- Email: [email protected]
General decomposition problem of fuzzy relations
Fuzzy Sets and Systems 54 (1993) 69-79 North-Holland
69
General decomposition problem of fuzzy relations J. Vrba Institute of Chemical Process Fundamentals, Czechoslovak Academy of Sciences, Praha, Czechoslovakia Received November 1991 Revised June 1992
Abstract: This paper deals with the problem of general max-min decomposition of binary fuzzy relations defined in the Cartesian product of finite spaces •" x R n where m and n as well as the resulting relations are not necessarily equal. The proposed algorithm guarantees decomposition within a finite sequence of steps related to the number of different non-zero membership levels in the decomposed relation. The concept of peak pattern analysis is used for this purpose. The paper also presents the potential role of the general decomposition of fuzzy relations for the needs of fuzzy logic in engineering and control modelling. Keywords: Fuzzy relation; decomposition of relations; peak pattern.
1. Introduction
Despite of many interesting works from the area of finding inverse solutions of fuzzy relation equations, devoted either to theoretical aspects [1, 2, 3] or to the needs of modelling in engineering [4] and control [5] theory and practice, the general decomposition problem of fuzzy relations has been only little attacked [6]. Di Nola et al. [6] solve the interesting problem of finding a relation A for a given binary relation C fulfilling the equation A oA = C
(1)
with the definition
~ ( X X X ) = (C: X x X---'~ L}
(C = [cij], cij = C(xi, xj), i, j e In)
(2)
where X = {Xl, x2 . . . . , Xn} denotes a finite set, L a linear lattice with universal bounds 0 and 1, In = {1, 2 . . . . . n} the set of the first n natural numbers and 'o' the max-min composition [7]. Then, the set o~ is the set of all fuzzy relations defined in the Cartesian product X x X. The result of the decomposition is thus the fuzzy relation A e ,~(X x X), A = [au] = [UA(Xi, X~)], i, j e I~, where
9 (aik A a~j)
=
cij ,
i, j, k e In,
(3)
k=l
with ' v ' and ' ^ ' denoting the usual maximum and minimum operators of L, respectively. The equation (1) need not always have a solution. It is possible, then, to look for its approximate solution, e.g., the problem of the 'least-square distance' as proposed by Pedrycz [2]. To utilize the decomposition of fuzzy relations in fuzzy logic control theory and practice (especially for modelling purposes), it is not very likely to expect always the structure of (1); one needs more general possibilities. Namely, our attention may be concentrated on the relation equation A oB = C
(4)
Correspondence to: Dr. J. Vrba, Institute of Chemical Process Fundamentals, Czechoslovak Academy of Sciences, Rozvojova 135, 165 02 Praha 6, Suchdol, Czechoslovakia. 0165-0114/93/$06.00 © 1993---Elsevier Science Publishers B.V. All rights reserved
J. Vrba / General decomposition problem of fuzzy relations
70
with the definition ~ ( S × Z) = {C : X x Z---~ L}
(C = [cij], ci~ = C(xi, zj), i • In, j • In)
(5)
where Im = {1, 2 . . . . , m}, A x ~:(X x Y), B • ~ ( Y x Z) and, consequently, C • ~ ( X × Z) with l
V (aikAbkj)=Cij,
k=l
i • l , , , j e l n , k•Ii,
(6)
and/~ = {1, 2 , . . . , l}. Obviously, Y = {Yl . . . . . Yt}, Z = {zl, • • •, zn} and the quantities a~k, bkj and c~j represent the membership degrees uA(x~, Yk), uB(yk, Zj) and Uc(Xi, zi), respectively. Moreover, A and B may be also only fuzzy subsets A x ~:(X) and B • ~ ( Z ) with It = {1). In the light of fuzzy-logic control modelling, the relations A, B and C may be understood as the time-invariant next-state relation, the control relation and the overall next-state relation, respectively. Usually, the relations B, C are supposed to be known so that the control relation A may be found as an inverse solution of (4) [5, 8]. In case we are informed only about the overall next-state relation C, the possibility of obtaining some information for the control model construction lays, possibly completely, on the mentioned general decomposition of C. Because of different forms of A and B due to the different generic variables of their universes, representing input, state and control of the system to be controlled, the decomposition (1) cannot be applied for the purpose. The decomposition (4) may seem rather artificial but it may sometimes be the only source to give some information about the composing structures for C and even some basic preliminaries for A and B. Therefore, there are no conditions put on them. As may be seen from (6), the main advantage of the decomposition (4) for a suitable algorithm is the free character of the index set It. As a rule, l ~
(7)
(a short discussion on the inequality (7) will be given later). Rather than the detailed membership structure of the relations A, B, their linguistic interpretation is of a more valuable descriptive character.
2. Notations and definitions
To simplify the task, the decomposition may be performed with advantage on individual membership levels of C. The concept of Tong's peak pattern analysis [8, 9] offers good properties and possibilities for it. A peak pattern, PP, of a fuzzy subset ~¢ on its finite universe of discourse (space) ~ is a binary map of the discrete membership function of sO, us(x), with P P , ( x ) = {~
ifu~(x)>'a' otherwise.
(8)
where a is a real number from (0, 1). Thus, PP~a is the map of ~¢, and PPaa(x) that of us(x). Obviously, a peak pattern 'covers' an element x • ~ iff PP~°(x)= 1; it is important to specify the 'a-covering' for PP~(x) = 1, x • suppa(M) = {x I us(x) >-a}, if necessary. A peak pattern may cover more than one element. A fuzzy subset with a peak pattern covering only one element in ~ is called singular. It is non-singular otherwise. The space ~ is considered multidimensional. Then, it is possible to speak about a fuzzy relation ~/. The formal presentation of a binary relation is a relation matrix with membership degrees as its elements. A row peak pattern, RPP~, of ~ is the binary map of a row of the matrix M in which a quantity ~>a occurs. So, the map entry is either 1 or 0 in the sense of the definition (8). The column peak pattern, cppa~t, is defined similarly by the binary map of a column. A relation is primary if it has only one distinct RPP.
