Belief functions on lattices

Belief functions on lattices Mic hel GRABISCH Universit ´ e P aris I – Pan th ´ eon-Sorb onne email Michel.Grab isch@lip6.fr Abstract W e extend the notion of b elief function to the case wher e the un derlying struc- ture is no more the Bo olean lattice of subsets of some u niv e rsal set, bu t a n y lattice, whic h we will end o w w ith a minimal set of prop erties according to our needs. W e sho w that all classica l constructions and definitions (e.g., mass allo cat ion, common- alit y function, p laus ib il it y fun c tions, necessit y measures with nested focal elements, p ossibilit y d istr i butions, Dempster rule of com bination, decomp osition w.r.t. sim- ple sup port fun c tions, etc.) remain v alid in this general setting. Moreo v er, our pro of of decomposition of b el ief functions into simple supp ort functions is m uc h simpler and general than the original one b y S hafer. Keyw ords: b elief function, lattice, plausibilit y , p ossibilit y , necessit y 1 In t r o duction The theory of evidence, as established by Shafer [16] after the w ork of Dempster [4], and brough t into a pra ctically usable form b y the works of Smets in particular [17, 18], has b ecome a popular to ol in artificial inte lligence for the represen tation of kno wledge and making decision. In particular, many applications in classification hav e b een done [5 , 6]. The main adv an tage ov er more t raditional mo dels based on probabilit y is that the mo del of Shaf er allo ws for a pro per represen tation o f ignor a nc e. On a mathematical p oin t of view, b elief functions, whic h are at the core of the the- ory of evidence, p osse ss remark a ble pro perties, in particular their links with the M¨ obius transform [15] and the co-M¨ obius transform [9, 10], called c o mmonality b y Shaf er. Re- marking that b elief functions a re non negativ e isotone functions defined on the Bo olean lattice of subsets, one ma y ask if all these prop erties remain v alid when more general lat- tices are considered. The aim of this pap er is precisely t o inv estigate t his question, and w e will show that amazingly they all remain v alid. A first inv estigation of this question w a s done b y Barth ´ elem y [1], and our w o rk will complete his results. W e are not a w a r e of other sim ilar w orks, exce pt the one of Kramosil [1 2 ], where b elief functions are defined on Bo olean latt ices but ta k e v alue in a partially ordered set, a n d the notion of bi-b elief prop osed b y Grabisch and Labreuc he [11], where the underlying lattice is 3 n . On an a pp lication p oin t of view, one may ask ab out the usefulness of suc h a general- ization, apart from its mathematical b eaut y . A general answ er to this is that the ob jects w e ma nipulat e ( e v en ts, logical prop ositions, etc.) may not f o rm a Bo olean lattice, i.e., 1 distributiv e and complem en ted. Th us, a study on a w eaker y et rich structure has its in terest. Let us give some examples. • Case where the univ ersal set Ω is the set of p os sible outcomes, states of nature, etc. In the classical case, all subse ts of Ω (called ev ents) are considered, but it ma y happ en that some ev en ts are not observ able or realizable, meaningful, etc. Then, the structure of the ev ents is no more the Bo olean lattice 2 Ω . • Case where the univ ersal set Ω is the set of propositiona l v ariables, eithe r true o r false. As argued b y Barth´ elem y [1], in non-classical logics, the set of prop ositions need not be 2 Ω , and as w e will see later, probabilit y t heory applies as far as the lattice in duced b y prop ositional calc ulus is distributiv e, and this co v ers intuitionistic logic and pa racons isten t logic. If distributivit y do es not hold, then b elief functions app ear a s a natural candidate, since as it will be sho wn, b elief functions can live on any lat t ice. • Case where the univ ersal set is the set of pla y ers/agen ts in some co operat ive game or m ultiagent situation. Subsets of Ω are called coalitions, and most of the time, it happ ens that some coalitions a re infeasible, .i.e., they cannot form, due to some inheren t imp oss ibilit y dep end ing on the con text. F or example, in v oting situations, clearly all coalitions of p olitical parties cannot form. The same holds fo r agen ts or pla y ers in general where some incompatibilities exist b et w een them. • Kno wledge extraction and mo deling: ob j ects under study a re often structured as lattices. F or example, t h e p opular F ormal Concept Analysis of Gan ter and Wille [8] build lattices of concepts, f r o m a matrix of ob jects described b y qualitativ e attributes. • Finally , in some cases, ob j ects of in terest are not subsets of some univers al set. This is the case f o r example when one is in