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Dempster's rule of conditioning translated into modal logic
l,
FUZZY
-~..~- ~ All
sets and systems ELSEVIER
Fuzzy Sets and Systems 102 (1999) 371-383
Dempster' s rule of conditioning translated into modal logic Elena Tsiporkova a, Bernard De Baets b'*'l, Veselka Boeva
c
aDepartment of Engineering Mathematics, University of Bristol, Queen's Building, University Walk, Clifton, Bristol BS8 1TR, UK bDepartment of Applied Mathematics and Computer Science, University of Gent, Krijgslaan 281 ($9), B-9000 Gent, Belgium CDepartment of Computer Systems, Technical University of Plovdiv, St. Petersburg Blvd. 61, 4000 Plovdiv, Bulgaria Received June 1998
Abstract
A modal logic interpretation of Dempster's rule of conditioning is developed. It is shown that by restricting a model of modal logic in a non-trivial way, the measures induced by this restricted model are, in fact, the conditional measures given the restricting set, corresponding to the measures induced by the original model. (~) 1999 Elsevier Science B.V. All rights reserved.
Keywords." Belief measure; Conditional measures; Modal logic; Multivalued mapping; Necessity measure; Plausibility measure; Possibility measure
1. Introduction
Modal logic is an extension of classical propositional logic endowed with modal operators of possibility and necessity. A model of modal logic is a triplet consisting of a set of possible worlds, a binary relation on this set of worlds called the accessibility relation, and a value assignment function, by which, in each world, truth or falsity is assigned to each atomic proposition. In various publications [8-12], Harmanec, Klir, Resconi, St. Clair and Wang have developed interpretations of Dempster-Shafer theory, and consequently of possibility theory, by employing syntactic and semantic structures of modal logic. However, a modal logic interpretation of conditioning in Dempster-Shafer theory has not yet been established. * Corresponding author. Tel.: 32 92644908; fax: 32 92644995; e-mail: [email protected]. I Post-Doctoral Fellow of the Fund for Scientific Research Flanders (Belgium).
In [ 14], we have reworked the modal logic interpretation of possibility and necessity measures on a finite universe by employing notions and concepts from multivalued analysis. Subsequently, we have developed a modal logic interpretation of conditional possibility and necessity measures [15]. Encouraged by the results obtained, we devoted [ 16] to the further development of the modal logic interpretation of plausibility and belief measures on an arbitrary universe of discourse, as proposed by Harmanec et al. [9], by applying again a multivalued approach. The motivation behind is the fact that the inception of DempsterShafer theory is closely related to the theory of multivalued mappings. Dempster [6] showed that any multivalued mapping propagates a probability measure defined over subsets of one universe into a system of upper and lower probabilities over subsets of another universe, which were called plausibility and belie f measure s (functions) by Shafer [ 13]. Therefore, involving multivalued mappings in the development
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E. Tsiporkova et al./ Fuzzy Sets and Systems 102 (1999) 371-383
of a modal logic interpretation of Dempster-Shafer theory seems to be the most natural approach. Moreover, the accessibility relation, an inherent feature of any model of modal logic, can be equivalently considered as a multivalued mapping. By constructing a multivalued mapping from the accessibility relation on an arbitrary set of possible worlds and a mapping determined by the value assignment function and thus establishing a correspondence between the set of possible worlds and the set of atomic propositions, we have been able to adopt Dempster's original idea and interpret plausibility and belief measures, and all their attributes, in terms of different types of images under this mapping [ 16]. In [6], Dempster also solved the problem of determining the appropriate upper and lower conditional probabilities when a system of upper and lower probabilities induced by a multivalued mapping is given, and moreover, managed to express these upper and lower conditional probabilities in terms of the unconditional ones. Recalling that Shafer's definitions [13] of conditional plausibility and belief measures are based on Dempster's expressions, known as Dempster's rule of conditioning, we are challenged here to show that the key to a successful modal logic interpretation of conditional plausibility and belief measures lies again in the multivalued approach.
2. Multivalued mappings In this section, we recall some basic concepts from the theory of multivalued mappings [1,2]. A multivalued mapping F from a universe X into a universe Y associates to each element x o f X a subset F(x) of Y. The domain o f F , denoted dom(F), is defined as dom(F) = {x t x E X A F(x) #
0}.
F is called non-void if (VxeX)(F(x)¢O), i.e. if d o m ( F ) = X . The composition of F and a second multivalued mapping G from Y into a universe Z is the multivalued mapping G o F from X into Z defined by, for any x E X:
(G o F)(x) = G(F(x)). Let us give an overview of the different direct and inverse images under a multivalued mapping needed
further on. Consider a subset A of X and a subset B of Y. (i) The direct image of A under F is the subset F(A) of Y, defined as
U F(x).
F ( A ) ----
xEA
(ii) The inverse image of B under F is the subset of X, defined as
F-(B)
F-(B)= {xIxEX AF(x)MB¢O }. (iii) The superinverse image of B under F is the subset F+(B) of X, defined as
F+(B) =
{x Ix E dom(F) A F(x) C_B}.
(iv) The pure inverse image of B under F is the subset F-I(B) of X, defined as
F-'(B)
=
{xlx e x /x F(x) =B}.
One easily verifies that F - ( 0 ) = F + ( 0 ) = 0 and F - I ( 0 ) = c o d o m ( F ) , and also F-(Y)=F+(Y)= dom(F). The following properties hold for the inverse and superinverse images: (i) the inclusion F+(B) C_F - ( B ) ; (ii) the complementation properties: coF - (B) = F + (co B) U codom(F), (1) coF+(B) = F - ( c o B ) U codom(F); obviously, if F is non-void then c o F - ( B ) F+(coB); (iii) the distributive laws:
F- ( UBj I \jEJ
/
VeJ /
= UF-(Bj), jEJ
jeJ
for any family (Bj)jej in ~(Y).
=
(2)
E. Tsiporkova et al./Fuzzy Sets and Systems 102 (1999) 371-383
373
(ii) A ~ ( X ) ~ [0, 1] mapping Bel is called a belief measure on ~ ( X ) if Bel(0)= 0, Bel(X)= 1 and
3. Dempster-Shafer theory 3.1. Plausibility and belief measures as upper and lower probabilities
Bel
UAi i=1
The development of a mathematical theory of evidence, nowadays referred to as Dempster-Shafer theory, was initiated by Dempster in 1967 with the study of upper and lower probabilities [6]. Considering a probability measure P on ~(X), he showed that a multivalued mapping F from X into Y induces upper and lower probabilities on ~(Y), as follows:
P(r-(A)) P*(A)- P(F-(Y))' (3) P(F+(A)) P,(A) = p ( r + ( y ) ) ' where F - ( Y ) = F + ( Y ) = dom(F). It is clear that P* and P. are only well defined ifP(dom(F)) > 0. In case of a finite universe Y, Dempster observed that these upper and lower probabilities are completely determined by the quantities P(F-1(C)), C E ~ ( Y ) . Note that the upper and lower probabilities (3) can be seen as conditional probabilities of inverse and superinverse images given the domain of the multivalued mapping [ 16]:
P*(A) = P ( F - ( A ) I dom(F) ),
~> /
~
(-1)lll+'Bel
1C{l,...,n}
NAi iE1
, /
" in ~(X). for any finite family (A i)i=l Note that plausibility and belief measures come in dual pairs. For any belief measure Bel on ~(X), the ~ ( X ) ~ [0, 1] mapping P1 defined by PI(A) -- 1 - Bel(co A) is a plausibility measure on ~@(X). For instance, P* and P. are dual. Furthermore, in case X is finite, Shafer introduced the concepts of a basic probability assignment and its focal elements [13]: (i) A ~@(X) ~ [0, 1] mapping m is called a basic probability assignment on ~@(X) if m(l~)--0 and Z
re(A) = 1.
(ii) A subset F of X for which r e ( F ) > 0 is called a focal element of m. There exists a one-to-one correspondence between belief measures, plausibility measures and basic probability assignments. Given a basic probability assignment m, the corresponding belief measure Bel and its dual plausibility measure P1 are given by
(4)
P,(A ) = P( F+ (A ) l dom( F) ).
m(F),
Bel(A) = Z FCA
Shafer reinterpreted upper and lower probabilities as degrees of plausibility and belief, abandoning Dempster's idea that they arise as upper and lower bounds over classes of Bayesian probabilities [13]. The formal definitions of plausibility and belief measures are given next: (i) A ~ ( X ) ~ [0, 1] mapping PI is called aplausibility measure on ~ ( X ) if PI(0)=0, PI(X)= 1 and P1
Ai
1_c{l. . . . . . } ~40 n for any finite family (Z i)i=l in ~(X).
PI(A)=
re(F).
Z FNA~O
Conversely, given a belief measure Bel, the corresponding basic probability assignment m is given by
re(A) = Z
( - 1)la\ClBel(C).
CCA
The basic probability assignment m corresponding to the upper and lower probabilities in (4) is given by [6]
m(A) = P(F-I(A) I dom(V)). For re(A) to be strictly positive it must hold that F - 1(A) ~ 0 and hence there should exist x E X such
E. Tsiporkova et al./Fuzzy Sets and Systems 102 (1999) 371-383
374
that A = F(x). Consequently, the focal elements are to be found in the set {F(x) Ix E dom(F)}.
xi ~ Xi, such that (xi ~ r - (A)) A (xi ~ r - (8)),
(xj ~ t - ( A ) ) A (xj ~ r-(~)).