J. Vrba / General decomposition problem of fuzzy relations
71
A relation is regular if it exactly represents a single statment of the corresponding linguistic model, i.e. if its peak pattern covers only one element of ~. It is non-singular otherwise. A necessary condition for M to be regular is that it is primary. The proof of this statement is straightforward and results directly from the min-operation on the subsets entering the relation M. The peak-pattern analysis is extremely important for the fuzzy-linguistic interpretation of fuzzy relations. The peak-pattern of their highest membership level detects the number and evaluating character of corresponding statements (implications); such a peak-pattern is unique for any possible way of decomposition. The concrete value PP~(xi, yj) is symbolized as PP,(i, j), for short. Example 1. Consider a binary relation C--
0.3 0.3 0 . 5 ) 0.4 1 0.5 . 0.4 0.4 0.5
The peak patterns on individual membership levels of C are ppL0 =
1
,
pp~5 =
0 pp~4 =
1
1
,
0 ,
pp~3 =
1
1
,
1
where individual rows and columns of the PPs represent the RPPs and CPPs, respectively. The relation C is singular; it may represent only a single statement. Its regularity may be determined by its decomposition into (m, 1)- and (1, n)-dimensional relations (subsets) A and B, respectively, fulfilling the equation (4) (m = n = 3). The regularity of C may be intermediated by the composing relations A, B themselves in case those relations are regular. Such a situation is illustrated by examples.
3. The method
Let us start with an example, again. Example 2. What is necessary to do to derive the PPs of max-min composing relations A and B for individual P~c from Example 1? First, one of the possible simplest forms of the highest membership may be, e.g., ppLO= (0, 1, 0) T and ppL0 = (0, 1, 0), so that
1.o,.ol.0
PPA o PP8 = PPA × ppLO =
(Z)
X (0
1 o)=
1
= pp~:O.
0 Now, comparing pp~5 with the ppLO and ppLO we know that the starting structure for pp~/S and pp~5 must be of the form ppO/5 =
x 1 ×
and pp 5_(
1 :)
72
J. Vrba / General decomposition problem of fuzzy relations
where the positions marked '.' can and '×' c a n n o t be occupied by '1' to protect and preserve the '0' positions in pp~5. Then, taking into account the PPs of A and B for the membership level a = 1.0 we m u s t accept the version ppO/5=
and
1 1
P~'/~ 01)
giving pp~5 =
1 . 0 Similarly, the starting structures ppO/4=
(x i) 1
and pp~4=
1
It
in common with the PPs for a = 0.5 allow o n l y the choice 1
and PP~4 t~ ° it giving pp~4 =
1 1
.
Finally, the starting structures for the lowest non-zero level are
PP3:(ii)and
1t,.
so that we m u s t accept 1 1 1)=pp~4 and PP~3= 0 0 1
giving pp~3 =
1 1
!/
Then, following, e.g., Kaufmann [7] and the membership level operations, we determine the relations A and B as unions of all ppas multiplied by the corresponding membership levels a, i.e. A = 1.0x (S i ) +0.5x (S
i ) +0.4x ( i
! ) +0.3× (11 i ) ( ~=. 3 1 \0.4
:.5 t 0.5/
73
J. Vrba / General decomposition problem of fuzzy relations
and B=I'0x
( 0 1 ~) + 0 . 5 X (0 1 1) + 0 . 4 X (1 1 1) + 0 . 3 X (1 1 ~ ) (=0 0 . 4 0 0 0 0 1 0 0 1 0 0
1 0.5 ) 0 0.5
(the symbol '+' denotes logical union and ' x ' algebraic multiplication). We can verify that
/o Oo/Oo .3
AoB=
\0.4
.5
°
.4
1 0.5 0 0.5 =
0.5/
0.4
1
0.5
0.4
0.4 0.5
=C.
The relations A, B, though primary, are non-regular. They cannot be decomposed into subrelations (subsets) of the (m, 1) and (1, n) types, respectively. Nevertheless, as pointed out by Tong [9], we can approximate the relation A as A' = (0.3, 1, 0.4) and B as B' = (0.4, 1, 0.5), i.e. the rows with the RPP and CPP of the highest level. Then, the relation
C'=A'oB'=
(01.3)
°(0.4
1 0.5)=
\0.4
(03 03 0.4 0.4
1 0.4
0.5 0.4
may be considered linguistically identical with the relation C and the subsets A', B' the fuzzyevaluating elements of the corresponding implication "if A' for x then B' for y". In the preceding example, all the rules which may form the decomposition algorithm are discussed (the symbol = stands for necessardy equal' and ~ stands for union): 1. The ordered pairs (aik, bkj ) from the created relations A, B must fulfil the rule of max-min composition a ' " ~ S P PaA (•t , k ) × P P aa( k , "1) =! PPc(l,/).
(9)
2. The situation cij = 0 indicates the fact that at least one of the composing elements of any pair (aik , bkj ) on Ii must equal zero, i.e. if PP~c(ir, is) = 0 then (a) for Vk e It such that PPaa(ir, k) = 1: P ~ ( k , is) -" 0, (b) for Vk e It such that PP,(k, j~) = 1: PP~(ir, k)--"0. 3. The situation ciy ~ 0 indicates the fact that there must exist at least one pair (aik =/:O, bkj ~ 0) on It fulfilling the conditions (6), (9), i.e. if PP~(ir, 1"~)= 1 then 3r ~ (a, 1), 3s ~ (a, 1~, 3 k ~ It:
PP~t(ir, k) X PiCa(k, j~) " 1.
A single choice of k e It for any pair (ir, j~) guarantees the resulting PP~ and PP~ have minimum number of l's. 4. The union of all peak patterns, multiplied by corresponding membership levels, gives the relation itself: A=
y o a
o
a x PPA,
B=
a x PP~.
