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Qualitative Possibility Theory and ItsApplications to Constraint Satisfactionand Decision under UncertaintyDidier Dubois and Henri Prade*

( )Institut de Recherche en Informatique de Toulouse IRIT ] CNRS, UniversitePaul Sabatier, 118 route de Narbonne, 31062 Toulouse Cedex 4, France

This paper provides a brief survey and an introduction to the modeling capabilities ofqualitati e possibility theory in decision analysis for the representation and the aggrega-tion of preferences, for the treatment of uncertainty and for the handling of situationssimilar to previously encountered ones. ‘‘Qualitative’’ here means that we restrict

Žourselves to linearly ordered valuation sets only the ordering of the grades is meaning-.ful for the assessment of preferences, uncertainty and similarity. Moreover, all the

Ž .evaluations refer to the same valuation set commensurability assumption . Such aqualitative structure is poor but not very demanding from an elicitation point of view;however, it is sufficient for giving birth to a valuable set of modeling tools. Q 1999 JohnWiley & Sons, Inc.

1. INTRODUCTION

Fuzzy sets and possibility theory provide a rich framework for the represen-tation of graded notions, which seem to be particularly useful in informationengineering.1 Among the applications in this field, decision-making problemsoffer a vast range of issues which can greatly benefit from the fuzzy sets andpossibility theory methodology, such as individual, or collective, preferencemodeling, decision under uncertainty, or case-based decision. Among manynoticeable contributions, let us especially mention works on

v 2 ] 4binary relation-based approaches to preference modeling;v Ž .aggregation of value functions representing criteria , including the case of fuzzy

utility grades;5] 9

v 10group decision-making;v 11,12decision under uncertainty with fuzzy probabilities.

� 4*E-mail: dubois, prade @irit.fr

Ž .INTERNATIONAL JOURNAL OF INTELLIGENT SYSTEMS, VOL. 14, 45]61 1999Q 1999 John Wiley & Sons, Inc. CCC 0884-8173r99r010045-17

DUBOIS AND PRADE46

All these different works exploit a numerical modeling of preferences andw xuncertainty by means of the valuation set embodied by the real interval 0, 1 , or

w xmore generally by means of fuzzy grades modeled by fuzzy subsets of 0, 1 .In the following we are going to consider representations which are based

on fuzzy sets and possibility distributions defined by means of a linearly orderedvaluation set, finite or not, where only the ordering between the grades ismeaningful. Let us mention the pioneering work of Yager13 in that respect.

�However, as an example, the finite ordered set 0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7,40.8, 0.9, 1 can be used as a qualitative valuation set, but the numerical encoding

is then just a matter of convenience. With such a valuation set S, fuzzy setintersection, union and complementation can be only defined by min, max, and

Žthe order-reversing operation of the valuation set that we shall continue toŽ .denote by 1 y ? in the following; 1 and 0 will still denote the top and the

.bottom elements of the valuation set, respectively .w xEven if this structure is clearly poorer than the interval 0, 1 w.r.t. modeling

issues, the potential of this simple framework is still worth investigating fordecision analysis, since it is also less demanding as to the quality of the datawhich are necessary to feed it.

However, as in the case of numerical membership functions, we can stilldistinguish between three basic semantics for gradedness: preference, uncer-tainty, and similarity.14 These three semantics are clearly useful in decisionanalysis and induce the structure of the paper made of three main sections, 3, 4,and 5, after a short background on qualitative possibility theory in the followingsection. This article is a revised and expanded version of an invited conferencepaper.15

2. QUALITATIVE POSSIBILITY THEORY

There are two distinct understandings of a possibility distribution p , i.e., amapping from a universe U to a bounded linearly ordered valuation set S, such

Ž . Ž Ž ..that p u* s 1 for at least one u* the most plausible, or usual value s ;Ž .p u s 0 means that u is an impossible situation. First, a possibility distribution

may encode a piece of imprecise knowledge about a situation, as in approximatereasoning.16 Another view is in terms of preference and leads to a formalframework for flexible constraint satisfaction17 based on the calculus of fuzzyrelations.18

Possibility theory is driven by the principle of minimal specificity. A possibil-ity distribution p is said to be at least as specific as another p 9 if and only if for

Ž . Ž .each state u, p u F p 9 u . Then, p is at least as restrictive and informative asp 9 in the sense that each value u is considered at least as plausible by p 9 as p ,so that p tends to eliminate more candidate values than p 9. Given a set ofconstraints restricting a feasible subset of possibility distributions, the bestrepresentative is the least specific feasible possibility distribution, which assignsthe highest degree of possibility to each state, since it is also the least committed

Ž .one. Hence, if a possibility distribution p resp., p is attached to variable xx yŽ . Ž .resp., y we have p s min p , p if no other constraints exist linking x andŽ x, y. x y

QUALITATIVE POSSIBILITY THEORY 47

y. Then x and y are said to be noninteractive. Generally, only the inequalityŽ .p F min p , p holds.Ž x, y . x y

Let x denote the variable fuzzily restricted by the possibility distribution p .Given a simple query of the form ‘‘does x belong to A?’’ where A is aprescribed subset, the response to the query can be obtained by computing towhat extent

v Ž . Ž .A is consistent with p , by the possibility degree Ł A s sup p u ,ug Av Ž . Ž .A is certainly implied by p , by the necessity degree N A s 1 yŁ A s

Ž . Ž .inf 1 y p u A is the complement of A .uf A

The basic axiom of possibility measures is

;A , ;B ,Ł A j B s max Ł A ,Ł BŽ . Ž . Ž .Ž .

Ž . Ž . Ž .By convention, ;A, Ł A g S, Ł U s 1, and Ł B s 0. Set functions Ł andN are simple qualitative representations of graded uncertainty. Their particularcharacter lies in their qualitative nature, i.e., the valuation set S is used only forrank-ordering the various possible situations in U, in terms of their compatibility

Žwith the normal course of things, or the expressed preferences depending on.the interpretation , as encoded by the possibility distribution p . Necessity

measures satisfy an axiom dual of the one of possibility measures, namely,

N A l B s min N A , N BŽ . Ž . Ž .Ž .

