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Composite fuzzy relational equations with non-commutative conjunctions
c ~
INFORMATION
SCIENCES AN I/4"I~RNATK)NALIOIJRNAL
Information Sciences 110 (1998) 113-125
ELSEVIER
Composite fuzzy relational equations with non-commutative conjunctions Mayuka F. Kawaguchi *, Masaaki Miyakoshi Division of Systems and Information Engineering, Graduate School of Engineering, Hokkaido University, Kita 13, Nishi 8, Kita-ku, Sapporo 060~8628, Japan
Received 28 May 1996; accepted 10 October 1997
Abstract
Through this study, the authors treat two kinds of fuzzy logical operators, that is, non-commutative conjunctions and their residual implications which include almost all of already-known functions in the field of fuzzy logic. Corresponding to such operators, two kinds of generalized compositions of fuzzy relations are introduced, and the solutions of new composite relational equations are given. This work reveals that the solutions include the results of the past researches in the field of fuzzy relational equations as special cases. Some specific numerical examples of the solutions of a given equation illustrate the mechanism of these compositions. © 1998 Published by Elsevier Science Inc. All rights reserved. Keywords." Composition of fuzzy relations; Relational equation; Non-commutative
conjunction; Residuation
1. Introduction
The concept of composite fuzzy relational equations and their solutions originated by Sanchez [17,18], have been developed by quite a few researchers and have given the mathematical foundation and an effective methodology to the application of fuzzy set theory to various fields [3]• Miyakoshi and Shimbo [13] have investigated in detail two kinds of relational compositions based on a t-norm and its residual operator called R-implication (i.e. T-composition and
*Corresponding author. 0020-0255/98/$19.00 © 1998 Published by ElsevierScienceInc. All rights reserved. PII: S 0 0 2 0 - 0 2 5 5 ( 9 7 ) 1 0 0 8 0 - 9
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M.F KawaguchL M. Miyakoshi / Information Sciences 110 (1998) 113-125
~r-composition) and have given the solutions of the equations regarding these compositions. Izumi et al. [12] have reached the same conclusion in the framework of L-fuzzy logic [10,11] and have pointed out that 3-composition (includes T-composition) and V-composition (includes ~r-composition) correspond to the modal concepts of possibility and necessity, respectively. On the other hand, we can see numerous kinds of implication functions which are not included in the class of R-implications, as in many works in the field of fuzzy logic. Dubois and Prade [4] have formulated the generation process of three classes of implications and two classes of non-commutative conjunctions from t-norms. Fodor [5,6] has extended the above generation process to broader classes of implications and conjunctions, and has shown that such classes include almost all of already-known implications. Furthermore, Fodor and Keresztfalvi [8] have suggested that non-commutative and non-associative conjunctions (i.e. less restricted ones than t-norms) are efficient in approximate reasoning from the axiomatic point of view. This work aims to formulate two generalized compositions of fuzzy relations, and to give the solutions of fuzzy relational equations regarding such compositions. For this purpose, firstly, the authors define a class of non-commutative and non-associative conjunctions and the class of the residual operators of such conjunctions. As the main result, it is shown that the composite equations with such operators can be solved in almost the same manner as the equations with t-norm and sT-operator. Several classes of functions which satisfy our new definitions of conjunction and implication are described. Numerical examples help us to compare the features of the equation regarding non-commutative conjunction with those of conventional ones.
2. Non-commutative conjunctions Through this paper, we treat a class of conjunctions defined by removing commutativity and associativity from the definition of t-norms and loosening the boundary condition of t-norms. Corresponding to such conjunctions, we derive the class of implications which satisfy monotonicity and a loose boundary condition. These two classes of functions are connected with each other via the procedure called residuation. Definition 1. A conjunction C : [0, 1]2 --~ [0, 1] is a function satisfying (C1)
C(0,1)=C(1,0)=0,
a#O=~C(1,a)¢O, (C2)
C(1,1)=l, C(a, 1) ¢ 0;
a <~b =~ C(a, c) <<.C(b, c), C(c, a) <~C(c, b) for any a, b, c E [0, 1].
M.F. Kawaguchi, M. Miyakoshi / Information Sciences 110 (1998) 113-125
115
Definition 2. An implication I: [0, 1] 2 ~ [0, 1] is a function satisfying
(I1)
I(0,0) = I(0, 1 ) = I(1, 1 ) = 1 , I ( 1 , 0 ) = 0 ,
a¢l~I(1,a)¢l, (I2)
a¢O~l(a,O)¢l;
a <~b ~ I(a, c) >~l(b, c), I(c, a) <~1(c, b) for any a, b, c E [0, 1].
Definition 3. The right-residual [1] of a conjunction C and its dual operator [2,13] associated with an implication I are defined by I(C)(a,b) -- sup{s IC(a,s)<~ b} s
and
C(I)(a,b) =_ inf{s I I(a,s) ~ b}, s
respectively. It is clear that the function K(C) defined from a conjunction C as b) - C(b, a)
is also a conjunction. The following theorem plays an essential role in this work. Theorem 1. The following statements (i)-(iv) are equivalent to one another."
(i) A conjunction C(a, s) is left-hand continuous with respect to s. (ii) The complete distributive law of C i.e. sup;. C(a,b).)= C(a, sup~ b;.)
holds Jor any family {bA}AEAof[0,1]. (iii) {s I C(a, s) <<.b} has a greatest element for any a, b E [0, 1]. (iv) A conjunction C and its residual implication I(C) satisfy the adjointness C(a,s)<.b
¢==~ s<.I(C)(a,b)
jor an), a, b, s E [0, 1]. Proof. (i) ~ (ii). Putting c = sup~ bj., we can derive limx~c_0 C(a,x) = C(a, c) from (i), thus we obtain sup~ C(a, b~) = C(a, c) = C(a, sup~ b~). (ii) ~ (iii). We can get C(a,I(C)(a, b))<<, b as follows:
116
M.F. KawaguchL M. Miyakoshi / Information Sciences 110 (1998) 113-125 C(a,I(C)(a,b)) = C(a, sup{s ] C(a,s) <~b}) =
sup C(a,s) <~b. s
C(a.s)~ b That is, l(C)(a, b) C {slC(a,s)<~ b} and I(C)(a, b) is the greatest element of
{s I C(a,s) <~hi. (iii) =~ (iv). Since l(C)(a,b) is the greatest s which satisfies C(a,s)<<, b, we have
C(a,s)<. b ~
s<~l(C)(a,b).