(10a,b)
a
The proofs of the above rules are trivial and follow directly from the concept of max-min composition. The rules convey the decomposition to the simplest (minimum) version of A and B with respect to the number of non-zero memberships and their values. That means that each nonzero element of A, B is m a x - r a i n active for C, i.e. for V k ~ It such that aik > O, 3(i, j):
aik = Cij
J. Vrba / General decomposition problem of fuzzy relations
74
and for Vk e It such that bkj > O, 3(i, j):
bkj = cij.
It is worth mentioning that the bounds m, n are predetermined by the dimensionality of the relation C, the quantity l for the number of rows in A and columns in B being governed by the condition (7). Really, from the point of the lowest dimensionality 1, the best case is with the same RPP for any row in C, differing only in membership level a. It is possible, then, to compose a unique RP~A with a unique C P ~ to obtain the corresponding PWc: the composition PP~t(i, 1) x PP,(l, ]) = PP,(i, j) is the same for any row of PP~ of dimensionality (m, 1) and any column of PP~ of dimensionality (1, n), i.e. RPP~ and CPP~. On the contrary, the worst situation with the highest value of 1 may occur when only the trivial decomposition is possible. This means that, at any membership level, PP~ equals either PWA or P ~ , the composition-complementary PWn or PP~ forming the (l x 1) diagonal matrix, l fulfilling the condition (7) as equality to n and m, respectively. Such a matrix represents a quasi-equivalent relation or ~ with non-zero diagonal elements and the properties or A o ~ = A = C .
alo~=B=C
(11),(12)
The non-specified number of rows in A as well as columns in B enables a free decomposition A(i, *)o B(*, j) that, actually, enables a decomposition of any C. Such a free dimensionality enables any pair RPP~, CPP~ to form a unique RPP~ and, consequently, a unique CPP~, too. If any of the RPPs of A would need a particular partner CPP of B (and vice versa), they cannot be greater in their number and dimensionality than m or n, respectively. These situations are illustrated by the following examples. Example 3. Applying the above rules we can decompose the relation
C=
'0.5
0.5
0.5/
0.7
0.8
0.5
0.7
1
0.5
0.6
0.6
0.5
0.3
0.3
0.3
into A = (0.5, 0.8, 1, 0.6, 0.3) T and B = (0.7, 1, 0.5), i.e. l = 1 < min[m, n]. Obviously, the relation C may be considered regular representing a single statement "if A for x then B for y". Example 4. The fuzzy subsets A = (1, 0.8, 0.4) T and B = (0.8, 1, 0.6) are the only subsets that may max-min compose the relation C=
0.8 1.0 0 . 6 ) 0.8 0.8 0.6 . 0.4 0.4 0.4
That may be detected by the analysis: a=1.0:
PPc=
0 0
,
PPA=
PPz=(O
1 0),
75
Vrba / General decomposition problem of fuzzy relations
J.
a=0.8:
PPc=
(i 1
,
PPA =
,
PPs=(1
1 0),
a=0.6:
PPc=
(i 1
,
PPA=
,
PPs=(1
1 1),
a=0.4:
PPc=
(i 1
,
PPA=
,
PP~=(1
1 1).
1 0
1 0
1 1
Example 5. The fuzzy relation
C=
0.4
0.8
1
0.8
0.4 /
0.8
1
0.8
0.8
1
1
0.8
0.5
0.9
1
0.8
1
0.8
1
0.9
1
0.8
0.6
0.1
0.6
is decomposable as
0
o o ol
0.4
0.8
1
0.8
0.4 /
0.8
1
0
0
0
0.8
1
0
0.8
1
0.5
0.8
1 0
0
1
0
0
0.9
1
0.8
0.9
0
1 0
0
1
0
1
0
0.1
0
0
0
1
0.8
0.6
0
0.6
fl A=
and
B=
1
In light of the linguistic structure of ppL0 and ppL0, the decomposition obtained is practically trivial, and the given relation C is decomposable in a trival manner only. The highest membership level peak-patterns of A and B are identical with those of the corresponding equivalence relation and C, respectively. The algorithm follows the decompositional possibilities on individual rows and columns in turn. Of course, it is possible to proceed in different ways but the difference obtained is rather formal and may be misleading. For example, the analysis of the last relation C allows the decomposition for a = 1, r0 0
rl
1 0 0
1
0 0
1 0 0 0
1
0 PPc=
0 0 0 01
o~
0
1
1 0
0
1 0
1 0 0 0 0
,
PPA =
0 0
1 1 0
1 0
0 0
tO 0
1 0
1 0
1
1 0 0 0
,
PPB =
1 0 01
1
0 0 0 0
0
1 0 0 0
0 0 0 0
1
0
0
0
0
1
Obviously, there are differences between this analysis and that of the previous example. Nevertheless, it is quite easy to check that the successive interchange of rows in PPB and columns in PPA gives the same results. In other words, both decompositions are linguistically identical. Example 6. An illustrative part of the fuzzy-linguistic model for the relation between citric acid (as the product- variable 1) and glucose (as the substrate- variable 3) concentrations is presented as MSC in
J. Vrba / General decomposition problem of fuzzy relations
76
Table 1 Statements Variable
1
2
3
4
5
6
7
8
9
10
11
1 3
9 1
8 1
7 2
6 3
5 4
4 5
4 6
3 7
3 8
2 9
1 9
the form of Table 1 [10]. The values within the table denote the indexes of the respective evaluating terms used for the variables. The fuzzy language for the substrate is X = (0, lO0)g/l,
m = K x = 9,
and for citric acid: Z = (0, 100)g/l,
n = K z = 9.
Supposing that in the citric acid fermentation the biomass concentration is another control variable (variable 2), we would like to decompose the model MSC to the models MSA and MSB which would express the relations of substrate/biomass and biomass/product concentrations, respectively. The respective fuzzy language for the biomass concentration is Y = (0, 9)g/1 and K y = 9. For the membership description of all variables x and their evaluating terms ~/, i.e. u~(x), the parabolic two-parameter formula [10]
Uzat(X) = { aO(X-- Xmin)(Xmax-- X)/(Xmax-- Xmin)2 if Xmin <~x <~X . . . . otherwise,
(13)
is used. Knowing the definitions x • ~, y • ~ for the respective variables we can construct any fuzzy relation of the problem, ~(x, y) • ~ × ~, for ~ and ~ evaluated by ~ / a n d ~ , respectively, as [7] V (x, y) • ~ x ~:
~(x, y) = min[ua(x), u~(y)].
