1)=0.21/[1 k12]=0.21/[1 0.65]=0.21/0.35=0.600000 mDS (θ2)=0 11/[1 k12]=0 11/[1 0 65]=0 11/0 35=0 314286

mDS (θ1 ∪ θ2)=0 03/[1 k12]=0 03/[1 0 65]=0 03/0 35=0 085714

•

Ifoneapplies Smets’rule[41,42](i.e.thenonnormalizedversionofDempster’srule withtheconﬂicting masstransferredontotheemptyset),onegets:

mS (∅)= m(∅)=0 65 (conﬂictingmass)

mS (θ1)=0 21

mS (θ2)=0 11 mS (θ1 ∪ θ2)=0.03

2.6.3ComparisonwithSmets,YagerandDubois&Praderules

Ifoneapplies Yager’srule [51–53],onegets:

mY (∅)=0

mY (θ1)=0.21

mY (θ2)=0 11 mY (θ1 ∪ θ2)=0 03+ k12 =0 03+0 65=0 68

• Ifoneapplies Dubois&Prade’srule [12],onegetsbecause θ3 M = ∅ :

mDP (∅)=0 (bydeﬁnitionofDubois&Prade’srule)

mDP (θ1)=[m1(θ1)m2(θ1)+ m1(θ1)m2(θ1 ∪ θ2) + m2(θ1)m1(θ1 ∪ θ2)] +[m1(θ1)m2(θ3)+ m2(θ1)m1(θ3)] =[0 1 · 0 5+0 1 · 0 1+0 5 · 0 3]+[0 1 · 0 3+0 5 · 0 2] =0.21+0.13=0.34

mDP (θ2)=[0 4 · 0 1+0 4 · 0 1+0 1 · 0 3]+[0 4 · 0 3+0 1 · 0 2] =0 11+0 14=0 25

mDP (θ1 ∪ θ2)=[m1(θ1 ∪ θ2)m2(θ1 ∪ θ2)] +[m1(θ1 ∪ θ2)m2(θ3)+ m2(θ1 ∪ θ2)m1(θ3)] +[m1(θ1)m2(θ2)+ m2(θ1)m1(θ2)]

=[0 30 1]+[0 3 · 0 3+0 1 · 0 2]+[0 1 · 0 1+0 5 · 0 4] =[0.03]+[0.09+0.02]+[0.01+0.20] =0 03+0 11+0 21=0 35

Nowifoneaddsupthemasses,onegets 0+0 34+0 25+0 35=0 94 whichislessthan1.Therefore Dubois&Prade’sruleofcombinationdoesnotworkwhenasingleton,oranunionofsingletons,becomes empty(inadynamicfusionproblem).Theproductsofsuchempty-elementcolumnsofthemassmatrix M arelost;thisproblemisﬁxedinDSmTbythesum S2(.) in(5)whichtransferstheseproductstothe totalorpartialignorances.

• Finally,ifoneapplies DSmHrule,onegetsbecause θ3 M = ∅ :

mDSmH (∅)=0 (bydeﬁnitionofDSmH)

mDSmH (θ1)=0.34 (sameas mDP (θ1))

mDSmH (θ2)=0.25 (sameas mDP (θ2)) mDSmH (θ1 ∪ θ2)=[m1(θ1 ∪ θ2)m2(θ1 ∪ θ2)] +[m1(θ1 ∪ θ2)m2(θ3)+ m2(θ1 ∪ θ2)m1(θ3)] +[m1(θ1)m2(θ2)+ m2(θ1)m1(θ2)]+[m1(θ3)m2(θ3)] =0 03+0 11+0 21+0 06=0 35+0 06=0 41 = mDP (θ1 ∪ θ2)

Wecaneasilyverifythat mDSmH (θ1)+ mDSmH (θ2)+ mDSmH (θ1 ∪ θ2)=1.Inthisexample,using thehybridDSmrule,onetransferstheproductoftheempty-element θ3 column, m1(θ3)m2(θ3)=0.2 0 3=0 06,to mDSmH (θ1 ∪ θ2),whichbecomesequalto 0 35+0 06=0 41.Clearly,DSmHrule doesn’tprovidethesameresultasDuboisandPrade’srule,butonlywhenworkingonstaticframesof discernment(restrictedcases).

•

2.7Fusionofimprecisebeliefs

Inmanyfusionproblems,itseemsverydifficult(ifnotimpossible)tohaveprecisesourcesofevidencegeneratingprecisebasicbeliefassignments(especiallywhenbelieffunctionsareprovidedbyhumanexperts),and amoreflexibleplausibleandparadoxicaltheorysupporting impreciseinformationbecomesnecessary.Inthe previoussections,wepresentedthefusionof precise uncertainandconflicting/paradoxicalgeneralizedbasic beliefassignments(gbba)inDSmTframework.Wemeanhereby precisegbba,basicbelieffunctions/masses m(.) definedpreciselyonthehyper-powerset DΘ whereeachmass m(X ),where X belongsto DΘ,isrepresentedbyonlyonerealnumberbelongingto [0, 1] suchthat X∈DΘ m(X )=1.Inthissection,wepresent theDSmfusionrulefordealingwith admissibleimprecisegeneralizedbasicbeliefassignments mI (.) defined asrealsubunitaryintervalsof [0, 1],orevenmoregeneralasrealsubunitarysets[i.e.sets,not necessarily intervals].

Animprecisebeliefassignment mI ( ) over DΘ issaid admissible ifandonlyifthereexistsforevery X ∈ DΘ atleastonerealnumber m(X ) ∈ mI (X ) suchthat X∈DΘ m(X )=1.Theideatoworkwith imprecisebeliefstructuresrepresentedbyrealsubsetintervalsof [0, 1] isnotnewandhasbeeninvestigated in[5,6,16]andreferencestherein.Theproposedworksavailableintheliterature,uponourknowledgewere limitedonlytosub-unitaryintervalcombinationintheframeworkofTransferableBeliefModel(TBM)developedbySmets[41,42].WeextendtheapproachofLamata&MoralandDenœuxbasedonsubunitary interval-valuedmassestosubunitaryset-valuedmasses;thereforetheclosedintervalsusedbyDenœuxtodenoteimprecisemassesaregeneralizedtoanysetsincludedin[0,1],i.e.inourcasethesesetscanbeunions of(closed,open,orhalf-open/half-closed)intervalsand/orscalarsallin [0, 1].Here,theproposedextension isdoneinthecontextofDSmTframework,althoughitcanalso applydirectlytofusionofimprecisebelief structureswithinTBMaswelliftheuserpreferstoadoptTBM ratherthanDSmT.

Beforepresentingthegeneralformulaforthecombinationofgeneralizedimprecisebeliefstructures,we remindthefollowingsetoperatorsinvolvedintheDSmfusionformulas.Severalnumericalexamplesaregiven inthechapter6of[31]. • Additionofsets

∈ S2}

• Divisionofsets:If 0 doesn’tbelongto S2, S1 S2 {x | x = s1/s2,s1 ∈ S1,s2 ∈ S2}

2.7.1DSmruleofcombinationforimprecisebeliefs

WepresentthegeneralizationoftheDSmrulestocombineany typeofimprecisebeliefassignmentwhichmay berepresentedbytheunionofseveralsub-unitary(half-)openintervals,(half-)closedintervalsand/orsetsof pointsbelongingto[0,1].Severalnumericalexamplesarealsogiven.Inthesequel,oneusesthenotation (a,b) foranopeninterval, [a,b] foraclosedinterval,and (a,b] or [a,b) forahalfopenandhalfclosedinterval.From thepreviousoperatorsonsets,onecangeneralizetheDSmrules(classicandhybrid)fromscalarstosetsinthe followingway[31](chap.6): ∀A = ∅∈ DΘ , m I (A)= X1,X2,...,Xk ∈DΘ (X1∩X2∩ ∩Xk )=A i=1,...,k

m I i (Xi)

1 ⊞ S2 = S2 ⊞ S1 {x | x = s1 + s2,s1 ∈ S1,s2 ∈ S2} • Subtractionofsets

1 ⊟ S2 {x | x = s1 s2,s1 ∈ S1,s2 ∈ S2} • Multiplicationofsets

1 S2 {x | x = s1 · s2,s1 ∈ S1,s2

(10)

where and representthesummation,andrespectivelyproduct,ofsets.

Similarly,onecangeneralizethehybridDSmrulefromscalarstosetsinthefollowingway: m I DSmH (A)= m I M(Θ)(A) φ(A) SI 1 (A) ⊞ SI 2 (A) ⊞ SI 3 (A) (11) whereallsetsinvolvedinformulasareinthecanonicalform and φ(A) isthe characteristicnonemptiness function oftheset A and SI 1 (A), SI 2 (A) and SI 3 (A) aredeﬁnedby SI 1 (A)

Inthecasewhenallsetsarereducedtopoints(numbers),the setoperationsbecomenormaloperationswith numbers;thesetsoperationsaregeneralizationsofnumericaloperations.Whenimprecisebeliefstructuresreducetoprecisebeliefstructure,DSmrules(10)and(11)reducetotheirpreciseversion(4)and(5)respectively.

2.7.2Example

Hereisasimpleexampleoffusionwithmultiple-intervalmasses.Forsimplicity,thisexampleisaparticular casewhenthetheoremofadmissibility(see[31]p.138fordetails)isveriﬁedbyafewpoints,whichhappento bejustonthebounders.Itisanextremeexample,becausewetriedtocompriseallkindsofpossibilitieswhich mayoccurintheimpreciseorveryimprecisefusion.So,let’sconsiderafusionproblemover Θ= {θ1,θ2}, twoindependentsourcesofinformationwiththefollowingimpreciseadmissiblebeliefassignments

[

∨[(

∅

∧

X1 ,X2,...,Xk ∈DΘ

1 ∩X2∩ ∩Xk=A i=1,...,k m I i (Xi) (12) SI 2 (A)

1 ,X2,...,Xk ∈∅

U =A]

U∈

)

(A=It)] i=1,...,k m I i (Xi) (13) SI 3 (A) X1 ,X2,...,Xk ∈DΘ X1 ∪X2∪ ∪Xk=A

1∩X2∩ ∩Xk∈∅ i=1,...,k m I i (Xi) (14)

A ∈ DΘ mI 1(A) mI 2(A) θ1 [0 1, 0 2] ∪{0 3} [0 4, 0 5] θ2 (0 4, 0 6) ∪ [0 7, 0 8] [0, 0 4] ∪{0 5, 0 6} Table3:Inputsofthefusionwithimprecisebba’s. UsingtheDSmclassic(DSmC)ruleforsets,onegets

I (θ1)=([0 1, 0 2] ∪{0 3}) [0 4, 0 5]=([0 1, 0 2] [0 4, 0 5]) ∪ ({0 3} [0 4, 0 5]) =[0 04, 0 10] ∪ [0 12, 0 15] m I (θ2)=((0 4, 0 6) ∪ [0 7, 0 8]) ([0, 0 4] ∪{0 5, 0 6}) =((0 4, 0 6) [0, 0 4]) ∪ ((0 4, 0 6) {0 5, 0 6}) ∪ ([0 7, 0 8] [0, 0 4]) ∪ ([0 7, 0 8] {0 5, 0 6}) =(0, 0.24) ∪ (0.20, 0.30) ∪ (0.24, 0.36) ∪ [0, 0.32] ∪ [0.35, 0.40] ∪ [0.42, 0.48] =[0, 0 40] ∪ [0 42, 0 48]

m I (

1 ∩ θ2)=[([0.1, 0.2] ∪{0.3}) ([0, 0.4] ∪{0.5, 0.6})] ⊞ [[0.4, 0.5] ((0.4, 0.6) ∪ [0.7, 0.8])] =[([0 1, 0 2] [0, 0 4]) ∪ ([0 1, 0 2] {0 5, 0 6}) ∪ ({0 3} [0, 0 4]) ∪ ({0 3} {0 5, 0 6})]

⊞ [([0.4, 0.5] (0.4, 0.6)) ∪ ([0.4, 0.5] [0.7, 0.8])] =[[0, 0 08] ∪ [0 05, 0 10] ∪ [0 06, 0 12] ∪ [0, 0 12] ∪{0 15, 0 18}] ⊞ [(0 16, 0 30) ∪ [0 28, 0 40]] =[[0, 0 12] ∪{0 15, 0 18}] ⊞ (0 16, 0 40] =(0.16, 0.52] ∪ (0.31, 0.55] ∪ (0.34, 0.58]=(0.16, 0.58]

HenceﬁnallythefusionadmissibleresultwithDSmCruleisgivenby:

A ∈ DΘ mI (A)=[mI 1 ⊕ mI 2](A)

θ1 [0 04, 0 10] ∪ [0 12, 0 15]

θ2 [0, 0 40] ∪ [0 42, 0 48] θ1 ∩ θ2 (0 16, 0 58] θ1 ∪ θ2 0

Table4:FusionresultwiththeDSmCrule.

Ifoneﬁndsout6 that θ1 ∩ θ2 M ≡∅ (thisisourhybridmodel M onewantstodealwith),thenoneusesthehybrid DSmrule(11)forsets: mI M(θ1 ∩ θ2)=0 and mI M(θ1 ∪ θ2)=(0 16, 0 58],theothersimprecisemassesare notchanged.

WiththehybridDSmrule(DSmH)appliedtoimprecisebeliefs,onegetsnowtheresultsgiveninTable5.

