Fuzzy relational equations and the inverse problem

Fuzzy Sets and Systems 15 (1985) 79-90 North-Holland

79

FUZZY RELATIONAL EQUATIONS AND THE INVERSE PROBLEM C.P. P A P P I S National Bank of Greece Ltd., Technical Seroices Dept., Stadiou 38, Athens, Greece

M. S U G E N O Dept. of Systems Science, 4259, Nagatsuta, Midori-ku, Yokohama 227, Japan

Received December 1983 Revised January 1984

The inverse problem concerned with fuzzy relations is investigated. The conditions for the existence of a solution axe shown and an analytical solution is given. A method for the improvement of the solution is proposed. Keywords: Fuzzy relations, Inverse problem.

1. Introduction This p a p e r is related to Sanchez' work [3] on fuzzy relational equations. H e dealt with the p r o b l e m " G i v e n two fuzzy relations O c U x V and S c U × W, find R = V × W such that R o O = S", where o denotes m a x - r a i n composition. H e showed an existence condition of the solutions by giving the least u p p e r b o u n d of the solutions. In general, the set of all the possible solutions for the a b o v e equation forms an u p p e r semi-lattice. Therefore, the greatest lower b o u n d does not always exist. T h e present p a p e r discusses the p r o b l e m called the 'inverse p r o b l e m ' , " G i v e n a fuzzy relation R = U x V and a fuzzy subset B = V, find all A = U such that A o R = B " . Although it is a special f o r m of Sanchez' equation, this fuzzy relational equation is widely used because of its simplicity and its usefulness in practical applications (see [4, 1, 2]). F o r example, a set of fuzzy implications (or fuzzy conditional statements) of the f o r m " I f At then B~, i ~ I " can b e conveniently expressed by the union of Cartesian products R = UiEtA~ xB~, At c U, B~ ~ V, i ~ L Given A c U, then B c V is induced, according to the fuzzy relational equation A oR = B. T h e p a p e r gives a different type of the existence conditions of the solutions, which is related to the lower bounds of the solutions. T h e lower bounds are analytically obtained by the m e t h o d presented in the paper. Thus, when a fuzzy 0165-0114/85/$3.30 © 1985, Elsevier Science Publishers B.V. (North-Holland)

C.P. Pappis, M. Sugeno

80

system is described by a relation matrix R associated with the m a x m i n c o m p o s i tional rule, the set of all the possible inputs A which give the s a m e o u t p u t B can be o b t a i n e d by c o m b i n i n g the least u p p e r b o u n d and a n u m b e r of the l o w e r bounds.

2. Statement of the problem D e n o t e a fuzzy subset A of U = {u~ [ i = 1 . . . . .

a = {(~, ~) I

i = 1 .....

m} by

m},

a f u z z y subset B of V = {vi [ J = 1 . . . . , n } by

B = {(v~, b,) I J = 1 . . . . . .}, and a fuzzy relation R of U x V by R = {((u~, vi) , r~j) [ i = 1 . . . . .

m, ] = 1 . . . . .

n},

w h e r e o4, bj and r~j are the grades of m e m b e r s h i p of t~, v i a n d (u~, v~), respectively. T h e c o m p o s i t i o n of A with R, d e n o t e d b y A o R, is defined to be a fuzzy subset B associated with the grades of m e m b e r s h i p

bi= V (ch^ni),

l~j~n.

O u r p r o b l e m can be stated as follows: " G i v e n R and B, find all A such that A o R = B ' .

3. Existence of a solution 3.1. Notations L e t a = (al, az . . . . . am), b = (bl, b2 . . . . . bin), e = (c~, Cz, •. •, c~) b e r o w vectors and R = [r~j], $ = [s~] m x n matrices. T h e following notations will b e used:

a>~b:

a~>~b~,Vi,

a<~b:

c~ <<-b~,Vi,

a^b:

(aiAbl, a2Ab2 . . . . . am A bin),

avb:

( a l V b l , a2vb2 . . . . . a~ v b,,),

a=O:

a~=O, Vi,

a:

V (a~), i

R >~$ :

r~j~ s~i,V i, V j,

R<~S:

r~i<~so,Vi, Vj,

aT:

transpose of a,

Fuzzy relational equations and the inverse problem

81

RT: t r a n s p o s e of R, A R:

flAP2

V-R:

rlvr2v"

^.

