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Fuzzy relational structures: The state-of-art
FUIIY
sets and systems Fuzzy Sets and Systems 75 (1995)241-262
ELSEVIER
Fuzzy relational structures: The state-of-art A n t o n i o D i N o l a a, W i t o l d P e d r y c z b, S a l v a t o r e S e s s a a'* . Universith di Napoli, Facoltit di Architettura, lstituto di Matematica, Via Monteoliveto 3, 80134 Napoli, Italy b University of Manitoba, Department of Electrical and Computer Engineering. Winnipeg, Manitoba, Canada R3T 2N2
Abstract
Fuzzy relational structures, built with the aid of fuzzy sets and fuzzy relations, constitute a fundamental class of constructs arising in fuzzy set theory. They give rise to fuzzy relation equations. In this paper we will develop the concepts of single and multilevel relational structures. Different logic operations will be discussed in detail. Aspects of the knowledge representation will be studied and illustrated with several schemes of applications. Keywords: Fuzzy relation; Fuzzy relation equation; Single and multilevel relational structure; Triangular norm
1. Introduction
Fuzzy relational calculus occupies a central place in fuzzy set theory as driven by the formalisms of logic and set theory. This calculus is conceptually well justified and contributes significantly to various models found in applications (cf. e.g. [1-3, 7, 23, 25]. These references outline clearly the role of this calculus in developing concepts of the theory of fuzzy sets and discuss general areas of applications. Generally speaking, by fuzzy relational structures we mean structures involving fuzzy sets and fuzzy relations summarized with the aid of different set-to-set (or relation-to-relation) composition operations. Each of them is characterized by their specific features and exhibits distinct representation aspects. Fuzzy relation equations emerge from fuzzy relational structures and in this context are viewed as containing some unknown fuzzy sets or relations. Most of the fuzzy relational structures and resulting equations currently studied can be classified as single-level structures. The aim of this paper is to introduce multilevel (multilayer) fuzzy relational structures, study their representation aspects, discuss the induced equations alongwith their solutions and highlight their applicational aspects. We will also generalize some single level structures. Within this area, we will comment on existing fuzzy relational equations and present how they are generated by the corresponding structures. In the remainder of the paper we will be treating fuzzy sets x, defined in a finite universe of discourse, as elements of the unit n-hypercube, i.e. x E [0, 1]". We will use PROLOG-Iike notation to provide a condensed
*Corresponding author. 0165-0114/95/$09.50 © 1995- ElsevierScienceB.V. All rights reserved SSDI 0165-01 14(95)00062-3
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notation of the structures. Fuzzy sets will be denoted by small letters (x, w, z), fuzzy relations by capital letters. It will be assumed that any triangular norm t and any conorm involved are continuous functions of their arguments. For any concept or formula of the theory of the triangular norms not explained in detail here, we remind to the famous monograph of Schweizer and Sklar [29] (see also [1] for further results). First we will study a variety of single-level fuzzy relational structures and discuss some types of relational equations emerging there. Some of them are well known. We will also comment on their possible generalizations and formulate the associated fuzzy relation equations. After we introduce multilevel relational structures and study their basic types. Applicational aspects will be summarized throughout the entire text of the paper.
2. Single-level fuzzy relational structures In this section we will take a thorough look at different fuzzy relation equations (structures). In spite of diversity they share a common property, that is their single-level architecture with the single composition operator. To get a better insight into the problem, let us notice that the standard form of fuzzy relation equation reads as y = x op R,
(1)
where x, y e [0, 1]" and the fuzzy relation R becomes an element defined over the Cartesian product [0, 1]" × [0, 1]". The operation "op" stands for any binary operation completed on the respective grades of membership o f x and y. Thus Yj = op(xl, Rift
for any j = 1,2, ..., n.
This form clearly underlines that (1) can be treated as a family of n equations indexed byj. The values of yj and Yk, J ~ k, are completely interrelated and can be determined independently from each other. As straightforward consequence of this observation one can drop the index j and study the equation: y = op(xi, Ri),
(2)
where y ~ [0, 1] as a single element of the system (1). The classes of fuzzy relation equations of this type have been studied in the existing literature. Some of them are summarized below; we provide their network interpretation to highlight essential and computational properties conveyed by them. Considering the grades of membership xi as individual input signals of the structure, Eq. (2) can be perceived as a certain model of distributed processing where xi are combined with the use of Ri and finally aggregated globally giving rise to the output signal y by means of the below compositions. s-t composition: By using a triangular norm t and a conorm s, the fuzzy relation equation realizing this operation, for x, w ~ [0, 1]", reads as y = x o w, i.e. n
y = S (w~tx~). i=1
(3)
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243
Xl WI
OR wi
Xi
~,[
N
Y
Wn Xn
Fig. 1. Network interpretation of the s--t composition.
In the logic-oriented notation one expresses it in the equivalent form: y = (xl AND wx)OR(x2 AND
w2)OR
"'"
OR(x, AND w,).
(4)
In the network notation one can portray (4) as shown in Fig. 1. t - s c o m p o s i t i o n : This is the dual form of the fuzzy relational structure (3),
(5)
y ~-X'W,
namely y =
t (wisxl), i=l
which in logic-oriented notation transforms into y = (xl OR wx)AND(x2 OR w2)AND ... AND(x, OR w,).
(6)
The network interpretation is similar to that shown in Fig. 1 and implies now a single multiple-input AND node. These two types of structures imply two general classes of fuzzy relation equations. The two originally developed by Sanchez [22] in 1976 involve sup-min (max-min) and inf-max (min-max) composition as quite specialized types of s - t and t - s compositions, respectively. The max-rain composition has played a key-role in many constructs of fuzzy sets starting from the extension principle proceeding with computations of possibility and necessity measures, etc. This operation has been used almost exclusively in many applications. The next step of generalization of this type of relational structure includes replacement of min and max operators by a triangular norm t and a conorm s, respectively. This gives rise to the sup-t (max-t) and inf-s (min s) compositions. These classes appeared as early as 1984, cf. I-3, 16]. The main motivation behind its introduction was to enhance the existing abilities by admittng a local level of interactivity between the grades of membership of the objects being composed under these compositions. Note that the lattice operations min and max are fully non-interactive making the obtained results fully dependent on a single argument. Nevertheless the "global" interactivity has not been achieved, i.e. the value o f y still depends on a single argument xi. In other words for the expression y = max(w1 t X1,
W 2 t X 2 .....
W n t Xn) ,
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the result becomes y z WiotXio,
while all the remaining arguments of the maximum operator are unessential and could be eliminated from the global expression. The global interactivity is achieved by replacing the max and min operators by s and t operations, respectively. Now the higher generality of the structure is achieved at expense of a lack of general solutions which cannot be derived in a general form. More formally, the solutions depend strongly from the type of t-norms and s-norms involved [25]. For s = max and t = rain we have definitive results for Eqs. (3) and (4), respectively ([3, 4, 6 9, 18, 19, 25]). 2. I. Solving Juzzv relation equations: optimization approach
In determining unknown fuzzy sets in fuzzy relation equations one can apply standard optimization techniques. The idea is to modify an unknown fuzzy set by a series of updates in such a way that the value y is made closer to the required value "u E [0, 1]". For a single equation, considered x and u as being available while w is looked, we define the performance index MSE (Mean-Squared Error) Q=(u-x
w)2
(7)
and our goal is to minimize it with respect to w, say min{u-x
wl 2.
H,
The Newton-like schemes of optimization are expressed as w(iter + 1) = w(iter) - a.c?Q/t?w with updates of w, w(iter + 1) and w(iter) calculated on a basis of the gradient of Q. The learning rate controls successive changes of w. The derivative taken with respect to w~, OQ/Ow~, is calculated as ~Q/c~wi = O(u - x o w)2/~wi = - 2(u - x,~ w). ~(x wj/~wi.
Once the s t composition has been defined the updates of wi can be derived. We will discuss these details in conjunction with a multi-level relational structure. The same method applies to fuzzy relation equations with the t-s composition. Since the equation is symmetrical with respect to x and w, the same formulas apply to c~Q/~xi. The system of relational equations Yl
=Xl
'W,
Y2
=X2~W,
...
,yn=Xn
'w
can be solved in a similar way. The criterion optimized now is again the MSE which takes on a form of the following sum: Q=
~It~-xk
w)2.
k=l
The fuzzy set w is adjusted to minimize (8).