terested in to the collection of partitions o f some set (again, this happ ens in game t h eory under the name “ g ame in partition function form” [20], and a lso in knowledge extraction where the fundamen t a l problem is to partition attributes), or when ob jects of in terest are “ bi- c oalitions” like for bi-b elief functions. A bi-coalition is a pair of subsets with empty intersec tion, and it ma y represen t the set of criteria whic h are satisfied and the one whic h ar e not satisfied. The pap er is organized as fo llows. Section 2 recalls necessary material on la t t ices and classical b elief functions. Section 3 giv es the main results o n b elief defined ov er lattices, while the last one examine the case of necessit y measures. Throughout the pap er, we will deal with finite lattices. 2 Bac kg r o und 2.1 Lattices W e b egin b y recalling necessary material on lattices (a go o d introduction on latt ic es can b e found in [3] and [14]), in a finite setting. A p oset is a set P endo w ed with a partial order ≤ (reflexiv e, an tisymmetric, transitiv e). A lattic e L is a poset suc h tha t f or any 2 x, y ∈ L their least upp er b ound x ∨ y and greatest low er b ound x ∧ y alw ays exist. F or finite lattices, the greatest ele men t of L (denoted ⊤ ) and least elemen t ⊥ alw a ys exist. x c overs y (denoted x ≻ y ) if x > y and there is no z suc h that x > z > y . Let P b e a p oset, Q ⊆ P is a downset if for any y ∈ P suc h that y ≤ x , x ∈ Q , then y ∈ Q . The set of a ll do wnsets of P is denoted b y O ( P ). A line ar lattic e , or chain , is suc h that ≤ is a to tal order. A chain C in L is maximal if no elemen t x ∈ L \ C can b e added so that C ∪ { x } is still a ch ain. Lattices can b e represen ted b y their Hasse diag r am , where no des a re elemen ts of t he lattice, and there is an edge b e t w een x and y , with x ab o v e y , if and only if x ≻ y . Fig. 1 sho ws three latt ices. The middle and right ones are tw o differen t diag rams of the la t tic e of subsets of { 1 , 2 , 3 } ordered by inclusion. c b a d e f ∅ 1 2 3 12 13 23 123 ∅ 1 2 3 12 13 23 123 Figure 1: Examples of lattices Let P , Q b e t w o posets, and consider f : P → Q . f is iso t one (resp. antitone ) if x ≤ y implies f ( x ) ≤ f ( y ) (resp. f ( x ) ≥ f ( y )). P and Q are isomorphic (resp. an t i- isomorphic ), denoted b y P ∼ = Q (resp. P ∼ = Q ∂ ), if it exists a bijection f fro m P to Q suc h that x ≤ y ⇔ f ( x ) ≤ f ( y ) (resp. f ( x ) ≥ f ( y )). Isomorphic p osets hav e same Hasse diagrams, up to the lab elling of elemen ts. F or any p ose t ( P , ≤ ), one can consider its dual b y in v erting the order relatio n, whic h is denoted by ( P , ≤ ∂ ) (or simply P ∂ if the order relation is not men tionned), i.e., x ≤ y if and only if y ≤ ∂ x . Au to dual p ose ts are suc h that P ∼ = P ∂ (i.e., they hav e the same Hasse diagram). The lattices of Fig . 1 are all auto dual, and F ig . 2 shows their dual. c f e d a b 123 12 13 23 1 2 3 ∅ 123 23 13 12 3 2 1 ∅ Figure 2: Dual of the lattices o f Fig. 1 A lattice L is lower semimo dular (resp. upp er semimo dular ) if for all x, y ∈ L , x ∨ y ≻ x and x ∨ y ≻ y imply x ≻ x ∧ y and y ≻ x ∧ y (r esp. x ≻ x ∧ y and y ≻ x ∧ y imply x ∨ y ≻ x and x ∨ y ≻ y ). A la t tic e b eing upp er and low er semimo dular is called mo dular . The lattice is distributive if ( x ∨ y ) ∧ z = ( x ∧ z ) ∨ ( y ∧ z ) holds for all x, y , z ∈ L . 3 a b c Figure 3: The la ttice s M 3 (left) and N 5 (righ t) ( L, ≤ ) is said to b e lower (upp er) lo c al ly distributive if it is low er (upp er) semimo dular, and it do es not con tain a sublattice isomorphic to M 3 . These are w eak er conditions than distributivit y , and if L is b oth low er and upp er lo cally distributiv e, then it is distributive . An elemen t j ∈ L is join-irr e ducible if j = x ∨ y implies either j = x or j = y , i.e., it cannot b e expressed as a suprem um of other elemen ts. Equiv alen tly j is join-irreducible if it cov ers only one elemen t. Join-irreducible elemen ts co vering ⊥ are called atoms , and the lattice is atomistic if all join-irreducible elemen ts a re atoms. The set of all join-ir r educible elemen ts of L is denoted J ( L ). On Fig. 1 and 2, they are figured as blac k no des. Similarly , me et-irr e ducible elements cannot b e written as an infimum of other ele- men ts, and are suc h that they are co v ered by a single elemen t. W e denote by M ( L ) the set of meet-irreducible elemen ts of L . Co-atoms are meet-irreducible elemen ts co v ered b y ⊤ . F or any x ∈ L , w e say that x has a c omp lement in L if there exists x ′ ∈ L such that x ∧ x ′ = ⊥ a nd x ∨ x ′ = ⊤ . The complemen t is unique if the lattice is distributive . L is said to b e c omplemen t e d if any elemen t has a complemen t. On Fig . 