When dealing with representation of uncertainty, a piece of fuzzy knowl-Ž .edge as given for instance in natural language pertaining to a given universe of

Ž .discourse U, a set of states other would say ‘‘set of possible worlds’’ , expressesan elastic constraint on this universe, that can be encoded as a possibilitydistribution p . When addressing queries in face of such pieces of knowledge,the corresponding responses can be given in terms of possibility-qualified andcertainty-qualified statements. In that case, no choice is at stake, that is, theactual situation is what it is, and p encodes a plausibility ordering on what the

Ž . Ž .real situation may be: p u ) p u9 means then that u is less surprising than u9Ž .as describing the real state of the world. Ł A qualifies the degree of possibility

of A, defined by assuming that, if it is only known that A occurs, then the mostplausible situation compatible with A prevails. For instance, the degree ofpossibility of the event ‘‘the light is off’’ is computed after the degree ofpossibility of ‘‘someone switched off the light’’ rather than ‘‘the bulb is broken,’’insofar as the latter is very unusual compared to the former.

A systematic assumption in possibility theory is that the actual situation isŽ .normal, i.e., it is any u such that p u is maximal given other known constraints.

Ž .It justifies the evaluation Ł A , and contrasts with the probabilistic evaluationof the likelihood of events. Here, ‘‘possible’’ means ‘‘unsurprising.’’ MoreoverŽ .N A ) 0 means that A holds in the most normal situations. Since the assump-

Ž .tion of normality is always made, N A ) 0 means that A is an accepted belief,i.e., one may act as if A were true. This assumption is always a default one and

DUBOIS AND PRADE48

Ž .can be revised if further pieces of evidence contradict it. N A G a ) 0 meansthat the most plausible situation where A is false is rather impossible, i.e., notpossible to a level greater than 1 y a .

When modeling controllable situations, the meaning of possibility has to doŽ . Žwith feasibility objective interpretation or preference subjective interpreta-

.tion . Then a possibility distribution encodes a flexible requirement, namely,Ž .how we would like the world to be, p u s 1 means that u is a preferred choice,

Ž . Ž .and p u ) p u9 means that u is a better choice than u9. The axiom ofpossibility measures means that if A or B is to be achieved, it is equivalent to

Ž .achieve the easiest of the two. In that context N A s a means that, accordingto the preference profile the satisfaction of the constraint ‘‘x g A’’ has a prioritylevel a , where x is the decision variable associated with the possibility distribu-tion defining N.

3. CONSTRAINTS AND PREFERENCES

3.1. Flexible Constraints

The notion of constraint is basic in operations research. A constraintŽdescribes what are the potentially acceptable decisions the solutions to a

.problem and what are the absolutely unacceptable ones: it is an all-or-nothingmatter. Moreover, no constraint can be violated, i.e., a constraint is classicallyconsidered as imperative. Especially the violation of a constraint cannot becompensated by the satisfaction of another one. If a solution violates a singleconstraint, it is regarded as unfeasible. The idea of flexible constraints is to keepthe noncompensatory property of constraints, while introducing intermediarylevels between feasibility and nonfeasibility as well as levels in the imperative-ness of constraints.

A classical hard constraint C is represented by a classical set of solutions,i.e., only using degrees of membership 0 or 1. However, since eventually a singlesolution will be picked up, the feasible solutions are mutually exclusive and thecharacteristic function m that is attached to the constraint is a binary possibil-City distribution p , where x is a vector of decision variables. Introducingxintermediary levels of feasibility on a linearly ordered valuation set S, C is then

Ž .called a fuzzy, or soft constraint: m u s 1 means that a solution u totallyCŽ . Ž .satisfies C while m u s 0 means that it totally violates C u is unfeasible . IfC

Ž . Ž . Ž .0 - m u - 1, u satisfies C only partially; m u ) m u9 indicates that C isC C Cmore satisfied by u than by u9. Hence, like an objective function, a fuzzyconstraint rank-orders the feasible decisions. However, contrary to an objective

Žfunction a fuzzy constraint also models a threshold represented by the bottom.level 0 beyond which a solution will be rejected. In fact, a fuzzy constraint can

Ž .be viewed as the association of a constraint defining the support of C and acriterion which rank-orders the solutions satisfying the constraints. In thisinterpretive framework, a membership function, construed as a possibility distri-bution, is similar to a qualitative utility function, or better, a value function. A


soft constraint C9 will be looser than another one, C if and only if m F m ,C C 9

that is, if any solution to C is at least as feasible for C9.

3.2. Prioritized and Discounted Constraints

Here, possibility distributions encode feasibility profiles of soft constraints.In the presence of very loose constraints, it is clear that the use of softconstraints can help breaking ties among feasible solutions, just as objectivefunctions do. On the contrary when hard constraints are tight, there is nofeasible solution. One way out is to relax the constraints, but automating thisprocess is not so easy and is usually time-consuming. What is usually done is toassign priorities to constraints. Solutions that satisfy all constraints are selectedif any. Otherwise the less prioritary constraints are dropped, and the chosensolution satisfies only higher priority constraints. This decision strategy can becaptured in the setting of possibility theory.

A priority level r for constraint C can be modeled by a degree of necessityŽ .estimating how imperative C is. Indeed, the pair C, r , where C is a crisp

constraint, can be represented by a special kind of fuzzy constraint C*,17

m u s 1 if u satisfies CŽ .C*

s 1 y r if u violates C

Ž .Then it can be checked that N C s r where N is defined from p s m . SinceC*r represents to what extent it is necessary to satisfy C, 1 y r indicates to whatextent it is possible to violate it. In other words, any potential solution u that

Ž .violates C satisfies C, r to a degree equal to 1 y r. When possibility degreesare interpreted in terms of feasibility, necessity degrees are thus levels ofpriority. If C is itself a soft constraint with priority r, it can be modeled by thefuzzy constraint C* defined by

m u s max 1 y r , m u 1Ž . Ž . Ž .Ž .C* C

by analogy with certainty qualification of fuzzy events.19

Ž . Ž .However, as indicated by 1 , compatibility of u with C, r is 0 only ifŽ .r s 1 and m u s 0, i.e., if constraint C is not satisfied at all but is fullyC

Ž . Ž .important. Moreover, if r s 0 C has no importance at all , 1 yields the topŽ .element 1 of the valuation set for any u, as expected. However, as shown by 1 ,

even if C is completely violated, the global level of satisfaction is only upperŽ .bounded by 1 y r because of this violation instead of 0 in case C is imperative .

