Download - Fuzzy Sets as a Basis for a Theory of Possibility-1978

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Fu z z y Se t s a nd Sys t e m s 1 ( 1978) 3 - 28 .

© N o r t h - H o l l a n d P u b l i sh i n g C o m p a n y

F U Z Z Y S E T S A S A B A S I S F O R

A T H E O R Y O F P O S S I B I L I T Y

L .A . Z A D E H

Computer Science Division Department of Electrical Engineering and Co mpu ter Sciences and the

Electronics Research Laboratory University of California Berkeley C A 94720 U.S.A.

Re c e ive d Fe br ua r y 1977

Revi,;ed June 1077

T he th e or y o f poss ib i l i t y de sc ribe d i n t h i s pa pe r i s r e l at e d t o t he t he o r y o f f uzz y se t s by de f in ing t he

c onc e p t o f a poss ib i l i t y d i s t r ibu t ion a s a f uz zy r e s t ri c t i on w hic h a c t s a s a n e l a st ic c ons t r a in t on t he

va lues tha t m ay be as s igned to a var iable . M or e spec i f ica l ly , i f F i s a fuzzy subse t of a universe of

d i sc our se U = {u} w hic h i s c ha r a c t e r iz e d by i t s m e m be r sh ip f unc t ion /~ r , t he n a p r opos i t i on o f t he

form "X is F ," where X is a var iable taking va lues in U, induces a poss ibi l i ty dis t r ibut ion Hx which

e qua t e s t he poss ib i l it y o f X t a k ing t he va lue u t o /~ r . ( u ) - - t he c om pa t ib i l i t y o f u w i th F . I n t h i s w a y , X

be c om e s a f uzz y va r i a b l e w hic h i s a s soc i a t e d w i th t he poss ib i l it y d i s t r i bu t ion F ix in m uc h the sa m e

w a y a s a r a ndo m va r i a b l e i s a s soc i a t ed w i th a p r ob a b i l i t y d i s t r i bu t ion . I n ge ne ra l , a va r i a b l e m a y be

a ssoc i a t e d bo th w i th a poss ib i l i t y d i s t r i bu t ion a nd a p r oba b i l i t y d i s t r i bu t ion , w i th t he w e a k

c onne c t ion be tw e e n the tw o e xpr e s se d a s t he poss ib i l i t y /p r oba b i l i t y c ons i s t e nc y p r inc ip l e .

A thes is advanced in this paper i s tha t the imprec is ion tha t i s in t r ins ic in na tura l languages i s , in

the m a in , poss ib i li s t ic r a the r t ha n p r oba b i l i s t ic i n na tu re . T hus , by e m plo y ing the c onc e p t o f a

poss ib i l i t y d i s t r i bu t ion , a p r opos i t i on , p , i n a na tu r a l l a ngua ge m a y be t r a ns l a t e d i n to a p r oc e dur e

w hic h c om pu te s t he p r ob a b i l i t y d i s t r i bu t ion o f a s e t o f a t t r i bu t e s w h ic h a r e im pl i e d by p . Se ve r a l t ype s

of c ond i t i ona l t r a ns l a t i o n r u le s a r e d isc usse d a nd , i n pa r ti c u l a r , a t r a ns l a t i on r u le r ,~ r p r opo s i t i ons o f

the form "X is F is ~-po ssible, "~whe re ~ i s a num be r in the inte rva l [0 , ] , i s formu la te~ and i l lus t ra ted by

examples .

1 . In troduct ion

T h e p i o n e e ri n g w o r k o f W i e n e r a n d S h a n n o n o n t h e s t a ti st ic a l t h e o r y o f

c o m m u n i c a t i o n h a s l e d t o a u n i v e r s a l a c c e p t a n c e o f t h e b e l ie f t h a t i n f o r m a t i o n i s

i n t r i n s i c a l l y s t a t i s t i c a l i n n a t u r e ~ n d , a s s u c h , m u s t b e d e a l t w i t h b y t h e m e t h o d s

p r o v i d e d b y p r o b a b i l i t y t h e o r y .

U n q u e s t i o n a b l y , t h e s t a ti s ti c a l p o i n t o f v i e w h a s c o n t r i b u t e d d e e p i n s ig h t s i n t o t h e

f u n d a m e n t a l p r o c e s s e s i n v o l v e d i n t h e c o d i n g , t r a n s m i s s i o n a n d r e c ep t io n o f d a t a , a n d

p l a y e d a k e y r o le in th e d e v e l o p m e n t o f m o d e r n c o m m u n i c a t i o n , d e t e c t io n a n d

t e le m e t e r i n g s y s t e m s . I n r e c en t y e a r s, h o w e v e r , a n u m b e r o f o th e r i m p o r t a n t

a p p l i c a t i o n s h a v e c o m e t o t h e f o r e i n w h i c h t h e m a j o r i s s u e s c e n t e r n o t o n t h e

T o P r o f e s s o r A r n o l d K a u f m a n n .

* R e s e ar c h s u p p o r t e d b y N a v a l E l e c tr o n ic S y s te m s C o m m a n d C o n t r a c t N 0 0 0 3 9 7 7 - C - 0 0 2 2 , U . S . A r m y

R e s e a rc h O f fi ce C o n t r a c t D A H C 0 4 - - 7 5- -G 0 0 5 6 a n d N a t i o n a l S c i en c e F o u n d a t i o n G r a n t M C S 7 6 - 0 6 6 9 3 .

3



4 L A Zadeh

t r a n s m i s s i o n o f i n f o r m a t i o n b u t o n i ts m e a n i n g . I n s u ch a p p l i c a ti o n s , w h a t m a t t e r s

i s t h e a b i l i t y t o a n sw e r q u e s t i o n s r e l a t i n g to i n f o r ma t io n th a t i s s t o r e d in a

d a t a b a s e - - - a s i n n a t u r a l l a n g u a g e p r o c e s s i n g , k n o w l e d g e r e p r e s e n t a t i o n , s p e e c h

r e c o g n i t i o n , r o b o t i c s , me d ic a l d i a g n o s i s , a n a ly s i s o f r a r e e v e n t s , d e c i s io n - ma k in g u n d e r

u n c e r t a in ty , p i c tu r e a n a ly s i s , i n f o r ma t io n r e t r i e v a l a n d r e l a t e d a r e a s .

A th e s is a d v a n c e d i n t h is p a p e r is t h a t w h e n o u r m a i n c o n c e r n is w it h t h e m e a n i n g o f

i n f o r m a t i o n - - r a t h e r t h a n w i t h i t s m e a s u r e - - t h e p r o p e r f r a m e w o r k f o r i n f o r m a t i o n

a n a ly s i s i s p o s s ib il i st i c t r a th e r t h a n p r o b a b i l i s t i c i n n a tu r e , t h u s imp ly in g th a t w h a t i s

n e e d e d f o r s u c h a n a n a l y s i s i s n o t p r o b a b i l i t y t h e o r y b u t a n a n a l o g o u s - a n d v e t

d i f f e r e n t - - t h e o r y w h i c h m i g h t b e c a ll e d th e t h eo r y o f p o s s i b i li t y 2

A s w i ll b e s e e n in t h e s e q u e l , t h e m a th e m a t i c a l a p p a r a tu s o f t h e t h e o r y o f f u z z y s e ts

p r o v id e s a n a tu r a l b a s i s f o r t h e t h e o r y o f p o s sib i li t y , p l a y in g a r o l e w h ic h i.s s imi l a r t o

th a t o f me a su r e t h e o r y i n r e la t i o n to t h e t h e o r y o f p r o b a b i l i t y . V ie w e d in t h i s

pe r spec t ive , a fuzzy re s t r ic t ion m ay be in te rpre ted a s a poss ib il f i y d is t r ibu t ion , wi th i t s

m e m b e r s h ip f u n c t io n p l a y in g th e r o le o f a p o s s ib i li t y d i s t r i b u t io n f u n c t io n , a n d a f u z z y

v a r i a b l e i s a s so c i a t e d w i th a p o s s ib i l i t y d i s t r i b u t io n in mu c h th e s a me ma n n e r a s a

r a n d o m v a r i a b l e i s a s so c i a t e d w i th a p r o b a b i l i t y d i s t r i b u t io n . I n g e n e r a l , h o w e v e r , a

v a r i a b l e ma y b e a s so c i a t e d b o th w i th a p o s s ib i l i t y d i s t r i b u t io n a n d a p r o b a b i l i t y

d i s t r i b u t io n , w i th t h e c o n n e c t io n b e tw e e n th e tw o e x p r e s s ib l e a s t i a e

poss ib i l i t y~probab i l i t y cons i s tency pr inc ip le

T h i s p r i n c i p l e - - w h i c h is a n e x p r e s si o n o f

a w e a k c o n n e c t io n b e tw e e n p o s s ib i l i t y a n d p r o b a b i l i t y - - w i l l b e d e sc r ib e d in g r e a t e r

de ta i l a t a la te r po in t in th i s pape r .

T h e i m p o r t a n c e o f t h e t h e o r y o f p o s s ib i li ty s t e m s fr o m t h e f a c t t h a t - - c o n t r a r y t o

w h a t h a s b e c o m e a w i d e l y a c c e p te d a s s u m p t i o n - - m u c h o f t h e i n fo r m a t i o n o n w h i c h

h u ma n d e c i s io n s a r e b a se d i s p o s s ib i l i s t i c r a th e r t h a n p r o b a b i l i s t i c i n n a tu r e . I n

p a r t i c u l a r , t h e i n t r in s i c f u z zin e ss o f n a tu r a l l a n g u a g e s - - w h ic h i s a lo g i c a l c o n s e q u e n c e

o f t h e n e c es s it y t o e x p r e s s i n f o r m a t io n in a su m ma r i z e d f o r m - - i s , i n t h e ma in ,

poss ib i l i s t ic in o r ig in . Based on th is p remise , J t i s poss ib le to cons t ruc t a un ive r sa l

l a n g u a g e a i n w h ic h th e t r a n s l a t i o n o f a p r o p o s i t i o n e x p r e s se d in a n a tu r a l l a n g u a g e

ta k e s t h e f o r m o f a p r o c e d u r e f o r c o m p u t in g th e p o s s ib il i ty d i s t r i b u t io n o f a s e t o f f u z z y

r e l a t i o n s i n a d a t a b a se . T h i s p r o c e d u r e , t h e n , m a y b e i n t e r p r e t e d a s th e me a n in g o f t h e

p r o p o s i t i o n i n q u e s t i o n , w i t h t h e c o m p u t e d p o s s ib i li ty d i s t r i b u t i o n p l a y in g t h e r o l e o f

t h e i n f o r m a t i o n w h i c h is c o n v e y e d b y t h e p r o p o s i t io n .

T h e p r e s e n t p a p e r h a s t h e l im i t e d o b j e ct i ve o f e x p l o r i n g s o m e o f t h e e l e m e n t a r y

p r o p e r t i e s o f t h e c o n c e p t o f a p o s s ib il i t y d i s t r i b u t io n , m o t iv a t e d p r in c ip a l ly b y th e

a p p l i c a t io n o f t h is c o n c e p t t o t h e r e p r e s e n t a t i o n o f m e a n i n g i n n a t u r a l l a n g u a g es . S i n c e

o u r i n tu i t i o n c o n c e r n in g th e p r o p e r t i e s o f p o s sib i li t y d i s t r i b u t io n s i s n o t a s y e t w e l l

d e v e lo p e d , so me o f t h e d e f in i ti o n s w h ic h a r e f o r mu la t e d i n t h e s e q u e l sh o u ld b e v i e w e d

a s p r o v i s io n a l i n n a tu r e .

