Processing in relational structures: Fuzzy relational equations

Fuzzy Sets and Systems 40 (1991) 77-106 North-Holland

77

Processing in relational structures: Fuzzy relational equations W. Pedrycz Department of Electrical and Computer Engineering, University of Manitoba, Winnipeg, Manitoba, Canada R3T 2N2 Received March 1990 Revised August 1990 Abstract: The paper provides an overview of methodology for processing fuzzy information in relational structures completed in terms of fuzzy relational equations. The origin and the central role of relational calculus in general, and fuzzy sets in particular, is explained with special emphasis paid to its role and representation capabilities. It is pointed out that fuzzy relational equations play a significant role as a platform for a uniform development of techniques in fuzzy sets. Many of the problems and frameworks developed in fuzzy sets so far can be easily translated into the language of fuzzy relational equations which immediately takes a significant advantage of their solid, well developed formalisms, techniques, and clarity of exposition. We will study the role of approximate solutions to fuzzy relational equations as a convenient tool to handle probabilistic type of uncertainty. Moreover those solutions can easily explain the origin and generate a way in which fuzzy sets of higher generality (such as for instance type-2 sets, interval-valued sets,...) can be algorithmically determined. Finally, we study the role of equations in diverse fields of applications making use of an ample framework of general systems theory. Keywords: Fuzzy relational equations; types of equations; solutions; approximate solutions; relational structures.

1. Introductory remarks: Why fuzzy relational calculus? T h e c o n c e p t u a l r o l e o f fuzzy sets in d e s c r i p t i o n a n d e m u l a t i o n o f h u m a n activity, e s p e c i a l l y in c r e a t i o n o f a r e l e v a n t c o g n i t i v e p e r s p e c t i v e is welld o c u m e n t e d a n d s t r o n g l y u n d e r l i n e d , see e.g. [55, 42, 41, 43]. T h e n o t i o n o f r e l a t i o n is a b a s i c o n e in m a t h e m a t i c s , s c i e n c e , a n d e n g i n e e r i n g . E v e r y t h i n g is r e l a t e d o r u n r e l a t e d ; for i n s t a n c e w e can c o n c l u d e t h a t s o m e v a r i a b l e s o f c o m p l e x s y s t e m s a r e r e l a t e d . O n e c a n also i n d i c a t e v a r i a b l e s w h i c h d o n o t influence e a c h o t h e r a n d t h e r e f o r e a r e t r e a t e d as u n r e l a t e d . W h e n t h e complexity of any situation increases the predicate 'being related' becomes more difficult to d e s c r i b e a n d h a n d l e in t h e s e n s e t h a t it b e c o m e s e x t r a o r d i n a r i l y artificial to f o l l o w t h e w e l l - k n o w n a n d w i d e l y a c c e p t a b l e n o t i o n o f this t w o - v a l u e d p a r t i t i o n ( n a m e l y i n v o l v i n g t h e t e r m ' r e l a t e d ' as o p p o s e d to ' u n r e l a t e d ' ) . R a t h e r t h a n a d h e r e to this r e s t r i c t e d p a t t e r n o f r e a s o n i n g o n e can t h i n k a b o u t a g r a d u a l t r a n s i t i o n b e t w e e n t h e t e r m s . T h i s a p p r o a c h b e c o m e s e s p e c i a l l y a t t r a c t i v e in s o - c a l l e d ' s o f t ' s y s t e m s w h i c h a r e n o t g o v e r n e d b y s t r a i g h t f o r w a r d p h y s i c a l laws 0165-0114/91/$03.50 © 1991--Elsevier Science Publishers B.V. (North-Holland)

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but are strongly tied with a significant unavoidable human component (manmachine systems in general, and manual control systems, intelligent computer interfaces, knowledge-based systems, etc. in particular). The complexity of the system itself may cause that even when its components can be adequately characterized, the vast number of relationships implies that only some of them can be viewed as significant. To highlight the generality of the concept of relations as well as point out their essential meaning, it is worthwhile to recall Goguen [18]: "the importance of relations is almost self-evident. Science is, in a sense, the discovery of relations between observations..." Zadeh and Desoer [53] made a strong statement on the importance of studies on relational equations in terms of general systems theory: on this ground any composition used in the equation yields a fuzzy output set associated with another fuzzy set viewed as the input of the system. Hence fuzzy sets are used to cope with the ambiguity of the concepts and form its appropriate representation, while relational calculus is involved in handling ambiguity at the level of dependencies (relationships) between the concepts. Fuzzy relational calculus makes the above mentioned objectives realizable at the level of processing individual grades of membership. Furthermore, fuzzy relational equations can be viewed as a tool to realize computational features of the calculus. Relational equations have quite a long history. Referring back to operations research, they were studied by Rudeanu [43] in the middle of the 1970's. They were exclusively devoted to the processing of Boolean variables in relational structures. An interesting contribution made in terms of multivalued logic applied to a digital circuits has been made in [27]. In the area of fuzzy sets, relational equations were introduced in 1976 by a pioneering research of Sanchez [45] and completed with a particular emphasis put on applicational issues in medical models of diagnosis. He introduced and studied equations with a basic sup-min composition (max-min in finite universes of discourse), while fuzzy sets were defined in Brouwerian lattices. From that date we can witness a steady development of this area of research. To underline selected streams of studies let us recall: • The theory of equations focussed on topics such as: -characterization of families of solutions of equations [1-7, 10, 28-30, 48, 49, 51]; -determination of solutions with specific algebraic properties [7, 22]; -building different structures of equations including a variety of composition operators and/or their topologies [8, 11, 12, 18, 33]; -studies on approximate solutions of equations [20, 21, 23, 25, 31-34, 36]; -equations defined for fuzzy sets of a higher order or interval-valued fuzzy sets. • Applicational issues of equations initiated by models of diagnosis, later including models in systems analysis [33-35, 37, 42], pattern recognition [38], knowledge-based systems [8, 9], etc., to name only a few selected fields. The equations can be also treated as a convenient platform for a general

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formulations and a uniform treatment of problems arising in the theory of fuzzy sets itself; see for instance an inverse problem in fuzzy arithmetic, discussed e.g. in [52], later finally solved in [13, 37] and also studied in [47]. The paper will provide an overall presentation of fuzzy relational equations. Of course it is impossible to cover all relevant topics; therefore we selectively address some core issues. First, we study types of equations available nowadays, consider their interpretations and give a sample of reasoning leading to solutions of some of the types of equations. Methods of approximate solutions form a bridge between the theory and most of the potential or existing fields of applications. It is our persuasion that these topics should be included in this paper. Finally, a detailed discussion covers applications referring to a general approach established within system theory. The content of the paper, which is structured into several sections, reflects the above composition. It is our objective to evince clearly that fuzzy relational equations can be perceived not only as a conceptual framework covering representation issues but are equally important as plausible computational structures in fuzzy set theory. To give a detailed insight, it is desirable to avoid some unnecessary algebraic burden. There is no doubt that analysis of algebraic properties forms an interesting fertile and significant field of studies, but it is decided here to avoid them as far as possible, referring to relevant references. It does not, by any means, entail that mathematical rigor of the exposition is going to be sacrificed in the overall presentation of the topics of this paper. Furthermore, for the reader's convenience, we will utilize a standard notation of fuzzy sets using capital letters to denote fuzzy sets and relations. Moreover, we will comply with the standard notation of discrete universes of discourse using bold letters, and denoting the family of fuzzy sets defined in them by F(.). Again, all operations will be denoted using common notation such as, e.g. ^ for minimum, and v for maximum.