(8)
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A. Di ?Cola et al. / Fuzzy Sets and Systems 75 (1995) 241-262 2.2. Further extensions o f single layer relational structures
The relational structure can be expanded by a c c o m m o d a t i o n s more combinations of inputs x1, x 2 . . . . . x n. By doing that one adds new representation abilities to the structure. In the first extension we include complemented values of xi, ffi = 1 - x~. A n o t h e r possibility could be to perform a polynomial-like approxim a t i o n of(xk, tk) in which the structure is augumented by second-order terms xi t xi (i.e. x~ A N D xi) as well as high-order ones (e.g. xi A N D x~ A N D xi, etc.). Example 1. F o r n = 1, the results of a p p r o x i m a t i o n are shown for the data set visualized in Fig. 2. The relational structure includes an s t composition with the probabilistic sum and Lukasiewicz A N D conjunction. Their form is the following: p = 1:
y = (wl A N D x) O R (w2 A N D .£),
p = 2:
y = (wl A N D x) O R (w2 A N D x A N D x) O R (w3 A N D ~) O R (w4 A N D ~ A N D ~).
p = 3:
y = (wl A N D x) O R (w2 A N D x A N D x) O R (w3 A N D x A N D x).
The learning process carried out with respect to wi's is displayed in Fig. 3. The best results (minimum of the MSE) are obtained for p = 3. The elements of w are summarized in Table 1. N o t e that higher orders imply higher accuracy of approximation. Neglecting low contributions (namely d r o p p i n g the terms with wi's close to 0), the emerging relational descriptions are the following: p = 1:
y = 0.98 A N D £ .
p = 2:
y = (xANDx)OR(0.866AND~)OR(0.95
p = 3:
y = (0.371ANDx)OR(0.999ANDfANDf)OR(0.874AND£).
AND2AND.f).
1.0 ~
+'--+ p-2 vmqp3 " " - = p-1 I . - .-e DATA
,,x t,='~
-'~- ,~,,
~
,
r"
.o
I'
.--- ....~,AI "-,,/ / 'J
\ "=.,
\
, %
/
\
I
\ 0
0
.25
.50
.75
x
Fig. 2. Data points (x~,),~) and successive approximation.
1.00
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3
2
1
0 I
0
200
400
600
800
1000
Fig. 3. Learning the fuzzy set w.
Table 1 Membership functions of w for different expansions of the structure. p = I p = 2
w = [ x / O , :U0.98] w = [ x / O , x A N D x~ 1.0, x/0.866, :? A N D 2/0.95]
p = 3
w = Ix~O, x A N D x / 0 . 3 7 1 ,
xANDxANDx/0.999,
.?/0.874, f A N D x / 0 . 0 , 2 A N D . f A N D . f / 0 . 0 ]
For comparison the results of a standard polynomial approximation p 3' =
2
ai Xi
i=o
are also given in Fig. 4. The performance is better for higher orders of p, however, the polynomial becomes multimodal with clearly introduced new nodes. It is also worth noting that the approximation of the same form as carried out by the s t relational structure is not feasible for max-min or min-max compositions (due to the idempotency of the max and min operators standing there). Example 2. We will study characteristics of the s t composition and its dual structure emerging in case of two variables xl and x2 and their complements. The structure reads as for that driven by the s - t composition: y = (wl A N D xl} OR (W2 A N D x2) OR (W3 A N D 21) OR (w4 A N D x2) and for that driven by the
t-s
composition:
y = (wl OR Xl) A N D (W2 OR x2) A N D (w3 OR 21) A N D (w4 OR "~2)' The two-dimensional contours of the output values of y are summarized in Figs. 5 and 6, respectively. The different level of interactivity between the arguments of the composition is clearly reflected in the obtained characteristics.
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10
.8
.4
.2
0
0
.9
.4
.8
.8
lo
Fig. 4. Polynomial approximation of data.
2.3. Referential fuzzy relational structures
The four following relational structures are of referential character. The idea is that to process firstly the x's with respect to some other fuzzy sets viewed as reference points and afterwards aggregated with the use of O R or A N D connectives. 2.3.1. Matching relational structure This relational structure is defined as follows n
Y = S [ ( x i - - rl)twi],
(9)
i=1
where the matching operation " - " is defined below and r = Jr1, r2 . . . . . (prototype) fuzzy set in [0, 1]". We write down (9) as y = (x - r ) o w ,
rn]
is treated as a reference
(10)
then x is compared with r and these pointwise results of matching are transformed with the use of wi's and finally contributing to y. The o p e r a t i o n " = " c a n be defined in many ways; here we apply a degree of equality as defined in [14, 17, 19, 24, 30]: (a --- b) = 1/2 [(aqSb) A (bcka) + (~tk6) A (~q~d)], where q5 denotes a pseudocomplement associated with a given t-norm, namely a4ab = s u p { c e [0, 1]: a t c ~< b} with a, b ~ [0, 1]. In P R O L O G - l i k e notation (9) translates into y-= [(xl E Q U A L r 0 A N D wl] O R [ ( x z E Q U A L r / ) A N D O R [(x, E Q U A L r , ) A N D w,].
Wz] O R ...
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0.3
0.4
0.5 0.6
X ~
.7
i
× t
Fig. 5a.
U///// X ~
o X
1
Fig. 5b.
A. Di Nola et al.
Fuzzy Sets and Systems 75 (1995) 241 262
249
O,
00~''" X
1
Fig. 5c. Fig. 5. T w o - d i m e n s i o n a l c o n t o u r s o f the s-t r e l a t i o n a l s t r u c t u r e s for different t r i a n g u l a r n o r m s : (a) s = m a x i m u m , t = m i n i m u m ; (b) s = p r o b a b i l i s t i c s u m , t = p r o d u c t ; (c) s = p r o b a b i l i s t i c s u m , t = L u k a s i e w i c z c o n n e c t i v e .
0.6
O.
X
1
Fig. 6a.
0.4
0.3
0.2 0.i
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A. Di Nola et al. / Fuzzy Sets and Systems 75 (1995) 241-262
0.4
0.3
0.2
0.I
N
:,<
0 x Fig. 6b.
AND
0.2
0.1 X ~
0.0
0 ×
Fig. 6c. Fig. 6. T w o - d i m e n s i o n a l c o n t o u r s of the t-s r e l a t i o n a l s t r u c t u r e s for different t r i a n g u l a r n o r m s : (a) t = m i n i m u m , s = m a x i m u m ; (bJ t = p r o d u c t , s = p r o b a b i l i s t i c s u m ; (c) t = L u k a s i e w i c z c o n n e c t i v e , s = p r o b a b i l i s t i c sum.
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251
The referential structure can be treated as a fuzzy relation equation of the form (10). Several problems can be formulated regarding their solutions: - for given x and y, determine r and w (identification problem). This problem could have several versions depending on a number of fuzzy sets to be determined such as (i) given x, y and r determine w, (ii) given x, y and w determine r; - for given y, w and r determine x (inverse problem). Assuming additionally that w = 1 (all w~'s are equal to 1), we get a class of fuzzy relation equations which have been already studied [1, 5]. Example 3. The relational structure with the equality operator w~ll be illustrated for a two-dimensional case (two inputs xl and x2): y = (xl E Q U A L 0.3) AND (x2 E Q U A L 0.6), where the operator E Q U A L is given, using the product, by
aEQUALb=
1/2.[b/a+(1-a)/(l-b)]
ifa>b,
1
if a = b ,
1/2.[a/b+(1-b)/(1-a)]
if a < b ,
a, b ~ [0, 1] while AND is the Lukasiewicz conjunction. The two-dimensional contours of the values of y are summarized in Fig. 7.
2.3.2. Referential structure with difference operator The associated relational structure includes the difference operator " =1" defined as a =1 b = 1 - (a --- b) for any a, b ~ [0, 1] Then n
y=
S [(xi~rAtw~]. i--I
Its P R O L O G - d r i v e n equivalent form reads accordingly y = [(xl D I F F E R rl) AND wl] OR [(x2 D I F F E R r2) AND w2 ] OR ... OR [(x, D I F F E R r,) AND w,], where the operator D I F F E R has an obvious meaning. Example 4. We will consider the relational structure y = (xl D I F F E R 0.3) AND (x2 D I F F E R 0.6) with the same t-norm (product) and the operator AND of Example 3. The corresponding contours of the values of y are in Fig. 8.
2.3.3. Relational structure with inclusion operation This relational structure is described as n
Y=
S [(xic~ri)twi], i=1
(11)
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A. Di Nola et al. / Fuzzy Sets and Systems 75 (1995) 241 262
X ~
× 4
Fig. 7. Two-dimensional plots of the matching relational structure.
0.3 0.15 0.0
0.0
0.3
0.45
0.0 X ~
0.15
0
0.30
0.15
0.45
0.30
0.60
Fig. 8. Two-dimensional plots of the difference relational structure.
