An introduction to DSmT

An intr oduction to DSmT Jean Dezert Florentin Smarandac he French Aerospace Resea rch Lab ., Chair of Math. & Science s Dept., ONERA/DTIM/SIF , Univ ersity of Ne w Mexico, 29 A venue de la Di vision Leclerc, 200 College Road, 92320 Ch ˆ atillon, France. Gallup, NM 87301, U.S.A. jean.dezert @onera.fr sma rand@unm.edu Abstract – The management and combination of uncertain, impr ecise, fuzzy and even paradoxical or h igh conﬂictin g sour ces of in formation has always been, and still remains today , of primal importan ce for the development of r eliable modern in formation systems involving artiﬁcial reasoning. In th is in tr oduction , we pr esent a survey of our recent the- ory of plausible and parado xical r easonin g, known as Dezert-Smaranda che T heory (DSmT), developed fo r d ealing with impr ecise, u ncertain an d con ﬂicting sou r ces of information. W e fo cus our p r esentatio n on the found ations of DSmT and on its most importa nt rules of comb ination, rather than on br owsing speciﬁc applica tions of DSmT availa ble in litera- tur e. Several simple e x amples ar e given thr ough out this pr esentatio n to s how the efﬁciency and the generality of this new appr o ach. Keywords: Dezert-Smar andache Theory , DS mT , quantitativ e and qualitati ve reasoning, information fusion. MSC 2000 : 68T37, 94A15, 94A17, 68T40. 1 Intr o duction The managemen t and combinat ion of uncertain , impreci se, fuzzy and e ven paradox ical or high conﬂicting source s of information has alw ays been, and still remains today , of primal importance for the de velo pment of reliabl e modern information systems in volv ing artiﬁcial reasoning. The combination (fusion) of information arises in man y ﬁelds of applic ations no wadays (especi ally in defense, med icine, ﬁnance, geo-scie nce, economy , etc). When se vera l sensors, observ ers or experts hav e to be combined togethe r to solv e a problem, or if one wants to update our current estimatio n of solutions for a giv en problem with some ne w information av ailable, we need po w erful and solid mathematical tools for the fusion, specially w hen the informati on one has to deal with is imprecise and uncert ain. In this paper , we presen t a surv ey of our recent theory of plausib le an d parado xical reasoning, kno wn as Dezert-Smarand ache Theory (DSmT) in the literature, de veloped for dealing with imprecis e, uncerta in and conﬂicting sources of informati on. Recent publication s hav e shown the interest and the ability of D SmT to solve problems where other approaches fail, especially when conﬂict between source s becomes high. W e focu s t his presen tation rather on the fou ndatio ns of DSmT , an d on the m ain important rules of combination , than on browsing speciﬁc applicat ions of DSm T av ailable in literatur e. Seve ral simple exa mples are giv en throughout the presentatio n to sho w the efﬁci ency and the genera lity of DSm T . 2 Foun datio ns of DSmT The de velop ment of DSmT (Dezert-Smaran dache Theory of pla usible and p aradox ical reasoning [8, 31]) arises from the necess ity to overco me the inherent limitations of DST (Dempster -Shafer Theory [24]) which are closel y related wit h the accep tance of S hafer’ s model for th e fusion pr oblem u nder conside ration (i.e. the frame of discern ment Θ is implicitly deﬁned as a ﬁnite set of exh austive and exclu sive hypotheses θ i , i = 1 , . . . , n since the masses of belief are deﬁned only on the po w er set of Θ - see sectio n 2.1 for details ), th e t hird m iddle ex- cluded pri nciple (i.e. the ex istence o f the comple ment for any elements/prop osition s belonging to th e po wer set of Θ ), and the a ccepta nce of Dempster’ s rule of combination ( in v olving normaliz ation) a s the frame work fo r the combina tion of indepe ndent sources of ev idence . Discussions on limitation s of DST and presentation of some This paper is based on the ﬁrst chapter of [36]. 1 alterna ti ve rules to Dempster’ s rule of combination can b e found in [11, 15, 17–19, 21, 23, 31, 38, 46, 49, 50, 53–56] and therefore they will be not reported in details in this introdu ction. W e argu e that these three fundament al condit ions of D ST can be remov ed and another ne w mathematic al approa ch for combi nation of evid ence is possib le. This is the purpose of DSm T . The bas is of DSmT i s the refu tation of the prin ciple of the third exclu ded middle and Shafer ’ s model, si nce for a wide class of fusion problems the intrinsi c nature of hypotheses can be only vague and imprecise in such a way that pre cise reﬁnement is just impossible to obta in in reality so that the exclusi ve elements θ i canno t be properly identiﬁed and precise ly separated. Many problems in volv ing fuzzy continuo us and relati ve con- cepts de scribe d in natur al langu age and ha ving no absolut e interpreta tion like tal lness/s mallness , pleasure/pai n, cold/h ot, Sorites parado xes , etc, enter in this categor y . DSmT starts with the notion of free DSm model , de- noted M f (Θ) , and consid ers Θ only as a frame of exha usti ve elements θ i , i = 1 , . . . , n which can potent ially ov erlap. This model is fre e because no other assumptio n is done on the hypotheses , but the weak exh austi vity constr aint which can alwa ys be satisﬁed according the closure principle expl ained in [31]. No other constrai nt is in v olved in the free DSm m odel. When the free DSm model holds, the classic commutativ e and associati ve classic al DS m ru le of combin ation, denoted DSmC, corre spond ing to the conj uncti ve consensus deﬁned on the free Dedekind ’ s lattice is performed. Dependin g on th e intrins ic natur e of the elements of the fusion p roblem under conside ration , it can ho wev er happe n that the free model does not ﬁt the reality becaus e some subsets of Θ can contain elements known to be truly excl usi ve but also truly non existing at all at a giv en time (specially w hen working on dynamic fusion proble m where the frame Θ varies with time with the revisio n of the knowled ge a vail able). These inte grity constr aints are then expl icitly and formally introduce d into the free DS m model M f (Θ) in order to adapt it proper ly to ﬁt as close a s possibl e with the reality an d permit to construc t a hybrid DSm model M (Θ) on which the combinat ion w ill b e efﬁcie ntly perfor med. Shafer’ s model, deno ted M 0 (Θ) , corresp onds to a very spe ciﬁc hybrid DSm model includ ing all possible exclu si vity cons traints . DST has been de velo ped for working only with M 0 (Θ) while D SmT has been dev eloped for workin g with any kind of hybrid m odel (including S hafer’ s model and the free D Sm model), to manage as ef ﬁ cientl y and precisely as possib le imprecise, uncertain and potent ially high conﬂicting sou rces of evid ence while keep ing in mind the possible dynamicity of the infor- mation fusion problematic. The foundations of DSmT are therefor e totally differe nt from those of all existi ng approa ches managin g uncer taintie s, imprecisions and conﬂicts. DSmT prov ides a ne w i nterest ing way to at tack the informati on fusion problematic w ith a genera l framewo rk in order to cov er a w ide v ariety of problems. DSmT ref utes also the idea th at sourc es of e vidence pro vide their be liefs with th e same ab solute interpreta- tion of elements of the same frame Θ and the conﬂict between sources arises not only becau se of the possibl e unreli ability of source s, but also because of possible dif ferent and relati ve interp retation of Θ , e.g. what is consid ered as good for somebody can be con sidered as bad for somebody else. T here is some una void able subjec ti vity in the belief assignments pro vided by the sources of e vidence, otherwise it would mean that all bodies of evid ence hav e a same objecti ve and uni versa l interp retation (or measure) of the phenomena under consid eration , which unfor tunate ly rarely occu rs in real ity , but when b asic beli ef assignmen ts (bba’ s) are ba sed on some objectiv e pr obab ilities transformati ons. But in this last case, probab ility theory can handle properly and ef ﬁ cientl y the informat ion, and DST , as well as DSmT , becomes useless. If we no w get out of the prob- abilist ic backgro und argu mentatio n for the constructio n of bba, we claim that in m ost of cases, the sources of e videnc e pro vide their beliefs about elements of the frame of the fusion problem only based on their own limited kno wledge and exper ience without reference to the (inaccessib le) absolute truth of the space of possibili ties. Sev eral success ful applica tions of DS mT (in tar get trackin g, satellite surv eillance, situation analys is, robotics, medicine , etc) can be found in [31, 3 4]. 2.1 The power set, hyper -power set and super-power se t In DSmT , we take ver y care of the model associated with the set Θ of hypothes es where the solution of the proble m is a ssumed to belong to. In parti cular , the thre e main sets (po w er se t, h yper -po wer set and s uper -po wer set) can be us ed dependi ng on their ability to ﬁt adequ ately with the nature of hyp othese s. In th e follo wing, we 2 assume that Θ = { θ 1 , . . . , θ n } is a ﬁnite set (called frame) of n exhausti ve elements 1 . If Θ = { θ 1 , . . . , θ n } is a priori not closed ( Θ is said to be an open world/frame), one can always include in it a closure element, say θ n +1 in such away that we can work with a new closed world/fra me { θ 1 , . . . , θ n , θ n +1 } . So without loss of general ity , we will alw ays assume that we work in a closed world by con siderin g the frame Θ as a ﬁ nite set of exhausti ve elements. Before introduc ing the power set, the hyper -power set and the super -power set it is necess ary to recall that su bsets are re garded as propos itions in Dempster -Shafer T heory (s ee Chapter 2 of [24 ]) and we adopt the same approa ch in DSmT . • Subsets as propositi ons : Glenn Shafer in pages 35–37 of [24] consider s the subset s as proposition s in the case we are concer ned with the true v alue of some quantity θ taking its possible v alues in Θ . T hen the propos itions P θ ( A ) of intere st are those of the form 2 : P θ ( A ) , T he true valu e of θ is in a subset A of Θ . Any propo sition P θ ( A ) is thus in one-to-o ne corresp ondenc e with the subset A of Θ . Such corresp on- dence is very usef ul since it tran slates the lo gical notio ns of conjun ction ∧ , disju nction ∨ , implicatio n ⇒ and nega tion ¬ into the set-theoret ic notions of interse ction ∩ , union ∪ , inclus ion ⊂ and complement a- tion c ( . ) . Indeed, if P θ ( A ) and P θ ( B ) are two proposition s correspon ding to subsets A and B of Θ , then the conjunct ion P θ ( A ) ∧ P θ ( B ) corresp onds to the intersection A ∩ B and th e disjunctio n P θ ( A ) ∨ P θ ( B ) corres ponds to the union A ∪ B . A is a subset of B if and only if P θ ( A ) ⇒ P θ ( B ) and A is the set- theore tic complement of B with respect to Θ (written A = c Θ ( B ) ) if and only if P θ ( A ) = ¬P θ ( B ) . In other words, the follo w ing equi valenc es are then used between the operat ions on the subsets and on the propo sitions : Operatio ns Subsets Proposit ions Interse ction/ conjunction A ∩ B P θ ( A ) ∧ P θ ( B ) Union/di sjunct ion A ∪ B P θ ( A ) ∨ P θ ( B ) Inclus ion/impli cation A ⊂ B P θ ( A ) ⇒ P θ ( B ) Complementa tion/ne gation A = c Θ ( B ) P θ ( A ) = ¬ P θ ( B ) T able 1: Correspon dence between operat ions on subs ets and on propositio ns. • Canonical form of a propos ition : In DS mT w e consid er all propos itions/ sets in a canonica l form. W e tak e the disju ncti ve normal form, w hich is a disjun ction of conjun ctions, and it is uniqu e in Boolean algebr a and simplest. For example, X = A ∩ B ∩ ( A ∪ B ∪ C ) it is not in a canonic al form, but we simplify the formul a and X = A ∩ B is in a canonica l form. • The power set : 2 Θ , (Θ , ∪ ) Aside Dempster’ s rule of combination, the power set is one of the corner stones of Dempster-Sha fer Theory (DST) s ince the basic beli ef assig nments to co mbine a re de ﬁned on the po wer set of the frame Θ . In math emat- ics, giv en a set Θ , the power set of Θ , written 2 Θ , is the set of all subsets of Θ . In ZF C axiomatic set theory , the exist ence of the po w er set of any set is postulate d by the axiom of po w er set. In other words, Θ generat es the po w er set 2 Θ with the ∪ (un ion) operator only . 1 W e do not assume here that elements θ i are necessary exclusiv e , un less speciﬁed. T here is n o restriction on θ i but the exhaustivity . 2 W e use the symbol , to mean eq uals by deﬁnition ; th e righ t-hand side of the equation is the deﬁnition of the lef t-hand side. 3 More precisely , th e power set 2 Θ is deﬁned as the set of all composite propositio ns/sub sets b uilt from elements of Θ with ∪ opera tor such that: 1. ∅ , θ 1 , . . . , θ n ∈ 2 Θ . 2. If A, B ∈ 2 Θ , then A ∪ B ∈ 2 Θ . 3. No other elements belong to 2 Θ , excep t those obtained by using rules 1 and 2. Examples of power sets : • If Θ = { θ 1 , θ 2 } , then 2 Θ= { θ 1 ,θ 2 } = { {∅} , { θ 1 } , { θ 2 } , { θ 1 , θ 2 }} which is commonly written as 2 Θ = {∅ , θ 1 , θ 2 , θ 1 ∪ θ 2 } . • Let’ s consider two frames Θ 1 = { A, B } and Θ 2 = { X, Y } , then their power sets are respecti vely 2 Θ 1 = { A,B } = {∅ , A, B , A ∪ B } and 2 Θ 2 = { X,Y } = {∅ , X, Y , X ∪ Y } . Let’ s consider a reﬁned frame Θ r ef = { θ 1 , θ 2 , θ 3 , θ 4 } . The granu les θ i , i = 1 , . . . , 4 are not necessari ly exhaust iv e, nor excl usi ve. If A and B are expre ssed m ore precisely in function of the granu les θ i by example as A , { θ 1 , θ 2 , θ 3 } ≡ θ 1 ∪ θ 2 ∪ θ 3 and B , { θ 2 , θ 4 } ≡ θ 2 ∪ θ 4 then the power sets can be expr essed from the granules θ i as follo ws: 2 Θ 1 = { A,B } = {∅ , A, B , A ∪ B } = {∅ , { θ 1 , θ 2 , θ 3 } | {z } A , { θ 2 , θ 4 } | {z } B , {{ θ 1 , θ 2 , θ 3 } , { θ 2 , θ 4 }} | {z } A ∪ B } = {∅ , θ 1 ∪ θ 2 ∪ θ 3 , θ 2 ∪ θ 4 , θ 1 ∪ θ 2 ∪ θ 3 ∪ θ 4 } If X and Y are expresse d m ore precis ely in function of the ﬁner granules θ i by ex ample as X , { θ 1 } ≡ θ 1 and Y , { θ 2 , θ 3 , θ 4 } ≡ θ 2 ∪ θ 3 ∪ θ 4 then: 2 Θ 2 = { X,Y } = {∅ , X , Y , X ∪ Y } = {∅ , { θ 1 } | {z } X , { θ 2 , θ 3 , θ 4 } | {z } Y , {{ θ 1 } , { θ 2 , θ 3 , θ 4 }} | {z } X ∪ Y } = {∅ , θ 1 , θ 2 ∪ θ 3 ∪ θ 4 , θ 1 ∪ θ 2 ∪ θ 3 ∪ θ 4 } W e see that one has natura lly: 2 Θ 1 = { A,B } 6 = 2 Θ 2 = { X,Y } 6 = 2 Θ r ef = { θ 1 ,θ 2 ,θ 3 ,θ 4 } e ven i f working from θ i with A ∪ B = X ∪ Y = { θ 1 , θ 2 , θ 3 , θ 4 } = Θ r ef . • The hyper -power set : D Θ , (Θ , ∪ , ∩ ) One of the cornerston es of DS mT is the free Dedekin d’ s lattice [4] denoted hyper -power set in DSmT frame work. Let Θ = { θ 1 , . . . , θ n } be a ﬁnite set (call ed frame) of n exhau sti ve elements. The hyper -power set D Θ is deﬁned as the s et of all co mposite proposi tions/ subsets b uilt from elements of Θ with ∪ and ∩ opera tors such that: 1. ∅ , θ 1 , . . . , θ n ∈ D Θ . 2. If A, B ∈ D Θ , then A ∩ B ∈ D Θ and A ∪ B ∈ D Θ . 3. No other elements belong to D Θ , excep t those obtained by using rules 1 or 2. 