~The term possib i l i s t ic-- in the sense used here--was coined by Gaines and Kohout in thei r paper on

poss ib l e au tom ata [1 ] .

2The in terpre ta t ion of the concept of possib i li ty in the theory of possib i l ity i s qui te d ifferent from th at of

mo dal logic [2] in which propo si t ions of the form "It i s possib le tha t . . . " and " I t is necessary tha t . . . " are

considered .

3Such a language, called PRUF (Possibil ist ic Relational Universal Fuzzy,), is described in [30].



F u z z y s e t s a s a b a s i s f o r a t h eo r y o f p o s s i b i l i t y

2 Th e concep t o f a pos s i b i l it y d i s t r ibut i on

W h a t i s a p o s s i b i li t y d i s t r i b u t i o n ? I t is c o n v e n i e n t t o a n s w e r t h is q u e s t i o n i n t e r m s o f

a n o t h e r c o n c e p t , n a m e l y , t h a t o f a fu z zy r es t r i c t ion [4 , 5 ], t o wh i c h t h e c o n c e p t o f a

p o s s i b i l i t y d i s t r i b u t i c a b e a r s a c l o s e r e l a t i o n .

L e t X b e a v a r ia l '~ e wh i c h t a k e s v a l u e s i n a u n i v e r s e o f d i s c o u r s e U , w i t h t h e g e n e r i c

e l e m e n t o f U d e n o t e d b y u a n d

X = u (2.1)

s i g n i fy i n g t h a t X i s a s s i g n e d t h e v a l u e u , u ~ U.

L e t F b e a f u z z y s u b s e t o f U w h i c h i s c h a r a c t e r i z e d b y a m e m b e r s h i p f u n c t i o n / i v .

T h e n F is

afi4zzy re str ict ion on X

(o r

associated w ith X )

i f F a c t s a s a n e l a st ic c o n s t r a i n t

o n t h e v a l u e s t h a t m a y b e a s si g n ed t o X - - i n t h e s en s e t h a t t h e a s s i g n m e n t o f a v a l u e u t o

X h a s t h e fo rm

X =u: / t v u )

(2.2)

w h e r e / ~ r ( u ) i s i n t e r p r e t e d a s t h e d e g r e e t o w h i c h t h e c o n s t r a i n t r e p r e s e n t e d b y F i s

s a t is f ie d wh e n u i s a s s i g n e d t o X . Eq u i v a l e n t l y , ( 2.2 ) im p l i e s t h a t 1 - l a y (u ) i s t h e d e g re e

t o w h i c h th e c o n s t r a i n t i n q u e s t i o n m u s t b e s t r e t c h e d in o r d e r t o a l lo w t h e a s s i g n m e n t

o f u t o X . *

L e t R (X) d e n o t e a fu z z y r e s t r i c t io n a s s o c i a t e d w i t h X . Th e n , t o e x p re s s t h a t F p l a y s

t h e ro l e o f a fu z z y r e s t r ic t i o n i n r e l a t i o n t o X , we w r i te

R X ) = F . t2 .3)

An e q u a t i o n o f t h i s fo rm i s c a l l e d a

relational assignm ent equation

b e c a u s e i t r e p re s e n t s

t h e a s s i g n m e n t o f a fu z z y s e t (o r a fu z z y r e l a t i o n ) t o t h e r e s t r i c t i o n a s s o c i a t e d w i t h X .

T o i l lu s t r a t e t h e c o n c e p t o f a fu z z y r e s t r i c t io n , c o n s i d e r a p ro p o s i t i o n o f t h e fo rm p

a _X i s F , 5 wh e re X i s t h e n a m e o f a n o b j e c t , a v a r i a b l e o r a p ro p o s i t i o n , a n d F i s t h e

n a m e o f a fu z z y s u b s e t o f U , a s in " J e s s i e is v e ry i a t e l l ig e n t , " "X is a s m a l l n u m b e r , "

" H a r r i e t is b l o n d e i s q u i t e t r u e ," e tc . A s s h o w n in [ 4 ] a n d [ 6 ] , t h e t r a n s l a t i o n o f s u c h a

p r o p o s i t i o n m a y b e e x p r e s se d a s

R A X ) ) = F

(2.4)

wh e re A (X) is a n i m p l i e d a t t r i b u t e o f X wh i c h t a k e s v a l u e s i n U , a n d (2.4 ) s ig n i f ie s t h a t

t h e p r o p o s i t i o n

pa--X

i s F h a s t h e e f f e ct o f a s s i g n i n g F t o t h e fu z z y r e s t r i c t io n o n t h e

va lue s o f A (X) .

As a s i m p l e e x a m p l e o f (2.4 ), l e t p b e t h e p ro p o s i t i o n " J o h n i s y o u n g , " i n w h i c h y o u n g

is a fu z z y s u b s e t o f U = [ 0 , 1 00 ] c h a r a c t e r i z e d b y t h e m e m b e l s h i p f u n c t i o n

#you,g(U) = 1 -

S u;

20, 30, 40)

t 2 . 5 )

'~ A p o im th a t mu s t b e s t r e s s e d i s t h a t e f u z z y se t per s~ i s no t a fuzzy re s t r ic t i on . To be a fuzzy re s t r ic t ion , i t

m u s t b e a c t i n g a s a c o n s t r a i n t o n t h e v a l u e s o f a v a r i ab l e .

SThe sym bo l a__. s ta nd s for "d eno tes" or " i s de f in ed to be" .



6 L A Zadeh

w h e r e u is t h e n u m e r i c a l a g e a n d th e S - f u n c t io n i s d e f in e d b y [ 4 ] .

S ( u ; ~ , f l, 7 ) = O f or u < ~

= 2 fo r 0 c < u < f l

\ 7 - 0 U

= l _ 2 ( u - - ' ~ 2

\ ~ - ~ / f o r f l < = u < ~

= 1 f o r u > ) , ,

(2.6)

in which the pa ram ete r fl -~ ( a + ) ,) /2 is the c ro ssov er po in t , th a t is , S ( fl ;~ , f l , ' t )= 0 .5 . In

th i s c a se, t h e imp l i e d a t t r i b u t e A ( X ) is A g e ( Jo h n ) a n d th e t r a n s l a t i o n o f " Jo h n is y o u n g "

a s s u m e s t h e f o r m :

J o h n is y o u n g - , R ( A g e (J o h n )) = y o u n g .

(2.7)

T o r e l a t e th e c o n c e p t o f a fu z z y r e s t r ic t i o n to t h a t o f a p o s s ib i l i ty d i s t r i b u t io n , w e

in t e r p r e t t h e r i g h t - h a n d m e m b e r o f 1 2 .7 ) i n th e f o l lo w in g m a n n e r .

Co n s id e r a n u m e r i c a l a g e , s a y u = 28 , w h o se g r a d e o f m e m b e r s h ip i n t h e fu z z y s et

y o u n g ( as d e fin e d b y ( 2 . 5) ) i s a p p r o x im a te ly 0 .7 . F i rs t , w e in t e r p r e t 0 .7 a s t h e d e g r e e o f

co mp a t ib i l i t y o f 2 8 w i th t h e c o n c e p t l a b e l e d y o u n g . T h e n , w e p o s tu l a t e t h a t t h e

p r o p o s i t i o n " J o h n is y o u n g " c o n v e r t s t h e m e a n i n g o f 0 .7 f r o m t h e d e g r ee o f

c o m p a t ib i l i t y o f 2 8 w i th y o u n g to t h e d e g r e e o f p o s s ib il i ty t h a t Jo h n i s 2 8 g iv e n th e

p r o p o s i t i o n " J o h n is y o u n g . " I n s h o r t , t h e c o m p a t i b i li t y o f a v a l u e o f u w i th y o u n g

b e c o m e s c o n v e r t e d i n to t h e p o s s ib i li t y o f t h a t v a lu e o f u g iv e n " Jo h n i s y o u n g . "

S ta t e d i n mo r e g e n e r a l t e r ms , t h e c o n c e p t o f a p o s s ib i l it y d i s t r i b u t io n m a y b e d e fin e d

a s f ol lo w s . ( F o r s imp l i c it y , w e a s su me th a t A ( X ) =X . )

ef init ion

2 .1 . Le t F be a fuzzy subs e t o f a un ive r se o f d isco urse U wh ich i s

c h a r a c t e r iz e d b y i ts m e m b e r s h i p f u n c t io n # r , w i th t h e g r a d e o f m e m b e r s h i p , p r ( u ) ,

i n t e r p r e t e d a s t h e c o m p a t ib i l i t y o f u w i th t h e c o n c e p t l a b e l e d F .

Le t X be a va r iab le tak ing va lues in U, and le t F ac t a s a fuzzy re s t r ic t ion , R(X) ,

a s so c i a t e d w i th X . T h e n th e p r o p o s i t i o n " X i s F , " w h ic h t r a n s l a t e s i n to

R ( X ) = F ,

(2.8)

assoc ia te s a

poss ib i l i t y d i s t r ibu t ion ,

l-Ix,

w i th X

w h ic h i s p o s tu l a t e d t o b e e q u a l t o R ( X ),

i.e.,

l-lv = R(X ). (2.9)

C o r r e s p o n d i n g l y , t h e

p o s s ib i l it y d i s t r ib u t i o n fu n c t i o n a s s o c ia ted w i th X

( o r t h e

p o s s ib i l i t y d i s t r i b u t io n f u n c t io n o f F ix ) i s d e n o te d b y n x a n d i s d e f in e d to b e

n u m e r i c a l ly e q u a l t o t h e m e m b e r sh ip f u n c t io n o f F , i.e .,

A

r r x= v . (2 .10)

T h u s , nx(U), t h e p o s s ib i l i t y t h a t X =u , i s p o s tu l a t e d t o b e e q u a l t o /~ r (u ) .



F u z z , s e t s a s a b a s i s f o r a t h e o r y O [ p o s s i b i l i ty 7

In v iew of (2.9), the re la t iona l ass ig nm ent equ at ion (2.8) ma y be expressed

equ i va l en t l y i n the fo rm

H x = F . (2.11)

_A_X

lacing in ev idence tha t the pro po s i t ion p i s F has the ef fec t o f associa t ing X wi th a

poss ib i l i ty d i s t r ibu t ion F ix which , by (2.9) , is equal to F . Whe n expressed in the form of

12.1 l ) , a rela t ional a ss ig nm en t e qu at io n wil l be referred to as a possibility assignment

equation

with the unders tanding tha t l - Ix is

induced by p.

As a s imple i l lus t ra t ion , l e t U be the u n iverse o fpos i t ive in tegers and let F be the fuzzy

set o f small integ ers def ined by ( 4 a__un i on )

s m a ll i n te g e r = 1/1 + 1 /2 + 0 . 8 / 3 + 0 . 6 / 4 + 0 . 4 / 5 + 0 . 2 / 6 .

Then , the propos i t ion "X i s a smal l in teger" associa tes wi th X the poss ib i l i ty

d i s t r i bu t i on

H x = 1 /1 + 1/2 + 0 . 8 / 3 + 0 . 6 / 4 + 0 . 4 / 5 + 0 . 2 / 6

{2.12)

in which a term such as 0 .8/3 s ignif ies that the possibi l i ty that X is 3 , given that X is a

sm all integ er, is 0.8.

