Possibility Theory

8 UNCERTAINTY MODELING

8.2 Possibility Theory

8.2. 1 Fuzzy Sets and Possibility Distributions

Possibility theory focuses primarily on imprecision, which is intrinsic in natural languages and is assumed to be "possibilistic" rather than probabilistic. There- fore the term variable is very often used in a more linguistic sense than in a strictly mathematical one. This is one reason why the terminology and the sym- bolism of possibility theory differ in some respects from those of fuzzy set theory.

In order to facilitate the study of possibility theory, we will therefore use the common possibilistic terminology but will always show the correspondence to fuzzy set theory.

Suppose, for instance, we want to consider the proposition "X is

i:

where X is the name of an object, a variable, or a proposition. For instance, in "X is a small integer," X is the name of a variable. In "John is young," John is the name of an object.fr(i.e., "small integer" or "young") is a fuzzy set characterized by its mem- bership function I1F'

One of the central concepts of possibility theory is that of a possibility distri-

bution (as opposed to a probability distribution). In order to define a possibility distribution, it is convenient first to introduce the notion of a fuzzy restriction. To visualize a fuzzy restriction, the reader should imagine an elastic suitcase that acts on the possible volume of its contents as a constraint. For a hardcover suit- case, the volume is a crisp number. For a soft valise, the volume of its contents depends to a certain degree on the strength that is used to stretch it. The variable in this case would be the volume of the valise; the values this variable can assume may be u E U,and the degree to which the variable (X) can assume different values of u is expressed by/lieU). Zadeh [Zadeh et al. 1975, p. 2; Zadeh 1978, p. 5] defines these relationships as follows.

Definition 8-2

Let

F

be a fuzzy set of the universe Ucharacterized by a membership function /l ieU).

F

is ajuzzy restriction on the variable X if

F

acts as an elastic constraint on the values that may be assigned to X, in the sense that the assignment of the values u to X has the form

X= u: /lp(u)

/li(U)is the degree to which the constraint represented by

F

is satisfied when u is assigned to X. Equivalently, this implies that 1 - /li(U)is the degree to which the constraint has to be stretched in order to allow the assignment of the values uto the variable X.

Whether a fuzzy set can be considered as a fuzzy restriction or not obviously depends on its interpretation: This is only the case if it acts as a constraint on the values of a variable, which might take the form of a linguistic term or a classical variable.

Letj?(X)b~a fuzzy restriction associated with X, as defined in definition 8-1.

Then R(X)=Fis called a relational assignment equation, which assigns the fuzzy set F to the fuzzy restriction R(X).

Let us now assume thatA(X) is an implied attribute of the variable X-for instance,A(X)= "age of Jack," and

F

is the fuzzy set "young." The proposition

"Jack is young" (or better "the age of Jack is young") can then be expressed as R(A(X» =

F

Example 8-1 [Zadeh 1978, p. 5]

Let p be the proposition "John is young," in which "young" is a fuzzy set of the universe U = [0, 100] characterized by the membership function

~young(u)= Stu;20, 30, 40) where u is the numerical age and the S-function is defined by

1

^{for u<a}

(u-a y _{for a~u~~}

1-2--

Siu;a,~,

y)

=

y-a

2(~y _y-a

^{for ~<u~y}

0 for

u>y

In this case, the implied attributeA(X) is Age (John), and the translation of "John is young" has the form

John is young~R(Age(John»=young

Zadeh [1978] related the concept of a fuzzy restriction to that of a possibility distribution as follows:

Consider a numerical age, sayu

=

28, whose grade of membership in the fuzzy set

"young" is approximately 0.7. First we interpret 0.7 as the degree of compatibility of 28 with the concept labelled young. Then we postulate that the proposition "John is young" converts the meaning of 0.7 from the degree of compatibility of 28 with young to the degree of possibility that John is 28 given the proposition "John is young." In short, the compatibility of a value of u with young becomes converted into the possi- bility of that value ofugiven "John is young" [Zadeh 1978, p. 6].