3.2. Possibility and necessity measures as particular plausibility and belief measures Let us now recall the definitions of possibility and necessity measures. The concept of 'possibility' was introduced by Zadeh [17]. The formal definition of possibility and necessity measures was given by Dubois and Prade [7]. Consider a universe X. (i) A ~ ( X ) --* [0, 1] mapping H is called a possibility measure on :~(X) if /-/(~)=0, /-/(X)= 1 and
(6)
Since all F(x), for x E X , are nested, we have that F(xi) c_ F(xj) or F(xj) C_F(xi). Suppose that F(xi) C_F(xj), then xi E F- (A) implies that X/E F-(A), which contradicts (6). Similarly, from F(xj) C_F(xi) it follows that xi E F-(B). Therefore F-(A) C_F-(B) or F-(B) c_ t-(A). Let us now, for instance, prove that the lower probability measure induced by F is inf-decomposable, or in other words, a necessity measure. Consider a family ( g ) j c J in ~(Y). If Y is finite then (2) and (5) imply that
H ( UA~)/ = iC1 for any family (Ai)ie~ in ~ ( X ) . (ii) A ~ ( X ) ~ [0, 1] mapping N is called a necessity measure on :~(X) if N ( 0 ) = 0 , N ( X ) = 1 and
and hence
N ( N A i l =infN(Ai),iEl
P* ( / 0 ~ )
\ iEl
= min P(F+(Bj)), jEJ
= min .ieJ P*(Bj)"
/
for any family (Ai)iEl in ~ ( X ) . It is well known that possibility and necessity measures are plausibility and belief measures whose focal elements can be ordered in a nested sequence [13]. For instance, if F is a multivalued mapping from a universe X into a universe Y, such that all F(x), for x C X, can be ordered in a nested sequence and P is a probability measure on ~'(X) then the upper and lower probabilities induced by F are, in fact, possibility and necessity measures. The latter is due to the following properties of the inverse and super/nverse images under F, for any two subsets A and B of Y:
The latter can also be guaranteed, in case of infinite Y, if it is supposed that the probability measure P is continuous from above on the set {F+(B)]B E ~(Y)}. Formally, a ~ ( X ) ~ [0, 1] mapping P is continuous from above on d C ~ ( X ) if and only if for any family (Ai)i~l in d , such that A1D_A2 D_... and Ni=1Ai E ~ , it holds that
P
Ai =
P(As).
Therefore, the fact that (F+(Bi))jej can be ordered in a nested sequence implies that
( r - ( A ) c r - ( a ) ) v ( r - ( a ) c_ t - ( A ) ) , (5)
(F+(A) c_ r+(B)) v (F+(B) c r+(A)). 3.3. Dempster's rule of conditioning Let us for instance prove the property for the inverse images. Suppose that F - ( A ) ~ F-(B) and F - ( B ) ~ F - ( A ) . Then there exist (xi,xj)EX 2,
In his first paper on upper and lower probabilities [6], Dempster also solved the problem of finding
E. Tsiporkovaet al./ Fuzzy Sets and Systems 102 (1999) 371-383
375
the appropriate upper and lower conditional probabilities when a system of upper and lower probabilities in a universe Y, induced by a multivalued mapping F from a universe X into Y, is given. He assumed it is known that Y\B is impossible, where B is a subset of Y such that F - ( B ) ¢ 0. Then F can be restricted to subsets of B by constructing a multivalued mapping FIB fromX into B, defined by FIB(X) = F(x) O B. Since dom(FlB) = F-(B), it has to be further assumed that P ( F - ( B ) ) > 0. Thus FIB induces upper and lower probabilities on ~(Y), as follows:
general approach naturally leads back, in the infinite case, to the algebraic product/probabilistic sum, up to a [0, 1]-automorphism, and, in the finite case, to a continuous triangular norm/conorm without zero divisors/unit multipliers. Our further considerations of conditional possibility and necessity measures in this paper are based on (7) and (8), since the modal logic interpretation of (unconditional) possibility and necessity measures is also based on the algebraic product.
P*(A IB)= P(FIb-(A)I dom(FlB)),
4. Modal logic
(7)
P,(AIB) = P(FI+(A)I dom(Fb)), called upper and lower conditional probabilities. Observe that the inverse and superinverse images of any A E ~ ( Y ) under FIB are given by
FIR(A)=F-(AnB), FI~(A ) - - / ~ (A L/co B) N coF+(co B). Due to the above properties, Dempster [6] managed to express the upper and lower conditional probabilities induced by Fie in terms of the unconditional upper and lower probabilities induced by F. These expressions are known as Dempster's rule of conditioning. Using this rule, Shafer defined conditional plausibility and belief measures [13], which are, of course, only well defined if PI(B) > 0 and Bel(co B) < 1: Pl(A NB) PI(A IB) - - - , PI(B)
Modal logic is an extension of classical propositional logic [3]. It has been developed to formalize arguments involving the notions of possibility and necessity. The language of modal logic consists of a set of atomic propositions, logical connectives A, V,--7, ---~,,--+, and modal operators of possibility 0 and necessity []. The propositions of the language are of the following form: • atomic propositions, • if p and q are propositions, then so are -np, p A q, pVq, p--+q, p ~-+q, Dp, Op. Modal logic is well developed syntactically. However, in order to provide semantic analyses of different systems of modal logic, the notion of a model of modal logic is used. Although it is impossible to give a definition of a model in general, we may view it as a structure of the form [3]
M=(W, .... V), (8)
Bel(A U coB) - Bel(co B) eel(A IB) = 1 - Bel(coB) Since possibility and necessity measures are particular plausibility and belief measures, it follows that conditional possibility and necessity measures can be defined as above. It is often argued that, for the latter purpose, Shafer's expressions, based on the algebraic product, carry too obviously the traces of probability theory, and that a more general, triangular norm/conorm based approach should be followed. However, in [4, 5] we have exposed the necessary and sufficient conditions under which conditional possibilities and necessities again establish a possibility and necessity measure and it turns out that this more
where W denotes a set of possible worlds. . . . indicates the possibility of additional components and V is the value assignment function by which truth T or falsity F of each atomic proposition p in each world w is assigned. Since a proposition may have different truth values in different worlds, the assignment function assigns the truth values not to atomic propositions alone, but to pairs consisting of a possible world and an atomic proposition, i.e. V(w,p)E {T,P}. Therefore V(w, p) is to be thought of as the truth value of p in w. The value assignment function is inductively extended to all non-modal propositions (propositions that do not contain O and [~) in the usual way. A proposition of the form Op holds in a possible world if and only if p holds in at least one possible world,
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E. Tsiporkova et al./ Fuzzy Sets and Systems 102 (1999) 371-383
and a proposition of the form D p is true in a possible world just in case p itself is true in all possible worlds. Note that we will say that a proposition is true in a model if and only if it is true in all possible worlds of the model. Furthermore, a proposition is valid in a class o f models if and only if it is true in all models in the class. The work presented in this paper is based on standard models of modal logic, a modification of the above description of a model introducing an element of relative possibility [3]. A standard model o f modal logic is a triplet M = (W,R, V}, where W and V are as above and R is a binary relation on W called the accessibility relation. We say that world v is accessible from world w when wRy. Thus the truth values of possibilitations, i.e. propositions of the type Op, and the truth values of necessitations, i.e. propositions of the type Dp, are defined as follows:
V(w, Op) = T ¢:~ (3v E W)(wRv A V(v, p ) = T), V(w, Dp) = T ¢:~ (Vv E W)(wRv ~ V(v, p ) = T), for any proposition p and any world w E W. A system of modal logic is any set of propositions closed under (contains the conclusion whenever it contains the hypotheses) all propositionally correct modes of inference. The theorems of a system are just the propositions in it. A system is sound with respect to a class of models just in case every theorem of the system is valid in the class of models, and the system is complete with respect to this class of models if and only if every proposition valid in the class is a theorem of the system. Thus a system is said to be determined by a class of models if and only if it is sound and complete with respect to this class. A system determined by a class of standard models is called normal [3]. Any normal system contains the theorem Op ~ ---~E]--qp
The accessibility relation in a standard model expresses the fact that some things may be possible from the standpoint of one world and impossible from the standpoint of another. When we say that v is accessible from w, we mean that the world v is considered as possible from the perspective of world w. Imposing various conditions on the accessibility relation, we obtain different classes of standard models that determine different normal systems. For instance, in a model M = (W,R, V}, R is called • serial if(Vu E W)(3v E W)(uRv); • reflexive if (Vu E W)(uRu); • transitive if (V(u, v, w) E W 3)((uRv A vRw) ~ uRw); • connected if (V(u, v) E W2)(u ¢ v ~ (uRv V vRu)); • strongly connected if (V(u, v) E WZ)(uRv V vRu). Note that a reflexive relation is strongly connected if and only if it is connected. A standard model is called, for instance, serial if its accessibility relation is serial. Let us now consider the following theorems [3]: (D) ff]p ~ Op; (T) Vqp ~ p; (4) [Ep--~[EDp; (H ++) [~(OpV Oq)--~([]OpVDOq). D is valid in any class of serial standard models, T is valid in any class of reflexive standard models and 4 is valid in any class of transitive standard models. Moreover, a normal system that contains T, 4 and H ++ is sound with respect to any class of reflexive, transitive and connected (and hence strongly connected) standard models. This system is known as $4.3. Let us use [[pllM to denote the truth set of a proposition p, i.e. the set of all worlds in which p is true:
Ilpll M = { w I w c m /~ V(w, p) = T}. Since from here on we will only deal with standard models and normal systems, we will omit the adjectives standard and normal.
(9)
and is closed under the rule of inference
5. A modal logic interpretation of Dempster-Shafer theory
(pl A . . . A pn)---~ p ( [ ] P 1 A " " ADPn)---*DP'
5.1. The approach o f Harmanec et al.
(n>~O),
which expresses a general rule of modal consequence that a proposition is necessary if it is a consequence of a collection of propositions each of which is necessary.
In this subsection, we recall the modal logic interpretation of Dempster-Shafer theory and possibility theory developed by Harmanec, Klir, Resconi,
E. Tsiporkova et al./ Fuzzy Sets and Systems 102 (1999) 371-383
St. Clair and Wang [8-12]. Harmanec et al. consider a model M = (W,R, V) and employ propositions of the form eA = "e is in A", where e E X and A E ~ ( X ) . The proposition eA means that a given incompletely characterized element e is classified in A. As atomic propositions, they consider the propositions e{x}, for all x E X. The propositions eA are then defined by the equations eo = A ---qe{x}, xEX
03) (V(A,B) ~ ~(X):)(Vw E W) (A MB = ~ =~V(W, eA ~ ---qeB)= T);
(c) (V(A,~) E ~(X)2)(Vw C W) (V(w, eA uB) = T ¢:~ V(w, em V e ~ ) = T). It is important to notice that it is not necessary to add these requirements, since they follow from the standard way of evaluating the value assignment function.
Theorem 5.1 (Harmanec et al. [9]). A serial model M = ( W, R, V, P) satisfyin9 ( S V A ) and requirements ( A ) - ( C ) induces a plausibility measure P1M and a belief measure BelM on ~ ( X ) , defined by P1M(A) = P( II<>eAIIM),
eA = V e{x},
VA E ~ ( X ) \ { O } .