A ∈ DΘ mI M(A)=[mI 1 ⊕ mI 2](A) θ1 [0 04, 0 10] ∪ [0 12, 0 15] θ2 [0, 0 40] ∪ [0 42, 0 48] θ1 ∩ θ2 M ≡∅ 0 θ1 ∪ θ2 (0.16, 0.58]

Table5:FusionresultwithDSmHrulefor M

Let’schecknowtheadmissibilitycondition.Forthesource 1,thereexisttheprecisemasses (m1(θ1)= 0 3) ∈ ([0 1, 0 2] ∪{0 3}) and (m1(θ2)=0 7) ∈ ((0 4, 0 6) ∪ [0 7, 0 8]) suchthat 0 3+0 7=1.Forthesource 2,thereexisttheprecisemasses (m1(θ1)=0 4) ∈ ([0 4, 0 5]) and (m2(θ2)=0 6) ∈ ([0, 0 4] ∪{0 5, 0 6}) suchthat 0 4+0 6=1.Thereforebothsourcesassociatedwith mI 1( ) and mI 2( ) areadmissibleimprecise sourcesofinformation.ItcanbeverifiedthatDSmCfusionof m1( ) and m2( ) yieldstheparadoxicalbba m(θ1)=[m1 ⊕m2](θ1)=0 12, m(θ2)=[m1 ⊕m2](θ2)=0 42 and m(θ1 ∩θ2)=[m1 ⊕m2](θ1 ∩θ2)=0 46. Oneseesthattheadmissibilityconditionissatisfiedsince (m(θ1)=0.12) ∈ (mI (θ1)=[0.04, 0.10] ∪ [0.12, 0.15]), (m(θ2)=0.42) ∈ (mI (θ2)=[0, 0.40]∪[0.42, 0.48]) and (m(θ1 ∩θ2)=0.46) ∈ (mI (θ1 ∩θ2)= (0.16, 0.58]) suchthat 0.12+0.42+0.46=1.Similarlyifonefindsoutthat θ1 ∩ θ2 = ∅,thenoneusesDSmH ruleandonegets: m(θ1 ∩ θ2)=0 and m(θ1 ∪ θ2)=0.46;theothersremainunchanged.Theadmissibility conditionstillholds,becauseonecanpickatleastonenumberineachsubset mI ( ) suchthatthesumofthese numbersis1.

3ProportionalConﬂictRedistributionrule

InsteadofapplyingadirecttransferofpartialconflictsontopartialuncertaintiesaswithDSmH,theideabehind theProportionalConflictRedistribution(PCR)rule[33,35]istotransfer(totalorpartial)conflictingmassesto non-emptysetsinvolvedintheconflictsproportionallywithrespecttothemassesassignedtothembysources asfollows: 6Weconsidernowadynamicfusionproblem.

1.calculationtheconjunctiveruleofthebeliefmassesofsources;

2.calculationthetotalorpartialconﬂictingmasses;

3.redistributionofthe(totalorpartial)conﬂictingmassestothenon-emptysetsinvolvedintheconﬂicts proportionallywithrespecttotheirmassesassignedbythe sources.

ThewaytheconflictingmassisredistributedyieldsactuallyseveralversionsofPCRrules.ThesePCRfusion rulesworkforanydegreeofconflict,foranyDSmmodels(Shafer’smodel,freeDSmmodeloranyhybrid DSmmodel)andbothinDSTandDSmTframeworksforstaticordynamicalfusionsituations.Wepresent belowonlythemostsophisticatedproportionalconflictredistributionruledenotedPCR5in[33,35].PCR5 ruleiswhatwefeelthemostefficientPCRfusionruledevelopedsofar.Thisruleredistributesthepartial conflictingmasstotheelementsinvolvedinthepartialconflict,consideringtheconjunctivenormalformof thepartialconflict.PCR5iswhatwethinkthemostmathematicallyexactredistributionofconflictingmassto non-emptysetsfollowingthelogicoftheconjunctiverule. Itdoesabetterredistributionoftheconflictingmass thanDempster’srulesincePCR5goesbackwardsonthetracks oftheconjunctiveruleandredistributesthe conflictingmassonlytothesetsinvolvedintheconflictandproportionallytotheirmassesputintheconflict. PCR5ruleisquasi-associativeandpreservestheneutralimpactofthevacuousbeliefassignmentbecausein anypartialconflict,aswellinthetotalconflict(whichisasumofallpartialconflicts),theconjunctivenormal formofeachpartialconflictdoesnotinclude Θ since Θ isaneutralelementforintersection(conflict),therefore Θ getsnomassaftertheredistributionoftheconflictingmass.Wehaveprovedin[35]thecontinuityproperty ofthefusionresultwithcontinuousvariationsofbba’stocombine.

3.1PCRformulas

ThePCR5formulaforthecombinationoftwosources(s =2)isgivenby: mPCR5(∅)=0 and ∀X ∈ GΘ \{∅} mPCR5(X )= m12(X )+ Y ∈GΘ \{X} X∩Y =∅

[ m1(X )2 m2(Y ) m1(X )+ m2(Y ) + m2(X )2m1(Y ) m2(X )+ m1(Y ) ] (15)

whereallsetsinvolvedinformulasareincanonicalformand where GΘ correspondstoclassicalpowerset 2Θ if Shafer’smodelisused,ortoaconstrainedhyper-powerset DΘ ifanyotherhybridDSmmodelisusedinstead, ortothesuper-powerset SΘ iftheminimalreﬁnement Θref of Θ isused; m12(X ) ≡ m∩(X ) correspondsto theconjunctiveconsensuson X betweenthe s =2 sourcesandwherealldenominatorsaredifferentfromzero. Ifadenominatoriszero,thatfractionisdiscarded.

AgeneralformulaofPCR5forthefusionof s> 2 sourceshasbeenproposedin[35],butamoreintuitivePCRformula(denotedPCR6)whichprovidesgoodresultsinpracticehasbeenproposedbyMartinand Osswaldin[35](pages69-88).WecanrewriteMartin-OsswaldPCR6formulainthefollowingsimpleway: mPCR6(∅)=0

(

) correspondstotheconjunctiveconsensuson X betweenthe s> 2 sources.Ifadenominatoriszero,thatfractionisdiscardedbecauseallmasses mi(Xi)=0 sothenumeratorisalsozero,i.e.noconﬂictingmass(nothingtoredistribute).Fortwosources(s =2),PCR5 andPCR6formulascoincide.TheimplementationofPCR6iseasierthanPCR5andcanbefoundin[50].

and ∀X ∈ GΘ \{∅} mPCR6(X )= m12...s(X )+ X1,X2 ,...,Xs∈GΘ \{∅} X1∩X2 ∩ ∩Xs=∅ [ s r=1 δX Xr mr(Xr )] m1(X1)m2(X2) ...ms(Xs) m1(X1)+ m2(X2)+ ... + ms

δX Xr 1, if X = Xr 0, if X = Xr

≡ m∩

(

) (16) where

(17) andwherethebeliefmassassignment m12...s(X )

3.2Examples

• Example1:Let’stake Θ= {A,B} ofexclusiveelements(Shafer’smodel),andthefollowingbba:

ABA ∪ B m1(.) 0.600.4 m2( ) 00.30.7 m∩( ) 0.420.120.28

Theconflictingmassis k12 = m∩(A ∩ B) andequals m1(A)m2(B)+ m1(B)m2(A)=0.18.Therefore A and B aretheonlyfocalelementsinvolvedintheconflict.HenceaccordingtothePCR5hypothesis only A and B deserveapartoftheconflictingmassand A ∪ B donotdeserve.WithPCR5,oneredistributestheconflictingmass k12 =0.18 to A and B proportionallywiththemasses m1(A) and m2(B) assignedto A and B respectively.

HerearetheresultsobtainedfromDempster’srule,DSmHand PCR5:

ABA ∪ B mDS 0.5120.1460.342 mDSmH 0.4200.1200.460 mPCR5 0.5400.1800.280

• Example2:Let’smodifyexample1andconsider

ABA ∪ B m1( ) 0.600.4 m2( ) 0.20.30.5 m∩( ) 0.500.120.20

Theconﬂictingmass k12 = m∩(A ∩ B) aswellasthedistributioncoefﬁcientsforthePCR5remains the sameasinthepreviousexamplebutonegetsnow

ABA ∪ B mDS 0.6090.1460.231 mDSmH 0.5000.1200.380 mPCR5 0.6200.1800.200

• Example3:Let’smodifyexample2andconsider

ABA ∪ B m1(.) 0.60.30.1 m2( ) 0.20.30.5 m∩( ) 0.440.270.05

Theconflictingmass k12 =0 24= m1(A)m2(B)+ m1(B)m2(A)=0 24 isnowdifferentfrom previousexamples,whichmeansthat m2(A)=0 2 and m1(B)=0 3 didmakeanimpactonthe conflict.Therefore A and B aretheonlyfocalelementsinvolvedintheconflictandthusonly A and B deserveapartoftheconflictingmass.PCR5redistributesthepartialconflictingmass0.18to A and B proportionallywiththemasses m1(A) and m2(B) andalsothepartialconflictingmass0.06to A and B proportionallywiththemasses m2(A) and m1(B).Afterallderivations(see[13]fordetails),onefinally gets:

ABA ∪ B

mDS 0.5790.3550.066 mDSmH 0.4400.2700.290 mPCR5 0.5840.3660.050

Oneclearlyseesthat mDS (A ∪ B) getssomemassfromtheconflictingmassalthough A ∪ B doesnot deserveanypartoftheconflictingmass(accordingtoPCR5hypothesis)since A ∪ B isnotinvolvedin theconflict(only A and B areinvolvedintheconflictingmass).Dempster’sruleappearstouslessexact thanPCR5andInagaki’srules[15].Itcanbeshowed[13]that Inagaki’sfusionrule(withanoptimal choiceoftuningparameters)canbecomeinsomecasesveryclosetoPCR5butuponouropinionPCR5 resultismoreexact(atleastlessad-hocthanInagaki’sone).

• Example4(Amoreconcreteexample):Threepeople,John(J),George(G),andDavid(D)aresuspectstoamurder.Sotheframeofdiscernmentis Θ {J,G,D}.Twosources m1( ) and m2( ) (witnesses)providethefollowinginformation: JGD m1 0.900.1 m2 00.80.2

WeknowthatJohnandGeorgearefriends,butJohnandDavidhateeachother,andsimilarlyGeorgeand David.

a)Freemodel,i.e.allintersectionsarenonempty: J ∩G = ∅, J ∩D = ∅, G∩D = ∅, J ∩G∩D = ∅. UsingtheDSmclassicruleonegets: JGDJ ∩ GJ ∩ DG ∩ DJ ∩ G ∩ D mDSmC 000.020.720.180.080 SowecanseethatJohnandGeorgetogether(J ∩ G)aremostlikelytohavecommittedthecrime, sincethemass mDSmC (J ∩ G)=0.72 isthebiggestresultingmassafterthefusionofthetwo sources.InShafer’smodel,onlyonesuspectcouldcommitthecrime,butthefreeandhybrid modelsallowtwoormorepeopletohavecommittedthesamecrime-whichhappensinreality.

b)Let’sconsiderthehybridmodel,i.e.someintersections areempty,andothersarenot.Accordingto theabovestatementabouttherelationshipsbetweenthethreesuspects,wecandeducethat J ∩ G = ∅,while J ∩ D = G ∩ D = J ∩ G ∩ D = ∅.ThenweﬁrstapplytheDSmClassicrule,andthen thetransferoftheconﬂictingmassesisdonewithPCR5:

JGDJ ∩ GJ ∩ DG ∩ DJ ∩ G ∩ D m1 0.900.1 m2 00.80.2 mDSmC 000.020.720.180.080

UsingPCR5nowwetransfer m(J ∩ D)=0 18,since J ∩ D = ∅,to J and D proportionallywith 0.9and0.2respectively,so J gets0.15and D gets0.03since: xJ/0 9= z1D/0 2=0 18/(0 9+0 2)=0 18/1 1

whence xJ =0 9(0 18/1 1)=0 15 and z1D =0 2(0 18/1 1)=0 03. AgainusingPCR5,wetransfer m(G ∩ D)=0 08,since G ∩ D = ∅,to G and D proportionally with0.8and0.1respectively,so G gets0.07and D gets0.01since: yG/0.8= z2D/0.1=0.08/(0.8+0.1)=0.08/0.9

whence yG =0.8(0.08/0.9)=0.07 and zD =0.1(0.08/0.9)=0.01.Addingwegetﬁnally:

JGDJ ∩ GJ ∩ DG ∩ DJ ∩ G ∩ D mPCR5 0.150.070.060.72000

SoonehasahighbeliefthatthecriminalsareJohnandGeorge (bothofthemcommittedthecrime) since m(J ∩ D)=0 72 anditisbyfarthegreatestfusionmass.

InShafer’smodel,ifwetrytoreﬁnewegetthedisjointparts: D, J ∩ G, J \ (J ∩ G),and G \ (J ∩ G), butthelasttwoareridiculous(whatisthereal/physicalnatureof J \ (J ∩ G) or G \ (J ∩ G) ?Halfof aperson(!)?),sothereﬁningdoesnotworkhereinreality.That’swhythehybridandfreemodelsare needed.

• Example5(ImprecisePCR5):ThePCR5formulacannaturallyworkalsoforthecombinationof imprecisebba’s.Thishasbeenalreadypresentedinsection 1.11.8page49of[35]withanumerical exampletoshowhowtoapplyit.Thisexamplewillthereforenotbereincludedhere.