2. Types of fuzzy relational equations. Terminology and interpretation This section will deal with different forms of equations which are found in theory and can be used in many applications. We will start our discussion with various composition operators utilized there, afterwards elucidate some interpretations of these equations from logical, set-theoretic and network-like standpoints. A basic type of equation will involve set-relation compositions. Relationrelation compositions can be introduced afterwards in a natural way as a straightforward extension of previously given operations. In the following discussion we will consider a fuzzy set X defined in X, and a fuzzy relation R is specified in X × ¥ involved in the process of composition. Types of composition operators frequently used in relational equations are listed below: M a x - r a i n composition. A fuzzy set X ~ F ( X ) is combined with a fuzzy relation R ~ F ( X × Y) yielding a fuzzy set Y ~ F(Y): Y=XoR

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W. Pedrycz Table 1. Basic types of fuzzy relational equations

X*R ~ XDR

X.R XIR

Adjoinmess I XtrR

I XeR

XtpR

XflR

with the membership function calculated by taking the maximum over respective minima of the involved membership functions, say

Y(y) = max[min(X(x), R(x, y))] = V [X(x) ^ R(x, y)]. xeX

x~X

Min-max composition. This composition implies the fuzzy relational equation dual to that obtained by means of max-min composition. Now we get

Y(y) =

min[max(X(x), R(x, y))] = / ~ [X(x) v x~X

R(x, y)].

x~X

Rewriting it in a compact form one has

Y=Y.R. Both types of equations can be generalized by studying t-norms as well as s-norms as generalized fuzzy set connectives. This implies

Y(y)=max[X(x)tR(x, y)]= V (X(x)tR(x, y)), xEX

Y=XOR,

x~X

and

Y(y) = min[(X(x) s R(x, y))] = /~ (X(x) s R(x, y)), x~X

Y = X • R.

x~X

Since the minimum is a maximal t-norm, namely a tb <-rain(a, b) for all t-norms,

and the maximum is the smallest one, we get the following bounds, characterizing the range of all possible results of composition: V [(X(x) tR(x, y))] ~< V [X(x) ^ R(x, y)], x~X

x~X

A [(X(x) s R(x, y))] 1> /~ [X(x) v R(x, Y)I. x~X

xEX

Adjoint fuzzy relation equation. This is introduced by taking a so-called pseudocomplement. Restricting to continuous t-norms, see also [10], we define the following operator: a cpb=sup{c e[O, 1] l atc<~b}.

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For the t-norm specified as minimum we have a so-called tr-operator studied in [451: 1 if a~b, atrb=sup{ce[O'l]la^c<~b}= b i f a > b . Thus the adjoint equation to that with max-min composition is expressed by re-composition, whereas generally for equations with t-norm we have the tp-composition.

Y=Xo:R,

Y(y) = min [X(x) o:R(x, y)] xEX

and

Y=XcpR,

Y(y) = min[X(x) ~ R(x, y)]. X EX

By analogy, an adjoint fuzzy relation equation to that with min-max (min-s) composition is defined as

Y=XeR,

Y(y)=max[X(x) eR(x, y)], xEX

Y=XflR,

Y(y)=max[X(x)flR(x, y)], x~-X

where the fl-operator is defined accordingly,

aflb = inf{c ~ [0, 1]lasc >~b}, and, in particular,

a eb = inf{c ~ [0, l l l a v c>~b}. Hence, by straightforward computations carried out in the above formula one derives an explicit expression,

aeb={bo ifa~b. Before proceeding with some other more complex structures of equations, it is worthwhile to look for some possible interpretations. They can be provided from quite distinct standpoints. We will consider logical and set-theoretic standpoints by using the apparatus of possibility measures. Afterwards we will concentrate on their network-like perspective. The first interpretation makes use of some basic logical terminology such as logical operators and, or, not and both universal and existential quantifiers V, 3. They are translated into well-known formulas making use of min and max operators, respectively (or t-norms and s-norms). The quantifiers are read then as

V p(x) = minp(x), x~X

xEX

:rl p(x) = maxp(x) x~X

x~X

with p(x) a certain logical expression with values taken from the universe X. Recall also that the tr(qg)-operator stands for implication operator ---> while the e(fl)-operator describes the dual implication <---.

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In set-theoretic terminology, we are using set intersection, union and complement. Furthermore the operators o~(e) (and correspondingly qg(fl)-operators) are used to represent the property of inclusion or its lack, respectively. To get a closer look let us refer now to a~-and e-operators. The first can be specified as an indicator of the degree of containment of one grade of membership (a) in another (b): {~

ao:b=acb=

ira>b,

ifa<-b.

Thus a t r b returns 1 if a ~ b then we get 0 whereas for a < b this definition returns the upper limit of this degree. Hence a e b equals 1 if and only if b is set to 1. Making a comparison of these two definitions we can observe that the o:-operator exhibits a positive like form of matching yielding a confirmatory type of evaluation of the property and returning 1 when this inclusion holds true. The e-operator has a complementary character to the it-operator providing values equal to 0 when the property of inclusion is not preserved; hence we can speak about the disconfirmatory type of evaluation. Furthermore, adopting notation essential for possibility theory we recall two basic definitions: (a) The possibility measure of two fuzzy sets A, B defined in the same universe of discourse is introduced as the maximal value of their intersection: Poss(A I B) = sr(m I B) = max[min(A(x), B(x))] x~iX

(b) The necessity measure is defined accordingly: Nec(A I B) = 1 - Poss(.,~ I B). Taking this into account, the equation with the m a x - m i n composition can be interpreted from different angles: - T h e logical-like characterization makes use of the existential quantifier taken over 'anding'. - T h e fuzzy set X and the constraint represented by the fuzzy relation R:

y(y) = ::1 [X(x) andR(x, y)]; x~fft"

thus Y(y) is the degree at which the composite expression X(x) and R(x, y) is satisfied (existential quantifiers taken over X), - I n the set-theoretic interpretation, since max and min stand for basic operations on fuzzy sets we write it down as

Y(Y) =

U

(X

n

Ry)

xEX

and then Y(y) is the height of intersection of X and R (obviously the second argument of R is fixed). - I n the possibility-measure interpretation,

Y(y) = P o s s ( X I gy)