(14)
With respect to the nature of the fermentation process, A must be expected as direct and B as reverse relation. First, the overall next-state fuzzy-linguistic model MSC is translated into the form of a fuzzy relation C(x, z): Z
0
C(x, z)=
0
0
0
0.6
0.6
0.6
0.9
1
0
0
0.6
0.6
0.6
0.9
0.9
0.9
1
0
0.6
0.6
0.9
0.9
0.9
1
1
0.~
0.6
0.6
0.9
0.9
1
1
0.9
0.9
0.9
0.6
0.9
0.9
1
0.9
0.9
0.9
0.6
0.6
0.9
0.9
1
0.9
0.9
0.6
0.6
0.6
0
0.9
1
0.9
0.9
0.6
0.6
0
0
0
1
0.9
0.9
0.6
0.6
0
0
0
0
1
0.9
0.6
0.6
0
0
0
0
0
X
J. Vrba / General decomposition problem of fuzzy relations
77
Applying the above ideas we perform the peak-pattern analysis shown in Table 2, with the results
A(x, y) =
B(y, z)=
1
0
0
Y 0
0
0
0
1
0.9
0
0
0
0
0
0.9
1
0.9
0
0
0
0
0.9
0.9
1
0.9
0
0
0
0.6
0.6
0.9
1
0.9
0
0
0
0.6
0.6
0.9
1
0.9
0
0
0
0
0.9
0.9
1
0
0
0
0
0.6
0.9
0.9
1
0
0
0
0
0.6
0.9
1
0
0
0
0
Y 0.6
0.6
0.6
0.9
1'
0
0
0.6
0.6
0.6
0.9
1
1
0
0
0.6
0
0.9
1
1
0.9
0
0
0.6
0.6
0.9
1
0.9
0.6
0
0
0
0.6
0.9
1
0.6
0
0
0
0
0
0.9
1
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
X
z
or, in the fuzzy-linguistic form, as shown in Table 3 (NSA = NSB = 9). Translating the above FL models into the form of fuzzy relations and composing them in the sense of equation (4), we obtain a fuzzy relation corresponding exactly to the given FL model MSC. The ex-post experiments confirm the results.
4. Concluding remarks It is unnecessary to speak about approximation in the general decomposition problem provided there is no specific claim on the form of the resulting relations in case the decomposed relation is drawn from other sources than from a FL model. It is more convenient, then, to look for their possible linguistic meaning than to reject them as mathematically uncorrect. A time series quantised on the defined universes and filtrated through the accepted term sets for respective variables forms the background for a FL model with the fuzzy terms as building units. These terms are the only objects a relation is constructed o f - its decomposition discovers that fact. Thus, the resulting relations must be evaluated from this point of view - see Example 6. As seen, the decomposition algorithm is suitable for on-line use as well. As soon as the overall next-state relation is constructed [5, 10] the decomposition step follows automatically. In that way, it is possible to model the process in its steady state as well as in its dynamic. Thus, the fuzzy-linguistic approach helps surmount the obstacles the conventional modeling technique sometimes cannot deal with.
78
J. Vrba / General decomposition problem of fuzzy relations Table 2 a
PPc
PPA
PPn
1.0
000000001 000000001 000000110 000011000 000100000 001000000 010000000 100000000 100000000
1000000 1000000 0100000 0010000 0001000 0000100 0000010 0000001 0000001
000000001 000000110 000011000 000100000 001000000 010000000 100000000
0.9
000000011 000001111 000111111 001111111 011111100 111110000 111100000 111000000 110000000
1000000 1100000 1110000 1111000 0011100 0001110 0001110 0000111 0000011
000000011 000001110 000111100 001110000 011000000 110000000 100000000
0.6
000011111 001111111 011111111 111111111 111111111 111111110 111111000 111110000 111100000
1000000 1100000 1110000 1111000 1111100 0111110 0001110 0001111 0000111
000011111 001111110 010111100 111111000 111100000 110000000 100000000
Table 3 Statement
MSA
MSB
Variable
1
2
3
4
1
1
2
3
4
2
1
1
2
3
2
1
2
2
3
3
9
8
7
6
5
6
7
8
9
5
6
4
5
7
8
9
6
7
7
3
5
4
5
6
7
4
3
2
1
References E. Czogala, J. Drewniak and W. Pedrycz, Fuzzy relation equations on a finite set, Fuzzy Sets and Systems 7 (1982) 89-101. W. Pedrycz, Numerical and applicational aspects of fuzzy relational equations, Fuzzy Sets and Systems 11 (1983) 1-18. E. Sanchez, Resolution of composite fuzzy relation equations, Information and Control 30 (1976) 38-48. Y. Tsukamoto and T. Terano, Failure diagnosis by using fuzzy logic, Proc. IEEE Conf. on Decision and Control, New Orleans (1977) 1340-1395. [5] J. Vrba, Peak-pattern concept and max-min inverse problem in fuzzy control modelling, Fuzzy Sets and Systems 47 (1992) 1-11.
[1] [2] [3] [4]
J. Vrba / General decomposition problem of fuzzy relations
79
[6] A. Di Nola, S. Sessa and W. Pedrycz, Decomposition problem of fuzzy relations, Internat. J. General Systems 10 (1985) 123-133. [7] A. Kaufmann, Introduction to the Theory of Fuzzy Subsets (Academic Press, New York, 1975). [8] R.M. Tong, Analysis and control of fuzzy systems using finite discrete relations, Internat. J. Control 3 (1978) 431-440. [9] R.M. Tong, Analysis of fuzzy control algorithms using the relation matrix, lnternat. Man-Machine Stud. 8 (1976) 679-686. [10] J. Vrba and J. Brozek, Fuzzy concept in modelling of biotechnoiogical processes, Systems Anal. Modelling Simulation (in press).