Conversely, if s <<.I(C)(a, b) holds, then we have
C(a, s) <<.C(a, I(C)(a, b))<~ b from (C2) monotonicity and (iii). (iv) ~ (i). Putting c = sup~b;., we have from (iv)
(
C(a,b;~) <~~c-olim C(a,x) e==~ b~ <~I(C) a, xfim ° e==~ c<~I(C)(a, limo.~- C(a,x)) e==~ C(a, c) <. lira C(a,x). X~C
0
On the other hand, since C is an increasing function, lim ..... o C(a,x) <<,C(a, c) holds in general. Therefore, C(a,s) is left-hand continuous with respect to s. [] The following corollaries can be derived directly from the above-mentioned definitions and theorem.
Corollary 1. I f C(a,s) is left-hand continuous with respect to s, then I( C) is an implication. If I(a,s) is right-hand continuous with respect to s, then C(I) is a conjunction.
Corollary 2. I f C(a,s) is left-hand continuous with respect to s, then I( C) (a,s) is right-hand continuous with respect to s. I f I(a,s) is right-hand continuous with respec t t ° s, then C(I) (a,s) is left-hand continuous with respect to s.
Corollary 3. I f C(a,s) is left-hand continuous with respect to s, then C(I( C) ) = C holds. IfI(a,s) is right-hand continuous with respect to s, then I(C(I)) holds.
= I
M.F. Kawaguchi, M. Miyakoshi / Information Sciences 110 (1998) 113-125
117
Hereafter, we assume that C is left-hand continuous with respect to both variables, and utilize the symbol Ctc instead of I(C) according to the notation by Sanchez [17,18] and Miyakoshi and Shimbo [13]. Obviously, when C is a left-hand continuous t-norm T, ~c is reduced to ~r-operator. Then, C and ~c have the following properties for any a, b E [0, 1]. Lemma 1. (p-l)
C(a, otc(a,b))
(p-1)'
C(~.(cl (a, b), a) <~b, ~c(a, C(a,b)) >>-b,
(p-2) (p-2)' (p-3)
~(c)(a,C(b,a)) >~b, ac(a~(c)(a,b),b) >~a.
Proof. (p-l) and (p-2) are directly derived from Theorem 1 (iii). We can get (p-l)' and (p-2)' from (p-l) and (p-2), respectively, replacing C with •(C) and noting that K(~(C)) = C. Moreover, we obtain (p-3) from (p-l)' and Theorem 1 (iv) as follows:
C(~.(c)(a,b),a) <~b ~=~ a<~c(Ot~(cl(a,b),b).
[]
3. Generalized compositions of fuzzy relations Now, the authors generalize T-composition and er-composition replacing T and ~r by C and ~c, and name them C-composition and ~c-composition, respectively. Let X, Y and Z be non-empty sets, ,~-(X x Y), ~ ( Y x Z) and ~ ( X x Z) be the sets of all fuzzy relations on X x Y, Y x Z and X x Z, respectively. Let R E ~ ( X x Y),P E ~ ( Y x Z) and Q E ~-(X x Z). Definition 4. The C-composition of R and P, denoted by R © P, is a fuzzy relation on X × Z whose grades of membership are defined by
(R © P)(x,z) =- supC(R(x,y),P(y,z)) yE Y
for all (x, z) E X × Z. Definition 5. The ~c-composition of R and P, denoted by R @c P, is a fuzzy relation on X × Z whose grades of membership are defined by
(R @c P)(x,z) =-- ~nf ~c(R(x,y),P(y,z)) for all (x, z) E X × Z.
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M.F. Kawaguchi,M. Miyakoshi I Information Sciences110 (1998) 113-125
The inverse of R, denoted by R -l, is a fuzzy relation on Y x X whose grades of membership are given by
R-l(y,x) -- R(x,y) for all (x,y) E X x Y. There exist the following connections among R, P and Q via C-composition and ~c-composition.
Lemma 2. P C_ R ' @ c (R © P),
(1)
R © (R-' (~)c Q) c_ Q,
(2)
R C_ (P(~.-(c) (R © P)-') 1
(3)
(P(~(ct Q-') ' © P C_Q.
(4)
Proof. Eqs. (1)-(4) are obtained directly from (p-2), (p-l), (p-2)' and (p-l)', respectively. [] Lemma 3. P_~R ' © (R®cP),
(5)
R(~)c (R ' © Q) 2 Q,
(6)
R C_ (R(~(c) P)@c P-',
(7)
(Q@~(c) P-l)@c P D_Q.
(8)
Proof. Eqs. (5)-(8) are obtained directly from (p-l), (p-2), (p-3) and (p-3), respectively. [] It should be noted that Eqs. (3), (4) and (7) and (8) in the above lemmas are the new results on composite fuzzy relations, which are caused by the lack of commutativity of C. If C is commutative i.e. ~(C) = C, then they are reduced to the same as the case of T-composition and er-composition. See Property 3 in [13].
4. Solutions of fuzzy relational equations Let us cope with two kinds of relational equations regarding C-composition and ere-composition as
M.F Kawaguchi, M. Miyakoshi / Information Sciences 110 (1998) 113-125
119
R© P=QandR(~)c P=Q for two cases in which: (I) R and Q are given, P is unknown, (II) P and Q are given, R is unknown. Theorem 2. (I) Let R E ~ ( X x Y) and Q E ~ ( X x Z) be given, and 3 £ - {P I P E ~ ( Y × Z ) and R © P = Q}. Then, ~ is not empty iff R -l @c Q E W, and in that case R -1 @c Q is the greatest element in Y(. (II) Let P E ~,~(Y x Z) and Q E ~ ( x x Z) be given, and 81=- {R [ R E ~ ( X x Y) and R @ P = Q}. Then, 81 is not empty iff (P @~(c) Q-l)-i E 81, and in that case (P @~(clQ-J )-I is the greatest element in 81.
