Fuzzy Sets and Systems 36 (1990) 45-53 45 North-Holland

T H E P R O B L E M R E D U C T I O N M E T H O D U N D E R

U N C E R T A I N T Y A N D D E C O M P O S I T I O N O F F U N C T I O N S

LI Zhong-Fu* Department of Mathematics, Sichuan University, Chengdu, China

Received May 1988 Revised October 1988

Abstract: The problem reduction method provides a useful tool for the design of expert systems. In the method, how to acquire and represent combination functions is an important issue. It is interesting to know whether or not the combination functions can be decomposed. If they do, a complex issue will be much simplified. We discuss the necessary and sufficient conditions under which functions of several variables can be represented by binary operations satisfying some conditions such as associativity and commutativity, especially by triangular norms or conorms.

Keywords: Expert system; combination function; decomposition of functions; operation for fuzzy sets.

1. Introduction

The problem reduction method provides a useful tool for the design of expert systems for complex problems [3, 6]. The problem reduction idea usually involves replacing a problem goal by a set of subgoals such that if the subgoals are solved, the main goal is also solved. Explaining problem reduction in terms of decomposable problem systems allows us to to indefinite about whether we are decomposing problem goals or problem states. If the problem and each subproblem are binary, or the main goal and subgoals are proved to be either true or false, then possible relations for the problem decomposition in this situation are AND and OR. The problem therefore can be represented by an AND/OR graph. However, in many real-word decision-making problems, situations are not so clear. Due to uncertainties, the states of main goal or subgoals may take intermediate truth values between absolutely true and false. Suppose that the truth values of the states of subgoals are Xx, x2 . . . . . xn, respectively, and the truth value of the state of main goal is y. Then y will be a function of the n variables xl, x2 . . . . , xn say y = F ( x l . . . . . xn). Such a function F is called combination function. As is often the case, human experts can assign a value to the combination function y when the values Xl, x2 . . . . . xn are given, but can not give out a precise formula for F. How to acquire and represent combination functions is an important issue in building knowledge-based systems.

It is interesting to know whether or not the function F can be decomposed as

F(Xl , x2 . . . . . xn) = x l * x 2 * • • .*xn, (1.1)

* Project Supported by the National Natural Science Foundation of China.

0165-0114/90/$3.50 © 1990, Elsevier Science Publishers B.V. (North-Holland)

46 L i Zhong-Fu

where * is a binary operation satisfying some conditions such as associativity and commutativity. If (1.1) holds, the complex problem of acquiring and representing the function F of n variables could be reduced to a much simpler one of acquiring and representing binary operation *, that is a function of two variables.

Actually, in many existing expert systems, it is often assumed that the combination functions can be represented in the form of (1.1). How can we check the resonability of these assumptions in a given system?

Fuzzy set theory [8] provides a natural setting for management of uncertainty (or fuzziness)in building expert systems. Applying fuzzy set theory to decision- making problems, combination functions are usually represented by some aggregative operations for fuzzy sets. Up to now, almost all investigations on fuzzy set-theoretic operations only deal with binary operations (see [2]). Thus a problem arose naturally: can all combination functions be represented as (1.1)? Obviously the answer is negative. Then, to make clear the necessary and sufficient conditions is an important issue not only in theory but also in its applications.

In this paper, we discuss the necessary and sufficient conditions under which (1.1) holds. In Section 2, a concept of utility independence at some point, which is crucial to our topic, is introduced. In Section 3, some necessary and sufficient conditions are given. Section 4 deals with the cases where the binary operation * in (1.1) is a triangular norm [7] or conorm. Triangular norms and conorms are considered as appropriate for representing fuzzy set-theoretic intersections and unions [1, 2, 4].

2. Utility independence at some point

Firstly, we give an example to illustrate that there are functions of several variables which can not be represented as (1.1).

Example 2.1. Let F ( x l , x2, x3) = m a x { x l , x2} + x3 - m a x { x 1 , x2}x3. Then F can not be represented as F ( x l , x2, x3)=x~*x2*x3 , where * is a binary operation satisfying associativity and commutativity. For if this is not true, we would have F(0.5, 0, 0.5) = 0 . 5 * 0 * 0 . 5 = 0 . 5 . 0 . 5 . 0 = F ( 0 . 5 , 0.5, 0). But F(0.5, 0, 0.5) = 0.75 and F(0.5, 0.5, 0) = 0.5, leading to a contradiction.