69
General decomposition problem of fuzzy relations J. Vrba Institute of Chemical Process Fundamentals, Czechoslovak Academy of Sciences, Praha, Czechoslovakia Received November 1991 Revised June 1992
Abstract: This paper deals with the problem of general max-min decomposition of binary fuzzy relations defined in the Cartesian product of finite spaces •" x R n where m and n as well as the resulting relations are not necessarily equal. The proposed algorithm guarantees decomposition within a finite sequence of steps related to the number of different non-zero membership levels in the decomposed relation. The concept of peak pattern analysis is used for this purpose. The paper also presents the potential role of the general decomposition of fuzzy relations for the needs of fuzzy logic in engineering and control modelling. Keywords: Fuzzy relation; decomposition of relations; peak pattern.
1. Introduction
Despite of many interesting works from the area of finding inverse solutions of fuzzy relation equations, devoted either to theoretical aspects [1, 2, 3] or to the needs of modelling in engineering [4] and control [5] theory and practice, the general decomposition problem of fuzzy relations has been only little attacked [6]. Di Nola et al. [6] solve the interesting problem of finding a relation A for a given binary relation C fulfilling the equation A oA = C
(1)
with the definition
~ ( X X X ) = (C: X x X---'~ L}
(C = [cij], cij = C(xi, xj), i, j e In)
(2)
where X = {Xl, x2 . . . . , Xn} denotes a finite set, L a linear lattice with universal bounds 0 and 1, In = {1, 2 . . . . . n} the set of the first n natural numbers and 'o' the max-min composition [7]. Then, the set o~ is the set of all fuzzy relations defined in the Cartesian product X x X. The result of the decomposition is thus the fuzzy relation A e ,~(X x X), A = [au] = [UA(Xi, X~)], i, j e I~, where
9 (aik A a~j)
=
cij ,
i, j, k e In,
(3)
k=l
with ' v ' and ' ^ ' denoting the usual maximum and minimum operators of L, respectively. The equation (1) need not always have a solution. It is possible, then, to look for its approximate solution, e.g., the problem of the 'least-square distance' as proposed by Pedrycz [2]. To utilize the decomposition of fuzzy relations in fuzzy logic control theory and practice (especially for modelling purposes), it is not very likely to expect always the structure of (1); one needs more general possibilities. Namely, our attention may be concentrated on the relation equation A oB = C
(4)
Correspondence to: Dr. J. Vrba, Institute of Chemical Process Fundamentals, Czechoslovak Academy of Sciences, Rozvojova 135, 165 02 Praha 6, Suchdol, Czechoslovakia. 0165-0114/93/$06.00 © 1993---Elsevier Science Publishers B.V. All rights reserved
J. Vrba / General decomposition problem of fuzzy relations
70
with the definition ~ ( S × Z) = {C : X x Z---~ L}
(C = [cij], ci~ = C(xi, zj), i • In, j • In)
(5)
where Im = {1, 2 . . . . , m}, A x ~:(X x Y), B • ~ ( Y x Z) and, consequently, C • ~ ( X × Z) with l
V (aikAbkj)=Cij,
k=l
i • l , , , j e l n , k•Ii,
(6)
and/~ = {1, 2 , . . . , l}. Obviously, Y = {Yl . . . . . Yt}, Z = {zl, • • •, zn} and the quantities a~k, bkj and c~j represent the membership degrees uA(x~, Yk), uB(yk, Zj) and Uc(Xi, zi), respectively. Moreover, A and B may be also only fuzzy subsets A x ~:(X) and B • ~ ( Z ) with It = {1). In the light of fuzzy-logic control modelling, the relations A, B and C may be understood as the time-invariant next-state relation, the control relation and the overall next-state relation, respectively. Usually, the relations B, C are supposed to be known so that the control relation A may be found as an inverse solution of (4) [5, 8]. In case we are informed only about the overall next-state relation C, the possibility of obtaining some information for the control model construction lays, possibly completely, on the mentioned general decomposition of C. Because of different forms of A and B due to the different generic variables of their universes, representing input, state and control of the system to be controlled, the decomposition (1) cannot be applied for the purpose. The decomposition (4) may seem rather artificial but it may sometimes be the only source to give some information about the composing structures for C and even some basic preliminaries for A and B. Therefore, there are no conditions put on them. As may be seen from (6), the main advantage of the decomposition (4) for a suitable algorithm is the free character of the index set It. As a rule, l ~
(7)
(a short discussion on the inequality (7) will be given later). Rather than the detailed membership structure of the relations A, B, their linguistic interpretation is of a more valuable descriptive character.
2. Notations and definitions
To simplify the task, the decomposition may be performed with advantage on individual membership levels of C. The concept of Tong's peak pattern analysis [8, 9] offers good properties and possibilities for it. A peak pattern, PP, of a fuzzy subset ~¢ on its finite universe of discourse (space) ~ is a binary map of the discrete membership function of sO, us(x), with P P , ( x ) = {~
ifu~(x)>'a' otherwise.
(8)
where a is a real number from (0, 1). Thus, PP~a is the map of ~¢, and PPaa(x) that of us(x). Obviously, a peak pattern 'covers' an element x • ~ iff PP~°(x)= 1; it is important to specify the 'a-covering' for PP~(x) = 1, x • suppa(M) = {x I us(x) >-a}, if necessary. A peak pattern may cover more than one element. A fuzzy subset with a peak pattern covering only one element in ~ is called singular. It is non-singular otherwise. The space ~ is considered multidimensional. Then, it is possible to speak about a fuzzy relation ~/. The formal presentation of a binary relation is a relation matrix with membership degrees as its elements. A row peak pattern, RPP~, of ~ is the binary map of a row of the matrix M in which a quantity ~>a occurs. So, the map entry is either 1 or 0 in the sense of the definition (8). The column peak pattern, cppa~t, is defined similarly by the binary map of a column. A relation is primary if it has only one distinct RPP.