BelM (A) = P( II[1eA IIM ).
xEA
Furthermore, they assume that each world in the model M gives its own unique answer to the classification question, i.e. the following holds: (SVA) Singleton Valuation Assumption: One and only one proposition e{x} is true in each world. The latter implies that the following two theorems: •
377
ex~
• e{x} ~ --n(Vycxe{y}) are always true in M. Moreover, as a consequence eA ~ ---lecoA
Ileco~U = colleAIIM. In order to be able to develop a modal logic interpretation of Dempster-Shafer theory on arbitrary universes, Harmanec et al. [9] add to a model a probability measure on the set of possible worlds. Hence, a model is meant to be a quadruplet M=(W,R,V,P}, where W, R and V are as above, and P is a probability measure on ~ ( W ) . As mentioned above, different systems are characterized by different additional properties of the accessibility relation R. In [9], it is assumed that R is serial. The following requirements on the value assignment function V are also imposed: (A C_B =~ V(w, eA--+es ) = T);
Theorem 5.2 (Harmanec et al. [9]). The interpretation of plausibility and belief measures introduced in Theorem 5.1 is complete, i.e. for every belief measure Bel (or plausibility measure P1) on ~ ( X ) , there exists a serial model M satisfyin9 (SVA) and requirements ( A ) - ( C ) such that Bel = BelM.
(10)
is also true in M, i.e.
(A) (V(A,B) E ~(X)2)(Vw ~ W)
The foregoing theorem expresses that P1M(A) can be viewed as the probability of the set of worlds in which OeA is true and that BelM(A) can be viewed as the probability of the set of worlds in which DeA is true.
Since possibility and necessity measures are particular plausibility and belief measures, Harmanec et al. [9] could apply the foregoing interpretation of Dempster-Shafer theory also to possibility theory. They consider a class of models with transitive and strongly connected accessibility relations and assume that P is continuous from above on the set
{ll[]eAII M I A E ~(X)}.
(11)
Recall that strong connectedness implies reflexivity. Formally, they consider system $4.3.
Theorem 5.3 (Harmanec et al. [9]). A transitive and strongly connected model M = (W, R, V, P) satisfyin 9 (SVA) and requirements ( A ) - ( C ) induces a possibility measure 1-IM and a necessity measure NM on ~ ( X ) , defined by IlM(A) = P( II(>cAItM ), NM(A) = P(II []eA IIM).
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E. Tsiporkova et al./Fuzzy Sets and Systems 102 (1999) 371-383
Theorem 5.4 (Harmanec et al. [9]). The interpretation o f possibility and necessity measures introduced in Theorem 5.3 is complete, i.e. for every necessity measure N ( or possibility measure 1I) on ~ ( X ), there exists a transitive and strongly connected model M satisfyin9 (SVA) and requirements ( A ) - ( C ) such that N = NM.
W. In this context, world v is accessible from world w if and only if v E R(w). The truth sets of possibilitations and necessitations can then be rewritten in terms of inverse and superinverse images under R of non-modal truth sets. For any proposition p it holds that [16] IJ<>pll M =R-(IIPlIM),
Remark 5.1. In [8, 11], a modal logic interpretation of Dempster-Shafer theory on finite universes was proposed in terms of finite models, i.e. models with a finite set of worlds. In this work, the accessibility relation R is assumed to be reflexive. In [10], the modal logic interpretation of plausibility and belief measures, developed in [11], is applied to possibility theory by assuming further that the accessibility relation is transitive and strongly connected. In fact, these interpretations can be obtained as particular cases of Theorems 5.1 and 5.3 by considering a finite model with a probability measure corresponding to a uniform probability distribution. 5.2. The set-va~ed approach
In [16], we further pursued the modal logic interpretation of Dempster-Shafer theory by employing notions and concepts from set-valued analysis and thus obtaining a natural analogy to the original approach of Dempster [6]. In a model, a multivalued mapping is constructed from the accessibility relation and a mapping determined by the value assignment function. This multivalued mapping induces a plausibility measure and a belief measure expressed in terms of conditional probabilities of inverse and superinverse images, or equivalently, in terms of conditional probabilities of truth sets of possibilitations and necessitations. In this subsection, we recall the modal logic interpretation of Dempster-Shafer theory, developed in [16], and show that, in case of a transitive and connected model, this interpretation can also be applied to possibility theory. Consider a model M = (W, R, V,P) with P a probability measure on ~ ( W ) . As atomic propositions we consider propositions of the form e{x}, for all x belonging to some universe X. Furthermore, we assume that exactly one formula e{x} is true in each world, i.e. (SVA) holds. From here on, we regard the accessibility relation R as a multivalued mapping from W into
(12)
IIDP[IM = R+(II ell M) u codom(R). We now construct a mapping f from W into X, associating to each world w E W the unique x C X for which V(w, e{x})= T. It is clear that f is non-void. Moreover, the inverse image of any A E ~ ( X ) under f equals IleAl]M, i.e. f-l(A)= IleAllm. Let F denote the composition of R and f , i.e. F = f o R . Note that, since f is non-void, it holds that dom(F) = dom(R). We also have that, for any w E W [16]:
F(w) = {x I x E X A w ~ II<>e~x~[rM}.
(13)
Then the inverse and superinverse images of any A C ~ ( X ) under F are given by [16]:
t-(A)
= R-(IIeAIfM),
r+(A) = R+(IIeAIIM).
(14)
One easily verifies that II<>exIIM = dom(F) = dom(R) and IIDexIIM= w. Thus, as an immediate consequence of the latter equalities and (12), we obtain that the inverse and superinverse images of any A E ~ ( X ) under F can be expressed as
F-(A) = II<>eAIIM,
(~5)
r+(A) = II[]eAII ~ n I[<>exU. The following theorem is then due to (4) and (15). Theorem 5.5 (Tsiporkova et al. [16]). A model M = ( IV, R, V, P) with P(dora(R) ) > 0 that satisfies ( SVA ) induces a plausibility measure PIM and a belief measure BelM on ~ ( X ) , defined by
P1M(A)=P(F-(A) pdom(r)), BelM(A)=P(r+(A) l dom(F)),
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E. Tsiporkova et al./ Fuzz)' Sets and Systems 102 (1999) 371-383
or equivalently, P1M(A)=P(IIOeAIIM l lIOex[IM), BelM(A)=P(I[DeAI]M I [I<>exI[M).
Proof. Since all F(w), for w E W, are nested, it follows from (5) that any decreasing sequence in {F+(A)IA E ~ ( X ) } is determined by some family (Ai)iE 1 in ~ ( X ) . For any family (Ai)iE1 in ~ ( X ) , it follows using (15) that
Remark 5.2. Notice that in case of a serial accessibility relation R it holds that
P n
= P { _~ic N (IIDeA,IIMNII<>exllM
II~ex II~ = IIDexl[ M = w, and hence also P(dom(R))-----P(W)= 1. The expressions in Theorem 5.5 then reduce to P1M(A) = P(II<>eAI1M ), BelM (A) = P(II []eA IIM), in which we clearly recognize the expressions in Theorem 5.1, showing that the approach of Harmanec et al. [9] is a special case of the one presented here. Hereafter, we apply the modal logic interpretation of plausibility and belief measures, presented in Theorem 5.5, to possibility and necessity measures. Our further considerations are based on models with a transitive and connected accessibility relation. Note that the class of models that we employ here differs from the one used by Harmanec et al. in [9], since connectedness does not imply reflexivity. Let us also consider the multivalued mapping F from W into X, constructed above, and assume further that P is continuous from above on ( 11 ).
Proposition 5.1. For any (v, w) E W 2, either R(v) C R(w) or R(w) C_R(v), i.e. the family (R(w))wEW can be ordered in a nested sequence.
=P((i~II~eA~IIM)NIIOexIIM)" Due to (12) and (14), for any i E I it holds that H[~eAi HM = F + ( A i ) U codom(F), and obviously, IIDeA~ IIM u II¢>exIIM = I[DeA, II'~t u
Corollary 5.1. For any (v,w)E W e, either F(v)C_ r(w) or r(w) C_r(v), i.e. the family (r(W))wEW can be ordered in a nested sequence. Proposition 5.2. P is continuous from above on the
set {r+(A) I A ~ ~(X)}.
= w.
Hence, it also holds that (N~tll[]eA,II M) u [[Oex[[M = W. Moreover, (F+(Ai))i~t is a decreasing sequence if and only if ([l[]eA,tlM)e~z is a decreasing sequence. Recall that for any two subsets U and V of W such that U U V = W it holds that P(U n V ) = P ( U ) - P(co V). Since P is continuous from above on ( 11 ), it holds that
= inf
P(IIDeAill M ) - P(coll<>exll M)
= inf
(P(II []eA, II~) -- P(collOex IIM))
iE1
= inf P (11Vle,, IIM n Proof. Consider (v, w) E W 2. I f v = w there is nothing to prove. So, let us assume that v ~ w. From the connectedness of R, we have that either v E R(w) or w c R ( v ) . Suppose, for instance, that vER(w). The transitivity of R implies that (Vu E R(v))(u E R(w)) and hence R(v) C_R(w). []
dom(r)
iEl
tl<>exIIM).
Again applying (15), we obtain that
P (NF+(Ai)) \iEl
=infP(F+(Ai)iE ',
[]
/
Theorem 5.6. A transitive and connected model M = (W,R, V,P I with P ( d o m ( R ) ) > 0 that satisfies (SVA) induces a possibility measure IIM and a necessity measure NM on ~(X), defined by //M (A) = P ( F - (A) I dom(F)), NM(A) = P( r+(A) l dom( r ) ),
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E Tsiporkovaet at/Fuzzy Sets and Systems 102 (1999) 371-383
or equivalently,
HM(A)----P(II<>eAIIMI II<>exIIM), NM(A)= P(II DeAIIM I II<>exI[M). Proof. According to Corollary 5.1, all F(w), for w C W, can be ordered in a nested sequence, and by Proposition 5.1, P is continuous from above on {F+(A)IA E ~'(X)}. Therefore, following the discussion in Subsection 3.2, 1IM is a possibility measure and NM is a necessity measure on N(X). []
for any w E WB,which means that only worlds in which eB is true can be accessed in Ms, i.e. it may happen that MB contains worlds in which es is not true, but these worlds cannot be accessed from anywhere. This means that eB V OeB is always true in Ms. Note that dom(Rs) = II(>exIIu~ = II<>esllu Further, V8 is the restriction of V to WB, and Ps is defined as follows:
PB(U) =P(UI ~), for any U E ~(WB).
Proposition 6.1. The followino properties hold: 6. A modal logic interpretation of Dempster's rule of conditioning Consider a model M = (W, R, V,P) with P a probability measure on ~'(W). As atomic propositions we consider again propositions of the form e{x}, for all x belonging to some universe X. We assume that (SVA) holds and consider the multivalued mapping F constructed above.