ˆIn practice, it may be useful in this case to add a ‘‘safeguard’’ constraint C incase some decision among the ones which violates C are really inacceptable,

Ž .thus changing 1 into

m u s min m u , max 1 y r , m u 2Ž . Ž . Ž . Ž .Ž .Ž .ˆC* C C

There is a different way of relaxing a constraint,20 namely, by consideringthat the constraint is sufficiently satisfied if the level of satisfaction for u

DUBOIS AND PRADE50

Ž .reaches some threshold u , i.e., m u G u . Then, we shall speak of a discountedCŽ . Ž .constraint. Then m u will be changed into an evaluation m u s 1. IfC C 9

Ž .m u - u we may consider that the constraint is satisfied at the level which isCŽ . Ž . wactually reached, i.e., m u s m u the use of a relati e level of satisfactionC 9 C

Ž . w xm u ru would require a numerical valuation set like 0, 1 and not a simpleCxlinearly ordered valuation set . Then we can define

m u s u ª m u 3Ž . Ž . Ž .C 9 C

Ž .where a ª b is Godel implication a ª b s 1 if a F b, a ª b s b if a ) b .¨Generally speaking, the relaxing of a fuzzy constraint C by an importance

degree a g S, can be viewed as the modification of m into a H m where HC CŽ .should be such that, letting m u s sC

Ž . Ž .a ;s, a H s G s relaxation ; thus a H 1 s 1;Ž . Ž .b 1 H s s s and 0 H s s 1 limit casesŽ .c a H s increases with s; thus ;s, a H s G a H 0Ž . Ž .d a H s decreases from 1 to s when a increases.

Ž . Ž . Ž .a and c entail a H s G max a H 0, s . This suggests an implication connectivefor H since a H 0 is like a negation function. On the kind of qualitativevaluation set considered here, we may thus either take

v Ž . Ž .a H s s max 1 y a , s Dienes implication , adding the further requirement thatŽ .a H 0 strictly decreases with a and thus a H 0 s 0 m a s 1, as in 1 , or

v Ž .a H s s a ª s when ª is Godel implication, as in 3 with the further require-¨ment that a F s « a H s s 1.

Yager 21 already noticed the role of implications in the weighing process ofconjunctive operations. We might also think of using the implication obtained by

Ž .contraposition from Godel implication, letting m u s 1 y u if the threshold¨ C 9

Ž Ž . .is not reached m u - u . However, this no longer satisfies the requirementCŽ . Ž . Ž .a stating that m u G m u , ;u. The same problem arises with a ª s sC 9 C

Ž Ž . .max min a , s , 1 y a .

3.3. Satisfying Most of the Constraints

The conjunctive aggregation of prioritized fuzzy constraints C , namely,i

m u s min max 1 y r , m uŽ . Ž .Ž .D i Ciis1, m

can be used as the starting point for modeling the idea of satisfying ‘‘most’’ ofthe constraints under consideration, in a qualitative way. By ‘‘delocalizing’’ the

Ž .weights r in the above expression we can turn it into a fuzzily quantifiediconjunction, corresponding to the requirement that a solution u satisfies ‘‘at

Žleast k,’’ or more generally most constraints rather than ‘‘all’’ the constraints or


.more generally all the important ones . This can be done in the following wayŽ .see, e.g., Ref. 22 :

Ž . Ž .i rank-order the degrees m u s c decreasingly, in order to only consider theC iibest satisfied constraints in the weighting process, i.e., c G c G ??? G cs Ž1. s Ž2. s Žm.

� 4where s is a permutation of 1, . . . , m ;Ž . � 4 Ž .ii let I be a fuzzy subset of the set of integers 0, 1, 2, . . . , m s.t. m 0 s 1,I

Ž . Ž .m i G m i q 1 . For instance, the requirement that at least k constraints areI IŽ .important will be modeled by k weights equal to 1, i.e., r s m i in thei I

Ž . Ž .prioritized conjunction with m i s 1 if 0 F i F k, m i s 0 for i G k q 1;I IŽ .iii the aggregation operation is then defined by

m u s min max 1 y m i , cŽ . Ž .Ž .ŽC , . . . , C . I s Ž i.1 m is1, m

Ž . Ž .When m i s 1 for 0 F i F m, it reduces to m u s c sI ŽC , . . . , C . s Žm.1 mŽ . Ž . Ž . Ž .min m u as expected. When m 1 s 1 and m 2 s ??? s m m s 0, itis1, m C I I Ii Ž . Ž .reduces to m u s c s max m u .ŽC , . . . , C . s Ž1. is1, m C1 m i

The above expression can be easily modified for accommodating relativeŽ . Ž .quantifiers Q like most, by changing 1 y m i q 1 into m irm for i s 0,I Q

Ž . Žm y 1 and m 1 s 1 where m is increasing a required proportion of at leastQ Q.krm amounts to have k nonzero weights among m . What has been computed

Ž .here is an ordered weighted minimum operation OWmin , or if we prefer, theŽ .median of the set of numbers made by the c s and the 1 y m i s. See Ref. 22.s Ž i. I

OWmin can thus be related to the idea of fuzzy cardinality. It is also a Sugeno44

integral. However, there is no compensatory effects as opposed to orderedweighted averages.8

3.4. Conjunctive Aggregation and Optimization

The notion of fuzzy set as viewed by Bellman and Zadeh23 means torepresent constraints as well as objective functions by fuzzy subsets C ofpossible decisions. If C , . . . , C denote m fuzzy constraints, the fuzzy decision1 mset is then defined by

m u s min m u 4Ž . Ž . Ž .D Ciis1, m

However, this definition of a decision set is not really in accordance withthe usual paradigm of multiple-criteria decision-making since an optimal solu-tion in the sense of Bellman and Zadeh does not make a trade-off between the

Ž .membership values m u . On the contrary an optimal solution is one that leastCi

violates the most violated constraint. Hence, Bellman and Zadeh’s proposal is aconstraint-directed view of problem-solving, where constraints are flexible: ifthere is a solution u that completely satisfies all constraints, this solution is

Ž .optimal. Otherwise l* s sup m u - 1, and this indicates that the constraintsu DŽ .are partially contradictory. To accept a solution u such that m u s l* meansD

wto partially relax some of the constraints C with respect to the constrainti� Ž . 4xcorresponding to the core of C , i.e., u, m u s 1 . In that sense constraints Ci C ii

are flexible.

