Information Sciences 125 (2000) 187±206
www.elsevier.com/locate/ins
Network of probabilities associated with a
capacity of order-2
pez *, Serafõn Moral
Juan F. Verdegay-Lo
Departamento de Ciencias de la Computaci
on e Inteligencia Arti®cial, E.T.S. de Ingenierõa,
Inform
atica, Universidad de Granada, 18071 Granada, Spain
Received 21 July 1998; revised 30 July 1999; accepted 3 December 1999
Abstract
This paper studies capacities of order-2 from a geometric point of view, that is, from
convex polytope of probabilities de®ned by them. This convex polytope is characterized
by means of a network of probabilities representing relationships among vertices de®ned
by the edges. Interesting properties may be obtained about the capacities from this
network. Ó 2000 Published by Elsevier Science Inc. All rights reserved.
Keywords: Approximate reasoning; Knowledge representation; Uncertainty; Capacities
of order-2
1. Introduction
Let us consider a variable X taking values on a ®nite set U ˆ fu1 ; . . . ; un g.
Suppose that we have information about the values taken for this variable
given by a set of possible probability distributions P. We can resume this information by calculating the system of intervals ‰l…A†; u…A†Š 8A U given by
the functions
g …A† ˆ Inf p…A†;
p2P
*
g …A† ˆ Sup p…A†:
p2P
…1†
Corresponding author.
E-mail address: [email protected] (J.F. Verdegay-LoÂpez).
0020-0255/00/$ - see front matter Ó 2000 Published by Elsevier Science Inc. All rights reserved.
PII: S 0 0 2 0 - 0 2 5 5 ( 9 9 ) 0 0 1 4 8 - 6
188
J.F. Verdegay-L
opez, S. Moral / Information Sciences 125 (2000) 187±206
This pair of functions is called a probability envelope [11]. A special kind of
probability envelopes are the lower and upper capacities of order-2 [7]; …g ; g †
is a lower and upper capacity of order-2 i€ …g ; g † is a pair of set mappings
g ; g : P…U † 7! ‰0; 1Š
verifying
1:
2:
3:
4:
g …;† ˆ g …;† ˆ 0; g …U † ˆ g …U † ˆ 1;
g …A [ B† P g …A† ‡ g …B† ÿ g …A \ B† 8A; B U ;
g …A [ B† 6 g …A† ‡ g …B† ÿ g …A \ B† 8A; B U ;
ˆ 1 8A U :
g …A† ‡ g …A†
These functions have been used, by a large group of authors, as a representation of uncertain knowledge: Chateauneuf [5,6], Ja€ray [12], de Campos [3],
de Campos and Bola~
nos [4], Wasserman and Kadane [17], Anger [1,2], Gilboa
and Schmeidler [10], Schmeidler [13], Huber [11], etc.
There is always a maximal set of probabilities de®ned by a probability envelope. The one given by
P ˆ fp 2 P …U † : g …A† 6 p…A† 6 g …A†g;
where P …U † is the set of all probabilities on U. We shall study the structure of
this convex set; characterizing vertices, support hyperplanes, relationships
between vertices connected by an edge, etc. The study of this convex set P may
be important because it provides tools for handling information represented by
order-2 capacities. In this work, this study is carried out by means of a network
of probabilities.
Therefore, only
The functions g and g are duals, that is, g …A† ˆ 1 ÿ g …A†.
one of these functions is needed for extracting all information contained in the
capacity. For this reason, the studies will be centered only on one of them, the
upper capacity. For simplicity, from now on, this upper capacity will be denoted by g.
In Section 2 the network will be de®ned. In Section 3, the equivalence between the relationships represented on the network and the ones given by the
edges of convex set P will be shown. In Section 4, some results obtained from
the network representation will be studied. Finally, in Section 5, conclusions
are presented.
2. Network of probabilities de®ned by a capacity of order-2
The network will represent relationships among vertices de®ned by edges on
the convex polytope. That is, we want to represent the following relationships:
J.F. Verdegay-L
opez, S. Moral / Information Sciences 125 (2000) 187±206
189
De®nition 1. Two vertices, p1 and p2 , of the convex polytope P, associated with
a capacity of order-2 g are related on the convex polytope i€ there exists an
edge of P joining them.
Vertices of the convex polytope P were determined by de Campos [4] as the
set of associated probabilities fprg ; r 2 Sn g.
De®nition 2. Let g be an upper capacity of order-2 de®ned on the universe
U ˆ fu1 ; u2 ; . . . ; un g. The probabilities associated with g are the probability
distributions prg given by:
prg …fur1 g† ˆ g…fur1 g†;
prg …fur2 g† ˆ g…fur1 ; ur2 g† ÿ g…fur1 g†;
..
.
g
pr …furi g† ˆ g…fur1 ; . . . ; uri g† ÿ g…fur1 ; . . . ; uriÿ1 g†;
..
.
…2†
prg …furn g† ˆ 1 ÿ g…fur1 ; . . . ; urnÿ1 g†
for each r ˆ …r1 ; r2 ; . . . ; rn † 2 Sn , where Sn is the set of all permutations of
1; 2; . . . ; n.