6.1. Restriction of the model Consider a subset B of X such that P(II<>eBII M) >0. This implies in particular that II<>eBIIM ¢ O, i.e.
(3w ~ w)(e(w) n Ilesll u ¢ 0).
(16)
Moreover, P(ll ()eB [Ig ) > 0 implies that P(l[ (>ex I[M) - P ( d o m ( R ) ) > 0 . In view of (15), the condition P(ll(>esllg)>0 corresponds to the condition P ( F - ( B ) ) > 0 imposed by Dempster in (7). Let us now restrict M = (W,R, V,P) to Ms = (WB,RB, Vs,Ps) in the following way. We define WB= Ilesl[ g u II<>esllM, or equivalently,
~={wlw~llesllMVR(w)nllesllM #O}.
(17)
In fact, ~ is the subset of W containing all worlds in which es is true or from which a world can be accessed in which es is true. Observe that in case of a reflexive accessibility relation the inclusion IleBIIu _c II<>esllM always holds, and hence (16) reduces to IlesllM ¢ 0 and, of course, WB= [[(>eeIIg. Then Rs is constructed, as follows:
RB(w) = R ( w ) n IleBIIM,
(18)
(i) I f R & transitive then RB & also transitive. (ii) I f R is serial then RB is not always serial. (iii) I f R is reflexive then RB is not always reflexire. (iv) I f R is connected then Rs is not always connected
Proof. (i) Consider (u,v,w)EW83, such that vE Rs(u) and w C Rs(v). The latter implies that v E R(u), w E R ( v ) and wE IlesIIg. Since R is transitive, it follows that w E R(u) and therefore w E RB(u). (ii) Due to (17), it is completely possible that (~u~)(R(u)nllesllU=O) and hence by (18) Rs(u) = O. (iii) If there exists u E W~ such that u ~ Ilesll u then it follows that u q~Rs(u) and hence Rs will not be reflexive in this case. (iv) Let us now show that Rs is not connected in general, i.e. it is not always true that for any (u, v) E W82, such that u ~ v, it holds that u E Rs(v) or v E Rs(u). Suppose that for some (u, v) E Wff and u ~ v, we have that u E R(v) but v q~R(u) (which does not contradict the connectedness of R), and u ~ IleBIIM. Therefore, we have that u~R(v)nlleBIIM=RB(v) and vq~R(u)N Ilesll M =Rs(u). [] Example 6.1. In this example, we illustrate the above restriction procedure. We consider a set of worlds with cardinality 6, i.e. W={wl,w2,w3,w4,ws,w6}, and corresponding accessibility relation depicted in Fig. 1. For each world also the element x E X = {a,b,c,d} is indicated for which the atomic proposition e{x} is true. As probability measure P on ~ ( W ) we consider the one corresponding to a uniform probability distribution, i.e. P( U ) = #U/6.
E. Tsiporkova et al. / Fuzzy Sets and Systems 102 (1999) 371-383
<
381
Since P(II OeB IIM) > o, it follows that PB(dom(RB)) = -
P(IIOeB[IM I ~ ) P(IIOeBIIM) >0. P(Ws)
Fig. 1.
<
Theorem 5.5 can then be applied immediately, and we have that the restricted model MB induces a plausibility measure P1MB and a belief measure BelM8 on ~ ( X ) , given by
P1MAA) = PB(Is-(A) I dom(is))
= PB(II<>eAIIM" I II<>exlIM"), BelMs(A) = PB(Is+(A) I dom(is))
= PB([I DeAIIM~IIIOexIIM"). Fig. 2.
Note that for any two subsets U and V of WB, with PB(V)>0, it holds that PB(U I V ) = P ( U I V). The above measures can then be expressed as follows: P1M~(A) = P ( I s - ( A ) I dom(is))
= P(II<>eAIIM~III<>exIIMs), BelMs(A) = P(FB+(A) I dom(is)) =
Fig. 3. • Consider the set Bi = {b,c}, then HeB~]IM = {wl,w2,ws} and [l<>eB,IIM = {W2,W3,W4,W5}• The restricted set of worlds is given by Ws~ = {wl, w2, w3, w4, w5 } and is depicted with its accessibility relation in Fig. 2. • Consider the set Bz = {a,d}, then IleB21lM = {W3,W4, W6} and [IOeB2IIM={w3,w5}. The restricted set of worlds is given by WB2= {w3, w4, ws, w6} and is depicted with its accessibility relation in Fig. 3.
P(ll DeAII ~ I IIOex[l'~).
(19)
We would like to express the measures induced by the restricted model in terms of the measures induced by the original model, and establish a modal logic interpretation of the expressions for conditional plausibility and belief measures. Note that the definition of WB implies that
(Vw ~ W\W~)(R(w) n IleBII~ -- O) and therefore Is can be extended to W by associating to each w C W\ WBthe empty set.
Proposition6.2. It holds that
(VwE W ) ( I s ( w ) =
F ( w ) N B ) , i.e. Is = FIe.
6.2. Measures induced by the restricted model
Proof. Let us first consider the case w E Ws:
Let us consider the restricted model MB = (WB,RB, Vs,PB) constructed in the previous subsection. We define Is as the composition of RB and fB, i.e. Is = fB oRB, where fB(w) = f ( w ) , for all w E Wa.
x 6 Is(w) = f ( R ( w ) A]l eB I[M)
¢, (3vER(w) nlleBllM)(V(v,e{~})= T) ¢* (3vER(w))(V(V, eB)= T A V(v,e{z})= T)
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E. Tsiporkova et al./Fuzzy Sets and Systems 102 (1999) 371-383
sion for BelMB in (20). Since P1M8 and BelMB are dual measures, it holds that
¢:~ (3v E R(w))( V(v, e{x}) : T A x E B) <:> ((qv E R(w))(V(v, e{~}) = T) A x E B) ¢~ ( V ( w , ( ) e { ~ } ) = T ) A ( x C B ) .
BelM~(A) = 1 - P1M~(CoA)
According to (13), it then follows that
=e(coll<>ecoAnBlIM l [lt>eellM).
x ~ G(w) ¢* (x c F(w) A x ~ B).
Finally, due to (10) and (9), we find that
Now consider the case wE W\Ws, i.e. Ye(w)=0. Assume that F ( w ) N B ¢ O , which implies that w E F - ( B ) . Recall that by (15) we have that Y - ( B ) = IIOee[[u. Thus we have obtained, due to (17), that w E Wn, a contradiction. []
BelM~,(A) = P([I-"I 0 ---lea ucoel[ M
I Il<>eellM)
= P(llDeAucoellg J ll<>eellg ).
[]
P1MB(A) = P1M(A I B),
Remark 6.1. If we assume that the original model M = (W,R, V,P) is transitive and connected and P is continuous from above on (11), then according to Theorem 5.6 the plausibility measure P1M and the belief measure BelM induced by M, and consequently also the corresponding conditional measures, are a possibility measure and a necessity measure, respectively. Therefore by Theorem 6.1 the restricted model Me = (WB,Re, VB,Pe) induces a possibility measure HMB and a necessity measure NM~ on ~ ( X ) , which are, in fact, the conditional measures corresponding to the original model M = (W, R, V, P). Since according to Proposition 6.1 the restricted accessibility relation is transitive, but not necessarily connected, the foregoing result does not seem to be so obvious as in the case of plausibility and belief measures. However, it can be shown, independently from Theorem 6.1 and only using the fact that F8 = FIB, that the subsets (YB(w))w c ~ are nested and that Pe is continuous from above on the set {Ye+(A) ]A E ~(X)}. From the discussion in Subsection 3.2 it then follows that the multivalued mapping FB corresponding to the restricted model MR induces a possibility measure HM~ and a necessity m e a s u r e NM8 on ~ ( X ) , given by
Belgt,(A) = BelM(A [B).
nMB (A) = P ( r , - (A) I dom(rs)),
Hence, the extension to W of the multivalued mapping F8 corresponding to the restricted model, is exactly the restricted mapping FIB employed in Dempster's rule of conditioning [6]. It then follows from (7) and (19) that the plausibility and belief measures induced by the restricted model Me are the upper and lower conditional probabilities induced by Fs. Moreover, the measures induced by the restricted model Me can be expressed in terms of conditional probabilities of truth sets of M. This is stated in the following theorem. Theorem 6.1. Given a model M = (W,R, KP) that satisfies (SVA) and a subset B o f X such that P([I <)ee][M) > 0, the restricted model Me = (Ws, RB, Vs,Pe) induces a plausibility measure P1MB and a belief measure BelM~ on ~ ( X ) , which are, in fact, the conditional plausibility and belief measures corresponding to the original model M = ( W, R, V, P), i.e.
Moreover, it holds that
NM~(A) = P(Fe+(A) ] dom(Fs)),
P1MAA)=P(]I<)eAneIIM I IIOeBI]M), (20)
BelMAA ) = P( l[~eA ucoe[[M [ ]lOeeJ[M). Proof. Consider a subset A of X. The first property in (20) can easily be checked using P1M~(A) =P1M(A ]B) and (8). Let us now show the expres-
and consequently, again due to the equality Ye = FIB, it holds that
nMB(A)=nM(A IB), NMB(A) = NM(A ]B).
E. Tsiporkova et al. / Fuzzy Sets and Systems 102 (1999) 371.-383
References [1 ] J.-P. Aubin, H. Frankowska, Set-Valued Analysis, Birkhiiuser, Boston, 1990. [2] C. Berge, Topological Spaces, Including a Treatment of Multivalued Functions, Vector Spaces and Convexity, Oliver & Boyd, London, 1963. [3] B. Chellas, Modal Logic, an Introduction, Cambridge University Press, Cambridge, 1980. [4] B. De Baets, E. Tsiporkova, R. Mesiar, The surprising possibilistic nature of the algebraic product, in: H~-J. Zimmerrnann (Ed.), Proc. 4th European Congress on Intelligent Techniques and Soft Computing, vol. 1, Aachen, Germany, 2-5 September 1996, ELITE, Aachen, 1996, pp. 549-553. [5] B, De Baets, E. Tsiporkova, R. Mesiar, Conditioning in possibility theory with strict order norms, Fuzzy Sets and Systems, to appear. [6] A. Dempster, Upper and lower probabilities induced by a multivalued mapping, Ann. Math. Statist. 38 (1967) 325 339. [7] D. Dubois, H. Prade, Possibility Theory, Plenum Press, New York, 1988. [8] D. Harmanec, G. Klir, G. Resconi, On modal logic interpretation of Dempster-Shafer theory of evidence, Internat, J. Intell. Systems 9 (1994) 941-951. [9] D. Harmanec, G. Klir, Z. Wang, Modal logic interpretation of Dempster-Shafer theory: an infinite case, Internat. J. Approx. Reason. 14 (1996) 81 93.