3.3Zadeh’sexample

Wecompareherethesolutionsforwell-knownZadeh’sexample[56,59]providedbyseveralfusionrules. Adetailedpresentationwithmorecomparisonscanbefoundin[31,35].Let’sconsider Θ= {M,C,T } as theframeofthreepotentialoriginsaboutpossiblediseasesofapatient(M standingfor meningitis, C for concussion and T for tumor),theShafer’smodelandthetwofollowingbeliefassignmentsprovidedbytwo independentdoctorsafterexaminationofthesamepatient.

m1(M )=0.9 m1(C )=0 m1(T )=0.1 m2(M )=0 m2(C )=0 9 m2(T )=0 1

Thetotalconﬂictingmassishighsinceitis

m1(M )m2(C )+ m1(M )m2 (T )+ m2(C )m1(T )=0 99

• withDempster’sruleandShafer’smodel(DS),onegetsthecounter-intuitiveresult(seejustiﬁcations in[11,31,48,53,56]): mDS (T )=1

• withYager’srule[53]andShafer’smodel: mY (M ∪ C ∪ T )=0 99 and mY (T )=0 01

• withDSmHandShafer’smodel: mDSmH (M ∪ C )=0 81 mDSmH (T )=0 01 mDSmH (M ∪ T )= mDSmH (C ∪ T )=0.09

• TheDubois&Prade’srule(DP)[11]basedonShafer’smodelprovidesinZadeh’sexamplethesame resultasDSmH,becauseDPandDSmHcoincideinallstaticfusionproblems7

• withPCR5andShafer’smodel: mPCR5(M )= mPCR5(C )=0 486 and mPCR5(T )=0 028.

Oneseesthatwhenthetotalconflictbetweensourcesbecomes high,DSmTisable(uponauthorsopinion)to managemoreadequatelythroughDSmHorPCR5rulesthecombinationofinformationthanDempster’srule, evenwhenworkingwithShafer’smodel-whichisonlyaspecifichybridmodel.DSmHruleisinagreement withDPruleforthestaticfusion,butDSmHandDPrulesdifferingeneral(fornondegeneratecases)fordynamicfusionwhilePCR5ruleisthemostexactproportionalconflictredistributionrule.Besidesthisparticular example,weshowedin[31]thatthereexistseveralinfiniteclassesofcounter-examplestoDempster’srule whichcanbesolvedbyDSmT.

Insummary,DSTbasedonDempster’sruleprovidescounter-intuitiveresultsinZadeh’sexample,orinnonBayesianexamplessimilartoZadeh’sandnoresultwhentheconﬂictis1.Onlyad-hocdiscountingtechniques

7IndeedDPrulehasbeendevelopedforstaticfusiononlywhileDSmHhasbeendevelopedtotakeintoaccountthe possibledynamicityoftheframeitselfandalsoitsassociatedmodel.

allowtocircumventtroublesofDempster’sruleorweneedto switchtoanothermodelofrepresentation/frame; inthelatercasethesolutionobtaineddoesn’tﬁtwiththeShafer’smodeloneoriginallywantedtoworkwith. Wewantalsotoemphasizethatindynamicfusionwhentheconﬂictbecomeshigh,bothDST[24]andSmets’ TransferableBeliefModel(TBM)[41]approachesfailtorespondtonewinformationprovidedbynewsources. Thiscanbeeasilyshowedbytheverysimplefollowingexample.

Example (whereTBMdoesn’trespondtonewinformation):

              

and



 



m(12)3 DSmH (B)=0.240 m(12)3 DSmH (C )=0 120 m(12)3 DSmH (A ∪ B)=0 224 m(12)3 DSmH (A ∪ C )=0.056 m(12)3 DSmH (A ∪ B ∪ C )=0.360 

4Uniformandpartiallyuniformredistributionrules TheprinciplesofUniformRedistributionRule(URR)andPartiallyUniformRedistributionRule(PURR)have beenproposedin2006withexamplesin[34].

toallfocalele-

Θ generatedbytheconsensusoperator.ThiswayofredistributingmassisverysimpleandURRis 8Weintroducehereexplicitlytheindexesofsourcesinthefusionresultsincemorethantwosourcesareconsideredin thisexample. 9ActuallyDempster’sruledoesn’trespondalsotonewcompatibleinformation/bbaassoonasatotalmassofbeliefis alreadycommittedbyasourcetoonlyonefocalelement.Forexample,ifoneconsiders Θ= {A,B} withShafer’smodel (A ∩ B = ∅)andwith m1(A)=1, m2 (A)=0 2 and m2(B)=0 8,thenDempster’srulealwaysprovides mDS (A)=1 whateverarethevaluestakenby m2(A) > 0 and m2 (B) > 0.

          

Let Θ= {A,B,C} withthe(precise)bba’s m1(A)=0 4, m1(C )=0 6 and m2(A)=0 7, m2(B)=0 3. Thenonegets8 withDempster’srule,Smets’TBM(i.e.thenon-normalizedversionofDempster’scombination),DSmHandPCR5: m12 DS (A)=1, m12 TBM (A)=0.28, m12 TBM (∅)=0.72,    

12 DSmH (A)=0.28

12 DSmH (A ∪ B)=0 12

12 DSmH (A ∪ C )=0 42 m12 DSmH (B ∪ C )=0.18

m12 PCR5(A)=0 574725 m12 PCR5(B)=0.111429 m12 PCR5(C )=0 313846

Nowlet’sconsideratemporalfusionproblemandintroducea thirdsource m3( ) with m3(B)=0 8 and m3(C )=0 2.Thenonesequentiallycombinestheresultsobtainedby m12 TBM ( ), m12 DS ( ), m12 DSmH ( ) and m12 PCR( ) withthenewevidence m3( ) andoneseesthat m(12)3 DS becomesnotdeﬁned(divisionbyzero)and m(12)3 TBM (∅)=1 while(DSmH)and(PCR5)provide

and 

m(12)3 PCR5(A)=0 277490 m(12)3 PCR5(B)=0 545010 m(12)3 PCR5(C )=0.177500

Whenthemasscommittedtoemptysetbecomesoneataprevious temporalfusionstep,thenbothDST andTBMdonotrespondtonewinformation9.Let’scontinuetheexampleandconsiderafourthsource m4(.) with m4(A)=0 5, m4(B)=0 3 and m4(C )=0 2.Thenitiseasytoseethat m((12)3)4 DS ( ) isnotdefined sinceatpreviousstep m(12)3 DS ( ) wasalreadynotdefined,andthat m((12)3)4 TBM (∅)=1 whatever m4( ) isbecause atthepreviousfusionsteponehad m(12)3 TBM (∅)=1.Thereforeforanumberofsources n ≥ 2,DSTandTBM approachesdonotrespondtonewinformationincominginthe fusionprocesswhileboth(DSmH)and(PCR5) rulesrespondtonewinformation.TomakeDSTand/orTBMworkingproperlyinsuchcases,itisnecessary tointroducead-hoctemporaldiscountingtechniqueswhich arenotnecessarytointroduceifDSmTisadopted. Iftherearegoodreasonstointroducetemporaldiscounting,thereisobviouslynodifficultytoapplytheDSm fusionofthesediscountedsources.Ananalysisofthisbehaviorfortargettypetrackingispresentedin[9,35].

TheUniformRedistributionRuleconsistsinredistributingthetotalconﬂictingmass k

mentsof G

differentfromDempster’sruleofcombination,becauseDempster’sruleredistributesthetotalconflictproportionallywithrespecttothemassesresultedfromtheconjunctiveruleofnon-emptysets.PCR5rulepresented previouslydoesproportionalredistributionsofpartialconflictingmassestothesetsinvolvedintheconflict. TheURRformulafortwosourcesisgivenby: ∀A = ∅ m12URR(A)= m12(A)+ 1 n12 X1 ,X2∈GΘ X1∩X2 =∅

m1(X1)m2(X2) (18) where m12(A) istheresultoftheconjunctiveruleappliedtobeliefassignments m1( ) and m2( ),and n12 = Card{Z ∈ GΘ,m1(Z ) =0 or m2(Z ) =0}. For s ≥ 2 sourcestocombine: ∀A = ∅,onehas m12...sURR(A)= m12...s(A)+ 1 n12...s X1,X2 ,...,Xs∈GΘ X1 ∩X2∩ ∩Xs=∅

s i=1 m1(Xi) (19) where m12...s(A) istheresultoftheconjunctiveruleappliedto mi(.),forall i ∈{1, 2,...,s} and n12...s = Card{Z ∈ GΘ ,m1(Z ) =0 or m2(Z ) =0 or or ms(Z ) =0}

Asalternative(modiﬁedversionofURR),wecanalsoconsiderthecardinaloftheensembleofsetswhose massesresultedfromtheconjunctiverulearenon-null,i.e.thecardinalityofthecoreofconjunctiveconsensus: n c 12...s = Card{Z ∈ GΘ ,m12...s(Z ) =0}

Itisalsopossibletodoauniformlypartialredistribution,i.e.touniformlyredistributetheconflictingmass onlytothesetsinvolvedintheconflict.Forexample,if m12(A ∩ B)=0.08 and A ∩ B = ∅,then0.08is equallyredistributedto A and B only,supposing A and B arebothnon-empty,so0.04assignedto A and0.04 to B. ThePartiallyUniformRedistributionRule(PURR)fortwosourcesisdefinedasfollows: ∀A = ∅ m12PURR(A)= m12(A)+ 1 2

}) s i=1 m1(Xi) (21) whereCardA({X1 ,...,Xs}) isthenumberof A’soccurringin {X1 ,X2,...,Xs} If A = ∅, m12PURR(A)=0 and m12...sPURR(A)=0 TheseruleshavealowcomputationcostwithrespecttoProportionalConﬂictRedistribution(PCR)rules developedintheDSmTframeworkandtheypreservetheneutralityofthevacuousbeliefassignment(VBA)

m1(.) combinedwithVBAdeﬁnedonanyframe Θ= {θ1,...,θn} by mVBA

θ1 ∪...∪θn

, 25

X1 ,X2∈GΘ X1 ∩X2=∅ X1=A or X2 =A m1(X1)m2(X2) (20) where m12(A) istheresultoftheconjunctiveruleappliedtobeliefassignments m1( ) and m2( ) For s ≥ 2 sourcestocombine: ∀A = ∅,onehas m12...sPURR(A)= m12...s(A) + 1 s X1 ,X2,...,Xs∈GΘ X1 ∩X2 ∩ ∩Xs =∅ atleatone Xj =A,j∈{1,...,s} CardA({X1 ,...,Xs

sinceanybba

(

)=1

usingtheconjunctiverule,gives m1( ),sonoconflictingmassisneededtotransfer.Ofcoursethese rules areveryeasytoimplementbutfromatheoreticalpointofviewtheyremainlesspreciseintheirtransferof conflictingbeliefssincetheydonottakeintoaccounttheproportionalredistributionwithrespecttothemassof eachsetinvolvedintheconflict.Reasonably,URRorPURRcannotoutperformPCR5buttheymayhopefully couldappearasgoodenoughinsomespecificfusionproblemswhentheleveloftotalconflictisnotimportant. PURRdoesamorerefinedredistributionthatURRandMURRbutitrequiresalittlemorecalculation.

5RSCFusionrules

Inthissection,webrieﬂyrecallanewclassoffusionrulesbasedonthebeliefredistributiontosubsetsorcomplementsanddenotedCRSC(standingforClassofRedistributionrulestoSubsetsorComplements)forshort. Thisclassispresentedindetailsin[37]withseveralexamples.

Let m1(.) and m2(.) betwonormalizedbasicbeliefassignments(bba’s)defined10 from SΘ to [0, 1].We usetheconjunctiveruletofirstcombine m1(.) with m2(.) toget m∩(.) andthenthemassofconflictsay m∩(X ∩ Y )=0,when X ∩ Y = ∅ orevenwhen X ∩ Y isdifferentfromtheemptysetisredistributedto subsetsorcomplementsinmanyways(see[37]fordetails).Thenewclassoffusionrule(denoted CRSCc) fortransferringtheconflictingmassesonlyisdefinedfor A ∈ SΘ \{∅,It} by: mCRSCc (A)= m∩(A)+[α m∩(A)+ β Card(A)+ γ f (A)] X,Y ∈ SΘ X ∩ Y = ∅ A ⊆ M

m1(X )m2(Y ) Z∈SΘ,Z⊆M [α m∩(Z )+ β Card(Z )+ γ f (Z )] (22) where It = θ1 ∪ θ2 ∪ ∪ θn representsthetotalignorancewhen Θ= {θ1,...,θn} M canbe c(X ∪ Y ) (thecomplementof X ∪ Y ),orasubsetof c(X ∪ Y ),or X ∪ Y ,orasubsetof X ∪ Y ; α,β,γ ∈{0, 1} but α + β + γ =0;inaweightedwaywecantake α,β,γ ∈ [0, 1] alsowith α + β + γ =0; f (X ) isafunctionof X,i.e.anotherparameterthatthemassof X isdirectlyproportionallywithrespectto; Card(X ) isthecardinal of X

Themassofbelief mCRSCc (It) committedtothetotalignoranceisgivenby: mCRSCc (It)= m∩(It)+ X,Y ∈ SΘ {X ∩ Y = ∅ and M = ∅} or

{X ∩ Y = ∅ and Den

} m

where Den(Z ) Z∈SΘ,Z⊆M [α · m∩(Z )+ β · Card(Z )+ γ · f (Z )] Amoregeneralformulafortheredistributionofconflictand non-conflicttosubsetsorcomplementsclass ofrulesforthefusionofmassesofbelieffortwosourcesofevidenceisdefined A ∈ (SΘ Snon∅ ∩ ) {∅, Θ} by: mCRSC (A)= m∩(A)+ X,Y ∈ SΘ {X ∩ Y = ∅,A ∈T (X,Y )} or {X ∩ Y ∈ Snon∅ ∩,r ,A ∈T ′ (X,Y )} f (A) m1(X )m2 (Y ) Z∈T (X,Y ) f (Z ) (24) 10Sincetheserulesuseexplicitelythecomplementationoperator c( ),theyapplyonlywiththesuper-powerset S Θ or on 2Θ dependingontheunderlyingmodelchosenfortheframe Θ. 26