DUBOIS AND PRADE52

This notion of optimization is not new in the literature of decision-making.It is known as bottleneck optimization. It is well-known in decision underuncertainty as the pessimistic Wald criterion, and in game theory as providingthe most cautious strategy for players. Despite these formal analogies, theproblem tackled by flexible constraint satisfaction is very different, since index ipertains neither to an opposing player nor to a state of the world. The paradigmmost akin to Bellman and Zadeh’s is to be found in social choice theory 24 and

Ž . Ž .min m u is then an egalitarist social welfare function, m u being theis1, m C Ci iŽ .welfare index of an individual i. Maximizing min m u tends to selectis1, m Ci

solutions which equally satisfy all the constraints.ŽLike averaging operations, the minimum operation as well as the DeMor-

.gan dual disjunctive connective ‘‘max’’ can be weighted. In the qualitativesetting, thresholding constraints in order to acknowledge their respective levels

wof importance may also help breaking ties between possible decisions especiallyŽ . xwhen ;u, min m u s 0 . Importance is here an attribute attached to eachi C i

constraint, either in terms of priority or of discount as seen in Section 3.2. Thus,relaxing constraints comes down to assuming that all the constraints do not havethe same importance; it can be done in different ways in the context ofconjunctive aggregation.

An example of thresholded min aggregation is thus obtained as the min-based conjunction of prioritized constraints,

m u s min max 1 y r , m u 5Ž . Ž . Ž .Ž .D i Cii

Ž . Ž .Note that now m u G min m u , which indeed expresses a relaxation.D is1, m Ci

The convention max r s 1 is assumed in order to have an appropriateis1, m iscaling of the levels of importance. Denoting by WW the fuzzy set of important

w Ž . x Ž .constraints i.e., m C s r , 5 expresses a strong type of inclusion of WW intoWW i iw Ž . Ž .xthe fuzzy set of criteria satisfied by choice u, SS where m C s m u , basedu SS i Cu i

Ž .on Dienes implication a ª b s max 1 y a, b .Ž . Ž . Ž . Ž . Ž .With 3 instead of 1 in 5 , we still have m u G min m u .D is1, m Ci

Ž . Ž .However, it contrasts with the use of Dienes implication in 5 , and m u s 1DŽ . wm ; i, m u G u expresses an ordinary fuzzy set inclusion of WW into SS withC i ui

Ž . xm C s u .WW i iThe maxmin ordering of solutions is coarse but can be refined on a

qualitative basis either by focusing on the least satisfied discriminating con-Ž .straint the ‘‘discrimin’’ ordering or using lexicographical techniques such as the

leximin-ordering.25

The Pareto partial ordering corresponds to the notion of fuzzy set inclusion:Ž . Ž .s G t m SS : SS where s s s , . . . , s with s s m u and t is similarlyPareto ¨ u 1 m i Ci

Ž Ž ..defined changing u into ¨ t s m ¨ . It is not a refinement of the min-orderingi C iŽ .) although there always exists a Pareto-maximal, maximin solution. More-minover the discrimination power of the min-ordering is low as already said, whilethe Pareto-ordering leads to incomparabilities. Keeping in mind the idea ofinclusion, a refinement of these two orderings can be defined by comparing the


decisions on the basis of level-cut inclusion. Namely, we consider a newordering, ) , defined in the finite case byDiscrimin

s ) t iff 'a g S s.t.Discrimin

Ž . Ž . Ž .i ;b g s.a. b - a , SS s SS ;u b ¨ bŽ . Ž . Ž .ii SS > SS ,u a ¨ a

Ž . � Ž . 4where SS s C g CCrm w G g is a level-cut of SS . The relation )w g i C w Discrimini

is irreflexive and transitive: it is a strict partial ordering. It can be verified thatany discrimin-maximal solution is both Pareto-optimal and min-optimal. The useof ) consists in practice in comparing pointwise satisfaction degrees likeDiscriminin ) , and focusing on the lowest satisfaction degrees among the con-Paretostraints satisfied at different degrees by the competing decisions: decisions arecompared on the basis of the least satisfied discriminating constraints. Thisseems to be in accordance with the intuition, since only the constraints making adifference between two alternatives can help make a final decision.

Ž . � 4Let us note DD s, t s C rs / t the set of constraints which are satisfiedi i iby s and t to a different extent. An equivalent characterization of ) isDiscrimin

s ) t m min s ) min t 6Ž .Discrimin i iŽ . Ž .C gDD s, t C gDD s, ti i

The ‘‘discri-min’’ refinement is based on the idea that the constraints on whichtwo decisions receive the same evaluation, have no importance when comparingthe decisions.

Noticeably, the discrimin ordering satisfies a cancellation law that theminimum ordering does not satisfy. Let s and t be two vectors of n components,and let r be a vector of k components. Denote rs and rt the vector of n q kcomponents obtained by concatenation of r and s, and r and t, respectively.Then s ) t m rs ) rt.Discrimin Discrimin

Another refinement of the egalitarist maximin ordering has been proposedŽ .e.g., Ref. 24 : the leximin-ordering. The idea is to keep all the informationpertaining to SS , instead of summarizing it by a collective utility function as withu) , by considering a ranked multiset of satisfaction degrees. Let s be a vectorminof satisfaction degrees, and denote by s* another such vector such that s* s si s Ž i.where s is a permutation such that s F s F ??? F s obtained bys Ž1. s Ž2. s Žn.reordering the components of s. Indeed, two multisets built on an orderedreferential can always be compared using the leximin-ordering ) :Lex

v s ) t iff 'k F m such that ; i - k, s* s t* and s* ) t* ,Lex i i k k

Ž . Ž .where s * resp., t * stands for the ith component of s* resp., t* . The twoi ipossible decisions are indifferent if the corresponding multisets are the same.