There is one associated probability for each permutation r 2 Sn , but we can
obtain the same probability for several possible permutations. In order to
avoid this problem, we shall consider the following equivalence relationship.
De®nition 3. Two permutations r and l belonging to Sn are related according
to an upper capacity of order-2, g, i€ the associated probabilities are equal, i.e.,
g
def
r  l () prg ˆ plg :
…3†
The equivalence classes are given by the sets
‰pŠ ˆ fr 2 Sn : prg ˆ pg:
We shall consider that each equivalence class de®nes a node on the network.
An associated probability prg comes from a permutation r ˆ r1 rn . If we
change this permutation by a transposition (l ˆ r1 r2 rsÿ1 rs‡1 rs rs‡2 rn )
we obtain a new permutation producing a new associated probability (this new
probability may be equal to prg ). The associated probability plg is closely related
with the previous probability prg . In fact, if prg is not equal to plg , we shall see
that they are joined by an edge of the convex polytope.
190
J.F. Verdegay-L
opez, S. Moral / Information Sciences 125 (2000) 187±206
This relationship seems appropriate to de®ne the network. Speci®cally
speaking, the transposition of a permutation r can be de®ned as follows.
De®nition 4. We de®ne the s-transposition as the function
Ts : Sn 7! Sn ;
Ts …r ˆ r1 rn † ˆ r1 r2 rsÿ1 rs‡1 rs rs‡2 rn :
…4†
Further, the relationship is formally given by:
De®nition 5. Two equivalence classes ‰p1 Š and ‰p2 Š are neighbors on the network i€ there exist r 2 ‰p1 Š; l 2 ‰p2 Š and k 2 f1; 2; . . . ; n ÿ 1g such that
l ˆ Tk …r†.
Example 6. Consider the upper capacity de®ned on U ˆ fu1 ; u2 ; u3 ; u4 g given
by:
g…fu1 g† ˆ 0:5;
g…fu2 g† ˆ 0:5;
g…fu3 g† ˆ 0:5;
g…fu4 g† ˆ 0:5;
g…A† ˆ 1 8A U ;
A 6ˆ fu1 g; fu2 g; fu3 g; fu4 g:
For this function, the equivalence classes are:
‰p1 Š ˆ ‰…0:5; 0:5; 0; 0†Š ˆ f1234; 1243; 2134; 2143g;
‰p2 Š ˆ ‰…0:5; 0; 0:5; 0†Š ˆ f1324; 1342; 3124; 3142g;
‰p3 Š ˆ ‰…0:5; 0; 0; 0:5†Š ˆ f1423; 1432; 4123; 4132g;
‰p4 Š ˆ ‰…0; 0:5; 0:5; 0†Š ˆ f2314; 2341; 3241; 3214g;
‰p5 Š ˆ ‰…0; 0:5; 0; 0:5†Š ˆ f2413; 2431; 4213; 4231g;
‰p6 Š ˆ ‰…0; 0; 0:5; 0:5†Š ˆ f3412; 3421; 4312; 4321g
and the associated network can be seen in Fig. 1.
3. Equivalence between the convex polytope and the network
Once the network has been de®ned, the equivalence between the relationships represented by the edges of the convex polytope and the network must be
proved. That is, the relationship expressed in De®nition 5 is equivalent to the
®rst idea given by De®nition 1.
J.F. Verdegay-L
opez, S. Moral / Information Sciences 125 (2000) 187±206
191
Fig. 1. Network associated with the upper capacity.
The initial idea is based on the concept of edge. This concept is dicult to
use. However, an easier interpretation can be obtained: we can translate the
concept of edge as intersection of non-dependent hyperplanes.
Proposition 7. The intersection of n ÿ 1 non-dependent support hyperplanes of a
convex polytope, going through extreme points p1 and p2 , is a line containing the
edge that joins these points.
Firstly, we shall prove that if two equivalence classes are neighbors on the
network the associated extreme points are joined by an edge on the convex
polytope. To obtain this proof, it is important to de®ne the concept of associated hyperplanes of an upper capacity of order-2.
De®nition 8. The set of hyperplanes associated with an upper capacity of order-2 is the set [r2Sn Hr , where
)
(
r
X
r
p…uri † ˆ g…fur1 ; ur2 ; . . . ; urr g† : r ˆ 1 n :
…5†
Hr ˆ Hr iˆ1
The probability associated with the permutation r ˆ r1 rn and the upper
capacity g, prg , satis®es
192
J.F. Verdegay-L
opez, S. Moral / Information Sciences 125 (2000) 187±206
prg …fur1 g† ˆ g…fur1 g†;
prg …fur1 g† ‡ prg …fur2 g† ˆ g…fur1 ; ur2 g†
..
.
prg …fur1 g† ‡ ‡ prg …furi g† ˆ g…fur1 ; . . . ; uri g†;
..
.
g
g
pr …fur1 g† ‡ ‡ pr …furn g† ˆ 1:
That is, prg belongs to all the hyperplanes of Hr . It is possible to prove that the
hyperplanes of Hr constitute a set of n non-dependent hyperplanes, so we
have:
Lemma 9. The probability associated with the permutation r ˆ r1 rn and the
upper capacity g is given by the intersection of the n non-dependent hyperplanes of
Hr .