A. Di Nola et aL / Fuzzy Sets and Systems 75 (1995) 241 262
253
where, as before, r is viewed as a reference point. F r o m a logical point of view, the pseudocomplement plays a role of inclusion relation; observe that xl 4~rl = 1 if xl ~< r~. The elements of w describe an influence of the inclusion xi4~r~ on y. In a vector notation (11) is written as y = (x I N C L r) o w. Again the structure (11) can be treated as a fuzzy relation equation: - with respect to the parameters (fuzzy sets r and w) for x and y given (here one can have subcases when r or
w is known and the second fuzzy set has to be determined), with respect to x for y given, of course the parameters of the structure being available. E x a m p l e 5. The relational structure, with the inclusion operator i N C L defined as (a I N C L b) = min(1, b/a) for any a, b ~ [0, 1] and the operator A N D of Example 3, has the form
y = (xl I N C L 0.3) A N D (x2 I N C L 0.6) and yields a plot as shown in Fig. 9. One can also study a modified version of (11) with the t-s composition n
y:
t [(xi(ori)swi],
(12)
i=1
i.e. y = (x I N C L r)" w. If all wi's are set to 0 and the t-norm specified as the min operator, then (12) is known as adjoint fuzzy relation equation [10, 23].
2.3.4. Reference structure with dominance relationship The fuzzy relational structure of this class reads as n
Y = S [(ri(axi)twi]
(13)
i=1
or more concisely as y = (r D O M x) o w. The relationship ri 4) x~ denotes a degree of dominance ofr~ by x~; the characteristics are complementary to those given previously. The dual version of (13) is given by y = (r D O M x ) . w . E x a m p l e 6. The two-dimensional plots of the relational structure
y = (0.3 I N C L x~) A N D (0.6 I N C L xz) is provided in Fig. 10.
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@
X ~
0.8
0.6 ×
0.4
I
Fig. 9. Contours of the referential relational structure with the operator of inclusion.
X ~
0.8 0.6 0.4 0.2
Fig. 10. Contours of the referential relational structure with the dominance operator.
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255
3. Multilevel relational structures
We now will expand the structure of fuzzy relation equations by incorporating intermediate variables. In the network terminology this means an addition of some auxiliary layers situated between the input and output ones. Their role is to enhance representation capabilities of the equations. Our primary interest will be in three-layer relational structures. This additional intermediate (a so-called hidden) layer permits to capture specificity of the problem. We will show how the additional layers could add contribute toward handling peculiarities of the problems under discussion.
3.1. Logic processors-approximation of logic-based relationships between variables The three-layer relational structure given and studied in [10, 11, 22, 26] is suitable for copying with approximation-like problems where one is interested in eliciting logical and usually nonlinear relationships between input grades of membership (input signals) x~, x2, ..., x, as well as their complements and the grade of membership y (output signal). Their generic components are constituted by the single dual and direct fuzzy relation equations as we have studied so far. It is worth noting that the dual fuzzy relation equation with variables x~ and Y~'s forms a generalized minterm, namely: z = (wl OR xl) AND (W2 OR X2) AND ... AND (w, OR x,) AND (w,+l OR xl) AND (w,+2 OR "~2) AND ..- AND (W2n OR ~,).
(14)
If we restrict ourselves to all xi's and wi's set either to 0 or 1, then (14) converts into a standard minterm known in the two-valued logic. Similarly the direct fuzzy relation equation is interpreted as a generalized maxterm. Referring to the fundamental representation result of the two-valued logic stating that any Boolean function can uniquely given as a sum of minterms (or a product of maxterms), we will follow this path in approximating any fuzzy function [-12, 13] by a sum of generalized minterms (product of the generalized maxterms). The emerging relational structure is visualized in Fig. 11.
xl
AND
Fig. 11. A N D - O R logic processor.
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The hidden layer is built with the A N D nodes. The output is computed using a single O R node. The size of the hidden layer determines a number of the generalized minterms revealing this approximation. The detailed formulas are given below: y = (vl A N D z l ) OR(v2 A N D z2) OR ... OR(v,, A N D zm)
(15)
and Zh = (Whl OR Xl ) A N D (Wh2 OR x2) A N D ". A N D (Wh, OR x,) A N D (Wh,, + 1, OR £1) A N D ... A N D (Whiz,, OR .%), where h = 1. . . . . m. In compact form we will write down, y =zcv
(16)
for the first fuzzy (single) relation equation, while the second one converts into the t-s equation z =£.w,
(17)
where £ includes both xi's as well as their complemented values. The dual structure approximates logical relationships between x and y in taking advantage of the generalized maxterms. It should be stressed that in contrast to the two-valued logic functions, this representation is not universal, however constitutes a legitimate approximation and a good starting point for further refinements if those are viewed necessary. The solution to this class of multilevel fuzzy relation equations can be posed with respect to the fuzzy set v and the fuzzy relation w: "given x and y, determine v and w". In general the analytical solution is not available. The solution does not specify a minimal value of h. For a collection of Xk'S, the corresponding parameters tk'S of the relational structure should be determined through solving a relevant optimization problem (cf, Section 2.1). The optimization in this case pertains to determining w and v for a given size of the hidden layer. This in fact constitutes a sort of parametric learning. The performance index is a standard one involving a sum of squared errors, Q = ~ [yIXh) -- th] 2, h=l
where y(Xh) denotes the value o f y obtained for Xh; th is associated with it as a part of data to be approximated. The elements of w and v are determined by considering successive adjustments driven by a gradient of Q. Symbolically we put it down as writer + 1) = w(iter) - :~. ~3Q/Ow, v(iter + 1) = voter) - ~' ~Q/Ov, where ~ e (0, 1) denotes a learning coefficient. All the detailed computations can be directly completed once the t-norms have been defined. Example 7. We will discuss the relational structures (16) and (17) with the s-norm defined as a probabilistic sum and any t-norm. This yields the following detailed formulas: n
n
t (WhiSXi)t t (w,,,.+l,sg,),
zh=
i=1
i
trl
Y=
S h=l
(LIhSZh) .
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257
Assuming an on-line adjustment, we can drop "h" and consider a single pair x, t. This refinement yields ~Q/OWh, j = ~Q/t?Vk
[y(X)
---- [ y ( x )
-- t]
-- t]"
"Oy/t~Wh, j], t?y/C?Vk.
Subsequently nl
t?y/~v k = (3 S (13hSZh)/~Uk = ~ [A S(VkSZk)]/~l.)k, h=l
where A=
S (VhSZh). h ~ak
In virtue of our specification, we derive i?[A + Vk'Zk -- A ' V k ' Z k ] / t ~ V k
= Zk'(1 -- A).
Then ~y/~Wh, j = ~ ~Y/~Zh'~Zh/~Wh,j" h=l
Note however that the above sum reduces to a single component since ~Zh/~Wh, j = 0 for all h :~ h i . In this way ~y/t~Wh, j =- ~y/OZh, " OZh,/ OWh,j. And finally Oy/~zh, = Vk'(1 -- B),
n =
S (VhSZh), h~hl
= c~3[ l~ (wh.i ~- Xi - wh~iXi). ~-I ~wh'~n+ ~} ~- Xi - ~h'~+ ~.~i~.~wh'j ~- Xj - wh'Xj)~/t3wh'j-= C(~ - ~j)~ i~-j
ivaj
where C abbreviates the two first factors in the above expression. The solutions to the above relation equations can be obtained in an approximate way by formulating and solving associated optimization tasks; the reader is referred to [20-22, 27]. Regarding the internal derivatives, one should be aware that for nondifferentiable t-norms such as maximum and minimum, they should be defined with a certain caution [15]. Take for instance min(x, w) and calculate its derivative with respect to w. Obviously we obtain 1 if x >/w and 0 otherwise. The optimization problem driven by this type of "on-off" mode of updates could be easily affected by traps caused by some configurations of the connections and the data points in course of updating. A prudent inspection suggests a slight modification, the basic idea of which is to replace the values 0 and 1 by its multivalued counterpart. Namely we relax the inequality x >~ w taking only two values "True(l)" if satisfied and "False(0)" otherwise, by accepting an inequality which is satisfied to a certain degree, say IIx t> wll, where IIx/> wll = 1 ifx >~ w and IIx/>. wll = w INCL x otherwise. Thus IIx >/wll e [0, 1] and this contributes significantly to a smooth character of updating the parameters of the relational structures. For more details, we refer to [15].
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xl __
x2
EOUAL
1.0 f21 ~m
x2
..... Y .
f22
xn
1.0 X!
Fig. 12. Multilevel fuzzy relational structure with matching.