J. Vrba / General decomposition problem of fuzzy relations
71
A relation is regular if it exactly represents a single statment of the corresponding linguistic model, i.e. if its peak pattern covers only one element of ~. It is non-singular otherwise. A necessary condition for M to be regular is that it is primary. The proof of this statement is straightforward and results directly from the min-operation on the subsets entering the relation M. The peak-pattern analysis is extremely important for the fuzzy-linguistic interpretation of fuzzy relations. The peak-pattern of their highest membership level detects the number and evaluating character of corresponding statements (implications); such a peak-pattern is unique for any possible way of decomposition. The concrete value PP~(xi, yj) is symbolized as PP,(i, j), for short. Example 1. Consider a binary relation C--
0.3 0.3 0 . 5 ) 0.4 1 0.5 . 0.4 0.4 0.5
The peak patterns on individual membership levels of C are ppL0 =
1
,
pp~5 =
0 pp~4 =
1
1
,
0 ,
pp~3 =
1
1
,
1
where individual rows and columns of the PPs represent the RPPs and CPPs, respectively. The relation C is singular; it may represent only a single statement. Its regularity may be determined by its decomposition into (m, 1)- and (1, n)-dimensional relations (subsets) A and B, respectively, fulfilling the equation (4) (m = n = 3). The regularity of C may be intermediated by the composing relations A, B themselves in case those relations are regular. Such a situation is illustrated by examples.
3. The method
Let us start with an example, again. Example 2. What is necessary to do to derive the PPs of max-min composing relations A and B for individual P~c from Example 1? First, one of the possible simplest forms of the highest membership may be, e.g., ppLO= (0, 1, 0) T and ppL0 = (0, 1, 0), so that
1.o,.ol.0
PPA o PP8 = PPA × ppLO =
(Z)
X (0
1 o)=
1
= pp~:O.
0 Now, comparing pp~5 with the ppLO and ppLO we know that the starting structure for pp~/S and pp~5 must be of the form ppO/5 =
x 1 ×
and pp 5_(
1 :)
72
J. Vrba / General decomposition problem of fuzzy relations
where the positions marked '.' can and '×' c a n n o t be occupied by '1' to protect and preserve the '0' positions in pp~5. Then, taking into account the PPs of A and B for the membership level a = 1.0 we m u s t accept the version ppO/5=
and
1 1
P~'/~ 01)
giving pp~5 =
1 . 0 Similarly, the starting structures ppO/4=
(x i) 1
and pp~4=
1
It
in common with the PPs for a = 0.5 allow o n l y the choice 1
and PP~4 t~ ° it giving pp~4 =
1 1
.
Finally, the starting structures for the lowest non-zero level are
PP3:(ii)and
1t,.
so that we m u s t accept 1 1 1)=pp~4 and PP~3= 0 0 1
giving pp~3 =
1 1
!/
Then, following, e.g., Kaufmann [7] and the membership level operations, we determine the relations A and B as unions of all ppas multiplied by the corresponding membership levels a, i.e. A = 1.0x (S i ) +0.5x (S
i ) +0.4x ( i
! ) +0.3× (11 i ) ( ~=. 3 1 \0.4
:.5 t 0.5/
73
J. Vrba / General decomposition problem of fuzzy relations
and B=I'0x
( 0 1 ~) + 0 . 5 X (0 1 1) + 0 . 4 X (1 1 1) + 0 . 3 X (1 1 ~ ) (=0 0 . 4 0 0 0 0 1 0 0 1 0 0
1 0.5 ) 0 0.5
(the symbol '+' denotes logical union and ' x ' algebraic multiplication). We can verify that
/o Oo/Oo .3
AoB=
\0.4
.5
°
.4
1 0.5 0 0.5 =
0.5/
0.4
1
0.5
0.4
0.4 0.5
=C.
The relations A, B, though primary, are non-regular. They cannot be decomposed into subrelations (subsets) of the (m, 1) and (1, n) types, respectively. Nevertheless, as pointed out by Tong [9], we can approximate the relation A as A' = (0.3, 1, 0.4) and B as B' = (0.4, 1, 0.5), i.e. the rows with the RPP and CPP of the highest level. Then, the relation
C'=A'oB'=
(01.3)
°(0.4
1 0.5)=
\0.4
(03 03 0.4 0.4
1 0.4
0.5 0.4
may be considered linguistically identical with the relation C and the subsets A', B' the fuzzyevaluating elements of the corresponding implication "if A' for x then B' for y". In the preceding example, all the rules which may form the decomposition algorithm are discussed (the symbol = stands for necessardy equal' and ~ stands for union): 1. The ordered pairs (aik, bkj ) from the created relations A, B must fulfil the rule of max-min composition a ' " ~ S P PaA (•t , k ) × P P aa( k , "1) =! PPc(l,/).
(9)
2. The situation cij = 0 indicates the fact that at least one of the composing elements of any pair (aik , bkj ) on Ii must equal zero, i.e. if PP~c(ir, is) = 0 then (a) for Vk e It such that PPaa(ir, k) = 1: P ~ ( k , is) -" 0, (b) for Vk e It such that PP,(k, j~) = 1: PP~(ir, k)--"0. 3. The situation ciy ~ 0 indicates the fact that there must exist at least one pair (aik =/:O, bkj ~ 0) on It fulfilling the conditions (6), (9), i.e. if PP~(ir, 1"~)= 1 then 3r ~ (a, 1), 3s ~ (a, 1~, 3 k ~ It:
PP~t(ir, k) X PiCa(k, j~) " 1.
A single choice of k e It for any pair (ir, j~) guarantees the resulting PP~ and PP~ have minimum number of l's. 4. The union of all peak patterns, multiplied by corresponding membership levels, gives the relation itself: A=
y o a
o
a x PPA,
B=
a x PP~.
(10a,b)
a
The proofs of the above rules are trivial and follow directly from the concept of max-min composition. The rules convey the decomposition to the simplest (minimum) version of A and B with respect to the number of non-zero memberships and their values. That means that each nonzero element of A, B is m a x - r a i n active for C, i.e. for V k ~ It such that aik > O, 3(i, j):
aik = Cij
J. Vrba / General decomposition problem of fuzzy relations
74
and for Vk e It such that bkj > O, 3(i, j):
bkj = cij.