Proposition 10 enables us to assure that there is a probability mass transference between two neighboring equivalence classes on the network.
Proposition 10. If two equivalence classes ‰p1 Š and ‰p2 Š are neighbors on the
network, the extreme points p1 and p2 verify
p1 ÿ p2 ˆ …0; 0; . . . ; 0; e; 0; 0; . . . ; 0; ÿe; 0; 0; . . . ; 0†:
Proof. If the classes are related, there exist r 2 ‰p1 Š, l 2 ‰p2 Š and
k 2 f1; 2; . . . ; n ÿ 1g such that l ˆ Tk …r†.
Let us see that
prg ÿ plg ˆ …0; 0; . . . ; 0; e; 0; 0; . . . ; 0; ÿe; 0; 0; . . . ; 0†:
8i < k, we have
prg …uri † ˆ g…fur1 ; . . . ; uri g† ÿ g…fur1 ; . . . ; uriÿ1 g†
ˆ g…ful1 ; . . . ; uli g† ÿ g…ful1 ; . . . ; uliÿ1 g†
ˆ plg …uli †;
because, until the component k, r and l are equal.
The permutations r and l are only di€erent in the k and k ‡ 1 components,
so
fur1 ; . . . ; urk ; urk‡1 ; . . . ; urr g ˆ ful1 ; . . . ; ulk ; ulk‡1 ; . . . ; ulr g 8r P k ‡ 1;
therefore 8i > k ‡ 1,
J.F. Verdegay-L
opez, S. Moral / Information Sciences 125 (2000) 187±206
193
prg …uri † ˆ g…fur1 ; . . . ; urk ; urk‡1 ; . . . ; uri g† ÿ g…fur1 ; . . . ; urk ; urk‡1 ; . . . ; uriÿ1 g†
ˆ g…ful1 ; . . . ; ulk‡1 ; ulk ; . . . ; uli g† ÿ g…ful1 ; . . . ; ulk‡1 ; ulk ; . . . ; uliÿ1 g†
ˆ plg …uli †:
If e ˆ prg …urk † ÿ plg …ulk †, then
pr …urk‡1 † ÿ pl …ulk‡1 † ˆ g…fur1 ; . . . ; urk‡1 g† ÿ g…fur1 ; . . . ; urk g†
ÿ g…ful1 ; . . . ; ulk‡1 g† ‡ g…ful1 ; . . . ; ulk g†
ˆ g…ful1 ; . . . ; ulk g† ÿ g…fur1 ; . . . ; urk g†
since fur1 ; . . . ; urk‡1 g ˆ ful1 ; . . . ; ulk‡1 g. Adding and subtracting the quantity
g…ful1 ; . . . ; ulkÿ1 g†, we obtain
prg …urk‡1 † ÿ plg …ulk‡1 † ˆ g…ful1 ; . . . ; ulk g† ÿ g…ful1 ; . . . ; ulkÿ1 g†
‡ g…ful1 ; . . . ; ulkÿ1 g† ÿ g…fur1 ; . . . ; urk g†
ˆ g…ful1 ; . . . ; ulk g† ÿ g…ful1 ; . . . ; ulkÿ1 g†
‡ g…fur1 ; . . . ; urkÿ1 g† ÿ g…fur1 ; . . . ; urk g†
ˆ plg …ulk † ÿ prg …urk † ˆ ÿe
because, until the component k, r and l are equal. So,
prg ÿ plg ˆ …0; 0; . . . ; 0; e; 0; 0; . . . ; 0; ÿe; 0; 0; . . . ; 0†:
…6†
As r 2 ‰p1 Š and l 2 ‰p2 Š, we ®nd that prg ˆ p1 and plg ˆ p2 . Hence, from (6) we
get
p1 ÿ p2 ˆ …0; 0; . . . ; 0; e; 0; 0; . . . ; 0; ÿe; 0; 0; . . . ; 0†:
Value of e could be interpreted as a transference of probability mass from
component rk to rk‡1 between p1 and p2 . In these conditions, concept of edge
can be characterized.
Suppose that ‰p1 Š and ‰p2 Š are two neighboring equivalence classes on the
network. Then, the set of points in Rn ,
(
)
n\
ÿ1
r
X
n
p2R :
p…uri † ˆ g…fur1 ; . . . ; urr g†
Aˆ
rˆ1
r6ˆk
\
iˆ1
(
p 2 Rn :
n
X
)
p…ui † ˆ 1
…7†
iˆ1
determines a line through prg and plg (r 2 ‰p1 Š, l 2 ‰p2 Š and l ˆ Tk …r†† such as it
is shown in Lemma 11.
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J.F. Verdegay-L
opez, S. Moral / Information Sciences 125 (2000) 187±206
Lemma 11. Let ‰p1 Š and ‰p2 Š be two neighboring equivalence classes on the
network, that is, there exist r 2 ‰p1 Š, l 2 ‰p2 Š and k 2 f1; 2; . . . ; n ÿ 1g such that
l ˆ Tk …r†. Then, the set
(
)
n\
ÿ1
r
X
p 2 Rn :
p…uri † ˆ g…fur1 ; . . . ; urr g†
Aˆ
rˆ1
r6ˆk
\
iˆ1
(
n
p2R :
n
X
)
p…ui † ˆ 1
iˆ1
is a line through prg and plg .