Proof. (I) Assume that £r _¢ O, and let P E f . Then, we have P c_ R 1@c Q by substituting R © P = Q into Eq. (1). From this equation and (C2), we have Q c_ R © (R-1 @c Q). Moreover, R © (R 1@c Q) = Q is obtained directly by denoting Eq. (2). Thus, it has been proved that R -1 @c Q is the greatest element in W iff W E 0. (II) It is proved similarly by using Eq. (3), Eq. (4) and (C2). [] Theorem 3. (I) Let R E ~ ( X × Y) and Q E ~ ( X x Z) be given, and T =- { P I P E ~ ( r × Z) and R(~ c P = O}. Then, tP is not empty iff R -l © Q E }P, and in that case R -1 © Q is the least element in tp. (II) Let P E ~ ( Y × Z) and Q E ~ ( X × Z) be given, and S=- {R I R E ~ ( X × Y) and R(~c P = Q}. Then, ~ is not empty iff Q(~K(c) p-1 E ~-, and in that case Q@~.(c) P 1 is the greatest element in ~.
Proof. (I) It is proved by using Eqs. (5), (6) and (12). (II) It is proved by using Eqs. (7), (8) and (12). [] 5. Examples of conjunctions and implications
There have been many functions in the numerous past works in multiplevalued logic especially in fuzzy logic, which are included in the sets of the
120
M.F. Kawaguchi, M. Miyakoshi / Information Sciences 110 (1998) 113-125 K
@
@
@
Fig. 1. The connectionsamong LSC t-norms, pseudo-conjunctionsand implications. above-mentioned conjunctions and implications, outside of the framework of a t-norm and its residual operator. We can easily verify that the following functions satisfy Definition 1 or Definition 2. Example 1. Dubois and Prade [4] have formulated the generation process of implications from t-norms and have defined pseudo-conjunctions. Now, for sake of convenience, we define the following mappings on the set of binary operations • : [0, 1]2 ~ [0, 1]: q~(*)(a, b) =- n(*(a, n(b))), q(*)(a,b) =- sup{s I * (a,s)<~b}, s
x ( * ) ( a , b ) -- . ( b , a ) ,
where n : [0, 1] ~ [0, 11 is a strong negation. Starting from the class of lower semicontinuous t-norms (i.e. left-hand continuous t-norms with respect to both variables), two classes of pseudo-conjunctions T1 and T2, and three well-known classes of implications i.e. S-implication Is, R-implication Ie and n-reciprocal R-implication/,R are associated with one another as shown in Fig. 1. Since I ( C ) = ~(C), we can see in Fig. 1 that C(I) = ~o o q o q~(1), C = q~ o q o ~ o q( C),
i.e. Corollary 3 holds. (See also [5,6] for another approach to this property, and [15] for the detailed properties and the explicit forms of these functions.) Therefore, we can substitute the pairs (T1 ,Is) and (T2,I,R) for (C, ~c) in our theoretical results. Example 2. It is possible to generate non-commutative conjunctions from already-known implications. Gaines' implication [9]: /Gaines(a,b) ~
1 0
(a<~b),
(a > b),
Yager's implication [20]:
=
a
(b > 0),
0
(b = 0).
M.F Kawaguchi, M. Miyakoshi / Information Sciences 110 (1998) 113-125
1
Iyager(a, b) =
(a=0),
b ~ (otherwise), 0
C(Iyager)(a, b) =
121
(a = b = 0),
bl/~ (otherwise).
Contrapositive symmetric R-implication [7]: lcPs-R(a, b) - max(IR(a, b), I,e(n(b), n(a))),
C(IcPs-e)(a, b) = min(T(a, b), T2(a, b)). Example 3. Sato and Sato [19] have proposed an asymmetric aggregation operator for their clustering method based on asymmetric similarity as follows: CAs(a, b) -- f - i (f(a) + p(a)f(b)), where f is an additive generator of any Archimedean t-norm, p is a continuous and monotone decreasing function from [0, 1] to [0, ~ ] satisfying p(1) = 1, and fI-~l is a pseudoinverse o f f . See [14] for the explicit forms of the additive generators. The residual operator of CAS can be obtained as
{f[-l](-f(a)+f(b)) I(CAs(a, b)) =
(a~b),
p(a)
1
(a < b).
Example 4. QL-implication (Zadeh [21])
IoL(a,b ) -- n(T(a,n(T(a,b)))),
T isan LSC t-norm,
C(IQL)(a,b) = C(T)(a, T,(a,b)) = inf{s I T(a,s) >>.r~(a,b)}. QL-implication is not monotone with respect to both variables i.e. it does not satisfy (I2). Thus, IoL and C(IoL ) are not included in our implications and conjunctions.
6. Numerical examples of fuzzy relational equations For instance, if we put n(a) = 1 - a and T = min, then T1, T2,Is, IR and InR which were mentioned in the previous section are obtained as follows:
T(a, b) = min(a, b), Tl(a,b)= { 0 b
T2(a,b)= { 0 a
(a+b~ 1), (a+b~ 1),
Is(a, b) = max(1 - a, b)
(Kleene-Dienes),
122
M.E Kawaguchi, M. Miyakoshi / Information Sciences 110 (1998) 113 125
(a<~ b), (a > b),
1
In(a,b) =
b
1
l,e(a,b)=
1-a
(G6del),
(a<~ b), (a > b).
Now, let us consider the simplest f o r m of fuzzy relational equation as numerical examples. A © R = B,
R=
(9)
Yl
Y2
x2
0.5
0.9
X3
0.9
0.4
'
B=
0.7
0.4
•
Here, A E ~ ( X ) , R ~ ~ ( X x Y), B E ~ ( Y ) , and A = (A(xl) A(x2) A(xs)) is unknown. Then, we can solve Eq. (9) applying T h e o r e m 2 (II). 1. When C = T, C~c = IR and e~-(c/= IR. =
(R@~_(c/ B i )'
© R=(0.7
= (0.4 0.4 0.7),
0.4)=B.
Therefore, A is the greatest solution of Eq. (9). 2. W h e n C = Tl, C~c = Is and ~ ( c / = I,R.
z~-- (R(~)h.(c)B-I) 1 (0.5 0.1
0.1),
A©R=(0.80)¢B. Therefore, Eq. (9) has no solution. 3. W h e n C = T2, ac = I,R and ~(c) = Is.
A= (R@~.(c)
B ' ) ' = (0.5 0.4 0.6),
,4© R=(0.6
0.4)¢B.