383
[10] G. Klir, D. Harmanec, On modal logic interpretation of possibility theory, lnternat. J. Uncertain. Fuzziness Knowledge-Based Systems 2 (1994) 237 245. [11] G. Resconi, G. Klir, U. St. Clair, Hierarchical uncertainty metatheory based upon modal logic, lnternat. J. Gen. Systems 21 (1992) 23-50. [12] G. Resconi, G. Klir, U. St. Clair, D. Harmanec, On the integration of uncertainty theories, lnternat. J. Uncertain. Fuzziness Knowledge-Based Systems 1 (1993) 1 18. [13] G. Shafer, A Mathematical Theory of Evidence, Princeton University Press, Princeton, 1976. [14] E. Tsiporkova, V. Boeva, B. De Baets, A modal logic interpretation of possibility theory in terms of multivalued mappings, Proc. IFSA'97, 7th Int. Fuzzy Systems Association World Congress, vol. I, Prague, Czech Republic, 25-29 June 1997, Academia, Prague, 1997, pp. 466-471. [15] E. Tsiporkova, B. De Baets, V. Boeva, Possibilistic conditioning in modal logic, in: H.-J. Zimmermann (Ed.), Proc. 5th European Congr. on Intelligent Techniques and Soft Computing, vol. 1, Aachen, Germany, 8--12 September 1997, ELITE, Aachen, 1997, pp. 86 90, [16] E. Tsiporkova, V. Boeva, B. De Baets, Dempster-Shafer theory framed in modal logic, submitted. [17] L. Zadeh, Fuzzy sets as basis for a theory of possibility, Fuzzy Sets and Systems I (1978) 3 28.
FUZZY
-~..~- ~ All
sets and systems ELSEVIER
Fuzzy Sets and Systems 102 (1999) 371-383
Dempster' s rule of conditioning translated into modal logic Elena Tsiporkova a, Bernard De Baets b'*'l, Veselka Boeva
c
aDepartment of Engineering Mathematics, University of Bristol, Queen's Building, University Walk, Clifton, Bristol BS8 1TR, UK bDepartment of Applied Mathematics and Computer Science, University of Gent, Krijgslaan 281 ($9), B-9000 Gent, Belgium CDepartment of Computer Systems, Technical University of Plovdiv, St. Petersburg Blvd. 61, 4000 Plovdiv, Bulgaria Received June 1998
Abstract
A modal logic interpretation of Dempster's rule of conditioning is developed. It is shown that by restricting a model of modal logic in a non-trivial way, the measures induced by this restricted model are, in fact, the conditional measures given the restricting set, corresponding to the measures induced by the original model. (~) 1999 Elsevier Science B.V. All rights reserved.
Keywords." Belief measure; Conditional measures; Modal logic; Multivalued mapping; Necessity measure; Plausibility measure; Possibility measure
1. Introduction
Modal logic is an extension of classical propositional logic endowed with modal operators of possibility and necessity. A model of modal logic is a triplet consisting of a set of possible worlds, a binary relation on this set of worlds called the accessibility relation, and a value assignment function, by which, in each world, truth or falsity is assigned to each atomic proposition. In various publications [8-12], Harmanec, Klir, Resconi, St. Clair and Wang have developed interpretations of Dempster-Shafer theory, and consequently of possibility theory, by employing syntactic and semantic structures of modal logic. However, a modal logic interpretation of conditioning in Dempster-Shafer theory has not yet been established. * Corresponding author. Tel.: 32 92644908; fax: 32 92644995; e-mail: [email protected]. I Post-Doctoral Fellow of the Fund for Scientific Research Flanders (Belgium).
In [ 14], we have reworked the modal logic interpretation of possibility and necessity measures on a finite universe by employing notions and concepts from multivalued analysis. Subsequently, we have developed a modal logic interpretation of conditional possibility and necessity measures [15]. Encouraged by the results obtained, we devoted [ 16] to the further development of the modal logic interpretation of plausibility and belief measures on an arbitrary universe of discourse, as proposed by Harmanec et al. [9], by applying again a multivalued approach. The motivation behind is the fact that the inception of DempsterShafer theory is closely related to the theory of multivalued mappings. Dempster [6] showed that any multivalued mapping propagates a probability measure defined over subsets of one universe into a system of upper and lower probabilities over subsets of another universe, which were called plausibility and belie f measure s (functions) by Shafer [ 13]. Therefore, involving multivalued mappings in the development
0165-0114/99/$ - see front matter (~) 1999 Elsevier Science B.V. All rights reserved. PIE S 0 1 6 5 - 0 1 1 4 ( 9 8 ) 0 0 2 1 2 - 7
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E. Tsiporkova et al./ Fuzzy Sets and Systems 102 (1999) 371-383
of a modal logic interpretation of Dempster-Shafer theory seems to be the most natural approach. Moreover, the accessibility relation, an inherent feature of any model of modal logic, can be equivalently considered as a multivalued mapping. By constructing a multivalued mapping from the accessibility relation on an arbitrary set of possible worlds and a mapping determined by the value assignment function and thus establishing a correspondence between the set of possible worlds and the set of atomic propositions, we have been able to adopt Dempster's original idea and interpret plausibility and belief measures, and all their attributes, in terms of different types of images under this mapping [ 16]. In [6], Dempster also solved the problem of determining the appropriate upper and lower conditional probabilities when a system of upper and lower probabilities induced by a multivalued mapping is given, and moreover, managed to express these upper and lower conditional probabilities in terms of the unconditional ones. Recalling that Shafer's definitions [13] of conditional plausibility and belief measures are based on Dempster's expressions, known as Dempster's rule of conditioning, we are challenged here to show that the key to a successful modal logic interpretation of conditional plausibility and belief measures lies again in the multivalued approach.
2. Multivalued mappings In this section, we recall some basic concepts from the theory of multivalued mappings [1,2]. A multivalued mapping F from a universe X into a universe Y associates to each element x o f X a subset F(x) of Y. The domain o f F , denoted dom(F), is defined as dom(F) = {x t x E X A F(x) #
0}.
F is called non-void if (VxeX)(F(x)¢O), i.e. if d o m ( F ) = X . The composition of F and a second multivalued mapping G from Y into a universe Z is the multivalued mapping G o F from X into Z defined by, for any x E X:
(G o F)(x) = G(F(x)). Let us give an overview of the different direct and inverse images under a multivalued mapping needed
further on. Consider a subset A of X and a subset B of Y. (i) The direct image of A under F is the subset F(A) of Y, defined as
U F(x).
F ( A ) ----
xEA
(ii) The inverse image of B under F is the subset of X, defined as
F-(B)
F-(B)= {xIxEX AF(x)MB¢O }. (iii) The superinverse image of B under F is the subset F+(B) of X, defined as
F+(B) =
{x Ix E dom(F) A F(x) C_B}.
(iv) The pure inverse image of B under F is the subset F-I(B) of X, defined as
F-'(B)
=
{xlx e x /x F(x) =B}.
One easily verifies that F - ( 0 ) = F + ( 0 ) = 0 and F - I ( 0 ) = c o d o m ( F ) , and also F-(Y)=F+(Y)= dom(F). The following properties hold for the inverse and superinverse images: (i) the inclusion F+(B) C_F - ( B ) ; (ii) the complementation properties: coF - (B) = F + (co B) U codom(F), (1) coF+(B) = F - ( c o B ) U codom(F); obviously, if F is non-void then c o F - ( B ) F+(coB); (iii) the distributive laws:
F- ( UBj I \jEJ
/
VeJ /
= UF-(Bj), jEJ
jeJ
for any family (Bj)jej in ~(Y).
=
(2)
E. Tsiporkova et al./Fuzzy Sets and Systems 102 (1999) 371-383
373
(ii) A ~ ( X ) ~ [0, 1] mapping Bel is called a belief measure on ~ ( X ) if Bel(0)= 0, Bel(X)= 1 and
3. Dempster-Shafer theory 3.1. Plausibility and belief measures as upper and lower probabilities
Bel
UAi i=1
The development of a mathematical theory of evidence, nowadays referred to as Dempster-Shafer theory, was initiated by Dempster in 1967 with the study of upper and lower probabilities [6]. Considering a probability measure P on ~(X), he showed that a multivalued mapping F from X into Y induces upper and lower probabilities on ~(Y), as follows:
P(r-(A)) P*(A)- P(F-(Y))' (3) P(F+(A)) P,(A) = p ( r + ( y ) ) ' where F - ( Y ) = F + ( Y ) = dom(F). It is clear that P* and P. are only well defined ifP(dom(F)) > 0. In case of a finite universe Y, Dempster observed that these upper and lower probabilities are completely determined by the quantities P(F-1(C)), C E ~ ( Y ) . Note that the upper and lower probabilities (3) can be seen as conditional probabilities of inverse and superinverse images given the domain of the multivalued mapping [ 16]:
P*(A) = P ( F - ( A ) I dom(F) ),
~> /
~
(-1)lll+'Bel
1C{l,...,n}
NAi iE1
, /
" in ~(X). for any finite family (A i)i=l Note that plausibility and belief measures come in dual pairs. For any belief measure Bel on ~(X), the ~ ( X ) ~ [0, 1] mapping P1 defined by PI(A) -- 1 - Bel(co A) is a plausibility measure on ~@(X). For instance, P* and P. are dual. Furthermore, in case X is finite, Shafer introduced the concepts of a basic probability assignment and its focal elements [13]: (i) A ~@(X) ~ [0, 1] mapping m is called a basic probability assignment on ~@(X) if m(l~)--0 and Z
re(A) = 1.
(ii) A subset F of X for which r e ( F ) > 0 is called a focal element of m. There exists a one-to-one correspondence between belief measures, plausibility measures and basic probability assignments. Given a basic probability assignment m, the corresponding belief measure Bel and its dual plausibility measure P1 are given by
(4)
P,(A ) = P( F+ (A ) l dom( F) ).
m(F),
Bel(A) = Z FCA
Shafer reinterpreted upper and lower probabilities as degrees of plausibility and belief, abandoning Dempster's idea that they arise as upper and lower bounds over classes of Bayesian probabilities [13]. The formal definitions of plausibility and belief measures are given next: (i) A ~ ( X ) ~ [0, 1] mapping PI is called aplausibility measure on ~ ( X ) if PI(0)=0, PI(X)= 1 and P1
Ai
1_c{l. . . . . . } ~40 n for any finite family (Z i)i=l in ~(X).