•

.Arm,

" "vrm,

w h e r e r~ is the ith r o w v e c t o r of N. 3.2. Definitions In the sequel, small letters x, y, etc. are used to d e n o t e scalars a n d w h e n p r i n t e d bold face, like a, b, etc., to d e n o t e vectors. Capital letters R, S, etc. are used to d e n o t e fuzzy subsets a n d w h e n p r i n t e d bold face, like N, S, etc., to d e n o t e matrices. A n y scalar and any e l e m e n t s of vectors or matrices are a s s u m e d to h a v e their values in the interval [0, 1].

o-composition. The o-composition of a v e c t o r a = (al, a2 . . . . . am) with a c o l u m n v e c t o r b = (bl, b2 . . . . . bin) T, d e n o t e d by aob, is defined b y the scalar a ° b a= V (a~ ^ b,). i

T h e o-composition of a r o w v e c t o r a = (al, a2 . . . . . am) with an m × n matrix R = [r~i], d e n o t e d by a oR, is defined by the r o w v e c t o r

a o R ~(aorl, a o r

2 .....

aor.)

w h e r e r i is the ]th c o l u m n v e c t o r of R.

a-composition. The a-composition of a scalar x with a scalar y, d e n o t e d by x a y, is defined by the scalar xay=a { 1 y

if x~>Y, if x < y .

G i v e n a c o l u m n v e c t o r a = (al, a2 . . . . . am) T and a scalar x, the a - c o m p o s i t i o n of a with x, d e n o t e d by a a x, is defined by the c o l u m n v e c t o r

a a x a=(al a x , a 2 a x . . . . . a m a x ) "r. G i v e n an m x n matrix R = [rii] and a r o w v e c t o r a = (al, a2 . . . . . an), let r i be the ]th c o l u m n v e c t o r of R. T h e n the a - c o m p o s i t i o n of R with a, d e n o t e d by R a a, is defined by the m x n matrix

Raa~[rl

a a l , r20ta2 . . . . .

r. ao.~].

T h u s R a a is a matrix, R a a = [wii], w h e r e

wij = rij a c~. ~3-composition. The/3-composition of a scalar x with a scalar y, d e n o t e d by x/3 y, is defined by the scalar x/3y~{0 y

if x < y , if x>~y.

82

C.P. Pap#s,

M. Sugeno

G i v e n a c o l u m n v e c t o r a = ( a l , a2 . . . . . am) T a n d a s c a l a r x, t h e / 3 - c o m p o s i t i o n o f a with x, d e n o t e d b y a / 3 x, is d e f i n e d b y t h e c o l u m n v e c t o r

a /3 x & ( a l f3 x, a 2 / 3 x . . . . .

a m / 3 x ) "r.

G i v e n an m x n m a t r i x JR = [r~i] a n d a r o w v e c t o r a = (a~, a2 . . . . . an), l e t r i b e t h e j t h c o l u m n v e c t o r of JR. T h e n t h e / 3 - c o m p o s i t i o n o f JR w i t h a, d e n o t e d b y JR/3 or, is d e f i n e d b y t h e m x n m a t r i x

R /3 a &[r~ /3 al, i"2/3 a2, • • •, rn /3 an]. T h u s JR/3 a is a m a t r i x , JR/3 a = [z~i], w h e r e z~j = r~i/3 a ~.