Therefore, Eq. (9) has no solution. F o r another example, let us assume that n(a) = 1 - a product, and solve Eq. (9).
T(a,b) = a × b,
0
Tl(a,b)=
(a+b-1)/a
(a+b~ 1),
and T is algebraic
M.F Kawaguchi, M. Miyakoshi / InJormation Sciences 110 (1998) 113 125 T2(a,b) = f~ 0 t (a+b-1)/b
f 1 / b/a
I,R(a,b)=
I), 1),
(a+b>
(Reichenbach),
Is(a, b) = 1 - a + ab IR(a,b) = {
123
(a > b),
1 (l-a)/(1-b)
(Goguen), (a<.b),
(a>b).
1. When C = T, :~c : IR and etc.(c) = IR. d :
A@ R=(0.7
B ')-'
:
(4;) 0.8
,
0.4)=B.
Therefore, A is the greatest solution of Eq. (9). 2. When C = Tl, ~c = Is and :~K(c) =/,R. ,~=(R@~.(c) B ,)_ 1
: d@ R=(0.7
(2 1 1) 563'
0.4)=B.
Therefore, A is the greatest solution of Eq. (9). 3. When C = 7"2, :~c = I,R and :q-(c) = Is. = (R@~.(c) B-') ~ = (0.7 0.46 0.73), d@ R=(0.7
0.4)=B.
Therefore, d is the greatest solution of Eq. (9).
7. Concluding remarks
The original study on fuzzy relational equation by Sanchez [17,18] corresponds to the framework of the complete Brouwerian lattice which is wellknown as the algebraic system of intuitionistic logic. The theory of fuzzy relational equations with triangular norms [13] is based on the concept of complete lattice-ordered monoid which is the algebraic system of L-fuzzy logic [10,1 l], On the other hand, in the framework of this study, ([0, 1], v, A, C) forms a lattice-ordered groupoid [1] (in short,/-groupoid) because C is monotone increasing but non-commutative and non-associative, besides not having even a unit element.
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M.F. Kawaguchi, M. Miyakoshi / Information Sciences 110 (1998) 113-125
Applications of the results here to fuzzy reasoning shall be considered for our next project. Furthermore, the effect of non-associativity of conjunctions needs to be considered from theoretical and practical points of view. Aside from these is another complicated problem which is how to select an adequate operator among various conjunctions and implications described here for each given equation, for example, a fuzzy input-output system represented by a relational composition. As an approach toward this problem, Ohtani et al. [16] have tried to search for an operator which makes a given equation solvable by using a genetic algorithm. Further detailed investigations along this line are needed in order to apply the theoretical results on fuzzy relational equations to the practical problems involving an analysis of fuzzy systems.
References [1] G. Birkhoff, Lattice Theory, 3rd ed., American Mathematical Society Colloquium Publications, 1967. [2] B. De Baets, Residual operators of implications, in: Proceedings of the Third European Congress on Fuzzy Techniques and Soft Computing (EUFIT'95), Aachen, 1995, pp. 136 140. [3] A. Di Nola, S. Sessa, W. Pedrycz, E. Sanchez, Fuzzy Relation Equations and Their Applications to Knowledge Engineering, Kluwer Academic, New York, 1989. [4] D. Dubois, H. Prade, A theorem on implication functions defined from triangular norms, Stochastica VIII (1984) 267 279. [5] J.C. Fodor, On fuzzy implication operators, Fuzzy Sets and Systems 42 (1991) 293-300. [6] J.C. Fodor, A new look at fuzzy connectives, Fuzzy Sets and Systems 57 (1993) 141 148. [7] J.C. Fodor, Contrapositive symmetry of fuzzy implications, Fuzzy Sets and Systems 69 (1995) 141 156. [8] J.C. Fodor, T. Keresztfalvi, Nonstandard conjunctions and implications in fuzzy logic, Int. J. Approximate Reasoning 12 (1995) 69-84. [9] B.R. Gaines, Foundations of fuzzy reasoning, Int. J. of Man-Machine Studies 6 (1976) 623 668. [10] J.A. Goguen, L-fuzzy sets, J. of Mathematical Analysis and Applications 18 (1967) 145-174. [l 1] J.A. Goguen, The logic of inexact concepts, Synthese 19 (1968-69) 325 373. [12] K. Izumi, H. Tanaka, K. Asai, Adjointness of fuzzy systems, Fuzzy Sets and Systems 20 (1986) 211-221. [13] M. Miyakoshi, M. Shimbo, Solutions of composite fuzzy relational equations with triangular norms, Fuzzy Sets and Systems 16 (1985) 53 63. [14] M. Mizumoto, Pictorial representations of fuzzy connectives, part I: Cases of t-norms, tconorms and averaging operators, Fuzzy Sets and Systems 31 (1989) 217-242. [15] T. Ohno, M.F. Kawaguchi, T. Da-te, Properties of implications and pseudo-conjunctions defined from t-norms, MVL Technical Report, Japan Research Group on Multiple-Valued Logic, MVL96-14 (1996) 96-103, in Japanese. [16] Sh. Ohtani, H. Kikuchi, R.R. Yager, Sh. Nakanishi, Diagnosis of a fuzzy system character, in: Proceedings of the 12th Fuzzy Systems Symposium, Tokyo, 1996, pp. 499-502, in Japanese. [17] E. Sanchez, Resolution of composite fuzzy relation equations, Information and Control 30 (1976) 38-48.
M.F KawaguchL M. M~vakoshi / Information Sciences 110 (1998) 113-125
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[18] E. Sanchez, Solutions in composite fuzzy relation equations: Application to medical diagnosis in Brouwerian logic, in: M.M. Gupta et al. (Eds.), Fuzzy Automata and Decision Processes, North-Holland, Amsterdam, 1977, pp. 221-234. [19] M. Sato, Y. Sato, Asymmetric aggregation operators and its applications, in: Proceedings of the 12th Fuzzy Systems Symposium, Tokyo, 1996, pp. 141-142, in Japanese. [20] R.R. Yager, An approach to inference in approximate reasoning, Int. J. of Man-Machine Studies 13 (1980) 323-338. [21] L.A. Zadeh, Outline of a new approach to the analysis of complex systems and decision processes, IEEE Trans. Systems, Man, and Cybernetics SMC-3 (1973) 28M4.