1 (left), no elemen t has a complemen t, except top and b ottom, while the tw o ot hers are complemen t ed lattices. Bo ole an la t tic es are distributiv e and compleme n ted lattices, and in a finite setting, they are of the t yp e 2 N for some set N , i.e. they are isomorphic to the lattice of subsets of some set, ordered b y inclusion (see Fig. 1 (middle,righ t)) . Bo olean lattices are atomistic, and atoms corresp ond to singletons, while co-ato ms are of the form N \ { i } fo r some i ∈ N . An imp ortan t prop ert y is tha t in a low er lo c ally distributive lattice, a n y elemen t x can be written as an ir r edundant suprem um of join-irreducible eleme n ts in a unique w ay (this is called the min i m al de c omp osi tion of x ). W e denote b y η ∗ ( x ) the set of join-irreducible elemen ts in the minimal decomp osition of x , and we denote b y η ( x ) the normal de c omp o sit ion of x , defined as the set o f jo in -irreducible elemen ts smaller or equal to x , i.e., η ( x ) := { j ∈ J ( L ) | j ≤ x } . Hence η ∗ ( x ) ⊆ η ( x ), and x = _ j ∈ η ∗ ( x ) j = _ j ∈ η ( x ) j. Put differen tly , the mapping η is an isomorphism of L o nto O ( J ( L )) ( Bir khoff ’s theorem). Lik ewise , an y elemen t in a upp er lo cally distributiv e la ttice can b e written as a unique irredundan t infim um o f meet-irreducible elemen ts. Th e decomp osition are denoted b y µ and µ ∗ . Sp ecifically , µ ( x ) := { m ∈ M ( L ) | m ≥ x } , and x = ^ m ∈ µ ( x ) m . The height function h on L giv es the length of a longest c hain from ⊥ to an y elemen t in L . A lattice is r anke d if x ≻ y implies h ( x ) = h ( y ) + 1 . A lattice is low er lo cally distributiv e if and only if it is rank ed and the length of any maximal c hain is |J ( L ) | . 4 2.2 The M¨ obius and c o -M¨ obius transforms W e follo w the general definition of Rota [15] (see a lso [2 , p. 102]). Let ( L, ≤ ) b e a p oset whic h is lo cally finite (i.e., an y interv al is finite) ha ving a b ottom elemen t. F or an y function f on ( L, ≤ ), the M¨ obius tr ansform of f is the function m : L − → R solution of the equation: f ( x ) = X y ≤ x m ( y ) . (1) This equation has alw ays a unique solution, and the expression of m is obtained through the M¨ obius function µ : L 2 → R b y: m ( x ) = X y ≤ x µ ( y , x ) f ( y ) (2) where µ is defined inductive ly b y µ ( x, y ) =    1 , if x = y − P x ≤ t 0. A b elie f function on Ω is a f unction bel : 2 Ω → [0 , 1] generated b y a mass allo cation function as follow s: b el( A ) := X B ⊆ A m ( B ) , A ⊆ Ω . (5) Note that b el( ∅ ) = 0 and b el(Ω) = 1. One recognizes m as b eing the M¨ obius transform of b el ( apply Eq. (1) to ( L, ≤ ) := (2 Ω , ⊆ )). The in v erse fo rm ula, obtained b y using (2 ) and (3), is: m ( A ) = X B ⊆ A ( − 1) | A \ B | b el( B ) . (6) Giv en a mass allo cation m , the plausibility function is defined by: pl( A ) := X B | A ∩ B 6 = ∅ m ( B ) = 1 − b el( A c ) , A ⊆ Ω . (7) 5 Similarly , the c o m monality function is defined by : q ( A ) := X B ⊇ A m ( B ) , A ⊆ Ω . (8) It is the co-M¨ obius transform o f b e l (see (4)). Remark that q ( ∅ ) = 1. A c ap acity on Ω is a set function v : 2 Ω → [0 , 1] suc h that v ( ∅ ) = 0, v (Ω) = 1, and A ⊆ B implies v ( A ) ≤ v ( B ) ( m onotonicity ). Plausibilit y and b elie f f un ctions are capacities. F or a n y capacit y v , its c onjugate is defined b y v ( A ) := 1 − v ( A c ). Henc e, plausibilit y functions are conjuga te of b elief functions (and vice vers a). A capacit y is k -monotone ( k ≥ 2) if for an y family of k subsets A 1 , . . . , A k of Ω, it ho ld s: v ( [ i ∈ K A i ) ≥ X I ⊆ K ,I 6 = ∅ ( − 1) | I | + 1 v ( \ i ∈ I A i ) , (9) with K := { 1 , . . . , k } . A capacit y is total ly mono t one if it is k -monotone for ev ery k ≥ 2. Shafer [16] has sho wn that a capacity is totally monotone if and only if it is a b elief function, hence there exists some mass allo cation generating it. Giv en tw o mass allo cations m 1 , m 2 , t he Dempster’s rule of c ombination computes a com bination of b oth masses in to a single one: m ( A ) =: ( m 1 ⊕ m 2 )( A ) := X B 1 ∩ B 2 = A m 1 ( B 1 ) m 2 ( B 2 ) , ∀ A ⊆ Ω , A 6 = ∅ , (10) and m ( ∅ ) := 0. N ote that m is no more a mass allo cation in general, unless some normalization is carried out. It is w ell known that the D empster rule of com bination can b e computed thro ugh the commonality functions m uc h more easily . Sp ec ifically , calling q , q 1 , q 2 the commona lity functions asso ciated to m, m 1 , m 2 , one has: q ( A ) = q 1 ( A ) q 2 ( A ) , ∀ A ⊆ Ω . (11) A simple supp ort function fo cuse d on A is a particular b elief function b el A whose ma ss allo cation is: m A ( B ) :=      1 − w A , if B = A w A , if B = Ω 0 , otherwise . (12) with 0 < w A < 1. Smets [1 9 ], using results of Shafer, has show n that any b elief function suc h that m (Ω) 6 = 0 can b e decomp osed using only simple supp ort f u nctions as follow s: b el = M A ⊆ Ω b el A (13) with w A = Y B ⊇ A q ( B ) ( − 1) | B \ A | +1 , ∀ A ⊆ Ω . (14) In the a bov e decomp osition, co efficie n ts w A ma y b e greater than 1. If t h is happ ens, the corresp onding b el A is no more a b elief function. 