Proof. It is clear that A is the intersection of n ÿ 1 non-dependent hyperplanes
of Rn and, therefore, a line of Rn . According to Lemma 9, prg 2 A. Let us see
that plg also belongs.
By the previous proposition prg and plg verify
rk‡1
rk
^
^
g
g
pr ÿ pl ˆ 0; 0; . . . ; 0; e ; 0; 0; . . . ; 0; ÿe; 0; 0; . . . ; 0 :
That is, 8r 6ˆ k; k ‡ 1; plg …fulr g† ˆ prg …furr g†, so,
r
X
iˆ1
plg …uli † ˆ
r
X
iˆ1
prg …uri † ˆ g…fur1 ; . . . ; urr g† 8r < k
and 8r ˆ k ‡ 1 n,
r
X
iˆ1
plg …uli † ˆ
ˆ
kÿ1
X
iˆ1
kÿ1
X
iˆ1
ˆ
r
X
iˆ1
plg …uli † ‡
prg …uri † ‡
r
X
iˆk‡2
r
X
iˆk‡2
plg …uli † ‡ plg …ulk † ‡ plg …ulk‡1 †
prg …uri † ‡ prg …urk † ‡ prg …urk‡1 † ‡ e ÿ e
prg …uri † ˆ g…fur1 ; . . . ; urr g†:
Hence,
plg
2
n\
ÿ1
rˆ1
r6ˆk
(
n
p2R :
r
X
iˆ1
)
p…urk † ˆ g…fur1 ; . . . ; urr g†
and, because plg is a probability distribution, condition
veri®ed. Pn
iˆ1
plg …uli † ˆ 1 is
J.F. Verdegay-L
opez, S. Moral / Information Sciences 125 (2000) 187±206
195
Corollary 12. If ‰p1 Š and ‰p2 Š are two neighboring equivalence classes on the
network then the segment joining points p1 and p2 is an edge of the convex
polytope defined by the capacity of order-2.
Proof. ‰p1 Š and ‰p2 Š are two neighboring equivalence classes, then, there exist
r 2 ‰p1 Š, l 2 ‰p2 Š and k 2 f1; 2; . . . ; n ÿ 1g such that, l ˆ Tk …r†. According to
Lemma 11, set A, given by (7), is a line through prg and plg . A will be a line
through p1 and p2 since p1 ˆ prg and p2 ˆ plg .
Furthermore, prg and plg are probabilities associated with an upper capacity
of order-2 and, therefore, they are extreme points of the convex polytope de®ned by the capacity. So, A is a line joining two extreme points which is
constituted by the intersection of n ÿ 1 non-dependent support hyperplanes of
the convex polytope. According to Proposition 7, A is a line containing the
edge of convex polytope joining p1 and p2 . Corollary 13. If ‰p1 Š and ‰p2 Š are two neighboring equivalence classes, the points
p1 and p2 belonging to Rn are neighbors on the convex polytope.
This corollary proves that any relationship among equivalence classes on the
network implies a relationship between extreme points on the convex polytope
de®ned by the capacity.
Let us show the converse implication. If two extreme points, p1 and p2 , of the
convex polytope de®ned by a capacity of order-2 are joined by an edge, the
respective equivalence classes, ‰p1 Š and ‰p2 Š, are neighbors on the network. In
order to obtain this proof we have to de®ne the concept of strict convex
combination of capacities.
De®nition 14. The capacity of order-2 g is said a strict convex combination of
the capacities g1 and g2 i€ there exists a; 0 < a < 1, so that,
8A U ;
g…A† ˆ ag1 …A† ‡ …1 ÿ a†g2 …A†:
…8†
From a geometric point of view, the convex decomposition of a capacity g
into g1 and g2 is characterized by the decomposition of the convex polytope
de®ned by g into the convex polytopes associated with g1 and g2 by means of
the extreme points decomposition.
Theorem 15. Let g, g1 and g2 be three upper capacities of order-2. g is a strict
convex combination of g1 and g2 if and only if 8r 2 Sn , prg is a strict convex
combination of prg1 and prg2 .
196
J.F. Verdegay-L
opez, S. Moral / Information Sciences 125 (2000) 187±206
Proof. The direct implication is trivial from de®nition of associated probabilities and the equality (8).