• -sets. G i v e n a c o l u m n v e c t o r a = ( a l , i = 1.....

a2 . . . . , am) "r, such t h a t a~ = , i m, t h e set (a) of c o l u m n v e c t o r s ~b(a) is d e f i n e d as follows:

o r 0,

qb(a) &{~b(a)}, where ~b(a) = (~bl, ~b2. . . . . ~b, = 0 o r fi,

~b,.)T,

i = 1 . . . . m,

~ thl = ¢i. i=l

T h u s , if t h e r e a r e k n o n z e r o e l e m e n t s in a, t h e r e a r e k v e c t o r s in qb(a). N o t e t h a t qb(a) is d e f i n e d iff a~ = 0 o r fi, Vi. ~ T ~ m p l e . L e t a = (0, 0.3, 0, 0, 0 . 3 ) T. T h e n d = 0.3 a n d a~ = 0.3 o r 0, Vi, t h u s ~ ( a ) is d e f i n e d a n d w e h a v e

• (a) = {(0, 0.3, 0, 0, 0) a', (0, 0, 0, 0, 0.3)T}. G i v e n an m x n m a t r i x R = [r~i], l e t r i b e its j t h c o l u m n v e c t o r a n d a s s u m e t h a t ~ ( r j ) is d e f i n e d f o r j = 1 . . . . . n. T h e n t h e set 4>(JR) of m a t r i c e s ~b(N) is d e f i n e d as follows:

• (R) =~{6(R)}, where 4)(R) = [(h(rl), 4)(r2) . . . . .

4)(r.)].

E,mmple. Let

(o o:t

R =0.20 0.2 o

o.51

We have

q~(N)=

0

0 1,10.2

080t(

0

0/\0

0

{(i o8 .2

0 0

O.

0.2

°-8 0 0

0.5)(0 08 0 0

,

0 0.2

0 0

O.

Fuzzy relational equations and the inverse problem

83

N o t e that t h e r e are z matrices in ~ ( R ) , Z = ~ ' I Zi iffil

where

{~

u m b e r of n o n z e r o e l e m e n t s in r i

zi =

if r i ¢ 0, if rj = 0.

3.3. S o m e properties o f o, a, {3 compositions a n d g - s e t s G i v e n scalars x, y we have OP1) x a y > ~ y , (I~) x/3 y ~ ~ b / 3 x , (P6) 3 a : a o b = x ¢~ 3 b i ~ b : b i ~ x . G i v e n an m × n matrix R and a r o w v e c t o r b = (bl, b2 . . . . . b,) we h a v e (PT) R a b > ~ R / 3 b . G i v e n R, b as a b o v e and a r o w v e c t o r a = (a~, a2 . . . . . am) we h a v e (P8) a o R = b ¢~ a o r i = bj, V] (ri: j t h c o l u m n v e c t o r of R ) . 3.4. The necessary a n d sufficient conditions L e m m a 1. G i v e n a c o l u m n vector b = (b:, b2 . . . . . 3a:aob=x

¢:, ( b o t x ) T o b = x

bin) T a n d a scalar x, w e h a v e

¢:~ ( b / 3 x ) r o b = x

¢~ (qb(b/3 x))Vob = x, Vqb(b 13 x) ~ q~(b 13x). L e m m a 2. G i v e n a row vector a = ( a i , a2 . . . . . (bl, b2 . . . . . bin)T a n d a scalar x, w e h a v e aob=x

~

aob = x ~

a~),

a

column

vector

b=

(bax)T>~a,

(1)

3cb(b {3 x) ~ ~ ( b / 3 x): (~k(b/3 x ) ) T ~ a .

(2)

L e m m ~ 3. G i v e n a n m x n matrix R = [rii ] a n d row vectors a = (al, a2 . . . . . b = (hi, b2 . . . . . b,), we h a v e a~A(Rab)

T,

aoR=b

~

aoR=b

~ 3qb(R~b)~(R~b): V (qb(R/3b))T~a.

am), (3)

(4)

Theorem 1. G i v e n an m x n matrix R = [rii] a n d a row vector b = (bl, b2 . . . . . b,), we h a v e

A(Rab)ToR=b,

3a: aoR=b

¢~

3a:aoR=b

¢~ 3 q b ( R ~ b ) ~ ( R / 3 b ) : V ( c k ( R / 3 b ) ) T o R = b .