4 Therefore by con ventio n, w e write D Θ = (Θ , ∪ , ∩ ) which means that Θ generates D Θ under operators ∪ and ∩ . The dual (obtaine d by switching ∪ and ∩ in expr ession s) of D Θ is itself. There are elements in D Θ which are self-du al (dual to themselves) , for example α 8 for the case when n = 3 in the follo wing example . The cardinality of D Θ is majored by 2 2 n when the cardinality of Θ equals n , i.e. | Θ | = n . The generatio n of hyper -po wer set D Θ is closely related with the famous Dedekind’ s problem [3, 4] on enumerating the set of isotone B oolea n functio ns. The generation of the hyper -power set is presente d in [31]. Since for any giv en ﬁnite set Θ , | D Θ | ≥ | 2 Θ | we call D Θ the hyp er -power set of Θ . Example of the ﬁrst hyper -power sets : • For t he degen erate case ( n = 0) where Θ = {} , one has D Θ = { α 0 , ∅} and | D Θ | = 1 . • When Θ = { θ 1 } , one has D Θ = { α 0 , ∅ , α 1 , θ 1 } and | D Θ | = 2 . • When Θ = { θ 1 , θ 2 } , one has D Θ = { α 0 , α 1 , . . . , α 4 } and | D Θ | = 5 with α 0 , ∅ , α 1 , θ 1 ∩ θ 2 , α 2 , θ 1 , α 3 , θ 2 and α 4 , θ 1 ∪ θ 2 . • When Θ = { θ 1 , θ 2 , θ 3 } , one has D Θ = { α 0 , α 1 , . . . , α 18 } and | D Θ | = 19 with α 0 , ∅ α 1 , θ 1 ∩ θ 2 ∩ θ 3 α 10 , θ 2 α 2 , θ 1 ∩ θ 2 α 11 , θ 3 α 3 , θ 1 ∩ θ 3 α 12 , ( θ 1 ∩ θ 2 ) ∪ θ 3 α 4 , θ 2 ∩ θ 3 α 13 , ( θ 1 ∩ θ 3 ) ∪ θ 2 α 5 , ( θ 1 ∪ θ 2 ) ∩ θ 3 α 14 , ( θ 2 ∩ θ 3 ) ∪ θ 1 α 6 , ( θ 1 ∪ θ 3 ) ∩ θ 2 α 15 , θ 1 ∪ θ 2 α 7 , ( θ 2 ∪ θ 3 ) ∩ θ 1 α 16 , θ 1 ∪ θ 3 α 8 , ( θ 1 ∩ θ 2 ) ∪ ( θ 1 ∩ θ 3 ) ∪ ( θ 2 ∩ θ 3 ) α 17 , θ 2 ∪ θ 3 α 9 , θ 1 α 18 , θ 1 ∪ θ 2 ∪ θ 3 The cardin ality of hyper -po w er set D Θ for n ≥ 1 follows the sequenc e of Dedekind’ s numb ers [26], i.e. 1,2,5,19,16 7, 7580,7828 353,... and analytica l expre ssion of D edeki nd’ s numbers has been obtaine d recen tly by T ombak in [45] (see [31] for details on generation and ordering of D Θ ). Interesting in ves tigatio ns on the progra mming of the gener ation of hyper -power sets for enginee ring applica tions hav e been done in Chapter 15 of [34] and in [36]. Examples of hyper -power set s : Let’ s consider the frames Θ 1 = { A, B } and Θ 2 = { X, Y } , then their correspond ing hyper -po wer sets are D Θ 1 = { A,B } = {∅ , A ∩ B , A, B , A ∪ B } and D Θ 2 = { X,Y } = {∅ , X ∩ Y , X , Y , X ∪ Y } . Let’ s cons ider a reﬁned frame Θ r ef = { θ 1 , θ 2 , θ 3 , θ 4 } where the granul es θ i , i = 1 , . . . , 4 are now considere d as truly exha ustive and e xclusi ve . If A and B are expres sed m ore precisely in functi on of the granules θ i by example as A , { θ 1 , θ 2 , θ 3 } and B , { θ 2 , θ 4 } then D Θ 1 = { A,B } = {∅ , A ∩ B , A, B , A ∪ B } = {∅ , { θ 1 , θ 2 , θ 3 } ∩ { θ 2 , θ 4 } | {z } A ∩ B = { θ 2 } , { θ 1 , θ 2 , θ 3 } | {z } A , { θ 2 , θ 4 } | {z } B , {{ θ 1 , θ 2 , θ 3 } , { θ 2 , θ 4 }} | {z } A ∪ B = { θ 1 ,θ 2 ,θ 3 ,θ 4 } } = {∅ , θ 2 , θ 1 ∪ θ 2 ∪ θ 3 , θ 2 ∪ θ 4 , θ 1 ∪ θ 2 ∪ θ 3 ∪ θ 4 } 6 = 2 Θ 1 = { A,B } 5 If X and Y are expre ssed more precisely in functi on of the ﬁner granules θ i by exampl e as X , { θ 1 } and Y , { θ 2 , θ 3 , θ 4 } then in assumin g that θ i , i = 1 , . . . , 4 are exh austi ve and exclusi ve, one gets D Θ 2 = { X,Y } = {∅ , X ∩ Y , X , Y , X ∪ Y } = {∅ , { θ 1 } ∩ { θ 2 , θ 3 , θ 4 } | {z } X ∩ Y = ∅ | {z } ∅ , { θ 1 } | {z } X , { θ 2 , θ 3 , θ 4 } | {z } Y , {{ θ 1 } , { θ 2 , θ 3 , θ 4 }} | {z } X ∪ Y } = {∅ , { θ 1 } | {z } X , { θ 2 , θ 3 , θ 4 } | {z } Y , {{ θ 1 } , { θ 2 , θ 3 , θ 4 }} | {z } X ∪ Y } ≡ 2 Θ 2 = { X,Y } Therefore , we see that D Θ 2 = { X,Y } ≡ 2 Θ 2 = { X,Y } becaus e the e xclusi vity constraint X ∩ Y = ∅ holds sin ce on e has assumed X , { θ 1 } and Y , { θ 2 , θ 3 , θ 4 } with exh austi ve and exclusi ve granules θ i , i = 1 , . . . , 4 . If the granules θ i , i = 1 , . . . , 4 are not assumed exclusi ve, then of course the expres sions of hyper -power sets cannot be simpliﬁed and one would ha ve: D Θ 1 = { A,B } = {∅ , A ∩ B , A, B , A ∪ B } = {∅ , ( θ 1 ∪ θ 2 ∪ θ 3 ) ∩ ( θ 2 ∪ θ 4 ) , θ 1 ∪ θ 2 ∪ θ 3 , θ 2 ∪ θ 4 , θ 1 ∪ θ 2 ∪ θ 3 ∪ θ 4 } 6 = 2 Θ 1 = { A,B } D Θ 2 = { X,Y } = {∅ , X ∩ Y , X, Y , X ∪ Y } = {∅ , θ 1 ∩ ( θ 2 ∪ θ 3 ∪ θ 4 ) , θ 1 , θ 2 ∪ θ 3 ∪ θ 4 , θ 1 ∪ θ 2 ∪ θ 3 ∪ θ 4 } 6 = 2 Θ 2 = { X,Y } Shafer’ s model of a frame : More generally , when all the elements of a gi ven frame Θ are kno w n (or are assumed to be) truly exclusi ve, then the hyper -po wer set D Θ reduce s to the classic al power set 2 Θ . Theref ore, worki ng on po wer set 2 Θ as Glenn Shafer has proposed in his Mathematical Theory of Evidence [24]) is equi val ent to work on hyper -power set D Θ with the assumption that all elements of the frame are exclus i ve. This is what w e call Shafer’ s model of the frame Θ , written M 0 (Θ) , e ven if such model/assumptio n has not been clearly stated expli citly by Shafer himself in his milestone book. • The super -power set : S Θ , (Θ , ∪ , ∩ , c ( . )) The notion of super -po wer set has been intro duced by Smaran dache in the Chapt er 8 of [34 ]. It correspo nds actual ly to the theoretica l constr uction of the power set of the minimal 3 reﬁned frame Θ r ef of Θ . Θ genera tes S Θ under operators ∪ , ∩ and complementation c ( . ) . S Θ = (Θ , ∪ , ∩ , c ( . )) is a Boolean algebra with respect to the u nion, i ntersec tion and complement ation. T herefo re work ing with the super -po wer set is equi vale nt to work with a minimal theoret ical reﬁned frame Θ r ef satisfy ing Shafer’ s model. More precisely , S Θ is deﬁned as the set of all composit e propos itions/ subsets built from elements of Θ with ∪ , ∩ and c ( . ) opera tors such that: 1. ∅ , θ 1 , . . . , θ n ∈ S Θ . 2. If A, B ∈ S Θ , then A ∩ B ∈ S Θ , A ∪ B ∈ S Θ . 3. If A ∈ S Θ , then c ( A ) ∈ S Θ . 4. No other elements belong to S Θ , ex cept those obtaine d by using rules 1, 2 and 3. 3 The minimality refers here to the cardinality of the reﬁned frames. 6 As reported in [32], a similar gen eraliza tion has bee n pre viously use d i n 1993 by Guan an d Bell [14] f or the Dempster -Shafer rule using pro positio ns in sequent ial logic and reintrod uced in 1994 by Paris in his bo ok [20] , page 4. Example of a super -power set : Let’ s consider t he frame Θ = { θ 1 , θ 2 } and le t’ s a ssume θ 1 ∩ θ 2 6 = ∅ , i .e. θ 1 and θ 2 are not disj oint acc ording to Fig. 1 where A , p 1 denote s the part o f θ 1 belong ing only to θ 1 ( p stands her e for p art ), B , p 2 denote s the part of θ 2 belong ing only to θ 2 and C , p 12 denote s the part of θ 1 and θ 2 belong ing to both. In this exampl e, S Θ= { θ 1 ,θ 2 } is then gi ven by S Θ = {∅ , θ 1 ∩ θ 2 , θ 1 , θ 2 , θ 1 ∪ θ 2 , c ( ∅ ) , c ( θ 1 ∩ θ 2 ) , c ( θ 1 ) , c ( θ 2 ) , c ( θ 1 ∪ θ 2 ) } where c ( . ) is the complement in Θ . Since c ( ∅ ) = θ 1 ∪ θ 2 and c ( θ 1 ∪ θ 2 ) = ∅ , the super -po w er set is actually gi ven by S Θ = {∅ , θ 1 ∩ θ 2 , θ 1 , θ 2 , θ 1 ∪ θ 2 , c ( θ 1 ∩ θ 2 ) , c ( θ 1 ) , c ( θ 2 ) } Let’ s no w conside r the minimal reﬁnement Θ r ef = { A, B , C } of Θ built by splitting the granules θ 1 and θ 2 depict ed on the prev ious V enn di agram into disjoin t parts (i.e. Θ r ef satisﬁes the Shafer’ s model) as follo ws: Θ θ 1 A , p 1 θ 2 B , p 2 C , p 12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fig. 1: V enn diagra m of a free DS m model for a 2D frame . θ 1 = A ∪ C , θ 2 = B ∪ C, θ 1 ∩ θ 2 = C Then the classica l po wer set of Θ r ef is gi ven by 2 Θ r ef = {∅ , A, B , C , A ∪ B , A ∪ C, B ∪ C , A ∪ B ∪ C } W e see that we can deﬁne easily a one-to-o ne correspon dence, writ ten ∼ , between all the elements of the super -po w er set S Θ and the elements of the po wer set 2 Θ r ef as follo ws: ∅ ∼ ∅ , ( θ 1 ∩ θ 2 ) ∼ C, θ 1 ∼ ( A ∪ C ) , θ 2 ∼ ( B ∪ C ) , ( θ 1 ∪ θ 2 ) ∼ ( A ∪ B ∪ C ) c ( θ 1 ∩ θ 2 ) ∼ ( A ∪ B ) , c ( θ 1 ) ∼ B , c ( θ 2 ) ∼ A Such one-to- one correspond ence between the elements of S Θ and 2 Θ r ef can be deﬁned for any cardina lity | Θ | ≥ 2 of the frame Θ and thus one can consid er S Θ as the mathematical constructio n of the po wer set 2 Θ r ef of the m inimal reﬁnement of the frame Θ . Of cou rse, when Θ already satisﬁes Shafer ’ s mod el, t he hyper -po w er set and the super -po wer set coincide w ith the classica l power set of Θ . It is worth to note that ev en if we hav e a mathematical tool to built the minimal reﬁned frame satisf ying Shafer’ s model, it doesn’ t mean necessary 7 that one must work with this super -po w er set in general in real appl ication s because most of the times the elements /granu les of S Θ ha ve no clear physic al meaning, not to mention the drastic increas e of the complex ity since one has 2 Θ ⊆ D Θ ⊆ S Θ and | 2 Θ | = 2 | Θ | < | D Θ | < | S Θ | = 2 | Θ r ef | = 2 2 | Θ | − 1 (1) T ypically , | Θ | = n | 2 Θ | = 2 n | D Θ | | S Θ | = | 2 Θ r ef | = 2 2 n − 1 2 4 5 2 3 = 8 3 8 19 2 7 = 128 4 16 167 2 15 = 32768 5 32 7580 2 31 = 21474 83648 T able 2: Cardinalit ies of 2 Θ , D Θ and S Θ . In summary , DSmT of fers truly the possibil ity to buil d and to wor k on reﬁned frames and to deal with the complemen t whenev er necessary , but in most of appli cations either the frame Θ is already b uilt/chosen to satisfy Shafer’ s m odel or the reﬁned granule s ha ve no clear physic al meaning w hich ﬁnally pre vent to be consid ered/a ssessed indiv idually so that working on the hyper -po wer set is usually sufﬁci ent for dealing with uncert ain imprecise (quanti tati ve or qualitati ve) and highly conﬂictin g source s of evide nces. W orking with S Θ is actually very similar to working with 2 Θ in the sense that in both cases w e work with classical po wer sets; the only diffe rence is th at when working with S Θ we h a ve implicitly switc hed from the or iginal frame Θ repre- sentat ion to a minimal reﬁne ment Θ r ef repres entatio n. T herefo re, in the sequel we focus ou r disc ussion s based mainly on hyp er -power set rath er than (super -) power s et which has alre ady been the basis for t he de velopmen t of D ST . But as already mentioned , DSmT can easily deal with belief functio ns deﬁned on 2 Θ or S Θ similarly as those deﬁned on D Θ . Generic notatio n : In the seq uel, w e use the ge neric notat ion G Θ for deno ting the se ts (po wer set, hype r -po wer set and super -po wer set) on which the belief functio ns are deﬁned. Remark on the logical r eﬁnement : The reﬁnement in logic theory presente d recently by Cholvy in [2] was actual ly proposed in nineties by a Guan and Bell [14] and by Paris [20]. T his reﬁnement is isomorph ic to the reﬁnement in set theory done by many researchers. If Θ = { θ 1 , θ 2 , θ 3 } is a language where the propositio nal v ariables are θ 1 , θ 2 , θ 3 , Cholvy conside rs all 8 possible logical combinati ons of propo sitions θ i ’ s or negatio ns of θ i ’ s (called interpreta tions), and deﬁnes the 8 = 2 3 disjoi nt parts/p ropos itions of the V enn diag ram in F ig. 2 [one also conside rs as a part the ne gation of the total ignoran ce] in the set theory , so that: i 1 = θ 1 ∧ θ 2 ∧ θ 3 i 2 = θ 1 ∧ θ 2 ∧ ¬ θ 3 i 3 = θ 1 ∧ ¬ θ 2 ∧ θ 3 i 4 = θ 1 ∧ ¬ θ 2 ∧ ¬ θ 3 i 5 = ¬ θ 1 ∧ θ 2 ∧ ∧ θ 3 i 6 = ¬ θ 1 ∧ θ 2 ∧ ¬ θ 3 i 7 = ¬ θ 1 ∧ ¬ θ 2 ∧ θ 3 i 8 = ¬ θ 1 ∧ ¬ θ 2 ∧ ¬ θ 3 where ¬ θ i means the neg ation of θ i . 8 Θ θ 1 θ 1 ∧ ¬ θ 2 ∧ ¬ θ 3 p 1 θ 3 θ 2 ¬ θ 1 ∧ θ 2 ∧ ¬ θ 3 p 2 ¬ θ 1 ∧ ¬ θ 2 ∧ θ 3 p 3 ¬ θ 1 ∧ θ 2 ∧ θ 3 p 23 θ 1 ∧ θ 2 ∧ θ 3 p 123 θ 1 ∧ θ 2 ∧ ¬ θ 3 p 12 θ 1 ∧ ¬ θ 2 ∧ θ 3 p 13 ¬ θ 1 ∧ ¬ θ 2 ∧ ¬ θ 3 p 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fig. 2: V enn diagram of the free DSm model for a 3D frame. Because of Shafer’ s equi vale nce of subset s and proposi tions, Cholvy’ s logical reﬁnement is strictly equiv - alent to the reﬁnement we did alrea dy in 2006 in deﬁning S Θ - see Chap. 8 of [34] - b ut in the set theory frame work. W e did it using Smarandache’ s codiﬁcatio n (easy to unders tand and read) in the followin g way: - each V enn diagram disjoint part p ij , or p ij k repres ents respecti vely the intersection of p i and p j only , or p i and p j and p k only , etc; while the complement of the total ignorance is considered p 0 [ p stands for part]. Thus, we hav e an easier and cle arer rep resenta tion in DSmT than in C holvy ’ s logical repr esenta tion. While the reﬁnement in DST using logica l a pproac h for n very lar ge is very hard, w e can simply consider in the DSmT the super -po wer set S Θ = (Θ , ∪ , ∩ , c ( . )) . So, in DSmT represe ntatio n the disjoin t parts are noted as follo ws: p 123 = θ 1 ∧ θ 2 ∧ θ 3 = i 1 p 12 = θ 1 ∧ θ 2 ∧ ¬ θ 3 = i 2 p 13 = θ 1 ∧ ¬ θ 2 ∧ θ 3 = i 3 p 23 = ¬ θ 1 ∧ θ 2 ∧ θ 3 = i 5 p 1 = θ 1 ∧ ¬ θ 2 ∧ ¬ θ 3 = i 4 p 2 = ¬ θ 1 ∧ θ 2 ∧ ¬ θ 3 = i 6 p 3 = ¬ θ 1 ∧ ¬ θ 2 ∧ θ 3 = i 7 p 0 = ¬ θ 1 ∧ ¬ θ 2 ∧ ¬ θ 3 = i 8 As s eeing, in Smaran dache’ s codiﬁcation a disj oint V enn dia gram part is equal to the i ntersec tion of single- tons whose inde xes sho w up as indexe s of the V enn part; for example in p 12 case inde xes 1 and 2, inters ected with the complement of the missing index es, in this case inde x 3 is m issing . Smarandach e’ s codiﬁcatio n can easily transf orm any set from S Θ into its canoni cal disjunc ti ve normal form. For e xample, θ 1 = p 1 ∪ p 12 ∪ p 13 ∪ p 123 (i.e. all V enn diagram disjo int parts that contai n the index “1” in the ir inde xes ; such inde xes from S Θ are 1, 12, 13, 123) can be expre ssed as θ 1 = ( θ 1 ∩ c ( θ 2 ) ∩ c ( θ 3 )) ∪ ( θ 1 ∩ θ 2 ∩ c ( θ 3 ))( θ 1 ∩ c ( θ 2 ) ∩ θ 3 ) ∪ ( θ 1 ∩ θ 2 ∩ θ 3 ) where the set va lues of each part was take n from the abov e table. 9 θ 1 ∧ θ 2 = p 12 ∪ p 123 (i.e. all V enn d iagram disjoi nt parts t hat co ntain th e index “12” i n their i nde xes) e quals to ( θ 1 ∧ θ 2 ∧ ¬ θ 3 ) ∨ ( θ 1 ∧ θ 2 ∧ θ 3 ) . The reﬁnement bas ed on V enn Dia gram, bec omes very hard and almost imp ossible when the cardinal of Θ , n , is larg e and all intersec tions are non-empty (the free m odel) . Suppose n = 20 , or ev en bigger , and we hav e the free model. Ho w can we construct a V enn D iagram where to show all possible intersection s of 20 sets? Its geometri cal ﬁgure would be very hard to design and very hard to read (you don’t identify well each disjoint part of a such V enn D iagram to what intersect ion of sets it belongs to). The lar ger is n , the more dif ﬁ cult is the reﬁnement. Fortunate ly , based on S maranda che’ s codiﬁcati on, w e can algebr aically design in an easy way for all such intersecti ons (for exampl e, if n is very big, we can use computer programs to make combinations of inde xes { 1 , 2 , ..., n } taken in groups or 1, of 2, ..., or of n elemen ts each), so the reﬁnement shou ld not be a big problem from the progra mming point of vie w , but we must alway s keep in mind if such reﬁnement is really necess ary and if it has (or not) a deep physical interpr etation and justiﬁcation for the problem under consid eration . The assertio n in [2], upo n Milan Daniel’ s, that hybrid D Sm rule is equi v alent to Dubois-Prade rule is untrue, since in dynamic fusion they giv e differ ent results. Such example has been already gi ven in [7] and is reported in secti on 2.6.3 for the sake of clariﬁcati on for the readers. The asser tion in [2 ] that “fro m an e xpressi vity point of vie w DSmT is equiv alent to DST” is par tially true si nce this i dea is true when the re ﬁnement is p ossible (not alw ays it is practicall y/phys ically possible), and ev en w hen the spaces we work on, S Θ = 2 Θ r ef , where the hypot heses are exclusi ve, DSm T offers the adva ntage that the reﬁnement is already done (it is not necessary for the user to do (or implicitly presuppose) it as in DST). Also, DSmT accepts from the very begi nning the possib ility to deal with non-exc lusi ve hypothe ses and of course it can a fortiori deal with sets of exclusi ve hy- pothes is and work either on 2 Θ or 2 Θ r ef whene ver necessary , while DST ﬁ rst requires implicitly to work w ith exc lusi ve hypoth eses only . The main disti nction s between DS mT and DST ar e summarized by the follo wing points: 1. The reﬁnement is not always (physically) p ossible, esp ecially for e lements from the frame of discern ment whose frontie rs are not clear , su ch as: colors , v ague sets, un clear hypotheses, etc. i n the frame of discer nment; D ST does not ﬁt well for wor king in such cases, while DSmT does; 2. Even in the case when the frame of discernment can be reﬁned (i.e. the atomic elements of the frame ha ve all a distinct physica l meaning), it is still easier to use DS mT than DS T since in DSmT frame work the reﬁnement is don e automatically by the mathematical construction of the super -po wer set; 3. DSmT off ers better fusion rules, for example Proport ional Conﬂict redistr ib ution Rule # 5 (PCR5) - presen ted in the s equel - is bette r than Dempste r’ s rule; hy brid DSm rule (DSmH) wor ks for the dy namic fusion , while D ubois -Prade fusion rule does not (DS mH is an ex tension of Dubois-Prad e rule); 4. DSmT off ers the best qualitat i ve opera tors (when working w ith label s) gi ving the most accurate and cohere nt results; 5. DSmT offe rs new interesti ng qua ntitati ve conditio ning rules (BCRs) and qualita ti ve con dition ing rul es (QBCRs), dif ferent from S hafer’ s conditioni ng rule (SCR ). SCR ca n be seen simply as a combination of a prior mass of belie f w ith the mass m ( A ) = 1 whenev er A is the condition ing ev ent; 6. DSmT propose s a new approach for workin g with imprecis e quanti tati ve or qualitati ve informatio n and not limited to inte rv al-valu ed belief structures as proposed generally in the literature [5, 6, 47]. 