Now we introduce a new concept that is very important to our topic.

Definition 2.1. Let n I> 3, S and Si be sets, el e S~, i = 1, 2 , . . . , n. A function F : X7=1 Si--> S is said to satisfy utility independence at point ( e l . . . . . e n ) iff

F ( a l . . . . . a i - 1 , ei , a i+! . . . . , an) = F ( b l , • • • , b i -1 , ei , b i + l . . . . . b n )

implies that

F ( a l . . . . . ai+l, Xi, a i + l , • • • , an ) = F ( b l . . . . . b i - l , xi, bi+l . . . . . b,,),

for all i = 1, 2 . . . . . n, all xl ~ S~ and all a i, bj e Sj, ] ~ i.

Problem reduction under uncertainty 47

Example 2.2. F : [ 0 , 1] 3--~ [0, 1], F ( x l , x2, x3) = x l + x2 + X 3 - - X 1 X 2 - - X 1 X 3 - - X 2 X 3

-t-XIX2X 3. Then function F satisfies utility independence at point (0, 0, 0). But F does not satisfy utility independence at point (1, 1, 1), since F(1, O, O) = 1 4= 0 = F(O, O, 0), while F(1, O, 1) = F(O, O, 1).

Example 2.3. F: [0 , 1]n--* [0, 1], F ( X l , x 2 , . . . , x , ) = min{xl , x2 . . . . . xn}. Then function F satisfies utility independence at point (1, 1 , . . . , 1), but does not satisfy utility independence at point (0, 0 . . . . . 0).

Example 2.4. F : [0, 1] n--* [0, 1], F(Xl . . . . . Xn) = a~xl +" • • + anX,,, where 0 ~< ai ~< 1, i = 1 . . . . . n, a~ + • • • + an = 1. Then function F satisfies utility independ- ence at point (Pl . . . . . Pn) for all (Pl . . . . . Pn) e [0, 1]".

3. Basic theorems

Theorem 3.1. L e t S be a set, e ~ S, n >i 3, and F : S" --~ S be a f u n c t i o n sat i s fy ing

the f o l l o w i n g condi t ions :

(A1) F ( e . . . . . e, xi, e, . . . , e) = x i , f o r all xi ~ S and i ~ {1 . . . . . n}; (A2) utility i ndependence ho lds at p o i n t (e, e . . . . . e).

Then F can be d e c o m p o s e d as

F ( x l , x2 . . . . . xn) = x l * x 2 * " " * x , , (3.1)

where * is a b inary opera t ion on S sat is fy ing the f o l l o w i n g cond i t ions f o r all

x , y , z e S:

(C1) x * e = x ; (C2) ( x * y ) * z = x * ( y * z ) ;

(C3) x * y = y * x . M o r e o v e r , such a b inary opera t ion is un ique .

Proof. For convenience, we define

Fi(xi) = F ( e . . . . . e, xi, e . . . . . e) ,

Fi,j(xi, x j) = F(e , . . . , e, xi , e . . . . . e, x j , e . . . . . e) ,

Fi,/.k(xi, x i, xk ) = F ( e . . . . . e, xi, e . . . . . e, xj , e . . . . . e, xk . . . . . e) ,

where i, j , k ~ {1 . . . . . n}, and i 4=j, i=/=k, j 4=k.

We show that

F~,i(x, y ) = F~,m(X, y ) (3.2)

for all i, j, l, m ~ { 1 . . . . . n }, i :/: j, l 4= m, and all x, y ~ S. Since n /> 3, there is k ~ {1 . . . . . n} such that k 4= i and k 4= l. According to (A1),

Fi(y ) = F , , ( y ) = Fk(y) = y and E(x) = E(x) = x.

Thus, by (A2),

F~j(x, y) = F~.g(x, y) = Ft, g(x, y) = F~,m(X, y).

48 L i Z h o n g - F u

So we can define a binary operation * on S by

x *y = F~a(x, y), (3.3)

for all x, y ~ S and i, j ~ { 1 , . . . , n}, i 4:j. By (3.2), the operat ions * is well defined. And by (A1), x * e = F~.j(x, e ) = x . ( C 1 ) follows. According to the definition,

x * y = Fi , j (x , y ) = Fj, i ( x , y ) = F i a (Y , x ) = y * x ,

that means (C3) holds. To show (C2), by (A1) and (3.3),

F,-j(x *y, e) = x *y = F/j(x, y), (3.4)

FS.k(e, y * z ) = y * z ---- FS, k ( y , z ) . (3.5)

Thus,

( x * y ) * z = F ~ . k ( x * y , z )

= Fi.j ,k(X * y , e, z )

= F~./,k(x, y , z ) [by (3.4) and (A2)]

= Fi j , k(X, e, y * z )

= F~.k(X, y * Z) [by (3.5) and (A2)]

= x * ( y * z ) .