DUBOIS AND PRADE54

The leximin-ordering is a refinement of the Pareto-ordering, the min-order-ing, and the discrimin ordering:

v s ) t « s ) tmin Lexv s G t « s ) tPareto Lexv s ) t « s ) t.Discrimin Lex

Leximin optimal decisions are always discrimin optimal decisions, and thusindeed min-optimal and Pareto-optimal: ) is the most selective among theseLexpreference relations. Converse implications are not verified. The leximin-order-ing can discriminate among equivalent decisions according to the min-preorder-

Ž .ing min s s min t and among incomparable ones with respect to ) andi i Pareto) . The leximin-ordering tends to favor solutions that violate as fewDiscrimin

Žfuzzy constraints as possible where ‘‘few’’ refers to a fuzzy cardinality evalua-.tion; see Ref. 25 . The leximin-ordering also satisfies the same cancellation

property as the discrimin ordering. Moreover the leximin-ordering on a finite setof vectors can always be represented by means of an ordered weighted averageŽ .in the sense of Ref. 8 as shown in Ref. 20. Leximin optimal solutions are alsoobtained as the limit of sequences of optimal solutions to aggregation problemsinvolving triangular norms, generalized means, and ordered weighted averages.26

See Dubois and Fortemps27 for classes of problems where the discriminŽ .optimal solution is unique and a general technique to compute it .

3.5. Logical Handling of Preferences

It is not always convenient to express preferences directly in terms offlexible constraints represented by fuzzy membership functions. Preferences maybe also expressed in terms of sets of crisp goals having different levels ofpriority. In the simple case of a unique fuzzy constraint C, defined by its

� 4membership function m ranging on a finite scale a s 0 - a - ??? - a s 1 ,C 0 1 nŽ .C is equivalently represented by the set of constraints N C G 1 y a fora iy1i

i s 1, n, where N is the necessity measure defined from p s m . In other words,Cthe goal of finding a solution in C has priority 1 y a , and the larger thea iy1i

a-cut, the more important the priority; in particular it is imperative to find asolution in the support of C. More generally, the conjunctive aggregation of

Ž .fuzzy prioritized or discounted constraints can be interpreted in terms ofconjunctions of crisp goals having different levels of priority, thus providing anexpression of preferences in a weighted logic form. Conversely, a set of crispgoals with different levels of priority can be always represented in terms of afuzzy set membership function as we are going to see on different examples.

The thresholded min combination discussed in Section 3.2 can be easilyadapted in order to handle context-dependent specifications by turning thepriority degrees into degrees of truth which are equal to 1 if the contextcondition is fulfilled and are 0 otherwise. Thus, if criterion C has to beconsidered only if proposition p is true, the evaluation function m is turnedC


Ž Ž . Ž .. Ž .into max m u , 1 y ¨ p where ¨ p denotes the true value of p whenC u udecision u is applied. This enables us to handle conditional prioritized require-

Žments of the form ‘‘C should be satisfied, and among the solutions to C if1 1.any the ones satisfying C are preferred, and among the ones satisfying both C2 1

and C , those satisfying C are preferred and so on,’’ where C , C , C ??? are2 3 1 2 3Ž .here supposed to be classical constraints i.e., m s 0 or 1 ; such requirementsCi

have been considered in the database setting by Lacroix and Lavency.28 Thus,Ž .one wish to express that C should hold with importance or priority r s 1 ,1 1

Žand that if C holds, C holds with priority r , C holds with priority r with1 2 2 3 3.r - r - r . Using the representation of conditional requirements presented3 2 1

above, this nested conditional requirement can be represented by the expres-sion,

m u s min m u , max m u , 1 y min m u , r ,Ž . Ž . Ž . Ž .Ž .ž /žD C C C 21 2 1

max m u , 1 y min m u , m u , r 7Ž . Ž . Ž . Ž .Ž .ž / /C C C 33 1 2

Ž Ž . .It reflects that we are completely satisfied if C , C , and C m u s 1 are1 2 3 DŽ Ž . .completely satisfied, we are less satisfied m u s 1 y r if C and C onlyD 3 1 2

Ž Ž . .satisfied, and we are even less satisfied if only C m u s 1 y r is satisfied.1 D 2Ž .The expression 7 for binary constraints can be interpreted as the seman-

�Ž . Žtics of the possibilistic propositional logic knowledge base K s C , 1 ; !C k1 1. Ž .4C , r ; !C k !C k C , r . Indeed, the semantics of a possibilistic logic2 2 1 2 3 3

�Ž . 4base K s p , r ; j s 1, n , where p is a classical proposition and r , a level inj j j ja totally ordered valuation set, is given by the function from the set ofinterpretations to the valuation scale defined by

m u s min max ¨ p , 1 y r 8Ž . Ž . Ž .Ž .K u j jjs1, n

Possibilistic logic29 has been developed for handling formulas pervaded withŽuncertainty, and can be used for encoding default knowledge of the form ‘‘if C1

.generally C ’’ . See Ref. 30 for details. Here, the practical interpretation is2rather in terms of priority, and K reads: a good decision u should satisfy C1

w Ž . x Ž .imperatively otherwise m u s 0 , also C preferably priority r , and ifD 2 2Ž .possible C with priority level r - r . Indeed K is semantically equivalent to3 3 2

�Ž . Ž . Ž .4 Ž .K 9 s C , 1 ; C , r ; C , r , and 7 simplifies into the min-aggregation of1 2 2 3 3Ž . Ž .the prioritized constraints C , r , i s 1, 3 in the sense of 1 , as it can bei i

checked. This provides a logical description of the more or less satisfyingdecisions.

Ž . Ž . ŽExpressions 7 or 8 correspond to conjunctive normal forms i.e., it is a. Ž .min of max . They can be turned into disjunctive normal forms max of min

which provides a description of the different classes of decisions ranked accord-ing to their level of preference. Let us consider, for instance, the followingthree-criteria based evaluation: ‘‘if u satisfies A and B, u is completelysatisfactory, if A is not satisfied, less acceptable solutions should at least satisfyC.’’ Such a evaluation function can be encountered in multicriteria decision-

Ž . Ž .making MCDM . It can be directly represented by the disjunctive form m uDŽ Ž Ž . Ž .. Ž Ž . Ž . ..s max min m u , m u , min m u , 1 y m u , 1]r with r - 1. Turned intoA B C A

DUBOIS AND PRADE56

�Ž . Ž .a conjunctive form, it corresponds to the base K s ! A ª C, 1 ; ! B ª C, 1 ;Ž . Ž . Ž .4A ª B, 1 ; A, r ; B, r where ª is the material implication, which providesa logical, equivalent description of the requirements specifying acceptablesolutions.

Ž . Ž .In addition, expressions 5 or 8 are formally similar to the expressionsthat are used in qualitative decision under uncertainty in the next section. Thisis not accidental; see Ref. 31 for a general discussion of formal similaritiesbetween approaches in MCDM and in decision under uncertainty.