Let us show the converse implication. If
8r 2 Sn ;
prg ˆ aprg1 ‡ …1 ÿ a†prg2 ;
we shall inductively prove that 8r 2 Sn ,
g…fur1 ; ur2 ; . . . ; uri g†
ˆ ag1 …fur1 ; ur2 ; . . . ; uri g† ‡ …1 ÿ a†g2 …fur1 ; ur2 ; . . . ; uri g†:
From the de®nition of associated probabilities, we know that prg …ur1 † ˆ
g…fur1 g†. So,
g…fur1 g† ˆ prg …ur1 † ˆ aprg1 …ur1 † ‡ …1 ÿ a†prg2 …ur1 †
ˆ ag1 …fur1 g† ‡ …1 ÿ a†g2 …fur1 g†:
The property is shown by fur1 g. Suppose it is satis®ed for fur1 ; ur2 ; . . . ; uriÿ1 g,
i.e.,
g…fur1 ; ur2 ; . . . ; uriÿ1 g†
ˆ ag1 …fur1 ; ur2 ; . . . ; uriÿ1 g† ‡ …1 ÿ a†g2 …fur1 ; ur2 ; . . . ; uriÿ1 g†:
Then, this property must be veri®ed for fur1 ; ur2 ; . . . ; uri g:
prg …uri † ˆ aprg1 …uri † ‡ …1 ÿ a†prg2 …uri †
ÿ
ˆ a g1 …fur1 ; ur2 ; . . . ; uri g† ÿ g1 …fur1 ; ur2 ; . . . ; uriÿ1 g†
ÿ
‡ …1 ÿ a† g2 …fur1 ; ur2 ; . . . ; uri g† ÿ g2 …fur1 ; ur2 ; . . . ; uriÿ1 g†
ˆ …ag1 …fur1 ; ur2 ; . . . ; uri g† ‡ …1 ÿ a†g2 …fur1 ; ur2 ; . . . ; uri g††
ÿ
ÿ ag1 …fur1 ; ur2 ; . . . ; uriÿ1 g† ‡ …1 ÿ a†g2 …fur1 ; ur2 ; . . . ; uriÿ1 g† :
From induction hypothesis, we know that
g…fur1 ; ur2 ; . . . ; uriÿ1 g†
ˆ ag1 …fur1 ; ur2 ; . . . ; uriÿ1 g† ‡ …1 ÿ a†g2 …fur1 ; ur2 ; . . . ; uriÿ1 g†:
Then,
prg …uri † ˆ …ag1 …fur1 ; ur2 ; . . . ; uri g† ‡ …1 ÿ a†g2 …fur1 ; ur2 ; . . . ; uri g††
ÿ g…fur1 ; ur2 ; . . . ; uriÿ1 g†:
On the other hand,
prg …uri † ˆ g…fur1 ; ur2 ; . . . ; uri g† ÿ g…fur1 ; ur2 ; . . . ; uriÿ1 g†
is obtained from de®nition of associated probabilities.
J.F. Verdegay-L
opez, S. Moral / Information Sciences 125 (2000) 187±206
197
Therefore,
g…fur1 ; ur2 ; . . . ; uri g† ÿ g…fur1 ; ur2 ; . . . ; uriÿ1 g†
ˆ …ag1 …fur1 ; ur2 ; . . . ; uri g† ‡ …1 ÿ a†g2 …fur1 ; ur2 ; . . . ; uri g††
ÿ g…fur1 ; ur2 ; . . . ; uriÿ1 g†
hence,
g…fur1 ; ur2 ; . . . ; uri g†
ˆ ag1 …fur1 ; ur2 ; . . . ; uri g† ‡ …1 ÿ a†g2 …fur1 ; ur2 ; . . . ; uri g†:
So, if
g…fur1 ; ur2 ; . . . ; uri g†
ˆ ag1 …fur1 ; ur2 ; . . . ; uri g† ‡ …1 ÿ a†g2 …fur1 ; ur2 ; . . . ; uri g†
8r 2 Sn , then
8A U ;
g…A† ˆ ag1 …A† ‡ …1 ÿ a†g2 …A†:
Theorem 16. Let p1 and p2 be two neighboring extreme points of convex polytope
P defined by a capacity of order-2 g. Then, there are two permutations r and
l 2 Sn given by r ˆ r1 r2 rn and l ˆ r1 riÿ1 ri‡1 ri ri‡2 rn such that
prg ˆ p1 and plg ˆ p2 .
Proof. Firstly, we shall obtain the proof for the case of non-degenerate polytopes. Later, we consider the general case.
If P is non-degenerate there is exactly one extreme probability for each
permutation r 2 Sn , i.e., if l and r are two di€erent permutations of Sn , then
prg 6ˆ plg .
On the other hand, if P is non-degenerate, we can prove that each extreme
point is the intersection of only a set of the non-dependent support hyperplanes.
Simplex algorithm [15] tells us that it is always possible to obtain an extreme
point from another by means of changing variables in the base. If the change
produces the same point, then the polytope is degenerate. So, if we have two
sets of support hyperplanes H1 and H2 such that p ˆ \fh : h 2 H1 g and
q ˆ \fh : h 2 H2 g are extreme points of convex polytope and H1 6ˆ H2 then
p 6ˆ q, if the convex polytope is not degenerate.
Consider adjacent extreme points p1 and p2 . Again from simplex algorithm,
we know that if two extreme points are adjacent and the polytope is non-degenerate, only one variable is the di€erence between the bases associated with
the extreme points. This means that if H1 and H2 are the sets of hyperplanes
198
J.F. Verdegay-L
opez, S. Moral / Information Sciences 125 (2000) 187±206
such that p1 ˆ \fh : h 2 H1 g and p2 ˆ \fh : h 2 H2 g, the di€erence between
H1 and H2 is only one hyperplane.
