D:\User\sbornik_38_pdf\28.DVI


Mathematical Problems of Computer Science 38, 70{71, 2012.

M ember ship Functions or ®-Cuts? Algor ithmic

(Constr uctivist) Analysis Justi¯es an I nter val

Appr oach

V la d ik K r e in o vic h

Department of Computer Science, University of Texas at El Paso, USA, vladik@utep.edu

In h is p io n e e r in g p a p e r s , Ig o r Za s la vs ky s t a r t e d a n a lg o r it h m ic ( c o n s t r u c t ivis t ) a n a lys is
o f fu z z y lo g ic . In t h is p a p e r , we e xt e n d t h is a n a lys is t o fu z z y m a t h e m a t ic s a n d fu z z y d a t a
p r o c e s s in g . S p e c i¯ c a lly, we s h o w t h a t t h e t wo m a t h e m a t ic a lly e qu iva le n t r e p r e s e n t a t io n s o f a
fu z z y n u m b e r { b y a m e m b e r s h ip fu n c t io n a n d b y ®-c u t s { a r e not a lg o r it h m ic a lly e qu iva le n t ,
a n d o n ly t h e ®-c u t r e p r e s e n t a t io n e n a b le s u s t o e ± c ie n t ly p r o c e s s fu z z y d a t a .

1 Fir s t R e s u lt : Two R e p r e s e n t a t io n s A r e N o t E qu iva le n t

De¯nition 1. B y a c -m e m b e r s h ip fu n c t io n , we mean a tuple consisting of two real numbers
¢ and ¢ and a function ¹ :

h
¢ ; ¢

i
! [0 ; 1 ] for which ¹ ( ¢ ) = ¹

³
¢

´
= 0 , m a x

x
¹ ( x) = 1 ,

and a · b · c implies that ¹( b ) ¸ m in ( ¹ ( a) ; ¹( c) ) .
De¯nition 2. W e say that a c-membership function h ¢ ; ¢ ; ¹i is c o m p u t a b le if both real
numbers ¢ and ¢ are computable and the function ¹ is computable.

De¯nition 3. B y a fa m ily o f ®-c u t s ( o r s im p ly ®-cuts, fo r s h o r t ) c o r r e s p o n d in g t o a c -
m e m b e r s h ip fu n c t io n ¹, we m e a n a p a ir o f m a p p in g s x : [0 ; 1 ] ! IR a n d x : [0 ; 1 ] ! IR fo r
wh ic h , fo r e ve r y ® 2 [0 ; 1 ], we h a ve fx : ¹( x ) ¸ ®g = [x ( ®) ; x( ® ) ].
De¯nition 4. W e say that ®-cuts are c o m p u t a b le if both mapping x and x are computable.

P r oposition 1. There exists a computable c-membership function for which the corre-
sponding ®-cuts are not computable.

P r oposition 2. There exist computable ®-cuts for which the corresponding c-membership
function is not computable.

2 On ly ®-Cu t s Gu a r a n t e e A lg o r it h m ic Fu z z y D a t a P r o c e s s in g

S in c e t h e t wo r e p r e s e n t a t io n s o f fu z z y a r e n o t c o m p u t a t io n a lly e qu iva le n t , it is d e s ir a b le t o
a n a lyz e wh ic h o f t h e m le a d s t o a n a lg o r it h m ic fu z z y d a t a p r o c e s s in g . H e r e a r e t h e r e s u lt s o f
t h is a n a lys is : fu z z y d a t a p r o c e s s in g is c o m p u t a b le fo r ®-c u t s b u t , in g e n e r a l, n o t c o m p u t a b le
fo r m e m b e r s h ip fu n c t io n s .