2.2 Notion of fr e e and hybrid DSm models Fre e DSm model : The elements θ i , i = 1 , . . . , n of Θ constitu te the ﬁnite set of hypothe ses/co ncepts charac- terizin g the fusion proble m under considerati on. When there is no constra int on the elements of the frame, we call this model the fr ee DSm m odel , written M f (Θ) . T his free DSm m odel allows to deal directly with fuzzy 10 concep ts which dep ict a co ntinuous and relativ e intrins ic natur e and which cannot be preci sely reﬁne d i nto ﬁne r disjoi nt information granules hav ing an absolute interp retatio n because of the unreac hable uni versal truth. In such case, the use of the hyp er -power set D Θ (without integri ty con straint s) is particula rly well adapted for deﬁning the belief function s one wants to combi ne. Shafer’ s m odel : In some fusion problems in vol ving discrete concepts, all the elements θ i , i = 1 , . . . , n of Θ can b e tru ly e xclusi ve. In su ch case, all the exc lusi vity constr aints on θ i , i = 1 , . . . , n hav e to be included in the pre vious model to characterize properly the true nature of the fusion problem and to ﬁ t it with the reality . By doing this, the hyper- po w er set D Θ as w ell as the super- po w er set S Θ reduce naturally to the classical po wer set 2 Θ and this consti tutes what we hav e called Shafer’ s m odel , denote d M 0 (Θ) . Shafer’ s model corresp onds actual ly to the most restrict ed hybrid DSm model. Hybrid DS m models : Bet ween the class of fusion problems corresp onding to the free DSm model M f (Θ) and the class of fusion problems correspo nding to Shafer’ s model M 0 (Θ) , there exists another wide class of hybrid fus ion probl ems in v olving in Θ both fuzzy contin uous concep ts and discrete hypoth eses. In such (hybri d) class, some exclu si vity constraints and possib ly some non-ex istenti al constraints (espec ially w hen worki ng on dynamic 4 fusion ) ha ve to be taken into account. Each hybrid fusion problem of this class will then be characteri zed by a proper hybrid DSm model denote d M (Θ) w ith M (Θ) 6 = M f (Θ) and M (Θ) 6 = M 0 (Θ) . In an y fusion proble ms, we consid er as primordia l at the v ery be ginning and before combin ing information exp ressed as be lief func tions to d eﬁne clear ly the pro per frame Θ of the g i ven p roblem an d to choose e xplici tly its correspon ding model one wants to wor k with. O nce this is done, the seco nd important point is to select the proper set 2 Θ , D Θ or S Θ on which the belief functions w ill be deﬁned. T he third important point will be the choice of an efﬁcient rule of combination of belief functions and ﬁ nally the criteria adopted for decision - making. In the seque l, w e fo cus our presentatio n mainly on hyper -power set D Θ (unles s speciﬁed) since it the most interes ting new aspect of DSmT for readers already familia r with DS T framew ork, but a fortiori w e can work similarly on classica l po wer set 2 Θ if S hafer’ s model holds, and ev en on 2 Θ r ef (the power set of the m inimal reﬁned frame) whene ver one wants to use it and if possibl e. Examples of models fo r a frame Θ : • L et’ s consi der the 2D problem w here Θ = { θ 1 , θ 2 } with D Θ = {∅ , θ 1 ∩ θ 2 , θ 1 , θ 2 , θ 1 ∪ θ 2 } and assume no w that θ 1 and θ 2 are truly exclusi ve (i.e. S hafer’ s model M 0 holds) , then becaus e θ 1 ∩ θ 2 M 0 = ∅ , one gets D Θ = {∅ , θ 1 ∩ θ 2 M 0 = ∅ , θ 1 , θ 2 , θ 1 ∪ θ 2 } = {∅ , θ 1 , θ 2 , θ 1 ∪ θ 2 } ≡ 2 Θ . • As another simple e xample of hy brid DSm mod el, let’ s consid er the 3D c ase with the frame Θ = { θ 1 , θ 2 , θ 3 } with the model M 6 = M f in w hich we force all possib le conjunc tions to be empty , bu t θ 1 ∩ θ 2 . This hybrid DSm mod el is then represen ted with the V enn diagram on F ig. 3 (where bounda ries of inter section of θ 1 and θ 2 are not precise ly deﬁned if θ 1 and θ 2 repres ent only fuzzy conc epts like smallness and tallnes s by example ). 2.3 Generalized belief functions From a genera l frame Θ , we deﬁne a map m ( . ) : G Θ → [0 , 1] associat ed to a gi ven body of evi dence B as m ( ∅ ) = 0 and X A ∈ G Θ m ( A ) = 1 (2) The quantity m ( A ) is called the gene rali zed basic belief assignment/mass (gbba) of A . 4 i.e. when the frame Θ and /or the mo del M is ch anging with time. 11 ✫✪ ✬✩ ✫✪ ✬✩ ✫✪ ✬✩ ❅ ❘ θ 1  ✠ θ 2 ✛ θ 3 12 3 2 1 Fig. 3: V enn diagra m of a DSm hybri d model for a 3D frame. The gener alized belief and plausibility functions are deﬁned in almost the same manner as within DS T , i.e. Bel ( A ) = X B ⊆ A B ∈ G Θ m ( B ) Pl ( A ) = X B ∩ A 6 = ∅ B ∈ G Θ m ( B ) (3) W e recall that G Θ is the generic notation for the set on which the gbba is deﬁned ( G Θ can be 2 Θ , D Θ or ev en S Θ depen ding on the mode l chosen for Θ ). These deﬁnitions are compatible with the deﬁnitions of the classic al beli ef functions in DST frame work w hen G Θ = 2 Θ for fusion proble ms w here Shafer’ s model M 0 (Θ) holds. W e still hav e ∀ A ∈ G Θ , Bel ( A ) ≤ Pl ( A ) . Note that when working with the free DSm model M f (Θ) , one has alwa ys Pl ( A ) = 1 ∀ A 6 = ∅ ∈ ( G Θ = D Θ ) which is normal . Example : Let’ s consider the simple frame Θ = { A, B } , then depending on the m odel we choose for G Θ , one will consider either: • G Θ as the po wer set 2 Θ and therefo re: m ( A ) + m ( B ) + m ( A ∪ B ) = 1 • G Θ as the hyper -po wer set D Θ and therefo re: m ( A ) + m ( B ) + m ( A ∪ B ) + m ( A ∩ B ) = 1 • G Θ as the super -po wer set S Θ and therefo re: m ( A ) + m ( B ) + m ( A ∪ B ) + m ( A ∩ B ) + m ( c ( A )) + m ( c ( B )) + m ( c ( A ) ∪ c ( B )) = 1 2.4 The classic DSm rule of combination When the free DS m model M f (Θ) holds for the fusion problem under consid eratio n, the classic DS m rule of combina tion m M f (Θ) ≡ m ( . ) , [ m 1 ⊕ m 2 ]( . ) of two independen t 5 source s of evi dences B 1 and B 2 ov er the same frame Θ with belief function s Bel 1 ( . ) and Bel 2 ( . ) associated w ith gbba m 1 ( . ) and m 2 ( . ) corresponds to the conjun cti ve consensus of the sources. It is giv en by [31]: ∀ C ∈ D Θ , m M f (Θ) ( C ) ≡ m ( C ) = X A,B ∈ D Θ A ∩ B = C m 1 ( A ) m 2 ( B ) (4) Since D Θ is closed under ∪ and ∩ set operato rs, this ne w rule of combinatio n guarantees that m ( . ) is a proper generalize d belief assign ment, i.e. m ( . ) : D Θ → [0 , 1] . This rule of combination is commutativ e and associ ati ve and can alwa ys be used for the fusion of sources in v olving fuzzy concepts when free DSm m odel holds for the problem under consid eration . This rule has been exten ded for s > 2 sources in [31]. 5 While independence is a dif ﬁcult concept to deﬁne in all theories ma naging epistemic uncertainty , we follo w here the interpretatio n of Smets in [37] and [38], p. 2 85 an d consider that two sources of evidence are in depende nt ( i.e distinct and noninter acting) if each leaves one totally igno rant abou t the par ticular value the other will take. 12 Accordin g to T able 2, thi s classic D Sm rule of combin ation looks very expen si ve in terms of computation s and memory size due to the huge number of elements in D Θ when the cardinal ity of Θ increases. This remark is howe ver valid only if the cores (the set of focal elements of gbba) K 1 ( m 1 ) and K 2 ( m 2 ) coincid e with D Θ , i.e. when m 1 ( A ) > 0 and m 2 ( A ) > 0 for all A 6 = ∅ ∈ D Θ . Fortun ately , it is important to note here that in most of the practi cal applicat ions the sizes of K 1 ( m 1 ) and K 2 ( m 2 ) are much smaller than | D Θ | because bod ies of evide nce generally allocate their basic belief assignments only ove r a subset of the hyper -po wer set. This makes things easier for the implementat ion of the classic DS m rule (4). The DSm rule is actually very easy to implement. It sufﬁces for each focal element of K 1 ( m 1 ) to multiply it with the focal elements of K 2 ( m 2 ) and then to pool all combinations which are equi vale nt under the algebra of sets. While very costly in term on memory storage in the worst case (i.e. when all m ( A ) > 0 , A ∈ D Θ or A ∈ 2 Θ r ef ), the DSm rule howe ver requir es much smaller memory sto rage than when wo rking with S Θ , i.e. working with a minimal reﬁne d frame satisfy ing Shafer’ s model. In most fusion applications only a small subset of elements of D Θ ha ve a non null basic belief mass be- cause all the commitments are just usually impossible to obtain precisely when the dimensio n of the problem increa ses. Thus, it is not necessary to generate and kee p in memory all elements of D Θ (or e vent ually S Θ ) but only those which ha ve a positi ve belief mass. Howe ver there is a real technic al challenge on how to manage ef ﬁ cientl y all elements of the hyper -power set. This problem is obviou sly much more difﬁcult w hen trying to work on a reﬁned frame of discern ment Θ r ef if one really prefers to use Dempster -Shafer theory and apply Dempster’ s rule of combination. It is important to keep in mind that the ultimate and minimal reﬁned frame consis ting in exha usti ve and ex clusi ve ﬁnite set o f reﬁned exclu sive hyp otheses is just imposs ible to justi fy and to deﬁne precisel y for all problems dealin g with fuzzy and ill-deﬁned continuo us concepts. A discussion on reﬁnement with an exampl e has be included in [31]. 2.5 The hybrid DSm rule of combination When the free DS m model M f (Θ) does not hold due to the true nature of the fusion problem under consid- eration w hich requires to take into account some kno wn integrity constraints , one has to work with a proper hybrid DSm model M (Θ) 6 = M f (Θ) . In such case, the hybrid DSm rule (DSm H) of combina tion based on the chosen hybrid DSm model M (Θ) for k ≥ 2 indepen dent sources of information is deﬁned for all A ∈ D Θ as [31]: m D S mH ( A ) = m M (Θ) ( A ) , φ ( A ) h S 1 ( A ) + S 2 ( A ) + S 3 ( A ) i (5) where all sets in volv ed in formulas are in the canonic al form and φ ( A ) is the cha racteristi c non-emptine ss functi on of a set A , i.e. φ ( A ) = 1 if A / ∈ ∅ and φ ( A ) = 0 otherwise , where ∅ , { ∅ M , ∅} . ∅ M is the set of all elements of D Θ which hav e been forced to be empty throug h the constr aints of the m odel M and ∅ is the classic al/uni versa l empty set. S 1 ( A ) ≡ m M f ( θ ) ( A ) , S 2 ( A ) , S 3 ( A ) are deﬁned by S 1 ( A ) , X X 1 ,X 2 ,...,X k ∈ D Θ X 1 ∩ X 2 ∩ ... ∩ X k = A k Y i =1 m i ( X i ) (6) S 2 ( A ) , X X 1 ,X 2 ,...,X k ∈ ∅ [ U = A ] ∨ [( U ∈ ∅ ) ∧ ( A = I t )] k Y i =1 m i ( X i ) (7) S 3 ( A ) , X X 1 ,X 2 ,...,X k ∈ D Θ X 1 ∪ X 2 ∪ ... ∪ X k = A X 1 ∩ X 2 ∩ ... ∩ X k ∈ ∅ k Y i =1 m i ( X i ) (8) with U , u ( X 1 ) ∪ u ( X 2 ) ∪ . . . ∪ u ( X k ) where u ( X ) is the union of all θ i that compose X , I t , θ 1 ∪ θ 2 ∪ . . . ∪ θ n is the total ignoranc e. S 1 ( A ) correspon ds to the classic DSm rule for k indepen dent source s based on the free DSm mod el M f (Θ) ; S 2 ( A ) represents the mass of all rel ati vely and absolu tely empty sets which is transferred 13 to the total or relati ve ignorances associat ed with non existenti al constraints (if any , like in some dynamic proble ms); S 3 ( A ) tran sfers the sum of relati vely empty sets dir ectly onto the canonical disjun cti ve form of non-emp ty sets. The hybrid DSm rule of combination general izes the clas sic DSm rule of combination and is no t equ i v alent to D empter’ s rule. It works for any models (the free DSm m odel, Shafer’ s model or any other hybrid models) when manipulatin g pr ecise generalize d (or eve ntually classica l) basic belief functions. An extension of this rule for the combinat ion of impr ecise generalized (or e ven tually classic al) basic belief functions is presented in next section . As already stated, in DS mT frame work it is also possib le to deal directly with complements if nece ssary dependin g on the problem under consideratio n and the info rmation prov ided by the sources of e videnc e themselv es. The ﬁrst an d simplest way is to work w ith S Θ on S hafer’ s mod el when a m inimal reﬁnement is possible and makes sense. The second way is to deal with partially kno wn frame and introduce directly the complementary hypot heses into the frame itself. By example, if one knows only two hypotheses θ 1 , θ 2 and their complements ¯ θ 1 , ¯ θ 2 , then w e can choose switch from original frame Θ = { θ 1 , θ 2 } to the ne w frame Θ = { θ 1 , θ 2 , ¯ θ 1 , ¯ θ 2 } . In such case, w e don’ t n ecessa rily assume that ¯ θ 1 = θ 2 and ¯ θ 2 = θ 1 becaus e ¯ θ 1 and ¯ θ 2 may include other u nkno wn hypot heses we ha ve no in formatio n about (case of partial kno wn frame). M ore general ly , in DSmT f rame work, it is not nece ssary that the frame is b uilt on pure/simpl e (possib ly vague ) hypothese s θ i as usu ally done in all theories managing uncerta inty . The frame Θ can also contain directly as elements conjunction s and/or disjun ctions (or mixed propositio ns) and negatio ns/complements of pure hypothese s as well. The DSm rules also work in such non-class ic frames because DS mT works on any distrib uti ve lattice built from Θ any where Θ is deﬁned. 2.6 Examples of combination rules Here are some numerical example s on results obtained by DSm rules of combinatio n. More examp les can be found in [31]. 2.6.1 Example with Θ = { θ 1 , θ 2 , θ 3 , θ 4 } Let’ s consider the frame of discern ment Θ = { θ 1 , θ 2 , θ 3 , θ 4 } , two independe nt experts, and the two follo wing bbas m 1 ( θ 1 ) = 0 . 6 m 1 ( θ 3 ) = 0 . 4 m 2 ( θ 2 ) = 0 . 2 m 2 ( θ 4 ) = 0 . 8 repres ented in terms of mass matrix M =  0 . 6 0 0 . 4 0 0 0 . 2 0 0 . 8  • Dempster’ s rule canno t be applied because: ∀ 1 ≤ j ≤ 4 , one gets m ( θ j ) = 0 / 0 (unde ﬁned!). • But the classi c DSm rule work s because one obtains: m ( θ 1 ) = m ( θ 2 ) = m ( θ 3 ) = m ( θ 4 ) = 0 , and m ( θ 1 ∩ θ 2 ) = 0 . 12 , m ( θ 1 ∩ θ 4 ) = 0 . 48 , m ( θ 2 ∩ θ 3 ) = 0 . 08 , m ( θ 3 ∩ θ 4 ) = 0 . 32 (partial para- dox es/con ﬂicts). • Suppose no w one ﬁ nds out that all intersection s are empty (Shafer’ s model), then one applies the hybrid DSm rule and one gets (index h stands here for hybrid rule): m h ( θ 1 ∪ θ 2 ) = 0 . 12 , m h ( θ 1 ∪ θ 4 ) = 0 . 48 , m h ( θ 2 ∪ θ 3 ) = 0 . 08 and m h ( θ 3 ∪ θ 4 ) = 0 . 32 . 