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where, as before, Ry stands for the original fuzzy relation R with one argument (y) fixed• -In the graphical network interpretation, the way of specification of the equation is contained in Figure 1 where two types of nodes (with min and max operations) are implemented. Yet another graphical illustration of this equation is characterized by looking at it from a perspective of the commonly used tool of Artificial Intelligence, namely AND--OR graphs. Then AND and OR nodes are serially connected as illustrated in Figure 2. All the above types of equations can be viewed as fundamental in the theory of equations. In existing studies a long list of equations has been investigated. We recall some of them; notice that they strongly rely on the previous types, being usually their generalizations. • A convex combination of max-min and min-max types of equations [30]: Y = A ( X oR) + A ( X . g ) ,

with A: y - o [0, 1] a constant fuzzy set playing the role of the factor controlling the influence of the two plain types (i.e. max-rain and min-max) of equations in the developed construct. Addition is treated as the regular algebraic operation completed on the respective membership functions. • Polynomial relational equations are of the form [29] 1

Y = [._3(A i [] X [] B i) i=1

where A
Y ( y ) = sup[X(x) & R(x, Y)I, xEX

and Y=X&R,

Y(y)=sup[X(x)&R(x,y)] xeX

X(xt)

X(x~)

0

Fig. 1. Network illustration of max-rain type of fuzzy relational equation; observe two types of nodes with rain and max operations.

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IV. Pedrycz Y(Y]) OR

yj)

X(xz)

Fig. 2. The AND-ORgraph interpretation of max-min fuzzy relation equations. where &, & are accordingly,

equality

and

a & b = ( a o c b ) A(btra),

inequality

(difference)

operators

defined

aSzb=(aeb) v(bea)

for a, b e [0, 1]. All of these equations can be expanded in the sense of objects being involved in composition operations. Instead of studying set-relation structures we can consider relation-relation ones. Take three relations S :X X Z---~ [0, 1], R : Z x g--> [0, 1], T :X X g--> [0, 1]. Then the max-min composition implies that

T=SoR where T is fuzzy relation with the membership function

T(x, y) = V IS(x, z) ^ R(z, Y)I, z~Z

which is nothing but a family of set-relation equation indexed by 'y'. The cardinality of this family depends upon the universe ¥.

3. A method of solving fuzzy relational equations This section will illustrate how solutions to fuzzy relational equations are derived. It may give a deeper insight into techniques and the way which one has to follow to handle computations in relational structures. To make the entire discussion more specific, it will be devoted to equations with max-t composition.

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Referring to the structure of the equation, two basic problems are formulated: - determine R for X and Y given, - determine X for R and Y given. From now on let us make a preliminary assumption that the corresponding families of solutions to the two above stated problems are nonempty, namely

~= {R IXDR= Y}=#O, ~={XIXDR=Y},O. The solution to the first question is achieved in the following way of reasoning. First of all, let us introduce set-set and set-relation compositions based upon the tp-operator. Taking X ~ F(X) and R ~ F ( X X Y) we define X tp Y and R q9 Y in a pointwise manner:

( X cp Y)(x, y) = X ( x ) cp Y ( y ) and

(R q) Y)(x) = min[R(x, y) cp Y(y)]. y~Y

We start with two propositions [32].

Proposition 1.

V

V R=Xq2(XDR).

X~F(X) Y~F(I")

Proposition 2.

V

V

X[](XqgY)cY.

X~F(X) Y~F(Y)

They give some estimates (subset and superset inequalities) which are preserved for all fuzzy sets and relations coupled in terms of m a x - t and tp-composition. Combining the results of these two propositions we get:

Proposition 3. If ~ 4=0 then the greatest element of ~ , denoted by /~ = max (where maximum is viewed in the sense of set-theoretic inclusion), is calculated as

t=xwY. Proof. The proof results as a direct combination of inclusions provided by the above propositions. The first one involves that R c / ~ with/~ = X q0 Y. In virtue of monotonicity of m a x - t composition, from the inclusion R c / ~ we obtain X [] R c X []/~, namely Y c X []/~. The second proposition implies that X D/~ c Y. Then combining these two containments (subset and superset conditions) we come up with the final equality X []/~ -- Y.

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Table 2. Solutions to different types of fuzzy relational equations max-min max-t min-max min-s

R = X o~Y, X = R ocY R = X q~Y, X = R cp Y

R = X e Y, X = R e Y R = X fl Y, X = R fl Y

Two further propositions create a basis for the solutions of the second problem:

Proposition 4.

V

V

(R q) Y ) M R c Y.

X~F(X) R~F(X×¥)

Proposition 5.

V

V

X c R cp ( X E] R )

XeF(X) R~F(XXY)

Following the same way as before, the subset and superset conditions given by the proposition merged together imply:

Proposition 6. I f ~ : / : ~t then the greatest element o f ~ , c o m p u t e d as cp-composition o f Y and R:

say ) ( = max ~,

/s

X=RqvY.

Solutions to the remaining types of equations can be obtained in a similar fashion by proving relevant subset and superset conditions and using them to derive relevant equality constraints. Table 2 forms a concise summary of all corresponding results associated with the remaining types of equations. As we have underlined, all of the results have been derived on the basis of a strong assumption about the existence of solutions. This forms an essential component of the entire theory. Further on in Section 5 we will investigate methods leading to approximate solutions.

4. Eigen fuzzy sets as solutions of homogeneous fuzzy relational equations We will recall here a special type of equation which leads to so-called eigen fuzzy sets. Let us discuss a fuzzy relation defined on the Cartesian product of the same space X x X, R :X X X---~ [0, 1]. Then consider the following equation: XoR =X,

to be solved with regard to X. The elements of the family of solutions to the problem will be called fuzzy eigen sets associated with the fuzzy relation R. It becomes obvious that they create invariants of the relation with respect to the

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max-min composition (notice that in this way we have restricted ourselves to idempotent t-norms). One can also observe that this family is nonempty since X = 0 satisfies the equation. In [46] Sanchez proposed a set of algorithms to determine the largest eigen fuzzy set. We recall one of them in which a finite sequence of iterations generates the solution. Starting from the set El,

El(x) = max R(x, y), Y

and iterating:

E2=EloR,

E a = E 2 o R = E l o R 2. . . . .

we come up with the fuzzy set E which becomes invariant with respect to R. In [14] an interesting method has been proposed to form the entire family of solutions by building so-called generators of the fuzzy relation R, G1, G2. . . . . Gn, dim(X)= n. Then all eigen fuzzy sets are expressed as a linear (max-min driven) combination of the generators, namely,

E' = (AI f'l G1) U (A2 f'l G2) U - . - U (An ('1 Gn) where A1,/12, • • •, An are fuzzy sets defined in X and representing the weights in this combination.