Fig. 13. Pattern classification with matching.
the fuzzy relational structure
3.2. Three level relational structures' with reference objects' The relational structure can be developed in such a way that it realizes operations in [0, 1]" which pertain, in particular, to matching, difference, inclusion and dominance. Matching: In this relational structure x~, x2 . . . . . x, match prototype I21 , 1"22, ..., I2,, and these results are further aggregated as shown in Fig. 12. The first relation equation carriers out a process of coordinatewise matching, Zh = [(Xl E Q U A L Whl ) OR Whl ] AND [(x2 E Q U A L Wh2 }OR Wh2] AND -.. AND (18)
[(x, E Q U A L Wh,) OR Wh,], where h = 1, 2 . . . . , m and The operator " E Q U A L " membership, The number a direct relation equation.
Oh = [Whl, Wh2 Whn] denote the grades of membership of the hth prototype. returns a degree of matching (a level of similarity) between two grades of of prototypes specifies the format of the hidden layer. The next level constitutes Strictly speaking, Eq. (18) will be expressed as
Zh = (X E Q U A L Wh)'Wh,
. . . . .
y = Z V,
where z = [zl, z2 . . . . . z,,] and y is given as in (15). The relational structure of this type can be found useful in problems of pattern classification. For instance, for patterns belonging to the same class and distributed around several disjoint regions in the feature space centered in 121, I22 . . . . . 12,., the above value of y denotes a degree of class membership of the pattern x to be classified (cf. Fig. 13). The relational structure implements a general statement: "the pattern x belongs to the class if it is similar to I2~ OR it is similar to 122 OR ... OR it is similar to 12,,". Note that each subcondition in this statement is realized by the corresponding node in the hidden layer while the overall aggregation is of the OR-like type (the third layer). Difference: The structure is driven by the difference levels determined between xfs and some landmark points (here playing a role of exceptions) ~'1, ~2 . . . . . ~,,. Thus we get: zh = [(x~ D I F F E R @hl) OR Whl ] AND [(x2 D I F F E R @h2)OR Wh2] AND ".. AND [ (x. D I F F E R g,h.) O R wh.],
(19)
A. Di Nola et al. / Fuzzy Sets and Systems 75 (1995) 241-262
259
where h = 1,2, ... ,m and the membership vectors 0h = [0hl, 0h2 . . . . . 0h,] are used to calculate the respective differences. The results of (19) are then combined by AND-ing zh's. This yields two formulas, Zh = ( x D I F F E R O h ) o W h ,
y = z.v,
where y is given by y=(zlORvl)AND(z2ORvz)AND
A N D (z,, O R v,,).
The difference relational structure can be easily interpreted in pattern recognition, fault analysis and machine learning. The vectors Oh represent exceptions; thus a pattern is viewed as belonging to a certain category if it is excluded from 0a and it is excluded from 02 A N D ... A N D it is excluded from 0,, (see Fig. 14). The quite dispersed regions of exceptions are now better approximated in this way rather than applying the relational structure equipped with some matching capabilities. Inclusion: The structure has a hidden layer consisting of the elements capturing the property of inclusion (say, modelling the statement "x is included in"). The partial results derived at this layer can be afterwards aggregated either in a conjuctive or disjunctive manner. Formally, this conjunctive form of the aggregation converts into the formula Zh = [(Xl I N C L 27hl ) O R whl ] A N D [(x2 I N C L rh2) OR
Wh2"]A N D
... [(x, I N C L rh.) OR Wh,],
where h = 1, 2, ... ,m and ~h = [27hl, 2 7 h 2 , - - . ,72hra] denotes a reference point with respect to which the inclusion relationship is defined. At the second stage we get y = z o v or y = z. v depending on the way in which the final aggregation should be carried out. The relational structure models situations in which one attempts to comply with several constraints (requirements) which should not be exceeded (e.g. the speed should not be higher than rl A N D the temperature should be kept below ~2, etc.). D o m i n a n c e : The relational structure is dual to the previous one where the statements are formulated as "x should dominate (be higher) then ... ". Thus we have Zh = [(7hl I N C L Xl ) O R Whl ] A N D [-(])h2 I N C L xz) O R Wh2 ] A N D "- A N D E(Vh,I N C L x,) O R Wh,], where h = 1, 2 . . . . . m while the reference point ~'h indicates an element which should be dominated by x. The aggregation can be then performed in a disjunctive or conjunctive manner.
X2
.0 Xl
Fig. 14. Pattern classification the fuzzy relational structure with difference operator.
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A. Di ?Cola et al. / Fuzzy Sets" and Systems 75 (1995) 24l 262 b
~j Fig. 15. Relational structure with three nodes.
OR
decision
AND
Fig. 16. Relational structure formalizing a decision-making problem.
A. Di Nola et al. / Fuzzy Sets and Systems 75 (I995) 241-262
261
3.3. Decision-making processes modelled with the use of multilevel relational structures
The multilevel relational structures can also involve all types of the processing elements depending on a particular application. They create an ideal tool for formalizing compound statements in decision-making processes. The transformation (formalization) is immediate, while the learning can furnish the structure with numerical connections. Below we list several examples along with the relevant relational structures being their formal translation: "if (x satisfies constraint b) and (x dominates c) and (x does not exceed a), then the decision D should be taken". The relational structure is then described as Q = (x EQUAL b) AND (c INCLx) AND (x INCL a) and, as becomes evident, consists of three nodes in the hidden layer and aggregated via the generalized AND operation in the output layer (cf. Fig. 15). A final numerical calibration (leading to the membership values of w and v) is completed on a basis of some training data pertaining to available decision scenarios. The following statement of if-then type: "if [(x satisfies a) and (x differs from b) and (x satisfies c)] or [(x differs from d) and (x is included in f ) ] , then the decision D should be taken" translates into D = [(x EQUAL a) AND (x DIFFER b) AND (x EQUAL c)] OR [(x DIFFER d) AND (x I N C L f ) ] and this formulation gives rise to a four-level relational structure shown in Fig. 16.
4. Conclusions It should be noted that these structures are usually heterogeneous in their nature including different basic logic operations (or their generalizations) used to develop their basic components. It is worth stressing that they could be utilized in many real-world problems due to their ability to transform the problem into straightforward logical structures and carry out final calibration by modifying all necessary relations.
Acknowledgement Support provided by the Natural Sciences and Engineering Research Council of Canada is gratefully acknowledged by the second author.
References [1] A. Di Nola, W. Pedrycz, E. Sanchez and S. Sessa, Fuzzy Relation Equations and Their Applications to Knowledge Engineering (Kluwer Academic Publishers, Dordrecht, 1989). [2] A. Di Nola, W. Pedrycz, E. Sanchez and S. Sessa, Fuzzy relation equations theory as a basis of fuzzy modelling: an overview, Fuzzy Sets and Systems 40 (1991) 415 429. [3] A. Di Nola, W. Pedrycz, S. Sessa and P.Z. Wang, Fuzzy relation equations under a class of triangular norms: a survey and new results, Stochastica 8 (1984) 99 145. [4] A. Di Nola, W. Pedrycz and S. Sessa, Fuzzy relation equations under LSC and USC t-norms and their Boolean solutions, Stochastica I1 (1987) 151 183.