6 A ne c e s s it y function or n e c essity me asur e is a b elief function whose fo cal elemen ts form a c hain in (2 Ω , ⊆ ), i.e., A 1 ⊆ A 2 ⊆ · · · ⊆ A n (Dub ois and Prade, [7]). The c haracteristic prop ert y of necessit y functions is that fo r a n y subsets A, B , N( A ∩ B ) = min (N( A ) , N( B )), where N denotes a necessit y function. Conjugates of necessit y functions are called p ossibility functions , denoted b y Π, and are particular plausibilit y functions. It is easy to see that their c haracteristic prop ert y is that for any subsets A, B , Π( A ∪ B ) = max(Π( A ) , Π( B )). This ch aracteristic prop ert y implies that Π is entirely determined by its v alue on singletons, i.e., Π( A ) = max ω ∈ A Π( { ω } ) for an y A ⊆ Ω. F or this reason, π ( ω ) := Π( { ω } ) is called the p os s i bility di s tribution asso ciated to Π. Note that necessarily there exists ω 0 ∈ Ω such t ha t π ( ω 0 ) = 1. Although this is generally not conside red, one may define as w ell a n e c essity distribution ν ( ω ) := N(Ω \ ω ), with the prop ert y that N( A ) = min ω ∈ A c ν ( ω ). Let π b e a p ossibilit y distribution on Ω := { ω 1 , . . . , ω n } , and assume that for some p erm utation σ on { 1 . . . . , n } , it holds π ( ω σ (1) ) ≤ π ( ω σ (2) ) ≤ · · · ≤ π ( ω σ ( n ) ) = 1. Then it can be sho wn that the fo cal elemen ts of the mass allo cation associated to Π are of the form A σ ( i ) := { ω σ ( i ) , . . . , ω σ ( n ) } , i = 1 , . . . , n , and m ( A σ ( i ) ) = π ( ω σ ( i ) ) − π ( ω σ ( i − 1) ), with the conv en tion π ( ω σ (0) ) = 0. 3 Belief fu n ctions and capacities o n lattic e s Let ( L, ≤ ) b e a finite lattice. A c ap acity on L is a function v : L → [0 , 1] suc h that v ( ⊥ ) = 0, v ( ⊤ ) = 1, and x ≤ y implies v ( x ) ≤ v ( y ) (isotonicit y). T o define the conjugate of a capacity , a natural w a y w ould b e to write v ( x ) := 1 − v ( x ′ ), where x ′ is the complemen t of x . But this w ould imp ose that L is complemen ted, whic h is v ery restrictiv e. F or example , the lattice 3 n underlying bi-b elief functions is not complemen ted. Moreov er, if distributivit y is imp osed in addition, then o nly Bo olean lattices are allow ed, a nd we are bac k t o the classical definition. W e adopt a more general definition. Definition 1 A lattic e L is of D e Morgan t yp e if it exists a bije ctive mapping n : L → L such that for any x, y ∈ L it hold s n ( x ∨ y ) = n ( x ) ∧ n ( y ) , and n ( ⊤ ) = ⊥ . We c al l such a mapping a ∨ - negation . The follow ing is immediate. Lemma 1 L e t L b e a De Mor gan lattic e, with n a ∨ -ne g a tion. Then: (i) n ( ⊥ ) = ⊤ . (ii) n − 1 ( x ∧ y ) = n − 1 ( x ) ∨ n − 1 ( y ) , fo r al l x, y ∈ L ( n − 1 is c al le d a ∧ -negatio n ). (iii) If j i s join - irr e ducible, then n ( j ) is me e t-irr e ducible, and if m is me et-irr e ducible, then n − 1 ( m ) is join-irr e ducible. Pro of: (i) n ( x ∨ ⊥ ) = n ( x ) = n ( x ) ∧ n ( ⊥ ), f o r all x ∈ L , whic h implies n ( ⊥ ) = ⊤ b ecause n is a bijection. (ii) Putting x ′ := n ( x ) and y ′ := n ( y ) , we ha v e n − 1 ( x ′ ∧ y ′ ) = n − 1 ( n ( x ∨ y ) ) = x ∨ y = n − 1 ( x ′ ) ∨ n − 1 ( y ′ ). 7 (iii) If j is join-irreducible, j = x ∨ y implies that j = x or j = y . Henc e, n ( j ) = n ( x ∨ y ) = n ( x ) ∧ n ( y ) is either n ( x ) o r n ( y ), which means t ha t n ( j ) is meet-irreducible.  A complemen ted la ttice with unique complemen t is of D e Morgan type with n ( x ) := x ′ . If L is isomorphic to its dual L ∂ , i.e. it is auto dual, then it is of De Morg an t yp e since it suffices to take for n ( x ) the elemen t in the Hasse diagram of L ∂ whic h takes the place of x in the Hass e diagram o f L . In this case, w e call n a horizon tal symmetry . In general, n is not unique since there is no unique w ay to dra w Hasse diagrams. T aking lattices of Fig. 1 as examples, for the left one, w e w ould ha v e n ( a ) = e , for the middle one n (12) = 1, and for the righ t o ne n (12) = 3 (see Fig. 2). Since middle and rig h t lattices are the same, this sho ws that sev eral n exist in general. Not e that n for the righ t lattice is no thing else than the usual complemen t. The following result sho ws that in fact the only De Morgan t yp e lattices are those whic h are auto dual. Prop osition 1 A lattic e L is of D e Mor gan typ e if and only if it is auto dual. Pro of: W e already kno w that if L is a uto dual, then it is of De Morgan t yp e. Con ve rsely , assuming it is of De Morgan t yp e, it suffices to sho w that n is an an ti-isomorphism. W e already know that n is a bijection. T a king x ≤ y implies that x ∨ y = y , hence n ( x ∨ y ) = n ( y ) = n ( x ) ∧ n ( y ) , whic h implies n ( y ) ≤ n ( x ). Conv ersely , n ( y ) ≤ n ( x ) implies n ( y ) ∧ n ( x ) = n ( y ) = n ( x ∨ y ), hence x ∨ y = y since n is a bijection, so that x ≤ y .  