INFORMATION
SCIENCES AN I/4"I~RNATK)NALIOIJRNAL
Information Sciences 110 (1998) 113-125
ELSEVIER
Composite fuzzy relational equations with non-commutative conjunctions Mayuka F. Kawaguchi *, Masaaki Miyakoshi Division of Systems and Information Engineering, Graduate School of Engineering, Hokkaido University, Kita 13, Nishi 8, Kita-ku, Sapporo 060~8628, Japan
Received 28 May 1996; accepted 10 October 1997
Abstract
Through this study, the authors treat two kinds of fuzzy logical operators, that is, non-commutative conjunctions and their residual implications which include almost all of already-known functions in the field of fuzzy logic. Corresponding to such operators, two kinds of generalized compositions of fuzzy relations are introduced, and the solutions of new composite relational equations are given. This work reveals that the solutions include the results of the past researches in the field of fuzzy relational equations as special cases. Some specific numerical examples of the solutions of a given equation illustrate the mechanism of these compositions. © 1998 Published by Elsevier Science Inc. All rights reserved. Keywords." Composition of fuzzy relations; Relational equation; Non-commutative
conjunction; Residuation
1. Introduction
The concept of composite fuzzy relational equations and their solutions originated by Sanchez [17,18], have been developed by quite a few researchers and have given the mathematical foundation and an effective methodology to the application of fuzzy set theory to various fields [3]• Miyakoshi and Shimbo [13] have investigated in detail two kinds of relational compositions based on a t-norm and its residual operator called R-implication (i.e. T-composition and
*Corresponding author. 0020-0255/98/$19.00 © 1998 Published by ElsevierScienceInc. All rights reserved. PII: S 0 0 2 0 - 0 2 5 5 ( 9 7 ) 1 0 0 8 0 - 9
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M.F KawaguchL M. Miyakoshi / Information Sciences 110 (1998) 113-125
~r-composition) and have given the solutions of the equations regarding these compositions. Izumi et al. [12] have reached the same conclusion in the framework of L-fuzzy logic [10,11] and have pointed out that 3-composition (includes T-composition) and V-composition (includes ~r-composition) correspond to the modal concepts of possibility and necessity, respectively. On the other hand, we can see numerous kinds of implication functions which are not included in the class of R-implications, as in many works in the field of fuzzy logic. Dubois and Prade [4] have formulated the generation process of three classes of implications and two classes of non-commutative conjunctions from t-norms. Fodor [5,6] has extended the above generation process to broader classes of implications and conjunctions, and has shown that such classes include almost all of already-known implications. Furthermore, Fodor and Keresztfalvi [8] have suggested that non-commutative and non-associative conjunctions (i.e. less restricted ones than t-norms) are efficient in approximate reasoning from the axiomatic point of view. This work aims to formulate two generalized compositions of fuzzy relations, and to give the solutions of fuzzy relational equations regarding such compositions. For this purpose, firstly, the authors define a class of non-commutative and non-associative conjunctions and the class of the residual operators of such conjunctions. As the main result, it is shown that the composite equations with such operators can be solved in almost the same manner as the equations with t-norm and sT-operator. Several classes of functions which satisfy our new definitions of conjunction and implication are described. Numerical examples help us to compare the features of the equation regarding non-commutative conjunction with those of conventional ones.
2. Non-commutative conjunctions Through this paper, we treat a class of conjunctions defined by removing commutativity and associativity from the definition of t-norms and loosening the boundary condition of t-norms. Corresponding to such conjunctions, we derive the class of implications which satisfy monotonicity and a loose boundary condition. These two classes of functions are connected with each other via the procedure called residuation. Definition 1. A conjunction C : [0, 1]2 --~ [0, 1] is a function satisfying (C1)
C(0,1)=C(1,0)=0,
a#O=~C(1,a)¢O, (C2)
C(1,1)=l, C(a, 1) ¢ 0;
a <~b =~ C(a, c) <<.C(b, c), C(c, a) <~C(c, b) for any a, b, c E [0, 1].
M.F. Kawaguchi, M. Miyakoshi / Information Sciences 110 (1998) 113-125
115
Definition 2. An implication I: [0, 1] 2 ~ [0, 1] is a function satisfying
(I1)
I(0,0) = I(0, 1 ) = I(1, 1 ) = 1 , I ( 1 , 0 ) = 0 ,
a¢l~I(1,a)¢l, (I2)
a¢O~l(a,O)¢l;
a <~b ~ I(a, c) >~l(b, c), I(c, a) <~1(c, b) for any a, b, c E [0, 1].
Definition 3. The right-residual [1] of a conjunction C and its dual operator [2,13] associated with an implication I are defined by I(C)(a,b) -- sup{s IC(a,s)<~ b} s
and
C(I)(a,b) =_ inf{s I I(a,s) ~ b}, s
respectively. It is clear that the function K(C) defined from a conjunction C as b) - C(b, a)
is also a conjunction. The following theorem plays an essential role in this work. Theorem 1. The following statements (i)-(iv) are equivalent to one another."
(i) A conjunction C(a, s) is left-hand continuous with respect to s. (ii) The complete distributive law of C i.e. sup;. C(a,b).)= C(a, sup~ b;.)
holds Jor any family {bA}AEAof[0,1]. (iii) {s I C(a, s) <<.b} has a greatest element for any a, b E [0, 1]. (iv) A conjunction C and its residual implication I(C) satisfy the adjointness C(a,s)<.b
¢==~ s<.I(C)(a,b)
jor an), a, b, s E [0, 1]. Proof. (i) ~ (ii). Putting c = sup~ bj., we can derive limx~c_0 C(a,x) = C(a, c) from (i), thus we obtain sup~ C(a, b~) = C(a, c) = C(a, sup~ b~). (ii) ~ (iii). We can get C(a,I(C)(a, b))<<, b as follows:
116
M.F. KawaguchL M. Miyakoshi / Information Sciences 110 (1998) 113-125 C(a,I(C)(a,b)) = C(a, sup{s ] C(a,s) <~b}) =
sup C(a,s) <~b. s
C(a.s)~ b That is, l(C)(a, b) C {slC(a,s)<~ b} and I(C)(a, b) is the greatest element of
{s I C(a,s) <~hi. (iii) =~ (iv). Since l(C)(a,b) is the greatest s which satisfies C(a,s)<<, b, we have
C(a,s)<. b ~
s<~l(C)(a,b).