The re are severa l im po r ta n t po in t s re la t ing to the abo ve def in it ion which are in need

o f c o m m e n t .

F i rs t , {2 .9) impl ies tha t the poss ib i l i ty d i s t r ibu t ion l- Ix m ay be rega rded as an

in terpr e ta t ion of the conc ept o f a fuzzy res t r i c tion and , con sequen t ly , tha t the

m a t hem at i ca l app a ra t u s o f t he t heo ry o f fuzzy s e t s - - a nd , e speci a ll y , t he ca l cu lu s o f

fuzzy r e s tr i c ti ons [4 ] - -p rov i de s a bas i s fo r the m an i pu l a t i on o f pos s ib i li t y d is t ri -

bu t ion s by the ru les o f th i s ca lcu lus.

Second , the def in i t ion im pl ies the assu m pt ion tha t ou r in tu i t ive percep t ion of the

wa ys in which poss ib i l it i es com bine i s in accord wi th the ru les o f com bina t ion of fuzzy

res tr i c ti ons . A l t ho ugh t he va l i d i ty c f t h is a s sum pt i on cann o t be p rov ed a t t h is j unc t u re ,

i t app ear s tha t the re i s a fa ir ly c lose agre em ent be tween such b as ic opera t ions as the

unio n and in tersec t ion of |uzzy set s, on the one hand , an d the po ss ib i l ity d i s t r ibu t ions

as soc i a t ed wi th t he d i s junc t ions and con junc t i ons o f p ropo s i t i ons o f t he fo rm "X is F . "

Ho wev er , s ince our in tu i t ion conc ern ing the beha viour of poss ib il i ti es i s no t very

re l iab le , a g rea t deal o f emp i r ica l work wo uld have to be don e to p rovide us wi th a

be t t e r unde r s t and i ng o f t he ways in wh i ch pos s ib i l it y d i s t r i bu t ions a re m an i p u l a t ed by

hum ans . S uch an under s t an d i ng wo u l d be enhanced by the deve l opm en t o f an

ax i om at i c app ro ach t o t he de f in i ti on o f sub jec ti ve pos s i b i l i t i e s - - an app roa ch wh i ch

m ight be in the sp i r i t o f the ax io ma t ic ap pro ach es to the def in i tion of sub jec t ive

proba bi l i t i es [7 , 8 ] .

Thi rd , the def in i t ion of ~rx(U) impl ies th a t the degree of poss ib i li ty ma y be any

num ber i n t he i n t e rva l [0 ,1 ] r a t he r t han j u s t 0 o r 1 . In t h i s connec t i on , i t s hou l d be

no ted tha t the ex i s tence of in term edia te degrees of poss ib i l i ty is impl ic i t in such

com m onl y encoun t e red p ropos i t i ons a s "There i s a s l i gh t pos s i b i l i t y t ha t M ar i l yn i s

ve ry r i ch ," " I t i s qu i t e pos s i b le t h a t J ean -P au l w i ll be p rom ot ed , . . . . I t is a l m os t

impo ss ib le to f ind a needle in a hay s tack ," e tc .



8 L A Z a d e h

I t c o u l d b e a r g u e d , o f c o u r s e, t h a t a c h a r a c t e r i z a t io n o f a n i n t e r m e d i a t e d e g r e e o f

p o s s ib i li t y b y a la b e l s u c h a s " s l i g h t p o s s i b il i ty " i s c o m m o n l y m e a n t t o b e i n t e r p r e t e d a s

" s l i g h t p r o b a b i l i t y . " U n q u e s t io n a b ly , t h i s i s f r e q u e n t ly t h e c a se in e v e r y d a y d i s c o u r se .

N e v e r th e l e s s , t h e r e i s a f u n d a me n ta l d i f f e r e n c e b e tw e e n p r o b a b i l i t y a n d p o s s ib i l i t y

w h ic h , o n c e b e t t e r u n d e r s to o d , w i l l l e a d to a mo r e c a r e f u l d i f f e r e n t i a t i o n b e tw e e n th e

c h a r a c t e r i z a t i o n s o f d e g r e e s o f p o s s ib i l i ty v s . d e g r e e s o f p r o b a b i l i t y - - - e sp e c i a l l y i n l e g a l

d i s c o u r se , me d ic a l d i a g n o s i s , sy n th e t i c l a n g u a g e s a n d , mo r e g e n e r a l l y , t h o se

a p p l i c a t i o n s i n w h ic h a h ig h d e g r e e o f p r e c i s io n o f m e a n in g is a n im p o r t a n t

d e s i d e r a t u m .

T o i l lu s t r a t e t h e d i ff e re n c e b e tw e e n p r o b a b i l i t y a n d p o s s ib i l i t y b y a simp le e x a m p le ,

c o n s id e r t h e s t a t e m e n t " H a n s a t e X e g g s fo r b r e a k fa s t , " w i th X t a k in g v a lu e s in U = { 1,

2 , 3 , 4 , . . . } . W e m a y a s so c i a t e a p o s s ib i l it y d i s t r i b u t io n w i th X b y in t e r p r e t i n g n x (U ) a s

t h e d e g r e e o f e a s e w i t h w h i c h H a n s c a n e a t u e gg s. W e m a y a l s o a s so c i at e a p r o b a b i l i t y

d i s t r i b u t io n w i th X b y in t e r p r e t i n g

P x t ~ )

a s th e p r o b a b i l i t y o f H a n s e a t i n g u e g g s f or

b r e a k f a s t . A ssu m in g th a t w e e mp lo y so m e e x p li c it o r imp l i c i t c r i t e r io n f o r a s se s s in g th e

d e g r e e o f e a se w i th w h ic h H a n s c a n e a t u eg g s f o r b r e a k f a s t , t h e v a lu e s o f rex U) a n d

p x lu ) m ig h t b e as sh o w n in T a b le 1.

Tab le 1

Th e possib i li ty and proba bi l i ty d is t r ibut ions associa ted wi th X

u 2 3 4 5 6 7 8

rex U) 1 1 1 1 0 8 0 6 0 4 0 2

P x u )

0 1 0 8 0 1 0 0 0 0 0

We o b se r v e t h a t , w h e r e a s t h e p o s s ib i l it y t h a t H a n s m a y e a t 3 e g g s f o r b r e a k f a s t is 1 ,

t h e p r o b a b i l i t y t h a t h e m a y d o so m ig h t b e q u i t e sma l l, e .g ., 0 .1 . T h u s , a h ig h d e g r e e o f

p o s s ib i li t y d o e s n o t imp ly a h ig h d e g r e e o f p r o b a b i l i t y , n o r d o e s a l o w d e g r e e o f

p r o b a b i l i t y imp ly a l o w d e g r e e o f p o s s ib i li t y . H o w e v e r , i f a n e v e n t i s imp o ss ib l e , i t is

b o u n d to b e imp r o b a b le . T h i s h e u r i s t i c c o n n e c t io n b e tw e e n p o s s ib i l i t i e s a n d

p r o b a b i l i t i e s m a y b e s t a t e d i n t h e f o r m o f w h a t mig h t b e c a ll e d t h e

p o s s i b i l it y / p ro b a b i l i ty c o n s i s t e n c y p r in c i p le , n a m e l y :

I f a v a r i a b l e X c a n t a k e t h e v a lu e s

Ul, . . . ,un

with re spect iv e possibi l i t ies H = (n ~, . . .. n~)

a n d p r o b a b i l i t i e s

P = p ~ , . . . , p , ) ,

t h e n th e d e g r e e o f c o n s i s t e n c y o f t h e p r o b a b i l i t y

d is t r ibu t ion P wi th the poss ib i l i ty d is t r ib u t io n H is expressed b y ~+ A= a r i th m et ic sum )

Y= ~ 1P l + "" " + rtnPn.

(2.13)

I t sh o u ld b e u n d e r s to o d , o f c o u r se , t h a t t h e p o s s ib i l i t y /p r o b a b i l i t y c o n s i s t e n c y

p r in c ip l e i s n o t a p r e c i s e l a w o r a r e l a t i o n sh ip t h a t i s i n t r i n s i c i n t h e c o n c e p t s o f

p o s s ib i li t y a n d p r o b a b i l i t y . R a th e r i t i s a n a p p r o x i m a te f o r m a l i z a t i o n o f t h e h e u r i s t i c

o b s e r v a t io n th a t a l e s se n in g o f t h e p o s s ib il i ty o f a n e v e n t t e n d s t o l e s se n i t s

p r o b a b i l i t y - - b u t n o t v i c e -v e r sa. I n t h i s s e n se , t h e p r in c ip l e is o f u se i n s i t u a t io n s i n

w h i c h w h a t i s k n o w n a b o u t a v a r i a b l e X i s i ts p o s s i b i l i t y m r a t h e r t h a n i ts p r o b a b i l i t y - -

d i s t~ : ib u t io n . I n su c h c a se s - - w h ic h o c c u r f a r mo r e f r e q u e n t ly t h a n th o se i n w h ic h th e



Fuzzy sets as a basis or a theory of possibil ity 9

r e ve r se i s t r ue - - the poss ib i l i ty /p roba b i l i ty c ons i s t e nc y p r inc ip le p rov ide s a ba s i s f o r

t h e c o m p u t a t i o n o f t h e p o s s i b il it y d i s t r i b u t i o n o f t h e p r o b a b i l i ty d i s t r i b u t io n o f X .

S u c h c o m p u t a t i o n s p l a y a p a r t i c u l a r l y i m p o r t a n t r o l e i n d e c i s i o n - m a k i n g u n d e r

u n c e r t a i n t y a n d i n t h e t h e o r i e s o f e v i d e n c e a n d b e li ef [ 9 - 1 2 ] .

I n the e xa m ple d i sc usse d a bove , the poss ib i l ity o f X a ssum ing a va lue u is in t e rp r e te d

a s the de gre e o f e a se w i th w hic h u m a y be a s s igne d to X , e .g ., t he de gre e o f e a se w i th

w h i c h H a n s m a y e a t u eg g s f o r b r e a k t a s t. I t s h o u l d b e u n d e r s t o o d , h o w e v e r , t h a t t h i s

" d e g r e e o f e a s e " m a y o r m a y n o t h a v e p h y s i c a l re a li ty . T h u s , t h e p r o p o s i t i o n " J o h n is

young" induc e s a poss ib i l i ty d i s t r ibu t ion whose poss ib i l i ty d i s t r ibu t ion func t ion i s

e xpre sse d by (2 .5 ). I n th i s c a se , the poss ib i l i ty tha t the va r i a b le Age ( Joh n) m a y t a ke the

va lue 28 is 0 .7 , w i th 0 .7 r e p re se n t ing the de g re e o f e a se wi th whic h 28 ma y be a s s igne d to

Age ( John) g ive n the e l a s t ic i ty o f the fuz z y r e s tr i c tion l a be le d yo ung . T hus , in th i s c a se

" the de gre e o f e a se " h a s a f igu ra tive r a th e r tha n phys ic a l s ign i fi c anc e .