It is worth mentioning that the bounds m, n are predetermined by the dimensionality of the relation C, the quantity l for the number of rows in A and columns in B being governed by the condition (7). Really, from the point of the lowest dimensionality 1, the best case is with the same RPP for any row in C, differing only in membership level a. It is possible, then, to compose a unique RP~A with a unique C P ~ to obtain the corresponding PWc: the composition PP~t(i, 1) x PP,(l, ]) = PP,(i, j) is the same for any row of PP~ of dimensionality (m, 1) and any column of PP~ of dimensionality (1, n), i.e. RPP~ and CPP~. On the contrary, the worst situation with the highest value of 1 may occur when only the trivial decomposition is possible. This means that, at any membership level, PP~ equals either PWA or P ~ , the composition-complementary PWn or PP~ forming the (l x 1) diagonal matrix, l fulfilling the condition (7) as equality to n and m, respectively. Such a matrix represents a quasi-equivalent relation or ~ with non-zero diagonal elements and the properties or A o ~ = A = C .
alo~=B=C
(11),(12)
The non-specified number of rows in A as well as columns in B enables a free decomposition A(i, *)o B(*, j) that, actually, enables a decomposition of any C. Such a free dimensionality enables any pair RPP~, CPP~ to form a unique RPP~ and, consequently, a unique CPP~, too. If any of the RPPs of A would need a particular partner CPP of B (and vice versa), they cannot be greater in their number and dimensionality than m or n, respectively. These situations are illustrated by the following examples. Example 3. Applying the above rules we can decompose the relation
C=
'0.5
0.5
0.5/
0.7
0.8
0.5
0.7
1
0.5
0.6
0.6
0.5
0.3
0.3
0.3
into A = (0.5, 0.8, 1, 0.6, 0.3) T and B = (0.7, 1, 0.5), i.e. l = 1 < min[m, n]. Obviously, the relation C may be considered regular representing a single statement "if A for x then B for y". Example 4. The fuzzy subsets A = (1, 0.8, 0.4) T and B = (0.8, 1, 0.6) are the only subsets that may max-min compose the relation C=
0.8 1.0 0 . 6 ) 0.8 0.8 0.6 . 0.4 0.4 0.4
That may be detected by the analysis: a=1.0:
PPc=
0 0
,
PPA=
PPz=(O
1 0),
75
Vrba / General decomposition problem of fuzzy relations
J.
a=0.8:
PPc=
(i 1
,
PPA =
,
PPs=(1
1 0),
a=0.6:
PPc=
(i 1
,
PPA=
,
PPs=(1
1 1),
a=0.4:
PPc=
(i 1
,
PPA=
,
PP~=(1
1 1).
1 0
1 0
1 1
Example 5. The fuzzy relation
C=
0.4
0.8
1
0.8
0.4 /
0.8
1
0.8
0.8
1
1
0.8
0.5
0.9
1
0.8
1
0.8
1
0.9
1
0.8
0.6
0.1
0.6
is decomposable as
0
o o ol
0.4
0.8
1
0.8
0.4 /
0.8
1
0
0
0
0.8
1
0
0.8
1
0.5
0.8
1 0
0
1
0
0
0.9
1
0.8
0.9
0
1 0
0
1
0
1
0
0.1
0
0
0
1
0.8
0.6
0
0.6
fl A=
and
B=
1
In light of the linguistic structure of ppL0 and ppL0, the decomposition obtained is practically trivial, and the given relation C is decomposable in a trival manner only. The highest membership level peak-patterns of A and B are identical with those of the corresponding equivalence relation and C, respectively. The algorithm follows the decompositional possibilities on individual rows and columns in turn. Of course, it is possible to proceed in different ways but the difference obtained is rather formal and may be misleading. For example, the analysis of the last relation C allows the decomposition for a = 1, r0 0
rl
1 0 0
1
0 0
1 0 0 0
1
0 PPc=
0 0 0 01
o~
0
1
1 0
0
1 0
1 0 0 0 0
,
PPA =
0 0
1 1 0
1 0
0 0
tO 0
1 0
1 0
1
1 0 0 0
,
PPB =
1 0 01
1
0 0 0 0
0
1 0 0 0
0 0 0 0
1
0
0
0
0
1
Obviously, there are differences between this analysis and that of the previous example. Nevertheless, it is quite easy to check that the successive interchange of rows in PPB and columns in PPA gives the same results. In other words, both decompositions are linguistically identical. Example 6. An illustrative part of the fuzzy-linguistic model for the relation between citric acid (as the product- variable 1) and glucose (as the substrate- variable 3) concentrations is presented as MSC in
J. Vrba / General decomposition problem of fuzzy relations
76
Table 1 Statements Variable
1
2
3
4
5
6
7
8
9
10
11
1 3
9 1
8 1
7 2
6 3
5 4
4 5
4 6
3 7
3 8
2 9
1 9
the form of Table 1 [10]. The values within the table denote the indexes of the respective evaluating terms used for the variables. The fuzzy language for the substrate is X = (0, lO0)g/l,
m = K x = 9,
and for citric acid: Z = (0, 100)g/l,
n = K z = 9.
Supposing that in the citric acid fermentation the biomass concentration is another control variable (variable 2), we would like to decompose the model MSC to the models MSA and MSB which would express the relations of substrate/biomass and biomass/product concentrations, respectively. The respective fuzzy language for the biomass concentration is Y = (0, 9)g/1 and K y = 9. For the membership description of all variables x and their evaluating terms ~/, i.e. u~(x), the parabolic two-parameter formula [10]
Uzat(X) = { aO(X-- Xmin)(Xmax-- X)/(Xmax-- Xmin)2 if Xmin <~x <~X . . . . otherwise,
(13)
is used. Knowing the definitions x • ~, y • ~ for the respective variables we can construct any fuzzy relation of the problem, ~(x, y) • ~ × ~, for ~ and ~ evaluated by ~ / a n d ~ , respectively, as [7] V (x, y) • ~ x ~:
~(x, y) = min[ua(x), u~(y)].