5. Approximate solutions of equations The theory of fuzzy relational equations developed so far strongly relies on a fundamental assumption that the family of solutions is a nonempty one. From an applicational standpoint, however, there is no guarantee at all that this demand can be easily satisfied. If, what is usually the case, this assumption is violated, then to make the solutions (or pseudo-solutions) feasible one has to construct an interface and establish some suitable links between the theory and practice of fuzzy relational computations. We will start from a single equation and afterwards study conditions and methods which can imply a higher solvability of their respective systems. For any single equation necessary and sufficient conditions which guarantee existence of solutions can be formulated in a rather straightforward manner. In the following we specify relevant conditions for existence of the family ~ . - for max-t equations the height of the fuzzy set X should be higher or equal to the height of Y (recall that the height of the fuzzy set is defined as its maximal value among its grades of membership):

V 71 X ( x ) >i Y(y); ye. Y x e X

-for min-s equations the condition can be viewed as a dual to that specified above for the first type of equation, namely V 3 X(x)<~ Y(y); y~¥ x~X

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- f o r both adjoint and convex combination types of equations we have more flexibility since there are no constraints imposed on X and Y which should be fulfilled to get these equations solvable. As single equations can be studied relatively easily we are not aware of the existence of powerful and simple straightforward conditions which can assure solvability of the entire system of equations. This stems from the fact that a vast number of quite complex constraints has to be satisfied in order to make the system solvable. A way which has been studied in the theory is the following: assuming that there exists a nonempty family of solutions to all of the equations (which by default assumes that each individual equation does possess a nonempty family of solutions), we are able to determine its extremal element. Its character is determined by the character of the composition operator utilized. Then these extremal solutions to the individual equations are aggregated. For easy reference all relevant results are summarized in Table 3. An interesting fact which deserves attention refers to a general classification of the solution we are faced with. The task of solving the system of equations as it stands now falls into a general category of interpolation problems. In other words, we can look at the system of equations as a set of constraints (loosely speaking, a collection of 'fuzzy' points) which have to be satisfied, in the way the 'fuzzy' surface (relation) should be constructed to accommodate all of the points. Unfortunately, an analogy one can draw between this problem and curve interpolation issue cannot be pursued very far since the context of the latter task is much more sophisticated. Hence no direct conclusions similar to that obtained Table 3. Solutions to construction problems for some selected types of fuzzy relational equations, k = l , 2. . . . . N Equation

Construction problem N

X~,oR=yk

R= N k=l

(XkaY~)

N

X k O 4 = Yk

k=l N

Xk.R=Y k

4 = U (xk/~ Y,~) k=l N

Xk O R = Yk

R= U (Xk e Yk) k=l

N

X ~ @ R = Yk

4 = U (x~ @ Y~)~ N

4 = N (xk or Yk)a k=l

a The Cartesian product and the 'or' operator are associated with the composition operator of the equation, namely t- and s-norms, respectively.

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e.g. for polynomial curve fitting can be derived. Moreover the problem of 'capacity' of fuzzy relations (expressed in terms of a certain composition operator) has not yet been studied. To get a better insight into how a lack of solutions can be manifested, let us consider the following simple example dealing with two relational equations. They are to be solved with respect to the fuzzy relation. Also different composition operations will be specified. Let us take into account two pairs of data X1, Y~ and X2, Y2 such that X1 = [1 0], X2 = [1 0], I"1 = [1 6], Y2 = [1 6] which are viewed as input and output fuzzy sets forming the system of equations. There are four basic structures investigated: (i) Max-rain composition: A solution to this problem with R unknown is taken as g = (X1 tr Y1) fq (X2 oc Y2), which implies the following result:

By simple calculations one checks that X 1 ° R = [6

6] c [1

6],

X2oR=

[6

6]c[6 1],

and thus in both cases we get underestimated results; especially when 6--~0 the results tend to be meaningless. (ii) min-max composition: In this case R = (XI e I11) U (X2 e Y2) and this type of aggregation produces significant deviations from the sets Y~ and Y2 yielding almost noninformative results:

X 1 •

R = [1

1] ~ [1

6],

X2" R =

[1

11~ [6

1];

hence we have obtained a fuzzy set conveying no useful information. (iii) Adjoint equation with tp-composition: This yields R = (X1 x Y0 U (X2 x Y2), namely,

and again we get meaningless supersets: X1 tp R = X 2 q0R = [1 1] ~ ([1 6] and [6 11). (iv) Adjoint equation with fl-composition: Here R is composed as R = X1 s II2 N X2 s Y2 ('s' being a suitable s-norm associated with the fl-operator). This yields

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W. Pedrycz

and furthermore X, fiR = [6

6] c [1

61

and

X 2 fiR

= [6

61~- [6

1].

We can observe that the inclusion of the two equations with disjoint families of solutions dramatically reduces the quality of results produced. Two basic approaches can be considered with respect to the lack of solutions. The first, having strictly a passive character, enables us to measure the property of 'being solvable' and treat it as a multivalued logic predicate, cf. [20, 21, 23-25]. The second, taking an active form, is concentrated on various modifications of the equations of the system. Referring to the first stream of investigations [24] a discussion on the solvability index of a single equation as well as the entire system becomes a central component of this approach. The solvability index has a logical background and measures the degree of equality between the original fuzzy set Y (or Yk in the system of equations) and that produced by the equation(s). In the active approach several 'active' means are undertaken whose role is to modify the constraints created by fuzzy sets Xk and Y~, k = 1, 2 . . . . . N, in order to achieve a higher degree of solvability. Several distinct ways of modifications have been explored. A general taxonomy can be reported in this regard: (a) Modification of fuzzy sets by a certain gradual transformation of their membership functions, cf. [23]. Its underlying idea is to modify Xk and Yk by their 'fuzzified' versions, denoted by X~ and Y~, where X~ (Y~', respectively) are computed on the basis of Xk as follows:

X~(x) = max(Xk(x), o:),

oc e [0, 1].

The introduced threshold level o~ underlies the fact that a membership function with grades smaller than ~r is less 'reliable' and should be 'filtered out' before this fuzzy set is entered into the computations of the fuzzy relation. Completing this modification for all Xk and Yk we have to work with the system of equations

X'~oR=Y~,

k = l , 2 . . . . ,N.