(5) (6)

84

C.P. Pappis, M. Sugeno

l[h~mf. (5): See Sanchez [3]. (6): ( ~ )

a oR = b ~

3oh(jR/3 b) • ~(JR/3 b): V (~b(jR/3 b))T~
(from L e m m a 3)

V (6(JR/3 b))To JR ~< a o JR = b.

Let r i be the ]th column vector of R and oh(jR/3 b) = [¢bl(r;/3 b,),
:=> V (4,(JR/3 b))T >~(,bj(rj /3 b~))L Vi :z> V (~b(R/3 b))Vori ~(~bi(r i/3 bi))Xorj = bj, V]

(from L e m m a 1)

=> V (,b(R/3 b))~ojR >I b. Thus b ~
,n, ] = 1 . . . . .

n}

and a fuzzy subset B of V, B = {(vi, bj) I J = 1 . . . . .

n},

let JR = [r~i] be the m × n matrix corresponding to R and b = (bl, b2 . . . . . b.) the vector corresponding to B. Then the necessary and sufficient conditions f o r the existence of a fuzzy subset A c U satisfying A oR = B are given by either Eq. (5) or Eq. (6). Obviously the two conditions are equivalent, i.e.

A (JR a b)WojR = b ¢0 3cb(R/3 b) • ~(JR/3 b): V (4)(///3 b))WojR = b. 4. Soludon of the inverse problem 4.1. ~-composifion a n d its properties Given an m × n matrix R =[rii] and a row vector b = (bl, b2 . . . . . b,), the -composition of R with b, denoted by R ~ b, is defined by the m × n matrix

JR 8 b ~ [s,~], sii =

bk) /3(r,~/3bj),

i=1 .....

m,j=l

.....

n.

Note that / ~ = 1 (r, k a bk) is the ith d e m e n t of the row vector A ( j R a b) T and r~j/3 bi is the (i, j)th element of the m x n matrix JR/3 b. Thus JR 8 b can be obtained from A (JR a b)r and JR/3b.

Fuzzy relational equations and the inverse problem

85

Example. Let R=

0.2 0.8

5

1 0.7 0.3

0.5) ~.6,

b=(0.7

0.5

0.9

0.6).

Then we have !

R~b=

1 1 .7 0.5 0 0.5

R/3b=

\0.7 and finally RSb=

0 0.5

0.9 1 1

i)

0 o

0.6 ,

0 0

0.6 . 0

,

/ k ( R ~ b ) T=(0.9

1 0.5),

o.9O)o 0.90)

Given R, b, we have (1~) R6b<~R/3b, (P10) q~(S 8 b) c q~(R/3 b), (]Pll/ V ((b(R 8 b))T~ < A (R oz b/T, V&(R 8 b) • qb(R 8 b), (]P1-2/ V (~b(R/3 b)) T ~
4.2. The solution Lemm~ 4. Given a column vector b = (bl, b2 3a: a o b = x. Then Va: a o b = x ~

. . . . .

bin) T and a scalar x, assume that

(7) (8)

3 ¢ b ( b / 3 x ) • c b ( b / 3 x ) : (4~(b/3x)T~a<---(bax) T,

Va, V 4 J ( b / 3 x ) • c b ( b / 3 x ) : ( c b ( b / 3 x ) ) T < a < ~ ( b a x ) w ~

aob=x.

Theorem 2. Given an m × n matrix R and a row vector b = (bl, b2. . . . . b,), assume that 3a: a o R = b. Then

Va: a oR = b :ff 3~b(R 8 b) • qb(R 8 b): ~/(~b(R 6 b)) T ~
(9) Va, V4~(R 8 b) • qb(R ~ b): V (~b(R 6 b))r ~
(10) ~

3cb(R/3b)•~(R/3b):

V(&(R/3 b))T~
(from Lemma 3)

=) 3 ~ b ( R ~ b ) • q~(R~b): V (& (R 8 b)) T <~a ~
(from (P121/.