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[5] A. Di Nola, W. Pedrycz and S. Sessa, Fuzzy relation equations with equality and difference operators, Fuzzy Sets and Systems 25 (1988) 205 215. [6] M.J. Fernandez, F. Suarez and P. Gil, Equations of fuzzy relations defined on fuzzy subsets, Fuzzy Sets and Systems 52 (1992) 319 336. [7] S. Gottwald, Fuzzy Sets and Fuzzy Lo.qie (Vieweg, Braunschweig and Taknea, Toulouse, 1992). [8] S. Gottwald, Approximately solving fuzzy relation equations: some mathematical results and some heuristic proposals, Fuzzy Sets and Systems 66 (1994) 175 193. [9] S. Gottwald and W. Pedrycz, Revealing logical dependencies in fuzzy data with the aid of multilevel fuzzy relational equations, lnJorm. Sci. 78 (1994) 311 324. [t0] K. Hirota and W. Pedrycz, Fuzzy logic neural networks: design and computations, Proceedinqs oflnternat. Joint ConJerence on Neural Networks, Singapore (199t). [11] K. Hirota and W. Pedrycz, Logic-based neural networks, Inlbrm. Sci. 71 (1993) 99 130. [12] A. Kandel, On minimization of fuzzy functions, IEEE Trans. Comput. 22 (1973) 826 832. [13] A. Kandel, Inexact switching logic, IEEE Trans. Systems Man Cybernet. 6 t1976) 215 219. [ 14] M. Miyakoshi and M. Shimbo, Solutions of composite fuzzy relational equations with triangular norms, Fuzzy Sets and Systems 16 (1985) 53-63. [15] W. Pedrycz, Numerical and applicational aspects of fuzzy relational equations, Fuzzy Sets and Systems 11 (1983) 1 18. [16] W. Pedrycz, On generalized fuzzy relational equations and their applications, J. Math. Anal. Appl. 107 (1985) 520-536. [17] W. Pedrycz, Direct and inverse problem in comparison of fuzzy data, Fuzzy Sets and Systems 34 (1990) 223 235. [18] W. Pedrycz, Processing in relational structures: fuzzy relational equations, Fuzzy Sets and Systems 40 (1991) 77 106. [19] W. Pedrycz, Fuzzy modelling: fundamentals, construction and evaluation, Fuzzy Sets and Systems 41 (1991) 1 15. [20] W. Pedrycz. Neurocomputations in relational systems, IEEE Trans. Pattern. Anal. Machine Intell. 13 (1991) 289-296. [21] W. Pedrycz, A scheme of fuzzy decision-making with neural net structure, IEEE Trans. Systems. Man Cybernet. 21 (1991) 1593 1604. [22] W. Pedrycz, Fuzzy neural networks with reference neurons as pattern classifier, IEEE Trans. Neural Networks 3 (1992) 770-795. [23] W. Pedrycz, Fuzzy Control and Fuzzy Systems (Research Studies Press Ltd./Wiley, Taunton/New York, 2nd extended edn., 1993). [24] W. Pedrycz, Fuzzy neural networks and neurocomputations, Fuzzy Sets and Systems 56 (1993) 1 28. [25] W. Pedrycz, s t fuzzy relational equations, Fuzzy Sets and Systems 59 (1993) 189 195. [26] W. Pedrycz, Reasoning by analogy in fuzzy controllers, in: A. Kandel and G. Langholz, Eds., Fuzzy Control Systems (CRC Press, Boca Raton, FL, 1993). [27] W. Pedrycz, Logic-based neurons: extensions, uncertainty representation and development of fuzzy controllers, Fuzzy Sets and Systems 66 (1994) 251 266. [28] E. Sanchez, Resolution of composite fuzzy relation equations, lnJorm, and Control 30 (1976) 38 48. [29] B. Schweizer and A. Sklar, Probabilistic Metric Spaces (North-Holland, Amsterdam, 1983). [30] J. Valente de Oliveira, Neuron inspired learning rules for fuzzy relational structures, Fuzzy Sets and Systems 57 (1993) 41-53.
sets and systems Fuzzy Sets and Systems 75 (1995)241-262
ELSEVIER
Fuzzy relational structures: The state-of-art A n t o n i o D i N o l a a, W i t o l d P e d r y c z b, S a l v a t o r e S e s s a a'* . Universith di Napoli, Facoltit di Architettura, lstituto di Matematica, Via Monteoliveto 3, 80134 Napoli, Italy b University of Manitoba, Department of Electrical and Computer Engineering. Winnipeg, Manitoba, Canada R3T 2N2
Abstract
Fuzzy relational structures, built with the aid of fuzzy sets and fuzzy relations, constitute a fundamental class of constructs arising in fuzzy set theory. They give rise to fuzzy relation equations. In this paper we will develop the concepts of single and multilevel relational structures. Different logic operations will be discussed in detail. Aspects of the knowledge representation will be studied and illustrated with several schemes of applications. Keywords: Fuzzy relation; Fuzzy relation equation; Single and multilevel relational structure; Triangular norm
1. Introduction
Fuzzy relational calculus occupies a central place in fuzzy set theory as driven by the formalisms of logic and set theory. This calculus is conceptually well justified and contributes significantly to various models found in applications (cf. e.g. [1-3, 7, 23, 25]. These references outline clearly the role of this calculus in developing concepts of the theory of fuzzy sets and discuss general areas of applications. Generally speaking, by fuzzy relational structures we mean structures involving fuzzy sets and fuzzy relations summarized with the aid of different set-to-set (or relation-to-relation) composition operations. Each of them is characterized by their specific features and exhibits distinct representation aspects. Fuzzy relation equations emerge from fuzzy relational structures and in this context are viewed as containing some unknown fuzzy sets or relations. Most of the fuzzy relational structures and resulting equations currently studied can be classified as single-level structures. The aim of this paper is to introduce multilevel (multilayer) fuzzy relational structures, study their representation aspects, discuss the induced equations alongwith their solutions and highlight their applicational aspects. We will also generalize some single level structures. Within this area, we will comment on existing fuzzy relational equations and present how they are generated by the corresponding structures. In the remainder of the paper we will be treating fuzzy sets x, defined in a finite universe of discourse, as elements of the unit n-hypercube, i.e. x E [0, 1]". We will use PROLOG-Iike notation to provide a condensed
*Corresponding author. 0165-0114/95/$09.50 © 1995- ElsevierScienceB.V. All rights reserved SSDI 0165-01 14(95)00062-3
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notation of the structures. Fuzzy sets will be denoted by small letters (x, w, z), fuzzy relations by capital letters. It will be assumed that any triangular norm t and any conorm involved are continuous functions of their arguments. For any concept or formula of the theory of the triangular norms not explained in detail here, we remind to the famous monograph of Schweizer and Sklar [29] (see also [1] for further results). First we will study a variety of single-level fuzzy relational structures and discuss some types of relational equations emerging there. Some of them are well known. We will also comment on their possible generalizations and formulate the associated fuzzy relation equations. After we introduce multilevel relational structures and study their basic types. Applicational aspects will be summarized throughout the entire text of the paper.
2. Single-level fuzzy relational structures In this section we will take a thorough look at different fuzzy relation equations (structures). In spite of diversity they share a common property, that is their single-level architecture with the single composition operator. To get a better insight into the problem, let us notice that the standard form of fuzzy relation equation reads as y = x op R,
(1)
where x, y e [0, 1]" and the fuzzy relation R becomes an element defined over the Cartesian product [0, 1]" × [0, 1]". The operation "op" stands for any binary operation completed on the respective grades of membership o f x and y. Thus Yj = op(xl, Rift
for any j = 1,2, ..., n.
This form clearly underlines that (1) can be treated as a family of n equations indexed byj. The values of yj and Yk, J ~ k, are completely interrelated and can be determined independently from each other. As straightforward consequence of this observation one can drop the index j and study the equation: y = op(xi, Ri),
(2)
where y ~ [0, 1] as a single element of the system (1). The classes of fuzzy relation equations of this type have been studied in the existing literature. Some of them are summarized below; we provide their network interpretation to highlight essential and computational properties conveyed by them. Considering the grades of membership xi as individual input signals of the structure, Eq. (2) can be perceived as a certain model of distributed processing where xi are combined with the use of Ri and finally aggregated globally giving rise to the output signal y by means of the below compositions. s-t composition: By using a triangular norm t and a conorm s, the fuzzy relation equation realizing this operation, for x, w ~ [0, 1]", reads as y = x o w, i.e. n
y = S (w~tx~). i=1
(3)
A. Di Nola et al. / Fuzzy Sets and Systems 75 (1995) 2 4 1 - 2 6 2
243
Xl WI
OR wi
Xi
~,[
N
Y
Wn Xn
Fig. 1. Network interpretation of the s--t composition.
In the logic-oriented notation one expresses it in the equivalent form: y = (xl AND wx)OR(x2 AND
w2)OR
"'"
OR(x, AND w,).
(4)
In the network notation one can portray (4) as shown in Fig. 1. t - s c o m p o s i t i o n : This is the dual form of the fuzzy relational structure (3),
(5)
y ~-X'W,
namely y =
t (wisxl), i=l
which in logic-oriented notation transforms into y = (xl OR wx)AND(x2 OR w2)AND ... AND(x, OR w,).
(6)
The network interpretation is similar to that shown in Fig. 1 and implies now a single multiple-input AND node. These two types of structures imply two general classes of fuzzy relation equations. The two originally developed by Sanchez [22] in 1976 involve sup-min (max-min) and inf-max (min-max) composition as quite specialized types of s - t and t - s compositions, respectively. The max-rain composition has played a key-role in many constructs of fuzzy sets starting from the extension principle proceeding with computations of possibility and necessity measures, etc. This operation has been used almost exclusively in many applications. The next step of generalization of this type of relational structure includes replacement of min and max operators by a triangular norm t and a conorm s, respectively. This gives rise to the sup-t (max-t) and inf-s (min s) compositions. These classes appeared as early as 1984, cf. I-3, 16]. The main motivation behind its introduction was to enhance the existing abilities by admittng a local level of interactivity between the grades of membership of the objects being composed under these compositions. Note that the lattice operations min and max are fully non-interactive making the obtained results fully dependent on a single argument. Nevertheless the "global" interactivity has not been achieved, i.e. the value o f y still depends on a single argument xi. In other words for the expression y = max(w1 t X1,
W 2 t X 2 .....