An interesting property of the method is its monotonicity, namely, the higher the threshold, the higher the degree of solvability of the system of equations. Of course one should consider a rational trade-off between the level of the solvability index and the induced specificity of the fuzzy sets (i.e. the degree to which the fuzzy set X ~ is similar to the set with membership function identically equal to 1 distributed over the entire universe of discourse). (b) The background of the second approach is to analyze the structure of the data set (Xk, Yk) and, on the basis of that, perform a clear distinction between its components with respect to their varying contribution to the formation of the fuzzy relation. A general rule discriminating between the data should take into account the level of consistency of the individual pair with respect to the remaining family of data. The higher the level of consistency, the stronger the contribution of this particular data to the computed relation. Few streams of investigation can be distinguished: (i) In the first one a probabilistic layer is constructed which combines the entire characterization of consistency in the data in terms of respective condi-

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tional probabilities- this generates an underlying probabilistic information about the structure of the relation, cf. [25, 33, 36, 40]. (ii) The second method is of purely deterministic character and its main objective is to rank all of the data with respect to their consistency. The structure can be determined with the aid of some hierarchical clustering methods as introduced in [30]. Then the most consistent part of the data ('core' data) is used towards formation of the fuzzy relation whereas all of the remaining ones are skipped and therefore do not deteriorate the estimation process. Of course, a reasonable compromise has to be achieved: too many inconsistent data can lower the solvability of the system of equations while, on the other hand, only few data can produce a poor estimation of the fuzzy relation because of the lack of representation power of the remaining fuzzy sets; for a detailed discussion refer to I36]. The hierarchy in the data set can be figured out by completing all pairwise comparisons. Let us organize the relevant algorithm in a sequence of steps: 1. Compare Xk and XI, k, l = 1, 2 , . . . , N, as well as Yk and Yt calculating averaged values of equality indices for all of them. Organize them in two N × N matrices, denoting them by ~ and ~ respectively. 2. Perform pointwise q0-composition on the matrices, which yields a matrix of inconsistencies ~ = ~q~ ~. Its (i,j)-th entry, ~(i, j), describes the degree to which the i-th and the ]-th pair of data are consistent. The definition of q~-composition implies that as far as the ~(i, j) is greater than ~(i, j) then ~(i, j) attains values lower than the second argument of the composition. This, however, indicates higher similarity of Xi and Xj with respect to the similarity existing between Yi and Yj, simultaneously pointing out a lack of consistency which is reflected by the low value of the (i, j)-th entry of ~ . 3. Select a row in ~ denoting it by i0, for which the sum of entries attains maximum, i.e. N

io = arg max ~ ~(i, j). i

j=l

This index pertains to the first element in the data set which has been detected as the most consistent one (in the light of the discussed criterion). Afterwards reduce ~ eliminating its i0-th column and row, thus obtaining ~ ' . 4. Repeat 3 for a sequence of reduced matrices until a 1 x 1 structure is derived. The data selected by a sequence of indices as determined on the basis ~ , L e ' , . . . are used in successive computations of the fuzzy relation. At the same time the values of the obtained cumulative consistency degree can serve as a suitable indicator of increasing inconsistency controlling the progress of further computations and serving as a stopping criterion.

6. Application considerations Fuzzy relational equations by their general nature were also studied in a vast number of applications. Originally they were considered in modelling of

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situations of medical diagnosis [45] by expressing dependences between spaces of symptoms, diagnoses, and patients in terms of two-argument relations defined on twofold Cartesian products of the respective spaces. The use of equations in other fields was studied in many papers; not pretending to give an exhaustive list of applications some selected examples are contained in Table 4. It is convenient to take a broad look at applications following general systems methodology as it has been discussed in [34]. In this framework basic problems are formulated, solved and discussed. Having this stage completed and faced with a certain domain of application we translate them to accommodate domainoriented specifications. Three problems are envisioned, namely [34]: - construction (which sometimes is referred to as identification problem), - direct problem, - inverse problem. The first one is centred around all construction issues which specify the type of fuzzy relational equation being of interest for the problem as well as all entries of the fuzzy relation. Adopting a type of terminology used in system identification we can distinguish between structure determination and parameter estimation. The latter task refers mainly to specification of a suitable type of relational ties being present in the system, a particular composition operator and fuzzy sets referring to state, control or output variables of the system. All of the above subtasks can be solved by taking a close look at the data and the problem itself. First, one has to make a clear statement that the level at which fuzzy sets operate and the way they are used to manipulate all the available data stipulate nonexistence of fundamental physical laws the system may adhere to. This is due to two major reasons: (a) First, the very nature of the systems studied is such that they are not governed by physical laws. For instance decision-making problems have nothing to do with physics. A vast number of Artificial Intelligence topics which relate to

Table 4. Selected examples of applications of theory fuzzy-relational equations fuzzy control:

-validation of control protocol, -identification of fuzzy models of the fuzzy controller, -reduction problem, -sensitivity analysis, -controllability and observability, -optimal control

decision-making:

-preference structures between goals and constraints, -evaluation of importance of objectives, -multistage decision making

knowledge-based systems: -knowledge acquisition, -intelligent interfaces, -inference mechanisms, -linguistic approximation

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problem-solving activity are again not well describable in terms of differential or difference equations having their origin in 'hard' sciences. (b) Secondly, due to the depth of cognitive perspective developed by fuzzy sets, cf. [55, 41], a description of physical systems is completed at its conceptual, or better to say, qualitative level by which we try to emulate the way a human being is able to cope with complexity, uncertainty, and ambiguity of the system, preserving his learning capabilities. Thus fuzzy sets can be also viewed as tools for qualitative modelling, cf. [41], with features of subsymbolic computations added to that original concept. Bearing this in mind, we can consider the search of a suitable structure of the model as completed over a family of sentences (linguistic description) characterizing the structure in terms of variables as well as its dynamics. This dynamics is usually not very high since it originates from the limited length of statements generated and accepted in natural language communication, in particular, in knowledge-based systems. Dynamical properties of the system induce a basic structure of the equations. A proper choice of the type of composition operator can be selected among finite types of them, constituting classes of equations. They can be determined according to e.g. some specific boundary situations occurring in the input-output data. Afterwards, for the given composition operator, the fuzzy relation is determined. At this point it is computed by direct application of formulas coming from theory or, what is usually the case, estimated and obtained as an approximate solution to the family of equations. Moreover, the use of methods studied in the previous section gives rise to confidence intervals [39] where so-called intervalvalued fuzzy sets are computed [44]. Further on we will comment on their importance in formulation of direct, and especially, inverse problems. Let us now focus attention on fuzzy data (XD Y1), (X2, Y2) . . . . . (XN, YN). This format implies a single-entry single-output structure of the equation. The direct problem, as its name suggests, deals with computations of the output for a given input fuzzy set. They are worked out by performing a specified composition operation. Because of the approximate form of solutions, the resulting fuzzy set is tied with confidence levels and finally, by straightforward transformations, associated with interval-valued fuzzy sets or fuzzy sets of type-2 (i.e. fuzzy sets with membership grades being again fuzzy sets defined in [0, 1]). This can be schematically put down as X~

Y--~ [Y_, Y+I

where both R, F come from the construction stage. Y is determined with respect to X and R while a nondegenerated confidence interval occurs due to values of the confidence interval usually lower than 1. In the direct problem we can discuss the question of propagation of uncertainty represented by confidence intervals. Let us start with a chain of equations with relations and associated confidence values R, F, G, ~, T, J . . . . such that their dimensions are consistent (treating the relation as two-dimensional matrices we

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assume that dim(R) = n × P l ,

dim(G) = P l xp2,

dim(T) =P2 xp3 . . . .

holds). If confidence values were neglected then for X provided a straightforward sequence of compositions yields: X

Y Z

opl ,R

op2,G op3,T

~Y, ~Z, ,W,

(of course each equation in row can possess a different composition operator). Finally the fuzzy set occurring at the end of the chain is solely dependent on a sequence of composition operators of the equations and the corresponding relations. Only few general results can be derived which are helpful in characterization of behavior of the output fuzzy set; they are restricted in such a sense that max-min composition is involved and the same fuzzy relation is iterated several times. This in fact refers to a series of powers of any square relation and addresses the process of convergence of this sequence, cf. [50]. The use of confidence values can enhance the final look at the resulting fuzzy set from the standpoint of its accuracy. Let us investigate the same sequence of compositions, Opl , R , / " ,

op2, G, q0, op3, T, J.