86

C.P. Pappis,

M. Sugem

(10): V(c#a(R6b))Ttiaa/j((Rab)T

+ V G#dRPbNTV *

a s /\ (R a b)=, tl+(R

/3 b) E @(R p b) (from (PlO))

(&j((rj 6 bj))TsU s(rj Cybj)T, Vj,

where rj is the jth column vector of R. Thus aorj = bj, Vj (from Lemma 4), i.e. aoR=b. The solution of the inverse problem is derived from Theorem 2 as follows: Given the fuzzy relation R and the fuzzy subset B, all fuzzy subsets A such that AoR =B are given by V&(R 6 b) E @(R S b),

V (c$(R S b))Ts Q s l\(R a! b)=.

provided that there exists at least one such A, where R is the matrix corresponding to R, and a, b are the vectors corresponding to A, B respectively. 4.3. Example Let

u\

v

Ul

R=

‘2 u3 u4

Vl

v2

u3

v4

vg

0.4 0.7 0.6 0.2

0 0.8 0.4 1

0.9 0.3 0.3 0.5

0.6 1 0.4 0.8

0.8 0.5 0.9 0.4

v

Vl

v2

v3

v4

%

b

0.6

0.5

0.9

0.6

0.8

B=

We have 11111 0.6 i6 1

Rab= Rab=(

0.5 ;;I 1 0.5

1i 0.6 ;:I 1 1 1 0.6

0

0

0.6 0.6 0

0.5 0 0.5

0.9 0 0 0

\l

0.6 0

1 i-8]) 1 / 0.8

A(Rab)T=(l A(Ra!b)T=(l

0.5

0.8

0.5)

Fuzzy relational equations and the inverse problem

I O 0 R /s b = 0.6 0

0 0.5 0 0.5

0.9 0 0 0

0.6 0 0 0

87

.81 0.8]" 0

It is easily seen that A (Rotb) T ° R = b , thus 3a: a o R = b . Consider, for exampie, ~bx(R 8 b) ~ q~(R/5 b) such that 00

00 . 5 0 0.9

0.6 0

~ t

0.6 0

0 0

0 0

0 0

0.8 0

V(~bl(R/sb)) T=(0.9

0.5

0.8

~bl(R/5 b) =

"

We have 0).

Any fuzzy subset A with grades of membership vector a such that (0.9

0.5

0.8

0)~
0.5

0.8

0.5),

has the property that A o R = B. Now consider (~2(R/5 b)E (/)(R/5 b), such that

~ 2 ( R /5 b) =

0.6 0

0 0.5 0 0

0.9 0 0 0

0.6 0 0 0

0.8) 0 0 " 0

We have V (~b2(R/5 b)) "r = (0.9

0.5

0.6

0).

Again, any fuzzy subset A with grades of membership vector a such that (0.9

0.5

0.6

0)~
0.5

0.8

0.5),

has the property that A o R =B. Note, however, that (0.9

0.5

0.6

0)~<(0.9

0.5

0.8

0).

4.4. Remark A (R a b)T is the least upper bound (1.u.b.) of the solution vectors of the inverse problem. The set {V (~b(R 8 b)) T I ~b(R 8 b) ~ q~(R/5 b)} includes the lower bounds of the solution vectors. Generally, we may have the case where V (~I(R/5 b))T~ < V (~2(R/5 b)) T as has been seen in the last example. In the next section, a class of non-greatest lower bounds is identified, the solution of the inverse problem being thus improved.

C.P. Pappis, M. Sugeno

88

5. Improvement of the solution 5.1. Definitions Let an m x n matrix R = [rii] and a row vector b = (bl, b2. . . . . b,) be given. In the sequel, the set {V(cb(RSb))Xl4~(RSb)~cI)(R~b)} will be denoted by

V (O(R a b))T. A vector V (4h(R 8 b))T~ V ( O ( R ~ b)) T is said to be redundant if there exists V (4~2(R ~ b))Xe V ( O ( R 8 b)) T such that

V (~b2(R c5b)) T ~ V (6~(R ~5b)) T. Let Sk, S~ be the kth, lth column vectors of matrix S respectively. If s, ¢: 0 ::> sik ~ 0 and sik ~
slk is arbitrary),

then sk is said to be dominated by sz. 5.2. The improvement of the solution of the inverse problem Let an m x n matrix R = [r~j] and a row vector b = (bl, b2 . . . . . bn) be given and assume that :la: u oR = b. Let S = R 8 b = [s~i]. If So is the matrix obtained from S by deleting all its zero column vectors, it can be shown that V (O(S)) T = V (O(So)) x.