The concept of a possibility distribution can now be defined as follows:

Definition8-3[Zadeh 1978, p. 6]

Let

i

be a fuzzy set in a universe of discourseUthat is characterized by its mem- bership function~F(U), which is interpreted as the compatibility ofu E Uwith the concept labeled

F.

Let X be a variable taking values in U,and let

F

act as a fuzzy restriction, R(X) , associated with X. Then the proposition "X is

i ,"

which translates into R(X) =

i ,_

associates apossibility distribution,1t_{x ,} with X that is postulated to be equal toR(X) .

The possibility distribution function, 1tiu), characterizing the possibility distribution 1t_x is defined to be numerically equal to the membership function

~F(U)of

i;

that is,

The symbol ~ will always stand for "denotes" or "is defined to be." In order to stay in line with the common symbol of possibility theory, we will denote a possibility distribution with 1t_x rather than with

it",

even though it is a fuzzy set.

Example 8-2[Zadeh 1978, p. 7]

Let Ube the universe of positive integers, and let

F

be the fuzzy set of small integers defined by

F

={(I, 1),(2,1),(3,.8), (4, .6), (5,.4),(6, .2)}

Then the proposition "X is a small integer" associates with X the possibility distribution

1t_x= F

in which a term such as (3, .8) signifies that the possibility that X is 3, given that X is small integer, is .8.

Even though definition 8-3 does not assert that our intuition of what we mean by possibility agrees with the min-max fuzzy set theory, it might help to realize their common origin. It might also make more obvious the difference between possibility distribution and probability distribution.

Zadeh [1978, p. 8] illustrates this difference by a simple but impressive example.

Example 8-3

Consider the statement "Hans ate X eggs for breakfast," X= {I, 2, . . .}. A pos- sibility distribution as well as a probability distribution may be associated with X. The possibility distribution1txCu) can be interpreted as the degree of ease with which Hans can eat u eggs while the probability distribution might have been determined by observing Hans at breakfast for 100 days . The values of1txCu) and PxCu) might be as shown in the following table :

u

PxCu)

1

.1

2

.8

3

.1

4 1

o

5 .8

o

6 .6

o

7 .4

o

8 .2

o

We observe that a high degree of possibility does not imply a high degree of prob- ability. If,however, an event is not possible, it is also improbable. Thus, in a way, the possibility is an upper bound for the probability. A more detailed discussion of this "possibility/probability consistency principle" can be found in Zadeh [1978].

This principle is not intended as a crisp principle, from which exact probabil- ities or possibilities can be computed, but rather as a heuristic principle, express- ing the principle relationship between possibilities and probabilities.

8.2.2 Possibility and Necessity Measures

Inchapter 4, a possibility measure was already defined (definition 4-2) for the case in whichA is a crisp set.If

A

is a fuzzy set, a more general definition of a possibility measure has to be given [Zadeh 1978, p. 9].

Definition 8-4

Let

A

be a fuzzy set in the universe U,and let 7t_x be a possibility distribution associated with a variable X that takes values inU.Thepossibility measure,7tiA), of

A

is then defined by

poss{X isA}~7t(A)

~sup min{Il A(u),7tx(u)}

ue U

Example8-4 [Zadeh 1978]

Let us consider the possibility distribution induced by the proposition "X is a small integer" (see example8-2):

n , ={(l,1),(2, 1), (3, .8), (4, .6), (5, .4), (6, .2)}

and the crisp setA= {3, 4, 5}.

The possibility measure7t(A)is then

7t(A)= max (.8, .6,.4) =.8

If

A,

on the other hand, is assumed to be the fuzzy set "integers which are not small," defined as

A

={(3, .2),(4, .4), (5, .6), (6, .8), (7,1) , . . .}

then the possibility measure of "X is not a small integer" is poss(X is not a small integer)

=

^{max{.2,A, A,.2}}

=

^A

Similar to probability theory, conditional possibilities also exist. Such a con- ditional possibility distribution can be defined as follows [Zadeh 1981b, p. 81].