The first iteration induces for X an interval-valued fuzzy set, X op,.R.r y - o (y_, y+).

Secondly the lower and the upper bound is processed separately yielding Y_ op2,C.,~ Z,---~ [Zl, Z2]. Completing the same composition for the upper bound we obtain Y+

op2,G,

, Z*----~ [Z3, Z4].

The pessimistic way of aggregation (i.e. leading to the broadest interval) is that in which the minimum of lower bounds Z1 and Z3 is realized. For the upper bounds Za and Z4 we take maximum. Finally, the second itertion generates a fuzzy set with bounds [min(Z~, Z3), max(Z2, Z4)]. Repeating the same procedure at all remaining steps of the iteration process we come up with a sequence of interval-valued fuzzy sets with increasing differences between their corresponding

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lower and upper bounds. Treating the difference between the bounds as an indicator of accuracy of achieved results in the direct problem we will consider a stopping criterion of the form: • Halt the process of solving a chain of direct problems when the above stated difference exceeds a predefined threshold. The inverse problem is formulated as follows: (i) Withour confidence values taken into account, for Y and R given, calculate the fuzzy set X satisfying the constraint X

op,R

~Y.

In terms of the problems discussed so far a solution can be immediately recalled from Section 3. (ii) With the confidence value included the inverse problem can be slightly reformulated. The way which is likely to be adopted in this situation is such that a strict equality constraint should be relaxed due to existing limited accuracy. Observe that for values of F lower than 1 the result of computation is always an interval-valued fuzzy set. Therefore it is too artificial to seek a fuzzy set X given Y; in the sense of all the arguments recalled one should look for X for which the generated output is covered by its interval-valued counterpart. In other words, for given F, Y_ and Y÷ are computed and afterwards a solution to a containment constraint problem is searched: X°P,g ~ [Y-, Y÷] (the containment as it stands above is meant in the standard set-theoretic language, namely Y_cYcY÷

where

X

op ,R

~Y).

Now we will concentrate on some selected fields of research which are typical examples of applciations of fuzzy sets. They are suitable on account of numerous experiments, with some industrial impacts already reported. Thus it is beneficial to take a look at them from the perspective of fuzzy relational equations trying to formulate (or reformulate) the problems existing therein and highlight how the methodology of the equations can contribute there by pointing out some overlooked aspects or improve existing approaches. Simultaneously we will keep in mind the idea of specification (particularization) of those three problems of genral system theory in all discussed fields.

6.1. Fuzzy control algorithms Fuzzy controllers (sometimes known as fuzzy logic controllers), born in the middle of the 70's, could be concisely described by a collection of control rules "if condition then control (action)". The construction of the fuzzy relation of the controller as well as a relevant inference scheme is well known. The major assumption, having in fact a default nature, is that all the rules create a consistent control algorithm and they perfectly describe the control policy applied to the system.

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Unfortunately, this is not always the case. Even if the control policy could be formalized by a relatively small number of rules, some of them could be induced by quite different if not contradictory control objectives. This, of course, immediately affects the property of their consistency. The control rules can be viewed as forming inputs (condition parts) and outputs (action) of the fuzzy relation equation. Then the fuzzy relation R of the controller can be formally determined. Various types of equations could be used and different triangular norms could be analyzed to fit the structure of the data (antecedents and actions of control rules). Now the mechanisms of the construction of the relation (which in fact corresponds to the aggregation of the control knowledge) and respective calculations of the fuzzy control are tied together in terms of the equation; see, e.g., [13, 37]. Some experiments performed towards construction of a fuzzy controller with the use of often cited data (rules) clearly indicated that this way of applying equations generates better results than the original one [36]. The experiments were exclusively completed with regard to mapping properties of the controller. These features, which in fact deal with its static character, are tackling the logical background of the controller. An adjustment of the fuzy controller to improve the dynamical properties is not directly handled within this framework. From a system theory standpoint we can provide the following interpretation: - T h e construction problem deals with all issues of building the fuzzy relation of the controller and the optimization process which turns out to be a search of the best-mapping property enabling us to develop and master the static structure of the control algorithm. - In the direct problem we determine the fuzzy control (action) which has to be applied to the system under control. - A n interpretation of the inverse problem can be given accordingly: for a certain control action we are looking backward to understand which fuzzy set of state could cause it. Thus it may be considered as a sort of post-mortem analysis which should generate an answer to the problem of applicability of some control actions. If for a certain control there is no corresponding state (calculated as the solutions to the inverse problem) we can treat it as irrelevant for this protocol. Fuzzy controllers and fuzzy control schemes are also closely associated with fuzzy models. Fuzzy models form a broad class of modelling tools for uncertain systems, cf. [54, 33, 37, 25]. The question of stability control in fuzzy models becomes then essential in the development of fuzzy control procedures. The notion of the eigen fuzzy set of control finds its direct utilization. To reveal this let us consider the first order model governed by the equation X k +l = Uk ° X k ° R

with Uk, X k standing for fuzzy control and state in the k-th time instant and Xk+l describing the state in the consecutive discrete time moment. Now in this formulation, we are interested in determining stabilizing properties of the given fuzzy control U, in particular, the collection of fuzzy states stabilized by this control should be determined. From a formal standpoint the task is

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defined accordingly: Express the family of fuzzy sets

( x l x = Uk oXoR), i.e. all sets being invariant under max-min composition for Uk and R provided. Let us now define another relation,

T=UkoR. Reformulating the original problem we come up with an equivalent statement:

X=XoT. Then it becomes obvious that the collection of states stabilized by Uk is nothing but the family of all eigen fuzzy sets of T. Referring to the method descussed in one of the previous sections, the generators of T are computed and all linear combinations spanned over them induce the family ~Y. 6.2. Fuzzy pattern recognition The use of fuzzy set techniques in pattern recognition has been well investigated [38]. In a very general setting, the process of pattern recognition, especially its classification part, can be conveniently described by pointing out all relevant relationships between the collection of features of the patterns and their class membership vectors. As indicated by Zadeh, the use of fuzzy sets can be viewed as an attempt to transform opaque classification schemes characteristic for human beings into transparent ones implementable on a computer. In such a sense we are dealing with a fuzzy relation equation possessing many input fuzzy sets forming the feature space and a single outpt being a fuzzy set of class membership. Then the fuzzy relation obtained on the basis of the learning set of patterns can be viewed as conveying the entire information useful for classification purposes. Moreover, a collection of interesting features of the relation of the classifier can be investigated, among the other, for instance, discrimination properties. They allow us to select a subset of features that are the most relevant when different classes have to be distinguished. Individual features are considered as well as their selected groups [38]. Both direct and reference classification schemes have been investigated. The reference classifier makes use of a fuzzy set of similarity (equality) or a fuzzy set of difference (dissimilarity) describing mutual relationships between the pattern being classified and the prototype of a particular class. This fuzzy set is viewed as input information of the classifier. The type of equation usually involved in this form of classifier, deals with an equality composition operator [11; 12, Chapter 13]. Returning to the methodology of general system theory, the three problems studied so far can be interpreted as follows: (a) The construction problem refers to studies on choices of recognition spaces (namely selection of features as well as determination of classes of patterns to be discriminated) and formation the fuzzy relation decribing ties (dependence) between the feature and the class spaces.