Lastly, let us mention that the possibilistic framework can be also useful inqualitative preference elicitation. Thus, preference about a binary property, B

Ž .over not B represented by a crisp subset of U can be expressed by a constraintŽ . Ž .of the form Ł B )Ł B which is equivalent to say that there exists at least

one decision value in B which is better than all the decision values in B. Such aconstraint can be made context dependent: if A is satisfied, B is preferred to B,

Ž . Ž .can be expressed by the constraint Ł A l B )Ł A l B . Thus, a collectionof such requirements gives birth to possibilistic constraints, whose least specificsolution p * can be computed and represents the preference profile agreeingwith the requirements. See Ref. 30 for the handling of similar constraints indefault reasoning. See Ref. 32 for such an approach to preference modeling,although not referring to possibility theory.

4. HANDLING QUALITATIVE UNCERTAINTY IN DECISION THEORY

In the general case, decisions are made in an uncertain environment. In theSavage33 framework, the consequence of a decision depends on the state of theworld in which it takes place. If U is a set of states and X a set of possibleconsequences, the decision-maker has some knowledge of the actual state andsome preference on the consequences of his decision. It makes sense if informa-tion is qualitative, to represent the incomplete knowledge on the state by apossibility distribution p on U with values in a plausibility valuation set L andthe decision-maker’s preference on X by means of another possibility distribu-tion m with values on a preference valuation set S. The utility of a decision d

Ž .whose consequence in state u is x s d u for u g U, can be evaluated byŽ . Ž .combining the plausibilities p u and the utilities m x in a suitable way. Two

quantitative criteria that evaluate the worth of decision d have been put forwardin the literature, and can be axiomatically justified34 provided that a commensu-rability assumption between plausibility and preference is made:

v A pessimistic criterion,

UU# d s inf max n p u , m d uŽ . Ž . Ž .Ž . Ž .Ž .ugU

which generalizes the max-min Wald criterion in the absence of probabilisticknowledge. Mapping n is order reversing from L to S.

v An optimistic criterion,

UU* d s sup min m p u , m d uŽ . Ž . Ž .Ž . Ž .Ž .ugU


which generalizes the maximax optimistic criterion. Mapping m is order-preserv-ing from L to S.

The optimistic criterion has been first proposed by Yager 35 and the pes-simistic criterion by Whalen,36 and also used in Inuiguchi et al.37 Maximizing

Ž .UU# d means to find a decision d whose possible consequences with a highplausibility are among the consequences with a high level of preference. Thedefinition of ‘‘highly plausible’’ is decision-dependent and reflects the compro-mise between high plausibility and low utility expressed by the order-reversing

Ž .map between the plausibility valuation set and the utility valuation set; UU# d issmall only if it exists a possible consequence of d which is both highly plausibleand bad with respect to preferences. This is clearly a uncertainty-averse and

Ž .thus a pessimistic attitude. UU# d corresponds to an optimistic attitude since itis high as soon as it exists a possible consequence of d which is both highlyplausible and highly prized. It is worth pointing out that from the computationof an optimal decision maximizing the pessimistic criterion we can easily

Ž .forecast the worst situation s from which we are protected by the decision.The pessimistic criterion has been axiomatically justified by Dubois and

Prade34 in the style of von Neumann and Morgenstern38 utility theory. The ideais that if the uncertainty on the state is represented by p , each decision induces

Ž .on the set of consequences X a possibility distribution such that p x sdŽ y1Ž ..Ł d x . So ranking decisions comes down to ranking possibility distributions

on X. Assume the decision-maker supplies an ordering between possibilitydistributions on X, thus expressing his attitude in front of uncertainty, that is, infront of various possibilities of happy and unhappy consequences in X . Let

Ž .p x be the plausibility of getting x under decision d. The question is to knowdwhat kind of axioms on the ordering between possibility distributions on Xmake it representable by the ranking of decisions according to the above

Ž .pessimistic or optimistic criteria. Let lrp , mrp 9 denote the ‘‘qualitativelottery’’ yielding p with plausibility l and p 9 with plausibility m. Of course,

Ž .max l, m s 1. The following are the pessimistic axioms:

Ž . Ž .1 The set of possibility distributions is equipped with a structure GGGGG , ;;;;; , ))))) ofcomplete preordering, where GGGGG , ;;;;; , ))))) are the weak preference, indifference,and strict preference.

Ž . Ž . Ž .2 Independence: p ;;;;; p « lrp , mrp ;;;;; lrp , mrp1 2 1 2Ž . Ž .3 Continuity: if p ))))) p 9 then 'l g L, p 9 ;;;;; 1rp , lrXŽ . Ž Ž .. Ž .4 Reduction of lotteries: lrx, mr arx, bry ;;;;; grx, dry

Ž Ž .. Ž .where g s max l, min m, a and d s min b , mŽ .5 Uncertainty a¨ersion: p G p 9 « p 9 GGGGG p .

The uncertainty-aversion axiom states that the less informative is p , themore uncertain is the situation; the worst epistemic state is total ignorance.Continuity says that the utility of p goes down if the uncertainty about p raises.It can be proved that if the knowledge is represented by a subset A of possiblestates, then ' x g A, x ; A, see Dubois et al.45 This property, violated by

DUBOIS AND PRADE58

expected utility, suggests that contrary to it, the pessimistic utility is not basedon the idea of average and repeated decisions, but makes sense for one-shotdecisions. The pessimistic criterion is an extension of Wald pessimistic criterion,evaluating decisions on the basis of their worst consequences, however unlikelythey are. However, the possibilistic criterion is less pessimistic. It focuses on theidea of usuality and relies on the worst plausible consequences induced by thedecision. Some unlikely states are neglected by a variable thresholding and thethreshold is determined by comparing the possibility distributions valued on Land S via the mapping n. A dual set of axioms can be devised for the optimisticcriterion. The latter can be used as a secondary criterion, for breaking tiesbetween decisions which are equivalent w.r.t. the pessimistic criterion. Thisapproach sounds realistic in settings where information about plausible statesand preferred consequences is poor and linguistically expressed, and wheredecisions are not repeated. Moreover, as already said, the approach relies on acommensurability assumption; see Ref. 39 on the problems encountered by aqualitative approach when such an assumption is not made.