Let r and l be the permutations of Sn verifying that prg ˆ p1 and plg ˆ p2 . The
set H1 is the set of hyperplanes associated with r, Hr ,
)
(
r
X
r
p…uri † ˆ g…fur1 ; ur2 ; . . . ; urr g†; r ˆ 1 n :
H1 ˆ Hr iˆ1
As H1 and H2 are di€erent only in one hyperplane, we ®nd that p2 belongs
to all hyperplanes minus one of H1 . Suppose that this hyperplane is the one
associated with r ˆ i, so, p1 and p2 belong to the intersection of the hyperplanes:
p…ur1 † ˆ g…fur1 g†;
p…ur1 † ‡ p…ur2 † ˆ g…fur1 ; ur2 g†;
..
.
p…ur1 † ‡ ‡ p…uriÿ1 † ˆ g…fur1 ; . . . ; uriÿ1 g†;
…9†
p…ur1 † ‡ ‡ p…uri‡1 † ˆ g…fur1 ; . . . ; uri‡1 g†;
..
.
p…ur1 † ‡ ‡ p…urn † ˆ 1:
Since the set of hyperplanes (9) is given by n ÿ 1 non-dependent hyperplanes,
the intersection is a line. So, the edge joining the extreme points p1 and p2 is
contained in this line.
The n ÿ 1 hyperplanes determine n ÿ 2 coordinates of the points belonging
to the edge. Let q be any point on the edge, then,
q…fur1 g† ˆ g…fur1 g†;
q…fur2 g† ˆ g…fur1 ; ur2 g† ÿ g…fur1 g†;
..
.
q…furiÿ1 g† ˆ g…fur1 ; . . . ; uriÿ1 g† ÿ g…fur1 ; . . . ; uriÿ2 g†;
q…furi‡2 g† ˆ g…fur1 ; . . . ; uri‡2 g† ÿ g…fur1 ; . . . ; uri‡1 g†;
..
.
q…furn g† ˆ 1 ÿ g…fur1 ; . . . ; urnÿ1 g†:
Furthermore, the coordinates uri and uri‡1 of q satisfy
q…uri † ‡ q…uri‡1 † ˆ g…fur1 ; . . . ; uri‡1 g† ÿ g…fur1 ; . . . ; uriÿ1 g†:
As q is a point on the convex polytope P,
J.F. Verdegay-L
opez, S. Moral / Information Sciences 125 (2000) 187±206
199
g…fur1 ; . . . ; uri g† P q…fur1 ; . . . ; uri g† ˆ q…fur1 g† ‡ ‡ q…furi g†
ˆ g…fur1 ; . . . ; uriÿ1 g† ‡ q…furi g†;
therefore,
q…furi g† 6 g…fur1 ; . . . ; uri g† ÿ g…fur1 ; . . . ; uriÿ1 g†:
In a similar way, we obtain
q…furi‡1 g† 6 g…fur1 ; . . . ; uriÿ1 ; uri‡1 g† ÿ g…fur1 ; . . . ; uriÿ1 g†:
So, points on the edge could be calculated by distributing the quantity
g…fur1 ; . . . ; uri‡1 g† ÿ g…fur1 ; . . . ; uriÿ1 g†
…10†
between the coordinates uri and uri‡1 , subject to the restrictions:
q…furi g† 6 g…fur1 ; . . . ; uri g† ÿ g…fur1 ; . . . ; uriÿ1 g†;
q…furi‡1 g† 6 g…fur1 ; . . . ; uriÿ1 ; uri‡1 g† ÿ g…fur1 ; . . . ; uriÿ1 g†:
…11†
Extreme points of the edge are obtained by assigning more possible of (10) to
coordinate uri or to coordinate uri‡1 . If we assign it to uri the following points
are obtained:
q…furi g† ˆ g…fur1 ; . . . ; uri g† ÿ g…fur1 ; . . . ; uriÿ1 g†;
q…furi‡1 g† ˆ …g…fur1 ; . . . ; uri‡1 g† ÿ g…fur1 ; . . . ; uriÿ1 g†† ÿ …g…fur1 ; . . . ; uri g†
ÿ g…fur1 ; . . . ; uriÿ1 g††
ˆ g…fur1 ; . . . ; uri‡1 g† ÿ g…fur1 ; . . . ; uri g†:
That is, the associated probability pr , with r ˆ r1 r2 rn . Whereas if we assign
this quantity to uri‡1 , we get:
q…fuli g† ˆ q…uri‡1 † ˆ g…fur1 ; . . . ; uriÿ1 ; uri‡1 g† ÿ g…fur1 ; . . . ; uriÿ1 g†;
q…fuli‡1 g† ˆ q…uri † ˆ …g…fur1 ; . . . ; uri‡1 g† ÿ g…fur1 ; . . . ; uriÿ1 g††
ÿ …g…fur1 ; . . . ; uriÿ1 ; uri‡1 g† ÿ g…fur1 ; . . . ; uriÿ1 g††
ˆ g…fur1 ; . . . ; uri‡1 g† ÿ g…fur1 ; . . . ; uriÿ1 ; uri‡1 g†;
i.e., the associated probability pl , with l ˆ r1 riÿ1 ri‡1 ri ri‡2 rn as we want
to prove.