Conversely, if s <<.I(C)(a, b) holds, then we have
C(a, s) <<.C(a, I(C)(a, b))<~ b from (C2) monotonicity and (iii). (iv) ~ (i). Putting c = sup~b;., we have from (iv)
(
C(a,b;~) <~~c-olim C(a,x) e==~ b~ <~I(C) a, xfim ° e==~ c<~I(C)(a, limo.~- C(a,x)) e==~ C(a, c) <. lira C(a,x). X~C
0
On the other hand, since C is an increasing function, lim ..... o C(a,x) <<,C(a, c) holds in general. Therefore, C(a,s) is left-hand continuous with respect to s. [] The following corollaries can be derived directly from the above-mentioned definitions and theorem.
Corollary 1. I f C(a,s) is left-hand continuous with respect to s, then I( C) is an implication. If I(a,s) is right-hand continuous with respect to s, then C(I) is a conjunction.
Corollary 2. I f C(a,s) is left-hand continuous with respect to s, then I( C) (a,s) is right-hand continuous with respect to s. I f I(a,s) is right-hand continuous with respec t t ° s, then C(I) (a,s) is left-hand continuous with respect to s.
Corollary 3. I f C(a,s) is left-hand continuous with respect to s, then C(I( C) ) = C holds. IfI(a,s) is right-hand continuous with respect to s, then I(C(I)) holds.
= I
M.F. Kawaguchi, M. Miyakoshi / Information Sciences 110 (1998) 113-125
117
Hereafter, we assume that C is left-hand continuous with respect to both variables, and utilize the symbol Ctc instead of I(C) according to the notation by Sanchez [17,18] and Miyakoshi and Shimbo [13]. Obviously, when C is a left-hand continuous t-norm T, ~c is reduced to ~r-operator. Then, C and ~c have the following properties for any a, b E [0, 1]. Lemma 1. (p-l)
C(a, otc(a,b))
(p-1)'
C(~.(cl (a, b), a) <~b, ~c(a, C(a,b)) >>-b,
(p-2) (p-2)' (p-3)
~(c)(a,C(b,a)) >~b, ac(a~(c)(a,b),b) >~a.
Proof. (p-l) and (p-2) are directly derived from Theorem 1 (iii). We can get (p-l)' and (p-2)' from (p-l) and (p-2), respectively, replacing C with •(C) and noting that K(~(C)) = C. Moreover, we obtain (p-3) from (p-l)' and Theorem 1 (iv) as follows:
C(~.(c)(a,b),a) <~b ~=~ a<~c(Ot~(cl(a,b),b).
[]
3. Generalized compositions of fuzzy relations Now, the authors generalize T-composition and er-composition replacing T and ~r by C and ~c, and name them C-composition and ~c-composition, respectively. Let X, Y and Z be non-empty sets, ,~-(X x Y), ~ ( Y x Z) and ~ ( X x Z) be the sets of all fuzzy relations on X x Y, Y x Z and X x Z, respectively. Let R E ~ ( X x Y),P E ~ ( Y x Z) and Q E ~-(X x Z). Definition 4. The C-composition of R and P, denoted by R © P, is a fuzzy relation on X × Z whose grades of membership are defined by
(R © P)(x,z) =- supC(R(x,y),P(y,z)) yE Y
for all (x, z) E X × Z. Definition 5. The ~c-composition of R and P, denoted by R @c P, is a fuzzy relation on X × Z whose grades of membership are defined by
(R @c P)(x,z) =-- ~nf ~c(R(x,y),P(y,z)) for all (x, z) E X × Z.
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M.F. Kawaguchi,M. Miyakoshi I Information Sciences110 (1998) 113-125
The inverse of R, denoted by R -l, is a fuzzy relation on Y x X whose grades of membership are given by
R-l(y,x) -- R(x,y) for all (x,y) E X x Y. There exist the following connections among R, P and Q via C-composition and ~c-composition.
Lemma 2. P C_ R ' @ c (R © P),
(1)
R © (R-' (~)c Q) c_ Q,
(2)
R C_ (P(~.-(c) (R © P)-') 1
(3)
(P(~(ct Q-') ' © P C_Q.
(4)
Proof. Eqs. (1)-(4) are obtained directly from (p-2), (p-l), (p-2)' and (p-l)', respectively. [] Lemma 3. P_~R ' © (R®cP),
(5)
R(~)c (R ' © Q) 2 Q,
(6)
R C_ (R(~(c) P)@c P-',
(7)
(Q@~(c) P-l)@c P D_Q.
(8)
Proof. Eqs. (5)-(8) are obtained directly from (p-l), (p-2), (p-3) and (p-3), respectively. [] It should be noted that Eqs. (3), (4) and (7) and (8) in the above lemmas are the new results on composite fuzzy relations, which are caused by the lack of commutativity of C. If C is commutative i.e. ~(C) = C, then they are reduced to the same as the case of T-composition and er-composition. See Property 3 in [13].
4. Solutions of fuzzy relational equations Let us cope with two kinds of relational equations regarding C-composition and ere-composition as
M.F Kawaguchi, M. Miyakoshi / Information Sciences 110 (1998) 113-125
119
R© P=QandR(~)c P=Q for two cases in which: (I) R and Q are given, P is unknown, (II) P and Q are given, R is unknown. Theorem 2. (I) Let R E ~ ( X x Y) and Q E ~ ( X x Z) be given, and 3 £ - {P I P E ~ ( Y × Z ) and R © P = Q}. Then, ~ is not empty iff R -l @c Q E W, and in that case R -1 @c Q is the greatest element in Y(. (II) Let P E ~,~(Y x Z) and Q E ~ ( x x Z) be given, and 81=- {R [ R E ~ ( X x Y) and R @ P = Q}. Then, 81 is not empty iff (P @~(c) Q-l)-i E 81, and in that case (P @~(clQ-J )-I is the greatest element in 81.
