Representing qualitative capacities as families of possibility measures

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OULOUSE

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Dubois, Didier and Prade, Henri and Rico, Agnés

Representing qualitative capacities as families of possibility measures.

(2014) International Journal of Approximate Reasoning, vol. 58. pp. 3-24.

ISSN 0888-613X

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Representing

qualitative

capacities

as

families

of

possibility

measures

Didier Dubois

a

,

Henri Prade

a,∗

,

Agnès Rico

b

a_IRIT,_Université_Paul_Sabatier,₁₁₈_route_de_Narbonne,₃₁₀₆₂_Toulouse_cedex_9,_France b_ERIC,_Université_Claude_Bernard_Lyon_1,₄₃_bld_du₁₁_novembre,₆₉₁₀₀_{Villeurbanne,}_France

a b s t r a c t

Keywords:

Fuzzymeasures Possibilitytheory Modallogic

Thispaperstudiesthestructureofqualitativecapacities,thatis,monotonicset-functions, when they range on a finite totally ordered scale equipped with an order-reversing map.Theseset-functions correspondtogeneralrepresentationsofuncertainty,aswell as importancelevelsofgroupsofcriteriainmultiple-criteriadecision-making.Weshowthat any capacityor fuzzy measure rangingon aqualitative scalecan beviewed bothas the lowerbound ofa setofpossibility measuresand theupper boundof aset ofnecessity measures(asituationsomewhat similartotheoneofquantitative capacitieswithrespect to imprecise probability theory). We show that any capacity is characterized by a non-empty class of possibility measures having the structure of an upper semi-lattice. The lower bounds of this class are enough to reconstruct the capacity, and the number of themischaracteristicofitscomplexity.Analgorithmisprovidedtocomputetheminimal set of possibility measures dominating a given capacity. This algorithm relies on the representationofthecapacitybymeansofitsqualitativeMöbiustransform,andtheuseof selectionfunctions ofthe corresponding focal sets. Weprovide theconnection between Sugeno integrals and lower possibility measures. We introduce a sequence of axioms generalizing the maxitivity property of possibility measures, and related to the number of possibility measures needed for this reconstruction. In the Boolean case, capacities arecloselyrelated tonon-regularmodal logicsand theirneighborhood semantics canbe describedintermsofqualitativeMöbiustransforms.

1. Introduction

Afuzzymeasure [38] (oracapacity[10])isaset-functionthatismonotonicunderinclusion.Inthispaper,thecapacity is saidto bequalitative (or q-capacity, forshort) if itsrangeisafinitetotally ordered set.Itmeans wedo notpresuppose addition isavailableinthecapacityrange,only minimumand maximum.Insuchacontexttheconnectionwithprobability measuresislost.Consequentlyanumberofnotions,meaningfulinthenumericalsetting,arelostaswell,suchastheMöbius transform[34],theconjugate,supermodularity[10] andthelike.Likewise,somenumericalcapacities(suchasconvexones, belieffunctions)canbeviewed asencodingaconvexfamilyofprobabilitydistributions[35,40].Thisconnectiondisappears if wegiveupusingadditionintherangeofthecapacity.

*

Correspondingauthor.

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Yet,itistemptingtocheckwhethercounterpartsofmanysuchquantitativenotionscanbedefinedforqualitative capac-ities,ifwereplaceprobabilitymeasuresbypossibilitymeasures.Conjugatenesscanberecoverediftherangeofthecapacity is equipped withan order-reversingmap. A qualitativecounterpart ofa Möbiustransformhas been introduced byMesiar [31] and Grabisch [25] in 1997 and further studiedby Grabisch[26]. The qualitativeMöbius transformcan be viewed as the possibilisticcounterpart to abasic probability assignment,wherebya capacityis defined withrespectto thelatterby a qualitative counterpart of thebelief functiondefinition, extending aswell thedefinition ofpossibility measures. In fact the process of generation of belieffunctions, introduced byDempster [9], was applied very early to possibility measures by Dubois and Prade [16,17] so as to generate upper and lower possibilities and necessities. It was noticed that upper possibilitiesand lower necessitiesarestill possibilityand necessitymeasures respectively,but uppernecessitiesand lower possibilitiesarenot.However,asweshallsee,theformalanalogybetweenbelieffunctionsandqualitativecapacitiesviathe qualitativeMöbiustransformcanbemisleadingattheinterpretivelevel.

Thissituationleadstonaturalquestions,namely,whetheraqualitativecapacitycanbeexpressedintermsofafamilyof possibility measures,and ifaqualitativeMöbiustransformcanencodesuchafamily.Previous recentworks[11,32] started addressingthisissue,takingupapioneeringworkbyBanon[4].Inthispaperweshowthatinthefinite(qualitative)setting, specialsubsetsofpossibilitymeasuresplayarolesimilartoconvexsetsofprobabilitymeasures.Weprovethatanycapacity can bedefinedinterms ofafiniteset ofpossibility measures,eitherasalower possibilityoranuppernecessityfunction. Thisresultshouldnotcomeasasurprise.Indeed,ithasbeenshownthatpossibilitymeasurescanberefinedbyprobability measures using alexicographic refinement of thebasic axiom of possibility measures, and that capacities on a finiteset canberefinedbybelieffunctions[13,14].Basedonthisfundamentalresultwecangeneralizethemaxitivityandminitivity axiom ofpossibility theoryso astodefine familiesofqualitativecapacities ofincreasing complexity.Finally, thisproperty enablesqualitativecapacities tobeseen asnecessitymodalitiesinanon-regularclassofmodallogics, extending thelinks betweenpossibilitytheoryandmodallogictoapotentiallylargerrangeofuncertaintytheories.

The structure of thepaper is asfollows.1 _{Section 2 provides} _basic _definitions_pertaining _to _capacities, _recalls _and dis-cusses thesimilaritybetweenbelieffunctionsand qualitativecapacities,indicating thelimitationofthisanalogy. Section 3 providesthemaincontributionofthispaper,namelyitshowstheformalanalogybetweenqualitativecapacitiesand impre-ciseprobabilities,provinganycapacitycomes downtoany oftwofamiliesofpossibilitydistributions,andcanbedescribed byfinitesetsthereof,eitherasalowerpossibilityoranuppernecessityfunction.Thissectionalsoextendstheseresults to Sugeno integrals. Section 4 providesanalgorithm that computesthe set ofminimalelements among possibility measures that dominate acapacity fromits qualitativeMöbiustransforms. Section 5 axiomaticallydefines subfamilies ofqualitative capacities of increasing complexity generalizing the maxitive and the minitivity axioms of possibility theory. Finally, in Section 6 we lay bare a connection between capacities and neighborhood semantics in non-regular modal logics, which suggestspotentialapplicationstoreasoningfromconflicting informationcomingfromseveralsources.

2. QualitativecapacitiesandMöbiustransforms

Consider a finite set S and a finite totally ordered scale L= {λ0=0< λ1< · · · < λℓ=1} with top 1 and bottom 0.

Moreover we assume that L is equipped with an order-reversing map, i.e., a strictly decreasingmapping

ν

:L→L with

ν

(1)=0 and

ν

(0)=1.Notethat

ν

isunique,and suchthat

ν

(λi)= λℓ−i.

Definition1. A capacity (or fuzzy measure) is a mapping

γ

:2S →L such that

γ

(∅)=0;

γ

(S)=1; and if A⊆B then

γ

(A)≤

γ

(B).Theconjugateof

γ

isthecapacity

γ

c defined as

γ

c(A)=

ν

(

γ

(Ac)),∀A⊆S,where Ac isthecomplementof set A.

The value

γ

(A) can be interpreted as the degreeof confidence in a propositionrepresented by the set A of possible states ofthe world, or, if S isa set ofcriteria, the degreeof importanceof thegroup of criteria A [27]. In thispaperwe basically use the first interpretation, unless specified otherwise. We here speak of qualitativecapacity (or q-capacity, for short) tomeanthatweonlyrelyonanordinalstructure,notanadditiveunderlying structure.