2 .1 . P o s s i b i l i ty m e a s u r e

Addi t iona l in s igh t in to the d i s t inc t ion be twe e n p roba b i l i ty a nd poss ib i l i ty ma y be

g a i n e d b y c o m p a r i n g t h e c o n c e p t o f a p o s s i b i l i t y m e a s u r e wi th th e f a mi l i ar c onc e p t o f a

p rob a b i l i ty me a su re . M or e spe ci fi ca l ly , l e t A be a nonfuz z y sub se t o f U a nd l e t F ix be a

poss ib i li ty d i s t r ibu t ion a s soc ia t e d wi th a va r i a b le X w hic h t a ke s va lue s in U . The n , the

p o s s i b i l i t y m e a s u r e , n ( A ) , of A is def ined as a n um ber in [0 , 1] g iven by 6

rc(A ) ~ Su pu ~A rcX(U) ,

(2.14)

wh ere nx(U) i s the po ss ib i l i ty d is t r ibu t ion func t ion of l-Ix . Th i s numbe r , the n , ma y be

in te rp r e te d a s the poss ib i l ity t h a t a v a l u e o f X b e l o n g s t o A , tha t i s

Poss{X ~ A} a - n ( A )

A

= S u p . ~ A rc x(U ).

t2.15)

W he n A is a f uz z y se t, t he be long ing o f a va lue o f X to A i s no t me a n ingfu l . A m ore

gen era l def in i t ion of poss ib i l i ty m easu re wh ich exten ds (2.15) to fuzzy se ts is the

fol lowing.

De f i n i t i on 2.2 . Le t A be a fuzzy subse t o f U and let I -Ix be a poss ib i l ity d is t r ibu t ion

a ssoc ia te d wi th a va r i a b le X whic h t a ke s va lue s in U . The p o s s i b i l i t y m e a s u r e , r c(A ) , o f A

is def ined by

Poss{X is

}

~ ( A )

£ Sup.~ t ,

.4(u) ^ rex(U),

(2.16)

~'The measure defined by {2.14)may be viewed as a particular case of the fuzzy measure defined by Sugeno

and Te rano [20 , 21]. Furthe rmo re, n(A ) as defined by (2.14) provides a possibilistic interpretation for tile

scale unction , a(A), which is defined by" Nah mias [2 2] as the supremum o f a membership function ove r a

nonfuzzy set A.



10 L A Zadeh

w h e r e " X i s A ' " r e p la c e s 'XeA ' i n (2 .15) , ttA i s the m em be rs h i p func t ion of A , an d ^

s t a n d s , a s u su a l, f o r m i n . I t sh o u l d b e n o t e d t h a t , i n t e r m s o f t h e height of fuzzy se t,

w h i c h is d e f in e d a s t h e s u p r e m u m o f i ts m e m b e r s h i p f u n c t i o n [ 2 3 ] , (2 .1 6 ) m a y b e

e x p r e s s e d c o m p a c t l y b y t h e e q u a t i o n

u(A)a= He igh t (A c~ f ix ) . (2 .17)

A s a s im p l e i l l u s t r a t io n , c o n s i d e r t h e p o s s i b i l i ty d i s t r i b u t i o n ( 2 .1 2 ) w h i c h i s i n d u c e d

by th e p r op os i l ion "X i s a sma l l i n t ege r . " In th i s case , l e t A be the se t {3, 4 , 5}. Th en

~z(A )= 0. 8 v ' 0 .6 v 0 .4 =0.8 ,

wh ere v s t and s , a s usua l , fo r max .

O n the o th e r b road , i f A i s the fuzzy se t o f in t ege rs w hich a re no t sma l l , i. e. ,

A ~ 0.2/3 + 0 .4/4 + 0 .6/5 + 0 .8/6 + 1/7 + . . . (2 .18)

t h e n

P o ss{ X ~s n o t a sm a l l i n te g e r} = H e i g h t ( 0 . 2 /3 + 0 . 4 / 4 + 0 . 4 / 5 + 0 . 2 / 6 )

= 0 . 4 .

(2 .19)

I t sh o u l d b e n o t e d t h a t ( 2 .1 9 ) i s a n i m m e d i a t e c o n se q u e n c e o f t h e a s se r t i o n

X is F ~ P o s s { X is A } = H e i g h t ( F n A ) ,

(2 .20)

wh ich i s impl i ed by (2 .11) and (2 .17). In pa r t i cu la r , i f A i s a no rm al fuzzy se t ( i. e.,

H e i g h t ( A ) = 1 ), t h e n , a s s h o u l d b e e x p e c te d

X i s A :~ Po ss {X i s A} = 1.

(2 .21)

Le t A a n d B b e a r b i t r a r y f u z zy su b se t s o f U . T h e n , f r o m t h e d e f in i t io n o f t h e

possibiht2~ m ea su re of a fuzzy se t (2 .16) , i t fol lows t ha t 7

x ( A w B ) = x ( A ) v r r(B ).

(2 .22)

B y c o m p a r i s o n , t h e c o r r e s p o n d i n g r e l a t io n f or p r o b a b i l i t y m e a s u r e s o f A , B a n d

A w B {if the y e xist) is

P ( A w B ) < P ( A ) + P ( B )

(2 .23)

an d, i f A and B are di s jo int ( i. e. , VA(U)#n(u)--O),

P ( A w B ) = P ( A ) + P (B ), (2 .24)

Tit is of interest that (2.22) is analogous to the extension principle for fuzzy sets [5], with + (union) in the

right-hand side of the st;atement of the principle replaced by v.



Fu zzy sets as a basis or tt theory of possibiiity 1

w h ic h e x p r e s ses t h e b a s ic a d d i t i v i t y p r o p e r ty o f p r o b a b i l i t y m e a su r e s . T h u s , i n c o n t r a s t

t o p r o b a b i l i t y m e a su r e , p o s s ib i l i ty m e a su r e i s n o t a d d i t i v e . I n s t e a d , it h a s t h e p r o p e r ty

e x p r e s se d b y ( 2. 22 ), w h ic h m a y b e v i e w e d a s a n a n a lo g o f ( 2. 24 t w i th + r e p l a c e d

b y v .

I n a s im i l a r f a sh io n , t h e p o s s ib i l i t y me a su r e o f t h e i n t e r s e c tio n o f A a n d B i s re l a t e d t o

t h o s e o f A a n d B b y

rc(A n B )< rc (A ) A re(B).

~2.25)

I n p a r t i c u l a r , i f A a n d B a r e non i n t e r ac t i v e , 8 (2 .25) holds with the equal i ty s ign, i .e . ,

n (A n B ) = n ( A J ^ n ( B ) . (2 .26)

B y c o m p a r i s o n , i n t h e c a s e o f p r o b a b i l i t y m e a s u r es , w e h a v e

P (A n B ) < P ( A ) ^ P ( B )

(2 .27)

a n d

P ( A ~ B ) = P ( A ) P ( B )

(2 .28)

i f A a n d B a r e i n d e p e n d e n t a n d n o n f u z z y . A s in th e c a se o f (2 .2 2 ), (2 .2 6 ) is a n a lo g o u s t o

( 2. 28 ), w i th p r o d u c t c o r r e sp o n d in g to m in .

2 . 2. P os s i b i l i t y and i n j b r m a t i on

I f p is a p r o p o s i t i o n o f t h e f o r m

p a=X

i s F which t r ans la te s in to the poss ib i l i ty

a s s i g n m e n t e q u a t i o n

I-IAIX = F, (2 .29)

wh ere F i s a fuzzy sub se t o f U and A (X) i s an im pl ied a t t r i bu te o f X tak in g v a lues in U,

th e n th e i n f o r m a t i on c onv e y e d by p , l ( p ) , may be iden t i f ied wi th the poss ib i l i ty

dis t r ibu t ion , l-lA(x~, of th e fuzzy v ar i ab le

A ( X ) .

T h u s , t h e c o n n e c t i o n b e t w e e n

l ( p ) ,

F la tx ~, R ( A ( X ) )

a n d F i s e x p r e s se d b y

w h e r e

l(p) a= -lmx~,

(2 .30)

I-IA tx~= R ( A ( X ) ) = F .

(2 .31)

A

F o r e x a mp le , i f t h e p r o p o s i t i o n p = J o h n is y o u n g t r a n s l a t e s i n to t h e p o s s ib i li t y

a s s i g n m e n t e q u a t i o n

l-lAge(Johnl"-young , (2 .32)

a N o n i n t e r a c t i o n i n t h e s e n s e d e f i n e d h e r e i s c l o s e l y r e l a t e d t o t h e c o n c e p t o f n o n i n t e r a c t i o n o f f u z z y

r e s t r ic t i o n s [ 5 , 6 ] . I t s h o u l d a l s o b e n o t e d t h a t ( 2 . 2 6 ) p r o v i d e s a p o s s i b i l i st i c in t e r p r e ta ' d o n f o r u n r e l a t e d n e s s '

a s d e fi n e d b y N a h m i a s [ 2 2 ] .



1 2 L . A . Z a d e h

w h e r e y o u n g i s d e f in e d b y ( 2. 5) , t h e n

/ ( J o h n i s y o u n g ) = H A g e 0 o h n )

i n w h ic h th e p o s s ib i l i ty d i s t r i b u t io n f u n c t io n o f A g e ( Jo h n ) is g iv e n b y

(2.33)

~ t A g e O o h n ) u ) = l - - S U ; 2 0 , 3 0 , 4 0 ) .

U E [0 , 100] .

(2 .34)

F r o m t h e d e fi n it io n o f 1 (p ) i t fo l lows tha t i f p ~ X i s F an d q A=X i s G , th e n p is a t l e a s

as in form a t ive a s q , exp resse d a s 1 (p ) > 1 (q ), if F ~ G. Thu s , we h ave a pa r t ia l o rd e r in g o1 '

the l (p ) de f ined b y

F ~ G ~ I X is F)>=I X is G)

(2 .35)

w h ic h imp l i e s t h a t t h e mo r e r e s t r i c t i v e a p o s s ib i l i t y d i s t r i b u t io n i s , t h e mo r e

in f o r m a t iv e i s t h e p r o p o s i t i o n w i th w h ic h i t i s a s so c i a te d . F o r e x a m p le , s in c e v e ry t a ll

c ta l l , we have

I (Luc y i s ve ry ta l l ) > l (L uc y i s ta l l) .

(2 .36)

3 . N - a r y p o s s ib i l it y d i s t r i b u t i o n s

I n a s se r t i n g th a t t h e t r a n s l a t i o n o f a p r o p o s i t i o n o f t h e fo r m p ~X i s F i s e x p r e s se d b y

X i s F - + R A X ) ) = F

o r , e q u iv a l e n t ly ,

X is F - ) H a x ) = F,

(3.1)

(3.2)

w e a r e t a c i t l y a s su min g th a t p c o n ta in s a s in g l e imp l i e d a t t r i b u t e

A X )

w h o s e

p o ss ib i l it y d i s t r i b u t io n i s g iv e n b y th e r i g h t - h a n d m e m b e r o f (3 .2 ).

M o r e g e n e r a l l y , p m a y c o n t a i n n i m p l i e d a t t r i b u t e s

A ~ X ) , . . . , A . X ) ,

w i th

A i l X )

t a k in g v a lu e s i n U z , i= 1 , . . . , n . I n t h i s c a se , t h e t r a n s l a t i o n o f

p ~ - X

is F , where F is a

f u z zy r e l a t i o n in t h e c a r t e s i a n p r o d u c t U = U j x . . - x U . , a s su m e s th e f o r m

X is F --+ R A I X ) , . . . , A . X ) ) = F

o r , e q u iv a l e n t ly ,

X is F - - + H A ~ X ) . . . . A. X)) F

(3.3)

(3.4)

w h e r e

R A I X ) , . . . , A . X ) )

is an n-a ry fuzzy restr ic t ion an d H(al(x) . .. . A.(X)) is an

n-ary

possibil i ty distr ibution

which i s

induce d by p.