Definition 8-5

Let X and Y be variables in the universes U and V, respectively. The conditional possibility distributionof X given Y is then induced by a proposition of the form

"If X is

F,

^{then Yis}

C;"

and is denoted by1t(YIxlv/u).

Proposition 8-1

Let 1t(YIX) be the conditional possibility distribution functions of X and Y, respectively. The joint possibility distribution function of X and Y,1t(X,I'), is then given by

1t(X,Y)(u,v)= min{1t x(u) , 1t(y/X) (v/u)}

Not quite settled yet seems to be the question of how to derive the conditional possibility distribution functions from the joint possibility distribution function.

Different views on this question are presented by Zadeh [1981b, p. 82], Hisdal [1978], and Nguyen [1978].

Fuzzy measures as defined in definition 4-2 express the degree to which a certain subset of a universe,Q ,or an event is possible. Hence, we have

g(O)

=

0 and g(Q)

=

1 As a consequence of condition 2 of definition 4-2, that is,

A~B==>g(A)S;g(B) we have

g(A U B)~max(g(A), g(B» and

g(A

n

B)~min(g(A), g(B» for A, B

c

^Q

Possibility measures (definition 4-2) are defined for the limiting cases:

1t(AUB)=max(1t(A),1t(B) 1t(A

n

B)=min(1t(A),1t(B)

(8.1) (8.2)

(8.3) (804)

Table 8-2. Possibility functions.

Grade

Student A B C D E

1 .8 1 .7 0 0

2 1 .8 .6 .1 0

3 .6 .7 .9 .1 0

4 0 .8 .9 .5 0

5 0 0 .3 1 .2

6 .3 1 .3 0 0

If etAis the complement ofAin

n,

^then

n(A U etA)= max(n(A), n(etA)) = 1 (8.5) which expresses the fact that eitherA oretAis completely possible.

In possibility theory, an additional measure is defined that uses the conjunc- tive relationship and, in a sense, is dual to the possibility measure:

N(A

n

B)

=

min(N(A), N(B» (8.6)

Nis called then necessity measure .N(A) = 1 indicates thatA is necessarily true (A is sure). The dual relationship of possibility and necessity requires that

n(A) = 1 - N(etA) ;VA k

n

Necessity measures satisfy the condition

min(N(A), N(etA)) = 0

(8.7)

(8.8) The relationships between possibility measures and necessity measures satisfy the following conditions [Dobois and Prade 1988, p. 10]:

n(A)~N(A), VAk

n

N(A)>0=>n(A)= 1 n(A)<1=>N(A)=0 Here

n

is always assumed to be finite.

(8.9)

(8.10)

Example 8-5

Let us assume that we know, from past experience, the performance of six stu- dents in written examinations. Table 8-1 exhibits the possibility functions for the grades A through E and students 1 through 6.

First we observe that the membership function for the grades of student 4 is not a possibility function, sinceg(Q) =1=1.

We can now ask different questions:

1. How reliable is the statement of student 1 that he will obtain a B in his next exam?

In this case,"A" is {B} and"etA" is {A ,C,D, E} . Hence,1t(A)= 1

N(A)= min {l-1t;}

=min{.2, .3, 1,1}=.2.

Hence, the possibility of student 1 getting a B is 1t= 1, the necessityN=0.2.

2. If we want to know the truth of the statement "Either student 1 or 2 will achieve an A or a B," ourQ has to be defined differently. It now contains the elements of the first two rows. The result would be

1t(A)=1t(student 1 A or B or Student 2 A or B)=1 N(A)=.3

3. Let us finally determine the credibility of the statement "student 1 will get a

c."

In this case

1t(A)= .7 N(A) =0.