2.6.2 Gener alization of Zadeh’ s exa m ple with Θ = { θ 1 , θ 2 , θ 3 } Let’ s consider 0 < ǫ 1 , ǫ 2 < 1 be two very tiny positi ve numbers (close to zero), the frame of discernment be Θ = { θ 1 , θ 2 , θ 3 } , ha ve two exp erts (independ ent sources of ev idence s 1 and s 2 ) gi ving the belief m asses m 1 ( θ 1 ) = 1 − ǫ 1 m 1 ( θ 2 ) = 0 m 1 ( θ 3 ) = ǫ 1 m 2 ( θ 1 ) = 0 m 2 ( θ 2 ) = 1 − ǫ 2 m 2 ( θ 3 ) = ǫ 2 14 From no w on, we pref er to use matrices to describe the masses, i.e.  1 − ǫ 1 0 ǫ 1 0 1 − ǫ 2 ǫ 2  • Using Dempster’ s rule of combin ation, one gets m ( θ 3 ) = ( ǫ 1 ǫ 2 ) (1 − ǫ 1 ) · 0 + 0 · (1 − ǫ 2 ) + ǫ 1 ǫ 2 = 1 which is absurd (or at least counter -intuiti ve). Note that whatev er positi ve value s for ǫ 1 , ǫ 2 are, Demp- ster’ s rule of combinati on provide s always the s ame result (on e) which is abno rmal. T he on ly accept able and correct result obtained by Dempster’ s rule is really obtained only in the trivi al case when ǫ 1 = ǫ 2 = 1 , i.e. when both source s agree in θ 3 with certainty which is obvi ous. • Using the D Sm rule of combinatio n based on free-DSm model, one gets m ( θ 3 ) = ǫ 1 ǫ 2 , m ( θ 1 ∩ θ 2 ) = (1 − ǫ 1 )(1 − ǫ 2 ) , m ( θ 1 ∩ θ 3 ) = (1 − ǫ 1 ) ǫ 2 , m ( θ 2 ∩ θ 3 ) = (1 − ǫ 2 ) ǫ 1 and t he others are zero w hich appears more relia ble/tru stable. • Going back to Shafer’ s model and using the hybrid DSm rule of combination, one gets m ( θ 3 ) = ǫ 1 ǫ 2 , m ( θ 1 ∪ θ 2 ) = (1 − ǫ 1 )(1 − ǫ 2 ) , m ( θ 1 ∪ θ 3 ) = (1 − ǫ 1 ) ǫ 2 , m ( θ 2 ∪ θ 3 ) = (1 − ǫ 2 ) ǫ 1 and the others are zero. Note that in the special case when ǫ 1 = ǫ 2 = 1 / 2 , one has m 1 ( θ 1 ) = 1 / 2 m 1 ( θ 2 ) = 0 m 1 ( θ 3 ) = 1 / 2 m 2 ( θ 1 ) = 0 m 2 ( θ 2 ) = 1 / 2 m 2 ( θ 3 ) = 1 / 2 Dempster’ s rule of combinations still yields m ( θ 3 ) = 1 while the hybrid DSm rule based on the same S hafer’ s model yields no w m ( θ 3 ) = 1 / 4 , m ( θ 1 ∪ θ 2 ) = 1 / 4 , m ( θ 1 ∪ θ 3 ) = 1 / 4 , m ( θ 2 ∪ θ 3 ) = 1 / 4 which is normal . 2.6.3 Comparis on with Smets, Y ager and Duboi s & Prade rules W e compare the results provided by DSmT rules and the main common rules of combinatio n on the follo w - ing very simple numerical example where only 2 independ ent sourc es (a priori assumed equally reliable) are in v olved and pro viding their belief initially on the 3D frame Θ = { θ 1 , θ 2 , θ 3 } . It is assumed in this exa mple that Shafer’ s model holds and thus the belief assignment s m 1 ( . ) and m 2 ( . ) do not commit belief to interna l conﬂictin g informat ion. m 1 ( . ) and m 2 ( . ) are chosen as follo ws: m 1 ( θ 1 ) = 0 . 1 m 1 ( θ 2 ) = 0 . 4 m 1 ( θ 3 ) = 0 . 2 m 1 ( θ 1 ∪ θ 2 ) = 0 . 3 m 2 ( θ 1 ) = 0 . 5 m 2 ( θ 2 ) = 0 . 1 m 2 ( θ 3 ) = 0 . 3 m 2 ( θ 1 ∪ θ 2 ) = 0 . 1 These belief masses are usually represe nted in the form of a belief mass matrix M gi ven by M =  0 . 1 0 . 4 0 . 2 0 . 3 0 . 5 0 . 1 0 . 3 0 . 1  (9) where inde x i for the ro w s correspo nds to the index of the source no. i and the index es j for columns of M corres pond to a giv en choice for enumeratin g the focal elements of all sources. In this particular example, in- dex j = 1 correspon ds to θ 1 , j = 2 correspon ds to θ 2 , j = 3 correspon ds to θ 3 and j = 4 correspo nds to θ 1 ∪ θ 2 . No w let’ s imagine that one ﬁnds out that θ 3 is actually truly empty becaus e some ext ra and certain kno wl- edge on θ 3 is recei ved by the fusion center . As example, θ 1 , θ 2 and θ 3 may correspon d to three suspect s (poten tial murders) in a police in vestiga tion, m 1 ( . ) and m 2 ( . ) correspo nds to two report s of independ ent w it- nesses , bu t it turns out that ﬁnally θ 3 has p rov ided a strong al ibi to the criminal police in vestiga tor once arrested 15 by the policemen . This situation correspon ds to set up a hybrid model M with the cons traint θ 3 M = ∅ . Let’ s examine the result of the fusion in such situation obtained by the Smets’, Y ager ’ s, Dubois & P rade’ s and hybrid DS m rules of combinations . First note that, based on the free DSm model, on e would get by applyi ng the classic DS m rule (den oted here by index D S mC ) the follo wing fusion result m D S mC ( θ 1 ) = 0 . 21 m D S mC ( θ 2 ) = 0 . 11 m D S mC ( θ 3 ) = 0 . 06 m D S mC ( θ 1 ∪ θ 2 ) = 0 . 03 m D S mC ( θ 1 ∩ θ 2 ) = 0 . 21 m D S mC ( θ 1 ∩ θ 3 ) = 0 . 13 m D S mC ( θ 2 ∩ θ 3 ) = 0 . 14 m D S mC ( θ 3 ∩ ( θ 1 ∪ θ 2 )) = 0 . 11 But becau se of the exclu si vity constraints (imposed here by the use of Shafer’ s model and by the non- exi stentia l constraint θ 3 M = ∅ ), the to tal co nﬂicting mas s is actually gi ven by k 12 = 0 . 06 + 0 . 21 + 0 . 13 + 0 . 14 + 0 . 11 = 0 . 65 . • If one applie s D empster’ s rule [24] (denote d here by inde x D S ), one gets: m D S ( ∅ ) = 0 m D S ( θ 1 ) = 0 . 21 / [1 − k 12 ] = 0 . 21 / [1 − 0 . 65] = 0 . 21 / 0 . 35 = 0 . 600000 m D S ( θ 2 ) = 0 . 11 / [1 − k 12 ] = 0 . 11 / [1 − 0 . 65] = 0 . 11 / 0 . 35 = 0 . 314286 m D S ( θ 1 ∪ θ 2 ) = 0 . 03 / [1 − k 12 ] = 0 . 03 / [1 − 0 . 65] = 0 . 03 / 0 . 35 = 0 . 085714 • If one applies S mets’ rule [39, 40] (i.e. the non normalize d v ersion of Dempster’ s rul e with the conﬂictin g mass trans ferred onto the empty set), one gets: m S ( ∅ ) = m ( ∅ ) = 0 . 65 ( conﬂictin g mass) m S ( θ 1 ) = 0 . 21 m S ( θ 2 ) = 0 . 11 m S ( θ 1 ∪ θ 2 ) = 0 . 03 • If one applie s Y ager’ s rule [48–50], one gets: m Y ( ∅ ) = 0 m Y ( θ 1 ) = 0 . 21 m Y ( θ 2 ) = 0 . 11 m Y ( θ 1 ∪ θ 2 ) = 0 . 03 + k 12 = 0 . 03 + 0 . 65 = 0 . 68 • If one applie s D ubois & Prade’ s rule [12], one gets because θ 3 M = ∅ : m D P ( ∅ ) = 0 (by deﬁnition of Dubois & Prade’ s rule) m D P ( θ 1 ) = [ m 1 ( θ 1 ) m 2 ( θ 1 ) + m 1 ( θ 1 ) m 2 ( θ 1 ∪ θ 2 ) + m 2 ( θ 1 ) m 1 ( θ 1 ∪ θ 2 )] + [ m 1 ( θ 1 ) m 2 ( θ 3 ) + m 2 ( θ 1 ) m 1 ( θ 3 )] = [0 . 1 · 0 . 5 + 0 . 1 · 0 . 1 + 0 . 5 · 0 . 3] + [0 . 1 · 0 . 3 + 0 . 5 · 0 . 2] = 0 . 21 + 0 . 13 = 0 . 34 m D P ( θ 2 ) = [0 . 4 · 0 . 1 + 0 . 4 · 0 . 1 + 0 . 1 · 0 . 3] + [0 . 4 · 0 . 3 + 0 . 1 · 0 . 2] = 0 . 11 + 0 . 14 = 0 . 25 16 m D P ( θ 1 ∪ θ 2 ) = [ m 1 ( θ 1 ∪ θ 2 ) m 2 ( θ 1 ∪ θ 2 )] + [ m 1 ( θ 1 ∪ θ 2 ) m 2 ( θ 3 ) + m 2 ( θ 1 ∪ θ 2 ) m 1 ( θ 3 )] + [ m 1 ( θ 1 ) m 2 ( θ 2 ) + m 2 ( θ 1 ) m 1 ( θ 2 )] = [0 . 30 . 1 ] + [0 . 3 · 0 . 3 + 0 . 1 · 0 . 2] + [0 . 1 · 0 . 1 + 0 . 5 · 0 . 4] = [0 . 03] + [0 . 0 9 + 0 . 02] + [0 . 01 + 0 . 20] = 0 . 03 + 0 . 11 + 0 . 21 = 0 . 35 No w if one adds up the masses, one gets 0 + 0 . 34 + 0 . 25 + 0 . 35 = 0 . 94 which is less than 1. T herefo re Dubois & Prade’ s rule of combinatio n d oes not work when a singleton, o r an union of singleto ns, becomes empty (in a dynamic fusion problem). The products of such empty-el ement columns of the mass matrix M are lost; this problem is ﬁxed in DSmT by the sum S 2 ( . ) in (5) which transfers these produc ts to the total or partial ignora nces. • Finally , if one applies DSm H rule , one gets bec ause θ 3 M = ∅ : m D S mH ( ∅ ) = 0 (by deﬁnitio n of DSmH) m D S mH ( θ 1 ) = 0 . 34 (same as m D P ( θ 1 ) ) m D S mH ( θ 2 ) = 0 . 25 (same as m D P ( θ 2 ) ) m D S mH ( θ 1 ∪ θ 2 ) = [ m 1 ( θ 1 ∪ θ 2 ) m 2 ( θ 1 ∪ θ 2 )] + [ m 1 ( θ 1 ∪ θ 2 ) m 2 ( θ 3 ) + m 2 ( θ 1 ∪ θ 2 ) m 1 ( θ 3 )] + [ m 1 ( θ 1 ) m 2 ( θ 2 ) + m 2 ( θ 1 ) m 1 ( θ 2 )] + [ m 1 ( θ 3 ) m 2 ( θ 3 )] = 0 . 03 + 0 . 11 + 0 . 21 + 0 . 06 = 0 . 35 + 0 . 06 = 0 . 41 6 = m D P ( θ 1 ∪ θ 2 ) W e can easily verify that m D S mH ( θ 1 ) + m D S mH ( θ 2 ) + m D S mH ( θ 1 ∪ θ 2 ) = 1 . In this exampl e, using the hybrid DS m rule, one transfers the produc t of the empty-element θ 3 column, m 1 ( θ 3 ) m 2 ( θ 3 ) = 0 . 2 · 0 . 3 = 0 . 06 , to m D S mH ( θ 1 ∪ θ 2 ) , which becomes equal to 0 . 35 + 0 . 06 = 0 . 41 . Clearly , DSmH rule doesn ’t pro vide the same result as Dubois and Prade’ s rule, but only when workin g on static frames of discer nment (restricted cases). 2.7 Fusion of imprecise beliefs In many fusion proble ms, it seems very difﬁcult (if not impossibl e) to hav e precise sources of evide nce gener - ating precise basic belief assignments (especiall y when belief functio ns are provi ded by human expe rts), and a more ﬂ exi ble plausible and paradoxi cal theory suppor ting imprecise information becomes necessary . In the pre vious sections, w e presente d the fusion of pr ecise uncertain and conﬂicting/p aradoxical gener alized basic belief assignments (gbba) in DSmT frame work. W e mean here by precise gbba, basic belief functi ons/mass es m ( . ) deﬁned precisely on the hyper -power set D Θ where each mass m ( X ) , where X belongs to D Θ , is repre- sented by only one real number belong ing to [0 , 1] such that P X ∈ D Θ m ( X ) = 1 . In this section, we present the DSm fusion rule for dealing with admissible impre cise gener alized basic belief assignmen ts m I ( . ) deﬁned as real subun itary interv als of [0 , 1] , or e ven more gen eral as real sub unitary sets [i.e. sets, not nece ssarily interv als]. An imprecise belief assignment m I ( . ) ov er D Θ is said admissib le if and only if there exist s for eve ry X ∈ D Θ at least one real number m ( X ) ∈ m I ( X ) such that P X ∈ D Θ m ( X ) = 1 . The idea to wor k with imprecis e belief structure s represen ted by real subset interv als of [0 , 1] is not new and has been in ves tigated in [5, 6, 16] and referen ces therein . The propos ed works av ailable in the literatu re, upon our kno wledge were limited only to sub-unitary interv al combinati on in the frame work of Tran sferab le Belief Model (TBM) de- vel oped by Smets [39, 40]. W e exten d the approa ch of Lamata & Moral and Denœux based on sub unitary interv al-v alued masses to subu nitary set-v alued masses; therefore the closed interv als used by Denœux to de- note imprecise masses are general ized to any sets includ ed in [0,1], i.e. in our case these sets can be unions 17 of (closed, open, or half-open/ half-cl osed) interv als and/ or scalars all in [0 , 1] . H ere, the propos ed extensi on is done in the context of DSmT frame work, although it can also apply directly to fusion of imprecise belief structu res within TBM as well if the user prefe rs to adop t T BM ra ther than DSmT . Before presentin g the general formula for the combination of generalize d imprecise belief structures, we remind the follo wing set o perators in v olve d in the DSm fusion f ormulas. Sev eral numerica l exa mples are gi ven in the chapter 6 of [31]. • Addition of sets S 1 ⊞ S 2 = S 2 ⊞ S 1 , { x | x = s 1 + s 2 , s 1 ∈ S 1 , s 2 ∈ S 2 } • Subtraction of sets S 1 ⊟ S 2 , { x | x = s 1 − s 2 , s 1 ∈ S 1 , s 2 ∈ S 2 } • Multiplication of sets S 1  S 2 , { x | x = s 1 · s 2 , s 1 ∈ S 1 , s 2 ∈ S 2 } • Divisi on of sets : If 0 doesn’ t belong to S 2 , S 1  S 2 , { x | x = s 1 /s 2 , s 1 ∈ S 1 , s 2 ∈ S 2 } 2.7.1 DSm rule of combin ation for impr ecise beliefs W e prese nt the gen eraliza tion of th e DSm rules to c ombine an y type of imprecise bel ief assignmen t which may be represented by the union of sev eral sub-uni tary (half-) open interv als, (half-)cl osed interv als and/or sets of points bel onging to [0,1]. Sev eral numeric al example s are a lso g iv en. In the se quel, one u ses the notation ( a, b ) for a n ope n interv al, [ a, b ] for a closed inter v al, and ( a, b ] or [ a, b ) for a half open and ha lf clo sed inte rv al. From the pre vious operators on sets , one can gen eralize the DSm rules ( classic and hyb rid) from sc alars to sets i n the follo wing way [31] (chap. 6): ∀ A 6 = ∅ ∈ D Θ , m I ( A ) = X X 1 ,X 2 ,...,X k ∈ D Θ ( X 1 ∩ X 2 ∩ ... ∩ X k )= A Y i =1 ,...,k m I i ( X i ) (10) where X and Y repres ent the summation, and respecti vely produ ct, of sets. Similarly , one can generaliz e the hybrid DSm rule from scalars to sets in the follo wing way: m I D S mH ( A ) = m I M (Θ) ( A ) , φ ( A )  h S I 1 ( A ) ⊞ S I 2 ( A ) ⊞ S I 3 ( A ) i (11) where all sets in volv ed in formulas are in the cano nical form and φ ( A ) is the char acteristic non emptiness functi on of the set A and S I 1 ( A ) , S I 2 ( A ) and S I 3 ( A ) are deﬁned by S I 1 ( A ) , X X 1 ,X 2 ,...,X k ∈ D Θ X 1 ∩ X 2 ∩ ... ∩ X k = A Y i =1 ,...,k m I i ( X i ) (12) S I 2 ( A ) , X X 1 ,X 2 ,...,X k ∈ ∅ [ U = A ] ∨ [ ( U ∈ ∅ ) ∧ ( A = I t )] Y i =1 ,...,k m I i ( X i ) (13) S I 3 ( A ) , X X 1 ,X 2 ,...,X k ∈ D Θ X 1 ∪ X 2 ∪ ... ∪ X k = A X 1 ∩ X 2 ∩ ... ∩ X k ∈ ∅ Y i =1 ,...,k m I i ( X i ) (14) In the case when all sets are reduced to points (numbers), the set operation s become normal operatio ns with numbers ; the sets operation s are generalizat ions of numerical operations. When imprecise belief structu res re- duce to precise be lief struc ture, DSm rules (10) a nd (1 1) re duce to their precise v ersion (4) and (5) respect i vely . 18 2.7.2 Example Here is a simple example of fusion with multiple-i nterv al masses. For simplicity , this example is a particula r case when the theore m of admiss ibility (see [31 ] p. 138 for details) is veri ﬁed by a fe w poin ts, which hap pen to be just on the bound ers. It is an e xtreme exa mple, because we tried to compris e all kinds of possibili ties which may occur in the imprecise or very imprecise fusion. So, let’ s consid er a fusion problem ov er Θ = { θ 1 , θ 2 } , two indep endent sources of information w ith the foll o wing imprecise admissible belief assignment s A ∈ D Θ m I 1 ( A ) m I 2 ( A ) θ 1 [0 . 1 , 0 . 2] ∪ { 0 . 3 } [0 . 4 , 0 . 5] θ 2 (0 . 4 , 0 . 6) ∪ [0 . 7 , 0 . 8] [0 , 0 . 4] ∪ { 0 . 5 , 0 . 6 } T able 3: Inputs of the fusion with imprecise bba’ s. Using the DSm classic (DSmC) rule for sets, one gets m I ( θ 1 ) = ([0 . 1 , 0 . 2] ∪ { 0 . 3 } )  [0 . 4 , 0 . 5] = ([0 . 1 , 0 . 2]  [0 . 4 , 0 . 5]) ∪ ( { 0 . 3 }  [0 . 4 , 0 . 5]) = [0 . 04 , 0 . 10] ∪ [0 . 12 , 0 . 15] m I ( θ 2 ) = ((0 . 4 , 0 . 6) ∪ [0 . 7 , 0 . 8])  ([0 , 0 . 4] ∪ { 0 . 5 , 0 . 6 } ) = ((0 . 4 , 0 . 6)  [0 , 0 . 4]) ∪ ((0 . 4 , 0 . 6)  { 0 . 5 , 0 . 6 } ) ∪ ([0 . 7 , 0 . 8]  [0 , 0 . 4]) ∪ ([0 . 7 , 0 . 8]  { 0 . 5 , 0 . 6 } ) = (0 , 0 . 24) ∪ (0 . 20 , 0 . 30) ∪ (0 . 24 , 0 . 36) ∪ [0 , 0 . 32] ∪ [0 . 35 , 0 . 40] ∪ [0 . 42 , 0 . 48 ] = [0 , 0 . 40] ∪ [0 . 42 , 0 . 48] m I ( θ 1 ∩ θ 2 ) = [([0 . 1 , 0 . 2] ∪ { 0 . 3 } )  ([0 , 0 . 4] ∪ { 0 . 5 , 0 . 6 } )] ⊞ [[0 . 4 , 0 . 5]  ((0 . 4 , 0 . 6) ∪ [0 . 7 , 0 . 8])] = [([0 . 1 , 0 . 2]  [0 , 0 . 4]) ∪ ([0 . 1 , 0 . 2]  { 0 . 5 , 0 . 6 } ) ∪ ( { 0 . 3 }  [0 , 0 . 4]) ∪ ( { 0 . 3 }  { 0 . 5 , 0 . 6 } )] ⊞ [([0 . 4 , 0 . 5]  (0 . 4 , 0 . 6)) ∪ ([0 . 4 , 0 . 5]  [0 . 7 , 0 . 8])] = [[0 , 0 . 08] ∪ [0 . 05 , 0 . 10] ∪ [0 . 06 , 0 . 12 ] ∪ [0 , 0 . 12] ∪ { 0 . 15 , 0 . 18 } ] ⊞ [(0 . 16 , 0 . 30) ∪ [0 . 28 , 0 . 40]] = [[0 , 0 . 12] ∪ { 0 . 15 , 0 . 18 } ] ⊞ (0 . 16 , 0 . 40] = (0 . 16 , 0 . 52] ∪ (0 . 31 , 0 . 55] ∪ (0 . 34 , 0 . 58] = (0 . 16 , 0 . 58] Hence ﬁnally the fusion admissib le result with DSmC rule is gi ven by: A ∈ D Θ m I ( A ) = [ m I 1 ⊕ m I 2 ]( A ) θ 1 [0 . 04 , 0 . 10] ∪ [0 . 12 , 0 . 15] θ 2 [0 , 0 . 40] ∪ [0 . 42 , 0 . 48] θ 1 ∩ θ 2 (0 . 16 , 0 . 58] θ 1 ∪ θ 2 0 T able 4: Fusion result with the DSmC rule. If on e ﬁnds out 6 that θ 1 ∩ θ 2 M ≡ ∅ (this is o ur hy brid model M one wants to deal wit h), the n one uses the hybrid DSm rule for sets (11) : m I M ( θ 1 ∩ θ 2 ) = 0 and m I M ( θ 1 ∪ θ 2 ) = (0 . 1 6 , 0 . 58] , the others imprecise m asses are not change d. 6 W e consider now a dynamic fusion problem. 19 W ith the hybrid DSm rule (DSmH) appli ed to imprecise beliefs, one gets now the resu lts giv en in T able 5. A ∈ D Θ m I M ( A ) = [ m I 1 ⊕ m I 2 ]( A ) θ 1 [0 . 04 , 0 . 10] ∪ [0 . 12 , 0 . 15] θ 2 [0 , 0 . 40] ∪ [0 . 42 , 0 . 48] θ 1 ∩ θ 2 M ≡ ∅ 0 θ 1 ∪ θ 2 (0 . 16 , 0 . 58] T able 5: Fusion result with DSm H ru le for M . Let’ s check no w the admissibi lity condition . For the source 1, there exist the precise masses ( m 1 ( θ 1 ) = 0 . 3) ∈ ([0 . 1 , 0 . 2] ∪ { 0 . 3 } ) and ( m 1 ( θ 2 ) = 0 . 7) ∈ ((0 . 4 , 0 . 6) ∪ [0 . 7 , 0 . 8]) such that 0 . 3 + 0 . 7 = 1 . For the source 2, there exist the precis e masses ( m 1 ( θ 1 ) = 0 . 4) ∈ ([0 . 4 , 0 . 5]) and ( m 2 ( θ 2 ) = 0 . 6) ∈ ([0 , 0 . 4] ∪ { 0 . 5 , 0 . 6 } ) such that 0 . 4 + 0 . 6 = 1 . Therefore both sources associate d with m I 1 ( . ) and m I 2 ( . ) are admissible imprecise source s of information . It can be v eriﬁed that DS mC fusion of m 1 ( . ) and m 2 ( . ) yields the paradoxic al bba m ( θ 1 ) = [ m 1 ⊕ m 2 ]( θ 1 ) = 0 . 