Remark1. In fact, even if thescale isencoded bymeans ofnumbersin [0,1], we do notassume these figures represent orders ofmagnitude,so thattheiraddition orsubtraction makeno sense.Ofcourse,wecouldconstructaq-capacityusing a probabilitymeasure P on S,and considering{P(A):A⊆S} asthetotallyordered set L byrenamingthenumbersusing symbols λi. It isalways possible todo so using any numericalcapacity on S.However, since thesymbols λi only encode

a ranking, we thenare unableto distinguish betweencapacities that yieldthe sameorderingof events (inparticular the probabilitymeasure P cannolongerbedistinguishedfromthemanynon-additivenumericalcapacitiesthatyieldthesame orderingofeventsas P ).Soqualitative herepresupposesthat the“distance” betweentwoconsecutiveλi’scanbearbitrary.

And inourview,anumericalset-functionisaspecialcaseofaqualitativeonewithadditionalstructureinitsrange. Importantspecialcasesofcapacityarepossibilityandnecessitymeasures.

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Definition2.Apossibilitymeasure is acapacitythatsatisfiesthecharacteristicaxiom ofmaxitivity:

γ

(

A

∪

B

) =

max

¡

γ

(

A

),

γ

(

B

)¢.

Apossibilitymeasure isusuallydenoted byΠ.In possibilitytheory[18,20],theavailableinformationisrepresentedby means ofapossibility distribution [42].This isa function,usually denoted by

π

,from theuniverse of discourse S tothe scale L,suchthat

π

(s)=1 forsomeelements∈S.Theassociatedpossibilitymeasureisdefinedby

Π (

A

) =

max s∈A

π

(

s

)

and in the finite case any possibility measure induces a unique possibility distribution

π

(s)= Π ({s}). The function

π

is supposedtorank-orderpotentialvaluesof(someaspectof)thestateoftheworld–accordingtotheirplausibility.Thevalue

π

(s)isunderstoodasthepossibilitythat s betheactualstateoftheworld.Preciseinformationcorrespondstothesituation where ∃s∗_,

_π

₍_s∗₎₌_1,_and _∀_s₆₌_s∗_,

_π

₍_s₎₌_0,_while_complete_ignorance _is_represented_by_the_vacuous_possibility_distribution

denoted by

π

?_,_such_that _∀_s_∈_S,

_π

?₍_s₎₌_1._A_possibility_distribution

_π

_is_said_to _be_more_{specific (in}_the_wide_sense)_than anotherpossibilitydistribution

ρ

if∀s∈S,

π

(s)≤

ρ

(s).Itdefinesapartialorderbetweenpossibilitydistributions,reflecting theirrelativeinformativeness.

Definition3.Anecessitymeasure isacapacitythatverifiesthecharacteristicaxiomofminitivity:

γ

(

A

∩

B

) =

min

¡

γ

(

A

),

γ

(

B

)¢.

A necessity measure is usually denoted by N. Function N is characterized by an “impossibility” distribution

ι

:S→L

as

ι

(s)=N(S\ {s}),since N(A)=mins/∈A

ι

(s). Notethat the conjugate N(A)= Πc(A)=

ν

(Π (Ac)) ofa possibilitymeasure

is a necessity measure such that

ι

(s)=

ν

(

π

(s)), which explains the interpretation of

ι

(s) as a degree of impossibility. It is also called a degree of potential surprise by Shackle [36], or a grade of disbelief by Spohn [37] (who use numerical representationsthereof).

Possibilitydistributionsseemtoplay inthequalitativesettingarolesimilartoprobabilitydistributionsinthenumerical setting, replacing sumby maximum.In thisscope, it iswell-known that, inthefinite case,the probabilitydistribution of a probability measure isgeneralizedbythe Möbiustransformof thecapacity. In thissectionwe recallqualitative Möbius transforms that were firstproposed by Grabisch[25] and Mesiar[31], and discuss the analogyit suggestsbetween belief functions andqualitativecapacities.

2.1. QualitativeMöbiustransforms

Definition4.Theinner(qualitative)Möbiustransformofacapacity

γ

isamapping

γ

#:2S→L definedby

γ

#

(

E

) =

n

_γ

₍

_E

₎

_if

_γ

₍

_E

_{) >}

_max

B(E

γ

(

B

)

0 otherwise

.

Aset E suchthat

γ

#(E)>0 is calledafocalset.

Intheabove definition,due to themonotonicitypropertyof

γ

, thecondition

γ

(E)>max_B(E

γ

(B) canbereplacedby

γ

(E)>maxx∈E

γ

(E\ {x}).Itiseasytocheck that

•

γ

#(∅)=0;maxA⊆S

γ

#(A)=1;

• If A⊂B,and

γ

#(B)>0,then

γ

#(A)<

γ

#(B).

Let F (

γ

)= {E,

γ

#(E)>0} be the family of focal sets associated to

γ

. The last property says that the inner qualitative Möbiustransformof

γ

isstrictlymonotonicwithrespecttoinclusionoffocalsets.

Example 1. Let S= {s1,s2,s3} and

γ

({s1})=

γ

({s1,s3})=0.3,

γ

({s1,s2})=0.7,

γ

({s2,s3})=

γ

({s1,s2,s3})=1 (so L is chosen arbitrarily as {0,0.3,0.7,1}), and

γ

(A)=0 otherwise. Then F (

γ

)= {{s1},{s1,s2},{s2,s3}}, with

γ

#({s1})=0.3,

γ

#({s1,s2})=0.7 and

γ

#({s2,s3})=1. ✷

ItisclearthattheinnerqualitativeMöbiustransformofapossibilitymeasure coincideswithitspossibilitydistribution: Π#(A)=

π

(s) if A= {s}and0otherwise.HencethesetoffocalsetsF (Π) ismade ofsingletonsonly.Thispropertymakes itclearthat

γ

#generalizesthenotion ofpossibilitydistributiontothepowersetof S.

Just like in the numerical setting, the focal sets associated to a necessitymeasure N form a collection ofnested sets

E1⊂E2⊂ · · · ⊂Ek suchthat N(A)=maxEi⊆AN#(Ei).If thenecessitymeasureisbasedonapossibilitydistribution

π

such

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N#(Ei)=

ν

(λi−1). A set of the form {s:

π

(s)≥ λi>0} is called a cut of

π

. So the focal sets of N are the cuts of the

possibility distributionofitsconjugate.

More generally, the inner (qualitative) Möbius transform contains the minimal information needed to reconstruct the capacity

γ

since,byconstruction[25,13]:

γ

(

A

) =

max

E⊆A

γ

#

(

E

).

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Thereadercancheckthatifoneofthevalues

γ

#(E)ischanged,thecorrespondingcapacitywillbedifferent,namelythe values

γ

(A) suchthat

γ

(A)=

γ

#(E).

Remark2. Anotherqualitativecounterpartof Möbiustransforms,namedouterqualitativeMöbiustransforms,canbeused to represent acapacity

γ

.Itisamapping

γ

#_:₂S_→_{L defined} _by

γ

#

(

A

) =

n

γ

(

A

)

if

γ

(

A

) <

minA⊂F

γ

(

F

)

1 otherwise

.

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Inparticular,

γ

#₍_S₎₌_1._Due_to_the_monotonicity_of

_γ

_,_the_condition

_γ

₍_A₎_<_min

A⊂F

γ

(F)canbeequivalentlyreplacedby

γ

(A)<mins∈A/

γ

(A∪ {s}).

Theoriginalcapacityisthenretrievedas[14]:

γ

(

A

) =

min

A⊆F

γ

#

(

F

).

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Notethat

γ

(A)isrecoveredfrom

γ

# _via_weights_assigned_to_supersets_of_set _A,_which_reminds_of_outer_measures,_while

γ

# canbecalledaninnerqualitativemassfunction.

It is worth noticing that the inner qualitative Möbius transform

γ

_#c of

γ

c is related to the outer qualitative Möbius transform

γ

# [13,14]:

γ

#

(

E

) =

ν

¡

γ

_#c

¡

Ec

¢¢

(4) since

γ

(A)<minA⊂F

γ

(F) alsowrites

γ

c(Ac)>maxFc_⊂Ac

γ

c₍Fc₎. So if we usecapacities and theirconjugates, usingthe

innerand theouterqualitativeMöbiustransformsconjointlyisredundant.Inthispaper,weonlyuseinnerones.