Now, we shall study the general case, the case when P could be degenerate.
Let P be any non-degenerate convex polytope de®ned by a capacity of order2. 8e > 0 we de®ne the convex polytope Pe given by the extreme points
pre ˆ …1 ÿ e†prg ‡ epr
8r 2 Sn ;
…12†
200
J.F. Verdegay-L
opez, S. Moral / Information Sciences 125 (2000) 187±206
where pr is the extreme probability associated with the permutation r and the
capacity de®ned by the convex polytope P .
Since P and P are convex polytopes de®ned by capacities, according to
Theorem 15, Pe is a convex polytope associated with a capacity. Furthermore,
as P is not degenerate and the extreme points of Pe verify (12) we have that Pe
is not degenerate. Then, 8e > 0 we have obtained a non-degenerate convex
polytope associated with a capacity of order-2 and such that
lim Pe ˆ P
e!0
and
lim pre ˆ prg
e!0
i.e., the vertices of P can be obtained as the limit of the vertices of the convex
polytopes Pe . In the same way, edges of P are the limit of the edges of the
convex polytopes sequence, Pe .
Suppose that pr and pl are two extreme points of P joined by an edge, a. If a
is an edge of P then,
a ˆ lim ae ;
e!0
where a is an edge of Pe . As Pe is non-degenerate the extreme points joined by
ae , pke pce , come from permutations which are equal except for a transposition,
c ˆ Ti …k†. As a ˆ lime!0 ae ,
e
pr ˆ lim pke
e!0
and
pl ˆ lim pce ;
e!0
so, r ˆ k, l ˆ c and, therefore, l ˆ Ti …r†. Corollary 17 can be immediately obtained from previous result. This corollary guarantees that the relationships given by the network are the only relationships de®ned by the edges on the convex polytope.
Corollary 17. If p1 and p2 are two extreme points on the convex polytope P
joined by an edge, the equivalence classes ‰p1 Š and ‰p2 Š are neighbors on the
network.
4. Applications
The network as a structure which is equivalent to the convex polytope enables us to make a study of such polytope on the basis of the relationships
existing on the network. From this study we have obtained the following results:
J.F. Verdegay-L
opez, S. Moral / Information Sciences 125 (2000) 187±206
201
4.1. Characterization of capacities of order-2
A ®rst application of this network is a characterization of the capacities of
order-2 on the basis of the geometric structure of the associated convex
polytope. This characterization is ®rstly based on the characterization given by
Huber [11]:
Theorem 18. Let …f ; g† be a probability envelope defined over a finite universe U.
Let P be the associated convex polytope of probabilities
P ˆ fp 2 P …U † : f …A† 6 p…A† 6 g…A†
8A U g:
Then the following declarations are equivalent:
1. …f ; g† is a capacity of order-2.
2. For all monotone succession A1 A2 Am there is a probability p^ 2 P
^ i † ˆ g…Ai † 8i.
such that p…A
^
^
3. 8A; B U : A \ B ˆ ;, 9p^ 2 P such that p…A†
ˆ g…A† and p…B†
ˆ f …B†.
And, secondly, on Proposition 10.
Theorem 19. Let C be a convex polytope given by the set of extreme points V.
Then, the probability envelope defined from this convex polytope is a capacity of
order-2 if and only if
p1 ÿ p2 ˆ …0; 0; . . . ; e; 0; . . . ; ÿe; 0; . . . ; 0†
for all pairs of vertices of V, p1 and p2 , joined by an edge.
Proof. Let …f ; g† be the probability envelope de®ned by C,
f …A† ˆ Inf p…A†;
p2C
g…A† ˆ Sup p…A†:
p2C
Let us see the direct implication. Assume that …f ; g† is a capacity of order-2. If
p1 and p2 are two extreme points of C joined by an edge, by Corollary 17, the
equivalence classes ‰p1 Š and ‰p2 Š are related on the network. So, from Proposition 10, we have
p1 ÿ p2 ˆ …0; 0; . . . ; e; 0; . . . ; ÿe; 0; . . . ; 0†
as we want to prove.
Inversely, if we consider the probability envelope …f ; g†. We have to prove
that …f ; g† is a capacity of order-2. Let P be the convex polytope de®ned by this
probability envelope
P ˆ fp 2 P …U †=f …A† 6 p…A† 6 g…A† 8A U g:
In order to show that …f ; g† is a capacity of order-2, we must only show that
^
8A; B : A \ B ˆ ; there exists p^ 2 P, such that p^…A† ˆ g…A† and p…B†
ˆ f …B†.
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J.F. Verdegay-L
opez, S. Moral / Information Sciences 125 (2000) 187±206
Let A and B be two subsets of U such that A \ B ˆ ;. We know that
g…A† ˆ Supp2C p…A†, therefore, there is q 2 C so that q…A† ˆ g…A†.
We denote by Vq the set of adjacent vertices to q into V. There are two cases:
1. 8p 2 Vq , p…B† P q…B† is veri®ed.