12 , m ( θ 2 ) = [ m 1 ⊕ m 2 ]( θ 2 ) = 0 . 42 and m ( θ 1 ∩ θ 2 ) = [ m 1 ⊕ m 2 ]( θ 1 ∩ θ 2 ) = 0 . 46 . One sees that the admissibil ity condition is satisﬁed since ( m ( θ 1 ) = 0 . 12) ∈ ( m I ( θ 1 ) = [0 . 04 , 0 . 10] ∪ [0 . 12 , 0 . 15]) , ( m ( θ 2 ) = 0 . 42) ∈ ( m I ( θ 2 ) = [0 , 0 . 40] ∪ [0 . 42 , 0 . 48]) and ( m ( θ 1 ∩ θ 2 ) = 0 . 46) ∈ ( m I ( θ 1 ∩ θ 2 ) = (0 . 16 , 0 . 58]) such th at 0 . 1 2 + 0 . 42 + 0 . 46 = 1 . Similarly if one ﬁnd s o ut th at θ 1 ∩ θ 2 = ∅ , t hen one uses DSmH rule and one gets: m ( θ 1 ∩ θ 2 ) = 0 and m ( θ 1 ∪ θ 2 ) = 0 . 46 ; the others remain unchang ed. The admissibi lity condit ion still holds, because one can pick at least one number in each subset m I ( . ) such that the sum of these numbers is 1. 3 Pr oportional Conﬂict Redistrib ution rule Instea d of applying a direct transfer of partial con ﬂicts onto partial uncertaintie s as with DSmH, t he idea beh ind the Proportio nal Conﬂict Redistrib ution (PCR) rule [33, 34] is to transfer (total or parti al) conﬂicting masses to non-emp ty sets in volv ed in the conﬂicts proportiona lly with respect to the masses assigned to them by sources as follo ws: 1. calcula tion the conjuncti ve rule of the belief masses of sour ces; 2. calcula tion the total or partial conﬂicting masses; 3. redistri b ution of the (total or partial) conﬂicting masses to the non-empty sets in v olved in the conﬂicts propo rtional ly w ith resp ect to their masses assigned by the sources. The way the conﬂicting mass is redistrib uted yields actually sev eral version s of PCR rules. T hese PC R fusion rules work for any degre e of con ﬂict, for any DSm models (Shafer’ s model, free D Sm model or any hybrid DSm model) and both in DST and DSmT frame works for static or dyn amical fusion situations. W e present belo w only the most sophi sticate d propor tional conﬂict redistr ib ution rule denoted PCR5 in [33, 34]. PCR 5 rule is w hat w e feel the most ef ﬁ cient P CR fusion rule dev eloped so far . This rule redistrib utes the partial conﬂictin g mass to the elements in volv ed in the partial conﬂict, conside ring the conjuncti ve normal form of the partial conﬂict. PCR5 is w hat we think the most mathematica lly exact redistr ib ution of conﬂicting mass to non-emp ty sets fo llo w ing the lo gic o f t he co njunct i ve rule. It doe s a better redist rib ution of the conﬂictin g mass than Dempster’ s rule since P CR5 goes backwar ds on the tracks of the conjuncti ve rule and redistrib utes the conﬂictin g mass only to the sets in vo lve d in the conﬂict and proportion ally to their masses put in the conﬂict. PCR5 rule is quasi-a ssocia ti ve and preserv es the neutra l impact of the vacu ous belief assignment bec ause in any partial conﬂict, as well in the total conﬂict (which is a sum of all partial conﬂicts), the conjuncti ve normal form of each partial con ﬂ ict d oes not incl ude Θ since Θ is a neutral element f or interse ction (conﬂic t), ther efore Θ gets no m ass after the redistr ib ution of the conﬂictin g mass. W e ha ve prov ed in [34] the continuity property of the fusion result with continu ous va riation s of bba’ s to combin e. 20 3.1 PCR f or mulas The PC R5 formula for the combinat ion of two sources ( s = 2 ) is gi ven by: m P C R 5 ( ∅ ) = 0 and ∀ X ∈ G Θ \ {∅} m P C R 5 ( X ) = m 12 ( X ) + X Y ∈ G Θ \{ X } X ∩ Y = ∅ [ m 1 ( X ) 2 m 2 ( Y ) m 1 ( X ) + m 2 ( Y ) + m 2 ( X ) 2 m 1 ( Y ) m 2 ( X ) + m 1 ( Y ) ] (15) where all s ets in v olved in formulas are in canonical f orm and where G Θ corres ponds to classical p o wer set 2 Θ if Shafer’ s m odel is us ed, or to a constrai ned hyper -po wer set D Θ if an y other hybri d DSm mod el is u sed inst ead, or to the super- po w er set S Θ if the minimal reﬁnement Θ r ef of Θ is used; m 12 ( X ) ≡ m ∩ ( X ) correspond s to the c onjunc ti ve consensus on X between th e s = 2 sources a nd wher e all denominato rs are dif ferent from zero. If a denominat or is zero, that fracti on is discarded. A general formula of PCR5 for the fusion of s > 2 sources has been propose d in [34], but a m ore intu- iti ve PCR formula (denote d PCR6) which provid es good results in practice has been proposed by Martin and Osswald in [34] (page s 69-88) and is giv en by: m P C R 6 ( ∅ ) = 0 and ∀ X ∈ G Θ \ {∅} m P C R 6 ( X ) = m 12 ...s ( X ) + s X i =1 m i ( X ) 2 X s − 1 ∩ k =1 Y σ i ( k ) ∩ X ≡∅ ( Y σ i (1) ,...,Y σ i ( s − 1) ) ∈ ( G Θ ) s − 1        s − 1 Y j = 1 m σ i ( j ) ( Y σ i ( j ) ) m i ( X ) + s − 1 X j = 1 m σ i ( j ) ( Y σ i ( j ) )        (16) where σ i counts from 1 to s a v oiding i :  σ i ( j ) = j if j < i, σ i ( j ) = j + 1 if j ≥ i, (17) Since Y i is a focal element of expe rt/sour ce i , m i ( X )+ s − 1 X j = 1 m σ i ( j ) ( Y σ i ( j ) ) 6 = 0 ; the belie f mass assignmen t m 12 ...s ( X ) ≡ m ∩ ( X ) corresponds to the conjun cti ve conse nsus on X between the s > 2 sources. For two source s ( s = 2 ), PCR5 and PCR6 formula s coincide. 3.2 Examples • Example 1 : Let’ s tak e Θ = { A, B } of exclusi ve elements (Shafer’ s model), and the follo wing bba: A B A ∪ B m 1 ( . ) 0.6 0 0.4 m 2 ( . ) 0 0.3 0.7 m ∩ ( . ) 0.42 0.12 0.28 The conﬂicting mass is k 12 = m ∩ ( A ∩ B ) and equa ls m 1 ( A ) m 2 ( B ) + m 1 ( B ) m 2 ( A ) = 0 . 18 . T herefo re A and B are the only focal elements in volv ed in the conﬂict. H ence accordin g to the PCR 5 hypothesis only A and B deserv e a part of the conﬂictin g mass and A ∪ B do not deserv e. W ith PCR 5, one redistr ib utes the conﬂicting mass k 12 = 0 . 18 to A and B proportio nally with the masses m 1 ( A ) and m 2 ( B ) assi gned to A and B respec ti vely . 21 Here are the results obtain ed from Dempster’ s rule, DS mH an d PCR 5: A B A ∪ B m D S 0.512 0.146 0.342 m D S mH 0.420 0.120 0.460 m P C R 5 0.540 0.180 0.280 • Example 2 : Let’ s modify exa mple 1 and conside r A B A ∪ B m 1 ( . ) 0.6 0 0.4 m 2 ( . ) 0.2 0 .3 0.5 m ∩ ( . ) 0.50 0.12 0.20 The con ﬂicting mass k 12 = m ∩ ( A ∩ B ) as well as the dist rib ution coef ﬁcients for the PCR5 remains the same as in the pre vious example b ut one gets no w A B A ∪ B m D S 0.609 0.146 0.231 m D S mH 0.500 0.120 0.380 m P C R 5 0.620 0.180 0.200 • Example 3 : Let’ s modify exa mple 2 and conside r A B A ∪ B m 1 ( . ) 0.6 0 .3 0.1 m 2 ( . ) 0.2 0 .3 0.5 m ∩ ( . ) 0.44 0.27 0.05 The conﬂictin g mass k 12 = 0 . 24 = m 1 ( A ) m 2 ( B ) + m 1 ( B ) m 2 ( A ) = 0 . 24 is no w dif ferent from pre vious example s, w hich means that m 2 ( A ) = 0 . 2 and m 1 ( B ) = 0 . 3 did make an impact on the conﬂict. Therefor e A and B are the only focal elements in v olv ed in the conﬂict and thus only A and B deserv e a part of the conﬂictin g mass. PCR5 redistrib utes the partial conﬂicting mass 0.18 to A and B propo rtional ly w ith the masses m 1 ( A ) and m 2 ( B ) and also the partial conﬂictin g mass 0.06 to A and B propo rtional ly with th e masses m 2 ( A ) and m 1 ( B ) . A fter all deri vati ons (see [13 ] for de tails), one ﬁnally gets: A B A ∪ B m D S 0.579 0.355 0.066 m D S mH 0.440 0.270 0.290 m P C R 5 0.584 0.366 0.050 One clearly sees that m D S ( A ∪ B ) gets some m ass from the conﬂictin g mass although A ∪ B does not deserv e any part of the conﬂicting mass (according to PCR5 hypothesis) since A ∪ B is not in vol ved in the con ﬂict (only A and B are in vol ved in the co nﬂicting mass). Dempster’ s rule appears to us l ess exac t than PCR5 and Inagaki’ s rules [15]. It can be showed [13] that Inagaki ’ s fusion rule (with an optimal choice of tuning parameters) can become in some cases ver y close to PCR5 but upon our opinion PCR5 result is more ex act (at least less ad-hoc than Inagak i’ s one). • Example 4 (A more concr ete example) : Three people, John ( J ), George ( G ), and Dav id ( D ) are sus- pects to a murder . So the frame of discernment is Θ , { J , G, D } . T wo sourc es m 1 ( . ) and m 2 ( . ) (witnesse s) provide the follo wing informat ion: 22 J G D m 1 0.9 0 0.1 m 2 0 0.8 0.2 W e kno w that John and G eor ge are friends, b ut John a nd D a vid hate each othe r , and similarly Geor ge and Dav id. a) Free model, i. e. all intersectio ns are nonempty : J ∩ G 6 = ∅ , J ∩ D 6 = ∅ , G ∩ D 6 = ∅ , J ∩ G ∩ D 6 = ∅ . Using the DSm class ic rule one gets: J G D J ∩ G J ∩ D G ∩ D J ∩ G ∩ D m D S mC 0 0 0.02 0.72 0 .18 0.08 0 So we can see that John and Geor ge together ( J ∩ G ) are most likely to ha ve committed the crime, since the mass m D S mC ( J ∩ G ) = 0 . 72 is the biggest resultin g mass after the fusion of the two source s. In Shafer’ s model, only one suspe ct could commit the crime, b ut the free and hybrid models allo w two or more people to ha ve committed the same crime - w hich happ ens in reality . b) Let’ s conside r the hybrid m odel, i. e. some interse ctions a re empty , and others are not. According to the above statement ab out th e relatio nships between the thre e sus pects, we can d educe that J ∩ G 6 = ∅ , while J ∩ D = G ∩ D = J ∩ G ∩ D = ∅ . Then we ﬁ rst apply the DSm Classic rule, and then the transfe r of the conﬂicting masses is done with PC R5: J G D J ∩ G J ∩ D G ∩ D J ∩ G ∩ D m 1 0.9 0 0.1 m 2 0 0.8 0. 2 m DS mC 0 0 0. 02 0.72 0.18 0.08 0 Using PCR5 no w we transfer m ( J ∩ D ) = 0 . 18 , since J ∩ D = ∅ , to J and D proport ionally with 0.9 and 0.2 resp ecti vely , so J gets 0.15 and D gets 0.03 since: xJ / 0 . 9 = z 1 D / 0 . 2 = 0 . 18 / (0 . 9 + 0 . 2) = 0 . 18 / 1 . 1 whence xJ = 0 . 9(0 . 1 8 / 1 . 1) = 0 . 15 and z 1 D = 0 . 2(0 . 18 / 1 . 1) = 0 . 03 . Again using PCR5, we transfer m ( G ∩ D ) = 0 . 08 , since G ∩ D = ∅ , to G and D proport ionally with 0.8 and 0.1 respec ti vely , so G gets 0.07 and D gets 0.01 since: y G/ 0 . 8 = z 2 D / 0 . 1 = 0 . 08 / (0 . 8 + 0 . 1) = 0 . 08 / 0 . 9 whence y G = 0 . 8(0 . 08 / 0 . 9) = 0 . 07 and z D = 0 . 1(0 . 08 / 0 . 9) = 0 . 01 . Adding w e get ﬁnally : J G D J ∩ G J ∩ D G ∩ D J ∩ G ∩ D m P C R 5 0.15 0.07 0.06 0.72 0 0 0 So o ne ha s a h igh belief th at the criminals ar e Joh n and Geor ge (both of the m committed t he cri me) since m ( J ∩ D ) = 0 . 72 and it is by far the great est fusion m ass. In Shafer’ s model, if we try to reﬁne we get the disjoint parts: D , J ∩ G , J \ ( J ∩ G ) , and G \ ( J ∩ G ) , b ut the last two are ridiculous (what is the real/ph ysical nature of J \ ( J ∩ G ) or G \ ( J ∩ G ) ? H alf of a person (!) ?), so the reﬁning does not work here in reality . That’ s why the hybrid and free models are needed . • Example 5 (Imprec ise PC R5) : The PCR5 formula can natura lly work also for the combination of imprecis e bba’ s. This has bee n already pre sented in sectio n 1.11.8 page 49 of [34] with a numerical exa mple to sho w how to a pply it. T his e xample will therefore not be reincluded here. 23 3.3 Zadeh’ s example W e compare here the solutions for well-kno w n Zadeh’ s example [53, 56] pro vided by se vera l fusion rules. A detailed presentation with more compariso ns can be found in [31, 34]. Let’ s con sider Θ = { M , C , T } as the frame of three potential origins about possible diseas es of a patien t ( M stan ding for meningitis , C for concu ssion and T for tumor ), the Shafer’ s model and the two follo wing beli ef assignmen ts pro vided by two indepe ndent doctors after exa mination of the same patient. m 1 ( M ) = 0 . 9 m 1 ( C ) = 0 m 1 ( T ) = 0 . 1 m 2 ( M ) = 0 m 2 ( C ) = 0 . 9 m 2 ( T ) = 0 . 1 The total conﬂicting mass is high since it is m 1 ( M ) m 2 ( C ) + m 1 ( M ) m 2 ( T ) + m 2 ( C ) m 1 ( T ) = 0 . 99 • with Dempster’ s rule and Shafer’ s model (DS), one gets the counter -intuiti ve result (see justiﬁcation s in [11, 31, 46, 50, 53]): m D S ( T ) = 1 • with Y ager’ s rule [50] and Shafer’ s model: m Y ( M ∪ C ∪ T ) = 0 . 99 and m Y ( T ) = 0 . 01 • with DSmH and Shafer’ s m odel: m D S mH ( M ∪ C ) = 0 . 81 m D S mH ( T ) = 0 . 01 m D S mH ( M ∪ T ) = m D S mH ( C ∪ T ) = 0 . 09 • The D ubois & Prade’ s rule (DP) [11] based on Shafer’ s model provid es in Z adeh’ s e xample the same result as DSmH, becau se DP and D SmH coinci de in all static fusion problems 7 . • with PCR5 and Shafer’ s model: m P C R 5 ( M ) = m P C R 5 ( C ) = 0 . 486 and m P C R 5 ( T ) = 0 . 028 . One sees that when the total conﬂict between sources becomes high, DSmT is able (upon authors opinion ) to manage m ore adequate ly through DSmH or PCR5 rules the combination of information than Dempster’ s rule, e ven w hen workin g w ith Shafer’ s model - which is only a speciﬁc hybrid model. DS mH rule is in agreement with DP rule for the static fusion, bu t DSmH and DP rules differ in general (for non dege nerate cases) for dy- namic fusi on while PCR5 rule is the most exact pro portional conﬂict redistrib ution rule. Besides th is partic ular exa mple, we sho wed in [31] tha t there e xist se veral inﬁnite classes of coun ter -examples to Dempster’ s rule which can be solve d by DSm T . In summary , DST based on Dempster’ s rul e provi des counter -intuiti ve resu lts in Z adeh’ s example, or in non- Bayesian e xamples similar t o Zadeh’ s and no resu lt when the co nﬂict is 1. Only ad-ho c discount ing techniques allo w to ci rcumv ent trou bles of Dempster’ s rule or we need to switch to another model of repres entatio n/frame; in the later case the solution obtained doesn’ t ﬁt with the Shafer’ s model one origina lly wanted to work w ith. W e want also to emphasi ze that in dynamic fusion when the conﬂict becomes high, both DST [24] and Smets’ T ransferable Beli ef Model (TBM) [39] approache s fail to re spond t o n e w inf ormation pro vided by ne w sources . This can be easily sho wed by the very simple follo wing exa mple. Example (where TBM doesn ’t respond to new inf ormation ): Let Θ = { A, B , C } with the (precis e) bba’ s m 1 ( A ) = 0 . 4 , m 1 ( C ) = 0 . 6 and m 2 ( A ) = 0 . 7 , m 2 ( B ) = 0 . 3 . Then one gets 8 with D empster ’ s rule, Sm ets’ TBM (i.e. the non-normali zed version of Dempster’ s combina- 7 Indeed DP ru le has b een developed for static fusion only while DSmH has been developed to take in to account the possible dynam icity of the frame itself and also its associated model. 8 W e introduce here e xplicitly the indexes o f sources in the fusion result since more than two sources are considered i n this example. 24 tion), DSmH and PCR5: m 12 D S ( A ) = 1 , m 12 T B M ( A ) = 0 . 28 , m 12 T B M ( ∅ ) = 0 . 72 ,            m 12 D S mH ( A ) = 0 . 28 m 12 D S mH ( A ∪ B ) = 0 . 12 m 12 D S mH ( A ∪ C ) = 0 . 42 m 12 D S mH ( B ∪ C ) = 0 . 18 and      m 12 P C R 5 ( A ) = 0 . 574725 m 12 P C R 5 ( B ) = 0 . 111429 m 12 P C R 5 ( C ) = 0 . 313846 No w let’ s consider a temporal fusion problem and introduce a third source m 3 ( . ) w ith m 3 ( B ) = 0 . 8 and m 3 ( C ) = 0 . 2 . Then one sequen tially combi nes the results obtained by m 12 T B M ( . ) , m 12 D S ( . ) , m 12 D S mH ( . ) and m 12 P C R ( . ) with the ne w evid ence m 3 ( . ) and one sees that m (12)3 D S becomes not deﬁned (di vision by zero) and m (12)3 T B M ( ∅ ) = 1 while (DSm H) and (PCR5) pr ovi de                m (12)3 D S mH ( B ) = 0 . 240 m (12)3 D S mH ( C ) = 0 . 120 m (12)3 D S mH ( A ∪ B ) = 0 . 224 m (12)3 D S mH ( A ∪ C ) = 0 . 056 m (12)3 D S mH ( A ∪ B ∪ C ) = 0 . 360 and      m (12)3 P C R 5 ( A ) = 0 . 277490 m (12)3 P C R 5 ( B ) = 0 . 545010 m (12)3 P C R 5 ( C ) = 0 . 17750 0 When the mass committed to empty set becomes one at a pre vious temporal fusion step, then both D ST and TBM do not respond to new information. L et’ s continu e the example and consider a fourth source m 4 ( . ) with m 4 ( A ) = 0 . 5 , m 4 ( B ) = 0 . 3 and m 4 ( C ) = 0 . 