In general, n and n − 1 differ, that is, n is not alwa ys involutive . T ak e fo r example the lattice M 3 of Fig. 3, and n defined by n ( ⊤ ) = ⊥ , n ( ⊥ ) = ⊤ , n ( a ) = b , n ( b ) = c and n ( c ) = a . Clearly , n is a ∨ -negatio n, but n ( n ( a )) = c 6 = a . The ∨ -negat io n is in v olutiv e whenev er n is a horizontal symmetry on the Hasse diagram. If n is inv olutive , it is simply called a ne gation . Definition 2 L et L b e an auto d ual lattic e, and n a ∨ -ne gation o n L . F or any c ap acity v , its ∨ -conjugate a nd ∧ -conjugate (w . r. t. n ) ar e define d r esp e ctively by ∨ v ( x ) := 1 − v ( n ( x )) ∧ v ( x ) := 1 − v ( n − 1 ( x )) , for any x ∈ L . If n is a ne gation, then v ( x ) := 1 − v ( n ( x )) is the conjugate of v . The follow ing is immediate. Lemma 2 L e t L b e an auto dual lattic e, and n a ∨ -ne gation on L . F or any c ap acity v , it h o lds (i) ∨ v and ∧ v ar e c ap acities on L . (ii) ∨ ∧ v = ∧ ∨ v = v . 8 Pro of: (i) ∨ v ( ⊤ ) = 1 − v ( n ( ⊤ )) = 1, similarly for ⊥ . Isotonicity of ∨ v follows from an titonicit y of n and isotonicit y of v . (ii) ∨ ∧ v ( x ) = 1 − ∧ v ( n ( x )) = 1 − (1 − v ( x )) = v ( x ).  The following definition of b elief f unctions is in the spirit of the orig inal one by Shafer. W e used it a lso f or defining bi- b elief functions [11]. Definition 3 A function b el : L → [0 , 1] is c al le d a b elief function if b el( ⊤ ) = 1 , b el( ⊥ ) = 0 , and its M¨ obius tr an s f o rm is no n ne gative. Referring to (1), w e recall that b el( x ) = X y ≤ x m ( y ) , ∀ x ∈ L. (15) Note t ha t bel( ⊤ ) = 1 is equiv alen t to P x ∈ L m ( x ) = 1 , and b el( ⊥ ) = 0 is equiv alen t to m ( ⊥ ) = 0. The in v erse formula, g iving m in terms of b el, ha s to b e computed from (3 ), and dep ends only on the structure of L . Remark that b el is an isotone function b y nonnegativity of m , and hence a capacit y . Thanks to the definition o f conjugatio n, if L is auto dual and n is a ∨ - negation, one can define plausibility functions as the ∨ - conjug ate of b elief functions, whic h are again capacities. 3.1 k - mon otone fu n ctions Barth ´ elem y defines belief function a s totally monotone functions. T o detail this p oint, w e define k - monotone functions. F or k ≥ 2, a function f : L → R is said to be k - monotone (called we ak l y k -monotone b y Barth´ elem y) if it satisfies, for an y family of elemen ts x 1 , . . . , x k ∈ L : f ( _ i ∈ K x i ) ≥ X I ⊆ K ,I 6 = ∅ ( − 1) | I | + 1 f ( ^ i ∈ I x i ) (16) where K := { 1 , . . . , k } . A function is said to b e total ly monotone if it is k -monotone for all k ≥ 2. One can pro v e that in fact, if | L | = n , total mono t onicit y is equiv alen t to ( n − 2 ) -monotonicity [1]. F or k ≥ 2, a function is said to b e a k -valuation if the inequalit y (16) degenerates in to an equalit y (called also Poinc ar´ e ’ s ine quality ). Similarly , a function is a n infin i te valuation or total valuation if it is a k -v aluation for all k ≥ 2. It is w ell known that monotone infinite v aluations satisfying f ( ⊤ ) = 1 and f ( ⊥ ) = 0 are probabilit y measures . The follow ing lemma, cited in [1], summarizes we ll-kno wn r esults from lat t ice theory (see Birkhoff [2]). Lemma 3 L e t L b e a lattic e. Then (i) L is mo dular if and only if it a dmits a strictly monotone 2-valuation. (ii) L is distributive if and only if it is m o dular and every strictly monotone 2-valuation on L is a 3-valuation. 9 (iii) L is distributive if and only if it admits a strictly monotone 3 -valuation. (iv) L is distributive if and only if it is mo dular and every s trictly monotone 2-valuation on L is an infinite valuation. Barth ´ elem y show ed in addition that any lat t ice admits a totally monotone function. In view of this result, Barth ´ elem y defines b elief functions as totally monotone function b eing monotone and satisfying f ( ⊤ ) = 1 and f ( ⊥ ) = 0. In summary , a b elief function can b e defined on any la t tice, while probabilit y measures can live only on distributiv e lattices. The following prop osition shows the relation betw een b oth definitions. Before, w e state a result from [1 ]. Lemma 4 F or any lattic e L and a n y function m : L → [0 , 1] such that m ( ⊥ ) = 0 and P x ∈ L m ( x ) = 1 , the function f m : L → [0 , 1 ] define d by f m ( x ) := P y ≤ x m ( y ) is total ly monotone and sa tisfies f m ( ⊤ ) = 1 and f m ( ⊥ ) = 0 . Prop osition 2 A ny b elief function is total ly monotone. Pro of: Let b el b e a belief function, and m its M¨ obius tra nsform. W e kno w that m ( ⊥ ) = 0 and P x ∈ L m ( x ) = 1. Henc e, by Lemma 4, b el is totally monotone.  