(14)
With respect to the nature of the fermentation process, A must be expected as direct and B as reverse relation. First, the overall next-state fuzzy-linguistic model MSC is translated into the form of a fuzzy relation C(x, z): Z
0
C(x, z)=
0
0
0
0.6
0.6
0.6
0.9
1
0
0
0.6
0.6
0.6
0.9
0.9
0.9
1
0
0.6
0.6
0.9
0.9
0.9
1
1
0.~
0.6
0.6
0.9
0.9
1
1
0.9
0.9
0.9
0.6
0.9
0.9
1
0.9
0.9
0.9
0.6
0.6
0.9
0.9
1
0.9
0.9
0.6
0.6
0.6
0
0.9
1
0.9
0.9
0.6
0.6
0
0
0
1
0.9
0.9
0.6
0.6
0
0
0
0
1
0.9
0.6
0.6
0
0
0
0
0
X
J. Vrba / General decomposition problem of fuzzy relations
77
Applying the above ideas we perform the peak-pattern analysis shown in Table 2, with the results
A(x, y) =
B(y, z)=
1
0
0
Y 0
0
0
0
1
0.9
0
0
0
0
0
0.9
1
0.9
0
0
0
0
0.9
0.9
1
0.9
0
0
0
0.6
0.6
0.9
1
0.9
0
0
0
0.6
0.6
0.9
1
0.9
0
0
0
0
0.9
0.9
1
0
0
0
0
0.6
0.9
0.9
1
0
0
0
0
0.6
0.9
1
0
0
0
0
Y 0.6
0.6
0.6
0.9
1'
0
0
0.6
0.6
0.6
0.9
1
1
0
0
0.6
0
0.9
1
1
0.9
0
0
0.6
0.6
0.9
1
0.9
0.6
0
0
0
0.6
0.9
1
0.6
0
0
0
0
0
0.9
1
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
X
z
or, in the fuzzy-linguistic form, as shown in Table 3 (NSA = NSB = 9). Translating the above FL models into the form of fuzzy relations and composing them in the sense of equation (4), we obtain a fuzzy relation corresponding exactly to the given FL model MSC. The ex-post experiments confirm the results.
4. Concluding remarks It is unnecessary to speak about approximation in the general decomposition problem provided there is no specific claim on the form of the resulting relations in case the decomposed relation is drawn from other sources than from a FL model. It is more convenient, then, to look for their possible linguistic meaning than to reject them as mathematically uncorrect. A time series quantised on the defined universes and filtrated through the accepted term sets for respective variables forms the background for a FL model with the fuzzy terms as building units. These terms are the only objects a relation is constructed o f - its decomposition discovers that fact. Thus, the resulting relations must be evaluated from this point of view - see Example 6. As seen, the decomposition algorithm is suitable for on-line use as well. As soon as the overall next-state relation is constructed [5, 10] the decomposition step follows automatically. In that way, it is possible to model the process in its steady state as well as in its dynamic. Thus, the fuzzy-linguistic approach helps surmount the obstacles the conventional modeling technique sometimes cannot deal with.
78
J. Vrba / General decomposition problem of fuzzy relations Table 2 a
PPc
PPA
PPn
1.0
000000001 000000001 000000110 000011000 000100000 001000000 010000000 100000000 100000000
1000000 1000000 0100000 0010000 0001000 0000100 0000010 0000001 0000001
000000001 000000110 000011000 000100000 001000000 010000000 100000000
0.9
000000011 000001111 000111111 001111111 011111100 111110000 111100000 111000000 110000000
1000000 1100000 1110000 1111000 0011100 0001110 0001110 0000111 0000011
000000011 000001110 000111100 001110000 011000000 110000000 100000000
0.6
000011111 001111111 011111111 111111111 111111111 111111110 111111000 111110000 111100000
1000000 1100000 1110000 1111000 1111100 0111110 0001110 0001111 0000111
000011111 001111110 010111100 111111000 111100000 110000000 100000000
Table 3 Statement
MSA
MSB
Variable
1
2
3
4
1
1
2
3
4
2
1
1
2
3
2
1
2
2
3
3
9
8
7
6
5
6
7
8
9
5
6
4
5
7
8
9
6
7
7
3
5
4
5
6
7
4
3
2
1
References E. Czogala, J. Drewniak and W. Pedrycz, Fuzzy relation equations on a finite set, Fuzzy Sets and Systems 7 (1982) 89-101. W. Pedrycz, Numerical and applicational aspects of fuzzy relational equations, Fuzzy Sets and Systems 11 (1983) 1-18. E. Sanchez, Resolution of composite fuzzy relation equations, Information and Control 30 (1976) 38-48. Y. Tsukamoto and T. Terano, Failure diagnosis by using fuzzy logic, Proc. IEEE Conf. on Decision and Control, New Orleans (1977) 1340-1395. [5] J. Vrba, Peak-pattern concept and max-min inverse problem in fuzzy control modelling, Fuzzy Sets and Systems 47 (1992) 1-11.
[1] [2] [3] [4]
J. Vrba / General decomposition problem of fuzzy relations
79
[6] A. Di Nola, S. Sessa and W. Pedrycz, Decomposition problem of fuzzy relations, Internat. J. General Systems 10 (1985) 123-133. [7] A. Kaufmann, Introduction to the Theory of Fuzzy Subsets (Academic Press, New York, 1975). [8] R.M. Tong, Analysis and control of fuzzy systems using finite discrete relations, Internat. J. Control 3 (1978) 431-440. [9] R.M. Tong, Analysis of fuzzy control algorithms using the relation matrix, lnternat. Man-Machine Stud. 8 (1976) 679-686. [10] J. Vrba and J. Brozek, Fuzzy concept in modelling of biotechnoiogical processes, Systems Anal. Modelling Simulation (in press).