2 . Then it is easy to see that m ((12)3 )4 D S ( . ) is not deﬁned since at pre vious step m (12)3 D S ( . ) was already not deﬁned, and that m ((12)3 )4 T B M ( ∅ ) = 1 whate ver m 4 ( . ) is becaus e at the previo us fusion step one had m (12)3 T B M ( ∅ ) = 1 . Therefore for a number of source s n ≥ 2 , DS T and TBM approa ches do not respon d to new in formatio n incoming in the fusion proce ss while both (DSmH) an d (PCR5) rules respond to new information. T o make DST and/or TBM working properly in such cases, it is necessar y to intro duce ad-hoc temporal discount ing techniques which are not neces sary to introdu ce if DSmT is adopte d. If there are good reason s to introduce temporal discounting , there is obv iously no difﬁcu lty to apply the DSm fusion of these discou nted source s. An analysi s of this beha vior for tar get type tracking is presente d in [9, 34]. 4 The generalized pignistic transf ormation (GPT) 4.1 The classical pignistic transformation W e follow here Philippe Smets’ vision which considers the management of information as a two 2-lev els proces s: credal (for combination of e videnc es) and pignisti c 9 (for decision-maki ng) , i.e ” when someone m ust tak e a decision , he/she must then c onstru ct a pr obabi lity func tion der ived fr om th e belief fun ction th at describ es his/he r cr edal state. This pr obabili ty function is then used to make decisions ” [38] (p. 284). One obvi ous way to b uild this p robabil ity function corre sponds to th e so- called Classic al Pignistic T ransformat ion (CPT) deﬁned in DST frame work (i.e. based on the Shafer’ s model assumption ) as [40]: B etP { A } = X X ∈ 2 Θ | X ∩ A | | X | m ( X ) (18) where | A | denote s the number of worlds in the set A (with con vent ion |∅| / |∅| = 1 , to deﬁne B etP {∅} ). Decision s are achie ved by computing the expec ted utilitie s of the acts using the subjec ti ve/pignis tic B etP { . } as the probability function needed to compute expe ctation s. Usuall y , one uses the maximum of the pignistic probab ility as decisio n criterion. T he m aximum of B etP { . } is often consid ered as a prudent betting decisio n criteri on between the two other alte rnati ves (max of plausi bility or max. of credi bility which appears to be respec ti vel y too optimistic or too pessimistic). It is easy to show that B etP { . } is indeed a probabilit y function (see [39]). 9 Pignistic terminolog y has been coined by Philippe Smets and comes from pignu s , a bet in Latin. 25 4.2 Notion of DSm cardinality One important notion in volv ed in the deﬁnition of the G enera lized P ignis tic Transf ormation (G PT) is the DSm car dinalit y . The DSm car dinality of any element A of hyper -po wer set D Θ , denoted C M ( A ) , correspond s to the number of parts of A in the correspond ing fuzzy /v ague V enn diagram of the proble m (model M ) taking into accou nt the set of integ rity constraints (if any), i.e. all the pos sible intersect ions due to the na ture of th e el- ements θ i . T his intrinsi c car dinality dep ends on the model M (free, hybrid or Shafer ’ s mod el). M is the model that contains A , which depends both on the dimension n = | Θ | and on the number of non-empty intersection s presen t in its associated V enn diagram (see [31] for details ). The DS m cardinality depends on the cardin al of Θ = { θ 1 , θ 2 , . . . , θ n } and on the model of D Θ (i.e., the number of intersec tions and between w hat eleme nts of Θ - in a word the st ructure ) at th e s ame time; it is n ot necessa rily that eve ry singlet on, say θ i , ha s t he same DSm cardin al, because each singleton has a dif ferent structure; if its struct ure is the simplest (no intersec tion of this elements with other elemen ts) then C M ( θ i ) = 1 , if the stru cture is more complic ated (many interse ctions ) then C M ( θ i ) > 1 ; let’ s consider a single ton θ i : if it h as 1 intersection only then C M ( θ i ) = 2 , fo r 2 int ersecti ons only C M ( θ i ) is 3 or 4 depend ing on the model M , for m intersecti ons it is between m + 1 and 2 m depen ding on the model; the maximum DSm cardinalit y is 2 n − 1 and occurs for θ 1 ∪ θ 2 ∪ . . . ∪ θ n in the fr ee model M f ; s imilarly for an y set from D Θ : the more compli cated structu re it has, the bigger is the DSm cardina l; thus the DSm cardin ality measures the complex ity of en element from D Θ , which is a nice characteriz ation in our opinio n; we may say that for the singleton θ i not ev en | Θ | counts, but only its structur e (= how many other singleton s interse ct θ i ). Simple illu strati ve examples are g i ven in Chapter 3 and 7 of [31 ]. One has 1 ≤ C M ( A ) ≤ 2 n − 1 . C M ( A ) m ust not be confus ed w ith the classica l cardinalit y | A | of a giv en set A (i.e. the number of its distinct elements ) - that’ s w hy a new notatio n is necessa ry here. C M ( A ) is very easy to co mpute by p rogramming from the algorit hm of genera tion of D Θ gi ven explicate d in [31]. Example : let’ s take back the example of the simple hybrid DSm m odel desc ribed in section 2.2, then one gets the followin g list of elements (with their DSm cardinal) for the restric ted D Θ taking into accoun t the integrity constr aints of this hybrid model: A ∈ D Θ C M ( A ) α 0 , ∅ 0 α 1 , θ 1 ∩ θ 2 1 α 2 , θ 3 1 α 3 , θ 1 2 α 4 , θ 2 2 α 5 , θ 1 ∪ θ 2 3 α 6 , θ 1 ∪ θ 3 3 α 7 , θ 2 ∪ θ 3 3 α 8 , θ 1 ∪ θ 2 ∪ θ 3 4 Example of DSm car dinal s : C M ( A ) for hybr id model M . 4.3 The Generalized Pignistic T ransformation T o tak e a rational decision within DSmT frame work, it is nec essary to generaliz e the Classic al P ignis tic Tr ans- formatio n in order to construct a pignistic probabil ity fun ction from any generalized basic belief assignment m ( . ) dra wn from the DSm rules of combi nation . Here is the simplest and dire ct exten sion of the CPT to deﬁne the Generaliz ed Pignistic T ransformation : ∀ A ∈ D Θ , B etP { A } = X X ∈ D Θ C M ( X ∩ A ) C M ( X ) m ( X ) (19) where C M ( X ) denote s the DS m cardinal of propos ition X for the DSm model M of the problem under con- sidera tion. 26 The decisi on about the solution of the problem is usually taken by the maximum of pignistic probabi lity functi on B etP { . } . Let’ s remark the clos e resse mblance of the two pignistic trans formation s (18) and (19). It can be sho wn that (19) reduces to (18) when the hyper- po wer set D Θ reduce s to classic al power set 2 Θ if we adopt S hafer’ s m odel. But (19) is a generali zation of (18) since it can be used for computing pignist ic probab ilities for any models (includi ng Shafer ’ s model). It has been prov ed in [31] (Chap. 7) that B etP { . } deﬁned in (19) is indeed a probab ility distrib ution. In the follo wing section , we introdu ce a ne w alterna ti ve to BetP which is presen ted in details in [36]. 5 The DSmP transf o rmation In the theorie s of belief functions , the mapping from the belief to the probab ility doma in is a contro versial is sue. The original purpose of such m appin gs was to make (hard) decision, but contr ariwise to erroneou s widesprea d idea/c laim, this is not the only interest for using such mappings no waday s. Actually the probab ilistic transfor - mations of be lief mas s assignments (a s t he pignistic tr ansformat ion mentione d pre viously ) are for e xample very useful in modern m ultita r get multisenso r trackin g systems (or in any other systems) where one deals w ith soft decisi ons (i.e. where all pos sible solutions are kept for sta te estimation with their lik elihoods). For exa mple, in a Multiple Hypothes es Tr acke r u sing both kinemati cal and attrib ute data, o ne need s to compute all probabiliti es v alues for deri ving the likeli hoods of data associat ion hypoth eses and then mixing them altogeth er to estimate states o f tar gets. T herefo re, it is very rel e v ant to use a map ping which p rovides a high probabi listic information conten t (PIC) for exp ecting better performanc es. In th is sec tion, we br ieﬂy recall a new pr obabil istic transformat ion, denoted D S mP and introdu ced in [10] which will be exp lained in details in [36]. DS mP is straight and dif ferent from other transformatio ns. T he basic idea o f DS mP cons ists in a ne w wa y of p roport ionaliz ations of the mass of each partial ignoran ce such as A 1 ∪ A 2 or A 1 ∪ ( A 2 ∩ A 3 ) or ( A 1 ∩ A 2 ) ∪ ( A 3 ∩ A 4 ) , etc. and th e mass o f the tota l i gnoran ce A 1 ∪ A 2 ∪ . . . ∪ A n , to the elements in v olv ed in the ignorances. This new transfo rmation take s into account both the v alues of the masses and the card inality of ele ments in the proportion al redistrib ution process. W e ﬁrst remind what P IC criteri a is and then s hortly pres ent the general formu la for DSmP t ransfo rmation with few numerical example s. More exampl es and comparisons with respect to other transformation s are giv en in [36]. 5.1 The Probabilistic Information Content (PIC) Follo wing Sudano’ s appro ach [41 , 42, 44], we adopt the Probabilis tic Informat ion Content (PIC) criterion as a metric depicting the strength of a critical decisio n by a speciﬁc probabil ity distrib ution. It is an essentia l measure in any threshold -dri ven automate d decision system. The PIC is the dual of the normalized S hanno n entrop y . A PIC val ue of one indicates the total knowle dge to make a correct decisio n (one hypothes is has a probab ility val ue of one and the rest of zero). A PIC va lue of zero indicates that the kno wledge to make a correc t decisio n does not exist (all the hypothese s hav e an equal probabili ty v alue), i.e. one has the maximal entrop y . T he PIC is used in our analysis to sort the per formance s of the dif ferent pignistic transfo rmations throug h se veral numerical examp les. W e ﬁrst r ecall what Shannon entrop y and PIC measure are and thei r tight relatio nship. • Shannon entr opy Shannon entrop y , usually expressed in bits (binar y digits), of a probab ility measure P { . } over a discret e ﬁnite set Θ = { θ 1 , . . . , θ n } is deﬁne d by 10 [25]: H ( P ) , − n X i =1 P { θ i } log 2 ( P { θ i } ) (20) H ( P ) is maximal for the uniform probability distrib ution over Θ , i.e. w hen P { θ i } = 1 /n for i = 1 , 2 , . . . , n . In that case, one gets H ( P ) = H max = − P n i =1 1 n log 2 ( 1 n ) = log 2 ( n ) . H ( P ) is minimal for a totally deter - ministic probab ility , i.e. for any P { . } such that P { θ i } = 1 for some i ∈ { 1 , 2 , . . . , n } and P { θ j } = 0 for j 6 = i . H ( P ) measures the randomnes s carried by an y discrete probability P { . } . 10 with common conv ention 0 lo g 2 0 = 0 . 27 • The PIC metric The Probabil istic Information Content (PIC) of a probabili ty measure P { . } associated with a proba bilistic source ov er a discrete ﬁ nite se t Θ = { θ 1 , . . . , θ n } is deﬁned by [42]: P I C ( P ) = 1 + 1 H max · n X i =1 P { θ i } log 2 ( P { θ i } ) (21) The PIC is nothing b ut the dual of the normalized Shannon entropy and thus is actually unit less. P I C ( P ) tak es its va lues in [0 , 1] . P I C ( P ) is maximum, i.e. P I C max = 1 with any determinis tic probability and it is minimum, i.e. P I C min = 0 , with the unifor m probab ility over the frame Θ . T he simple relation ships between H ( P ) and P I C ( P ) are P I C ( P ) = 1 − ( H ( P ) /H max ) and H ( P ) = H max · (1 − P I C ( P )) . 5.2 The DSmP f ormula Let’ s c onside r a discre te frame Θ with a gi ven mo del (f ree DSm model, hy brid DSm m odel or Shafe r’ s m odel) , the D S mP mapping is deﬁned by D S mP ǫ ( ∅ ) = 0 and ∀ X ∈ G Θ \ {∅} by D S mP ǫ ( X ) = X Y ∈ G Θ X Z ⊆ X ∩ Y C ( Z )=1 m ( Z ) + ǫ · C ( X ∩ Y ) X Z ⊆ Y C ( Z )=1 m ( Z ) + ǫ · C ( Y ) m ( Y ) (22) where ǫ ≥ 0 is a tuning parameter and G Θ corres ponds to the generi c set ( 2 Θ , S Θ or D Θ includ ing ev entually all the integrit y constr aints (if any) of the model M ); C ( X ∩ Y ) and C ( Y ) denot e the D Sm cardinals 11 of the sets X ∩ Y and Y respecti vely . ǫ allows to reach the maximum P IC v alue of the approximat ion of m ( . ) into a subjec ti ve probabil ity m easur e. The smalle r ǫ , the better /bigger P IC v alue. In some particu lar deg enerate cases ho weve r , the D S mP ǫ =0 v alues cannot be der i ved , bu t the D S mP ǫ> 0 v alues can ho wev er al ways be deri ved by choos ing ǫ as a very small positi ve number , say ǫ = 1 / 1000 for example in order to be as close as we want to the maximum of the PIC. When ǫ = 1 and when the m asses of all elements Z ha ving C ( Z ) = 1 are zero, (22) reduce s to (19), i.e. D S mP ǫ =1 = B etP . The pass age from a free DSm model to a Shafer’ s model in volv es the passag e from a struc ture to another one, and the cardinals change as well in the formula (22). D S mP works for all m odels (free, hybrid and Shafer’ s). In order to apply classical transfo rmation (Pig- nistic, C uzzoli n’ s one, Sudano’ s ones, etc - see [36]), w e need at ﬁrst to reﬁne the frame (on the cases when it is possi ble!) in order to wor k with Shafer’ s model, and then appl y their formulas. In the case where reﬁne ment makes sense, then one can apply the other subjecti ve proba bilities on the reﬁned frame. D S mP works on the reﬁned frame as well and gi ves the same result as it does on the non-reﬁned frame. Thus D S mP with ǫ > 0 works o n an y models and s o is v ery gener al and appea ling. D S mP does a redistrib ution of the ig noranc e mass with respect to both the singleton masses and the singletons’ cardina ls in the same time. Now , if all m asses of single tons in vo lved in all ig noranc es are dif ferent fro m zero, then we can ta ke ǫ = 0 , a nd DS mP giv es the best result, i.e. the best PIC value. In summary , D S mP does an ’impro vemen t’ ove r prev ious known probabilist ic transfo rmations in the sense that D S mP mathematical ly makes a more accurat e redistr ib ution of the ignora nce masses to the singletons in volv ed in ignora nces. DS mP and B etP wo rk in both theories : DST (= S hafer’ s model) and DSmT (= free or hybrid models) as well. 11 W e have omitted the index of the model M f or the notation con venience. 28 5.3 Examples f or DSmP a nd BetP The examples brieﬂy presented here are detailed in [36] which includes also addition al results based on C uz- zolin’ s and Sudano’ s transfor mations. • Wi th Shafer’ s model and a n on-Bay esian mass Let’ s consid er the frame Θ = { A, B } and let’ s assume Shafer’ s model and the non-Bayesi an mass (more precis ely the simple supp ort mass) gi ven in T able 6. W e summari ze in T able 7, the results obtain ed with DSmP and BetP . One sees that P I C ( D S mP ǫ → 0 ) is maximum among all PIC v alues. A B A ∪ B m ( . ) 0.4 0 0.6 T able 6: Quantitati ve inputs. A B P I C ( . ) B etP ( . ) 0.7000 0.3000 0.1187 D S mP ǫ =0 . 001 ( . ) 0.9985 0.0015 0.9838 D S mP ǫ =0 ( . ) 1 0 1 T able 7: Results of the probabil istic transfo rmations. The best result is an adequa te pr obabilit y , not the bigg est PIC in this case. This is because P ( B ) deserves to re cei ve some mass from m ( A ∪ B ) , so th e most correct r esult is done by DS mP ǫ =0 . 001 in T able 7 (of cou rse we can choose any other very small positi ve v alue for ǫ if we want). Always when a single ton whose mass is zero, b ut it is in volv ed in an ignoran ce whose mass is not zero, th en ǫ (in D S mP formula (2 2 )) should be taken dif ferent from zero. • Wi th a Hybrid DSm model Let’ s consider the frame Θ = { A, B , C } and let’ s c onside r the hybrid DSm mo del in which all in tersect ions of elemen ts of Θ are empty , but A ∩ B correspo nding to ﬁgure 4. In this case, G Θ reduce s to 9 elemen ts {∅ , A ∩ B , A, B , C, A ∪ B , A ∪ C, B ∪ C, A ∪ B ∪ C } . The input masses of focal elements are giv en by m ( A ∩ B ) = 0 . 20 , m ( A ) = 0 . 10 , m ( C ) = 0 . 20 , m ( A ∪ B ) = 0 . 30 , m ( A ∪ C ) = 0 . 10 , and m ( A ∪ B ∪ C ) = 0 . 