A totally monotone f unction do es not hav e necessarily a non negativ e M¨ obius function. Simple examples sho w that monotonicit y is a nece ssary conditio n. The question to know whether monotonicit y and total monoto nicit y imply non negativit y of the M¨ obius function is still o p en. 3.2 Prop erties of b elief functions A first result show n b y Barth ´ elemy shows that capacities collapse to b elief functions when L is linear [1 ]. Prop osition 3 A ny c ap acity on L is a b eli e f function if and only if L is a line ar lattic e. In the sequ el, we address the com bina t io n of b elief f unctions a nd their decomp osition in terms o f simple supp or t functions. W e will see that classical results generalize. Definition 4 L et b el 1 , b el 2 b e two b elief functions on L , with M¨ obius tr ansforms m 1 , m 2 . The D empster’s rule of combination of b el 1 , b el 2 is define d thr ough its M¨ obius tr ans f o rm m b y: m ( x ) =: ( m 1 ⊕ m 2 )( x ) := X y 1 ∧ y 2 = x m 1 ( y 1 ) m 2 ( y 2 ) , ∀ x ∈ L. Since m defines unamb iguously the b elief function, we ma y write as w ell b el = b el 1 ⊕ b el 2 to denote the com bination. Prop osition 4 L et bel 1 , b el 2 b e two b elief functions on L , with c o-M¨ obius tr an sforms q 1 , q 2 , and c onsider their Dempster c ombination. Then, if q de n otes the c o-M¨ obius tr ans- form of bel := bel 1 ⊕ b el 2 , q ( x ) = q 1 ( x ) q 2 ( x ) , ∀ x ∈ L. 10 Pro of: W e ha v e: q ( x ) = X y ≥ x X y 1 ∧ y 2 = y m 1 ( y 1 ) m 2 ( y 2 ) = X y 1 ∧ y 2 ≥ x m 1 ( y 1 ) m 2 ( y 2 ) . One can decomp o se the ab ov e sum since if y 1 ≥ x and y 2 ≥ x , then y 1 ∧ y 2 ≥ x and recipro cally . Th us, q ( x ) = X y 1 ≥ x m ( y 1 ) X y 2 ≥ x m ( y 2 ) = q 1 ( x ) q 2 ( x ) .  The ab ov e prop osition g eneralizes (11), and giv es a simple means to compute the Demp- ster combination. Remark 1 : In D ef. 4, one may put as in the classical case m ( ⊥ ) = 0. This do es not a ff ect the v a lidity of Prop. 4, except for x = ⊥ . Indeed, by Prop. 4, one obtains q ( ⊥ ) = 1 , but q ( ⊥ ) = P x ∈ L m ( x ) < 1 in general if one puts m ( ⊥ ) = 0 in Def. 4 . Definition 5 L et y ∈ L . A simple supp ort function fo cused on y , denote d by y w , is a function on L such that its M¨ obius tr ans f o rm satisfie s: m ( x ) =      1 − w , if x = y w , if x = ⊤ 0 , otherwise. The decomposition of some b elief function b el in terms of simple supp ort functions is th us to write b el under the form: b el( x ) = M y ∈ L y w y ( x ) . (17) The follow ing result generalizes the decomp osition in the classical case (see Sec. 2.3). Theorem 1 L et b el b e a b elief function such that its M¨ obius tr ansform m satisfies m ( ⊤ ) 6 = 0 . The c o effi c ients w y of the de c omp osition (17) w ri te w y = Y x ≥ y q ( x ) − µ ( x,y ) wher e µ ( x, y ) is the M¨ obius function o f L . Pro of: W e try to find w y suc h that b el( x ) = M y ∈ L y w y . This expression can b e written in terms of the co-M¨ o bius tr a nsform: q ( x ) = Y y ∈ L q y ( x ) , x ∈ L, (18) 11 where q y is the co- M¨ obius transform of y w y : q y ( x ) = ( 1 , if x ≤ y w y , otherwise. F rom (18), w e obtain: log q ( x ) = X y ∈ L log q y ( x ) = X y 6≥ x log w y = X y ∈ L log w y − X y ≥ x log w y . On the other hand, q ( ⊤ ) = Y y ∈ L q y ( ⊤ ) = Y y ∈ L w y . W e supp osed that q ( ⊤ ) = m ( ⊤ ) 6 = 0 , hence: log q ( x ) = log q ( ⊤ ) − X y ≥ x log w y . W e set Q ( x ) := log q ( x ) and W ( y ) := log w y . The last equality b ecomes: Q ( x ) = Q ( ⊤ ) − X y ≥ x W ( y ) . If w e define Q ′ ( x ) = Q ( ⊤ ) − Q ( x ), w e finally obtain: Q ′ ( x ) = X y ≥ x W ( y ) . W e recognize here the equation defining the M¨ o bius transform of Q ′ , up to an in v ersion of the o r der ( dua l order)(see (1 ) ) . Hence, using ( 2): W ( y ) = X x ≥ y µ ( x, y ) Q ′ ( x ) with µ defined by (3) . Rewriting t his with original notation, w e obtain: log w y = X x ≥ y µ ( x, y )[log q ( ⊤ ) − log q ( x )] . Remarking that P x ≥ y µ ( x, y ) log q ( ⊤ ) is ze ro, since it corresp onds to the M¨ obius trans- form o f a constant function, we finally get: w y = Y x ≥ y q ( x ) − µ ( x,y ) .  Note that the abov e pro of is m uc h shorter a nd general t ha n the o riginal o ne b y Sha f er [16]. As in the classical case, these co efficien ts ma y b e strictly greater than 1, hence cor- resp onding simple supp ort functions ha v e negative M¨ obius transform and a r e no more b elief functions. 