Recently, a justification for the pessimistic and optimistic criteria have beenproposed by Dubois et al.,46 starting from axioms on preferences between acts,which are qualitative counterparts to Savage’s33 axioms. Especially, this requiresto weaken Savage’s sure thing principle which expresses that if an act f ispreferred or is equivalent to another act g and these two acts result in identicalconsequences when applied in a subset A of situations, then if f and g aremodified in the same way on A, the two modified acts remain ordered as f andg. The weakening of the sure thing principle which is used acknowledges the fact

Ž .that extreme consequences either very good or very bad are allowed to blurminor differences between f and g when applied outside A.

These qualitative counterparts of the expected utility theory nicely fit thesetting of flexible constraint propagation.17 This approach sounds realistic insettings where information about plausible states and preferred consequences ispoor and linguistically expressed, and where decisions are not repeated. It canbe easily implemented in the framework of a possibilistic logic machineryhandling both a possibilistic base describing the available knowledge about thestate of the world, and another possibilistic base specifying the preferences.40

5. TOWARD CASE-BASED DECISION

Preferences can be similarity-based. Here we illustrate the use of qualita-tive similarity degrees in decision only by briefly suggesting an approach tocase-based decision.

Recently, Gilboa and Schmeidler 41 have advocated an approach to deci-sion, based on the notion of similarity. In this approach, the decision-maker uses

Ž .a memory M of cases, viewed as triples u, d, x where u is a state of the world,d an action, and x its result in state u. Let Sim be a nonnegative function whichestimates the similarity of two situations.


When the decision-maker is faced with a situation u , he chooses an action0d which maximizes the following criterion, counterpart of classical expectedutility,

UU d s Sim u , u ? ¨ xŽ . Ž . Ž .Ýu , M 00Ž .u , d , x gM

where ¨ is a nonnegative utility function. Note that contrary to classicalŽ .expected utility, Ý Sim u , u / 1.Žu, d, x .g M 0

In Ref. 41, an axiomatic justification of this criterion is given. The intuitiveinterpretation of UU is to give preference to acts which in similar situationsu , M0

led to results with a high utility. However, with this approach an act which givesa low, but nonzero, utility in many similar cases can be preferred to an act whichgave a very high utility in one very similar situation. This approach also requiresa numerical setting.

The previous study suggests a pessimistic qualitative counterpart to theabove case-based utility, namely,42

UU# d s min max ¨ x , 1 y Sim u , uŽ . Ž . Ž .Ž .u , M 00 Ž . Ž .u , x : u , d , x gM

where ¨ and Sim take their values into two bounded commensurate valuationŽ .sets, and 1 y ? is the order-reversing function from one to the other. According

to this criterion, a decision is considered good only if it always gives good resultsŽin similar situations provided that the fuzzy set of similar situations is normal-

.ized . An optimistic counterpart can be easily imagined.

6. CONCLUDING REMARKS

This paper has advocated the interest of qualitative possibility theory formodeling both preference and uncertainty in a not too demanding way, indecision analysis. Existing results, as well as some new lines of research havebeen also suggested. In particular, the computation of discrimin or leximinsolutions, the logical handling of preferences, or case-based decisions seemworth developing. Moreover, existing extensions of possibilistic logic for dealingwith inconsistent knowledge bases, stratified into layers corresponding to cer-

43 Ž .tainty levels, might also lead to fruitful treatments of partially inconsistentsets of prioritized constraints representing preferences.

References

1. Dubois, D.; Prade, H.; Yager, R. R., Eds., Fuzzy Information Engineering}AGuided Tour of Applications; Wiley: New York, 1997.

2. Roubens, M.; Vincke, P. Preference Modelling, Lecture Notes in Economics andMathematical Systems; Springer-Verlag: Berlin, 1985; Vol. 250.

3. Kitainik, L. Fuzzy Decision Procedures with Binary Relations. Towards a UnifiedTheory; Kluwer Acad.: Dordrecht, The Netherlands, 1993.

4. Fodor, J.; Roubens, M. Fuzzy Preference Modelling and Multicriteria DecisionSupport; Kluwer Academic: Dordrecht, The Netherlands 1994.

DUBOIS AND PRADE60

Ž .5. Dubois, D.; Prade, H. Inf Sci 1985, 36 1]2 , 85]121.6. Zimmermann, H. J. Fuzzy Sets, Decision Making, and Expert Systems; Kluwer

Academic: Dordrecht, 1987.Ž .7. Chen, S. J.; Hwang, C. L. with the collaboration of F.P. Hwang Fuzzy multiple

attribute decision making}methods and applications, Lecture Notes in Economicsand Mathematical Systems; Springer-Verlag: Berlin, 1992; Vol. 375.

8. Yager, R. R. IEEE Trans Syst, Man, Cybern 1988, 18, 183]190.9. Grabisch, M.; Nguyen, H. T.; Walker, E. A. Fundamentals of Uncertainty Calculi

with Applications to Fuzzy Inference; Kluwer Academic: Dordrecht, 1995.10. Kacprzyk, J.; Fedrizzi, M., Eds.; Multiperson Decision Making Models Using Fuzzy

Sets and Possibility Theory; Kluwer Academic: Dordrecht, 1990.11. Dubois, D.; Prade, H. Decision evaluation methods under uncertainty and impreci-

sion, in Combining Fuzzy Imprecision and Probabilistic Uncertainty in DecisionMaking, Kacprzyk, J.; Fedrizzi, M., Eds.; Lectures Notes in Economics and Mathe-matical Systems; Springer-Verlag: Berlin, 1988; Vol. 310, pp. 48]65.

12. Kacprzyk, J.; Fedrizzi, M., Eds.; Combining Fuzzy Imprecision with ProbabilisticUncertainty in Decision Making, Lecture Notes in Economics and MathematicalSystems; Springer-Verlag: Berlin, 1988; Vol. 310.

13. Yager, R. R. Decision Sci 1981, 12, 589]600.14. Dubois, D.; Prade, H. Fuzzy Sets Syst 1997, 90, 141]150.15. Dubois, D.; Prade, H. Constraint satisfaction and decision under uncertainty based

Žon qualitative possibility theory, Proc of the 6th Int Conf on Fuzzy Sytems FUZZ-.IEEE’97 ; 1997; pp. 23]30.

16. Zadeh, L. A. Fuzzy Sets Syst 1978, 1, 3]28.17. Dubois, D.; Fargier, H.; Prade, H. Appl Intell 1996, 6, 287]309.18. Zadeh, L. A. Calculus of fuzzy restrictions, in Fuzzy Sets and their Applications to

Cognitive and Decision Processes, Zadeh, L. A.; Fu, K. S.; Tanaka, K.; Shimura; M.,Eds.; Academic Press: New York, 1975; pp. 1]39.