(Z)=0

1(X )m2(Y ) (23)

andfor A = It:

m1(X )m2 (Y ) (25) where S∩ = {X ∈ SΘ|X = Y ∩ Z, where Y,Z ∈ SΘ {∅}},allpropositionsareexpressedintheircanonical formandwhere Xcontainsatleastan ∩ symbolinitsexpression; S∅ ∩ bethesetofallemptyintersectionsfrom S∩ (i.e.thesetofexclusivityconstraints),and Snon∅ ∩ thesetofallnon-emptyintersectionsfrom S∩ Snon∅ ∩,r isthesetofallnon-emptyintersectionsfrom Snon∅ ∩ whosemassesareredistributedtoothersets/propositions Theset Snon∅ ∩,r highlydependsonthemodelfortheframeoftheapplicationunderconsideration. f ( ) isa mappingfrom SΘ to R+.Forexample,wecanchoose f (X )= m∩(X ), f (X )= |X|, f T (X )= |X| |T (X,Y )| ,or f (x)= m∩(X )+ |X|,etc.Thefunction T speciﬁesasubsetof SΘ,forexample T (X,Y )= {c(X ∪ Y )},or T (X,Y )= {X ∪ Y } orcanspecifyasetofsubsetsof SΘ.Forexample, T (X,Y )= {A ⊂ c(X ∪ Y )},or T (X,Y )= {A ⊂ X ∪ Y }.Thefunction T ′ isasubsetof SΘ,forexample T ′(X,Y )= {X ∪ Y },or T ′ isa subsetof X ∪ Y ,etc.

mCRSC (It)= m∩(It)+ X,Y ∈ SΘ X ∩ Y = ∅, {T (X,Y )= ∅ or X Z∈T (X,Y ) f (Z)=0}

Itisimportanttohighlightthatinformulas(22)-(23)onetransfersonlytheconflictingmasses,whereasthe formulas(24)-(25)aremoregeneralsinceonetransfersthe conflictingmassesorthenon-conflictingmasses aswelldependingonthepreferencesofthefusionsystemdesigner.Thepreviousformulashavebeendirectly extendedforany s ≥ 2 sourcesofevidencein[37].AlldenominatorsintheseCRSCformulasarenaturally supposeddifferentfromzero.Itisworthtonotealsothattheextensionsoftheserulesforincludingthe reliabilitiesofthesourcesarealsopresentedin[37].

6Thegeneralizedpignistictransformation(GPT)

6.1Theclassicalpignistictransformation

WefollowherePhilippeSmets’visionwhichconsidersthemanagementofinformationasatwo2-levels process:credal(forcombinationofevidences)andpignistic11 (fordecision-making),i.e”whensomeonemust takeadecision,he/shemustthenconstructaprobabilityfunctionderivedfromthebelieffunctionthatdescribes his/hercredalstate.Thisprobabilityfunctionisthenusedtomakedecisions”[40](p.284).Oneobviousway tobuildthisprobabilityfunctioncorrespondstotheso-calledClassicalPignisticTransformation(CPT)deﬁned inDSTframework(i.e.basedontheShafer’smodelassumption)as[42]:

(26) where |A| denotesthecardinalityof A (withconvention |∅|/|∅| =1,todeﬁne BetP {∅}).Decisionsare achievedbycomputingtheexpectedutilitiesoftheactsusingthesubjective/pignistic BetP { } astheprobabilityfunctionneededtocomputeexpectations.Usually,oneusesthemaximumofthepignisticprobability asdecisioncriterion.Themaximumof BetP { } isoftenconsideredasaprudentbettingdecisioncriterion betweenthetwootheralternatives(maxofplausibilityormax.ofcredibilitywhichappearstoberespectively toooptimisticortoopessimistic).Itiseasytoshowthat BetP {.} isindeedaprobabilityfunction(see[41]).

6.2NotionofDSmcardinality

OneimportantnotioninvolvedinthedeﬁnitionoftheGeneralizedPignisticTransformation(GPT)isthe DSm cardinality.The DSmcardinality ofanyelement A ofhyper-powerset DΘ,denoted CM(A),correspondsto thenumberofpartsof A inthecorrespondingfuzzy/vagueVenndiagramoftheproblem(model M)taking intoaccountthesetofintegrityconstraints(ifany),i.e. allthepossibleintersectionsduetothenatureoftheelements θi.This intrinsiccardinality dependsonthemodel M (free,hybridorShafer’smodel). M isthemodel 11PignisticterminologyhasbeencoinedbyPhilippeSmetsand comesfrom pignus,abetinLatin.

BetP

A} = X∈2Θ |X ∩ A| |X| m

{

(X )

thatcontains A,whichdependsbothonthedimension n = |Θ| andonthenumberofnon-emptyintersections presentinitsassociatedVenndiagram(see[31]fordetails ).TheDSmcardinalitydependsonthecardinalof Θ= {θ1,θ2,...,θn} andonthemodelof DΘ (i.e.,thenumberofintersectionsandbetweenwhatelementsof Θ -inawordthestructure)atthesametime;itisnotnecessarilythateverysingleton,say θi,hasthesameDSm cardinal,becauseeachsingletonhasadifferentstructure;ifitsstructureisthesimplest(nointersectionofthis elementswithotherelements)then CM(θi)=1,ifthestructureismorecomplicated(manyintersections) then CM(θi) > 1;let’sconsiderasingleton θi:ifithas1intersectiononlythen CM(θi)=2,for2intersectionsonly CM(θi) is3or4dependingonthemodel M,for m intersectionsitisbetween m +1 and 2m dependingonthe model;themaximumDSmcardinalityis 2n 1 andoccursfor θ1 ∪ θ2 ∪ ... ∪ θn inthefreemodel Mf ;similarly foranysetfrom DΘ:themorecomplicatedstructureithas,thebiggeristheDSm cardinal;thustheDSm cardinalitymeasuresthecomplexityofanelementfrom DΘ,whichisanicecharacterizationinouropinion; wemaysaythatforthesingleton θi noteven |Θ| counts,butonlyitsstructure(=howmanyothersingletons intersect θi).SimpleillustrativeexamplesaregiveninChapter3and7of[31].Onehas 1 ≤CM(A) ≤ 2n 1 CM(A) mustnotbeconfusedwiththeclassicalcardinality |A| ofagivenset A (i.e.thenumberofitsdistinct elements)-that’swhyanewnotationisnecessaryhere. CM(A) isveryeasytocomputebyprogrammingfrom thealgorithmofgenerationof DΘ givenexplicatedin[31].

Example:let’stakebacktheexampleofthesimplehybridDSmmodeldescribedinsection2.2,thenonegets thefollowinglistofelements(withtheirDSmcardinal)for therestricted DΘ takingintoaccounttheintegrity constraintsofthishybridmodel: A ∈ DΘ CM(A) α0 ∅ 0 α1 θ1 ∩ θ2 1 α2 θ3 1 α3 θ1 2 α4 θ2 2 α5 θ1 ∪ θ2 3 α6 θ1 ∪ θ3 3 α7 θ2 ∪ θ3 3 α8 θ1 ∪ θ2 ∪ θ3 4

ExampleofDSmcardinals: CM(A) forhybridmodel M

6.3TheGeneralizedPignisticTransformation

TotakearationaldecisionwithinDSmTframework,itisnecessarytogeneralizetheClassicalPignisticTransformationinordertoconstructapignisticprobabilityfunctionfromanygeneralizedbasicbeliefassignment m( ) drawnfromtheDSmrulesofcombination.Hereisthesimplest anddirectextensionoftheCPTtodeﬁne theGeneralizedPignisticTransformation:

(27) where CM(X ) denotestheDSmcardinalofproposition X fortheDSmmodel M oftheproblemunderconsideration.

Thedecisionaboutthesolutionoftheproblemisusuallytakenbythemaximumofpignisticprobability function BetP {.}.Let’sremarkthecloseressemblanceofthetwopignistictransformations(26)and(27). Itcanbeshownthat(27)reducesto(26)whenthehyper-power set DΘ reducestoclassicalpowerset 2Θ if weadoptShafer’smodel.But(27)isageneralizationof(26) sinceitcanbeusedforcomputingpignistic probabilitiesforanymodels(includingShafer’smodel).Ithasbeenprovedin[31](Chap.7)that BetP { } deﬁnedin(27)isindeedaprobabilitydistribution.Inthefollowingsection,weintroduceanewalternativeto BetPwhichispresentedindetailsin[37].

∀A ∈ DΘ,BetP {A} = X∈DΘ CM(X ∩ A) CM(X ) m

(

)

7TheDSmPtransformation

Inthetheoriesofbelieffunctions,themappingfromthebelieftotheprobabilitydomainisacontroversialissue. Theoriginalpurposeofsuchmappingswastomake(hard)decision,butcontrariwisetoerroneouswidespread idea/claim,thisisnottheonlyinterestforusingsuchmappingsnowadays.Actuallytheprobabilistictransformationsofbeliefmassassignments(asthepignistictransformationmentionedpreviously)areforexamplevery usefulinmodernmultitargetmultisensortrackingsystems (orinanyothersystems)whereonedealswithsoft decisions(i.e.whereallpossiblesolutionsarekeptforstateestimationwiththeirlikelihoods).Forexample,in aMultipleHypothesesTrackerusingbothkinematicalandattributedata,oneneedstocomputeallprobabilities valuesforderivingthelikelihoodsofdataassociationhypothesesandthenmixingthemaltogethertoestimate statesoftargets.Therefore,itisveryrelevanttouseamappingwhichprovidesahighlyprobabilisticinformationcontent(PIC)forexpectingbetterperformances.

Inthissection,webrieﬂyrecallanewprobabilistictransformation,denoted DSmP andintroducedin[10] whichisexplainedindetailsin[37]. DSmP isstraightanddifferentfromothertransformations.Thebasicidea of DSmP consistsinanewwayofproportionalizationsofthemassofeachpartialignorancesuchas A1 ∪ A2 or A1 ∪ (A2 ∩ A3) or (A1 ∩ A2) ∪ (A3 ∩ A4),etc.andthemassofthetotalignorance A1 ∪ A2 ∪ ... ∪ An, totheelementsinvolvedintheignorances.Thisnewtransformationtakesintoaccountboththevaluesofthe massesandthecardinalityofelementsintheproportionalredistributionprocess.WeﬁrstremindwhatPIC criteriaisandthenshortlypresentthegeneralformulafor DSmPtransformationwithfewnumericalexamples. Moreexamplesandcomparisonswithrespecttoothertransformationsaregivenin[37].

7.1TheProbabilisticInformationContent(PIC)

FollowingSudano’sapproach[43,44,46],weadopttheProbabilisticInformationContent(PIC)criterionas ametricdepictingthestrengthofacriticaldecisionbyaspeciﬁcprobabilitydistribution.Itisanessential measureinanythreshold-drivenautomateddecisionsystem.ThePICisthedualofthenormalizedShannon entropy.APICvalueofoneindicatesthetotalknowledgetomakeacorrectdecision(onehypothesishasa probabilityvalueofoneandtherestofzero).APICvalueofzeroindicatesthattheknowledgetomakea correctdecisiondoesnotexist(allthehypotheseshaveanequalprobabilityvalue),i.e.onehasthemaximal entropy.ThePICisusedinouranalysistosorttheperformancesofthedifferentpignistictransformations throughseveralnumericalexamples.WeﬁrstrecallwhatShannonentropyandPICmeasureareandtheirtight relationship.

• Shannonentropy

Shannonentropy,usuallyexpressedinbits(binarydigits),ofaprobabilitymeasure

{ } overadiscrete ﬁniteset

• ThePICmetric

Θ= {θ1,...,θn} isdeﬁnedby12 [25]: H (P ) n i=1 P {θi} log2(P {θi}) (28) H (P ) ismaximalfortheuniformprobabilitydistributionover Θ,i.e.when P {θi} =1/n for i =1, 2,...,n Inthatcase,onegets H (P )= Hmax = n i=1 1 n log2( 1 n )=log2(n) H (P ) isminimalforatotally deterministic probability,i.e.forany P { } suchthat P {θi} =1 forsome i ∈{1, 2,...,n} and P {θj } =0 for j = i H (P ) measurestherandomnesscarriedbyanydiscreteprobability P { }

12withcommonconvention

TheProbabilisticInformationContent(PIC)ofaprobabilitymeasure P { } associatedwithaprobabilistic sourceoveradiscreteﬁniteset Θ= {θ1,...,θn} isdeﬁnedby[44]: PIC (P )=1+ 1 Hmax n i=1 P {θi} log2(P {θi}) (29)

0log2 0=0.