12 4 Necessi ty functi ons Definition 6 A function N : L → [0 , 1] is c al le d a necessit y f unction if it satisfies N( x ∧ y ) = min(N( x ) , N( y )) , for al l x, y ∈ L , and N( ⊥ ) = 0 , N( ⊤ ) = 1 . The follow ing result is due to Bart h´ elem y [1]. Prop osition 5 N is a ne c essity function if and on l y if it is b el i e f function whose M¨ obius tr ansform m is such that its fo c al elements form a chain in L . W e define p ossibilit y f unctions as ∨ -conjug a tes o f necessit y functions. Definition 7 L et L b e an auto dual lattic e, and n a ∨ -ne gation on L . F or a ny ne c essity function N on L , its ∨ -c o n jugate is c al le d a p ossibilit y function . Let Π be a p o ssibilit y function. The n ∧ Π is its corresp onding necessit y function b y Lemma 2 (ii). Prop osition 6 L et L b e an auto dual lattic e, and n a ∨ -ne gation on L . The m a pping Π : L → [0 , 1] is a p ossibility function if and only if Π( x ∨ y ) = max(Π( x ) , Π( y )) , ∀ x, y ∈ L. (19) Pro of: Let Π be a p ossibilit y function being the ∨ -conjugate of some neces sit y function N . Then: Π( x ∨ y ) = 1 − N( n ( x ∨ y )) = 1 − N( n ( x ) ∧ n ( y )) = 1 − min(N( n ( x )) , N( n ( y ))) = max(1 − N( n ( x )) , 1 − N( n ( y ))) = max(Π( x ) , Π( y )) . Con v ersely , let Π satisfy (19) and consider its ∧ -conjugate ∧ Π. W e hav e: ∧ Π( x ∧ y ) = 1 − Π( n − 1 ( x ∧ y )) = 1 − Π( n − 1 ( x ) ∨ n − 1 ( y )) = 1 − max (Π( n − 1 ( x )) , Π( n − 1 ( y ))) = min( ∧ Π( x ) , ∧ Π( y )) . Hence ∧ Π is a nece ssit y function, whic h implies that Π is a p ossibilit y function since ∨ ∧ Π = Π b y Lemma 2 (ii).  The next topic w e address concerns distributions. Since we need the prop erty of decomp osition of elemen ts into suprem um of jo in-irreducible elemen ts, we imp o se that L is low er locally distributiv e. Since L has to b e auto dual, then it is also upp er locally distributiv e, and so it is distributiv e. W e prop ose the following definition. Definition 8 L et L b e an auto dual distributive lattic e, some ∨ -ne gation n on L , and N a ne c es s i ty function. The p ossibilit y distribution π : J ( L ) → [0 , 1] a s s o ciate d to N is define d by π ( j ) := Π( { j } ) , j ∈ J ( L ) , with Π the p ossibility function which is ∨ -c onjugate of N . The nec essit y distribution ν : M ( L ) → [0 , 1] a sso ciate d to N is d e fine d by ν ( m ) := N( { m } ) , m ∈ M ( L ) . 13 Then, the v alue of Π and N a t any x ∈ L can b e computed as f ollo ws: Π( x ) = max( π ( j ) | j ∈ η ∗ ( x )) , N( x ) = min( ν ( m ) | m ∈ µ ∗ ( x )) . Remark that due to isotonicit y of Π and N, and hence of π and ν , one can replace as w ell η ∗ , µ ∗ b y η , µ . The ab ov e formulas are w ell-defined since the decomp o sition is unique for distributiv e lattices. Lastly , remark that necessarily there exists j 0 ∈ J ( L ) suc h that π ( j 0 ) = 1, a nd m 0 ∈ M ( L ) suc h that ν ( m 0 ) = 0, since Π( ⊤ ) = 1 and N( ⊥ ) = 0. π and ν are related t hr o ugh conjugation since n maps join-irreducible elemen ts to meet-irreducible elemen t s and vice-v ersa for n − 1 (see Lemma 1 (iii)). Hence, for j ∈ J ( L ) and m ∈ M ( L ): π ( j ) = 1 − N( n ( j )) = 1 − ν ( m j ) ν ( m ) = 1 − Π( n − 1 ( m )) = 1 − π ( j m ) , where m j := n ( j ), and j m := n − 1 ( m ). Giv en a mass allo cation defining some necessit y function, it is easy to deriv e the corresp onding p ossibilit y distribution. The con v erse problem, i.e., giv en a p ossibilit y distribution, find (if p ossible) the corresp onding c hain of fo cal elemen ts and mass allo ca- tion giving rise to this p ossibilit y distribution, is less simple. In terestingly enough, this problem ha s a lw a ys a unique solution, whic h is v ery close to the classical case. Theorem 2 L et L b e auto d ual, distributive, and n b e a ∨ -ne gation on L . L et π b e a p ossibility distribution, an d assume that the jo in-irr e ducibl e elements of L ar e numb er e d such that π ( j 1 ) < · · · < π ( j n ) = 1 . Then ther e is a unique maximal chai n of fo c al e lements gener ating π , given by the fol lowi n g pr o c e dur e: Going fr om j n to j 1 , at e ach step k = n, n − 1 , . . . , 1 , sele ct the uniq ue join- irr e ducible element ι k such that: ι k 6∈ η ( n ( j k )) , ι k ∈ k − 1 \ l =1 η ( n ( j l )) . ( 2 0) Then the maximal chain is define d by C π := { ι n , ι n ∨ ι n − 1 , . . . , ι n ∨ · · · ∨ ι 2 , ⊤} , and m ( ι n ∨ ι n − 1 ∨ · · · ∨ ι k ) = π ( j k ) − π ( j k − 1 ) , k = 1 , . . . , n, (21) with π ( j 0 ) := 0 . Mor e over, at e ach step k , it is e quiva lent to cho ose ι k as the smal lest in η ( n ( j k − 1 )) \ η ( n ( j k )) . Pro of: F or ease of notation, denote n ( j k ) b y m k (meet-irreducible). W e first sho w that such a pro cedure can alw a ys work and leads to a unique solution for C π . Assume that the p oset J ( L ) has q connecte d comp onen ts J 1 , . . . , J q . By definition, j n is one o f the maximal elemen ts of one of the connected comp onen ts, sa y J q 0 . Clearly , V n k =1 m k = n ( W n k =1 j k ) = n ( ⊤ ) = ⊥ . But W n − 1 k =1 j k 6 = ⊤ , otherwise  S l =1 ,...,q ,l 