W n t Xn) ,
A. Di Nola et al. / Fuzzy Sets and Systems 75 (1995) 241 262
244
the result becomes y z WiotXio,
while all the remaining arguments of the maximum operator are unessential and could be eliminated from the global expression. The global interactivity is achieved by replacing the max and min operators by s and t operations, respectively. Now the higher generality of the structure is achieved at expense of a lack of general solutions which cannot be derived in a general form. More formally, the solutions depend strongly from the type of t-norms and s-norms involved [25]. For s = max and t = rain we have definitive results for Eqs. (3) and (4), respectively ([3, 4, 6 9, 18, 19, 25]). 2. I. Solving Juzzv relation equations: optimization approach
In determining unknown fuzzy sets in fuzzy relation equations one can apply standard optimization techniques. The idea is to modify an unknown fuzzy set by a series of updates in such a way that the value y is made closer to the required value "u E [0, 1]". For a single equation, considered x and u as being available while w is looked, we define the performance index MSE (Mean-Squared Error) Q=(u-x
w)2
(7)
and our goal is to minimize it with respect to w, say min{u-x
wl 2.
H,
The Newton-like schemes of optimization are expressed as w(iter + 1) = w(iter) - a.c?Q/t?w with updates of w, w(iter + 1) and w(iter) calculated on a basis of the gradient of Q. The learning rate controls successive changes of w. The derivative taken with respect to w~, OQ/Ow~, is calculated as ~Q/c~wi = O(u - x o w)2/~wi = - 2(u - x,~ w). ~(x wj/~wi.
Once the s t composition has been defined the updates of wi can be derived. We will discuss these details in conjunction with a multi-level relational structure. The same method applies to fuzzy relation equations with the t-s composition. Since the equation is symmetrical with respect to x and w, the same formulas apply to c~Q/~xi. The system of relational equations Yl
=Xl
'W,
Y2
=X2~W,
...
,yn=Xn
'w
can be solved in a similar way. The criterion optimized now is again the MSE which takes on a form of the following sum: Q=
~It~-xk
w)2.
k=l
The fuzzy set w is adjusted to minimize (8).
(8)
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A. Di ?Cola et al. / Fuzzy Sets and Systems 75 (1995) 241-262 2.2. Further extensions o f single layer relational structures
The relational structure can be expanded by a c c o m m o d a t i o n s more combinations of inputs x1, x 2 . . . . . x n. By doing that one adds new representation abilities to the structure. In the first extension we include complemented values of xi, ffi = 1 - x~. A n o t h e r possibility could be to perform a polynomial-like approxim a t i o n of(xk, tk) in which the structure is augumented by second-order terms xi t xi (i.e. x~ A N D xi) as well as high-order ones (e.g. xi A N D x~ A N D xi, etc.). Example 1. F o r n = 1, the results of a p p r o x i m a t i o n are shown for the data set visualized in Fig. 2. The relational structure includes an s t composition with the probabilistic sum and Lukasiewicz A N D conjunction. Their form is the following: p = 1:
y = (wl A N D x) O R (w2 A N D .£),
p = 2:
y = (wl A N D x) O R (w2 A N D x A N D x) O R (w3 A N D ~) O R (w4 A N D ~ A N D ~).
p = 3:
y = (wl A N D x) O R (w2 A N D x A N D x) O R (w3 A N D x A N D x).
The learning process carried out with respect to wi's is displayed in Fig. 3. The best results (minimum of the MSE) are obtained for p = 3. The elements of w are summarized in Table 1. N o t e that higher orders imply higher accuracy of approximation. Neglecting low contributions (namely d r o p p i n g the terms with wi's close to 0), the emerging relational descriptions are the following: p = 1:
y = 0.98 A N D £ .
p = 2:
y = (xANDx)OR(0.866AND~)OR(0.95
p = 3:
y = (0.371ANDx)OR(0.999ANDfANDf)OR(0.874AND£).
AND2AND.f).
1.0 ~
+'--+ p-2 vmqp3 " " - = p-1 I . - .-e DATA
,,x t,='~
-'~- ,~,,
~
,
r"
.o
I'
.--- ....~,AI "-,,/ / 'J
\ "=.,
\
, %
/
\
I
\ 0
0
.25
.50
.75
x
Fig. 2. Data points (x~,),~) and successive approximation.
1.00
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A. D i Nola et al. /' F u z z y Sets a n d S y s t e m s 75 (1995) 2 4 1 - 2 6 2
3
2
1
0 I
0
200
400
600
800
1000
Fig. 3. Learning the fuzzy set w.
Table 1 Membership functions of w for different expansions of the structure. p = I p = 2
w = [ x / O , :U0.98] w = [ x / O , x A N D x~ 1.0, x/0.866, :? A N D 2/0.95]
p = 3
w = Ix~O, x A N D x / 0 . 3 7 1 ,
xANDxANDx/0.999,
.?/0.874, f A N D x / 0 . 0 , 2 A N D . f A N D . f / 0 . 0 ]
For comparison the results of a standard polynomial approximation p 3' =
2
ai Xi
i=o
are also given in Fig. 4. The performance is better for higher orders of p, however, the polynomial becomes multimodal with clearly introduced new nodes. It is also worth noting that the approximation of the same form as carried out by the s t relational structure is not feasible for max-min or min-max compositions (due to the idempotency of the max and min operators standing there). Example 2. We will study characteristics of the s t composition and its dual structure emerging in case of two variables xl and x2 and their complements. The structure reads as for that driven by the s - t composition: y = (wl A N D xl} OR (W2 A N D x2) OR (W3 A N D 21) OR (w4 A N D x2) and for that driven by the
t-s
composition:
y = (wl OR Xl) A N D (W2 OR x2) A N D (w3 OR 21) A N D (w4 OR "~2)' The two-dimensional contours of the output values of y are summarized in Figs. 5 and 6, respectively. The different level of interactivity between the arguments of the composition is clearly reflected in the obtained characteristics.
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A. Di Nola et al. / Fuzzy Sets and Systems 75 (1995) 241 262
10
.8
.4
.2
0
0
.9
.4
.8
.8
lo
Fig. 4. Polynomial approximation of data.
2.3. Referential fuzzy relational structures
The four following relational structures are of referential character. The idea is that to process firstly the x's with respect to some other fuzzy sets viewed as reference points and afterwards aggregated with the use of O R or A N D connectives. 2.3.1. Matching relational structure This relational structure is defined as follows n
Y = S [ ( x i - - rl)twi],
(9)
i=1
where the matching operation " - " is defined below and r = Jr1, r2 . . . . . (prototype) fuzzy set in [0, 1]". We write down (9) as y = (x - r ) o w ,
rn]
is treated as a reference
(10)
then x is compared with r and these pointwise results of matching are transformed with the use of wi's and finally contributing to y. The o p e r a t i o n " = " c a n be defined in many ways; here we apply a degree of equality as defined in [14, 17, 19, 24, 30]: (a --- b) = 1/2 [(aqSb) A (bcka) + (~tk6) A (~q~d)], where q5 denotes a pseudocomplement associated with a given t-norm, namely a4ab = s u p { c e [0, 1]: a t c ~< b} with a, b ~ [0, 1]. In P R O L O G - l i k e notation (9) translates into y-= [(xl E Q U A L r 0 A N D wl] O R [ ( x z E Q U A L r / ) A N D O R [(x, E Q U A L r , ) A N D w,].
Wz] O R ...
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A. Di Nola et al. / Fuzzy Sets and Systems 75 (1995) 241 262
0.3
0.4
0.5 0.6
X ~
.7
i
× t
Fig. 5a.
U///// X ~
o X
1
Fig. 5b.
A. Di Nola et al.
Fuzzy Sets and Systems 75 (1995) 241 262
249
O,
00~''" X
1
Fig. 5c. Fig. 5. T w o - d i m e n s i o n a l c o n t o u r s o f the s-t r e l a t i o n a l s t r u c t u r e s for different t r i a n g u l a r n o r m s : (a) s = m a x i m u m , t = m i n i m u m ; (b) s = p r o b a b i l i s t i c s u m , t = p r o d u c t ; (c) s = p r o b a b i l i s t i c s u m , t = L u k a s i e w i c z c o n n e c t i v e .
0.6
O.
X
1
Fig. 6a.
0.4
0.3
0.2 0.i
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A. Di Nola et al. / Fuzzy Sets and Systems 75 (1995) 241-262
0.4
0.3
0.2
0.I
N
:,<
0 x Fig. 6b.
AND
0.2
0.1 X ~
0.0
0 ×
Fig. 6c. Fig. 6. T w o - d i m e n s i o n a l c o n t o u r s of the t-s r e l a t i o n a l s t r u c t u r e s for different t r i a n g u l a r n o r m s : (a) t = m i n i m u m , s = m a x i m u m ; (bJ t = p r o d u c t , s = p r o b a b i l i s t i c s u m ; (c) t = L u k a s i e w i c z c o n n e c t i v e , s = p r o b a b i l i s t i c sum.