ThePICisnothingbutthedualofthenormalizedShannonentropyandthusisactuallyunitless. PIC (P ) takesitsvaluesin [0, 1] PIC (P ) ismaximum,i.e. PICmax =1 withany deterministic probabilityanditis minimum,i.e. PICmin =0,withtheuniformprobabilityovertheframe Θ.Thesimplerelationshipsbetween H (P ) and PIC (P ) are PIC (P )=1 (H (P )/Hmax) and H (P )= Hmax · (1 PIC (P ))

7.2TheDSmPformula

Z⊆X∩Y C(Z)=1

m(Z )+ ǫ ·C(X ∩ Y ) Z⊆Y C(Z)=1

m(Z )+ ǫ ·C(Y ) m(Y ) (30)

Let’sconsideradiscreteframe Θ withagivenmodel(freeDSmmodel,hybridDSmmodelorShafer’smodel), the DSmP mappingisdeﬁnedby DSmPǫ(∅)=0 and ∀X ∈ GΘ \{∅} by DSmPǫ(X )= Y ∈GΘ

where ǫ ≥ 0 isatuningparameterand GΘ correspondstothegenericset(2Θ , SΘ or DΘ includingeventually alltheintegrityconstraints(ifany)ofthemodel M); C(X ∩ Y ) and C(Y ) denotetheDSmcardinals13 ofthe sets X ∩ Y and Y respectively. ǫ allowstoreachthemaximumPICvalueoftheapproximationof m(.) intoa subjectiveprobabilitymeasure.Thesmaller ǫ,thebetter/biggerPICvalue.Insomeparticulardegeneratecases however,the DSmPǫ=0 valuescannotbederived,butthe DSmPǫ>0 valuescanhoweveralwaysbederivedby choosing ǫ asaverysmallpositivenumber,say ǫ =1/1000 forexampleinordertobeascloseaswewantto themaximumofthePIC.When ǫ =1 andwhenthemassesofallelements Z having C(Z )=1 arezero,(30) reducesto(27),i.e. DSmPǫ=1 = BetP .ThepassagefromafreeDSmmodeltoaShafer’smodelinvolvesthe passagefromastructuretoanotherone,andthecardinalschangeaswellintheformula(30).

DSmP worksforallmodels(free,hybridandShafer’s).Inorderto applyclassicaltransformation(Pignistic,Cuzzolin’sone,Sudano’sones,etc-see[37]),weneedatfirsttorefinetheframe(onthecaseswhenit ispossible!)inordertoworkwithShafer’smodel,andthenapplytheirformulas.Inthecasewhererefinement makessense,thenonecanapplytheothersubjectiveprobabilitiesontherefinedframe. DSmP worksonthe refinedframeaswellandgivesthesameresultasitdoesonthe non-refinedframe.Thus DSmP with ǫ> 0 worksonanymodelsandsoisverygeneralandappealing. DSmP doesaredistributionoftheignorancemass withrespecttoboththesingletonmassesandthesingletons’cardinalsinthesametime.Now,ifallmassesof singletonsinvolvedinallignorancesaredifferentfromzero,thenwecantake ǫ =0,and DSmP givesthebest result,i.e.thebestPICvalue.Insummary, DSmP doesan’improvement’overpreviousknownprobabilistic transformationsinthesensethat DSmP mathematicallymakesamoreaccurateredistributionofthe ignorance massestothesingletonsinvolvedinignorances. DSmP and BetP workinboththeories:DST(=Shafer’s model)andDSmT(=freeorhybridmodels)aswell.

7.3ExamplesforDSmPandBetP

Theexamplesbrieﬂypresentedherearedetailedin[37]whichincludesalsoadditionalresultsbasedonCuzzolin’sandSudano’stransformations.

• WithShafer’smodelandanon-Bayesianmass

Let’sconsidertheframe Θ= {A,B} andlet’sassumeShafer’smodelandthenon-Bayesianmass(more preciselythesimplesupportmass)giveninTable6.

WesummarizeinTable7,theresultsobtainedwithDSmPandBetP.Oneseesthat PIC (DSmPǫ→0) is maximumamongallPICvalues.Thebestresultisan adequateprobability,not thebiggestPIC inthiscase. Thisisbecause P (B) deservestoreceivesomemassfrom m(A ∪ B),sothemostcorrectresultisdoneby DSmPǫ=0 001 inTable7(ofcoursewecanchooseanyotherverysmallpositivevaluefor ǫ ifwewant). Alwayswhenasingletonwhosemassiszero,butitisinvolved inanignorancewhosemassisnotzero,then ǫ (in DSmP formula(30))shouldbetakendifferentfromzero.

13Wehaveomittedtheindexofthemodel M forthenotationconvenience.

•

A B A ∪ B m( ) 0.4 0 0.6

Table6:Quantitativeinputs.

A B PIC (.)

BetP ( ) 0.7000 0.3000 0.1187 DSmPǫ=0 001( ) 0.9985 0.0015 0.9838 DSmPǫ=0( ) 1 0 1

Table7:Resultsoftheprobabilistictransformations.

WithaHybridDSmmodel

Let’sconsidertheframe Θ= {A,B,C} andlet’sconsiderthehybridDSmmodelinwhichallintersections ofelementsof Θ areempty,but A ∩ B correspondingtoﬁgure4.Inthiscase, GΘ reducesto9elements {∅,A ∩ B,A,B,C,A ∪ B,A ∪ C,B ∪ C,A ∪ B ∪ C}.Theinputmassesoffocalelementsaregivenby m(A ∩ B)=0 20, m(A)=0 10, m(C )=0 20, m(A ∪ B)=0 30, m(A ∪ C )=0 10,and m(A ∪ B ∪ C )= 0 10 andgivenintheTable8.

D′ A′ ∪ D′ C′ m( ) 0.2 0.1 0.2 A′ ∪ B′ ∪ D′ A′ ∪ C′ ∪ D′ A′ ∪ B′ ∪ C′ ∪ D′ m( ) 0.3 0.1 0.1

Table8:Quantitativeinputs.

Fig.4:Hybridmodelfor Θ= {A,B,C}.

ApplyingBetPandDSmPtransformations,onegets: A′ B′ C′ D′ PIC ( ) BetP (.) 0.2084 0.1250 0.2583 0.4083 0.0607 DSmPǫ=0 001( ) 0.0025 0.0017 0.2996 0.6962 0.5390

Table9:Resultsoftheprobabilistictransformations.

• WithafreeDSmmodel

Let’sconsidertheframe Θ= {A,B,C} andlet’sconsiderthefreeDSmmodeldepictedonFigure5with theinputmassesgiveninTable10.ToapplySudano’sandCuzzolin’smappings,oneworksontherefined frame Θref = {A′,B′,C′,D′,E′,F ′,G′} wheretheelementsof Θref areexclusive(assumingsuchrefinement hasaphysicalmeaning)accordingtoFigure5.Thisrefinementstepisnotnecessarywhenusing DSmP since itworksdirectlyonDSmfreemodel.ThePICvaluesobtainedwithDSmPandBetParegiveninTable11.One seesthat DSmPǫ→0 provideshereagainthebestresultsintermofPIC.

AnextensionofDSmP(denotedqDSmP)forworkingwithqualitativelabelsinsteadofnumbersispossible andhasbeenproposedandpresentedin2008in[10]usingapproximateoperatorsonlabels.Asimpleexample forqDSmPbasedonpreciseoperatorsonreﬁnedlabelsdevelopedin[37]ispresentedinthenextsection.

✫✪ ✬✩ ✫✪ ✬✩ ✫✪ ✬✩

❘

✠

✛

❅

C D′ C′ B′ A′

m( ) 0.1 0.2 0.3 0.1 0.3

Table10:Quantitativeinputs.

Fig.5:FreeDSmmodelfora3Dframe.

Transformations PIC (.)

BetP ( ) 0.1176 DSmPǫ=0 001( ) 0.8986

Table11:Resultsoftheprobabilistictransformations.

8Fusionofqualitativebeliefs

Werecallherethenotionofqualitativebeliefassignmenttomodelbeliefsofhumanexpertsexpressedin naturallanguage(withlinguisticlabels).Weshowhowqualitativebeliefscanbeefﬁcientlycombinedusing anextensionofDSmTtoqualitativereasoning.Amoredetailedpresentationcanbefoundin[35,37].The derivationsarebasedonanewarithmeticonlinguisticlabelswhichallowsadirectextensionofallquantitative rulesofcombinationandconditioning.ThequalitativeversionofPCR5ruleandDSmPisalsopresentedinthe sequel.

8.1QualitativeOperators

Computingwithwords(CW)andqualitativeinformationismorevague,lessprecisethancomputingwith numbers,butitofferstheadvantageofrobustnessifdonecorrectly.Hereisageneralarithmeticwepropose forcomputingwithwords(i.e.withlinguisticlabels).Let’sconsiderafiniteframe Θ= {θ1,...,θn} of n (exhaustive)elements θi, i =1, 2,...,n,withanassociatedmodel M(Θ) on Θ (eitherShafer’smodel M0(Θ),free-DSmmodel Mf (Θ),ormoregeneralanyHybrid-DSmmodel[31]).Amodel M(Θ) isdefined bythesetofintegrityconstraintsonelementsof Θ (ifany);Shafer’smodel M0(Θ) assumesallelementsof Θ trulyexclusive,whilefree-DSmmodel Mf (Θ) assumesnoexclusivityconstraintsbetweenelementsofthe frame Θ.Let’sdefineafinitesetoflinguisticlabels ˜ L = {L1,L2,...,Lm} where m ≥ 2 isaninteger. ˜ L isendowedwithatotalorderrelationship ≺,sothat L1 ≺ L2 ≺ ≺ Lm.Toworkonacloselinguistic setunderlinguisticadditionandmultiplicationoperators,weextends ˜ L withtwoextremevalues L0 and Lm+1 where L0 correspondstotheminimalqualitativevalueand Lm+1 correspondstothemaximalqualitativevalue, insuchawaythat L0 ≺ L1 ≺ L2 ≺ ... ≺ Lm ≺ Lm+1

where ≺ meansinferiorto,orless(inquality)than,orsmaller(inquality)than,etc.hencearelationoforder fromaqualitativepointofview.Butifwemakeacorrespondencebetweenqualitativelabelsandquantitative valuesonthescale [0, 1],then Lmin = L0 wouldcorrespondtothenumericalvalue0,while Lmax = Lm+1 wouldcorrespondtothenumericalvalue1,andeach Li wouldbelongto [0, 1],i.e. Lmin = L0 <L1 <L2 <...<Lm <Lm+1 = Lmax

Fromnowon,weworkonextendedorderedset L ofqualitativevalues

L = {L0, ˜ L,Lm+1} = {L0,L1,L2,...,Lm,Lm+1}

Inourpreviousworks,wedidproposeapproximatequalitativeoperators,butin[37]weproposetouse betterandaccurateoperatorsforqualitativelabels.Sincethesenewoperatorsaredeﬁnedindetailsinthe

A ∩ B ∩ C A ∩ B A A ∪ B A ∪ B ∪ C

✫✪ ✬✩ ✫✪ ✬✩ ✫✪ ✬✩ ❅ ❘ A ✠ B ❅ ■ C D′ G′ C′ E′ F ′ B′ A′

chapterof[37]devotedontheDSmFieldandLinearAlgebraof ReﬁnedLabels(FLARL),wejustbrieﬂy introducehereonlythethemainones(i.e.theaccuratelabeladdition,multiplicationanddivision).InFLARL, wecanreplacethe”qualitativequasi-normalization”ofqualitativeoperatorsweusedinourpreviouspapersby ”qualitativenormalization”sinceinFLARLwehaveexactqualitativecalculationsandexactnormalization.

• Labeladdition:

La + Lb = La+b (31) since a m+1 + b m+1 = a+b m+1

• Labelmultiplication:

La × Lb = L(ab)/(m+1) (32) since a m+1 b m+1 = (ab)/(m+1) m+1 .

• Labeldivision(when Lb = L0):

La ÷ Lb = L(a/b)(m+1) (33) since a m+1 ÷ b m+1 = a b = (a/b)(m+1) m+1 .

Moreaccuratequalitativeoperations(substraction,scalarmultiplication,scalarroot,scalarpower,etc)can befoundin[37].Ofcourse,ifonereallyneedstostaywithin theoriginalsetoflabels,anapproximationwill benecessaryattheveryendofthecalculations.

8.2QualitativeBeliefAssignment

Aqualitativebeliefassignment14 (qba)isamappingfunction qm(.): GΘ → L where GΘ correspondseither to 2Θ,to DΘ orevento SΘ dependingonthemodeloftheframe Θ wechoosetoworkwith.Inthecasewhen thelabelsareequidistant,i.e.thequalitativedistancebetweenanytwoconsecutivelabelsisthesame,weget anexactqualitativeresult,andaqualitativebasicbelief assignment(bba)isconsiderednormalizedifthesum ofallitsqualitativemassesisequalto Lmax = Lm+1.Ifthelabelsarenotequidistant,westillcanuseall qualitativeoperatorsdeﬁnedintheFLARL,butthequalitativeresultisapproximate,andaqualitativebbais consideredquasi-normalizedifthesumofallitsmassesisequalto Lmax.Usingthequalitativeoperatorof FLARL,wecaneasilyextendallthecombinationandconditioningrulesfromquantitativetoqualitative.Inthe sequelwewillconsider s ≥ 2 qualitativebeliefassignments qm1(.),...,qms(.) deﬁnedoverthesamespace GΘ andprovidedby s independentsources S1,...,Ss ofevidence.

Note:Theadditionandmultiplicationoperatorsusedinallqualitativefusionformulasinnextsectionscorrespondto qualitativeaddition and qualitativemultiplication operatorsandmustnotbeconfusedwithclassical additionandmultiplicationoperatorsfornumbers.

8.3QualitativeConjunctiveRule

ThequalitativeConjunctiveRule(qCR)of s ≥ 2 sourcesisdeﬁnedsimilarlytothequantitativeconjunctive consensusrule,i.e.