(C2) follows. We show inductively that (3.1) holds. Obviously,

F ( x l , e, . . . , e ) = x l = x l * e * . . . * e .

And if

F ( x l . . . . . x~,, e . . . . . e ) = x l * " .*x~ ,

=Xl*" • "*Xk*e*" • .*e ,

k < n, then, by (A1),

F ( x l . . . . . xl , , e , . . . , e ) = F ( x l * . . " * Xk, e, . . . , e ) .

And, by (A2),

F ( x l . . . . . Xk, X k + l , e . . . . . e ) = F ( x l * . . . * X k , e . . . . . e , X k + l , e . . . . , e )

= ( x , , . . .

~ X 1 * " " " * X k * X k + 1 * e * • • • * e .

The result follows. To show that the binary operation is unique, it is sufficient to notice that if F

can also be represented as

F ( x l , x z . . . . . x , , ) = x l ° x 2 . . . . . x , , ,

where o is a binary operation satisfying (C1)-(C3), then

x o y = x oy oe . . . . . e = F l , z ( x , y ) = x * y ,

for all x, y ~ S.

Problem reduction under uncertainty 49

Note. If a binary operation * on S satisfies the conditions (C1), (C2) and (C3) in Theorem 3.1, it means that (S, *) is a commutative semigroup with identity e.

Theorem 3.2 (the inverse of Theorem 3.1.). Suppose that

F(x, . . . . . x , ) = x l * " • . * x , ,

where n >i 3 and * is a binary operation on S satisfying conditions (C1), (C2) and (C3) o f Theorem 3.1. Then F satisfies conditions (A1) and (A2) o f Theorem 3.1.

Proof. Since operation * satisfies (C1) and (C3),

e*x = x * e =x,

for all x • S. Thus, it is obvious that F satisfies (A1). To show that F satisfies (A2), suppose

F(a l . . . . . a i - l , e, ai+l . . . . . a,,) = F ( b l . . . . . bi-1, e, bi+l . . . . . bn),

that is

a l* • • • * a i - i * e * a i + l * • • • * a n - - b l * • • • * b i _ l * e * b i + l * • • • * b n.

Or, by (C1) and (C2),

a l , . . . , a i _ l , a i + l , - . . , a n = b l , . . . , b i _ l , b i + l , . . . , b n .

Then, by (C2) and (C3),

F ( a l , • • • , a i - 1 , x i , a i + l , • • • , a n ) = a l * • • • * a i - 1 * x i * a i + l * • • • * a n

= ( a I * . . . , a i _ 1 * a i + l * . . . * a n ) * x i

= (bl*" ' "*bi- l*bi÷l*" " " * b , ) * x i

= b l * " • " * b i _ l * X i * b i + l * . • " * b n

= F ( b l . . . . , b i + l , x l , b i + l . . . . . b n ) .

The result follows.

Combining Theorem 3.1 and Theorem 3.2, we get:

T h e o r e m 3.3. Le t S be a set, e • S, n >i 3, and F : S n--~ S be a funct ion. Then F can be decomposed as

F(Xl , X2, . . . , x , ) = x l * x 2 * " " * x , ,

where * is a binary operation on S satisfying the fo l lowing conditions f o r all x, y • S :

(C1) x * e = x ; (C2) ( x * y ) * z = x * ( y * z ) ; (C3) x * y = y * x ,

i f f F satisfies the fo l lowing conditions: (A1) F(e, . . . , e, x~, e . . . . . e) = x , f o r all xi • S and i • {1 . . . . . n}; (A2) utility independence holds at po in t (e, e . . . . . e). Moreover , i f such a binary operation * exists, it is unique.

50 Li Zhong-Fu

4. Triangular norm and conorm

In Theorem 3.1, the binary operation * is on S and satisfies (C1)-(C3). If we let S = [0, 1], e = 0 (or 1) and suppose * is nondecreasing, then * becomes a triangular conorm (or a triangular norm, respectively).