Proof. (I) Assume that £r _¢ O, and let P E f . Then, we have P c_ R 1@c Q by substituting R © P = Q into Eq. (1). From this equation and (C2), we have Q c_ R © (R-1 @c Q). Moreover, R © (R 1@c Q) = Q is obtained directly by denoting Eq. (2). Thus, it has been proved that R -1 @c Q is the greatest element in W iff W E 0. (II) It is proved similarly by using Eq. (3), Eq. (4) and (C2). [] Theorem 3. (I) Let R E ~ ( X × Y) and Q E ~ ( X x Z) be given, and T =- { P I P E ~ ( r × Z) and R(~ c P = O}. Then, tP is not empty iff R -l © Q E }P, and in that case R -1 © Q is the least element in tp. (II) Let P E ~ ( Y × Z) and Q E ~ ( X × Z) be given, and S=- {R I R E ~ ( X × Y) and R(~c P = Q}. Then, ~ is not empty iff Q(~K(c) p-1 E ~-, and in that case Q@~.(c) P 1 is the greatest element in ~.
Proof. (I) It is proved by using Eqs. (5), (6) and (12). (II) It is proved by using Eqs. (7), (8) and (12). [] 5. Examples of conjunctions and implications
There have been many functions in the numerous past works in multiplevalued logic especially in fuzzy logic, which are included in the sets of the
120
M.F. Kawaguchi, M. Miyakoshi / Information Sciences 110 (1998) 113-125 K
@
@
@
Fig. 1. The connectionsamong LSC t-norms, pseudo-conjunctionsand implications. above-mentioned conjunctions and implications, outside of the framework of a t-norm and its residual operator. We can easily verify that the following functions satisfy Definition 1 or Definition 2. Example 1. Dubois and Prade [4] have formulated the generation process of implications from t-norms and have defined pseudo-conjunctions. Now, for sake of convenience, we define the following mappings on the set of binary operations • : [0, 1]2 ~ [0, 1]: q~(*)(a, b) =- n(*(a, n(b))), q(*)(a,b) =- sup{s I * (a,s)<~b}, s
x ( * ) ( a , b ) -- . ( b , a ) ,
where n : [0, 1] ~ [0, 11 is a strong negation. Starting from the class of lower semicontinuous t-norms (i.e. left-hand continuous t-norms with respect to both variables), two classes of pseudo-conjunctions T1 and T2, and three well-known classes of implications i.e. S-implication Is, R-implication Ie and n-reciprocal R-implication/,R are associated with one another as shown in Fig. 1. Since I ( C ) = ~(C), we can see in Fig. 1 that C(I) = ~o o q o q~(1), C = q~ o q o ~ o q( C),
i.e. Corollary 3 holds. (See also [5,6] for another approach to this property, and [15] for the detailed properties and the explicit forms of these functions.) Therefore, we can substitute the pairs (T1 ,Is) and (T2,I,R) for (C, ~c) in our theoretical results. Example 2. It is possible to generate non-commutative conjunctions from already-known implications. Gaines' implication [9]: /Gaines(a,b) ~
1 0
(a<~b),
(a > b),
Yager's implication [20]:
=
a
(b > 0),
0
(b = 0).
M.F Kawaguchi, M. Miyakoshi / Information Sciences 110 (1998) 113-125
1
Iyager(a, b) =
(a=0),
b ~ (otherwise), 0
C(Iyager)(a, b) =
121
(a = b = 0),
bl/~ (otherwise).
Contrapositive symmetric R-implication [7]: lcPs-R(a, b) - max(IR(a, b), I,e(n(b), n(a))),
C(IcPs-e)(a, b) = min(T(a, b), T2(a, b)). Example 3. Sato and Sato [19] have proposed an asymmetric aggregation operator for their clustering method based on asymmetric similarity as follows: CAs(a, b) -- f - i (f(a) + p(a)f(b)), where f is an additive generator of any Archimedean t-norm, p is a continuous and monotone decreasing function from [0, 1] to [0, ~ ] satisfying p(1) = 1, and fI-~l is a pseudoinverse o f f . See [14] for the explicit forms of the additive generators. The residual operator of CAS can be obtained as
{f[-l](-f(a)+f(b)) I(CAs(a, b)) =
(a~b),
p(a)
1
(a < b).
Example 4. QL-implication (Zadeh [21])
IoL(a,b ) -- n(T(a,n(T(a,b)))),
T isan LSC t-norm,
C(IQL)(a,b) = C(T)(a, T,(a,b)) = inf{s I T(a,s) >>.r~(a,b)}. QL-implication is not monotone with respect to both variables i.e. it does not satisfy (I2). Thus, IoL and C(IoL ) are not included in our implications and conjunctions.
6. Numerical examples of fuzzy relational equations For instance, if we put n(a) = 1 - a and T = min, then T1, T2,Is, IR and InR which were mentioned in the previous section are obtained as follows:
T(a, b) = min(a, b), Tl(a,b)= { 0 b
T2(a,b)= { 0 a
(a+b~
Is(a, b) = max(1 - a, b)
(Kleene-Dienes),
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M.E Kawaguchi, M. Miyakoshi / Information Sciences 110 (1998) 113 125
(a<~ b), (a > b),
1
In(a,b) =
b
1
l,e(a,b)=
1-a
(G6del),
(a<~ b), (a > b).
Now, let us consider the simplest f o r m of fuzzy relational equation as numerical examples. A © R = B,
R=
(9)
Yl
Y2
x2
0.5
0.9
X3
0.9
0.4
'
B=
0.7
0.4
•
Here, A E ~ ( X ) , R ~ ~ ( X x Y), B E ~ ( Y ) , and A = (A(xl) A(x2) A(xs)) is unknown. Then, we can solve Eq. (9) applying T h e o r e m 2 (II). 1. When C = T, C~c = IR and e~-(c/= IR. =
(R@~_(c/ B i )'
© R=(0.7
= (0.4 0.4 0.7),
0.4)=B.
Therefore, A is the greatest solution of Eq. (9). 2. W h e n C = Tl, C~c = Is and ~ ( c / = I,R.
z~-- (R(~)h.(c)B-I) 1 (0.5 0.1
0.1),
A©R=(0.80)¢B. Therefore, Eq. (9) has no solution. 3. W h e n C = T2, ac = I,R and ~(c) = Is.
A= (R@~.(c)
B ' ) ' = (0.5 0.4 0.6),
,4© R=(0.6
0.4)¢B.