7 0



V. Kreinovich 7 1

De¯nition 5. L et ¹1; : : : ; ¹n be membership functions, and let f ( x1; : : : ; xn ) be a function.
B y the r e s u lt of applying f to fuzzy sets ¹1; : : : ; ¹n, we mean a membership function de¯ned
by the formula ¹( y ) = m a x

x1;:::;xn:y=f (x1;:::;xn)
m in ( ¹1 ( x1 ) ; : : : ; ¹n ( xn ) ) :

P r oposition 3. There exists a computable c-membership function ¹1 ( x1 ) and a computable
function f ( x1 ) for which the result ¹ of applying f to ¹1 is not computable.

P r oposition 4. There exists an algorithm that, given n computable families of ®-cuts cor-
responding to the membership functions ¹1; : : : ; ¹n and a computable function f ( x1; : : : ; xn ) ,
returns computable ®-cuts for the result ¹ of applying f to ¹1; : : : ; ¹n.

3 A u xilia r y R e s u lt : W h y m in a n d N o t A n y Ot h e r A n d -Op e r a t io n

W e wa n t a ll t h e p r o p e r t y t o s a t is fy t h e \ c o n ve xit y" c o n d it io n , t h a t if a · b · c, t h e n ¹ ( b) ¸
m in ( ¹ ( a) ; ¹( c) ) . S o m e t im e s , we kn o w t h a t t h e a c t u a l va lu e x s a t is ¯ e s two p r o p e r t ie s S0 a n d
S00 c h a r a c t e r iz e d b y m e m b e r s h ip fu n c t io n s ¹0 ( x ) a n d ¹00 ( x ) ; t h e n , t h e d e g r e e ¹( x ) t o wh ic h a
r e a l n u m b e r x is c o n s is t e n t wit h t h is in fo r m a t io n c a n b e d e s c r ib e d a s ¹( x ) = f& ( ¹

0 ( x) ; ¹00 ( x) ) .
It is r e a s o n a b le t o r e qu ir e t h a t t h is c o m b in e d p r o p e r t y s h o u ld a ls o b e \ c o n ve x" ( in t h e a b o ve
s e n s e ) .

De¯nition 6. A function ¹ : IR ! [0 ; 1 ] is called f-c o n ve x if a · b · c implies that
¹( b ) ¸ m in ( ¹( a ) ; ¹ ( c) ) .
De¯nition 7. B y a g e n e r a liz e d a n d -o p e r a t io n , we mean a function f : [0 ; 1 ]£[0 ; 1 ] ! [0 ; 1 ]
which satis¯es the following two properties:

² for all a, a0, b, and b0, if a · a0 and b · b0, then f ( a; b ) · f ( a0; b0 ) (monotonicity);

² for all a, we have f ( a; 1 ) = f ( 1 ; a ) = a.

P r oposition 5. L et f ( a; b) be a generalized and-operation. Then, the following two condi-
tions are equivalent to each other:

² for every two f-convex functions ¹0 ( x ) and ¹00 ( x) , the function ¹( x ) = f ( ¹0 ( x ) ; ¹00 ( x ) )
is also f-convex;

² f ( a; b ) = m in ( a; b ) .

R e fe r e n c e s

[1 ] B . A . K u s h n e r , L ectures on Constructive M athematical Analysis, A m e r . Ma t h . S o c .,
P r o vid e n c e , R h o d e Is la n d , 1 9 8 4 .

[2 ] H . T. N g u ye n a n d E . A . W a lke r , F irst Course In F uzzy L ogic, CR C P r e s s , B o c a R a t o n ,
Flo r id a , 2 0 0 6 .

[3 ] I. D . Za s la vs ky, \ Fu z z y c o n s t r u c t ive lo g ic " , In : Studies in constructive mathematics and
mathematical logic. P art XI, Zap. Nauchn. Sem. P OM I, 2 0 0 8 , V o l. 3 5 8 , p p . 1 3 0 { 1 5 2 ;
E n g lis h t r a n s la t io n in J ournal of M athematical Sciences, 2 0 0 9 , V o l. 1 5 8 , N o . 5 , p p . 6 7 7 {
6 8 8 .