A. Di Nola et al. / Fuzzy Sets and Systems 75 (1995) 241 262
251
The referential structure can be treated as a fuzzy relation equation of the form (10). Several problems can be formulated regarding their solutions: - for given x and y, determine r and w (identification problem). This problem could have several versions depending on a number of fuzzy sets to be determined such as (i) given x, y and r determine w, (ii) given x, y and w determine r; - for given y, w and r determine x (inverse problem). Assuming additionally that w = 1 (all w~'s are equal to 1), we get a class of fuzzy relation equations which have been already studied [1, 5]. Example 3. The relational structure with the equality operator w~ll be illustrated for a two-dimensional case (two inputs xl and x2): y = (xl E Q U A L 0.3) AND (x2 E Q U A L 0.6), where the operator E Q U A L is given, using the product, by
aEQUALb=
1/2.[b/a+(1-a)/(l-b)]
ifa>b,
1
if a = b ,
1/2.[a/b+(1-b)/(1-a)]
if a < b ,
a, b ~ [0, 1] while AND is the Lukasiewicz conjunction. The two-dimensional contours of the values of y are summarized in Fig. 7.
2.3.2. Referential structure with difference operator The associated relational structure includes the difference operator " =1" defined as a =1 b = 1 - (a --- b) for any a, b ~ [0, 1] Then n
y=
S [(xi~rAtw~]. i--I
Its P R O L O G - d r i v e n equivalent form reads accordingly y = [(xl D I F F E R rl) AND wl] OR [(x2 D I F F E R r2) AND w2 ] OR ... OR [(x, D I F F E R r,) AND w,], where the operator D I F F E R has an obvious meaning. Example 4. We will consider the relational structure y = (xl D I F F E R 0.3) AND (x2 D I F F E R 0.6) with the same t-norm (product) and the operator AND of Example 3. The corresponding contours of the values of y are in Fig. 8.
2.3.3. Relational structure with inclusion operation This relational structure is described as n
Y=
S [(xic~ri)twi], i=1
(11)
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X ~
× 4
Fig. 7. Two-dimensional plots of the matching relational structure.
0.3 0.15 0.0
0.0
0.3
0.45
0.0 X ~
0.15
0
0.30
0.15
0.45
0.30
0.60
Fig. 8. Two-dimensional plots of the difference relational structure.
A. Di Nola et aL / Fuzzy Sets and Systems 75 (1995) 241 262
253
where, as before, r is viewed as a reference point. F r o m a logical point of view, the pseudocomplement plays a role of inclusion relation; observe that xl 4~rl = 1 if xl ~< r~. The elements of w describe an influence of the inclusion xi4~r~ on y. In a vector notation (11) is written as y = (x I N C L r) o w. Again the structure (11) can be treated as a fuzzy relation equation: - with respect to the parameters (fuzzy sets r and w) for x and y given (here one can have subcases when r or
w is known and the second fuzzy set has to be determined), with respect to x for y given, of course the parameters of the structure being available. E x a m p l e 5. The relational structure, with the inclusion operator i N C L defined as (a I N C L b) = min(1, b/a) for any a, b ~ [0, 1] and the operator A N D of Example 3, has the form
y = (xl I N C L 0.3) A N D (x2 I N C L 0.6) and yields a plot as shown in Fig. 9. One can also study a modified version of (11) with the t-s composition n
y:
t [(xi(ori)swi],
(12)
i=1
i.e. y = (x I N C L r)" w. If all wi's are set to 0 and the t-norm specified as the min operator, then (12) is known as adjoint fuzzy relation equation [10, 23].
2.3.4. Reference structure with dominance relationship The fuzzy relational structure of this class reads as n
Y = S [(ri(axi)twi]
(13)
i=1
or more concisely as y = (r D O M x) o w. The relationship ri 4) x~ denotes a degree of dominance ofr~ by x~; the characteristics are complementary to those given previously. The dual version of (13) is given by y = (r D O M x ) . w . E x a m p l e 6. The two-dimensional plots of the relational structure
y = (0.3 I N C L x~) A N D (0.6 I N C L xz) is provided in Fig. 10.
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@
X ~
0.8
0.6 ×
0.4
I
Fig. 9. Contours of the referential relational structure with the operator of inclusion.
X ~
0.8 0.6 0.4 0.2
Fig. 10. Contours of the referential relational structure with the dominance operator.
A. Di Nola et aL / Fuzzy Sets and Systems 75 (1995) 241 262
255
3. Multilevel relational structures
We now will expand the structure of fuzzy relation equations by incorporating intermediate variables. In the network terminology this means an addition of some auxiliary layers situated between the input and output ones. Their role is to enhance representation capabilities of the equations. Our primary interest will be in three-layer relational structures. This additional intermediate (a so-called hidden) layer permits to capture specificity of the problem. We will show how the additional layers could add contribute toward handling peculiarities of the problems under discussion.
3.1. Logic processors-approximation of logic-based relationships between variables The three-layer relational structure given and studied in [10, 11, 22, 26] is suitable for copying with approximation-like problems where one is interested in eliciting logical and usually nonlinear relationships between input grades of membership (input signals) x~, x2, ..., x, as well as their complements and the grade of membership y (output signal). Their generic components are constituted by the single dual and direct fuzzy relation equations as we have studied so far. It is worth noting that the dual fuzzy relation equation with variables x~ and Y~'s forms a generalized minterm, namely: z = (wl OR xl) AND (W2 OR X2) AND ... AND (w, OR x,) AND (w,+l OR xl) AND (w,+2 OR "~2) AND ..- AND (W2n OR ~,).
(14)
If we restrict ourselves to all xi's and wi's set either to 0 or 1, then (14) converts into a standard minterm known in the two-valued logic. Similarly the direct fuzzy relation equation is interpreted as a generalized maxterm. Referring to the fundamental representation result of the two-valued logic stating that any Boolean function can uniquely given as a sum of minterms (or a product of maxterms), we will follow this path in approximating any fuzzy function [-12, 13] by a sum of generalized minterms (product of the generalized maxterms). The emerging relational structure is visualized in Fig. 11.
xl
AND
Fig. 11. A N D - O R logic processor.
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The hidden layer is built with the A N D nodes. The output is computed using a single O R node. The size of the hidden layer determines a number of the generalized minterms revealing this approximation. The detailed formulas are given below: y = (vl A N D z l ) OR(v2 A N D z2) OR ... OR(v,, A N D zm)
(15)
and Zh = (Whl OR Xl ) A N D (Wh2 OR x2) A N D ". A N D (Wh, OR x,) A N D (Wh,, + 1, OR £1) A N D ... A N D (Whiz,, OR .%), where h = 1. . . . . m. In compact form we will write down, y =zcv
(16)
for the first fuzzy (single) relation equation, while the second one converts into the t-s equation z =£.w,
(17)
where £ includes both xi's as well as their complemented values. The dual structure approximates logical relationships between x and y in taking advantage of the generalized maxterms. It should be stressed that in contrast to the two-valued logic functions, this representation is not universal, however constitutes a legitimate approximation and a good starting point for further refinements if those are viewed necessary. The solution to this class of multilevel fuzzy relation equations can be posed with respect to the fuzzy set v and the fuzzy relation w: "given x and y, determine v and w". In general the analytical solution is not available. The solution does not specify a minimal value of h. For a collection of Xk'S, the corresponding parameters tk'S of the relational structure should be determined through solving a relevant optimization problem (cf, Section 2.1). The optimization in this case pertains to determining w and v for a given size of the hidden layer. This in fact constitutes a sort of parametric learning. The performance index is a standard one involving a sum of squared errors, Q = ~ [yIXh) -- th] 2, h=l
where y(Xh) denotes the value o f y obtained for Xh; th is associated with it as a part of data to be approximated. The elements of w and v are determined by considering successive adjustments driven by a gradient of Q. Symbolically we put it down as writer + 1) = w(iter) - :~. ~3Q/Ow, v(iter + 1) = voter) - ~' ~Q/Ov, where ~ e (0, 1) denotes a learning coefficient. All the detailed computations can be directly completed once the t-norms have been defined. Example 7. We will discuss the relational structures (16) and (17) with the s-norm defined as a probabilistic sum and any t-norm. This yields the following detailed formulas: n
n
t (WhiSXi)t t (w,,,.+l,sg,),
zh=
i=1
i
trl
Y=
S h=l
(LIhSZh) .
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Assuming an on-line adjustment, we can drop "h" and consider a single pair x, t. This refinement yields ~Q/OWh, j = ~Q/t?Vk
[y(X)
---- [ y ( x )
-- t]
-- t]"
"Oy/t~Wh, j], t?y/C?Vk.