C o r r e s p o n d i n g l y , t h e

n-ary possibility

distr ibution u nctio n induced by p i s g iven by

7t al X) .... A. x, Ut,...,Un)= lar Ul,...,U:,), u l, .. - ,u .) ¢ U ,

(3.5)

w h e r e / t r i s t h e me m b e r s h ip f u n c t io n o f F . I n p a r t i c u l a r , i f F i s a c a r t e s i a n p r o d u c t o f n



F u zzy se t s as a bas i s f o r a t he ory o f poss ib i l i ty 3

u n a r y f uz z y r e la t io n s F ~ , . . . , F . , t h e n t h e r i g h t h a n d m e m b e r o f (3 .3 ) d e c o m p o s e s i n t o a

sys t em o f n una ry r e l a t i ona l a s s i gnm en t equa t i ons , i.e .,

X is

F - R ( A 1

(X ) = F ,

3 . 6 )

R A2 X))=F2

k ( A . ( x ) )

C or re spond i ng l y ,9

a n d

where

I I ( A , ( X ) A . X ) ) ----- A t ( X ) X • • • X I - I A . ( X )

n A , x ) . . . . . A n X ) ) U , , . . . , U , , ) = r U , x ) U , ) ^ . . . A r C A . x ) U , , ) ,

3 . 7 )

(3.8)

7 rA ,( x) (U i )= l a F ,(U i) , u i e U i , i = 1 . . . , n

(3.9)

an d A denote s min ( in infix form).

A

As a s imple i llus t ra t ion , co ns ider the pro po s i t ion p = carpet i s l a rge , in which large is

a fuzzy re la t ion w hose tab leau i s o f the form show n in Table 2 (wi th length and wid th

expressed in metric uni ts)•

Table 2

Tab leau o f l a rge

Large W idth Leng th /~

250 300 0.6

250 350 0.7

300 400 0.8

400 600 1

In th i s case , the t ra ns la t ion 13 .3) l eads

t o

t he pos s i b i l i t y a s s i gnm en t equa t i on

l I ( w i d t h l c a r p e t } , l e n g t h ( c a r p e t } - -

large, (3.10)

which impl ies tha t i f the co mp at ib i l i ty o f a carpet whose w id th is , say , 250 cm and length

i s 350 cm wi th " large car pet " i s 0 .7 , then the p oss ib i li ty tha t the wid th of the ca rpet is

A

2 5 0 c m a n d i ts l e n g th i s 3 50 c m - - g i v e n t h e p r o p o s i ti o n p = c a r p e t is l a r g e - i s 0 .7 .

No w , i f l a rge is def ined as

large = wid e x lon g (3.11 )

91fF and G are fuzzy relat ions in U and V, resp ectiv dy, then the ir cartes ian pro du ct F x G is a fuzzy relat ion

in [r ~ Vwhnse membership function is given by l ,)(u)~', p,~lt ,).

B



14

L A Z a d e h

whe r e long a nd w ide a r e una r y f uz z y r e l a t i ons , t he n ( 3 . 10) de c ompose s i n to t he

poss ib i l it y a s soc i a tion e qu a t ion s

nd

t'[width(carpet ) ---"

w i d e

l" l e n g t h c a r p e t ) - - l o n g

whe r e the t a b l e a ux o f l ong a nd w ide a re o f t he f o r m shown in T a b le 3 .

T a b l e 3

T a b l e a u x o f w i d e a n d l o n g

W i d e W i d t h /~ L o n g L e n g t h * /t

250 0 .7 . 300 0 .6

3 0 0 0 . 8 3 5 0 0 . 7

3 5 0 0 . 8 4 0 0 0 . 8

4 0 0 1 5 0 0 1

3 .1 . M a r g i n a l p o s s i b i l i t y d i s t r ib u t i o n s

T h e c on c e p t o f a ma r g in a l poss ib i li ty d i s t r i bu t ion be a r s a c lose re l a ti on to t he

con cep t of a m argina l fuzzy res t r ic t ion 1-4], which in turn i s an a lo go us to the conce pt of

a ma r g in a l p r oba b i l i t y d i s t r i bu t ion .

M ore spec if ica lly , l e t X = (X t , . . . ,X , ) be an n-a ry fuzzy var iab le takin g va lues in U

= U t x - . - x U , , a nd l e t Hx be a poss ib i li t y d i s t r i bu t ion a s soc i a t e d w i th X , w i th

nx( Ut , . .. , u , ) d e no t ing the poss ib i li t y d i s t r i bu t ion f unc t ion o f [I x .

e t

A

q = ( i t , . . . , i k) be a subse quen ce of the index sequence (1 , . . . , n ) and le t X(q) be th e

q-ary fuzzy var iab le X(~) a= (Xi, , .. . ,Xi~) . T he m a r g i n a l p o s s i b i l i t y d i s t r i b u t i o n [ix~q) is a

poss ib i l i ty d is t r ibut io n asso c ia ted wi th X(a) wh ich i s

i n d u c e d

by l -Ix as the projec t ion of

H x o n U (o a__Ui t x . . . x Uic Thu s , by def in i t ion ,

H x,~, a=Projv(~)l-lx, (3.12)

which impl ies tha t the pro bab i l i ty d is t r ibu t ion func t ion of X(q) i s r e la ted to tha t o f X by

7rx,~,(u(q)= V r t x ( U ) (3.13)

U ~ q ,

whe r e u,q,,,=a uq , . . ., u~j) , q,a= i t , . . ",Jm) is a sub sequ ence of (1 , , . . , n) wh ich is

A

c om ple m e nta r y to q (e .g ., i f n = 5 a nd q = ( i t i 2 ) = (2 , 4) , t he n q = ~ j t , j 2 , j a ) = ( l , 3 , 5 ) ,

A -

u~¢) = (u~,, .. .,z.9, ) an d v , , , den otes th e su pr em um ove r ( u ~ , , . . . , u j . ) E U ~ , x . . . x U j , , .

• . . q '

A s a s imple i l lus t ra t ion, assu me tha t U 1 = U2 - U 3 = {a , b} an d the table au of r lx i s

g ive n by



F u z z y s e t s a s a b a s i s f o r a t h eo r y o f p o s s i b i l i ty 1 5

T h e n ,

T a b l e 4

T a b l e a u o f l ' x

H x X l X 2 X a n

a a a 0 .8

a a b 1

b a a 0 .6

b a b 0 .2

b b b 0 .5

H(x, ,x , ) = P ro jv~ , , u 2 H x - ~ l / a , a ) + O . 6 / b , a ) + O . 5 / b , b )

w h i c h i n t a b u l a r f o r m r e a d s

T a b l e 5

T a b l e a u o f n ~ x ,.

g21

l-l(x v x2) X1

12

b

b

(3 .14)

X2 n

a

, t 0 .6

b 0 .5

Th en , f rom H~: i t fo l lows tha t the poss ib i l i t y t ha t X 1 = b , X2 = a an d X 3 = b is 0 .2 , whi l e

f r o m F l t x t . x 2 ~ i t f o l lo w s t h a t t h e p o s s i b i l it y o f X l = b a n d X 2 = a i s 0 .6 .

B y a n a l o g y w i th t h e c o n c e p t o f i n d e p e n d e n c e o f r a n d o m v a r ia b le s , th e f u z zy

v a r i a b l e s

a n d

A

X ~ q ) = X i ~ , . . . , X ~ )

A

a r e n o n i n t e r a c t i v e [ 5- 1 i f a n d o n l y i f t h e p o s s i b i li t y d i s t r i b u t i o n a s so c i a t e d w i t h X

= ( X I , . . . , X , ) i s t h e c a r t e s i a n p r o d u c t o f t h e p o s s ib i l it y d i s t r i b u t i o n s a s so c i a t e d w i~h

Xtq) an d Xt¢), i .e .,

H x =Hx~q~ x Flx,q.~ (3. 15 )

o r , e q u i v a l e n t l y ,

nx(U~ .. .. , u,, ) = nx,,,,(ui, . . . . . u ~k ) ^ n x , , , , u j t , . . . , u j , ,, ) .

(3 .16)

I n p a r t i c u l a r , t h e v a r i a b l e s X 1 , . . . , X , a r e n o n i n t e r a c t i v e i f a n d o n l y i f

l ' Ix =l -Ix1 x Fix2 x " '" x l ' Ix . (3 .17)



16 L A Zadeh

T h e in tu i ti v e s ig n if i ca n c e o f n o n in t e r a c t io n m a y b e c la r if i ed b y a s imp le e x a m p le .

Su pp os e tha t X a_. (X ~ ,X2 ), an d X t an d X 2 a re non in te rac t iv e , i .e .,

nx(u l , u2 )= n x , (u i ) /x nx ,(u21. (3 .18)

F u r t h e r m o r e , s u p p o s e t h a t f o r s o m e p a r t i c u l a r v a lu e s o f u t a n d u 2 ,

n x , ( U t

) = ~ t , n x2 ( u 2)

=or2 < o q a n d h e n c e

n x ( U t , U 2 ) = o t 2 .

N o w , i f t h e v a lu e o f

n x ~ ( u t )

i s inc reas ed to oft + f i t ,

61 > 0 , i t i s no t po ss i lqe to dec reas e the v a lue of ~ tx2(u2) by a p os i t ive am ou nt , say 62 ,

s u c h t h a t t h e v a l u e o f

r t x ( U l , U 2 )

r e ma in s u n c h a n g e d . I n t h i s s e n se , a n i n c r e a se i n t h e

p os s ~ . ii ty o f u l c a n n o t b e c o m p e n sa t e d b y a d e c r e a se in t h e p o s s ib i l it y o f u2 , a n d v i c e-

v e r s t , h u s , i n e s sen c e , n o n in t e r a c t io n m a y b e v i e w e d a s a f o r m o f n o n c o m p e n sa t io n in

w h i c h a v a r i a t io n i n o n e o r m o r e c o m p o n e n t s o f a p o ss i b il it y d i s t r ib u t i o n c a n n o t b e

c o m p e n s a t e d b y v a r i a ti o n s i n t h e c o m p l e m e n t a r y c o m p o n e n t s .

I n t h e m a n ip u la t i o n o f p o s s ib i l it y d i s t ri b u t io n s , i t i s c o n v e n ie n t t o e m p lo y a t y p e o f

sy m b o l i c r e p r e se n t a t i o n w h ic h i s c o m m o n ly u se d in th e c a se o f f u z z y s e ts . S p ec i fi c al ly ,

a ssu m e , fo r s impl ic i ty , th a t U 1, . . . , U . a re f in i te se t s, and le t r ~ = ( r ] , . . . , ? . ) den ote an n-

tu p l e o f v a lu e s d r a w n f r o m U t , . . . , U . , r e sp ec t iv ely . F u r th e r m o r e , l e t n~ d e n o te t h e

poss ib i l i ty o f r ~an d le t the n - tup le ( r ] , . . . , r~) be wr i t ten a s the s t r ing r ] . . . r~..

U s in g th i s n o t a t i o n , a p o s s ib i l i t y d i s t r i b u t io n H x ma y b e e x p r e s se d in t h e sy mb o l i c

f o r m

N

= ~ 1 . i ( 3 . 1 9 )

I-Ix ni r] r~ "'" ~.

i=

o r , in c a se a s e p a r a to r sy m b o l i s n e e d e d , a s

N

H x = E

n,/r~r~2 r~,

(3.20)

i= 1

w h e r e N i s t h e n u m b e r o f n - tu p l e s in t h e t a b l e a u o f l-Ix , a n d th e su m m a t io n sh o u ld b e

in t e r p r e t e d a s t h e u n io n o f t h e f u z zy s in g l e to n s

n i / ( r ] , . . . , r ~ ) .