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98

(b) The direct problem refers to any classification problem solved, namely the feature vector is given and grades of class membership are computed. (c) The inverse problem in pattern classification has an interesting interpretation pertaining to the problem of prototype search. For a given class membership we are looking for its representative (prototype) which represents this class to the highest possible extent. 6.3. Models o f reasoning processes

Models of reasoning in presence of fuzzy information have been studied extensively, resulting in the treatment of a significant number of real-world problems solved with knowledge based systems. A first issue, that has been already raised in the case of the fuzzy controllers, is that the way of knowledge acquisition and the way of inferences (reasoning) are strongly coupled in a straightforward logical fashion. This becomes even more strongly articulated in knowledge-based systems. Let us study rule-based systems to illustrate the models of reasoning completed there. If different rules are combined together (assuring the best mapping of the knowledge could be structured in terms of rules), then the inference operator (composition operator) is uniquely determined. [34] Following this assumption by working with fuzzy relation equations we eliminate a significant portion of ad-hockery present in various existing approaches. The second aspect of reasoning, usually requiring long inference chains concerns the problem of propagation of uncertainty along the chain of reasoning (i.e. expressing it in successive stages of inference). The individual element in the chain of reasoning will be described by a fuzzy relation R and a relevant composition operator. Moreover each stage is characterized by a vector of equality indices specifying the relevance of this inference layer. All these components are schematically visualized in Table 5 where simultaneously we summarize the conceptual level of reasoning as well as indicate its implementation as far as this propagation aspect is concerned. Observe that at the conceptual level of reasoning (as it is visible at the left-hand side of the scheme), the length of the reasoning scheme is not restricted. This might be, however, highly misleading. The fact that the initial premise A is not usually equal to one of the antecedents at the corresponding stage of reasoning Table 5. Propagation of uncertainty Rules (concepts)

Implementation

A Ai---~B~

A R,F

B Bk---~ Ck

C

[B-, B +] G,

[C-, C +]

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and the collection of the rules is not perfectly consistent (which is reflected by the vector of equality indices, F) gives rise to the interval-valued fuzzy set [B-, B ÷] as being the consequence of the reasoning process. In further steps of reasoning, this sort of imprecision is even enhanced. For example, when the max-min composition operator is involved, the right-hand side of the implementation scheme reads as follows:

A AoR =B

r--C-~[B-, B÷]

and at the second stage of reasoning one gets [B-, B*] o G = [G o B - , G o B +l = [C', C"]. Observe that the max-rain composition is monotonic with respect to the arguments involved. This implies that

[c', C"l

[c-, C+l,

where C'>~ C- and C " ~< C ÷. Thus the uncertainty of the reasoning process increases as it proceeds along the chain. It also becomes evident that the interval-valued sets give information about the precision of the inferred conclusion, enabling us to control the length of the reasoning process. These and related topics are carefully investigated in [29, 31, 13]. Summarizing, the construction problem is focussed on building the knowledge base, the direct problem pertains to forward chaining, whereas the inverse problem describes another popular inference procedure known as backward chaining.

7. Selected aspects of fuzzy relation equations in development of foundations and techniques of fuzzy sets In this section we will highlight the direct impact of the framework created by these equations on the development of some foundations of the theory and relevant techniques. We will indicate how they could be made more algorithmicoriented and tied with the existing empirical data processed in terms of fuzzy sets, therefore constituting a suitable feedback between the concepts arising from them and their justification in the light of the established conceptual platform and empirical findings. It is our perception that this sort of link is rather nonexistent at the actual stage of development of the theory. The section will be devoted to the problem of algorithmic ways of establishing more complex concepts in fuzzy sets such as interval-valued sets or fuzzy sets of higher order or probabilistic-type of sets. Moreover we will address the question of how inverse problems in fuzzy

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arithmetic can be handled by transforming the problem into the format of a suitable relational structure.

7.1. Algorithmic issues of determination of high-order constructs in fuzzy set theory A need for studying of more complex constructs containing fuzzy sets appeared at a relatively early stage of development of fuzzy sets. Its purpose was to accommodate a request to lower the precision attached to pointwise values of grades of membership. One straightforward and logically plausible form is to relocate the existing single-valued characterization by a more realistic intervallike specification. It gives rise to the concept of a so-called interval-valued fuzzy set. In other words, the interval-valued set [44] A~ defined in X is defined by a couple of membership mappings A_, A÷, A~,= (A_, A÷), such that A_, A÷ :X---> [0, 1] and satisfying the inclusion property A_(x)<~A÷(x) for all x. Moving one step forward we can define an infinite sequence of so-called m-th fuzzy sets. Following the recursive definition we have: - a type-1 fuzzy set is a fuzzy set, - a type-m fuzzy set is a fuzzy set whose membership values are t y p e - ( m - 1) fuzzy sets. This extension reflects a situation in which it is more suitable to model the grade of membership by another fuzzy set. Unfortunately, this concept being plausible at a first glance is not very convincing mainly due to the infinite regress paradox. By this paradox we mean the phenomenon in which the imprecision is put on a second level; again it is observed that at the next stage we cannot avoid it and therefore a third level might be necessary. In this way this procedure could continue ad infinitum without any mechanism which may complete the chain of nested levels or at least diminish uncertainty at higher levels. Usually type-2 fuzzy sets can be considered as a practical upper limit of the construction. Our statement is such that whenever a fuzzy model is constructed its performance should be directly related with some empirical facts used towards its determination. Furthermore, a rational point of view should be such that when fuzzy sets are involved the results produced by the model should be received as being of higher order of complexity simultaneously conveying useful information about the precision (relevance) of the model itself. This directly implies that interval-valued fuzzy sets, sets of higher type, etc. should be determined in a purely algorithmic way and their occurrence should be justified by a lack of perfection of the framework (e.g. model) which has been developed for processing purposes. We have already shown some cases of this in the previous section while now will discuss selected methods by which a probabilistic form of information contributes towards formation of all those higher-order constructs. Let us deflart from the point in which we possess the results produced by the model Y~, Y2, . . . , I?N and corresponding fuzzy sets creating the elements of the empirical data set, Y1, Y2. . . . . YN used in its formation. Assuming that the respective universe of discourse consists of m discrete elements, all values of