Example 1 (continued). In the previous example, it can be checked that

γ

#_({_s

2})=

γ

#({s3})=0,

γ

#({s1,s2})=0.7,

γ

#_({_s

1,s3})=0.3 (and

γ

#(E)=1 otherwise). ✷

2.2. Mayqualitativecapacitiesbeinterpretedasqualitativebelieffunctions?

The similarity between capacities and belief functions is striking on Eq. (1). A belief function [34] is a set function 2S_{→ [}₀_,₁_]_defined_by

Bel

(

A

) =

X

E:E⊆A

m

(

E

),

∀

A

⊆

S (5)

wherem isabasicprobabilityassignment,i.e.aprobabilitydistributionover2S\ {∅}.ThedegreeBel(A)isadegreeofbelief or certaintyinthesensethat Bel(A)=1 means A iscertain, whileBel(A)=0 means A isatbestunknown. Theconjugate set function

Pl

(

A

) =

1

−

Bel

¡

Ac

¢ =

X

E:E∩A6=∅

m

(

E

),

∀

A

⊆

S (6)

expresses theidea ofpossibility or plausibility, as Pl(A)=0 means that A isimpossible. The mass assignment m canbe reconstructedfromBel byinvertingthesystemofequationsdefiningthelatter:thisistheMöbiustransform[34].InEq. (1), max replacesthesumintheexpressionofBel.

Hence function

γ

# stands as the qualitative counterpart to a basic probability assignment in the theory of evidence, obtained via a kind of Möbius transform [26]. By analogy with the numerical case,

γ

# can be called a basicpossibility

assignment [32] associatedwith

γ

since

γ

#isapossibilitydistribution onthepowerset of S.Thesubsets E thatreceivea positive supportplay thesameroleasthefocalsetsinDempster–Shafertheory:theyaretheprimitiveitemsofknowledge. Theparallelbetweenpossibilityandprobabilityisconfirmed: thefocalsetsofaqualitativepossibilitymeasureinthesense of theinner Möbius massfunctionare singletons, just asthe focalsets of aprobability measure for thestandard Möbius transformaresingletons.

HowevertherearedifferencesbetweennumericalandqualitativeMöbiustransforms.WhileaninnerqualitativeMöbius transform canbeviewed asapossibility distribution onthepower set of S, theconverseis nottrue. Indeed, apossibility distribution

π

m on a space 2S is not requested to satisfy the monotonicity constraint of a qualitative Möbius transform.

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It means that several basic possibility assignments may yield thesame capacity

γ

(A)=maxE⊆A

π

m(E), in contrast with

thesetting ofevidence theory, wherethereisaone-to-one correspondencebetweenbelieffunctions and basic probability assignments.

Remark3. If

π

m isapossibilitydistribution over2S, monotonicitycanberestoredbydeletingtheterms

π

m(F) whenever ∃E, E⊂F ,

π

m(E)>

π

m(F). Clearly,if for some F ⊆A,there is E1⊂F ,

π

m(E1)>

π

m(F), then

γ

(A)=maxE⊆A,E6=F

π

m(E).

So we define a unique qualitative Möbiusmass from

π

m as follows:

γ

#(E)=

π

m(E)>0 if and only if ∀F ⊂E,

π

m(E)>

π

m(F), otherwise

γ

#(E)=0. The inner qualitative Möbius transform is the canonical (monotonic) possibility assignment associated withthecapacityinthesensethatallotherpossibilityassignmentscontainredundantinformationwithrespect toreconstructingthecapacityandonly

γ

#canberecoveredinanon-ambiguouswayfrom

γ

.

It is also possible to introduce the counterpart of the contourfunction [34]2 of a belief function (it is

µ

:S→ [0,1] defined by

µ

(s)=P

E∋sm(E)).Namely,

Definition5.Thequalitativecontourfunction ofacapacity

γ

isthepossibilitydistribution

π

γ

(

s

) =

max

E∋s

γ

#

(

E

).

Notethatitisapossibilitydistributionsince

π

γ(s)=1 forsomes∈S.Finally,thequalitativecounterpartofaplausibility

function,inconformitywithEq. (6) is:

Π

γ

(

A

) =

max

A∩E6=∅

γ

#

(

E

).

It is easyto seethat Πγ is just apossibility measure such that Πγ(A)≥

γ

(A), ∀A.In contrast, inthe numerical setting,

replacingmax byasum,and usingamassfunction,one getsaplausibilitymeasure,i.e.aspecialcaseofupperprobability functionmoregeneralthanapossibilitymeasureandaprobabilitymeasure.Itcanbeprovedthatthepossibilitydistribution associated withΠγ ispreciselythequalitativecontourfunction:

Proposition1.∀A,Πγ(A)=maxs∈A

π

γ(s).

Proof. To seeitjustnotethat{E:A∩E6= ∅}= {E,∃s∈A∩E}.SoΠγ(A)=maxs∈AmaxE∋s

γ

#(E). ✷

Inthenumericalsetting,wedonothavethatPl(A)=maxs∈A

µ

(s)(norhavewePl(A)=Ps∈A

µ

(s)).Themeaningofthe

qualitativecontourfunctionasaspecial upperpossibilisticapproximationof

γ

willbelaidbareattheendofSection 4.

Example1 (continued). ForthecapacityinExample 1,thereadercancheckthat

π

γ(s1)=max(

γ

#({s1}),

γ

#({s1,s2}))=0.7,

π

γ(s2)=max(

γ

#({s2,s3}),

γ

#({s1,s2}))=1,and

π

γ(s3)=

γ

#({s2,s3})=1. ✷

In the construction of Dempster [9], a belief function is actually a special case of a lower probability bound induced by a probability space (Ω,P) and a multi-valued mapping, Γ : Ω →2S asBel(A)=P({

ω

∈ Ω : Γ (

ω

)⊆A}). Considering the selection functions f of Γ (i.e.,

ω

∈ Ω, f(

ω

)∈ Γ (

ω

)) and the probability functions Pf induced by (Ω,P) on S as

Pf(A)=P(f−1(A)),itholds(inthefinitecase)thatBel(A)=minf∈Γ Pf(A)[8].

Incontrast,inthequalitativecase,any capacitycanbegeneratedbythisprocess,replacingprobabilitybypossibility.The possibilisticcounterpartofDempsterprocesshasbeenstudiedatlengthbyDuboisandPrade[17] andmorerecentlybyDe Baets and Tsiporkova[39].Given apossibility distribution

π

on Ω,we canthus definean upperpossibility functionΠ =

maxf∈ΓΠf(A) and a lower possibility function Π =minf∈ΓΠf(A) where Πf(A)= Π (f−1(A)) and Π is the possibility

measure associatedtoapossibilitydistribution

π

: Ω → [0,1].

Sinceanycapacitycanbeexpressedbyamonotonicpossibilityassignment,itcanalwaysbeviewedasalowerpossibility, as weshall see inthe nextsection. Indeed if we choose Ω = F,the set of focalsets of

γ

,and let

π

=

γ

#, then

γ

(A)=

minf∈ΓΠf(A)= Π, while Π = Πγ. As a consequence, Π and Π are not conjugateset-functions, contrary to the caseof

belieffunctions.Especially,wecannotusetheconjugatefunctionΠc,thatis,thenecessitymeasureinducedbythecontour functionof

γ

asalower boundforthelatter.Indeed

Nγ

(

A

) =

ν

¡Π

γ

¡

Ac

¢¢ =

min s∈/A

ν

³

max E∋s

γ

#

(

E

)

´

=

min E∩Ac_6=∅

ν

¡

γ

#

(

E

)¢,

whichcannotbecomparedwith

γ

(A).