As an i l lus t r a t ion , in the

n o ta t i o n o f ( 3. 19 ), t h e p o s s ib i l it y d i s t r i b u t io n d e f in e d in T a b le 4 r e a d s

F i x = 0 . 8 a a a + l a a b + 0 .6 b a a + 0 .2 b a b + 0 . 5 bb b .

(3 .21)

T h e a d v a n ta g e o f t h i s n o t a t i o n i s t h a t i t a l l o w s th e p o s s ib il i t y d i s t r i b u t io n s t o b e

m a n i p u l a t e d i n m u c h t h e s a m e m a n n e r a s l i n e a r f o r m s i n n v a r i a b l e s , w i t h t h e

u n d e r s t a n d in g th a t , i f r a n d s a r e tw o tu p l e s a n d ~ a n d f l a r e t h e i r r e sp e c t iv e

poss ib i l i ti e s , then

~ r+ / l r = qu. v / /b r (3 .22)

u.r ,-~/It= 1~ ~ 11~" (3.23)

a n d

~r x / J s = ( ~ ^ f l it s . ( 3 .2 4 )



F u z z y s e t s a s a b a s i s f o r a t h e o r y o f p o s s i b i l i ty 1 7

w h e r e r s d e n o t e s t h e c o n c a t e n a t i o n o f r a n d s. F o r e x a m p l e , if

a n d

l-Ix = 0 .8a a + 0 . 5 a b + l b b

t h e n

a n d

(3 .25)

H r = O . 9 b a + O . 6 b b

(3 .26)

l - I x + H r = O . 8 a a + O . 5 a b + O . 9 b a + l b b

(3 .27)

l-lx c~ l-Iv = 0. 6b b (3.2 8)

H x x H v = 0 . 8 a a b a + 0 .5 a b b a + 0 . 9 b b b a + 0 . 6 a a b h + 0 . 5 a b b b + 0 . 6 b b b b .

(3.29 j

A

T o o b t a i n t h e p r o j e c t i o n o f a p o s s i b i li t y d i s t r i b u t i o n H x o n Utq~= (Ui, , . . . , Uik), i t is

su ff ic ie n t to s e t t h e v a l u e s o f X i , , . . . , X j , , i n e a c h t u p l e in H x e q u a l t o t h e n u l l s t r i n g A

( i. e. , mu l t ip l i ca t ive ide n t i ty ) . As an i l l us t ra t io n , t he p r o je c t ion of the poss ib i l i t y

d i s t r ibu t ion de f ined by Table 4 on U~ x U2 i s g iven by

Pro j v , × u2Hx = 0 . 8 a a + 1a a + 0 . 6 b a + 0 . 2 b a + 0 . 5 b b

(3 .30)

= l a a + O . 6 b a + O . 5 b b

w h i c h a g r e e s w it h T a b l e 5 .

3 . 2 . C o n d i t i o n e d p o , s s ib i li t) , d i s t r i b u t i o n s

I n t h e t h e o r y o f p o s s ib i li ti e s , t h e c o n c e p t o f a c o n d i t i o n e d p o s s i b i li t y d i s t r i b u t i o n

p l ay s a ro le t h a t is a n a l o g o u s - - t h o u g h n o t c o m p l e t e l y - - t o t h a t o f a c o n d i t io n a l

p o s s i b i l it y d i s t r i b u t i o n i n t h e t h e o r y o f p r o b a b i l i t i e s .

M o r e c o n c r e t e l y , l e t a v a r i a b l e X = X ~ , . . . , X , ) be assoc ia t ed wi th a poss ib i J i ty

d i s t r ibu t ion I -Ix , w i th I -Ix cha rac te r i ze d by a poss ib i l i t y d i s t r ib u t ion func t ion

nx Ul , . . . ,Un)

which ass igns to each n - tup le (u l , . . . . u , ) in U~ x . . . x U , i t s poss ib i l i t y

r C x U l , . . . , u , ) .

L e t q = i ~ , . . . , i k ) a n d s = (j'~ . . . , i s ) b e s u b s e q u e n c e s o f t h e i n d e x s e q u e n c e ( 1 , . . . , n ) ,

a n d l e t a j , , . . . , a i m ) b e a n n - t u p l e o f v a l u e s a s s i g n e d t o X W } = X i , , . . . , X i m ) . B y

def in i t ion , t he c o n d i t i o n e d p o s s i b i l i t y d i s t r i b u t i o n o f

g iven

A

X q ) = X i l , . . . , X i k )

X ~ , ~ = a i , , . . . , a i . )

is a p o s s ib i l it y d i s t r i b u t i o n e x p r e s se d a s

1-Ix,~ EX , = a j , ; . . . ; X i = a ~ . ]



18 L A Zadeh

who se poss ib i li t y d i s t r i bu t ion f unc t ion i s g ive n b y ~°

n x , ,, (u ~ , , .. ., u ~ , X ~ , = ; . . . ; X j . = a j . ) (3.31)

7rx(Ut,. •., u n) [

uj , = . . . . . u j =a~ .

A s a s im p l e e x a m p l e , i n th e c a s e o f 3 .2 1 ), w e h a v e

I l t x , . x 3 ) [ X l = a ] = 0 . 8 a a + l a b

(3.32)

as the ex press ion for the con di t io ned poss ib i l i ty d is t r ibut ion of (X , ,X a ) g iven X 1 = a .

An e q u iva l e n t e xpr e ssion f o r t he c ond i t i one d po ss ib i li ty d i s t r i bu t ion w hic h ma ke s

c l e a r e r t he c onne c t ion b e twe e o

l-lx,,,[Xj, = % ;.. .;X ~= aj .]

a nd l ' lx m a y be de r ive d a s fo l lows .

Le t

l -lx [X h = a j , ; . . . ; X j = a J

de no te a poss ib i l i ty d is t r ibu t ion which cons is t s of those te rms in (3 .19) in which the j t th

e l e me nt i s a i ,, t he j , t h e l e me nt i s a ye, .. ., a nd the jmth e le me nt is a j . Fo r e xa mple , in t he

case of (3.21)

H x ( X 1

= a ]

= 0 . 8 a a a + l a a b .

(3.33)

E xpr e sse d in t he a b ove no ta t ion , t he c o nd i t i one d poss ib i li t y d i s t r i bu t ion o f X tq}

= ( X~ ,, .. ., X~) g iveh X j , = a j ~ , . . . , X j = a j m a y be wr i t te n a s

rlx,,,Exh

=

; . . . ;X~ =aj

= Projv~,~Hx[Xh = a h ; . . . ; X j = a J

(3.34)

wh ich p laces in evid ence th at l -Ix, ~ con dit io ned on X{s)=a{~)) is a m arg ina l po ssibi l i ty

distr ibut ion reduced by 1-1x ( c ond i t ione d on

X ~ s ) = a t e } ) .

T hus , by e mp loy ing ( 3 .33) a nd

(3,34), we o bta in

I I~ x 2 . x 3 ) [ X 1 = a ] = 0 . 8 a a + I a b

(3.35)

wh ich agrees w ith (3.32).

I n t he f o r e go ing d i sc ussion , we ha ve a s su me d tha t t he poss ib i li t y d i s t ri bu t ion o f X

= ( X 1 , . . . , X n ) i s con di t io ned on the va lues ass igned to a spec i fied subse t , X{s), of the

const i tuent var iables of X. In a more genera l se t t ing , what might be spec i f ied i s a

~°In some appl icat ions , i t ma y be a pprop r iate to n orma l ize the express ion for the condi t ioned poss ibi l i ty

dis t r ibut ion funct ion by dividing the r ight-ha nd mem ber of (3.31) by i ts supre mu m over Ui, . - - x Uic



F u z z y s e t s a s a b a s i s f o r a t h e o r y o f p o s s i b i l i ty 1 9

poss ib i l it y d i s t r ibu t ion a s soc i a t e d w i th X{~) r a the r t ha n the va lue s o f X j~ , . . . , X j . I n suc h

cases, we shal l say th at 1-Ix is p a r t i c u l a r i z e d ~~ by spe cify ing th a t rlx{~}= G, wh ere G is a

given m-ary poss ib i l ity d is t r ibu t ion. I t shou ld be no ted th a t in the present con text l'Ix,~

i s a g ive n poss ib il it y d i s t r i bu t io n r a the r t h a n a m a r g ina l d i s t r i bu t ion tha t i s i nduc e d by

l -I x .

T o a n a ly z e t h is ca se , i t is co n v e n i e n t t o a s s u m e - - i n o r d e r t o s i m p li fy th e n o t a t i o n ~

t h a t Xj~ = X ~ , ? : h = X z , . . . , X j . = X ~ , m < n . L e t G d e n o t e th e c y l i n d r i c a l e x t e n s i o n of G,

tha t i s, the poss ib i l i ty d is t r ibu t ion def ined by

G ~ G x U s + ~ x x U n (3.36)

wlalch implies that

A

• -

# r . ( u l , . . , u n ) # ~ ( u : , . . . , u ~ ) ,

u : e U j ,

j = l , . . . , n , t 3 . 3 7 )

wh ere #~ is the mem bers hip func t ion o f the fuzzy re la t ion G.

T h e a s sum pt ion tha t we a r e g ive n l' Ix a nd G is e qu iva l e n t t o a s suming tha t we a r e

given the in te r sec t ion Flx n G. Fr om this in te r sec t ion, then, we can dedu ce the

pa r t i c u l a r iz e d poss ib i l it y d i s t r i bu t ion I lx , , , [ Hx , , = G by p r o je c t ion on U,o . T h u s

Hx,q,[l 'Ix,~ = G ] = Projv,q,H x n G.

(3.38)

E qu iva l e n t ly , the l e f t -ha nd me m be r o f (3 .38) ma y be r e ga r de d a s t he c om pos i t i on o f I-Ix

and G [5] .

As a s imple i l lus t ra t ion, cons ider the poss ib i l i ty d is t r ibut ion def ined by (3 .21) and

a ssume tha t

G = 0 .4aa + 0 .8ba +

l b b .

(3.39)

T h e n

( i = 0 . 4 a a a + 0 . 4 a a b + 0 . 8 b a a + 0 . 8 b a b + l b b a + I b b b

(3.40)

a n d

l-Ix

c~ G = 0 . 4 a a a + 0 . 4 a a b + 0 . 6 b a a + 0 . 2 b a b + 0 . 5 b b b

Flx3[Il tx, .x~) = G] = 0. 6a + 0.5b.

(3.41)

(3.42)

A

As a n e l e me nta r y a pp l i c a t ion o f (3 .38), c ons ide r t he p r op os i t i on p = Joh n i s b ig ,

whe r e b ig i s a r e l a t ion who se t a b l e a u is o f t he f o r m show n in T a b le 6 (w i th he igh t a nd

weigh t expressed in m et r ic uni t s ) .