10 and gi ven in the T able 8. D ′ A ′ ∪ D ′ C ′ m ( . ) 0.2 0.1 0.2 A ′ ∪ B ′ ∪ D ′ A ′ ∪ C ′ ∪ D ′ A ′ ∪ B ′ ∪ C ′ ∪ D ′ m ( . ) 0.3 0.1 0.1 T able 8: Quantitati ve inputs. ✫✪ ✬✩ ✫✪ ✬✩ ✫✪ ✬✩ ❅ ❘ A  ✠ B ✛ C D ′ C ′ B ′ A ′ Fig. 4: Hybrid model for Θ = { A, B , C } . 29 Applying BetP and DSmP tran sformatio ns, one gets: A ′ B ′ C ′ D ′ P I C ( . ) B etP ( . ) 0.20 84 0.1250 0.2583 0.4083 0.0607 D S mP ǫ =0 . 001 ( . ) 0.002 5 0.0017 0.2996 0.6962 0.5390 T able 9: Results of the probabil istic transfo rmations. • Wi th a fr ee DSm model Let’ s consid er the frame Θ = { A, B , C } and let’ s consider the free DSm m odel depicte d on F igure 5 w ith the input masse s giv en in T able 10. T o apply Sudano ’ s and C uzzol in’ s mappings, one works on the reﬁned frame Θ ref = { A ′ , B ′ , C ′ , D ′ , E ′ , F ′ , G ′ } where the elements of Θ ref are exclusi ve (assumin g such reﬁnement has a physical ly sense) accordi ng to F igure 5. This reﬁnement step is not necessary when using D S mP since it works directl y on D Sm free model. The PIC va lues obtaine d with DSmP and Bet P are gi ven in T able 11. One sees that D S mP ǫ → 0 pro vides here again the best results in term of PIC. ✫✪ ✬✩ ✫✪ ✬✩ ✫✪ ✬✩ ❅ ❘ A  ✠ B ❅ ■ C D ′ G ′ C ′ E ′ F ′ B ′ A ′ Fig. 5: Free DSm model for a 3D frame. A ∩ B ∩ C A ∩ B A m ( . ) 0.1 0.2 0.3 A ∪ B A ∪ B ∪ C m ( . ) 0.1 0.3 T able 10: Quantitati ve inputs. T ransformations P I C ( . ) B etP ( . ) 0.1176 D S mP ǫ =0 . 001 ( . ) 0.8986 T able 11: Results of the probabilis tic transfor mations. An extensio n of DSmP ( denote d qDSmP) f or w orking w ith qualitati ve labels instead o f n umbers is possibl e and has bee n pro posed a nd presen ted in 2008 i n [10]. A si mple exa mple for qDSmP is gi ven in the next section. 6 Fusion of qualitative beliefs W e recall here the notion of qualitati ve belief assignment to model beliefs of human exp erts expre ssed in natura l language (with linguis tic labels). W e s ho w how qua litati ve beliefs can be ef ﬁciently combine d using an ext ension o f D SmT to q ualitat iv e reasonin g. A more det ailed present ation can be found in [34]. The de ri vation s are based on a ne w arithmetic on linguist ic labels w hich allo ws a direct exten sion of all quantita ti ve rule s of combina tion and conditionin g. The qu alitati ve version of PCR5 rul e and DSm P is a lso presen ted in the sequel. 6.1 Qualitati ve Operators Computing with words (C W) and qualitat i ve informatio n is more va gue, less precise than computin g w ith numbers , but it of fers the adva ntage of rob ustnes s if done correctly . Here is a general arithmeti c we propos e for computing w ith word s (i.e. with lingu istic labels) . Let’ s consider a ﬁ nite frame Θ = { θ 1 , . . . , θ n } of n (exh austi ve) elements θ i , i = 1 , 2 , . . . , n , with an associate d model M (Θ) on Θ (either Shafer’ s model M 0 (Θ) , free-DSm model M f (Θ) , or m ore general any Hybrid-DSm model [31]). A model M (Θ) is deﬁned 30 by the set of integrity constraints on elements of Θ (if any); Shafer’ s model M 0 (Θ) assumes all elements of Θ truly exclusi ve, w hile free-DSm model M f (Θ) assumes no exclusi vity constrain ts between elements of the frame Θ . Let’ s deﬁne a ﬁnite set of linguistic labels ˜ L = { L 1 , L 2 , . . . , L m } w here m ≥ 2 is an integ er . ˜ L is endo wed w ith a total order relationshi p ≺ , so that L 1 ≺ L 2 ≺ . . . ≺ L m . T o work on a close linguisti c set under lingu istic addition and multiplic ation operators, we extend s ˜ L w ith two e xtreme v alues L 0 and L m +1 where L 0 corres ponds to the minimal qu alitati ve v alue an d L m +1 corres ponds to the maxi mal q ualitat i ve v alue, in such a way that L 0 ≺ L 1 ≺ L 2 ≺ . . . ≺ L m ≺ L m +1 where ≺ means inferior to, or less (in quality) than, or smaller (in quality) than, etc. hence a relatio n of order from a qualit ati ve point of view . But if we make a correspond ence between qualitati ve labels and quantitati ve v alues on the scale [0 , 1] , then L min = L 0 would correspon d to the numerical v alue 0, while L max = L m +1 would corres pond to the numerical value 1, and each L i would bel ong to [0 , 1] , i. e. L min = L 0 < L 1 < L 2 < . . . < L m < L m +1 = L max From no w on, we wor k on extende d ordered set L of qualitati ve value s L = { L 0 , ˜ L, L m +1 } = { L 0 , L 1 , L 2 , . . . , L m , L m +1 } In our prev ious works, we did prop ose approximate qual itati ve operato rs, b ut in [36] we pro pose to use better and accura te opera tors for qualitati ve labels. Since these new operato rs are deﬁned in details in the chapte r of [36] dev oted on the DSm Field and L inear Algeb ra of Reﬁned Labels (FLARL), we just brieﬂy introd uce here only the the main ones (i.e. the accurate label addi tion, multiplicati on and div ision). In FLARL , we can re place the ”q ualitati ve quasi-no rmalization” of quali tati ve operators we used in our pre vious papers by ”quali tati ve normalizatio n” since in FLAR L we ha ve ex act qualitati ve calculatio ns and exact normaliza tion. • Label addi tion : L a + L b = L a + b (23) since a m +1 + b m +1 = a + b m +1 . • Label multip lication : L a × L b = L ( ab ) / ( m +1) (24) since a m +1 · b m +1 = ( ab ) / ( m +1) m +1 . • Label di vision (when L b 6 = L 0 ): L a ÷ L b = L ( a/b )( m +1) (25) since a m +1 ÷ b m +1 = a b = ( a/b )( m +1) m +1 . More accurat e qualitat i ve operations (substractio n, scalar multiplica tion, scalar root, scalar power , etc) can be foun d in [3 6]. Of cour se, if on e really n eed to stay within the origin al set of la bels, an approxi mation w ill be necess ary at the very end of th e calculatio ns. 6.2 Qualitati ve Belief Assignment A qualit ati ve belief assignment 12 (qba) is a mapping functio n q m ( . ) : G Θ 7→ L where G Θ corres ponds either to 2 Θ , to D Θ or ev en to S Θ depen ding on the m odel of the frame Θ we choose to work with. In the case when the labels are equidi stant, i.e. the qualitati ve distanc e between any two consec uti ve labels is the same, we get an exact qualitati ve result, and a qualitat i ve basic belief assignment (bba) is considere d normalized if the sum of all its qualitati ve masses is equa l to L max = L m +1 . If the labels are not equidistant, we still can use all qualit ati ve ope rators deﬁned in the FL ARL, but the qualita ti ve result is approximate , and a qualita ti ve bba is consid ered quas i-normali zed if the sum of all its masses is equal to L max . Using the qualitati ve ope rator of 12 W e call it also qualita tive belief mass or q-ma ss for short. 31 FLARL , we c an easily e xtend all the combi nation and conditio ning rules from quantitati ve to qualita ti ve. In t he sequel we will consider s ≥ 2 qualitati ve belief assign ments q m 1 ( . ) , . . . , q m s ( . ) deﬁned ov er the same space G Θ and pro vided by s indepe ndent sources S 1 , . . . , S s of ev idence . Note : T he addition and multiplica tion operator s used in all qualitati ve fusion formulas in next sections corre- spond to qualitativ e addition and qualitative multiplic ation operators and must not be confuse d with classical additi on and multipli cation operators for numbers. 6.3 Qualitati ve Conjunctive Rule The qualitati ve Conjuncti ve R ule (qCR) of s ≥ 2 sources is deﬁned similarly to the quantitati ve conjunc ti ve consen sus rule, i.e. q m q C R ( X ) = X X 1 ,...,X s ∈ G Θ X 1 ∩ ... ∩ X s = X s Y i =1 q m i ( X i ) (26) The total qualitat i ve conﬂicting mass is giv en by K 1 ...s = X X 1 ,...,X s ∈ G Θ X 1 ∩ ... ∩ X s = ∅ s Y i =1 q m i ( X i ) 6.4 Qualitati ve DSm Classic rule The qualit ati ve D Sm Classic ru le (q-DSmC) for s ≥ 2 is deﬁned similar ly to DSm Classic fusi on rule (DSmC) as follo ws : q m q D S mC ( ∅ ) = L 0 and for all X ∈ D Θ \ {∅} , q m q D S mC ( X ) = X X 1 ,,...,X s ∈ D Θ X 1 ∩ ... ∩ X s = X s Y i =1 q m i ( X i ) (27) 6.5 Qualitati ve hybrid DSm rule The qualitati ve hy brid DS m rule (q-DSmH) is de ﬁned similarly t o quantita ti ve hy brid DSm rule [31] as follo ws: q m q D S mH ( ∅ ) = L 0 (28) and for all X ∈ G Θ \ {∅} q m q D S mH ( X ) , φ ( X ) · h q S 1 ( X ) + q S 2 ( X ) + q S 3 ( X ) i (29) where all sets in v olved in formulas are in the canonical form and φ ( X ) is the char acteristic non-emptine ss functi on of a set X , i.e. φ ( X ) = L m +1 if X / ∈ ∅ and φ ( X ) = L 0 otherwis e, where ∅ , { ∅ M , ∅} . ∅ M is the set of all e lements of D Θ which ha ve been forced to be e mpty thro ugh the constra ints of the model M and ∅ is the classic al/uni versa l empty set. q S 1 ( X ) ≡ q m q D S mC ( X ) , q S 2 ( X ) , q S 3 ( X ) are deﬁned by q S 1 ( X ) , X X 1 ,X 2 ,...,X s ∈ D Θ X 1 ∩ X 2 ∩ ... ∩ X s = X s Y i =1 q m i ( X i ) (30) q S 2 ( X ) , X X 1 ,X 2 ,...,X s ∈ ∅ [ U = X ] ∨ [( U ∈ ∅ ) ∧ ( X = I t )] s Y i =1 q m i ( X i ) (31) 32 q S 3 ( X ) , X X 1 ,X 2 ,...,X k ∈ D Θ X 1 ∪ X 2 ∪ ... ∪ X s = X X 1 ∩ X 2 ∩ ... ∩ X s ∈ ∅ s Y i =1 q m i ( X i ) (32) with U , u ( X 1 ) ∪ . . . ∪ u ( X s ) where u ( X ) is the union o f a ll θ i that compose X , I t , θ 1 ∪ . . . ∪ θ n is th e tota l ignora nce. q S 1 ( X ) is nothing but the qDSmC rule for s independ ent sources based on M f (Θ) ; q S 2 ( X ) is the qualit ati ve mass of all relativ ely and absolu tely empty sets which is transferred t o the total or relativ e ig norances associ ated with non existe ntial constrain ts (if any , like in some dynamic problems); q S 3 ( X ) transfers the sum of relati vely empty sets directly onto the canoni cal disjuncti ve form of non-empty sets. qDSmH generalize s qDSmC works for any models (free DSm model, Shafer’ s model or any hybrid models) when manipulati ng qualit ati ve belief assignment s. 6.6 Qualitati ve PCR5 rule (qPCR5) In classical (i.e. quantitati ve) DSmT frame work, the Proportiona l Conﬂict Redistrib ution rule no. 5 (PCR5) deﬁned in [34] has been prov en to prov ide very good and cohere nt results for combining (quantitati ve) belief masses, see [9, 33]. When dealing with qu alitati ve beliefs within the DSm Field an d Linear Algebra of Reﬁne d Labels [36] we g et an exac t qualit ati ve result no matter what fus ion rule is used (DSm fusi on rules, Dempster ’ s rule, Smets’ s rule, Dubois-Prad e’ s rule, etc.). The exact qualitati ve result will a reﬁned label (bu t the user can round it up or do wn to the closest intege r index label). 6.7 A simple example of qualitativ e fusion of qba’ s Let’ s consid er the follo wing set of ordered linguistic labels L = { L 0 , L 1 , L 2 , L 3 , L 4 , L 5 } (for example, L 1 , L 2 , L 3 and L 4 may represent the valu es: L 1 , very poo r , L 2 , poor , L 3 , good and L 4 , very good , where , symbol means by deﬁni tion ). Let’ s consider no w a simple two-source case with a 2D frame Θ = { θ 1 , θ 2 } , Shafer’ s model for Θ , and qba’ s expr essed as follo ws: q m 1 ( θ 1 ) = L 1 , q m 1 ( θ 2 ) = L 3 , q m 1 ( θ 1 ∪ θ 2 ) = L 1 q m 2 ( θ 1 ) = L 2 , q m 2 ( θ 2 ) = L 1 , q m 2 ( θ 1 ∪ θ 2 ) = L 2 The two qualit ati ve masses q m 1 ( . ) and q m 2 ( . ) are normal ized since: q m 1 ( θ 1 ) + q m 1 ( θ 2 ) + q m 1 ( θ 1 ∪ θ 2 ) = L 1 + L 3 + L 1 = L 1+3+1 = L 5 and q m 2 ( θ 1 ) + q m 2 ( θ 2 ) + q m 2 ( θ 1 ∪ θ 2 ) = L 2 + L 1 + L 2 = L 2+1+2 = L 5 W e ﬁrst deri ve the result of the conjunct i ve conse nsus. T his yields: q m 12 ( θ 1 ) = q m 1 ( θ 1 ) q m 2 ( θ 1 ) + q m 1 ( θ 1 ) q m 2 ( θ 1 ∪ θ 2 ) + q m 1 ( θ 1 ∪ θ 2 ) q m 2 ( θ 1 ) = L 1 × L 2 + L 1 × L 2 + L 1 × L 2 = L 1 · 2 5 + L 1 · 2 5 + L 1 · 2 5 = L 2 5 + 2 5 + 2 5 = L 6 5 = L 1 . 2 q m 12 ( θ 2 ) = q m 1 ( θ 2 ) q m 2 ( θ 2 ) + q m 1 ( θ 2 ) q m 2 ( θ 1 ∪ θ 2 ) + q m 1 ( θ 1 ∪ θ 2 ) q m 2 ( θ 2 ) = L 3 × L 1 + L 3 × L 2 + L 1 × L 1 = L 3 · 1 5 + L 3 · 2 5 + L 1 · 1 5 = L 3 5 + 6 5 + 1 5 = L 10 5 = L 2 33 q m 12 ( θ 1 ∪ θ 2 ) = q m 1 ( θ 1 ∪ θ 2 ) q m 2 ( θ 1 ∪ θ 2 ) = L 1 × L 2 = L 1 · 2 5 = L 2 5 = L 0 . 4 q m 12 ( θ 1 ∩ θ 2 ) = q m 1 ( θ 1 ) q m 2 ( θ 2 ) + q m 1 ( θ 2 ) q m 2 ( θ 1 ) = L 1 × L 1 + L 2 × L 3 = L 1 · 1 5 + L 2 · 3 5 = L 1 5 + 6 5 = L 7 5 = L 1 . 4 Therefore we get: • for the fusio n w ith qDSmC, when assumin g θ 1 ∩ θ 2 6 = ∅ , q m q D S mC ( θ 1 ) = L 1 . 2 q m q D S mC ( θ 2 ) = L 2 q m q D S mC ( θ 1 ∪ θ 2 ) = L 0 . 4 q m q D S mC ( θ 1 ∩ θ 2 ) = L 1 . 4 • for the fusion w ith qDSmH, when assuming θ 1 ∩ θ 2 = ∅ . The mass of θ 1 ∩ θ 2 is transferred to θ 1 ∪ θ 2 . Hence: q m q D S mH ( θ 1 ) = L 1 . 2 q m q D S mH ( θ 2 ) = L 2 q m q D S mH ( θ 1 ∩ θ 2 ) = L 0 q m q D S mH ( θ 1 ∪ θ 2 ) = L 0 . 4 + L 1 . 4 = L 1 . 8 • for the fusion with qPC R5, when assumin g θ 1 ∩ θ 2 = ∅ . The mass q m 12 ( θ 1 ∩ θ 2 ) = L 1 . 4 is transferr ed to θ 1 and to θ 2 in the follo wing way: q m 12 ( θ 1 ∩ θ 2 ) = q m 1 ( θ 1 ) q m 2 ( θ 2 ) + q m 2 ( θ 1 ) q m 1 ( θ 2 ) Then, q m 1 ( θ 1 ) q m 2 ( θ 2 ) = L 1 × L 1 = L 1 · 1 5 = L 1 5 = L 0 . 2 is redistrib uted to θ 1 and θ 2 propo rtional ly with respe ct to their qualitati ve m asses put in the con ﬂict L 1 and respec ti vel y L 1 : x θ 1 L 1 = y θ 2 L 1 = L 0 . 2 L 1 + L 1 = L 0 . 2 L 1+1 = L 0 . 2 L 2 = L 0 . 2 2 · 5 = L 1 2 = L 0 . 5 whence x θ 1 = y θ 2 = L 1 × L 0 . 5 = L 1 · 0 . 5 5 = L 0 . 5 5 = L 0 . 1 . Actually , we could easier see that q m 1 ( θ 1 ) q m 2 ( θ 2 ) = L 0 . 2 had in this case to be equally split between θ 1 and θ 2 since the mass put in the conﬂict by θ 1 and θ 2 was the same for each of them: L 1 . Therefore L 0 . 2 2 = L 0 . 2 2 = L 0 . 1 . Similarly , q m 2 ( θ 1 ) q m 1 ( θ 2 ) = L 2 × L 3 = L 2 · 3 5 = L 6 5 = L 1 . 2 has to be redis trib uted to θ 1 and θ 2 propo rtional ly w ith L 2 and L 3 respec ti vel y : x ′ θ 1 L 2 = y ′ θ 2 L 3 = L 1 . 2 L 2 + L 3 = L 1 . 2 L 2+3 = L 1 . 2 L 5 = L 1 . 2 5 · 5 = L 1 . 2 whence ( x ′ θ 1 = L 2 × L 1 . 2 = L 2 · 1 . 2 5 = L 2 . 4 5 = L 0 . 48 y ′ θ 2 = L 3 × L 1 . 2 = L 3 · 1 . 2 5 = L 3 . 6 5 = L 0 . 72 No w , add all these to the qualitati ve masses of θ 1 and θ 2 respec ti vel y: q m q P C R 5 ( θ 1 ) = q m 12 ( θ 1 ) + x θ 1 + x ′ θ 1 = L 1 . 2 + L 0 . 1 + L 0 . 48 = L 1 . 2+0 . 1+0 . 48 = L 1 . 78 q m q P C R 5 ( θ 2 ) = q m 12 ( θ 2 ) + y θ 2 + y ′ θ 2 = L 2 + L 0 . 1 + L 0 . 72 = L 2+0 . 1+0 . 72 = L 2 . 82 q m q P C R 5 ( θ 1 ∪ θ 2 ) = q m 12 ( θ 1 ∪ θ 2 ) = L 0 . 4 q m q P C R 5 ( θ 1 ∩ θ 2 ) = L 0 The qualitati ve mass results us ing a ll f usion rules (qDSmC,qDSmH,qPC R5) remain no rmalized in FLARL . Naturally , if one prefers to exp ress the ﬁnal result s w ith qua litati ve labels belonging in the origin al discrete set of labels L = { L 0 , L 1 , L 2 , L 3 , L 4 , L 5 } , some approximat ions will be necess ary to round contin uous inde xed labels to their closest inte ger/di screte index v alue; by example, q m q P C R 5 ( θ 1 ) = L 1 . 78 ≈ L 2 , q m q P C R 5 ( θ 2 ) = L 2 . 82 ≈ L 3 and q m q P C R 5 ( θ 1 ∪ θ 2 ) = L 0 . 4 ≈ L 0 . 34 6.8 A simple example f or the qDSmP transformation W e ﬁrst recall that the qualita ti ve exte nsion of (22), den oted q D S mP ǫ ( . ) is giv en by q D S mP ǫ ( ∅ ) = 0 and ∀ X ∈ G Θ \ {∅} by q D S mP ǫ ( X ) = X Y ∈ G Θ X Z ⊆ X ∩ Y C ( Z )=1 q m ( Z ) + ǫ · C ( X ∩ Y ) X Z ⊆ Y C ( Z )=1 q m ( Z ) + ǫ · C ( Y ) q m ( Y ) (33) where all opera tions in (33) are referre d to labels, that is q -operators on linguist ic labels and not classical oper - ators on numbers. Let’ s consider the simple frame Θ = { θ 1 , θ 2 } (here n = | Θ | = 2 ) with Shafer’ s mode l (i.e. θ 1 ∩ θ 2 = ∅ ) and the followin g set of lingui stic labels L = { L 0 , L 1 , L 2 , L 3 , L 4 , L 5 } , with L 0 = L min and L 5 = L max = L m +1 (here m = 4 ) and t he fo llo wing qualitati ve belief assignmen t: q m ( θ 1 ) = L 1 , q m ( θ 2 ) = L 3 and qm ( θ 1 ∪ θ 2 ) = L 1 . q m ( . ) is quasi-no rmalized since P X ∈ 2 Θ q m ( X ) = L 5 = L max . In this ex ample and with D S mP transfo rmation, q m ( θ 1 ∪ θ 2 ) = L 1 is redistrib uted to θ 1 and θ 2 propo rtional ly w ith respect to their qualitati ve masses L 1 and L 3 respec ti vel y . S ince both L 1 and L 3 are diffe rent from L 0 , we can take the tuning parameter ǫ = 0 for the best transfer . ǫ is tak en dif ferent fro m zero when a mass of a set in vol ved in a partial or total ignora nce is zero (for qualit ati ve masses, it means L 0 ). Therefore using (25), one has x θ 1 L 1 = x θ 2 L 3 = L 1 L 1 + L 3 = L 1 L 4 = L 1 4 · 5 = L 5 4 = L 1 . 