Therefore, Eq. (9) has no solution. F o r another example, let us assume that n(a) = 1 - a product, and solve Eq. (9).
T(a,b) = a × b,
0
Tl(a,b)=
(a+b-1)/a
(a+b~
and T is algebraic
M.F Kawaguchi, M. Miyakoshi / InJormation Sciences 110 (1998) 113 125 T2(a,b) = f~ 0 t (a+b-1)/b
f 1 / b/a
I,R(a,b)=
I), 1),
(a+b>
(Reichenbach),
Is(a, b) = 1 - a + ab IR(a,b) = {
123
(a > b),
1 (l-a)/(1-b)
(Goguen), (a<.b),
(a>b).
1. When C = T, :~c : IR and etc.(c) = IR. d :
A@ R=(0.7
B ')-'
:
(4;) 0.8
,
0.4)=B.
Therefore, A is the greatest solution of Eq. (9). 2. When C = Tl, ~c = Is and :~K(c) =/,R. ,~=(R@~.(c) B ,)_ 1
: d@ R=(0.7
(2 1 1) 563'
0.4)=B.
Therefore, A is the greatest solution of Eq. (9). 3. When C = 7"2, :~c = I,R and :q-(c) = Is. = (R@~.(c) B-') ~ = (0.7 0.46 0.73), d@ R=(0.7
0.4)=B.
Therefore, d is the greatest solution of Eq. (9).
7. Concluding remarks
The original study on fuzzy relational equation by Sanchez [17,18] corresponds to the framework of the complete Brouwerian lattice which is wellknown as the algebraic system of intuitionistic logic. The theory of fuzzy relational equations with triangular norms [13] is based on the concept of complete lattice-ordered monoid which is the algebraic system of L-fuzzy logic [10,1 l], On the other hand, in the framework of this study, ([0, 1], v, A, C) forms a lattice-ordered groupoid [1] (in short,/-groupoid) because C is monotone increasing but non-commutative and non-associative, besides not having even a unit element.
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M.F. Kawaguchi, M. Miyakoshi / Information Sciences 110 (1998) 113-125
Applications of the results here to fuzzy reasoning shall be considered for our next project. Furthermore, the effect of non-associativity of conjunctions needs to be considered from theoretical and practical points of view. Aside from these is another complicated problem which is how to select an adequate operator among various conjunctions and implications described here for each given equation, for example, a fuzzy input-output system represented by a relational composition. As an approach toward this problem, Ohtani et al. [16] have tried to search for an operator which makes a given equation solvable by using a genetic algorithm. Further detailed investigations along this line are needed in order to apply the theoretical results on fuzzy relational equations to the practical problems involving an analysis of fuzzy systems.
References [1] G. Birkhoff, Lattice Theory, 3rd ed., American Mathematical Society Colloquium Publications, 1967. [2] B. De Baets, Residual operators of implications, in: Proceedings of the Third European Congress on Fuzzy Techniques and Soft Computing (EUFIT'95), Aachen, 1995, pp. 136 140. [3] A. Di Nola, S. Sessa, W. Pedrycz, E. Sanchez, Fuzzy Relation Equations and Their Applications to Knowledge Engineering, Kluwer Academic, New York, 1989. [4] D. Dubois, H. Prade, A theorem on implication functions defined from triangular norms, Stochastica VIII (1984) 267 279. [5] J.C. Fodor, On fuzzy implication operators, Fuzzy Sets and Systems 42 (1991) 293-300. [6] J.C. Fodor, A new look at fuzzy connectives, Fuzzy Sets and Systems 57 (1993) 141 148. [7] J.C. Fodor, Contrapositive symmetry of fuzzy implications, Fuzzy Sets and Systems 69 (1995) 141 156. [8] J.C. Fodor, T. Keresztfalvi, Nonstandard conjunctions and implications in fuzzy logic, Int. J. Approximate Reasoning 12 (1995) 69-84. [9] B.R. Gaines, Foundations of fuzzy reasoning, Int. J. of Man-Machine Studies 6 (1976) 623 668. [10] J.A. Goguen, L-fuzzy sets, J. of Mathematical Analysis and Applications 18 (1967) 145-174. [l 1] J.A. Goguen, The logic of inexact concepts, Synthese 19 (1968-69) 325 373. [12] K. Izumi, H. Tanaka, K. Asai, Adjointness of fuzzy systems, Fuzzy Sets and Systems 20 (1986) 211-221. [13] M. Miyakoshi, M. Shimbo, Solutions of composite fuzzy relational equations with triangular norms, Fuzzy Sets and Systems 16 (1985) 53 63. [14] M. Mizumoto, Pictorial representations of fuzzy connectives, part I: Cases of t-norms, tconorms and averaging operators, Fuzzy Sets and Systems 31 (1989) 217-242. [15] T. Ohno, M.F. Kawaguchi, T. Da-te, Properties of implications and pseudo-conjunctions defined from t-norms, MVL Technical Report, Japan Research Group on Multiple-Valued Logic, MVL96-14 (1996) 96-103, in Japanese. [16] Sh. Ohtani, H. Kikuchi, R.R. Yager, Sh. Nakanishi, Diagnosis of a fuzzy system character, in: Proceedings of the 12th Fuzzy Systems Symposium, Tokyo, 1996, pp. 499-502, in Japanese. [17] E. Sanchez, Resolution of composite fuzzy relation equations, Information and Control 30 (1976) 38-48.
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[18] E. Sanchez, Solutions in composite fuzzy relation equations: Application to medical diagnosis in Brouwerian logic, in: M.M. Gupta et al. (Eds.), Fuzzy Automata and Decision Processes, North-Holland, Amsterdam, 1977, pp. 221-234. [19] M. Sato, Y. Sato, Asymmetric aggregation operators and its applications, in: Proceedings of the 12th Fuzzy Systems Symposium, Tokyo, 1996, pp. 141-142, in Japanese. [20] R.R. Yager, An approach to inference in approximate reasoning, Int. J. of Man-Machine Studies 13 (1980) 323-338. [21] L.A. Zadeh, Outline of a new approach to the analysis of complex systems and decision processes, IEEE Trans. Systems, Man, and Cybernetics SMC-3 (1973) 28M4.