Definition 4.1. A triangular conorm * is a binary operation on [0, 1], which satisfies the following axioms for all x, y, z, u • [0, 1]:

(C1) x * O = x ; (C2) ( x * y ) * z = x * ( y * z ) ;

(C3) x * y = y * x ;

(C4) if x /> y then x * u 1> y * u.

Defmition 4.2. A triangular norm * is a binary operation on [0, 1], which satisfies the following axioms for all x, y, z, u • [0, 1]:

(N1) x * l = x ; (N2) ( x * y ) * z - - x * ( y * z ) ;

(N3) x * y = y * x ; (N4) if x/> y then x * u 1> y * u.

The triangular norms arose in the study of generalized triangle inequalities for statistical metric spaces [7]. Recently, triangular norms and their dual conorms are considered appropriate for representing fuzzy set-theoretic intersections and unions [2]. But there is a basic problem to be solved. That is: under what conditions can F(Xl . . . . . x~), n I> 3, be represented as

F ( x l , x2, • • • , x , ) ~-x 1 * X 2 * " " " * X n ,

where * is a triangular norm or conorm?

In this section, we discuss the necessary and sufficient conditions.

Theorem 4.1. Let F : [0, 1] n ~ [0, 1], n /> 3, represented as

F ( x l , x2, • • • , x . ) - Xl * X2 * " " " * x . ,

where * is a tr iangular c o n o r m , i f f F satisfies the f o l l o w i n g condi t ions:

(A1) F(0 . . . . . 0, Xi , 0 . . . . . O) = X i , f o r all xi • [0, 1] and i ~ {1 . . . . . n}; (A2) utility i ndependence holds at p o i n t (0, 0 . . . . ,0) ; (A3) it is nondecreas ing with respect to each variable.

be a f unc t ion . Then F can be

(4.1)

Proof. To show the sufficiency, since F satisfies (A1) and (A2), by Theorem 3.1 (let S = [ 0 , 1] and e = 0 ) , we have (4.1) with the operation -k satisfying (C1)-(C3). To show (C4), for any x ~ y and u • [0, 1], by (A3), FL2(x, u) >t F1.2(y, u). Since x * u = F1,2(x, u) (see the Proof of Theorem 3.1) and y * u = FI,2(y, u), we have x * u ~>y * u . (C4) follows.

To show the necessity, since F can be represented as (4.1), where the operation

Problem reduction under uncertainty 51

satisfies (C1) - (C3) , by T h e o r e m 3.2 (let S = [0, 1] and e = 0), F satisfies (A1) and (A2). T o prove A3, we show that if x /> y then

F ( a l . . . . , a i _ l , x , a i + 1 . . . . , an) > F ( a l , . • • , a i - 1 , y , ai+l . . . . . a n ) ,

for all sets of aj's where aj • [0, 1], j • {1 . . . . , n} and j :/: i. Le t

a = F ( a l . . . . . a i_ l , O, ai+ 1 . . . . . an)

and j :/: i. By (A1), Fj(a) = a. Thus by (A2) , ( E l ) and (C2),

F ( a l , . . . , a i -1 , x , ai+l, . . . , an) = Fi j (x , a) = x "Ira,

and

F ( a l . . . . , a i -1 , y , a i + l . . . . . an) = F i j ( y , a) = y ~r a.

If x ~> y, then, by (C4), x ~r a >/y -k a. T h e r e f o r e

F ( a l . . . . , ai-1, x , ai+ 1 . . . . . an) > F ( a l , . . . , ai-1, y , ai+l, • • • , an).

This is what was required.

Theorem 4.2 (dual of T h e o r e m 4.1). L e t F : [0, 1] n ~ [0, 1], n 1> 3, be a f u n c t i o n .

T h e n F can be d e c o m p o s e d as

F ( x l , x2 . . . . . xn) = x l * x2 * " " * x n , (4.2)

where * is a t r iangular n o r m , i f f F satisfies the f o l l o w i n g cond i t ions :

(B1) F(1 . . . . . 1, Xi, 1 . . . . . 1) = x i , f o r a l l x i • [0, 1] and i • {1, . . . , n}; (B2) utility i n d e p e n d e n c e ho lds at p o i n t (1, 1, . . . , 1); (B3) it is nondecreas ing wi th respect to each variable .