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membership functions are arranged into the following two-dimensional table:

Y~(Yl)

Y~(Y2)

Y,(Ym)

~'l(Yl)

I?,(y2)

YI(Ym)

Y2(Y~)

YE(Y2)

YE(Ym)

I?2(y,)

~'2(Y2)

IT'2(Ym)

Yu(yO

Yu(yd

Yu(ym)

I)N(Y~)

~'N(Y2)

"'"

"'"

l?u(y,~)

Then the construction of higher-order fuzzy sets is carried out pointwise. First for a fixed element of the universe, namely yj, j -- 1, 2 . . . . , m, we compute the value of the equality index [39] Yk(Yr) =--Yk(Yr), and average it over all elements of the data set, k = 1, 2 . . . . . N. This generates the number )'J = av(I?l(Yr) ~- Y~(Yr), I?2(YJ)--- Y 2 ( Y r ) , . . . , YN(Yr) =-- YN(Yr)). Then for any single value of grade of membership obtained from the relational equation (forming the equation of the model), say Y(yj), we derive the confidence interval solving the following inequality:

r(yj) =- ~ > Yr. As shown in [39] this inequality always possesses a nonempty solution set being an interval in [0, 1]. The lower and upper bounds of the interval denoted by Y-(YJ), Y+(Yr) contribute towards the values of grades of membership of the resulting interval-valued fuzzy set at the point yj. Carrying out the above process for all elements of Y, the set [Y_, Y÷] is obtained. In the above attempt, the threshold value of 7r is chosen somewhat arbitrarily as the mean (average) among the series of values of the equality index. We can incorporate a more complete way of examination by studying relationships between the value of Yr and the probability that a certain grade of membership Y is contained in the confidence interval induced by that level Yr- More precisely, one has to work with relationship g,

yr=g(pj),

j=l,

2.....

m,

with

pj = Prob(Yk(yr) ~ F, k = 1, 2 . . . . .

N

t

N) = ~ ,

while N' denotes the number of cases among the N fuzzy sets satisfying the inclusion

Yk(Yr) ~ Fj where F/ stands for the confidence interval induced by y and centered around gk(Yr), Fj = (~ e [0, 1] ] ~'k(Yr) =- ~ > Y}.

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Concisely speaking, N' denotes the cardinality of the family of fuzzy sets covered by the confidence interval. In virtue of the relationships between yj and pj we can conclude that g becomes a nonincreasing function of probability p. In the limit, pj = 1.0 implies the lowest possible value of y whereas pj ~ 0 implies the highest possible level of the index Yr. In this fashion we can derive a straightforward relationship between the level of y and the associated mass of probability. Since the values of probability can be converted into appropriate levels of possibility :%, see [17], by a sort of monotonic transformation (and vice-versa) we obtain yj = g(pj) = g'(a%). This enables us to determine type-2 fuzzy sets by specifying some of its a-levels induced by yj. For further details refer to [39]. 7, 2. Inverse problems in fuzzy arithmetic: formulation and problem handling in relational structures Computations with fuzzy numbers have become a popular approach towards handling uncertainty conveyed by numerical but yet imprecisely defined quantities; refer e.g. to pioneering research reported in [15, 16, 47]. The current status of this field of investigation is such that almost all efforts have been focussed on representation and analysis issues concerning defining and expressing ambiguous variables of the specified problem and proposing new more efficient computational schemes overcoming the significant numerical overhead imposed directly by the techniques of fuzzy sets, see e.g. L - R representation of fuzzy numbers [16]. On the other hand, almost nothing has been done with respect to synthesis aspects. Its essence is to compute one among fuzzy numbers in the model (structure problem formation) involving fuzzy quantities. We will refer to this task as the inverse problem. This, in fact, creates an essential component of the usage of fuzzy models with fuzzy numbers. First findings [52], already being available in interval analysis, were quite discouraging stating that straightforward inverse operations known for real numbers are simply nonexistent for calculus involving sets. This is caused by a lack of group structure for fuzzy numbers under relevant operations. It means for instance that the solution to the equation containing fuzzy numbers A~X=B with A and B known cannot be obtained by applying a straightforward subtraction. Of course in general the equality A ~) (B oA) = B does not hold. The way fuzzy relational equations apply here deals with a suitable interpretation and re-formulation of the problem at hand [13]. To illustrate this we will start with a straightforward example. Let the form of the structure put into account be given as C=A .XE)B with A, B, C fuzzy numbers playing the role of parameters while X is a fuzzy

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number one is looking for. From the extension principle we derive

C(c) =

[A(a) ^ X(x) A B(b)],

sup

c e ~.

a,x,b:c=ax+b

Now we will incorporate linear constraints in the above optimization problem in the form of a suitable Boolean relation T : ~3----~ [0, 1],

T(a,b,x)={lo

if c = a x + b, otherwise.

This yields

C(c) =

[A(a) ^ S ( x ) ^ B(b)]

sup a,x,b~R:T(a,b,x)=l

= sup [X(x) A A(a) A B(c - ax)] a,x~R

= sup[X(x) A sup(A(a) ^ B(c - ax)]. x

a

Furthermore we will introduce another fuzzy relation G : R2---~ [0, 1] as follows:

G(x, c) = sup[A(a) ^ B(c - ax)], aER

which implies that

C(c) = sup[X(x) ^ G(x, c)]. xqR

Finally, if there exists a nonempty family of solutions of the above equations, the greatest element of it is computed as the c~-composition of G and C.

X = G trC.

8. Conclusions In this paper we have exposed the methodology, theory and applicational areas of fuzzy relational equations. The results of the entire discussion can be concisely summarized accordingly: - T h e theory of fuzzy relational equations is well-developed, containing a significant portion of fundamental results. It provides a solid background for further theoretical studies including more advanced structures such as e.g. fuzzy sets of higher order. - The equations can be viewed as a platform for a creative analysis of existing topics studied in fuzzy sets. It is of interest to observe that the approach directed by them is a constructive one strongly adhering to the way in which some constructs can be carried out. We can refer e.g. to fuzzy arithmetic where the equations enable us to derive relevant solutions or the theory can be utilized to figure out the lack of their existence.

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It is difficult to project directions of progress in this area. It seems, however, that some particular topics will be under specific investigation. It is likely that this may refer to: • approximate solutions of equations with a particular emphasis on aspects of handling probabilistic types of uncertainty, • characterizations of families of solutions in relational equations, determination of extremal elements in their structures as well as their careful interpretation. • development of the paradigm of relational computations and embedded computational structures.

Acknowledgment Support provided by the National Sciences and Engineering Research Council of Canada is greatly appreciated.

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