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Summarizing, the situation of qualitative capacities with respect to possibility measures is different from the one of numericalcapacitieswithrespecttoprobabilitiesinseveralrespects:

• Anyqualitativecapacityisalowerpossibility,whilenotany numericalcapacityisalower probability.Someareupper probabilitiesonly (forinstance thevacuouspossibilityfunctionΠ?)and someareneitheruppernorlower probability bounds.Thereisnoprobabilityfunctionactingasthecounterpartofthevacuouspossibilityfunction.Theinterpretation of a belief function as measuring belief or certainty,3 _as _opposed _to _{plausibility,} _does _not _carry _over _to _qualitative capacities,even ifthedefinitionofacapacityvia

γ

# bearsaformalsimilaritywithbelieffunctions, andcapacities can begeneratedaslower boundsofpossibilityfunctions.

• Inparticular,sincetheset-function

γ

# canalsobeviewedasapossibilitydistributionoverthepowerset 2S,expression (1) isalsoageneralizationofthedefinitionofthedegreeofpossibilityofasetintermsofapossibilitydistributionon

S aspointedoutearlier.

• Theone-to-onecorrespondencebetweenrandomsets(usingMöbiusmassesm)andbelieffunctionsdoesnotcarryover toqualitative capacities:theset of possibilitydistributions on 2S canbe partitionedintoequivalence classes, one per capacity

γ

,eachcontainingtheinnerqualitativeMöbiustransformofsomecapacity,andallpossibilitydistributionsin theclassgeneratingthesame

γ

. Itisclearthat

γ

# istheleastelement (eventwisely)oftheequivalenceclassattached to

γ

.

• Thequalitativecounterparttoaplausibilityfunctionisapossibilityfunctionthatdominatesthecapacity,andits possi-bilitydistributionisthequalitativecontourfunctionofthecapacity.

Due to this state of facts, there is a big mathematical and semantic difference between belief functions and qualitative capacities.

Remark 4. Wong et al. [41] have shown that, in order to be representable by a numerical belief function, a qualitative capacitymustsatisfytheaxiom

BEL:

∀

A

,

B

,

C disjoint, if

γ

(

A

∪

B

) >

γ

(

A

)

then

γ

(

A

∪

B

∪

C

) >

γ

(

A

∪

C

).

Thisisaweakeningofthewell-known DeFinettiaxiom ofcomparativeprobability: If

∀

A

∩ (

B

∪

C

) = ∅

then

,

γ

(

B

) ≥

γ

(

C

)

if and only if

γ

(

A

∪

B

) >

γ

(

A

∪

C

),

which is known to beinsufficientto characterize numerical probabilityon finite sets [24].Curiously, even if a capacity

γ

failstosatisfyaxiomBEL,italmostsatisfiesitinthesensethatthefollowingpropertyalwaysholds[14]:

∀

A

,

B

,

C disjoint, if

γ

(

A

∪

B

) >

γ

(

A

)

then

γ

(

A

∪

B

∪

C

) ≥

γ

(

A

∪

C

).

However, usingcontraposition,thefollowingdualaxiom,anotherspecialization ofthelatterproperty(distinctfromBEL):

PL:

∀

A

,

B

,

C disjoint, if

γ

(

A

∪

B

∪

C

) >

γ

(

A

∪

C

)

then

γ

(

A

∪

B

) >

γ

(

A

)

makes

γ

representable byaplausibility function. Itmeans that the weakordering ofevents inducedbyany capacity can berefinedbyaweakorderingrepresentablebyabelieffunctionorbyaplausibility function[14],whichconfirmsthatthe formalanalogybetweencapacitiesandbelieffunctionsproperismisleading.

3. Capacitiesaslowerpossibilityanduppernecessityfunctions

It is well-known that a belief function can be equivalently represented by a non-empty convex set of probabilities, namelyP(Bel)= {P,P(A)≥Bel(A),∀A⊆S}.Then itholdsthatBel(A)=minP∈P(Bel)P(A). Inotherwords abelieffunction

is one example of coherent lower probability in the sense of Walley [40] (exact capacity after Schmeidler [35]). The set P(Bel)iscalledthecredalsetrepresenting Bel.4 Iftherangeoftheset-functionisnotequippedwithadditionand product, thisconstructionisimpossible.Thenaturalquestionisthenwhetherasimilarconstructionmaymakesensewithqualitative possibility measures in place of probability measures, using min and max instead of product and sum. From previous preliminary results [17,39,4] we knowtheanswer ispositive. In thispaper,we investigate theproblem indepth, without resortingtoDempstersetting.

3 _Bel₍_A₎₌_{1 means}_that _{A is}_totally_certain.

(8)

3.1. Capacitiesaslowerpossibilities

Thereisalwaysatleastonepossibilitymeasure thatdominates anycapacity:thevacuouspossibilitymeasure,basedon thedistribution

π

? _expressing_ignorance,_since_for_any_capacity

_γ

_,_∀_A_{6= ∅}_⊂_S,_Π?₍_A₎₌₁_≥

_γ

₍_A₎_,_and _Π?_(∅)₌

_γ

_(∅)₌_0.

Let R(

γ

)= {

π

: Π (A)≥

γ

(A),∀A⊆ S} be the non-empty set of possibility distributions whose corresponding set-functions Π dominate

γ

.Inanalogytothecorrespondingnotioningametheory[35],wecallR(

γ

) thepossibilisticcore of

the capacity

γ

. Clearly, R(

γ

), equipped withthepartial orderofspecificity (introducedat thebeginning ofSection 2), is an uppersemi-lattice,that isif

π

1,

π

2∈ R(

γ

), thenmax(

π

1,

π

2)∈ R(

γ

). Ofcourse,themaximal element ofR(

γ

) isthe vacuouspossibilitydistribution

π

?,butithasseveral minimalelements.Wedenote byR_∗₍

_γ

₎thesetofminimalelements inR(

γ

).

Any capacity

γ

can bereconstructed usingpossibility measures inducedbypermutations.Let uspresent this resultin details. Let

σ

be a permutation of the n= |S| elements in S. The ith element of the permutation is denoted by s_σ(i).

Moreoverlet Si

σ = {sσ(i),. . . ,sσ(n)}.Definethepossibilitydistribution

π

σγ asfollows:

∀

i

=

1

, . . . ,

n

,

π

σγ

(

sσ(i)

) =

γ

¡

Si_σ

¢.

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Thereareatmostn!(numberofpermutations)suchpossibilitydistributions,that arecalledthemarginals of

γ

.

Example 1 (continued). If we take the permutation

σ

= (1,2,3), we get

π

σγ(s3)=

γ

({s3})=0,

π

σγ(s2)=

γ

({s2,s3})=1,

π

σγ(s1)=

γ

(S)=1.

If we take the permutation

τ

= (3,2,1), we get

π

τγ(s1) =

γ

({s1})= 0.3,

π

τγ(s2) =

γ

({s1,s2})=0.7,

π

τγ(s3) =

γ

(S)=1. ✷

This is similar to the case of a numerical capacity g from which a set of probability distributions p_σg of the form

p_σg(s_σ(i))=g(Sσi )−g(Sσi+1) canbeextracted. If g isabelieffunction,5 theconvexhullofthese probabilitiescoincide with

the(non-empty)credalset P(g).

Similarly,itcanbecheckedthatthemarginalsof

γ

lieinR(

γ

)and enable

γ

tobereconstructed(alreadyin[4]):

Lemma1.Π_σγ(A)≥

γ

(A),∀A⊆S.

Proof. Consideragivenpermutation

σ

andanevent A.LetiA=max{i,A⊆S_σi }.ThenSσiA isthesmallestsetinthesequence {Si_σ}i that contains A andΠσγ(A)= Πσγ(SiσA)=

π

σγ(sσ(i)) since A contains sσ(iA) and otherelements in A areof theform sσ(j), j≥iA byconstruction.Hence,byconstruction,Πσγ(A)=

γ

(SσiA)≥

γ

(A). ✷

Proposition2.∀A⊆S,

γ

(A)=min_σΠσγ(A).