Subsequently nl
t?y/~v k = (3 S (13hSZh)/~Uk = ~ [A S(VkSZk)]/~l.)k, h=l
where A=
S (VhSZh). h ~ak
In virtue of our specification, we derive i?[A + Vk'Zk -- A ' V k ' Z k ] / t ~ V k
= Zk'(1 -- A).
Then ~y/~Wh, j = ~ ~Y/~Zh'~Zh/~Wh,j" h=l
Note however that the above sum reduces to a single component since ~Zh/~Wh, j = 0 for all h :~ h i . In this way ~y/t~Wh, j =- ~y/OZh, " OZh,/ OWh,j. And finally Oy/~zh, = Vk'(1 -- B),
n =
S (VhSZh), h~hl
= c~3[ l~ (wh.i ~- Xi - wh~iXi). ~-I ~wh'~n+ ~} ~- Xi - ~h'~+ ~.~i~.~wh'j ~- Xj - wh'Xj)~/t3wh'j-= C(~ - ~j)~ i~-j
ivaj
where C abbreviates the two first factors in the above expression. The solutions to the above relation equations can be obtained in an approximate way by formulating and solving associated optimization tasks; the reader is referred to [20-22, 27]. Regarding the internal derivatives, one should be aware that for nondifferentiable t-norms such as maximum and minimum, they should be defined with a certain caution [15]. Take for instance min(x, w) and calculate its derivative with respect to w. Obviously we obtain 1 if x >/w and 0 otherwise. The optimization problem driven by this type of "on-off" mode of updates could be easily affected by traps caused by some configurations of the connections and the data points in course of updating. A prudent inspection suggests a slight modification, the basic idea of which is to replace the values 0 and 1 by its multivalued counterpart. Namely we relax the inequality x >~ w taking only two values "True(l)" if satisfied and "False(0)" otherwise, by accepting an inequality which is satisfied to a certain degree, say IIx t> wll, where IIx/> wll = 1 ifx >~ w and IIx/>. wll = w INCL x otherwise. Thus IIx >/wll e [0, 1] and this contributes significantly to a smooth character of updating the parameters of the relational structures. For more details, we refer to [15].
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xl __
x2
EOUAL
1.0 f21 ~m
x2
..... Y .
f22
xn
1.0 X!
Fig. 12. Multilevel fuzzy relational structure with matching.
Fig. 13. Pattern classification with matching.
the fuzzy relational structure
3.2. Three level relational structures' with reference objects' The relational structure can be developed in such a way that it realizes operations in [0, 1]" which pertain, in particular, to matching, difference, inclusion and dominance. Matching: In this relational structure x~, x2 . . . . . x, match prototype I21 , 1"22, ..., I2,, and these results are further aggregated as shown in Fig. 12. The first relation equation carriers out a process of coordinatewise matching, Zh = [(Xl E Q U A L Whl ) OR Whl ] AND [(x2 E Q U A L Wh2 }OR Wh2] AND -.. AND (18)
[(x, E Q U A L Wh,) OR Wh,], where h = 1, 2 . . . . , m and The operator " E Q U A L " membership, The number a direct relation equation.
Oh = [Whl, Wh2 Whn] denote the grades of membership of the hth prototype. returns a degree of matching (a level of similarity) between two grades of of prototypes specifies the format of the hidden layer. The next level constitutes Strictly speaking, Eq. (18) will be expressed as
Zh = (X E Q U A L Wh)'Wh,
. . . . .
y = Z V,
where z = [zl, z2 . . . . . z,,] and y is given as in (15). The relational structure of this type can be found useful in problems of pattern classification. For instance, for patterns belonging to the same class and distributed around several disjoint regions in the feature space centered in 121, I22 . . . . . 12,., the above value of y denotes a degree of class membership of the pattern x to be classified (cf. Fig. 13). The relational structure implements a general statement: "the pattern x belongs to the class if it is similar to I2~ OR it is similar to 122 OR ... OR it is similar to 12,,". Note that each subcondition in this statement is realized by the corresponding node in the hidden layer while the overall aggregation is of the OR-like type (the third layer). Difference: The structure is driven by the difference levels determined between xfs and some landmark points (here playing a role of exceptions) ~'1, ~2 . . . . . ~,,. Thus we get: zh = [(x~ D I F F E R @hl) OR Whl ] AND [(x2 D I F F E R @h2)OR Wh2] AND ".. AND [ (x. D I F F E R g,h.) O R wh.],
(19)
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259
where h = 1,2, ... ,m and the membership vectors 0h = [0hl, 0h2 . . . . . 0h,] are used to calculate the respective differences. The results of (19) are then combined by AND-ing zh's. This yields two formulas, Zh = ( x D I F F E R O h ) o W h ,
y = z.v,
where y is given by y=(zlORvl)AND(z2ORvz)AND
A N D (z,, O R v,,).
The difference relational structure can be easily interpreted in pattern recognition, fault analysis and machine learning. The vectors Oh represent exceptions; thus a pattern is viewed as belonging to a certain category if it is excluded from 0a and it is excluded from 02 A N D ... A N D it is excluded from 0,, (see Fig. 14). The quite dispersed regions of exceptions are now better approximated in this way rather than applying the relational structure equipped with some matching capabilities. Inclusion: The structure has a hidden layer consisting of the elements capturing the property of inclusion (say, modelling the statement "x is included in"). The partial results derived at this layer can be afterwards aggregated either in a conjuctive or disjunctive manner. Formally, this conjunctive form of the aggregation converts into the formula Zh = [(Xl I N C L 27hl ) O R whl ] A N D [(x2 I N C L rh2) OR
Wh2"]A N D
... [(x, I N C L rh.) OR Wh,],
where h = 1, 2, ... ,m and ~h = [27hl, 2 7 h 2 , - - . ,72hra] denotes a reference point with respect to which the inclusion relationship is defined. At the second stage we get y = z o v or y = z. v depending on the way in which the final aggregation should be carried out. The relational structure models situations in which one attempts to comply with several constraints (requirements) which should not be exceeded (e.g. the speed should not be higher than rl A N D the temperature should be kept below ~2, etc.). D o m i n a n c e : The relational structure is dual to the previous one where the statements are formulated as "x should dominate (be higher) then ... ". Thus we have Zh = [(7hl I N C L Xl ) O R Whl ] A N D [-(])h2 I N C L xz) O R Wh2 ] A N D "- A N D E(Vh,I N C L x,) O R Wh,], where h = 1, 2 . . . . . m while the reference point ~'h indicates an element which should be dominated by x. The aggregation can be then performed in a disjunctive or conjunctive manner.
X2
.0 Xl
Fig. 14. Pattern classification the fuzzy relational structure with difference operator.
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A. Di ?Cola et al. / Fuzzy Sets" and Systems 75 (1995) 24l 262 b
~j Fig. 15. Relational structure with three nodes.
OR
decision
AND
Fig. 16. Relational structure formalizing a decision-making problem.
A. Di Nola et al. / Fuzzy Sets and Systems 75 (I995) 241-262
261
3.3. Decision-making processes modelled with the use of multilevel relational structures
The multilevel relational structures can also involve all types of the processing elements depending on a particular application. They create an ideal tool for formalizing compound statements in decision-making processes. The transformation (formalization) is immediate, while the learning can furnish the structure with numerical connections. Below we list several examples along with the relevant relational structures being their formal translation: "if (x satisfies constraint b) and (x dominates c) and (x does not exceed a), then the decision D should be taken". The relational structure is then described as Q = (x EQUAL b) AND (c INCLx) AND (x INCL a) and, as becomes evident, consists of three nodes in the hidden layer and aggregated via the generalized AND operation in the output layer (cf. Fig. 15). A final numerical calibration (leading to the membership values of w and v) is completed on a basis of some training data pertaining to available decision scenarios. The following statement of if-then type: "if [(x satisfies a) and (x differs from b) and (x satisfies c)] or [(x differs from d) and (x is included in f ) ] , then the decision D should be taken" translates into D = [(x EQUAL a) AND (x DIFFER b) AND (x EQUAL c)] OR [(x DIFFER d) AND (x I N C L f ) ] and this formulation gives rise to a four-level relational structure shown in Fig. 16.
4. Conclusions It should be noted that these structures are usually heterogeneous in their nature including different basic logic operations (or their generalizations) used to develop their basic components. It is worth stressing that they could be utilized in many real-world problems due to their ability to transform the problem into straightforward logical structures and carry out final calibration by modifying all necessary relations.
Acknowledgement Support provided by the Natural Sciences and Engineering Research Council of Canada is gratefully acknowledged by the second author.
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