PI(A)=
re(F).
Z FNA~O
Conversely, given a belief measure Bel, the corresponding basic probability assignment m is given by
re(A) = Z
( - 1)la\ClBel(C).
CCA
The basic probability assignment m corresponding to the upper and lower probabilities in (4) is given by [6]
m(A) = P(F-I(A) I dom(V)). For re(A) to be strictly positive it must hold that F - 1(A) ~ 0 and hence there should exist x E X such
E. Tsiporkova et al./Fuzzy Sets and Systems 102 (1999) 371-383
374
that A = F(x). Consequently, the focal elements are to be found in the set {F(x) Ix E dom(F)}.
xi ~ Xi, such that (xi ~ r - (A)) A (xi ~ r - (8)),
(xj ~ t - ( A ) ) A (xj ~ r-(~)).
3.2. Possibility and necessity measures as particular plausibility and belief measures Let us now recall the definitions of possibility and necessity measures. The concept of 'possibility' was introduced by Zadeh [17]. The formal definition of possibility and necessity measures was given by Dubois and Prade [7]. Consider a universe X. (i) A ~ ( X ) --* [0, 1] mapping H is called a possibility measure on :~(X) if /-/(~)=0, /-/(X)= 1 and
(6)
Since all F(x), for x E X , are nested, we have that F(xi) c_ F(xj) or F(xj) C_F(xi). Suppose that F(xi) C_F(xj), then xi E F- (A) implies that X/E F-(A), which contradicts (6). Similarly, from F(xj) C_F(xi) it follows that xi E F-(B). Therefore F-(A) C_F-(B) or F-(B) c_ t-(A). Let us now, for instance, prove that the lower probability measure induced by F is inf-decomposable, or in other words, a necessity measure. Consider a family ( g ) j c J in ~(Y). If Y is finite then (2) and (5) imply that
H ( UA~)/ = iC1 for any family (Ai)ie~ in ~ ( X ) . (ii) A ~ ( X ) ~ [0, 1] mapping N is called a necessity measure on :~(X) if N ( 0 ) = 0 , N ( X ) = 1 and
and hence
N ( N A i l =infN(Ai),iEl
P* ( / 0 ~ )
\ iEl
= min P(F+(Bj)), jEJ
= min .ieJ P*(Bj)"
/
for any family (Ai)iEl in ~ ( X ) . It is well known that possibility and necessity measures are plausibility and belief measures whose focal elements can be ordered in a nested sequence [13]. For instance, if F is a multivalued mapping from a universe X into a universe Y, such that all F(x), for x C X, can be ordered in a nested sequence and P is a probability measure on ~'(X) then the upper and lower probabilities induced by F are, in fact, possibility and necessity measures. The latter is due to the following properties of the inverse and super/nverse images under F, for any two subsets A and B of Y:
The latter can also be guaranteed, in case of infinite Y, if it is supposed that the probability measure P is continuous from above on the set {F+(B)]B E ~(Y)}. Formally, a ~ ( X ) ~ [0, 1] mapping P is continuous from above on d C ~ ( X ) if and only if for any family (Ai)i~l in d , such that A1D_A2 D_... and Ni=1Ai E ~ , it holds that
P
Ai =
P(As).
Therefore, the fact that (F+(Bi))jej can be ordered in a nested sequence implies that
( r - ( A ) c r - ( a ) ) v ( r - ( a ) c_ t - ( A ) ) , (5)
(F+(A) c_ r+(B)) v (F+(B) c r+(A)). 3.3. Dempster's rule of conditioning Let us for instance prove the property for the inverse images. Suppose that F - ( A ) ~ F-(B) and F - ( B ) ~ F - ( A ) . Then there exist (xi,xj)EX 2,
In his first paper on upper and lower probabilities [6], Dempster also solved the problem of finding
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375
the appropriate upper and lower conditional probabilities when a system of upper and lower probabilities in a universe Y, induced by a multivalued mapping F from a universe X into Y, is given. He assumed it is known that Y\B is impossible, where B is a subset of Y such that F - ( B ) ¢ 0. Then F can be restricted to subsets of B by constructing a multivalued mapping FIB fromX into B, defined by FIB(X) = F(x) O B. Since dom(FlB) = F-(B), it has to be further assumed that P ( F - ( B ) ) > 0. Thus FIB induces upper and lower probabilities on ~(Y), as follows:
general approach naturally leads back, in the infinite case, to the algebraic product/probabilistic sum, up to a [0, 1]-automorphism, and, in the finite case, to a continuous triangular norm/conorm without zero divisors/unit multipliers. Our further considerations of conditional possibility and necessity measures in this paper are based on (7) and (8), since the modal logic interpretation of (unconditional) possibility and necessity measures is also based on the algebraic product.
P*(A IB)= P(FIb-(A)I dom(FlB)),
4. Modal logic
(7)
P,(AIB) = P(FI+(A)I dom(Fb)), called upper and lower conditional probabilities. Observe that the inverse and superinverse images of any A E ~ ( Y ) under FIB are given by
FIR(A)=F-(AnB), FI~(A ) - - / ~ (A L/co B) N coF+(co B). Due to the above properties, Dempster [6] managed to express the upper and lower conditional probabilities induced by Fie in terms of the unconditional upper and lower probabilities induced by F. These expressions are known as Dempster's rule of conditioning. Using this rule, Shafer defined conditional plausibility and belief measures [13], which are, of course, only well defined if PI(B) > 0 and Bel(co B) < 1: Pl(A NB) PI(A IB) - - - , PI(B)
Modal logic is an extension of classical propositional logic [3]. It has been developed to formalize arguments involving the notions of possibility and necessity. The language of modal logic consists of a set of atomic propositions, logical connectives A, V,--7, ---~,,--+, and modal operators of possibility 0 and necessity []. The propositions of the language are of the following form: • atomic propositions, • if p and q are propositions, then so are -np, p A q, pVq, p--+q, p ~-+q, Dp, Op. Modal logic is well developed syntactically. However, in order to provide semantic analyses of different systems of modal logic, the notion of a model of modal logic is used. Although it is impossible to give a definition of a model in general, we may view it as a structure of the form [3]
M=(W, .... V), (8)
Bel(A U coB) - Bel(co B) eel(A IB) = 1 - Bel(coB) Since possibility and necessity measures are particular plausibility and belief measures, it follows that conditional possibility and necessity measures can be defined as above. It is often argued that, for the latter purpose, Shafer's expressions, based on the algebraic product, carry too obviously the traces of probability theory, and that a more general, triangular norm/conorm based approach should be followed. However, in [4, 5] we have exposed the necessary and sufficient conditions under which conditional possibilities and necessities again establish a possibility and necessity measure and it turns out that this more
where W denotes a set of possible worlds. . . . indicates the possibility of additional components and V is the value assignment function by which truth T or falsity F of each atomic proposition p in each world w is assigned. Since a proposition may have different truth values in different worlds, the assignment function assigns the truth values not to atomic propositions alone, but to pairs consisting of a possible world and an atomic proposition, i.e. V(w,p)E {T,P}. Therefore V(w, p) is to be thought of as the truth value of p in w. The value assignment function is inductively extended to all non-modal propositions (propositions that do not contain O and [~) in the usual way. A proposition of the form Op holds in a possible world if and only if p holds in at least one possible world,
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and a proposition of the form D p is true in a possible world just in case p itself is true in all possible worlds. Note that we will say that a proposition is true in a model if and only if it is true in all possible worlds of the model. Furthermore, a proposition is valid in a class o f models if and only if it is true in all models in the class. The work presented in this paper is based on standard models of modal logic, a modification of the above description of a model introducing an element of relative possibility [3]. A standard model o f modal logic is a triplet M = (W,R, V}, where W and V are as above and R is a binary relation on W called the accessibility relation. We say that world v is accessible from world w when wRy. Thus the truth values of possibilitations, i.e. propositions of the type Op, and the truth values of necessitations, i.e. propositions of the type Dp, are defined as follows:
V(w, Op) = T ¢:~ (3v E W)(wRv A V(v, p ) = T), V(w, Dp) = T ¢:~ (Vv E W)(wRv ~ V(v, p ) = T), for any proposition p and any world w E W. A system of modal logic is any set of propositions closed under (contains the conclusion whenever it contains the hypotheses) all propositionally correct modes of inference. The theorems of a system are just the propositions in it. A system is sound with respect to a class of models just in case every theorem of the system is valid in the class of models, and the system is complete with respect to this class of models if and only if every proposition valid in the class is a theorem of the system. Thus a system is said to be determined by a class of models if and only if it is sound and complete with respect to this class. A system determined by a class of standard models is called normal [3]. Any normal system contains the theorem Op ~ ---~E]--qp
The accessibility relation in a standard model expresses the fact that some things may be possible from the standpoint of one world and impossible from the standpoint of another. When we say that v is accessible from w, we mean that the world v is considered as possible from the perspective of world w. Imposing various conditions on the accessibility relation, we obtain different classes of standard models that determine different normal systems. For instance, in a model M = (W,R, V}, R is called • serial if(Vu E W)(3v E W)(uRv); • reflexive if (Vu E W)(uRu); • transitive if (V(u, v, w) E W 3)((uRv A vRw) ~ uRw); • connected if (V(u, v) E W2)(u ¢ v ~ (uRv V vRu)); • strongly connected if (V(u, v) E WZ)(uRv V vRu). Note that a reflexive relation is strongly connected if and only if it is connected. A standard model is called, for instance, serial if its accessibility relation is serial. Let us now consider the following theorems [3]: (D) ff]p ~ Op; (T) Vqp ~ p; (4) [Ep--~[EDp; (H ++) [~(OpV Oq)--~([]OpVDOq). D is valid in any class of serial standard models, T is valid in any class of reflexive standard models and 4 is valid in any class of transitive standard models. Moreover, a normal system that contains T, 4 and H ++ is sound with respect to any class of reflexive, transitive and connected (and hence strongly connected) standard models. This system is known as $4.3. Let us use [[pllM to denote the truth set of a proposition p, i.e. the set of all worlds in which p is true:
Ilpll M = { w I w c m /~ V(w, p) = T}. Since from here on we will only deal with standard models and normal systems, we will omit the adjectives standard and normal.
(9)
and is closed under the rule of inference
5. A modal logic interpretation of Dempster-Shafer theory
(pl A . . . A pn)---~ p ( [ ] P 1 A " " ADPn)---*DP'
5.1. The approach o f Harmanec et al.
(n>~O),
which expresses a general rule of modal consequence that a proposition is necessary if it is a consequence of a collection of propositions each of which is necessary.