Proof. Fromthelemma,

γ

(A)≤minσΠσγ(A). Conversely, ∀A⊆S, thereis apermutation

σ

A such that A=Si_σ_A. By

con-struction

γ

(A)= ΠσγA(A)≥minσΠ

γ

σ(A). ✷

Asconsequences, wehavethefollowingresults:

Proposition3.ForallA⊆S,

γ

(A)=min_π_∈R(γ₎Π (A).

Proof. For all

π

∈ R(

γ

), Π ≥

γ

so

γ

(A)≤min_π∈R(γ₎Π (A). Moreover there exists

σ

such that

γ

(A) = Πσγ(A)≥

min_π_∈R(γ)Π (A),and, byLemma 1,weknowthat

π

σγ ∈ R(

γ

). ✷

Proposition4.∀

π

∈ R(

γ

),

π

(s)≥

π

σγ(s),∀s∈S forsomepermutation

σ

ofS.

Proof. Justconsiderapermutation

σ

inducedby

π

,that is

σ

(i)≥

σ

(j) ⇐⇒

π

(si)≤

π

(sj).Forthispermutation,Π (S_σi)=

π

(si)≥

γ

(Si_σ)=

π

σγ(si), ∀i=1,. . . ,n. ✷

This resultsays that theset of marginals

π

σγ includes theset R∗(

γ

) of leastelements of R(

γ

), i.e., themost specific

possibility distributions dominating

γ

. In other terms, R(

γ

)= {

π

: ∃

σ

,

π

≥

π

σγ}. Not all the n! possibility distributions

π

σγ are leastelements of R(

γ

). As atrivialexample, if

γ

= Π, thisleast element isunique and isprecisely

π

.But other

permutations yieldotherless specificpossibilitydistributions.AmoreefficientmethodforgeneratingR_∗₍

_γ

₎ isprovidedin Section 4.

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Table 1

Leastelementsofthepossibilisticcore(R_∗(γ))for Ex-ample 1.

S s1 s2 s3

π1 0.3 1 0

π2 0.7 0 1

π3 0.3 0.7 1

By constructionR_∗(

γ

) isa finiteset ofpossibility distributionsnone ofwhich ismore specificthat another, that is, if

π

,

ρ

∈ R∗(

γ

),∃s16=s2∈S,

π

(s1)>

ρ

(s1) and

π

(s2)<

ρ

(s2).Wethus canprovethefollowingbasicresult:

γ

(A)=min_π∈R∗(γ)Π (A).

Proof. As ∀

π

∈ R∗(

γ

),Π ≥

γ

, it holdsthat

γ

(A)≤minπ∈R∗(γ)Π (A). Nowsuppose ∃A,

γ

(A)<minπ∈R∗(γ)Π (A). Then

∀

π

∈ R∗(

γ

), Π (A)>

γ

(A). However there is

π

′∈ R(

γ

)\ R∗(

γ

) such that Π′(A)=

γ

(A)< Π (A). Hence

π

′ is either

more specific than some

π

∈ R∗(

γ

) (which is impossible since R∗(

γ

) contains themost specific elements in R(

γ

)) or

incomparable withall

π

∈ R∗(

γ

),whichwouldmeaninsideit,whichiscontrarytotheassumption. ✷

Example1 (continued).Forinstance,thethreeleastspecificpossibilitydistributionsthatdominate

γ

inExample 1 aregiven inTable 1.Theyarenotcomparableand onecancheckthat

γ

(A)=min(Π1(A),Π2(A),Π3(A)). ✷

Thesefindingsalsoshowthatany capacitycanberepresentedbyafinitesetofpossibilitymeasures.Conversely,forany set T ofpossibility distributions,theset-function

γ

(A)=min_π∈T Π (A) isacapacity,and itiseasytoseethat T ⊆ R(

γ

).

If T onlycontainspossibilitydistributionsthatarenotcomparablewithrespecttospecificity,T = R∗(

γ

),themostspecific

elements ofR(

γ

).

Remark5.Note that theset function max_π∈T Π (A) isnot onlya capacity, butalso apossibility measure with possibility distribution

π

max₍_s₎₌_max

π∈T

π

(s)[19].

Itisinterestingtoconsider theclosureofR_∗₍

_γ

₎underthequalitativecounterpartofaconvexcombination[15]:

Definition6.If

π

1,. . . ,

π

k arepossibilitydistributionsand∀

α

1,. . . ,

α

k∈L,suchthatmax_i=k ₁

α

i=1,thequalitativemixture

ofthe

π

i’sismax_i=k ₁min(

α

i,

π

i).Theq-convexclosureC(T )oftheset T ofpossibilitydistributionsisthesetofqualitative

mixturesofelementsofT.

Note that the qualitative mixture of possibility distributions is a possibility distribution (it is normalized) and maxk_i=₁min(

α

i,Πi) is a possibility measure with distribution maxki=1min(

α

i,

π

i) [19]. We then define the q-convexcore

of the capacity

γ

by C(R∗(

γ

)), which possesses a maximal element, i.e., the possibility measure with distribution

max_π∈R_∗_(γ₎

π

. ClearlyC(R∗(

γ

))is in general aproper q-convex subsetof R(

γ

), that is analogousto anumerical credal

set.6 _The _minimal_set _R

∗(

γ

) plays thesamerole asextreme probabilitiesincredal sets(the verticesofa convex

polyhe-dron), fromwhichthelattercanbereconstructedviaconvexclosure.

3.2. Capacitiesasuppernecessities

We can dually describe capacity functions as uppernecessities by means ofa family of necessityfunctions that stem from thelower possibilitydescriptionof theirconjugates.Clearly,possibility measuresthat dominate

γ

c _are_conjugates_of

necessitymeasuresdominatedby

γ

.Inotherwords

γ

isalsoanuppernecessitymeasureinthesense that

Proposition6.

γ

(

A

) =

max

π∈R∗(γc)

N

(

A

).

Proof.

γ

c(A) = min_π∈R∗(γc)Π (A). Hence,

γ

(A) can also be expressed as:

γ

(A) =

ν

(minπ∈R∗(γc)Π (A

c₎₎ ₌

max_π∈R_∗_(γc₎

ν

(Π (Ac)). ✷

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However, this result can be directly obtained from the expression (1) of

γ

. Indeed, if E is a focal set of

γ

, define the necessity measure NE by ∀A6=S, NE(A)=

γ

#(E) if E⊆ A and0 otherwise (its focal sets are E, S, with qualitative Möbiusweights NE#(A)=

γ

#(E) andNE#(S)=1).Itisclearthat

γ

(A)=maxE∈F (γ)NE(A).Thisisnottheminimalformof

course.Togettheminimalformonemayconsiderallmaximalchainsofnestedfocalsubsets C_i= {E1

i ⊂ · · · ⊂E ki

i }⊆ F (

γ

),

i=1,. . . ,n:eachsuchchainC_i definesanecessitymeasure Ni whosenestedfocalsetsareasfollows: • if

γ

#(E_iki)=1 thenF (Ni)= Ci and Ni#(Ek_i)=

γ

#(Ek_i)

• otherwiseF (Ni)= Ci∪ {S} and Ni#(E_ik)=

γ

#(Ek_i), Ni#(S)=1.

Itiseasytocheck thatiftherearen maximalchainsofsubsetsinF (

γ

),

γ

(

A

) =

maxn i=1 maxE∈C_i

γ

#

(

E

) =

n max i=1 Ni

(

A

)

(8) and Ni(E)=

γ

#(E), ∀E∈ Ci.

Example1 (continued). The set of focal sets of

γ

in Example 1 containstwo maximal chains: ({s1},{s1,s2}) and {s2,s3} withrespectiveweights(0.3,0.7),and1.Henceweneedtwonecessitymeasurestorepresent

γ

:N1 withfocalsetsdefined by N1#({s1})=0.3, N1#({s1,s2})=0.7, N1#(S)=1; and a Boolean necessity measure N2 with one focal set defined by

N2#({s2,s3})=1.Itcanbecheckedthat

γ

(A)=max(N1(A),N2(A)). ✷

By construction, the cuts of possibility distributions

π

i underlying Ni and different from S are focal sets of

γ

. These

possibilitydistributionsarenotcomparablewithrespecttospecificity:iftheywere,theywouldnotcorrespondtomaximal nestedchains offocalsets.Inotherwords:

Proposition7.ThesetofpossibilitydistributionswhosecutsarebuiltfromthemaximalnestedchainsinF (

γ

)isthesetR_∗₍

_γ

c₎_of

possibilitydistributionsdominatingtheconjugateof

γ

.