The p roof is omit ted. A tr iangular conorm ~- is said to be strict if it satisfies the condi t ion: if x > y

then x ~-u > y ,ku , for all x, y, u • [0, 1). Dually, a t r iangular no rm * is said to be strict if it satisfies the condit ion: if

x > y then x * u > y * u, for all x, y, u e (0, 1]. For strict t r iangular conorms or norms, the following theo rems hold.

Theorem 4.3. L e t F:[O, 1]"-->[0, 1], n I> 3, be a f u n c t i o n . T h e n F can be d e c o m p o s e d as

F ( X l , X2 . . . . . X n ) = X 1 • X2 • " " " ~t X n ,

where ~ is a c o n t i n u o u s strict t r iangular c o n o r m , i f f F satisfies the f o l l o w i n g cond i t ions :

(A1) F(O . . . . . O, xi , 0 . . . . . O) = xi, f o r all xi e [0, 1] and i e {1 . . . . . n}; (A2) utility i n d e p e n d e n c e ho lds at p o i n t (0, 0 . . . . . 0); (A3) it is strictly increas ing wi th respect to each var iable in the d o m a i n [0, 1)"; (A4) it is con t inuous .

52 Li Zhong-Fu

Theorem 4.4 (dual of Theorem 4.3). Let F: [0, 1] n ---* [0, 1], n/> 3, be a function. Then F can be decomposed as

F ( x 1 , x2 . . . . . Xn) = X 1 * X 2 * " • . * X n ,

where * is a continuous strict triangular norm, iff F satisfies the following conditions:

(B1) F(1 . . . . ,1 , xi, 1 . . . . . 1) =xi, for all xi ~ [0, 1] and i ~ {1 . . . . . n}; (B2) utility independence holds at point (1, 1 . . . . . 1); (B3) it is strictly increasing with respect to each variable in the domain (0, 1]~; (B4) it is continuous.

The proofs of Theorems 4.3 and 4.4 are trivial and are omitted. As is well known [5], the continuous strict triangular conorm has the form

x ~ty = f - l ( f ( x ) + f ( y ) ) ,

where f is a continuous strictly increasing function from [0, 1] to [0, +o0] with f(0) = 0, f (1) = +~. And dually, the continuous strict triangular norm * has the form

x *y = h - l (h (x )h (y ) ) ,

where h is a continuous strictly increasing function from [0, 1] to [0, 1] with h(0) = 0, h(1) = 1. So, according to Theorem 4.3 and 4.4, we have:

Theorem 4.5. Let F: [0, 1] n--~ [0, 1], n >I 3, be a function. Then F can be represented as

F(xl . . . . . xn) = f - 1 f (x i (4.3) l 1

where f is a continuous strictly increasing function with f (O) = 0, iff F satisfies the conditions (A1), (A2), (A3) and (A4) of Theorem 4.3.

Theorem 4.6. Let F:[0, 1]n---~ [0, 1], n />3, be a function. Then F can be represented as

F(X1 . . . . . Xn) = h -1 h(xi (4.4) i = 1

where h is a continuous strictly increasing function with h(1) = 1, iff F satisfies the conditions (B1), (B2), (B3) and (B4) of Theorem 4.4.

Acknowledgements

The author wishes to express his sincere thanks to the late Professor K.S. Fu for his invitation to visit Purdue University and for his advice prior to his untimely death on April 29, 1985. In addition, the author wishes to thank Professor J.T.P. Yao for his discussions and suggestions.

Problem reduction under uncertainty 53

References

[1] C. Alsina, E. Trillas and L. Valverde, On some logical connectives for fuzzy set theory, J. Math. Anal. Appl. 93 (1983) 15-26.

[2] D. Dubois and H. Prade, A review of fuzzy set aggregation connectives, Inform. Sci. 36 (1985) 58-121.

[3] M. Ishizuka, K.S. Fu, and J.T.P. Yao, Inference procedure under uncertainty for the problem-reduction method, Inform. Sci. 28 (1982) 179-206.

[4] E.P. Klement, Operations on fuzzy sets: An axiomatic approach, Inform. Sci. 27 (1982) 221-232. [5] C.H. Ling, Representation of associate functions, Publ. Math. Debrecen 12 (1965) 189-212. [6] N.J. Nilsson, Principles of Artificial Intelligence (Tiogo, Palo Alto, CA, 1980). [7] B. Schweizer and A. Sklar, Associative functions and statistical triangle inequalities, Publ. Math.

Debrecen 8 (1961) 169-186. [8] L.A. Zadeh, Fuzzy sets, Inform. and Control 8 (1965) 338-353.