In this case there are no points, p 2 Vq such that p…B† < q…B†. According to
the optimality criterion of simplex algorithm [15], q…B† is minimum. So, q is a
probability verifying that q…A† ˆ g…A† and q…B† ˆ f …B†.
2. There is p 2 Vq such that p…B† < q…B†.
Since p is adjacent to q, from hypothesis, we have
q ÿ p ˆ …0; 0; . . . ; e; 0; . . . ; ÿe; 0; . . . ; 0†;
we can assume, without losing generality, that e > 0.
Suppose that e and ÿe are in the components i and j, respectively. Since
q…B† ÿ p…B† > 0 then q…B† ÿ p…B† ˆ e, so, ui 2 B and uj 62 B. As A \ B ˆ ; and
ui 2 B, then ui 62 A.
On the other hand, we know that q…A† is the supremum over C. We have
p…A† 6 q…A† because p 2 C, i.e., p…A† ÿ q…A† P 0.
But,
q ÿ p ˆ …0; 0; . . . ; e; 0; . . . ; ÿe; 0; . . . ; 0†
with e in the component i and ui 62 A, therefore, the only possibilities is
q…A† ÿ p…A† ˆ ÿe
or q…A† ÿ p…A† ˆ 0:
But we have supposed that e > 0 and q…A† ÿ p…A† P 0, then it is not possible
that q…A† ÿ p…A† ˆ ÿe. So, q…A† ÿ p…A† ˆ 0, hence uj 62 A. Thus,
p…A† ˆ q…A† ˆ g…A†;
i.e., the only possibility is p…A† ˆ q…A† ˆ g…A†, therefore p is still maximum for
A.
If we move through the probabilities maintaining the maximum value for A
we will reach the moment when the probability is maximum for A and minimum for B. Then, we will consider this probability as p^.
^
Therefore, we have found a probability p^ 2 C such that p…A†
ˆ g…A† and
p^…B† ˆ f …B†.
^
^
Because C P, the probability p^ 2 P, p…A†
ˆ g…A† and p…B†
ˆ f …B†,
therefore, …f ; g† is a capacity of order-2. 4.2. Decomposition of capacities
A new application of the network is the decomposition of a capacity of
order-2 into a strict convex combination of capacities using the network of
probabilities.
J.F. Verdegay-L
opez, S. Moral / Information Sciences 125 (2000) 187±206
203
According to Theorem 15, if g is a strict convex combination of g1 and g2 ,
the extreme probabilities of the convex polytope associated with g are strict
convex combinations of extreme probabilities of the convex polytopes associated with g1 and g2 . This relationship is also obtained on the networks of
probabilities associated with the three capacities.
Corollary 20. If the capacity of order-2 g is a strict convex combination of the
capacities g1 and g2 , then
g
g1
r  l () r  l
and
g2
r  l:
…13†
From Theorem 15, this corollary is straightforward.
Example 21. Let g be an upper capacity of order-2 de®ned on U ˆ fu1 ; u2 ; u3 g
given by
g…fu1 g† ˆ 0:8;
g…A† ˆ 1
8A U ;
A 6ˆ fu1 g;
the associated network can be seen in Fig. 2.
This network could be decomposed into the networks represented in Fig. 3,
networks which are associated with capacities of order-2, g1 and g2 , given by
Fig. 2. Network associated with the capacity g.
204
J.F. Verdegay-L
opez, S. Moral / Information Sciences 125 (2000) 187±206
Fig. 3. Networks associated with the capacities g1 and g2 .
g1 …fu1 g† ˆ 0;
g1 …A† ˆ 1
8A U ;
A 6ˆ fu1 g
and
g2 …A† ˆ 1
8A U :
We can easily verify that
g ˆ 0:2g1 ‡ 0:8g2 :
5. Conclusions
The network representation enables us for studying a convex polytope de®ned by a capacity of order-2 by means of a graphical representation. Thus,
some problems with dicult understanding on Rn becomes easier on the network. In Example 6, we have a capacity de®ned on R4 , therefore, we could not
see its convex polytope, however, we can obtain an equivalent representation,
the network, which can be already observed.
Also, the network permits us to approach the study of capacities from a
geometric point of view instead of the measure one. This feature causes that we
could reinterpret geometric properties as measure properties and conversely.
On the other hand, this representation provides a better understanding of
order-2 capacities and can be the basis to obtain results about how to calculate
a convex decomposition of a capacity of order-2. Using the properties proved
in this paper, algorithms on the network may be constructed, in the future, for
solving problems referred to capacities as the problem of maximizing functions
J.F. Verdegay-L
opez, S. Moral / Information Sciences 125 (2000) 187±206
205
over the convex polytope associated with the capacity, combination and
marginalization of a capacity, and so on. Now, we are trying to characterize
the networks of probabilities that are associated to plausibility and possibility
measures. We also plan to construct an algorithm that in the case of a network
associated to a plausibility measure obtains its associated basic probability
assignment directly from the network by decomposing it into a convex combination of elementary subnetworks.
These results can be applied to several, well-known ways of representing the
uncertain knowledge such as the evidence theory [8,14,16], the possibility
theory [9,18], etc.
Acknowledgements
This work is supported by DGICYT in the project PB95-1181.
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