Proof. The

π

i’sinducedbymaximalchains Ci dominate

γ

c sincethecorresponding Ni isdominatedby

γ

.These

distribu-tionsarenotcomparableand theΠi’sgenerate

γ

c duetoEq. (8). ✷

The conjugatecapacity isthus such that

γ

c(A)=minn_i=₁Πi(A) whereΠi are theconjugatesof the Ni inducedbythe

maximalchainsinF (

γ

).LetC(

π

) bethesetofcutsof

π

.Ofcourse,C(

π

i)= F (Ni).Wecanshowthatthefocalsetsof

γ

arepreciselythecutsofall

π

i’sinR∗(

γ

c)(uptoset S).

Proposition8.F (

γ

)⊆S

π∈R∗(γc)C(

π

)⊆ F (

γ

)∪ {S}.

Proof. If E is focal for

γ

thenit appears in one maximal chainC_i henceit is acut of

π

i∈ R∗(

γ

c). Conversely, if E6=S

is acutof

π

∈ R∗(

γ

c), itisafocalset of some N induced byamax-chain C offocalsets of

γ

. If E=S isacut ofsome

π

∈ R∗(

γ

c),

π

correspondstoamaximalchainwhoselargest setG maydifferfrom S (when

γ

#(G)<1). ✷

Soitisinterestingtonoticethatthereisalmostanidentitybetweenthefocalsetsofacapacityandthecutsofthemost specificpossibilitydistributionsthat generateitsconjugate.

Example1 (continued).In Example 1,

γ

(A)=max(N1(A),N2(A))where N1(A) is inducedby

π

1(s)=1 if s=s1,

ν

(0.3)= 0.7 ifs∈ {s1,s2}\ {s1},i.e.,s=s2 and

ν

(0.7) otherwise,thatiss=s3.Wecheck thatF (N1)= {{s1},{s1,s2},S}arethecuts of

π

1.AndN2 isinducedby

π

2(s)=1 ifs∈ {s2,s3}and 0 otherwise. ✷

Oneoftherepresentationsof

γ

bymeans ofR_∗(

γ

) orR_∗(

γ

c₎ _may_contain_less _elements_than_the_other._For_instance,

if

γ

isanecessitymeasurebasedonpossibilitydistribution

π

,thenR_∗₍

_γ

c₎_{= {}

_π

_}_while_R

∗(

γ

)containsseveralpossibility

distributions(whosefocalsetsaresingletonsor S). NotethatΠ (A)≥N(A)= Πc(A),sothatitlooksmorenaturaltoreach

N frombelowandΠ fromabove.ForthecapacityinExample 1,weseethatitneedsthreepossibilitymeasurestorepresent it, but only two necessitymeasures areneeded. The results of thissection generalizeto several possibility measures (the leastonesthatdominate

γ

)theobviousremarkthatthecutsofapossibilitydistributioninducingapossibilitymeasureare thefocalsets ofthecorrespondingnecessitymeasure.

3.3. Sugenointegralsasupperorlowerpossibilisticexpectations

Thefact that aconvexcapacity (orabelieffunction)coincides withalower probabilitycan becarriedout tointegrals. Namely, theChoquet integral of areal-valued function with respectto a numericalconvex capacity is equalto the lower

(11)

expectationwithrespecttoprobabilitiesinthecredalsetinducedbythisconvexcapacity[10].However,thisisnottruefor any capacityand anyconvexprobability set.Inthissection,weexamine thesituationwithqualitativecapacitiesforwhich acounterpartofChoquetintegralexists,namelySugenointegral (see[29] foranoverviewofthisoperator).

Let f :S→L beafunctionthatmayserveasautilityfunctionif S isaset ofattributes.Sugenointegralisoftendefined asfollows[38]:

S_γ

₍

f

) =

max

λ∈L min

¡λ,

γ

(

Fλ

)

¢

(9) where Fλ= {s: f(s)≥ λ} isthe set of attributeshaving best ratingsfor some object, above threshold λ, and

γ

(A) isthe

degreeofimportanceoffeatureset A.

When

γ

isapossibilitymeasure Π,Sugenointegralsimplifies intothepossibilityintegral[23]: S_Π

₍

f

) =

max

s∈S min

¡

π

(

s

),

f

(

s

)

¢

(10)

whichistheprioritizedmax operator.

AnequivalentexpressionofSugenointegralis[29]: S_γ

₍

f

) =

max A⊆Smin

³

γ

(

A

),

min s∈A f

(

s

)

´

.

In thisdisjunctive form, the set-function

γ

can be replaced without loss of information by the inner qualitative Möbius transform

γ

# definedearlier.

S_γ

₍

f

) =

max A∈F (γ)

min

¡

γ

#

(

A

),

fA

¢

(11)

where fA=mins∈A f(s). This form makesit clear that itsimplifies into the possibility integral (10) if

γ

= Π. The above

expressionofSugenointegralhasthestandardmaxminformofapossibilityintegral,viewing

γ

#asapossibilitydistribution over2S.ThesimilarityofSugenointegralintheform(11) withthediscreteformofChoquetintegralusingMöbiustransform [27] ispatent.Andindeed,ithasbeenprovedelsewhere[13,14] thatitispossibletorefinetheorderinginducedbySugeno integral with respectto a qualitativecapacityon the set offunctions f by means ofa Choquet integral withrespectto a belieffunctionthat refinestheorderinginducedbythecapacity.

YetanotherequivalentexpressionofSugenointegralis[29]: S_γ

₍

_f

_{) =}

_min A⊆Smax

¡

γ

¡

Ac

¢,

fA

¢

(12) where fA=maxs∈A f(s). Likewise when

γ

is a necessity measure N = Πc, based on possibility distribution

π

, Sugeno

integral simplifiesintoanecessityintegral[23],asclearintheform(12): S_N

₍

f

) =

min

s∈S max

¡

ν

¡

π

(

s

)¢,

f

(

s

)

¢

(13)

whichistheprioritizedmin operator.Besidesitiseasytocheckthat theconjugacypropertyextendstoSugenointegralas S_γ

₍

f

) =

ν

¡

_S

γc

¡

ν(

f

)¢¢.

(14)

Asaconsequenceofresultsinthissection,itcanbeprovedthatSugenointegralisalowerprioritizedmaximumandan upperprioritizedminimum:

Proposition9.S_γ₍f)=inf_π∈R_∗_(γ₎S_Π₍f)=sup_π_∈R

∗(γc)SN(f).

Proof. Viewing

γ

as a lower possibility (Proposition 5) and using the fact that maxxminy f(x,y) is never greater than

minxmaxy f(x,y),itcomes: S_γ

₍

f

) =

max A⊆Smin

³

min π∈R∗(γ)

Π (

A

),

fA

´

=

max A⊆Sπ∈Rmin∗(γ) min

¡Π(

A

),

fA

¢

≤

min π∈R∗(γ) max A⊆Smin

¡Π(

A

),

fA

¢.

HencewehaveS_γ₍f)≤inf_π_∈R_∗_(γ₎S_Π₍f).

Conversely, let

π

f be the marginal of

γ

obtained from the nested sequence of sets Fλ induced by function f , then

it is clear that Πf(Fλ)=

γ

(Fλ), and thus Sγ(f)= SΠf(f). As ∃

π

∈ R∗(

γ

),

π

f ≥

π

, by definition, SΠf(f)≥ SΠ(f)≥

inf_π∈R_∗_(γ)SΠ(f). Finally, using conjugacy property (14), the other property showing that Sugeno integral is an upper

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Remark6. The obtained equality is not surprising because Sugeno integral with respect to a functioncoincides with the possibilityintegral withrespecttoaspecificmarginalpossibilitydistributioncomputedfromthepermutation

σ

f suchthat

f(s_σ_f₍1))≤ f(sσf(2))≤ · · · ≤ f(sσf(|S|)),just asthediscreteChoquet integralisaweighted arithmeticmeanwithrespectto

theprobabilitydistributioncomputedfromthecapacitywiththesamepermutation.