~ n t h e c a s e o f n o n f u z z y r e l a t i o n s , p a r t i c u l a r i z a t i o n i s c l o se l y re l a t e d t o w h a t is c o m m o n l y r e f e rr e d t o a s

r e s t r i c t i o n W e a r e n o t e m p l o y i n g th i s m o r e c o n v e n t i o n a l te r m h e r e b e c a u s e o f o u r u s e o f t h e t e r m " f u zz y

r e s t r i c t i o n " t o d e n o t e a n e l a s t ic c o n s t r a i n t o n t h e v a l u e s t h a t m a y b e a s s i g n e d t o a v a r i a b l e .



20 L A Zadeh

Table 6

Tableau of big

Big Height Weight /~

170 70 0.7

170 80 0.8

180 80 0.9

190 90 1

N o w , s u p p o s e t h a t i n a d d i t i o n t o k n o w i n g i h a t J o h n i s b i g , w e a l s o k n o w t h a t q

a= Jo h n i s t a ll , w h e r e t h e t a b l e a u o f ta l l is g iv e n ( i n p a r t i a l l y t a b u la t e d f o r m) b y T a b le 7 .

Table 7

Tableau of tall

Tall Height /a

170 0.8

180 0.9

190 1

T h e q u e s t io n is " W h a t i s t h e w e ig h t o f Jo h n ? B y ma k in g u se o f ( 3. 38 ), th e p o s s ib i l it y

d i s t r i b u t io n o f t h e w e ig h t o f Jo h n m a y b e e x p r e s se d a s

I' ] weight =

P r o j

weight 1"1 (height. weight 1[ f l height - -

t a i l ]

(3.39)

= 0 . 7 / 7 0 + 0 . 9 / 8 0 + 1 / 9 0 .

A n a c c e p ta b l e l in g u i s ti c a p p r o x im a t io n [ 5 ] , [ 1 3 ] t o t h e r i g h t - h a n d s id e .o f ( 3. 39 ) mig h t

b e " so m e w h a t h e a v y , " w h e r e " s o m e w h a t " i s a mo d i f i e r w h ic h h a s a sp e c if ie d ef fe ct o n

t h e fu z zy se t la b e l ed " h e a v y . " C o r r e s p o n d i n g l y , a n a p p r o x i m a t e a n s w e r t o t h e q u e s t i o n

w o u l d b e " J o h n is s o m e w h a t h e a v y . "

4 Poss ib i l i ty d is tribut ions o f com pos i te and qual if ied propos i tions

A s w a s s t a t e d i n t h e I n t r o d u c t io n , t h e c o n c e p t o f a p o s sib i li t y d i s t r i b u t io n p r o v id e s a

n a t u r a l w a y f o r d e fi n in g t h e m e a n i n g a s w e ll a s th e i n f o r m a t i o n c o n t e n t o f a

p r o p o s i t i o n i n a n a tu r a l l a n g u a g e . T h u s , i f p i s a p r o p o s i t i o n i n a n a tu r a l l a n g u a g e N L

a n d M i s i t s me a n in g , t h e n M ma y b e v i e w e d a s a p r o c e d u r e w h ic h a c t s o n a s e t o f

r e l a t i o n s i n a u n iv e r se o f d i s c o u r se a s so c i a t e d w i th N L a n d y i e ld s t h e p o s s ib i li t y

d i s t r i b u t io n o f a s e t o f v a r i a b l e s o r r e l a t io n s w h ic h a r e e x p l ic i t o r im p l i c i t i n p .

I n c o n s t r u c t in g th e me a n in g o f a g iv e n p r o p o s i t i o n , i t i s c o n v e n ie n t t o h a v e a

c o l l e c t io n o f w h a t m ig h t b e c a l l e d

condition l tr nsl tion rules

[ 3 0 ] w h ic h r e l a t e t h e



F u z z y s e t s a s a b a s i s o r a t h eo O , o f p o s s i b i l i ty 2 1

m e a n i n g o f a p r o p o s i t i o n t o t h e m e a n i n g o f i ts m o d i f ic a t io n s o r c o m b i n a t i o n s w i t h

o t h e r p r o p o s i t i o n s . I n w h a t f o l lo w s , w e sh a l l d i s c u s s b r i e fl y so m e o f t h e b a s i c r u l es o f

t h i s t y p e a n d , i n p a r t i c u l a r , w i l l f o r m u l a t e a r u l e g o v e r n i n g t h e m o d i f i c a t i o n o f

p o s s i b i li t y d i s tr i b u t i o n s b y t h e

poss ib i l i t y qua l i f ica t ion

o f a p r o p o s i t i o n .

4 .1 . Ru les o f t ype I

Le t p b e a p r o p o s i t i o n o f t h e f o r m X i s F , a n d l e t m b e a m o d i f i e r su c h a s v e r y , q u i t e ,

r a t h e r , e t c . T h e so - c a l l e d

modif ier rule

[ 6 1 w h i c h d e f i n e s t h e m o d i f i c a t i o n i n t h e

p o s s i b i li t y d i s t r i b u t i o n i n d u c e d b y p m a y b e s t a t e d a s fo ll ow s .

I f

t h e n

X is

F--,rlaix~ = F

(4 .1)

X is

m F - - + H a x ) =

F +

4 . 2 }

w h e r e

A X )

is a n i m p l i e d a t t r i b u t e o f X a n d F + is a m o d i f i c a t i o n o f F d e f i n e d b y m . ~2

~

. F F 2 a

o r e x a m p l e , i f m - v e r y , t h e n = ; i f m = m o r e o r l es s t h e n F + = \ / F " a n d if m

a= n o t t h e n F + = F ' a= c o m p l e m e n t o f F . A s a n i l lu s t r a t i o n "

I f

Jo h n is y o u n g - - * H

Age(Joh,,J-'-

y o u n g

14.3)

t h e n

Jo hn is ve ry y o un g ~ H A~,,h ,,I = y ou n g 2 .

I n p a r t i c u l a r , i f

y o u n g = 1 - S ( 2 0 , 30 , 4 0 )

1 4 . 4 )

t h e n

yo un g 2 = (1 - S (20 , 30 , 40) )2 ,

w h e r e t h e S - f u n c t i o n ( w i t h i ts a r g u m e n t su p p r e s se d ) is d e f i n e d b y (2 .6 ).

4 .2 . Ru les o f t ype I I

A

I f p a n d q a re p r o p o s i t i o n s , t h e n

r = p * q

d e n o t e s a p r o p o s i t i o n w h i c h i s a

co m p o s i t i o n

o f p a n d q . T h e t h r e e m o s t c o m m o n l y u s e d m o d e s o f c o m p o s i t i o n a r e ( i)

c o n j u n c t i v e , i n v o l v i n g t h e c o n n e c t i v e " a n d " ; (ii) d i s ju n c ti v e , i n v o l v i n g t h e c o n n e c t i v e

" o r " ; a n d ( i i i ) c o n d i t i o n a l , i n v o l v i n g t h e c o n n e c t i v e " i f . . . t h e n . " T h e c o n d i t i o n a l

t r a n s l a t i o n r u le s re l a t in g t o t h e s e m o d e s o f c o m p o s i t i o n a r e s t a te d b e l o w .

12A m ore de ta i led discuss ion of the e f fec t of modif ie r s (or hedg es) ma y be fou nd in [ 15, 16,17, 8 , 6 ,13 and

18].



22 L A Zadeh

a n d

the n

Co n j u n c t i ve ( non in t e r a c t ive ) : I f

X i s F - - , II A x ~ = F

(4.5)

Y i s G - , H s t r~ = G 4 . 6 )

X is F and Y is G~l'lcatx~,n~r~ = F x G

(4.7)

wh,~re A (X) a nd B (Y ) are the im pl ied at t r ib ut es of X an d Y, respect ively, I 'I(A(x),B(r)) s

the po ss ib il i ty d i s t r i bu t ion o f t he va r i a b le s A ( X ) a n d B ( Y ) , and F x G is the ca r tes ian

pr od uc t o f F a nd G . I t shou ld be no te d tha t F x G ma y be e xpr e sse d e qu iva l e n t ly a s

F x G = F n CJ

(4.8)

whe re F an d Cr a re the cy l indr ic a l extens ions of F and G, respect ively .

Dis junc t i ve

(noni .nteract ive) : I f (4.5) and (4.6) hold, th en

X is F or Y is G--*l-ltA~x~.B(r~= F +¢~ ( 4. 9)

where the symb ols have the same m eanin g as in (4.5) and (4 .6), and + deno tes the

union.

Co n d i t i o n a l

~noninteract ive)" I f (4.5) and (4.6) hold, th en

I fX i s F the n Y i s

G~II~A~x~.n{r}~= F ~ C ,

whe r e F ' is t he c om ple m e nt o f F a nd ~ is t he bou nd e d sum de f ine d by

(4.10)

/#.~c,= 1 ^ {1--1tr+l.tc,),

(4.11)

in whic h + a nd - de no te the a r i thm e t i c a dd i t i on a nd sub t r a c t ion , a nd #v a nd #6 a r e

the m e mb e r sh ip f unc t ions o f F a n d G , r e spe c tively . I l l us t r a t i ons o f t he se r u l e s - -

e xpr e sse d in t e rms o f f uz zy r e s tr i ct i ons r a the r t ha n poss ib i li ty d i s t r i b u t io ns - - m a y be

fou nd in I -6 an d 14].

4 .3 . Tru th qua l~ca t ion , p roba bi l i t y qua l if i ca tion and poss ib i l i t y qua l i f ica t ion

I n na tu r a l l a ngua ge s , a n im po r t a n t m e c ha n i sm f o r t he mo di f i c a t ion o f t he me a n in g

of a prop osi t ion i s pro vid ed by the adju c t ion of three types of qua l i f ie r s : ( i) i s X, wh ere

is a l ing uist ic t ruth -va lue , e .g. , t rue, v ery t rue, m ore or less t rue, fa lse, e tc . ; ( ii ) is 2, w he re

2 is a l ingu ist ic prob ab il i ty -va lue (or l ike l ihoo d) , e.g. , l ikely, very l ikely, very unlikely ,

etc.; a n d (iii) is ~c, w he re n is a lingu istic possi bility -valu e, e.g. , po ssible , qu ite po ssib le,

sh~h t ly poss ib le, imposs ible , e tc. These m od es of qua l i f ica t ion wi l l be re fe r red to ,



Fuzzy sets as a basis.for

a theory of

possibility

23

respectively, as t ruth .qual i j i cat i on, probabi l i t y qual i j i cat i on and possib i l i t y quai+

ication. The rules governing these qualifications may be stated as follows.

Truth quali f i cati on: If

4.12)

.

x sF+II x, -F

then

X is F isr+IIA(xj=F+,

where

PF+@)=P,(P&))9

UEUU;

(4.13)

~1, nd PF are the membership functions of r and

F,

respectively, and U is the universe of

discourse associated with A(X). As an illustration, if young is defined by (4.4); z = very

true is defined by

very true = S2 (0.6,0.&l )

(4.14)

then

John is young+IIAge(John) young

John is young is very true-+nApe(,ahn)=youngt

where

PI

p&ll)=S’(l --WC 20, 30, 40); 0.6, 0.8, l ,

LIEU

It should be noted that for the unitary truth-value, u-true, defined by

Pu-true(V)v,

4-0911

(4.15)

(4.13) reduces to

PF+(U)=PF(U)~

UdJ

and hence

X is F is u-true+& =F.

(4.16)

Thus, the possibility distribution induced by any proposition is invariant under unitary

truth qualification.

Probabi l i t y qual i f icat i on:

If

then

X is F+I I A X I=F

X is F is ~-‘n,,,,,,,,,,,,,,,,,=E.