(11)

Denoting by Sk the matrix obtained from So by deleting a column vector sk, it can also be shown that if and only if sk is dominated, then

V (O(Sk)) ~

V (O(S)) ~

(12)

and :Itb(Sk) e O(Sk): V (tb(s))r>~ V(tb(Sk)) r, Vtb(S)e O(S).

(13)

The significance of (12) and (13) is that, by deleting a dominated vector sk from So, a (possibly empty) class of redundant vectors is excluded from the solutions which are obtained from

V (4~('-.~))T~
Zt~--

Zk

where zk is the number of nonzero elements in the column vector sk. If $* is the matrix obtained from So by deleting all its dominated column vectors, we further obtain from (12) and (13) that and

V (O(S*)) x c V (O(S)) r 3~b(S*) e O(S*):

V (cb(S))T~> V (~b(S*))T, W b ( S ) • O(S).

(14) (15)

Fuzzy relationalequations and the inverseproblem

89

Thus all the solutions of the inverse problem are given by

V (d~(S*))"r<~a ~ A (Rot b)T, V~b(S*)~ ~(S*). It can be shown that any reduction of the dimensions of S* would result in some nonredundant vectors of ~/(~(S)) T being eliminated. However, some redundant vectors may still be included in ~/(~(S*)) T. 5.3. Example In the last example (Section 4) we had

S=R~b

=

0 0.5 0 0.5

0.6 0

A(Rotb)T=(1

0.5

0.9 0 0 0 0.8

0.6 0 0 0

0.8) 0 0.8 ' 0

0.5).

Thus $. =

00 0.6 0

0 0.5 0 0.5

i.9)

Since the 4th and 5th column vectors of $ are dominated by the 3rd column vector, we have V(qb(S*))'r={(0.9

0.5

0.6

0),(0.9

0

0.6

0.5)}.

All a such that a oR = b are given by (0.9

0.5

(0.9

0

0.6

0)~
0.5

0.8

0.5)

0.5)~
0.5

0.8

0.5).

or 0.6

These are shown by the tree of Figure 1. /~0(1 0.5 0.8 0.5) .9 0.5 0.6 0.5) (0.9 0.5 0.6 o) (0.9 0 0.6 0.5) Fig. 1. Solution tree. 6. Conclusions

In this paper, the inverse problem of fuzzy relational equations has been investigated. A different type of the conditions for the existence of a solution have been derived, and the problem is given an analytical solution. A method has been proposed, by means of which a class of redundant lower bounds of the solutions are readily eliminated; the general solution is thus improved.

90

C.P. Pappis, M. Sugeno

Acknowledgements This work was c a r d e d out while the authors were at Q u e e n Mary College, D e p a r t m e n t of Electrical and Electronic Engineering, University of London, England. These results were presented at the 1976 workshop on Discrete Systems and Fuzzy Reasoning, held at Q u e e n Mary College, and appeared in the proceedings, published by D e p a r t m e n t of Electrical Engineering Science, University of Essex.

Rderences [1] E.H. Mamdani and S. Assilian, An experiment in linguistic synthesis with a fuzzy logic controller, Internat. J. Man-Machine Studies 7 (1974) 1-13. [2] C.P. Pappis and E.H. Mamdani, A fuzzy logic controller for a traffic junction, Dept. Electr. Engineering, Queen Mary College, University of London (1976). [3] E. Sanchez, Resolution of composite fuzz), relation equations, Inform. Control 30 (1) (1976) 38-48. [4] L.A. Zadeh, Outline of a new approach to the analysis of complex systems and decision processes, IEEE Trans. Systems Man Cybernet. 3 (1) (1973) 28-44.