Thesignificanceoftheabove resultstems fromthefactthat iftheset ofminimalpossibility (resp.necessity)measures dominating (resp.dominated by)a qualitativecapacityis very small,thecomputationalcomplexity ofthecapacityis low. Namely, representing acapacityistheoreticallyexponential inthenumberofelements |S| of S,whilerepresentinga pos-sibility oranecessitymeasure islinear(wejust needtoknowthe|S|possibilityvaluesofsingletons).If R_∗(

γ

) containsk

distributions, onlyk|S| valuesare neededto represent

γ

.Adirect consequenceis tocut downthecomplexity theSugeno integral computation with respectto suchcapacities accordingly (fromexponential to linear),which may beinstrumental inoptimization problemsusingSugenointegralasacriterion.

4. Computingtheminimaldominatingpossibilitymeasuresinthequalitativeconvexcore

Inthissectionweconsidertheproblemofdeterminingtheminimalsetofn possibilitydistributions

π

ithataresufficient

togenerateagivenqualitativecapacity

γ

intheform

γ

=minn_i=₁Πi.Theaimofthissectionistoshowthat thequalitative

Möbiustransformisinstrumentalinfindingtheseleastelements.

4.1. Selectionfunctions

We needthe notion ofa selection functionfrom F (

γ

), theset of focalsets of

γ

. Aselection function sel: F (

γ

)→S

assigns to eachfocalset A anelement s=sel(A)∈A.Wedenote by Σ (F (

γ

)) theset ofselection functionswith domain F (

γ

). Given a capacity

γ

for any selection function on F (

γ

), one can define a possibility distribution

π

_selγ by letting max∅=0 and

π

_selγ

(

s

) =

max

E:sel(E)=s

γ

#

(

E

),

∀

s

∈

S

.

(15)

Note that itparallels a similarconstruction inthe theoryof belieffunctions, wherethe probability assignments pBel_sel(s)= P

A:sel(A)=sm(A), ∀s∈S turn out to bethevertices ofthe credalset inducedby Bel.In fact, similarresults for

π

γ sel asfor

then! possibilitydistributions

π

σγ canbeobtained. Note that if

γ

= Π,thenthere isonly onepossible selectionfunction

(sincefocalsets aresingletons)and

π

Π

sel=

π

(asrecalledabove inSection 2.1).

Proposition10.Foranyselectionfunctionsel inΣ (F (

γ

)),Π_selγ(A)≥

γ

(A),∀A⊆S. Proof.

Π

_selγ

(

A

) =

max s∈A

π

γ

sel

(

s

) =

max_s_∈_A _E_:_selmax₍_E₎₌_s

γ

#

(

E

)

≥

max

s∈A E⊆Amax:sel(E)=s

γ

#

(

E

) =

γ

(

A

)

sinceforafocalset E⊆A,sel(E)∈A too,then{E∈ F (

γ

)⊆A: ∃s∈A:sel(E)=s}= {E∈ F (

γ

):E⊆A}. ✷ Infact,notallpossibilitydistributions

π

_selγ areoftheform

π

σγ forapermutation

σ

.

Example2 (Counterexample). Suppose S= {s1,s2,s3},

γ

#({s1,s3})=1,

γ

#({s2,s3})= λ<1. Then consider sel({s1,s3})=s3 and sel({s2,s3})=s2, so that

π

_selγ(s3)=1>

π

_selγ(s2)= λ>

π

_selγ(s1)=0. Using the corresponding permutation, note that

π

σγ(s3)=1 but

π

σγ(s2)=

γ

({s1,s2})=0. However if we choose another selection function sel′ such that sel′({s1,s3})=

sel′({s2,s3})=s3,

π

_selγ′ correspondstotheabove

π

γ σ. ✷

Neverthelesswedohavethat theset {

π

_selγ :sel∈ Σ(F (

γ

))}alsoreconstructs

γ

.

γ

(A)=min_{sel∈Σ (F (γ}))Πselγ(A).

Proof. Π_selγ(A)=maxs∈AmaxE:sel(E)=s

γ

#(E).Togettheequality,let E⊆A besuchthat

γ

(A)=

γ

#(E).Choosetheselection function as follows: assign

γ

#(E) to an element sE ∈E; then if

γ

#(F)>

γ

#(E), assign

γ

#(F) to some element not in A. Thisispossible,becauseif

γ

#(F)>

γ

#(E),thenF *E,and since

γ

#(F)>

γ

(A), F*A either.Forsuchaselectionfunction,

Π_selγ(A)=

π

_selγ(sE) holds since the only elements in A to which a possibility weight is assigned are sE ∈E, and possibly

other sC∈C∩A6= ∅,suchthat

γ

#(C)<

γ

#(E). So

γ

(A)≥minsel∈Σ (F (γ))Πselγ(A) anddue toProposition10,theconverseis

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4.2. Analgorithmbasedonusefulselectionfunctions

DuetoProposition 11,thesetofminimalelements(maximallyspecific)ofR(

γ

)isalsoincludedin{

π

γ

))}. Butnotallselectionfunctionsareofinterestsincewecanhaveredundantpossibilitydistributions.

Example3.We consider acapacity

γ

suchthat the set offocalsets has acommon intersection I. Wedenote E thefocal set suchthat

γ

#(E)=1 and F afocalset suchthat

γ

#(F)<1.

Wecandefineapossibilitysuchthat

π

_selγ(s∗₎₌_{1 for}_some _s∗_∈_I_⊆_{E and}

_π

γ

sel(s)=

γ

#(F) forsome s∈F \I. Butwecan

defineamorespecificpossibilitydistributiondominating

γ

:

π

_selγ(s∗₎₌_{1 for}_some_s∗_∈_I_⊆_{E and}_{0 otherwise.} _✷

Definition7.Aselectionfunctionsel issaidtobeuseful foracapacity

γ

ifitisaminimalelementin{

π

γ

))}. Letusconsiderthefollowingalgorithmtocalculateselectionfunctionssel andtheassociatedpossibilitydistribution

π

_selγ. Algorithm MSUP:Maximalspecificupperpossibilitygeneration

Input: n focalsets Ek k=1,. . . ,n rankedindecreasingvaluesof

γ

#(Ek)

i.e.,

γ

#(E1)≥ · · · ≥

γ

#(En) F ← {E1,· · · ,En}

foralls∈S let

π

(s)=0

repeat

E←thefirstelementofF

Define sel(E)=s forsomes∈E

Let

π

(s)=

γ

#(E)

Delete E fromF

T ← {G∈ F such that s∈G}

repeat

G←thefirstelement ofT Definesel(G)=s

DeleteG fromT DeleteG fromF.

untilT = ∅;

untilF = ∅;

Result: Apossibilitydistribution

π

_selγ.

In fact theabove algorithm appliesthe trick inthe proof ofProposition 11:the value

γ

(A)=

γ

#(Ej) is retrievedbya

possibility distribution wheretheweights

γ

#(Ek),k< j areassignedoutside A (which isoneoption offeredbythe above

procedure). A selection function sel thus generated satisfies the following property: If sel(E)=s for some E∈ F (

γ

) and

π

sel(s)=

γ

#(E),then∀F∈ F (

γ

),suchthats∈F :

• if

γ

#(E)≥

γ

#(F)thensel(F)=s

• if

γ

#(E)<

γ

#(F) thensel(F)6=s.

Conversely, apossibilitydistribution

π

selsatisfyingtheabove propertiesisapossiblesolutionforthealgorithmMSUP.

Let Σ∗(F (

γ

))betheset ofselectionfunctionsgeneratedbyrepeatedapplicationsofalgorithm MSUP,and MSUP(

γ

)be

thecorrespondingsetofpossibilitydistributions.