In this subsection, we recall the modal logic interpretation of Dempster-Shafer theory and possibility theory developed by Harmanec, Klir, Resconi,
E. Tsiporkova et al./ Fuzzy Sets and Systems 102 (1999) 371-383
St. Clair and Wang [8-12]. Harmanec et al. consider a model M = (W,R, V) and employ propositions of the form eA = "e is in A", where e E X and A E ~ ( X ) . The proposition eA means that a given incompletely characterized element e is classified in A. As atomic propositions, they consider the propositions e{x}, for all x E X. The propositions eA are then defined by the equations eo = A ---qe{x}, xEX
03) (V(A,B) ~ ~(X):)(Vw E W) (A MB = ~ =~V(W, eA ~ ---qeB)= T);
(c) (V(A,~) E ~(X)2)(Vw C W) (V(w, eA uB) = T ¢:~ V(w, em V e ~ ) = T). It is important to notice that it is not necessary to add these requirements, since they follow from the standard way of evaluating the value assignment function.
Theorem 5.1 (Harmanec et al. [9]). A serial model M = ( W, R, V, P) satisfyin9 ( S V A ) and requirements ( A ) - ( C ) induces a plausibility measure P1M and a belief measure BelM on ~ ( X ) , defined by P1M(A) = P( II<>eAIIM),
eA = V e{x},
VA E ~ ( X ) \ { O } .
BelM (A) = P( II[1eA IIM ).
xEA
Furthermore, they assume that each world in the model M gives its own unique answer to the classification question, i.e. the following holds: (SVA) Singleton Valuation Assumption: One and only one proposition e{x} is true in each world. The latter implies that the following two theorems: •
377
ex~
• e{x} ~ --n(Vycxe{y}) are always true in M. Moreover, as a consequence eA ~ ---lecoA
Ileco~U = colleAIIM. In order to be able to develop a modal logic interpretation of Dempster-Shafer theory on arbitrary universes, Harmanec et al. [9] add to a model a probability measure on the set of possible worlds. Hence, a model is meant to be a quadruplet M=(W,R,V,P}, where W, R and V are as above, and P is a probability measure on ~ ( W ) . As mentioned above, different systems are characterized by different additional properties of the accessibility relation R. In [9], it is assumed that R is serial. The following requirements on the value assignment function V are also imposed: (A C_B =~ V(w, eA--+es ) = T);
Theorem 5.2 (Harmanec et al. [9]). The interpretation of plausibility and belief measures introduced in Theorem 5.1 is complete, i.e. for every belief measure Bel (or plausibility measure P1) on ~ ( X ) , there exists a serial model M satisfyin9 (SVA) and requirements ( A ) - ( C ) such that Bel = BelM.
(10)
is also true in M, i.e.
(A) (V(A,B) E ~(X)2)(Vw ~ W)
The foregoing theorem expresses that P1M(A) can be viewed as the probability of the set of worlds in which OeA is true and that BelM(A) can be viewed as the probability of the set of worlds in which DeA is true.
Since possibility and necessity measures are particular plausibility and belief measures, Harmanec et al. [9] could apply the foregoing interpretation of Dempster-Shafer theory also to possibility theory. They consider a class of models with transitive and strongly connected accessibility relations and assume that P is continuous from above on the set
{ll[]eAII M I A E ~(X)}.
(11)
Recall that strong connectedness implies reflexivity. Formally, they consider system $4.3.
Theorem 5.3 (Harmanec et al. [9]). A transitive and strongly connected model M = (W, R, V, P) satisfyin 9 (SVA) and requirements ( A ) - ( C ) induces a possibility measure 1-IM and a necessity measure NM on ~ ( X ) , defined by IlM(A) = P( II(>cAItM ), NM(A) = P(II []eA IIM).
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Theorem 5.4 (Harmanec et al. [9]). The interpretation o f possibility and necessity measures introduced in Theorem 5.3 is complete, i.e. for every necessity measure N ( or possibility measure 1I) on ~ ( X ), there exists a transitive and strongly connected model M satisfyin9 (SVA) and requirements ( A ) - ( C ) such that N = NM.
W. In this context, world v is accessible from world w if and only if v E R(w). The truth sets of possibilitations and necessitations can then be rewritten in terms of inverse and superinverse images under R of non-modal truth sets. For any proposition p it holds that [16] IJ<>pll M =R-(IIPlIM),
Remark 5.1. In [8, 11], a modal logic interpretation of Dempster-Shafer theory on finite universes was proposed in terms of finite models, i.e. models with a finite set of worlds. In this work, the accessibility relation R is assumed to be reflexive. In [10], the modal logic interpretation of plausibility and belief measures, developed in [11], is applied to possibility theory by assuming further that the accessibility relation is transitive and strongly connected. In fact, these interpretations can be obtained as particular cases of Theorems 5.1 and 5.3 by considering a finite model with a probability measure corresponding to a uniform probability distribution. 5.2. The set-va~ed approach
In [16], we further pursued the modal logic interpretation of Dempster-Shafer theory by employing notions and concepts from set-valued analysis and thus obtaining a natural analogy to the original approach of Dempster [6]. In a model, a multivalued mapping is constructed from the accessibility relation and a mapping determined by the value assignment function. This multivalued mapping induces a plausibility measure and a belief measure expressed in terms of conditional probabilities of inverse and superinverse images, or equivalently, in terms of conditional probabilities of truth sets of possibilitations and necessitations. In this subsection, we recall the modal logic interpretation of Dempster-Shafer theory, developed in [16], and show that, in case of a transitive and connected model, this interpretation can also be applied to possibility theory. Consider a model M = (W, R, V,P) with P a probability measure on ~ ( W ) . As atomic propositions we consider propositions of the form e{x}, for all x belonging to some universe X. Furthermore, we assume that exactly one formula e{x} is true in each world, i.e. (SVA) holds. From here on, we regard the accessibility relation R as a multivalued mapping from W into
(12)
IIDP[IM = R+(II ell M) u codom(R). We now construct a mapping f from W into X, associating to each world w E W the unique x C X for which V(w, e{x})= T. It is clear that f is non-void. Moreover, the inverse image of any A E ~ ( X ) under f equals IleAl]M, i.e. f-l(A)= IleAllm. Let F denote the composition of R and f , i.e. F = f o R . Note that, since f is non-void, it holds that dom(F) = dom(R). We also have that, for any w E W [16]:
F(w) = {x I x E X A w ~ II<>e~x~[rM}.
(13)
Then the inverse and superinverse images of any A C ~ ( X ) under F are given by [16]:
t-(A)
= R-(IIeAIfM),
r+(A) = R+(IIeAIIM).
(14)
One easily verifies that II<>exIIM = dom(F) = dom(R) and IIDexIIM= w. Thus, as an immediate consequence of the latter equalities and (12), we obtain that the inverse and superinverse images of any A E ~ ( X ) under F can be expressed as
F-(A) = II<>eAIIM,
(~5)
r+(A) = II[]eAII ~ n I[<>exU. The following theorem is then due to (4) and (15). Theorem 5.5 (Tsiporkova et al. [16]). A model M = ( IV, R, V, P) with P(dora(R) ) > 0 that satisfies ( SVA ) induces a plausibility measure PIM and a belief measure BelM on ~ ( X ) , defined by
P1M(A)=P(F-(A) pdom(r)), BelM(A)=P(r+(A) l dom(F)),
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or equivalently, P1M(A)=P(IIOeAIIM l lIOex[IM), BelM(A)=P(I[DeAI]M I [I<>exI[M).
Proof. Since all F(w), for w E W, are nested, it follows from (5) that any decreasing sequence in {F+(A)IA E ~ ( X ) } is determined by some family (Ai)iE 1 in ~ ( X ) . For any family (Ai)iE1 in ~ ( X ) , it follows using (15) that
Remark 5.2. Notice that in case of a serial accessibility relation R it holds that
P n
= P { _~ic N (IIDeA,IIMNII<>exllM
II~ex II~ = IIDexl[ M = w, and hence also P(dom(R))-----P(W)= 1. The expressions in Theorem 5.5 then reduce to P1M(A) = P(II<>eAI1M ), BelM (A) = P(II []eA IIM), in which we clearly recognize the expressions in Theorem 5.1, showing that the approach of Harmanec et al. [9] is a special case of the one presented here. Hereafter, we apply the modal logic interpretation of plausibility and belief measures, presented in Theorem 5.5, to possibility and necessity measures. Our further considerations are based on models with a transitive and connected accessibility relation. Note that the class of models that we employ here differs from the one used by Harmanec et al. in [9], since connectedness does not imply reflexivity. Let us also consider the multivalued mapping F from W into X, constructed above, and assume further that P is continuous from above on ( 11 ).
Proposition 5.1. For any (v, w) E W 2, either R(v) C R(w) or R(w) C_R(v), i.e. the family (R(w))wEW can be ordered in a nested sequence.
=P((i~II~eA~IIM)NIIOexIIM)" Due to (12) and (14), for any i E I it holds that H[~eAi HM = F + ( A i ) U codom(F), and obviously, IIDeA~ IIM u II¢>exIIM = I[DeA, II'~t u
Corollary 5.1. For any (v,w)E W e, either F(v)C_ r(w) or r(w) C_r(v), i.e. the family (r(W))wEW can be ordered in a nested sequence. Proposition 5.2. P is continuous from above on the
set {r+(A) I A ~ ~(X)}.
= w.
Hence, it also holds that (N~tll[]eA,II M) u [[Oex[[M = W. Moreover, (F+(Ai))i~t is a decreasing sequence if and only if ([l[]eA,tlM)e~z is a decreasing sequence. Recall that for any two subsets U and V of W such that U U V = W it holds that P(U n V ) = P ( U ) - P(co V). Since P is continuous from above on ( 11 ), it holds that
= inf
P(IIDeAill M ) - P(coll<>exll M)
= inf
(P(II []eA, II~) -- P(collOex IIM))
iE1
= inf P (11Vle,, IIM n Proof. Consider (v, w) E W 2. I f v = w there is nothing to prove. So, let us assume that v ~ w. From the connectedness of R, we have that either v E R(w) or w c R ( v ) . Suppose, for instance, that vER(w). The transitivity of R implies that (Vu E R(v))(u E R(w)) and hence R(v) C_R(w). []
dom(r)
iEl
tl<>exIIM).