14Wecallitalso qualitativebeliefmass or q-mass forshort.

qmqCR(X )= X1 ,...,Xs∈GΘ X1 ∩ ∩Xs =X s i=1 qm

K1...s = X1 ,...,Xs∈GΘ X1 ∩ ∩Xs=∅ s i=1 qmi(Xi)

i) (34) Thetotalqualitativeconﬂictingmassisgivenby

8.4QualitativeDSmClassicrule

ThequalitativeDSmClassicrule(q-DSmC)for s ≥ 2 isdeﬁnedsimilarlytoDSmClassicfusionrule(DSmC) asfollows: qmqDSmC (∅)= L0 andforall X ∈ DΘ \{∅},

qmqDSmC (X )= X1 ,,...,Xs∈DΘ X1∩ ∩Xs=X

8.5QualitativehybridDSmrule

s i=1 qmi(Xi) (35)

ThequalitativehybridDSmrule(q-DSmH)isdeﬁnedsimilarlytoquantitativehybridDSmrule[31]asfollows: qmqDSmH (∅)= L0 (36) andforall X ∈ GΘ \{∅}

qmqDSmH (X ) φ(X ) qS1(X )+ qS2(X )+ qS3(X ) (37)

whereallsetsinvolvedinformulasareinthecanonicalform and φ(X ) isthe characteristicnon-emptiness function ofaset X,i.e. φ(X )= Lm+1 if X/ ∈ ∅ and φ(X )= L0 otherwise,where ∅ {∅M, ∅} ∅M isthe setofallelementsof DΘ whichhavebeenforcedtobeemptythroughtheconstraintsof themodel M and ∅ is theclassical/universalemptyset. qS1(X ) ≡ qmqDSmC (X ), qS2(X ), qS3(X ) aredeﬁnedby qS1(X ) X1,X2 ,...,Xs∈DΘ X1 ∩X2 ∩ ∩Xs=X

s i=1 qmi(Xi) (38) qS2(X ) X1,X2 ,...,Xs∈∅ [U =X]∨[(U∈∅)∧(X=It )]

s i=1 qmi(Xi) (39) qS3(X ) X1 ,X2,...,Xk∈DΘ X1 ∪X2 ∪ ∪Xs=X X1∩X2 ∩ ∩Xs∈∅

s i=1 qmi(Xi) (40) with U u(X1) ∪ ∪ u(Xs) where u(X ) istheunionofall θi thatcompose X, It θ1 ∪ ∪ θn isthetotal ignorance. qS1(X ) isnothingbuttheqDSmCrulefor s independentsourcesbasedon Mf (Θ); qS2(X ) isthe qualitativemassofallrelativelyandabsolutelyemptysetswhichistransferredtothetotalorrelativeignorances associatedwithnonexistentialconstraints(ifany,likeinsomedynamicproblems); qS3(X ) transfersthesum ofrelativelyemptysetsdirectlyontothecanonicaldisjunctiveformofnon-emptysets.qDSmHgeneralizes qDSmCworksforanymodels(freeDSmmodel,Shafer’smodelor anyhybridmodels)whenmanipulating qualitativebeliefassignments.

8.6QualitativePCR5rule(qPCR5)

Inclassical(i.e.quantitative)DSmTframework,theProportionalConflictRedistributionruleno.5(PCR5) definedin[35]hasbeenproventoprovideverygoodandcoherentresultsforcombining(quantitative)belief masses,see[9,33].WhendealingwithqualitativebeliefswithintheDSmFieldandLinearAlgebraofRefined Labels[37]wegetanexactqualitativeresultnomatterwhat fusionruleisused(DSmfusionrules,Dempster’s rule,Smets’srule,Dubois-Prade’srule,etc.).Theexactqualitativeresultwillbearefinedlabel(buttheuser canrounditupordowntotheclosestintegerindexlabel).

8.7Asimpleexampleofqualitativefusionofqba’s

Let’sconsiderthefollowingsetoforderedlinguisticlabels

L = {L0,L1,L2,L3,L4,L5} (forexample, L1, L2, L3 and L4 mayrepresentthevalues: L1 verypoor, L2 poor, L3 good and L4 verygood,where symbolmeans bydeﬁnition).

Let’sconsidernowasimpletwo-sourcecasewitha2Dframe Θ= {θ1,θ2},Shafer’smodelfor Θ,and qba’sexpressedasfollows:

qm1(θ1)= L1,qm1(θ2)= L3,qm1(θ1 ∪ θ2)= L1 qm2(θ1)= L2,qm2(θ2)= L1,qm2(θ1 ∪ θ2)= L2

Thetwoqualitativemasses qm1(.) and qm2(.) arenormalizedsince: qm1(θ1)+ qm1(θ2)+ qm1(θ1 ∪ θ2)= L1 + L3 + L1 = L1+3+1 = L5 and qm2(θ1)+ qm2(θ2)+ qm2(θ1 ∪ θ2)= L2 + L1 + L2 = L2+1+2 = L5

Weﬁrstderivetheresultoftheconjunctiveconsensus.This yields:

qm12(θ1)= qm1(θ1)qm2(θ1)+ qm1(θ1)qm2(θ1 ∪ θ2)+ qm1(θ1 ∪ θ2)qm2(θ1) = L1 × L2 + L1 × L2 + L1 × L2 = L 1 2 5 + L 1 2 5 + L 1 2 5 = L 2 5 + 2 5 + 2 5 = L 6 5 = L1 2

qm12(θ2)= qm1(θ2)qm2(θ2)+ qm1(θ2)qm2(θ1 ∪ θ2)+ qm1(θ1 ∪ θ2)qm2(θ2)

= L3 × L1 + L3 × L2 + L1 × L1 = L 3 1 5 + L 3 2 5 + L 1 1 5 = L 3 5 + 6 5 + 1 5 = L 10 5 = L2

qm12(θ1 ∪ θ2)= qm1(θ1 ∪ θ2)qm2(θ1 ∪ θ2)= L1 × L2 = L 1 2 5 = L 2 5 = L0 4

qm12(θ1 ∩ θ2)= qm1(θ1)qm2(θ2)+ qm1(θ2)qm2(θ1) = L1 × L1 + L2 × L3 = L 1 1 5 + L 2 3 5 = L 1 5 + 6 5 = L 7 5 = L1 4

Thereforeweget:

• forthefusionwithqDSmC,whenassuming θ1 ∩ θ2 = ∅, qmqDSmC (θ1)= L1 2 qmqDSmC (θ2)= L2 qmqDSmC (θ1 ∪ θ2)= L0 4 qmqDSmC (θ1 ∩ θ2)= L1 4

• forthefusionwithqDSmH,whenassuming θ1 ∩ θ2 = ∅.Themassof θ1 ∩ θ2 istransferredto θ1 ∪ θ2 Hence:

qmqDSmH (θ1)= L1 2 qmqDSmH (θ2)= L2 qmqDSmH (θ1 ∩ θ2)= L0 qmqDSmH (θ1 ∪ θ2)= L0 4 + L1 4 = L1 8

• forthefusionwithqPCR5,whenassuming θ1 ∩ θ2 = ∅.Themass qm12(θ1 ∩ θ2)= L1 4 istransferred to θ1 andto θ2 inthefollowingway: qm12(θ1 ∩ θ2)= qm1(θ1)qm2(θ2)+ qm2(θ1)qm1(θ2)

Then, qm1(θ1)qm2(θ2)= L1 × L1 = L 1 1 5 = L 1 5 = L0 2 isredistributedto θ1 and θ2 proportionally withrespecttotheirqualitativemassesputintheconﬂict L1 andrespectively L1: xθ1 L1 = yθ2 L1 = L0 2 L1 + L1 = L0 2 L1+1 = L0 2 L2 = L 0 2 2 5 = L 1 2 = L0 5

whence xθ1 = yθ2 = L1 × L0 5 = L 1 0 5 5 = L 0 5 5 = L0 1.

Actually,wecouldeasierseethat qm1(θ1)qm2(θ2)= L0 2 hadinthiscasetobeequallysplitbetween θ1 and θ2 sincethemassputintheconﬂictby θ1 and θ2 wasthesameforeachofthem: L1.Therefore L0 2 2 = L 0 2 2 = L0 1.

Similarly, qm2(θ1)qm1(θ2)= L2 × L3 = L 2 3 5 = L 6 5 = L1 2 hastoberedistributedto θ1 and θ2 proportionallywith L2 and L3 respectively: x′ θ1 L2 = y′ θ2 L3 = L1 2 L2 + L3 = L1 2 L2+3 = L1 2 L5 = L 1 2 5 5 = L1 2 whence x′ θ1 = L2 × L1 2 = L 2 1 2 5 = L 2 4 5 = L0 48 y′ θ2 = L3 × L1 2 = L 3 1 2 5 = L 3 6 5 = L0 72

Now,addallthesetothequalitativemassesof θ1 and θ2 respectively:

qmqPCR5(θ1)= qm12(θ1)+ xθ1 + x ′ θ1 = L1 2 + L0 1 + L0 48 = L1 2+0 1+0 48 = L1 78 qmqPCR5(θ2)= qm12(θ2)+ yθ2 + y ′ θ2 = L2 + L0 1 + L0 72 = L2+0 1+0 72 = L2 82 qmqPCR5(θ1 ∪ θ2)= qm12(θ1 ∪ θ2)= L0.4 qmqPCR5(θ1 ∩ θ2)= L0

Thequalitativemassresultsusingallfusionrules(qDSmC,qDSmH,qPCR5)remainnormalizedinFLARL. Naturally,ifonepreferstoexpresstheﬁnalresultswithqualitativelabelsbelongingintheoriginaldiscrete setoflabels L = {L0,L1,L2,L3,L4,L5},someapproximationswillbenecessarytoroundcontinuous indexed labelstotheirclosestinteger/discreteindexvalue;byexample, qmqPCR5(θ1)= L1 78 ≈ L2, qmqPCR5(θ2)= L2 82 ≈ L3 and qmqPCR5(θ1 ∪ θ2)= L0 4 ≈ L0. 8.8AsimpleexamplefortheqDSmPtransformation Weﬁrstrecallthatthequalitativeextensionof(30),denoted qDSmP

ǫ( ) isgivenby qDSmPǫ(∅)=0 and ∀X ∈ GΘ \{∅} by qDSmPǫ(X )= Y ∈GΘ Z⊆X∩Y C(Z)=1 qm(Z )+ ǫ ·C(X ∩ Y ) Z⊆Y C(Z)=1 qm(Z )+ ǫ ·C(Y ) qm(Y )

wherealloperationsin(41)arereferredtolabels,thatis q-operatorsonlinguisticlabelsandnotclassicaloper-

θ1,θ2}

(41)

atorsonnumbers. Let’sconsiderthesimpleframe Θ= {

(here

= |

| =2)withShafer’smodel(i.e. θ1 ∩θ2 = ∅)and thefollowingsetoflinguisticlabels L = {

,L4,L5},with L0 = Lmin and L5 = Lmax = Lm+1

(here m =4)andthefollowingqualitativebeliefassignment: qm(θ1)= L1, qm(θ2)= L3 and qm(θ1 ∪ θ2)= L1 qm( ) isquasi-normalizedsince X∈2Θ qm(X )= L5 = Lmax.Inthisexampleandwith DSmP transformation, qm(θ1 ∪ θ2)= L1 isredistributedto θ1 and θ2 proportionallywithrespecttotheirqualitative masses L1 and L3 respectively.Sinceboth L1 and L3 aredifferentfrom L0,wecantakethetuningparameter ǫ =0 forthebesttransfer. ǫ istakendifferentfromzerowhenamassofasetinvolvedinapartialortotal ignoranceiszero(forqualitativemasses,itmeans L0). Thereforeusing(33),onehas xθ1 L1 = xθ2 L3 = L1 L1 + L3 = L1 L4 = L 1 4 5 = L 5 4 = L1 25 andthususing(32),onegets xθ1 = L1 × L1 25 = L 1 (1 25) 5 = L 1 25 5 = L0 25 xθ2 = L3 × L1 25 = L 3 (1 25) 5 = L 3 75 5 = L0 75 Therefore, qDSmPǫ=0(θ1 ∩ θ2)= qDSmPǫ=0(∅)= L0 qDSmPǫ=0(θ1)= L1 + xθ1 = L1 + L0 25 = L1 25 qDSmPǫ=0(θ2)= L3 + xθ2 = L3 + L0 75 = L3 75

Naturallyinourexample,onehasalso qDSmPǫ=0(θ1 ∪ θ2)= qDSmPǫ=0(θ1)+ qDSmPǫ=0(θ2) qDSmPǫ=0(θ1 ∩ θ2) = L1 25 + L3 75 L0 = L5 = Lmax

Since Hmax =log2 n =log2 2=1,usingthequalitativeextensionofPICformula(29),oneobtainsthe followingqualitativePICvalue: PIC =1+ 1 1 · [qDSmPǫ=0(θ1)log2(qDSmPǫ=0(θ1)) + qDSmPǫ=0(θ2)log2(qDSmPǫ=0(θ2))] =1+ L1 25 log2(L1 25)+ L3 75 log2(L3 75) ≈ L0 94 sinceweconsideredtheisomorphictransformation Li = i/(m +1) (inourparticularexample m =4 interior labels).

9BeliefConditioningRules

9.1Shafer’sConditioningRule(SCR)

Untilveryrecently,themostcommonlyusedconditioningruleforbeliefrevisionwastheoneproposedby Shafer[24]andreferredhereasShafer’sConditioningRule (SCR).TheSCRconsistsincombiningtheprior bba m( ) withaspecificbbafocusedon A withDempster’sruleofcombinationfortransferringtheconflicting masstonon-emptysetsinordertoprovidetherevisedbba.In otherwords,theconditioningbyaproposition A,isobtainedbySCRasfollows: mSCR(.|A)=[m ⊕ mS ](.) (42) where m( ) isthepriorbbatoupdate, A istheconditioningevent, mS ( ) isthebbafocusedon A definedby mS (A)=1 and mS (X )=0 forall X = A and ⊕ denotesDempster’sruleofcombination[24].