Notethatwhilenotallselectionfunctions

π

selyieldapermutation

σ

suchthat

π

sel=

π

σ,thisistrueforusefulselection

functions. Thisis due to the orderingof elements generatedby theprocedure from great to small masses

γ

#(Ej):define

σ

from the sequence of elements sj obtained by one application of the algorithm constructing sel. For

π

sel, it is clear

that

π

sel(sσ(i))≤

π

sel(sσ(j)),∀i> j.Hence

π

σ(i)=

γ

(Sσi )= Πsel(Sσi )=

π

sel(i). Moreover,if two possibilitydistributionsare

generated bythealgorithm,theycorrespondtodifferentpermutations (sinceeachtime,differentelements receivepositive masses).So weconclude thatMSUP(

γ

)isasubsetof{

π

σ :

σ

permutation of S}.

Proposition12.If

π

6=

ρ

∈MSUP(

γ

),thenneither

π

>

ρ

nor

π

>

ρ

hold.

Proof. Suppose

π

>

ρ

generated byalgorithm MSUP, i.e.,

π

≥

ρ

and lets be such that

π

(s)>

ρ

(s). Let sel_π and sel_ρ be the selection functions associated with

π

and

ρ

, respectively. Let E be such that sel_π(E)=s and

π

(s)=

γ

#(E)>

ρ

(s). By construction, sel_ρ(E)=s′₆₌_s. _We _have

_ρ

₍_s′₎_≥

_γ

#(E)>

ρ

(s), so

π

(s′)≥

ρ

(s′)>0 and ∃G, such that selπ(G)=s′ and

π

(s′₎₌

_γ

#(G). But since by construction, s′∈E∩G, it follows that

γ

#(G)≥

γ

#(E) and the algorithm MSUP would en-force sel_π(E) =s′_, _which _contradicts _the _assumption _that _sel

π(E)=s. So

π

≥

ρ

with

π

(s)>

ρ

(s) is in conflict with the

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SothepossibilitydistributionsobtainedwiththealgorithmMSUParenotpairwisecomparable(theyarebasedonuseful selections).

Example 4. Let S= {s1,s2,s3,s4} and the focal sets F (

γ

)= {{s1,s2},{s1,s3},{s2,s3},{s2,s4}} where

γ

#({s1,s2})=1>

γ

#({s1,s3})= .8>

γ

#({s2,s3})= .4>

γ

#({s2,s4})= .2. We show allpossible applications of the MSUP algorithm, starting

withF = F (

γ

).

• sel({s1,s2})=s1,

π

(s1)=

γ

#({s1,s2})=1,T = {{s1,s3}},and F = {{s2,s3},{s2,s4}}.

– sel({s2,s3})=s2,

π

(s2)=

γ

#({s2,s3})= .4,T = {{s2,s4}},and F = ∅.

So,

π

(s1)=1,

π

(s2)= .4,

π

(s3)=

π

(s4)=0.Thepermutation

σ

canbe(1,2,3,4).

– sel({s2,s3})=s3,

π

(s3)=

γ

#({s2,s3})= .4,T = ∅and F = {{s2,s4}}.

∗ sel({s2,s4})=s2,

π

(s2)=

γ

#({s2,s4})= .2,F = ∅.

So,

π

(s1)=1,

π

(s2)= .2,

π

(s3)= .4,

π

σ

is(1,3,2,4).

∗ sel({s2,s4})=s4,

π

(s4)=

γ

#({s2,s4})= .2,F = ∅.

So

π

(s1)=1,

π

(s2)=0,

π

(s3)= .4,

π

(s4)= .2 Thepermutation

σ

is(1,3,4,2).

• sel({s1,s2})=s2,

π

(s2)=

γ

#({s1,s2})=1,T = {{s2,s3},{s2,s4}}and F = {{s1,s3}}.

– sel({s1,s3})=s1,

π

(s1)=

γ

#({s1,s3})= .8,F = ∅.

Wehave

π

(s1)= .8,

π

(s2)=1,

π

(s3)=

π

σ

canbe(2,1,3,4).

– sel({s1,s3})=s3,

π

(s3)=

γ

#({s1,s3})= .8,F = ∅.

Wehave

π

(s1)=0,

π

(s2)=1,

π

(s3)= .8,

π

σ

canbe(2,3,4,1). ✷

Moreoverthefamilyofpossibilitydistributionswhich canbeobtained withthealgorithmMSUPisnecessaryand suffi-cienttoreconstructthecapacity.Inordertoprovethismainresultthefollowingoneisneeded.

Proposition13.Foranypermutation

σ

,thereexistsaselectionfunctionsel correspondingtoanotherpermutation

τ

suchthat

π

σ ≥

π

τ =

π

sel.

Proof. Let

σ

beapermutationof S,andwedenotethecorrespondingelementsassi forsimplicity.Let

π

(si)=

γ

(Si)where

Si= {si,. . . ,sn}.Letusshowthat wecanderive ausefulselectionfunctionfrom

σ

.For eachsi weshalldecidewhetheror

nottomakeitaselectionfrom E∈ F (

γ

).Let U bethesetofelementsnotyetdecided,andF thecurrentset offocalsets.

• Let U =S and F = F (

γ

). If

π

σ(sj)=0 for j=i,. . . ,n, theelements sj must be discarded from S and will not be

selectedfromfocalsets.SetU=: {s1, . . . ,si−1}.

• For j=i−1 downto1

– Select E∈ F such that sj∈E, E⊆Sj,

γ

#(E) ismaximal,and letsel(E)=sj.If thereisnone, let

π

sel(sj)=0.Delete

sj from U .

– Letsel(F)=sj forall F suchthat sj∈F and F⊆Sj and

γ

#(F)≤

γ

#(E),anddeletesuch F fromF.

– Set j=: j−1.

Itisclearthat theobtainedselectionfunctionsel is useful.Moreover, itisalsoobviousthat

π

sel(si)≤

π

(si) since

π

sel(si)=

γ

#(Ei)forsome Ei⊆Si.Nowifweusethepermutation

τ

suchthat

π

sel(sτ(i))≤

π

sel(sτ(j))wheneveri> j,thenwehave

that

π

sel=

π

τ (this isbecause

π

sel(sτ(i))<

π

sel(sτ(j))ifandonlyif

γ

#(Ei)<

γ

#(Ej)). ✷

Example5(Generationofausefulselectionfromapermutation).Consider S= {s1,· · · ,s6}.Let

γ

besuchthat

γ

#({s1,s2})=1>

γ

#({s2,s3})= .8>

γ

#({s5,s6})= .7>

γ

#({s3,s4,s5})= .6>

γ

#({s1,s6})= .5.

Consider the permutation

σ

= (s6,s5,s4,s3,s2,s1). It is clear that

π

σ(si)=1, ∀i<6 and

π

σ(s6)=0. This possibility

distributionisaverylooseupperboundof

γ

.Letusfindamoreinformativepermutationviaaselection. WehaveU=S andF = {{s1,s2},{s2,s3},{s5,s6},{s3,s4,s5},{s1,s6}}.

1. Firstwecandelete s1 fromU and

π

sel(s1)=0; U= {s2,· · · ,s6}.

2. Thenweselect E= {s1,s2} andsel({s1,s2})=s2.Wealsoletsel({s2,s3})=s2.

NowU= {s3,s4,s5,s6} andF = {{s5,s6},{s3,s4,s5},{s1,s6}}.

3. Wecanselectnofocalset in S3= {s1,s2,s3},norin S4= {s4,s3,s2,s1}.So

π

sel(s3)=

π

sel(s4)=0.

NowU= {s5,s6}and F = {{s5,s6},{s3,s4,s5},{s1,s6}}.

4. Thenweselect E= {s3,s4,s5}⊂S5= {s1,· · · ,s5} andsel({s3,s4,s5})=s5.

NowU= {s6}andF = {{s5,s6},{s1,s6}}.

5. Finallyselect E= {s5,s6} andletsel({s5,s6})=sel({s1,s6})=s6.