6 = q 0 J l  ∪ J ′ q 0 , where J ′ q 0 is a maximal do wnset of J q 0 \ { j n } , w ould be another do wnset corresp o nding to ⊤ , whic h is imp o ssible since L is distributiv e (Birkhoff ’s theorem). This implies that there exists ι n ∈ T n − 1 k =1 η ( m k ), and ι n 6∈ η ( m n ). Let us show that ι n is unique. Since L is 14 distributiv e, it is r ank ed and any maximal chain has length |J ( L ) | = n . Hence, W n − 1 k =1 j k has heigh t n − 1 (it is a co-atom), and V n − 1 k =1 m k is an atom. Therefore, T n − 1 k =1 η ( m k ) is a singleton. F or ι n − 1 and subseque n t ones , w e apply the same reasoning on the lattice O ( J ( L ) \ { ι n } ), then on O ( J ( L ) \ { ι n , ι n − 1 } ), etc., instead of L = O ( J ( L )). Hence, there will b e n steps, and at eac h step o ne j oin-irreducible elemen t is c hosen in a unique w a y . W e pro ve now that the sequence { ι n , ι n ∨ ι n − 1 , . . . , ι n ∨ · · · ∨ ι 2 , ⊤} is a maximal c hain, denoted C π . It suffices to pro v e that ι n ∨ ι n − 1 ∨ · · · ∨ ι k ≻ ι n ∨ ι n − 1 ∨ · · · ∨ ι k +1 , k = 1 , . . . , n − 1. The f a ct that the former is greater or equal to the latter is obvious, hence C π is a c hain. T o prov e t ha t it is maximal, w e hav e to sho w that equalit y cannot o ccur among an y t w o subsequen t elemen ts. T o see this, observ e that at eac h step k : ι k 6≤ m k , ι k ≤ m k − 1 , ι k ≤ m k − 2 , . . . , ι k ≤ m 1 . (22) Hence ι k − 1 6≤ ι k , otherwise ι k − 1 ≤ m k − 1 w o uld hold, a con tradiction. Hence, the sequence ι n , ι n − 1 , . . . , ι 1 is no n decreasing, and equality cannot o ccur. Let us prov e that it suffices to c ho ose ι k as the smallest in η ( n ( j k − 1 ) \ η ( n ( j k )). If at step k , a smallest ι k is not chos en in η ( n ( j k − 1 ) \ η ( n ( j k )), it will b e tak en after, a nd the sequence ι n , ι n − 1 , . . . , ι 1 will b e no more non decreasing, a con tradiction. It remains to pro ve that π is strictly increasing and to ve rify the expression of m . Let us prov e b y induction that π ( j k ) = 1 − m ( ι n ) − m ( ι n ∨ ι n − 1 ) − · · · − m ( ι n ∨ · · · ∨ ι k +1 ) , k = n, . . . , 1 . (23) W e sho w it fo r k = n . W e hav e π ( j n ) = 1 − ν ( m n ) = 1 − X x ≤ m n x ∈ C π m ( x ) . Since ι n 6∈ η ( m n ), no x in C π can b e smaller t ha n m n . Hence π ( j n ) = 1. Let us assume (23) is true from n up to some k , and prov e it is still true for k − 1. Using (22), w e ha v e: π ( j k − 1 ) = 1 − ν ( m k − 1 ) = 1 − X x ≤ m k − 1 x ∈ C π m ( x ) = 1 − X x ≤ m k x ∈ C π m ( x ) − m ( ι n ∨ · · · ∨ ι k ) = π ( j k ) − m ( ι n ∨ · · · ∨ ι k ) , whic h pro ves (23). Lastly , remark that the linear system of n equations (23) is tria ngular, with no zero on the diagonal. Hence it has a unique solution, whic h is easily seen to b e (21).  As illustratio n of the theorem, w e giv e an example. Example 1: Let us consider the distributiv e auto dual lattice given on Fig. 4. Join-irreducible elemen ts a re a, b, c, d , e, f , while meet-irreducible ones are α, b, γ , δ, ǫ, f . W e prop ose a s ∨ -negation the f o llo wing: 15 a b c d e f a b c d e f α γ δ ǫ Figure 4: Example o f auto dual distributiv e lattice L (right), with J ( L ) (left) x n ( x ) x n ( x ) a α d ǫ b f e δ c γ f b Let us consider a p ossibilit y distribution satisfying π ( c ) < π ( d ) < π ( e ) < π ( a ) < π ( f ) < π ( b ) = 1 . (observ e that the sequence c, d, e, a, f , b is non decreasing, as requested). W e apply the pro cedure of Th. 2. F or b , w e hav e n ( b ) = f = c ∨ d ∨ e ∨ f , and for f , w e ha v e n ( f ) = b = a ∨ b . Hence the first join-irr educible elemen t of the sequence, ι 6 , is a (not in η ( f ), and minimal in η ( b )). T able 1 summarizes all the steps. The maximal c hain is in g ra y on Fig. 4. W e deduce that: step k x n ( x ) η ( n ( x )) ι k c hain 6 b f c, d, e, f a a 5 f b a, b c a ∨ c 4 a α a, c, d , e, f b a ∨ c ∨ b 3 e δ a, b, c, d e a ∨ c ∨ b ∨ e 2 d ǫ a, b, c, e d a ∨ c ∨ b ∨ e ∨ d 1 c γ a, b, c, d , e f ⊤ T able 1: Computation o f C π π ( b ) = 1 π ( f ) = 1 − m ( a ) π ( a ) = 1 − m ( a ) − m ( a ∨ c ) π ( e ) = 1 − m ( a ) − m ( a ∨ c ) − m ( a ∨ c ∨ b ) π ( d ) = 1 − m ( a ) − m ( a ∨ c ) − m ( a ∨ c ∨ b ) − m ( a ∨ c ∨ b ∨ e ) π ( c ) = 1 − m ( a ) − m ( a ∨ c ) − m ( a ∨ c ∨ b ) − m ( a ∨ c ∨ b ∨ e ) − m ( a ∨ c ∨ b ∨ e ∨ d ) 16 from which w e deduce m ( a ) = π ( b ) − π ( f ) m ( a ∨ c ) = π ( f ) − π ( a ) m ( a ∨ c ∨ b ) = π ( a ) − π ( e ) m ( a ∨ c ∨ b ∨ e ) = π ( e ) − π ( d ) m ( a ∨ c ∨ b ∨ e ∨ d ) = π ( d ) − π ( c ) and m ( ⊤ ) = 1 − m ( a ) − m ( a ∨ c ) − m ( a ∨ c ∨ b ) − m ( a ∨ c ∨ b ∨ e ) − m ( a ∨ c ∨ b ∨ e ∨ d ) = π ( c ). 5 Ac kn o w ledgment The author addresses all his thanks to Bruno Leclerc and Bernard Monjardet for fruitful discussions on negations. 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