TheSCRapproachbasedonDempster’sruleofcombinationofthepriorbbawiththebbafocusedonthe conditioningeventremains subjective sinceactuallyinsuchbeliefrevisionprocessbothsources aresubjective andinouropinionsSCRdoesn’tmanagesatisfactorilytheobjectivenature/absolutetruthcarriedbytheconditioningterm.Indeed,whenconditioningapriormass m( ), knowing (orassuming)thatthetruthisin A,means thatwehaveinhandsanabsolute(notsubjective)knowledge,i.e.thetruthin A hasoccurred(orisassumedto haveoccurred),thus A isrealized(orisassumedtoberealized)andthisis(oratleastmustbeinterpretedas) anabsolutetruth.Theconditioningterm”Given A”mustthereforebeconsideredasanabsolutetruth,while mS (A)=1 introducedinSCRcannotrefertoanabsolutetruthactually,butonlytoa subjectivecertainty onthepossibleoccurrenceof A froma virtual secondsourceofevidence.TheadvantageofSCRremains undoubtedlyinitssimplicityandthemainargumentinitsfavorisitscoherencewiththeconditionalprobabilitywhenmanipulatingBayesianbeliefassignment.Butinouropinion,SCRshouldbetterbeinterpreted asthefusionof m( ) withaparticularsubjectivebba mS (A)=1 ratherthananobjectivebeliefconditioning rule.ThisfundamentalremarkmotivatedustodevelopanewfamilyofBCR[35]basedonhyper-powerset decomposition(HPSD)explainedbrieﬂyinthenextsection. ItturnsoutthatmanyBCRarepossiblebecause theredistributionofmassesofelementsoutsideof A (theconditioningevent)tothoseinside A canbedonein n-ways.Thiswillbebrieﬂypresentedrightafterthenextsection.

9.2Hyper-PowerSetDecomposition(HPSD)

Let Θ= {θ1,θ2,...,θn}, n ≥ 2,amodel M(Θ) associatedfor Θ (freeDSmmodel,hybridorShafer’s model)anditscorrespondinghyper-powerset DΘ.Let’sconsidera(quantitative)basicbeliefassignment (bba) m( ): DΘ → [0, 1] suchthat X∈DΘ m(X )=1.Supposeoneﬁndsoutthatthetruthisintheset A ∈ DΘ \{∅}.Let PD(A)=2A ∩ DΘ \{∅},i.e.allnon-emptyparts(subsets)of A whichareincluded in DΘ.Let’sconsiderthenormalcaseswhen A = ∅ and Y ∈PD (A) m(Y ) > 0.Forthedegeneratecase whenthetruthisin A = ∅,weconsiderSmets’open-world,whichmeansthatthereareotherhypotheses Θ′ = {θn+1,θn+2,...θn+m}, m ≥ 1,andthetruthisin A ∈ DΘ′ \{∅}.If A = ∅ andweconsideracloseworld,thenitmeansthattheproblemisimpossible.Foranotherdegeneratecase,when Y ∈PD (A) m(Y )=0, i.e.whenthesourcegaveusatotally(100%)wronginformation m( ),then,wedeﬁne: m(A|A) 1 and, asaconsequence, m(X|A)=0 forany X = A.Let s(A)= {θi1 ,θi2 ,...,θip }, 1 ≤ p ≤ n,bethe singletons/atomsthatcompose A (forexample,if A = θ1 ∪ (θ3 ∩ θ4) then s(A)= {θ1,θ3,θ4}).TheHyperPowerSetDecomposition(HPSD)of DΘ \∅ consistsinitsdecompositionintothethreefollowingsubsets generatedby A:

• D1 = PD(A),thepartsof A whichareincludedinthehyper-powerset,excepttheemptyset;

• D2 = {(Θ \ s(A)), ∪, ∩}\{∅},i.e.thesub-hyper-powersetgeneratedby Θ \ s(A) under ∪ and ∩, withouttheemptyset.

• D3 =(DΘ \{∅}) \ (D1 ∪ D2);eachsetfrom D3 hasinitsformulasingletonsfromboth s(A) and Θ \ s(A) inthecasewhen Θ \ s(A) isdifferentfromemptyset. D1, D2 and D3 havenoelementincommontwobytwoandtheirunionis DΘ \{∅}.

SimpleexampleofHPSD:Let’sconsider

9.3Quantitativebeliefconditioningrules(BCR)

Sincethereexistsactuallymanywaysforredistributingthemassesofelementsoutsideof A (theconditioning event)tothoseinside A,severalBCR’shavebeenproposedin[35].Inthisintroduction,wewillnotbrowseall thepossibilitiesfordoingtheseredistributionsandallBCR’sformulasbutonlyone,theBCRnumber17(i.e. BCR17)whichdoesinouropinionthemostreﬁnedredistributionsince: -themass m(W ) ofeachelement W in D2 ∪ D3 istransferredtothose X ∈ D1 elementswhichareincluded

Θ= {θ1,θ2,θ3} withShafer’smodel(i.e.allelementsof Θ are exclusive)andlet’sassumethatthetruthisin θ2 ∪ θ3,i.e.theconditioningtermis θ2 ∪ θ3.Thenonehasthe followingHPSD: D1 = {θ2,θ3,θ2 ∪ θ3}, D2 = {θ1} and D3 = {θ1 ∪ θ2,θ1 ∪ θ3,θ1 ∪ θ2 ∪ θ3}.Morecomplex anddetailedexamplescanbefoundin[35].

in W ifanyproportionallywithrespecttotheirnon-emptymasses;

-ifnosuch X exists,themass m(W ) istransferredinapessimistic/prudentwaytothe k-largestelementfrom D1 whichareincludedin W (inequalparts)ifany; -ifneitherthiswayispossible,then m(W ) isindiscriminatelydistributedtoall X ∈ D1 proportionallywith respecttotheirnonzeromasses.

Note: TheauthorsmentionedinanErratumtotheprintedversionof thesecondvolumeofDSmTbookseries(http://fs.gallup.unm.edu//Erratum.pdf)andtheyalsocorrectedtheonlineversionofthe aforementionedbook(seepage240in http://fs.gallup.unm.edu//DSmT-book2.pdf thatalldenominatorsoftheBCR’sformulasarenaturallysupposedtobedifferentfromzero.Ofcourse,Shafer’sconditioningruleasstatedinTheorem3.6,page67of[24]doesnot workwhenthedenominatoriszeroandthat’s whyShaferhasintroducedthecondition Bel(B) < 1 (orequivalently Pl(B) > 0)inhistheoremwhenthe conditioningtermis B.

AsimpleexampleforBCR17:Let’sconsider Θ= {θ1,θ2,θ3} withShafer’smodel(i.e.allelementsof Θ are exclusive)andlet’sassumethatthetruthisin θ2 ∪ θ3,i.e.theconditioningtermis A θ2 ∪ θ3.Thenonehas thefollowingHPSD: D1 = {θ2,θ3,θ2 ∪ θ3},D2 = {θ1} D3 = {θ1 ∪ θ2,θ1 ∪ θ3,θ1 ∪ θ2 ∪ θ3} Let’sconsiderthefollowingpriorbba: m(θ1)=0 2, m(θ2)=0 1, m(θ3)=0 2, m(θ1 ∪ θ2)=0 1, m(θ2 ∪ θ3)=0 1 and m(θ1 ∪ θ2 ∪ θ3)=0 3 WithBCR17,for D2, m(θ1)=0 2 istransferredproportionallytoallelementsof D1,i.e. xθ2 0 1 = yθ3 0 2 = zθ2 ∪θ3 0 1 = 0 2 0 4 =0.5 whencethepartsof m(θ1) redistributedto θ2, θ3 and θ2 ∪ θ3 arerespectively xθ2 =0.05, yθ3 =0.10,and zθ2 ∪θ3 =0.05.For D3,thereisactuallynoneedtotransfer m(θ1 ∪θ3) because m(θ1 ∪θ3)=0 inthisexample;whereas m(θ1 ∪θ2)=0 1 istransferredto θ2 (nocaseof k-elementsherein); m(θ1 ∪θ2 ∪θ3)= 0 3 istransferredto θ2, θ3 and θ2 ∪ θ3 proportionallytotheircorrespondingmasses: xθ2 /0.1= yθ3 /0.2= zθ2 ∪θ3 /0.1=0.3/0.4=0.75 whence xθ2 =0 075, yθ3 =0 15,and zθ2∪θ3 =0 075.Finally,onegets mBCR17 (θ2|θ2 ∪ θ3)=0 10+0 05+0 10+0 075=0 325 mBCR17 (θ3|θ2 ∪ θ3)=0 20+0 10+0 15=0 450 mBCR17 (θ2 ∪ θ3|θ2 ∪ θ3)=0.10+0.05+0.075=0.225

BCR17isdeﬁnedbythefollowingformula(see[35],Chap.9fordetailedexplanationsandexamples): mBCR17(X|A)= m(X ) SD1 + W ∈D2∪D3 X⊂W S(W )=0 m(W ) S(W ) + W ∈D2∪D3 X⊂W, Xis k-largest S(W )=0 m(W )/k (43) where”X is k-largest”meansthat X isthe k-largest(withrespecttoinclusion)setincludedin W and S(W ) Y ∈D1 ,Y ⊂W m(Y ) SD1 Z∈D1, or Z∈D2 | ∄Y ∈D1 with Y ⊂Z m(Z ) Y ∈D1 m(Y )

whichisdifferentfromtheresultobtainedwithSCR,sinceonegetsinthisexample:

mSCR(θ2|θ2 ∪ θ3)= mSCR(θ3|θ2 ∪ θ3)=0 25

mSCR(θ2 ∪ θ3|θ2 ∪ θ3)=0.50

Morecomplexanddetailedexamplescanbefoundin[35].

9.4Qualitativebeliefconditioningrules

Inthissectionwepresentonlythequalitativebeliefconditioningruleno17whichextendstheprinciplesofthe previousquantitativeruleBCR17inthequalitativedomain usingtheoperatorsonlinguisticlabelsdeﬁnedpreviously.Weconsiderfromnowonageneralframe Θ= {θ1,θ2,...,θn},agivenmodel M(Θ) withitshyperpowerset DΘ andagivenextendedorderedset L ofqualitativevalues L = {L0,L1,L2,...,Lm,Lm+1}.The priorqualitativebasicbeliefassignment(qbba)takingitsvaluesin L isdenoted qm( ).Weassumeinthesequel thattheconditioningeventis A = ∅, A ∈ DΘ,i.e.theabsolutetruthisin A.Theapproachwepresenthereis adirectextensionofBCR17usingFLARLoperators.SuchextensioncanbedonewithallquantitativeBCR’s rulesproposedin[35],butonlyqBCR17ispresentedherefor thesakeofspacelimitations.

9.4.1QualitativeBeliefConditioningRuleno17(qBCR17)

SimilarlytoBCR17,qBCR17isdeﬁnedbythefollowingformula: qmqBCR17 (X|A)= qm(X ) · qSD1 + W ∈D2∪D3 X⊂W qS(W )=0

qm(W ) qS(W ) + W ∈D2∪D3 X⊂W, Xis k-largest qS(W )=0

qm(W )/k (44) where”X is k-largest”meansthat X isthe k-largest(withrespecttoinclusion)setincludedin W and qS(W ) Y ∈D1 ,Y ⊂W qm(Y ) SD1

qm(Z ) Y ∈D1 qm(Y ) Naturally,alloperators(summation,product,division,etc)involvedintheformula(44)aretheoperators deﬁnedinFLARLworkingonlinguisticlabels.Itisworthtonotethattheformula(44)requiresalsothedivisionofthelabel qm(W ) byascalar k.Thisdivisionisdeﬁnedasfollows: Let

Z∈D1, or Z∈D2 | ∄Y ∈D1 with Y ⊂Z

r ∈ R,r =0.Thenthelabeldivisionbyascalarisdeﬁnedby La r = La/r (45)

9.4.2AsimpleexampleforqBCR17 Let’sconsider L = {L0,L1,L2,L3,L4,L5,L6} asetoforderedlinguisticlabels.Forexample, L1, L2, L3, L4 and L5 mayrepresentthevalues: L1 verypoor, L2 poor, L3 medium, L4 good and L5 verygood Let’sconsideralsotheframe Θ= {A,B,C,D} withthehybridmodelcorrespondingtotheVenndiagramon Figure6. Weassumethatthepriorqualitativebba qm( ) isgivenby: qm(A)= L1,qm(C )= L1,qm(D)= L4

Fig.6:VennDiagramforthehybridmodelforthisexample. andthequalitativemassesofallotherelementsof GΘ taketheminimal/zerovalue L0.Thisqualitativemassis normalizedsince L1 + L1 + L4 = L1+1+4 = L6 = Lmax

Ifweassumethattheconditioningeventistheproposition A ∪ B,i.e.theabsolutetruthisin A ∪ B,the hyper-powersetdecomposition(HPSD)isobtainedasfollows: D1 isformedbyallpartsof A ∪ B, D2 isthe setgeneratedby {(C,D), ∪, ∩}\∅ = {C,D,C ∪ D,C ∩ D},and D3 = {A ∪ C,A ∪ D,B ∪ C,B ∪ D,A ∪ B ∪ C,A ∪ (C ∩ D),...}.Becausethetruthisin A ∪ B, qm(D)= L4 istransferredinaprudentwayto (A ∪ B) ∩ D = B ∩ D accordingtoourhybridmodel,because B ∩ D isthe1-largestelementfrom A ∪ B whichisincludedin D.While qm(C )= L1 istransferredto A only,sinceitistheonlyelementin A ∪ B whosequalitativemass qm(A) isdifferentfrom L0 (zero);hence: qmqBCR17(A|A ∪ B)= qm(A)+ qm(C )= L1 + L1 = L1+1 = L2. Therefore,oneﬁnallygets: qmqBCR17(A|A ∪ B)= L2 qmqBCR17 (C|A ∪ B)= L0 qmqBCR17 (D|A ∪ B)= L0 qmqBCR17 (B ∩ D|A ∪ B)= L4 whichisanormalizedqualitativebba. MorecomplicatedexamplesbasedonotherqBCR’scanbefound in[36].