Again applying (15), we obtain that
P (NF+(Ai)) \iEl
=infP(F+(Ai)iE ',
[]
/
Theorem 5.6. A transitive and connected model M = (W,R, V,P I with P ( d o m ( R ) ) > 0 that satisfies (SVA) induces a possibility measure IIM and a necessity measure NM on ~(X), defined by //M (A) = P ( F - (A) I dom(F)), NM(A) = P( r+(A) l dom( r ) ),
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or equivalently,
HM(A)----P(II<>eAIIMI II<>exIIM), NM(A)= P(II DeAIIM I II<>exI[M). Proof. According to Corollary 5.1, all F(w), for w C W, can be ordered in a nested sequence, and by Proposition 5.1, P is continuous from above on {F+(A)IA E ~'(X)}. Therefore, following the discussion in Subsection 3.2, 1IM is a possibility measure and NM is a necessity measure on N(X). []
for any w E WB,which means that only worlds in which eB is true can be accessed in Ms, i.e. it may happen that MB contains worlds in which es is not true, but these worlds cannot be accessed from anywhere. This means that eB V OeB is always true in Ms. Note that dom(Rs) = II(>exIIu~ = II<>esllu Further, V8 is the restriction of V to WB, and Ps is defined as follows:
PB(U) =P(UI ~), for any U E ~(WB).
Proposition 6.1. The followino properties hold: 6. A modal logic interpretation of Dempster's rule of conditioning Consider a model M = (W, R, V,P) with P a probability measure on ~'(W). As atomic propositions we consider again propositions of the form e{x}, for all x belonging to some universe X. We assume that (SVA) holds and consider the multivalued mapping F constructed above.
6.1. Restriction of the model Consider a subset B of X such that P(II<>eBII M) >0. This implies in particular that II<>eBIIM ¢ O, i.e.
(3w ~ w)(e(w) n Ilesll u ¢ 0).
(16)
Moreover, P(ll ()eB [Ig ) > 0 implies that P(l[ (>ex I[M) - P ( d o m ( R ) ) > 0 . In view of (15), the condition P(ll(>esllg)>0 corresponds to the condition P ( F - ( B ) ) > 0 imposed by Dempster in (7). Let us now restrict M = (W,R, V,P) to Ms = (WB,RB, Vs,Ps) in the following way. We define WB= Ilesl[ g u II<>esllM, or equivalently,
~={wlw~llesllMVR(w)nllesllM #O}.
(17)
In fact, ~ is the subset of W containing all worlds in which es is true or from which a world can be accessed in which es is true. Observe that in case of a reflexive accessibility relation the inclusion IleBIIu _c II<>esllM always holds, and hence (16) reduces to IlesllM ¢ 0 and, of course, WB= [[(>eeIIg. Then Rs is constructed, as follows:
RB(w) = R ( w ) n IleBIIM,
(18)
(i) I f R & transitive then RB & also transitive. (ii) I f R is serial then RB is not always serial. (iii) I f R is reflexive then RB is not always reflexire. (iv) I f R is connected then Rs is not always connected
Proof. (i) Consider (u,v,w)EW83, such that vE Rs(u) and w C Rs(v). The latter implies that v E R(u), w E R ( v ) and wE IlesIIg. Since R is transitive, it follows that w E R(u) and therefore w E RB(u). (ii) Due to (17), it is completely possible that (~u~)(R(u)nllesllU=O) and hence by (18) Rs(u) = O. (iii) If there exists u E W~ such that u ~ Ilesll u then it follows that u q~Rs(u) and hence Rs will not be reflexive in this case. (iv) Let us now show that Rs is not connected in general, i.e. it is not always true that for any (u, v) E W82, such that u ~ v, it holds that u E Rs(v) or v E Rs(u). Suppose that for some (u, v) E Wff and u ~ v, we have that u E R(v) but v q~R(u) (which does not contradict the connectedness of R), and u ~ IleBIIM. Therefore, we have that u~R(v)nlleBIIM=RB(v) and vq~R(u)N Ilesll M =Rs(u). [] Example 6.1. In this example, we illustrate the above restriction procedure. We consider a set of worlds with cardinality 6, i.e. W={wl,w2,w3,w4,ws,w6}, and corresponding accessibility relation depicted in Fig. 1. For each world also the element x E X = {a,b,c,d} is indicated for which the atomic proposition e{x} is true. As probability measure P on ~ ( W ) we consider the one corresponding to a uniform probability distribution, i.e. P( U ) = #U/6.
E. Tsiporkova et al. / Fuzzy Sets and Systems 102 (1999) 371-383
<
381
Since P(II OeB IIM) > o, it follows that PB(dom(RB)) = -
P(IIOeB[IM I ~ ) P(IIOeBIIM) >0. P(Ws)
Fig. 1.
<
Theorem 5.5 can then be applied immediately, and we have that the restricted model MB induces a plausibility measure P1MB and a belief measure BelM8 on ~ ( X ) , given by
P1MAA) = PB(Is-(A) I dom(is))
= PB(II<>eAIIM" I II<>exlIM"), BelMs(A) = PB(Is+(A) I dom(is))
= PB([I DeAIIM~IIIOexIIM"). Fig. 2.
Note that for any two subsets U and V of WB, with PB(V)>0, it holds that PB(U I V ) = P ( U I V). The above measures can then be expressed as follows: P1M~(A) = P ( I s - ( A ) I dom(is))
= P(II<>eAIIM~III<>exIIMs), BelMs(A) = P(FB+(A) I dom(is)) =
Fig. 3. • Consider the set Bi = {b,c}, then HeB~]IM = {wl,w2,ws} and [l<>eB,IIM = {W2,W3,W4,W5}• The restricted set of worlds is given by Ws~ = {wl, w2, w3, w4, w5 } and is depicted with its accessibility relation in Fig. 2. • Consider the set Bz = {a,d}, then IleB21lM = {W3,W4, W6} and [IOeB2IIM={w3,w5}. The restricted set of worlds is given by WB2= {w3, w4, ws, w6} and is depicted with its accessibility relation in Fig. 3.
P(ll DeAII ~ I IIOex[l'~).
(19)
We would like to express the measures induced by the restricted model in terms of the measures induced by the original model, and establish a modal logic interpretation of the expressions for conditional plausibility and belief measures. Note that the definition of WB implies that
(Vw ~ W\W~)(R(w) n IleBII~ -- O) and therefore Is can be extended to W by associating to each w C W\ WBthe empty set.
Proposition6.2. It holds that
(VwE W ) ( I s ( w ) =
F ( w ) N B ) , i.e. Is = FIe.
6.2. Measures induced by the restricted model
Proof. Let us first consider the case w E Ws:
Let us consider the restricted model MB = (WB,RB, Vs,PB) constructed in the previous subsection. We define Is as the composition of RB and fB, i.e. Is = fB oRB, where fB(w) = f ( w ) , for all w E Wa.
x 6 Is(w) = f ( R ( w ) A]l eB I[M)
¢, (3vER(w) nlleBllM)(V(v,e{~})= T) ¢* (3vER(w))(V(V, eB)= T A V(v,e{z})= T)
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sion for BelMB in (20). Since P1M8 and BelMB are dual measures, it holds that
¢:~ (3v E R(w))( V(v, e{x}) : T A x E B) <:> ((qv E R(w))(V(v, e{~}) = T) A x E B) ¢~ ( V ( w , ( ) e { ~ } ) = T ) A ( x C B ) .
BelM~(A) = 1 - P1M~(CoA)
According to (13), it then follows that
=e(coll<>ecoAnBlIM l [lt>eellM).
x ~ G(w) ¢* (x c F(w) A x ~ B).
Finally, due to (10) and (9), we find that
Now consider the case wE W\Ws, i.e. Ye(w)=0. Assume that F ( w ) N B ¢ O , which implies that w E F - ( B ) . Recall that by (15) we have that Y - ( B ) = IIOee[[u. Thus we have obtained, due to (17), that w E Wn, a contradiction. []
BelM~,(A) = P([I-"I 0 ---lea ucoel[ M
I Il<>eellM)
= P(llDeAucoellg J ll<>eellg ).
[]
P1MB(A) = P1M(A I B),
Remark 6.1. If we assume that the original model M = (W,R, V,P) is transitive and connected and P is continuous from above on (11), then according to Theorem 5.6 the plausibility measure P1M and the belief measure BelM induced by M, and consequently also the corresponding conditional measures, are a possibility measure and a necessity measure, respectively. Therefore by Theorem 6.1 the restricted model Me = (WB,Re, VB,Pe) induces a possibility measure HMB and a necessity measure NM~ on ~ ( X ) , which are, in fact, the conditional measures corresponding to the original model M = (W, R, V, P). Since according to Proposition 6.1 the restricted accessibility relation is transitive, but not necessarily connected, the foregoing result does not seem to be so obvious as in the case of plausibility and belief measures. However, it can be shown, independently from Theorem 6.1 and only using the fact that F8 = FIB, that the subsets (YB(w))w c ~ are nested and that Pe is continuous from above on the set {Ye+(A) ]A E ~(X)}. From the discussion in Subsection 3.2 it then follows that the multivalued mapping FB corresponding to the restricted model MR induces a possibility measure HM~ and a necessity m e a s u r e NM8 on ~ ( X ) , given by
Belgt,(A) = BelM(A [B).
nMB (A) = P ( r , - (A) I dom(rs)),
Hence, the extension to W of the multivalued mapping F8 corresponding to the restricted model, is exactly the restricted mapping FIB employed in Dempster's rule of conditioning [6]. It then follows from (7) and (19) that the plausibility and belief measures induced by the restricted model Me are the upper and lower conditional probabilities induced by Fs. Moreover, the measures induced by the restricted model Me can be expressed in terms of conditional probabilities of truth sets of M. This is stated in the following theorem. Theorem 6.1. Given a model M = (W,R, KP) that satisfies (SVA) and a subset B o f X such that P([I <)ee][M) > 0, the restricted model Me = (Ws, RB, Vs,Pe) induces a plausibility measure P1MB and a belief measure BelM~ on ~ ( X ) , which are, in fact, the conditional plausibility and belief measures corresponding to the original model M = ( W, R, V, P), i.e.
Moreover, it holds that
NM~(A) = P(Fe+(A) ] dom(Fs)),
P1MAA)=P(]I<)eAneIIM I IIOeBI]M), (20)
BelMAA ) = P( l[~eA ucoe[[M [ ]lOeeJ[M). Proof. Consider a subset A of X. The first property in (20) can easily be checked using P1M~(A) =P1M(A ]B) and (8). Let us now show the expres-
and consequently, again due to the equality Ye = FIB, it holds that
nMB(A)=nM(A IB), NMB(A) = NM(A ]B).
E. Tsiporkova et al. / Fuzzy Sets and Systems 102 (1999) 371.-383
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