19. Dubois, D.; Prade, H. Fuzzy rules in knowledge-based systems}Modelling graded-ness, uncertainty and preference}, in An Introduction to Fuzzy Logic Applicationsin Intelligent Systems, Yager, R. R.; Zadeh, L. A., Eds.; Kluwer Academic: Dor-drecht, The Netherlands, MA, 1992, pp. 45]68.

20. Dubois, D.; Fargier, H.; Prade, H. Ordered weighted operations, ‘‘discri-min’’ andleximin in multicriteria decision, in Soft Computing with Industrial ApplicationsŽ Ž .Proc of the 2nd World Automation Congress WAC’96 , Montpellier, France, May

.27]30, 1996 ; Jamshidi, M.; Fathi, M.; Pierrot, F., Eds.; TSI Press Series: Albu-querque, NM, 1996; pp. 167]172.

21. Yager, R. R. Int J Man-Mach Stud 1984, 21, 389]400.22. Dubois, D.; Prade, H.; Testemale, C. Fuzzy Sets Syst 1988, 28, 313]331.23. Bellman, R.; Zadeh, L. A. Manage Sci 1970, 17, 141]164.24. Moulin, H. Axioms of Cooperative Decision-Making; Cambridge Univ. Press: Cam-

bridge, U.K., 1988.25. Dubois, D.; Fargier, H.; Prade, H. Fuzzy Sets Syst 1996, 81, 103]122.26. Dubois, D.; Fargier, H.; Fortemps, P.; Prade, H. Leximin optimality and fuzzy set

theoretic operations, Proc of 7th Int Fuzzy Systems Association World CongressŽ .IFSA’97 ; Academia: Prague, June 25]29, 1997; Vol. 1, pp. 55]60.

27. Dubois, D.; Fortemps, P. Improved solutions to fuzzy constraint satisfaction prob-lems, Proc of the 5th Europ Cong on Intelligent Techniques and Soft ComputingŽ .EUFIT’97 ; Aachen, Germany, Sept. 8]12, 1997. Extended version to appear in EurJ Operation Res.

28. Lacroix, M.; Lavency, P. Preferences: Putting more knowledge into queries, Proc ofthe 13th Int Conf on Very Large Data Bases, Brighton, U.K., 1987; pp. 215]225.

Ž .29. Dubois, D.; Lang, J.; Prade, H. IEEE Trans Data Knowledge Eng 1994, 6 1 , 64]71.


30. Benferhat, S.; Dubois, D.; Prade, H. Representing default rules in possibilistic logic,Proc of the 3rd Int Conf on Principles of Knowledge Representation and ReasoningŽ .KR’92 , Cambridge, MA, Oct. 1992; pp. 673]684.

31. Dubois, D.; Grabisch, M.; Modave, F.; Prade, H. Relating decision under uncertaintyand MCDM models, Working Notes of the AAAI’97 Fall Symp Series on Frontiers inSoft Computing and Decision Systems; Medsker, L., Ed.; Boston, Nov. 8]10, 1997;pp. 6]15.

32. Boutilier, C.; Brafman, R.; Geib, C.; Poole, D. A constraint-based approach topreference elicitation and decision making, Working Notes of the AAAI’97 SpringSymp Series on Qualitative Preferences in Deliberation and Practical Reasoning;Stanford, CA, Mar. 24]26, 1997; pp. 19]28.

33. Savage, L. J. The Foundations of Statistics, 2nd ed., 1972, Dover, New York, 1954.34. Dubois, D.; Prade, H. Possibility theory as a basis for qualitative decision theory,

Ž .Proc of the 14th Int Joint Conf on Artificial Intelligence IJCAI’95 , Montreal,´Canada, Aug. 20]25, 1995; pp. 1924]1930.

35. Yager, R. R. IEEE Trans Syst, Man, Cybern 1979, 9, 388]392.36. Whalen, T. IEEE Trans Syst, Man, Cybern 1984, 14, 888]900.

Ž .37. Inuiguchi, M.; Ichihashi, H.; Tanaka, H. ORSA J Comput 1989, 1 3 , 146]158.38. von Neumann, J.; Morgenstern, O. Theory of Games and Economic Behavior;

Princeton Univ Press: Princeton, NJ, 1944.39. Dubois, D.; Fargier, H.; Prade, H. Decision-making under ordinal preferences and

comparative uncertainty, Proc of the 13th Conf on Uncertainty in Artificial Intelli-gence, Geiger, D.; Shenoy, P. P., Eds.; Morgan Kaufmann, San Francisco, CA, 1997;pp. 157]164.

40. Dubois, D.; Le Borre, D.; Prade, H.; Sabbadin, R. Logical representation andcomputation of optimal decisions in a qualitative setting. Proc Nat AAAI Conferenceon Artificial Intelligence, Madison, WI, 1998.

41. Gilboa, I.; Schmeidler, D. Quart J Econom Aug. 1995, 607]639.42. Dubois, D.; Esteva, F.; Garcia, P.; Godo, L.; Lopez de Mantaros, R.; Prade, H. Fuzzy

set modeling in case-based reasoning Int J Intell Syst 1998, 13, 345]373.43. Benferhat, S.; Dubois, D.; Prade, H. Reasoning in inconsistent stratified knowledge

Ž .bases, Proc of the 26 Int Symp on Multiple-Valued Logic ISMVL’96 , Santiago deCompostela, Spain, May 29]31, 1996; pp. 184]189.

44. Sugeno, M. Theory of fuzzy integrals and its applications, Doctoral Thesis, TokyoInst of Technology, 1974.

45. Dubois, D.; Godo, L.; Prade, H.; Zapico, A. Possibilistic representation of qualitativeutility: an improved representation. Proc Int Conf on Information Processing and

Ž .Uncertainty Management IPMU98 , Paris, 1998, pp. 180]187.46. Dubois, D.; Prade, H.; Sabbadin, R. Qualitative decision theory with Sugeno Inte-

grals, Proc 14th Conf on Uncertainty in AI, Madison, WI, Cooper, G.; Morol, S.,Eds.; Morgan Kaufman, San Francisco, CA, 1998, pp. 121]128.

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