25 and thus using (24), one gets x θ 1 = L 1 × L 1 . 25 = L 1 · (1 . 25) 5 = L 1 . 25 5 = L 0 . 25 x θ 2 = L 3 × L 1 . 25 = L 3 · (1 . 25) 5 = L 3 . 75 5 = L 0 . 75 Whence q D S mP ǫ =0 ( θ 1 ∩ θ 2 ) = q D S mP ǫ =0 ( ∅ ) = L 0 q D S mP ǫ =0 ( θ 1 ) = L 1 + x θ 1 = L 1 + L 0 . 25 = L 1 . 25 q D S mP ǫ =0 ( θ 2 ) = L 3 + x θ 2 = L 3 + L 0 . 75 = L 3 . 75 Naturally in our exampl e, one has also q D S mP ǫ =0 ( θ 1 ∪ θ 2 ) = q D S mP ǫ =0 ( θ 1 ) + q D S m P ǫ =0 ( θ 2 ) − q D S m P ǫ =0 ( θ 1 ∩ θ 2 ) = L 1 . 25 + L 3 . 75 − L 0 = L 5 = L max Since H max = log 2 n = log 2 2 = 1 , using the qualitati ve extensio n of PIC formula (21), one obtain s the follo wing qualitati ve PIC val ue: P I C = 1 + 1 1 · [ q D S mP ǫ =0 ( θ 1 ) log 2 ( q D S mP ǫ =0 ( θ 1 )) + q D S mP ǫ =0 ( θ 2 ) log 2 ( q D S mP ǫ =0 ( θ 2 ))] = 1 + L 1 . 25 log 2 ( L 1 . 25 ) + L 3 . 75 log 2 ( L 3 . 75 ) ≈ L 0 . 94 since we considered the isomorph ic transf ormation L i = i/ ( m + 1) (in our particul ar example m = 4 interior labels ). 35 7 Belief Conditioning Rules 7.1 Shafer’ s Conditioning Rule (SCR) Until v ery recent ly , the most commonly used condition ing rule for belief re vision was the one propose d by Shafer [24] and referred here as S hafer’ s Condition ing Rule (SCR). The SCR consists in combining the prior bba m ( . ) with a spe ciﬁc bba focus ed on A with Dempster’ s rule of combination for transfer ring the conﬂicting mass to non-empty sets in order to prov ide the rev ised bba. In other words, the conditionin g by a propos ition A , is obtain ed by S CR as follo ws : m S C R ( . | A ) = [ m ⊕ m S ]( . ) (34) where m ( . ) is the prior bba to update, A is the conditio ning ev ent, m S ( . ) is the bba focuse d on A deﬁned by m S ( A ) = 1 and m S ( X ) = 0 for all X 6 = A and ⊕ denotes Dempster’ s rule of combination [24]. The SCR appro ach based on Dempster’ s rule of combination of the prior bba with the bba focuse d on the condit ioning e ven t remains subjective since actually in such belief revisi on process both sources are subjecti ve and SCR does n’ t manage prope rly the objecti ve nature/a bsolut e truth carrie d by the cond itionin g term. Indeed, when condit ioning a prior mass m ( . ) , knowing (or assu ming) that the truth is in A , means that we hav e in hands an absolu te (not subje cti ve) knowled ge, i.e. the truth in A has oc curred (or is assumed to ha ve occ urred) , thus A is realized (or is assumed to be realize d) and this is (or at least must be inte rprete d as) an abs olute truth. The condition ing term ”Give n A ” must therefo re be consid ered as an absolut e truth, while m S ( A ) = 1 introd uced in SCR cannot refer to an absolute truth actually , but only to a subjec tive certai nty on the possibl e occurr ence of A from a virtual second source of evi dence. The advan tage of S CR remains undoubtedl y in its simplicit y and the main ar gument in its fa vor is its cohere nce with condit ional probabi lity when m anipu lating Bayesian belief assignment. But in our opinion, SC R should better be interpreted as the fusion of m ( . ) with a particular subject i ve bba m S ( A ) = 1 rather than an objecti ve belief conditionin g rule. This funda mental remark m oti vate d us to de velop a new family of BC R [34] based on hyper -po w er set decompositio n (HP SD) exp lained brieﬂy in the ne xt section. It turns out that many BCR are pos sible because the redistrib ution of masses of e lements outsi de of A (the condi tioning ev ent) to those in side A can be don e in n -ways . This will b e brieﬂy presented right after the next sec tion. 7.2 Hyper-P ower Set Secomposition (HPSD) Let Θ = { θ 1 , θ 2 , . . . , θ n } , n ≥ 2 , a model M (Θ) associated for Θ (free DSm model, hybrid or Shafer’ s model) and its corres pondin g hype r -po wer set D Θ . L et’ s consider a (quan titati ve) basic belief assignmen t (bba) m ( . ) : D Θ 7→ [0 , 1] such that P X ∈ D Θ m ( X ) = 1 . Suppose one ﬁnds out that the tru th is in the set A ∈ D Θ \ {∅} . Let P D ( A ) = 2 A ∩ D Θ \ {∅} , i.e. all non-empty parts (subsets) of A which are include d in D Θ . Let’ s consider the normal cases when A 6 = ∅ and P Y ∈P D ( A ) m ( Y ) > 0 . For the de generate case when the truth is in A = ∅ , we consider Smets’ open-w orld, which means that there are oth er hypothese s Θ ′ = { θ n +1 , θ n +2 , . . . θ n + m } , m ≥ 1 , and the truth is in A ∈ D Θ ′ \ {∅} . If A = ∅ and we conside r a close- world , then it mea ns th at the pro blem is imp ossibl e. For a nother de generate case, when P Y ∈P D ( A ) m ( Y ) = 0 , i.e. when the source gav e us a totally (100%) wrong informati on m ( . ) , then, we deﬁne: m ( A | A ) , 1 and, as a consequen ce, m ( X | A ) = 0 for any X 6 = A . Let s ( A ) = { θ i 1 , θ i 2 , . . . , θ i p } , 1 ≤ p ≤ n , be the single tons/at oms that compose A (for example , if A = θ 1 ∪ ( θ 3 ∩ θ 4 ) then s ( A ) = { θ 1 , θ 3 , θ 4 } ). The Hyper - Power S et Decompositi on (HPSD ) of D Θ \ ∅ cons ists in its dec ompositio n into the three followin g subsets genera ted by A : • D 1 = P D ( A ) , the parts of A which are incl uded in the hyper -power set, except the empty set; • D 2 = { (Θ \ s ( A )) , ∪ , ∩} \ {∅} , i.e. the sub-hyper -po wer set generated by Θ \ s ( A ) under ∪ and ∩ , without the empty set. • D 3 = ( D Θ \ {∅} ) \ ( D 1 ∪ D 2 ) ; each set from D 3 has in its formula singl etons from both s ( A ) and Θ \ s ( A ) in the case when Θ \ s ( A ) is differ ent from empty set. 36 D 1 , D 2 and D 3 ha ve no element in common two by two and their union is D Θ \ {∅} . Simple e xample of HPSD : Let’ s consid er Θ = { θ 1 , θ 2 , θ 3 } with S hafer’ s model (i.e. all elements of Θ are exc lusi ve) and let’ s assume that the truth is in θ 2 ∪ θ 3 , i.e. the conditio ning term is θ 2 ∪ θ 3 . Then one has the follo wing HPS D: D 1 = { θ 2 , θ 3 , θ 2 ∪ θ 3 } , D 2 = { θ 1 } and D 3 = { θ 1 ∪ θ 2 , θ 1 ∪ θ 3 , θ 1 ∪ θ 2 ∪ θ 3 } . More comple x and detaile d exa mples can be found in [34]. 7.3 Quantitativ e belief conditioning rules (BCR) Since there exis ts actual ly many ways for redistrib uting the m asses of elements outsid e of A (the conditionin g e ven t) to thos e inside A , se vera l BCR ’ s ha ve been pro posed in [34]. In this int roduct ion, we will n ot bro w se all the possibi lities for doing these redistrib utions and all BCR’ s formulas bu t only one, the BCR number 17 (i.e. BCR17) which does in our opinion the most reﬁned redistri b ution since: - the mass m ( W ) of each element W in D 2 ∪ D 3 is trans ferred to those X ∈ D 1 elements which are included in W if any pro portio nally with respect to their non-empty masses; - if no such X exists, the mass m ( W ) is tran sferred in a pessimistic/p rudent way to the k -larg est element from D 1 which are include d in W (in equal parts) if any; - if neither this way is possible, then m ( W ) is indiscri minately distrib uted to all X ∈ D 1 propo rtional ly with respec t to their nonz ero m asses . BCR17 is deﬁne d by the follo wing formula (see [34], Chap. 9 for detaile d explanati ons and exampl es): m B C R 17 ( X | A ) = m ( X ) · " S D 1 + X W ∈ D 2 ∪ D 3 X ⊂ W S ( W ) 6 =0 m ( W ) S ( W ) # + X W ∈ D 2 ∪ D 3 X ⊂ W, X is k -larges t S ( W )=0 m ( W ) /k (35) where ” X is k -lar gest ” means that X is the k -lar gest (with respect to inclusi on) set included in W and S ( W ) , X Y ∈ D 1 ,Y ⊂ W m ( Y ) S D 1 , X Z ∈ D 1 , or Z ∈ D 2 | ∄ Y ∈ D 1 with Y ⊂ Z m ( Z ) P Y ∈ D 1 m ( Y ) Note: The authors mentioned in an Erratum to the printed version of the second volume of DSmT book se- ries ( h ttp:// fs.ga llup.unm.edu//Erratum.pdf ) and the y also corrected the online version of the aforemen tioned book (see page 240 in h ttp:/ /fs.ga llup.unm.edu//DSmT-book2.pdf that all de- nominato rs of the BCR’ s formulas are naturally suppose d to be differe nt from zero. O f course, Shafer’ s con- dition al rule as stated in Theorem 3.6, page 67 of [24] does not work when the denominator is zero and that’ s why S hafer has introduced the conditi on B el ( ¯ B ) < 1 (or equi v alently P l ( B ) > 0 ) in his theorem when the condit ioning term is B . A simple example for B CR17 : Let’ s con sider Θ = { θ 1 , θ 2 , θ 3 } with Shafer’ s model (i.e. all el ements of Θ are exc lusi ve) and let’ s assume that the truth is in θ 2 ∪ θ 3 , i.e. the conditio ning term is A , θ 2 ∪ θ 3 . Then one has the follo wing H PSD: D 1 = { θ 2 , θ 3 , θ 2 ∪ θ 3 } , D 2 = { θ 1 } D 3 = { θ 1 ∪ θ 2 , θ 1 ∪ θ 3 , θ 1 ∪ θ 2 ∪ θ 3 } . Let’ s consider the followin g prior bba: m ( θ 1 ) = 0 . 2 , m ( θ 2 ) = 0 . 1 , m ( θ 3 ) = 0 . 2 , m ( θ 1 ∪ θ 2 ) = 0 . 1 , m ( θ 2 ∪ θ 3 ) = 0 . 1 and m ( θ 1 ∪ θ 2 ∪ θ 3 ) = 0 . 3 . 37 W ith BCR17, for D 2 , m ( θ 1 ) = 0 . 2 is transferred proportion ally to all elements of D 1 , i.e. x θ 2 0 . 1 = y θ 3 0 . 2 = z θ 2 ∪ θ 3 0 . 1 = 0 . 2 0 . 4 = 0 . 5 whence the parts of m ( θ 1 ) redistrib uted to θ 2 , θ 3 and θ 2 ∪ θ 3 are respecti vely x θ 2 = 0 . 05 , y θ 3 = 0 . 10 , and z θ 2 ∪ θ 3 = 0 . 05 . For D 3 , there is actu ally no need to trans fer m ( θ 1 ∪ θ 3 ) because m ( θ 1 ∪ θ 3 ) = 0 in this e xample; where as m ( θ 1 ∪ θ 2 ) = 0 . 1 is transfer red to θ 2 (no case of k -elements herein) ; m ( θ 1 ∪ θ 2 ∪ θ 3 ) = 0 . 3 is transfer red to θ 2 , θ 3 and θ 2 ∪ θ 3 propo rtional ly to their correspond ing masses: x θ 2 / 0 . 1 = y θ 3 / 0 . 2 = z θ 2 ∪ θ 3 / 0 . 1 = 0 . 3 / 0 . 4 = 0 . 75 whence x θ 2 = 0 . 075 , y θ 3 = 0 . 15 , and z θ 2 ∪ θ 3 = 0 . 075 . Finally , one gets m B C R 17 ( θ 2 | θ 2 ∪ θ 3 ) = 0 . 10 + 0 . 05 + 0 . 10 + 0 . 075 = 0 . 325 m B C R 17 ( θ 3 | θ 2 ∪ θ 3 ) = 0 . 20 + 0 . 10 + 0 . 15 = 0 . 450 m B C R 17 ( θ 2 ∪ θ 3 | θ 2 ∪ θ 3 ) = 0 . 10 + 0 . 05 + 0 . 075 = 0 . 225 which is dif ferent from the result obtaine d with SCR, since one gets in this examp le: m S C R ( θ 2 | θ 2 ∪ θ 3 ) = m S C R ( θ 3 | θ 2 ∪ θ 3 ) = 0 . 25 m S C R ( θ 2 ∪ θ 3 | θ 2 ∪ θ 3 ) = 0 . 50 More comple x and detailed examples can be found in [34]. 7.4 Qualitati ve belief conditioning rules In this section we p resent only the qu alitati ve belief con ditioni ng rule no 17 which ex tends the princ iples of the pre vious quantitati ve rule BCR17 in the qualitat i ve domain using the op erators on ling uistic labels deﬁned pre- viousl y . W e con sider from no w on a general frame Θ = { θ 1 , θ 2 , . . . , θ n } , a gi ven model M (Θ) with its hyper - po wer set D Θ and a gi ven ex tended ordered set L of qualitati ve value s L = { L 0 , L 1 , L 2 , . . . , L m , L m +1 } . The prior qualitati ve basi c belief ass ignment (qbba) taking its va lues in L is d enoted q m ( . ) . W e assume in th e sequel that the conditi oning ev ent is A 6 = ∅ , A ∈ D Θ , i.e. the absolut e truth is in A . The approach we present here is a direct extensio n of BCR17 using F LARL operato rs. Such extensio n can be done with all quanti tati ve BCR’ s rules propose d in [34], b ut only Q BCR17 is pre sented here for the sake of space limitations. 7.4.1 Qualita tive Belief Conditioning Rule no 17 (QBC R17) Similarly to BCR17, QBCR17 is deﬁned by the follo wing formula: q m B C R 17 ( X | A ) = q m ( X ) · " q S D 1 + X W ∈ D 2 ∪ D 3 X ⊂ W q S ( W ) 6 =0 q m ( W ) q S ( W ) # + X W ∈ D 2 ∪ D 3 X ⊂ W, X is k -larges t q S ( W )=0 q m ( W ) /k (36 ) where ” X is k -lar gest ” means that X is the k -lar gest (with respect to inclusi on) set included in W and q S ( W ) , X Y ∈ D 1 ,Y ⊂ W q m ( Y ) S D 1 , X Z ∈ D 1 , or Z ∈ D 2 | ∄ Y ∈ D 1 with Y ⊂ Z q m ( Z ) P Y ∈ D 1 q m ( Y ) Naturally , all operators (summation, product, di vision, etc) in v olved in the formula (36) are the operator s deﬁned in FLARL working on linguistic labels. It is worth to note that the formula (36) requires also the divi- sion of the label q m ( W ) by a scalar k . T his di vision is deﬁned as follo ws: Let r ∈ R , r 6 = 0 . Then the label di vision by a scalar is deﬁned by L a r = L a/r (37) 38 7.4.2 A simple e xample for QBCR17 Let’ s c onside r L = { L 0 , L 1 , L 2 , L 3 , L 4 , L 5 , L 6 } a set of ord ered lingu istic labels. For example, L 1 , L 2 , L 3 , L 4 and L 5 may rep resent the values : L 1 , very poor , L 2 , poor , L 3 , medium , L 4 , good an d L 5 , very good . Let’ s consider also the frame Θ = { A, B , C, D } with the hybrid model correspond ing to the V enn diagram on Figure 6. ✫✪ ✬✩ ✫✪ ✬✩ ✫✪ ✬✩ ✫✪ ✬✩ ❅ ❘ A  ✠ B ✛ C ✛ D Fig. 6: V enn Diagram for the hybrid model for this example . W e assume that the prior qualitati ve bba q m ( . ) is gi ven by: q m ( A ) = L 1 , q m ( C ) = L 1 , q m ( D ) = L 4 and the qua litati ve masses of all other element s of G Θ tak e the minimal/zero v alue L 0 . This qua litati ve m ass is quasi- normalize d since L 1 + L 1 + L 4 = L 1+1+4 = L 6 = L max . If we assume that the condit ioning ev ent is the proposi tion A ∪ B , i.e. the absolu te truth is in A ∪ B , the hyper -po w er set decompositi on (HPSD) is obtained as follo ws: D 1 is formed by all parts of A ∪ B , D 2 is the set generat ed by { ( C, D ) , ∪ , ∩} \ ∅ = { C, D , C ∪ D , C ∩ D } , and D 3 = { A ∪ C , A ∪ D , B ∪ C, B ∪ D , A ∪ B ∪ C , A ∪ ( C ∩ D ) , . . . } . Because the truth is in A ∪ B , q m ( D ) = L 4 is transferred in a prudent way to ( A ∪ B ) ∩ D = B ∩ D according to our hybrid model, because B ∩ D is the 1-lar gest element from A ∪ B which is included in D . While q m ( C ) = L 1 is transferr ed to A only , since it is the only element in A ∪ B whose qualit ati ve mass q m ( A ) is differ ent from L 0 (zero); hence: q m QB C R 17 ( A ) = q m ( A ) + q m ( C ) = L 1 + L 1 = L 1+1 = L 2 . Therefore , one ﬁ nally gets: q m QB C R 17 ( A | A ∪ B ) = L 2 q m QB C R 17 ( C | A ∪ B ) = L 0 q m QB C R 17 ( D | A ∪ B ) = L 0 q m QB C R 17 ( B ∩ D | A ∪ B ) = L 4 which is a normalized qualitati ve bba. More complicat ed exa mples based on other QBC R’ s c an be found in [35]. 8 Conclusion A general presentati on of the foundatio ns of DS mT has been propose d in this introduct ion. DSmT propos es ne w quantitati ve and qualitat i ve rules of combinat ion for uncerta in, imprec ise and highly conﬂicting sources of infor mation. Sev eral applicatio ns o f DSmT hav e been proposed recently in the literatur e and sho w the potent ial and the efﬁcien cy of this new theory . DSmT offe rs the possib ility to work in dif ferent fusion spaces depen ding on the nature of problem under considera tion. Thus, one can work either in 2 Θ = (Θ , ∪ ) (i.e. in the classical po wer set as in DST frame work), in D Θ = (Θ , ∪ , ∩ ) (the hyper -power set — also kno wn as Dedekind ’ s lattice ) or in the super -po wer set S Θ = (Θ , ∪ , ∩ , c ( . )) , which includes 2 Θ and D Θ and which repres ents the power set of the m inimal reﬁ nement of the frame Θ when the reﬁnement is possible (because for vagu e elements whose frontier s are not well known the reﬁnement is not possible ). W e hav e enriched the DSmT with a subj ecti ve probabi lity ( D S mP ǫ ) that gets the best Probabilistic Informatio n Content (PIC) in comparis on with other exist ing subject i ve probabi lities. Also, we hav e deﬁned and de velo ped the DSm Field and Linear Algebra of Reﬁned Labels that permit the transformation of an y fusio n rule to a corr espond ing qualit ati ve fusion rule which gi ves an ex act qualitat i ve result (i.e. a reﬁned label), so far the best in literature. 39 Refer ences [1] J. Bolanos, L.M . 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