(4.17)



2 4 L A Zadeh

where

p ( u ) d u

i s the p rob abi l i ty tha t the va lue of A (X) fal ls in the in te rva l (u , u + d u) ; the

integra l

~v p(u)laF(u)du

is the prob abi l i t y of the fuzzy even t F [19] ; and 2 is a linguis t ic prob abi l i ty-v a lue . Thus ,

(4.17) def ines a poss ib i l i ty d is t r ib ut io n of pro bab i l i ty d is t r ibut ions , wi th th e poss ib i l i ty

of a pro bab i l i ty d ens i ty p( - ) g iven impl ic i tly by

n [ j v p ( u ) l ( u

)du] = #x[ jv

p(u

)#r(u )du] . (4.18)

As an i l lus t ra t ion, cons ide r th e pro pos i t ion p a= Jo hn is youn g i s very l ike ly , in which

yo un g i s def ined by (4.4) and

T h e n

/ -/very l ikely -- $ 2

(0.6, 0.8, 1 ).

2 1

rc[~v p(u) v(u)du ] = S

[ jo p(u)(1 - S(u ;20 , 30, 40 ))du;0 .6, 0.8, 1].

(4.19)

I t shou ld be no te d tha t t he p r ob a b i l i t y qua l i fi c a t ion r u le is a c onse que nc e o f t he

a ssum pt ion tha t t he p r op os i t i ons "X i s F i s 2" a nd "Pr ob{ X is F} = 2" a r e

semantical ly

equivalent

( i.e ., induce ident ica l poss ib i l i ty d is tr ibut ions) , w hich i s expressed in sym bols

a s

X i s F i s 2 o P r o b { X is F } = 2 .

(4.20)

Thus , s ince the pro bab i l i ty of the fuzzy event F i s g iven by

Pro b{X is F} = ~v

p(u)l~v(u)du,

i t fo l lows f rom (4.20) tha t we ca n asse r t the sem ant ic equ iva lence

X i s F i s 2 ~ v

p(u)lav(u)du

is 2,

which by (2 .11) leads to the r igh t -han d m em ber of (4 .17).

Possibility qualification:

O ur c onc e r n he r e i s w i th t he f o llowing qu e s t io n :G ive n tha t

" 'X i s F" t r ans la tes in to the poss ib i l i ty ass ignment equa t ion I IA~x}=F, what i s the

tran slat ion of "X is F is re ," wh ere rc is a l ingu ist ic poss ibi l i ty-valu e such as qu i te

poss ib le , very poss ib le , mo re or less possib le , etc . ? S ince ou r in tu i t ion regardin g the

beh avio r of poss ib i l i ty d is t r ibu t ions i s not wel l -deve loped a t th i s jun c ture , the answer

sugge s t e d in t he f o llowing shou ld be v i e we d a s t e n t a t ive i n na tu r e .

Fo r s im pl ic ity , we sha l l in te rp re t the qua l i fie r "poss ib le" as " l -po ss ib le , " th a t i s , a s

the ass ignm ent of the po ss ib i l i ty-va lue 1 to the p rop osi t ion which i t qua li f ies . Wi th th is

unde r s t a nd ing , t he t r a ns l a t i on o f " X is F is poss ib l e " w il l be a s sume d to be g ive n by

X is F is possible ~ l ' la (x )= F +, (4.21)



F u z z , s e t s a s a 9 a s i s. fo r a t h e o r y o f p o s s i b i li t y

25

i n w h ic h

F + =FO I-I

w here I I i s a fuzzy se t o f Ty pe 2 la de f ined b y

(4.22)

/Zn(U) = [0, 1],

u e U ,

(4 .23)

a n d ~ ) is t h e b o u n d e d su m d e f in e d b y (4 .1 1 ). E q u iv a l e n t ly ,

/ t r + ( u ) = [ # r ( u ) , 1 ], u ~ U ,

(4 .24)

w h i c h d e f i n e s / # + a s a n i n t e r v a l - v a l u e d m e m b e r s h i p f u n ct io n .

In e f fec t , the ru le in ques t ion s ign i f ie s tha t poss ib i l i ty qua l i f ica t ion has the e f fec t o f

w e a k e n in g th e p r o p o s i t i o n w h ic h i t q u a l if i es t h r o u g h th e a d d i t i o n to F o f a p o s s ib i l i ty

d i s t r i b u t io n H w h ic h r e p r e se n t s t o t a l i n d e t e r m in a c y 14 in t h e s e n se t h a t t h e d e g r e e o f

p o s s ib i l i ty w h ic h it a s so c i a t e s w i th e a c h p o in t i n U m a y b e a n y n u m b e r i n th e i n t e r v a l

[ 0 , 1 ] . A n i l l u s t ra t i o n o f t h e a p p l i c a t i o n o f t h i s r u l e t o t h e p r o p o s i t i o n pa=X is sm all is

shown in Fig. 1 .

/ s m a l l

F s m o l l +

0

Fig. 1. Th e possibil i ty dis tribution ol "'X is small is possible".

A s a n e x t e n s io n o f t h e a b o v e r u l e , w e h a v e : I f

X is

F-oI-la~x =F

t h e n , f o r 0 < 0 t < 1,

X is F is ~t-possible-- , Ha(x) = F +

(4.25)

(4.26)

w h e r e F + is a f u zz y s e t o f T y p e 2 w h o se i n t e r v a l - v a lu e d m e m b e r s h ip f u n c t io n i s g iv e n

b y

# r + ( u ) = [~ . ^ l , r ( u ) , ~ ) ( l - p , ~ ( u ) ) ] ,

u e U .

(4 .27)

t3The mem bership funct ion of a fuzzy se t of Typ e 2 takes values in the se t of luzzy subsets of the uni t

in terval [5 .6] .

J41-1 ma y be interp reted as the possibil ist ic co un terp art of white noise.



26

L A Zadeh

As a n i l lus t r a t ion , t he r e su l t o f t he a p p l i c a t ion o f t h i s r u l e to t he p r op os i t i on p ~ X i s

smal l is sho w n it~ Fig . 2. No te t ha t th e ru le expressed by (4 .24) m ay be reg arde d as a

spec ia l ease of (4 .27) co r resp on din g to ct = 1 .

/ - F / - F

I - - - -

0

u

Fig. 2." Th e possibili ty dis t r ibut ion of" X is small is e-possible".

A fur ther ex tens ion of the ru le expressed by t4 .25) to l inguis t ic poss ib i l i ty-va lues may

be ob ta ine d by a n a pp l i c a t ion o f t he e x t e ns ion p r inc ip l e , l e a d ing to t he

l i ngu i s t i c

poss ib i l i t y qua l i f i ca t ion ru le

I f

X i s F - - , H a ~ x = F

then

X is F is

n - + H A x ) = F

+

(4.28)

where F ÷ i s a fuzzy se t of Ty pe 2 whose m em bersh ip func t ion i s g iven by

/@ (u ) - { =>c ( r rA~r(u) ) n ( ~ o ( r r~ (1- -gF(U)) ) )} , (4 .29)

wh ere rc is the l inguist ic possib i l i ty (e .g., qu i te pos sible , a lm ost imp ossib le , e tc .) and o

de no te s t he c om pos i t i on o f f uz zy r e l at i ons. T h i s r u l e shou ld be r e ga r d e d a s spe c u la t ive

in na ture s ince the imp l ica t ions of a l inguis tic poss ib i l ity qua l i f ica tion a re no t as ye t wel l

l l l l de : ' q lTr~ d

An a l te rna t iv e ap pro ach to the t r an s la t ion of "X i s F i s n" i s to in te rp re t th i s

p r opo s i t i on a s

X is F i s ~ ,- : Poss{X isF} =n , (4 .30)

which is in the spir i t of (4.20), an d then form ulate a ru le of the form (4.28) in wh ich I IA(x)

is the la rges t ( i .e . , l eas t r es t r ic t ive) poss ib i l i ty d is t r ibut io n sa t i s fy ing the con s t ra in t

Poss{X is F} =n . A com pl ica t ing fac tor in th is case is tha t the pro po si t io n "X is F i s n"

ma y be a s soc i a t e d w i th o the r impl i c i t p r opos i t i ons suc h a s "X i s no t F i s [ 0 , 1J -

poss ib le, " or "X i s not F i s no t im poss ible ," which a ffec t the t r ans la t io n of" X is F i s n . "

In th is connec t io n, i t wou ld be useful to dedu ce th e t r an s la t ion ru les (4.21), (4.26) an d

(4.29) (or the i r var iants ) f rom a conju nc t ion of "X is F i s n" w i th o ther impl ic i t

p r o p o s i t i o n s i n v o lv i n g t h e n e g a t i o n o f " X is F . "

An in te res t ing aspec t of poss ib i l i ty qua l i f ica t ion re la tes to the invar iance of



u z z y s e t s a s a b a s is f o r a t h e o r y o f p o s s i b i l it y 27

i m p l i ca t i on under t h is m ode o f qua l if i ca ti on . Thu s , f rom t he de f i n i ti on o f im p l i ca t i on

[6] , i t fo l lows a t once tha t

X i s F = ~ X is G i f F c G

No w, i t can r ead i l y be shown t ha t

F c G= ~F c G

(4.31)

where c i n t he r igh t -hand m em ber o f (4.31 ) shou l d be i n t e rp re t ed a s the r e la t i on o f

co nta inm en t for fuzzy se t s o f Ty pe 2 . In con sequ enc e of (4.31), then , w e can asser t tha t

X is F is possible= , ,X is G is poss ible i f F ~ G.

(4.32)

5 C o n c l u d i n g r e m a r k s

The expos i t i on o f t he t heo ry o f pos s i b il i ty in t he p resen t pap e r t ouch es upon on l y a

few o f t he m any face ts o f t h i s - - a s ye t l a rge ly une xp l o red - - t he o ry . C l ea r l y , t he i n tu i ti ve

concep t s o f pos s ib i l it y and p rob ab i l i t y p l ay a cen t r a l ro l e in hum an dec i s i on -m ak i ng

and unde r l ie m uc h o f t he hum an ab i li t y t o r eason i n app rox i m a t e t e rm s . C onse quen t l y ,

i t wi ll be essen t ia l to develop a be t ter u nde rs tan din g of th e in terp lay between poss ib i l i ty

and p rob ab i l i t y - - e spec i a l l y i n r e la t i on t o t he ro l e s wh ich t hese concep t s p l ay in na t u ra l

l ang uag es - - i n o rde r t o er, hance ou r ab i li t y t o deve l op m ach i nes wh i ch can s i m u l a t e the

rem arka b l e hum an ab i l it y to a t t a i n i m prec is e l y de f ined goa l s in a fuzzy env i ronm en t .

A c k n o w l e d g m e n t

Th e ide a of em ploy ing the theo ry of fuzzy se t s as a bas is fo r the th eo ry of poss ib i li t3~

was i n sp i r ed by the pape r o f B .R . Ga i nes and L . Koh ou t on pos s i b l e au t om at a [ 1 ]. In

add i t i on , ou r work was s t i m u l a t ed by d i s cus s i ons wi t h H . J . Z i m m erm ann rega rd i ng

t he i n t e rp re t a t i on o f t he op e ra t i ons o f con jun c t i on and d i s junc t i on ; t he r e su lt s o f a

psycho l og i ca l s t udy conduc t ed by E l eano r R osch and L ou i s Gom ez" and d i s cus s i ons

wi t h Barba ra ( ' e : nv .

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