10Conclusion

AgeneralpresentationofthefoundationsofDSmThasbeenproposedinthisintroduction.DSmTproposes newquantitativeandqualitativerulesofcombinationforuncertain,impreciseandhighlyconflictingsources ofinformation.SeveralapplicationsofDSmThavebeenproposedrecentlyintheliteratureandshowthe potentialandtheefficiencyofthisnewtheory.DSmToffersthepossibilitytoworkindifferentfusionspaces dependingonthenatureofproblemunderconsideration.Thus,onecanworkeitherin 2Θ =(Θ, ∪) (i.e.in theclassicalpowersetasinDSTframework),in DΘ =(Θ, ∪, ∩) (thehyper-powerset—alsoknownas Dedekind’slattice)orinthesuper-powerset SΘ =(Θ, ∪, ∩,c( )),whichincludes 2Θ and DΘ andwhich representsthepowersetoftheminimalrefinementoftheframe Θ whentherefinementispossible(because forvagueelementswhosefrontiersarenotwellknowntherefinementisnotpossible).Wehaveenrichedthe DSmTwithasubjectiveprobability(DSmPǫ)thatgetsthebestProbabilisticInformationContent(PIC)in comparisonwithotherexistingsubjectiveprobabilities. Also,wehavedefinedanddevelopedtheDSmField andLinearAlgebraofRefinedLabelsthatpermitthetransformationofanyfusionruletoacorresponding qualitativefusionrulewhichgivesanexactqualitativeresult(i.e.arefinedlabel),sofarthebestinliterature.

References

[1]J.Bolanos,L.M.DeCampos,S.Moral, Propagationoflinguisticlabelsincausalnetworks,Proc.of2nd IEEEInt.Conf.onFuzzySystems,Vol.2,pp.863–870,28March–1April1993.

[2]L.Cholvy UsinglogictounderstandrelationsbetweenDSmTandDempster-ShaferTheory,inpp.312–317oftheProceedingsoftheRencontresfrancophonessurla LogiqueFloueetsesApplications(LFA 2008),Lens,France,October16-17,2008.

✫✪ ✬✩ ✫✪ ✬✩ ✫✪ ✬✩✫✪ ✬✩ ❅ ❘ A ✠ B ✛

✛

[3]L.Comtet, SpernerSystems,Sec.7.2inAdvancedCombinatorics:TheArtofFiniteandInﬁniteExpansions,D.ReidelPubl.Co.,pp.271-273,1974.

[4]R.Dedekind, UberZerlegungenvonZahlendurchihregr¨osstengemeinsammenTeiler,InGesammelte Werke,Bd.1.pp.103-148,1897.

[5]T.Denœux, Reasoningwithimprecisebeliefstructures,TechnicalReportHeudiasys97/44,availableat http://www.hds.utc.fr/ tdenoeux/

[6]T.Denœux, Reasoningwithimprecisebeliefstructures,InternationalJournalofApproximateReasoning, 20,pp.79-111,1999.

[7]J.DezertJ.,F.Smarandache, PanelDiscussiononDSmT,Fusion2004InternationalConference,Stockholm,Sweden(andalsoinTutorialsonDSmTgiveninFusion2005-2008Conferences).

[8]J.DezertJ.,F.Smarandache, DSmT:ANewParadigmShiftforInformationFusion,inProceedingsof Cogis’06Conference,Paris,March2006.

[9]J.Dezert,A.Tchamova,F.Smarandache,P.Konstantinova, TargetTypeTrackingwithPCR5andDempster’srules:AComparativeAnalysis,inProceedingsofFusion2006Internationalconferenceon InformationFusion,Fusion2006,Firenze,Italy,July10-13,2006.

[10]J.Dezert,F.Smarandache, Anewprobabilistictransformationofbeliefmassassignment,inProceedings ofFusion2008Conference,Cologne,Germany,July2008.

[11]D.Dubois,H.Prade, OntheunicityofDempsterruleofcombination,InternationalJournalofIntelligent Systems,Vol.1,pp133-142,1986.

[12]D.Dubois,H.Prade, Representationandcombinationofuncertaintywithbelief functionsandpossibility measures,ComputationalIntelligence,4,pp.244-264,1988.

[13]M.C.Florea,Combinaisond’informationshétérogènesdanslecadreunificateurdesensemblesaléatoires :Approximationsetrobustesse,Ph.D.Thesis,LavalUniversity,Canada,July2007.

[14]J.W.Guan,D.A.Bell, GeneralizingtheDempster-ShaferruleofcombinationtoBooleanalgebras,Proc. ofIEEE/ACMInt.Conf.onDevelopingandManagingProjects, Washington,March,pp.229–236,1993.

[15]T.Inagaki, Interdependencebetweensafety-controlpolicyandmultiple-sensorschemesviaDempsterShafertheory,IEEETrans.onreliability,Vol.40,no.2,pp.182-188,1991.

[16]M.Lamata,S.Moral, Calculuswithlinguisticprobabilitiesandbeliefs,InR.R.Yager,M.Fedrizzi,and J.Kacprzyk,editors,AdvancesinDempster-ShaferTheoryofEvidence,pp.133-152,Wiley.

[17]E.Lefevre,O.Colot,P.Vannoorenberghe, Belieffunctionscombinationandconﬂictmanagement,InformationFusionJournal,ElsevierPublisher,Vol.3,No.2,pp.149-162,2002.

[18]E.Lefevre,O.Colot,P.Vannoorenberghe, ReplytotheCommentsofR.Haennionthepaper”Belief functionscombinationandconﬂictmanagement”,InformationFusionJournal,ElsevierPublisher,Vol.4, pp.63-65,2003.

[19]C.K.Murphy,Combiningbelieffunctionswhenevidenceconﬂicts,DecisionSupportSystems,Elsevier Publisher,Vol.29,pp.1-9,2000.

[20]J.B.Paris, Theuncertainreasoner’scompanion,amathematicalperspective,CambridgeUniversityPress, 1994.

[21]J.Pearl, ProbabilisticreasoninginIntelligentSystems:Networks ofPlausibleInference,MorganKaufmannPublishers,SanMateo,CA,1988.

[22]A.Robinson, Non-StandardAnalysis,North-HollandPubl.Co.,1966.

[23]K.Sentz,S.Ferson, CombinationofevidenceinDempster-ShaferTheory,SANDIATech.Report, SAND2002-0835,96pages,April2002.

[24]G.Shafer, AMathematicalTheoryofEvidence,PrincetonUniv.Press,Princeton,NJ,1976.

[25]C.E.Shannon, AMathematicalTheoryofCommunication,BellSyst.Tech.J.,27,pp.379-423and623656,1948.

[26]N.J.A.Sloane, TheOn-lineEncyclopediaofIntegerSequences2003,(SequenceNo.A014466), http://www.research.att.com/˜njas/sequences/

[27]F.Smarandache, AUnifyingFieldinLogics:NeutrosophicLogic.Neutrosophy,NeutrosophicSet,Probability,andStatistics,(4thEd.),Amer.ResearchPress,Rehoboth,2005.

[28]F.Smarandache, AUnifyingFieldinLogics:NeutrosophicLogic,Multiple-valuedlogic,Aninternational journal,Vol.8,No.3,pp.385-438,2002.

[29]F.Smarandache, Neutrosophy:Anewbranchofphilosophy,Multiple-valuedlogic,Aninternationaljournal,Vol.8,No.3,pp.297-384,2002.

[30]F.Smarandache(Editor), ProceedingsoftheFirstInternationalConferenceonNeutrosophics,Univ.of NewMexico,GallupCampus,NM,USA,1-3Dec.2001,Xiquan,Phoenix,2002.

[31]F.Smarandache,J.Dezert(Editors), ApplicationsandAdvancesofDSmTforInformationFusion,Am. Res.Press,Rehoboth,2004. http://www.gallup.unm.edu/ smarandache/DSmT-book1.pdf

[32]F.Smarandache, UniﬁcationofFusionTheories(UFT),PresentedatNATOAdvancedStudyInstitute, Albena,Bulgaria,16-27May2005alsoinInternationalJournalofAppliedMathematics&Statistics, Vol.2,1-14,2004andon http://arxiv.org/abs/cs/0409040

[33]F.Smarandache,J.Dezert, InformationFusionBasedonNewProportionalConﬂictRedistributionRules, ProceedingsofFusion2005Conf.,Philadelphia,July26-29,2005.

[34]F.Smarandache,J.Dezert, UniformandPartiallyUniformRedistributionRules,publishedin”Advances andApplicationsofDSmTforPlausibleandParadoxicalReasoningforInformationFusion”,InternationalWorkshoporganizedbytheBulgarianISTCentreofCompetencein21stCentury,December14, 2006,Bulg.Acad.ofSciences,Soﬁa,Bulgaria. http://xxx.lanl.gov/abs/cs/070202

[35]F.Smarandache,J.Dezert(Editors), ApplicationsandAdvancesofDSmTforInformationFusion,Vol.2, AmericanResearchPress,Rehoboth,August2006. http://www.gallup.unm.edu/˜smarandache/DSmT-book2.pdf

[36]F.Smarandache,J.Dezert, QualitativeBeliefConditioningRules,ProceedingsofFusion2007Conf., Qu´ebec,Canada,July2007.

[37]F.Smarandache,J.Dezert(Editors), ApplicationsandAdvancesofDSmTforInformationFusion,Vol.3, AmericanResearchPress,Rehoboth,2009. http://www.gallup.unm.edu/ smarandache/DSmT-book3.pdf

[38]F.Smarandache, DSmTwebpage, http://fs.gallup.unm.edu//DSmT.htm

[39]Ph.Smets, Combiningnondistinctevidence,Proc.NorthAmericanFuzzyInformationProcessing (NAFIP1986),NewOrleans,LA,1986.

[40]Ph.Smets,E.H.Mamdani,D.Dubois,H.Prade(Editors), Non-StandardLogicsforAutomatedReasoning, AcademicPress,1988.

[41]SmetsPh.,KennesR., Thetransferablebeliefmodel,Artif.Intel.,66(2),pp.191-234,1994.

[42]Ph.Smets, DataFusionintheTransferableBeliefModel,Proceedingsofthe3rdInternationalConference onInformationFusion,Fusion2000,Paris,July10-13,2000,ppPS21-PS33.

[43]J.Sudano,“PignisticProbabilityTransformsforMixesofLow-andHigh-ProbabilityEvents”, Proc.of Fusion2001,Montreal,August2001.

[44]J.Sudano,“Thesystemprobabilityinformationcontent(PIC)...”, Proc.ofFusion2002,Annapolis,July 2002.

[45]J.Sudano,“EquivalenceBetweenBeliefTheoriesandNaiveBayesianFusionforSystemswithIndependentEvidentialData-PartI,TheTheory”, Proc.ofFusion2003 ,Cairns,July2003.

[46]J.Sudano,“YetAnotherParadigmIllustratingEvidenceFusion(YAPIEF)”, Proc.ofFusion2006,Florence,July2006.

[47]M.Tombak,A.Isotamm,T.Tamme, OnlogicalmethodforcountingDedekindnumbers,Lect.Noteson Comp.Sci.,2138,p.424-427,Springer-Verlag,2001.

[48]F.Voorbraak, OnthejustiﬁcationofDempster’sruleofcombination,ArtiﬁcialIntelligence,48,pp.171197,1991. http://turing.wins.uva.nl/˜fransv/#pub

[49]Y.-M.Wang,J.-B.Yang,D.-L.Xu,K.-S.Chin, Onthecombinationandnormalizationofinterval-valued beliefstructures,InformationSciences177,pp.1230–1247,2007.

[50] http://bfas.iutlan.univ-rennes1.fr/wiki/index.php/Toolboxs

[51]R.R.Yager, Hedginginthecombinationofevidence,JournalofInformationandOptimizationScience, Vol.4,No.1,pp.73-81,1983.

[52]R.R.Yager, Ontherelationshipsofmethodsofaggregationofevidenceinexpertsystems,Cybernetics andSystems,Vol.16,pp.1-21,1985.

[53]R.R.Yager, OntheDempster-Shaferframeworkandnewcombinationrules,InformationSciences,Vol. 41,pp.93–138,1987.

[54]L.Zadeh, Fuzzysets,Inform.andControl8,pp.338-353,1965.

[55]L.Zadeh, FuzzyLogicandApproximateReasoning,Synthese,30,407-428,1975.

[56]L.Zadeh, OnthevalidityofDempster’sruleofcombination,MemoM79/24,Univ.ofCalifornia,Berkeley,1979.

[57]L.Zadeh, ReviewofMathematicaltheoryofevidence,byGlennShafer,AIMagazine,Vol.5,No.3,pp. 81-83,1984.

[58]L.Zadeh, AsimpleviewoftheDempster-Shafertheoryofevidenceanditsimplicationsfortheruleof combination,BerkeleyCognitiveScienceReportNo.33,UniversityofCalifornia,Berkeley,CA,1985.

[59]L.Zadeh, AsimpleviewoftheDempster-Shafertheoryofevidenceanditsimplicationfortheruleof combination,AIMagazine7,No.2,pp.85-90,1986.

[60]L.Zhang, Representation,independence,andcombinationofevidenceintheDempster-ShaferTheory, AdvancesintheDempster-ShaferTheoryofEvidence,R.R.Yager,J.KacprzykandM.Fedrizzi,Eds., JohnWileyandSons,Inc.,NewYork,pp.51-69,1994.