Multi_dimensional Fuzzy Reasoning

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Fuzzy Sets and Systems 9 (1983) 313-325North-Holland Publishing Company 313

MULTI-DIMENSIONAL FUZZY REASONINGMichio SUGENO and Tomohiro TAKAGIDepartment of Systems Science, Graduate Schoo l of Scienc e and Engineering, Tokyo Institute ofTechnology, 4259 Nagatsuta, Midori-ku, Yokohama 227, JapanReceived May 1981Revised December 1981

This paper suggests a method of multi-dimensional fuzzy reasoning concerned with bothmodus ponens and modus tollens. It also discusses an example to show how the method works.

Keywords: Lukasiewiczs infinite valued logic, Fuzzy logic, Fuzzy implication, Multi-dimensional fuzzy reasoning.

1. IntroductionA considerable number of studies on fuzzy reasoning, have been reported, e.g.[l, 2,3,5], since Zadeh presented the compositional rule of inference. Fuzzyreasoning has been especially applied for fuzzy control and diagnosis, and latelyfor fuzzy modelling. However there have been few discussions on it in themulti-dimensional case. This paper suggests a method of multi-dimensional fuzzyreasoning which is of great importance particularly when we deal with fuzzymodelling of systems. It also shows numerical examples which claim the validity ofthe proposed method.In general there are two methods, as is well known, on fuzzy reasoning: one isbased on the compositional rule of inference and the other on fuzzy logic withsuch and such a base logic, e.g. Tsukamotos method [4].Since a multi-dimensional implication such as if (x is A, y is B) then *z is Cwhere A, B and C are fuzzy sets is not merely a collection of one-dimensionalimplications, a conventional interpretation is usually taken in the multi-dimensional case. For example in the compositional rule of inference, the abovetwo-dimensional implication is translated into(1) if x is A and y is B then z is C,or (2) if x is A then if y is B then z is C.In Tsukamotos method, it is decomposed into

(1) i f x is A then z is C,and (2) if y is B then z is C,and the intersection C fl C, where C is the inferred value from the firs timplication and C that from the second implication, is taken for the consequenceof reasoning. It is clear that those methods do not always express real situations,01650114/83/0000-0000/$03.00 0 1983 North-Holland


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314 M. Sugeno, T. Takagi

since a pair of propositions (x is A, y is B) could be any fuzzy relation (x, y) isR with respect to x and y. For example there might be a functional relation suchthat z = x*y behind a fuzzy implication.

Further, in the multi-dimensional case we have some difficulties in applying thecompositional rule of inference, because we have to deal with multi-dimensionalfuizy relations.So we use in this paper Tsukamotos method by which we can do without thefuzzy relational matrix and present an algorithm of fuzzy reasoning in the generalcase. The algorithm is very similar to that we use to approximate a function f(x , y)by a linear function.

2. Fuzzilied Lukasiewiczs logic and its reasoningIn this section we shall briefly introduce Tsukamotos method of fuzzyreasoning.The truth value of implication in Lukasiewieczs infinite-valued logic is expres-sed as

/A-,B/=(l-/A/+/B/)/\l, (1)where /A/E [0, l] is the truth value of a proposition A. Let P, Q be fuzzypropositions and 1, Q be linguistic truth values that are fuzzy subsets of [0, l] asusual. Then P + Q z derived by the extension principle from (l),

P-,Q=(l-E+Q)/\l. (2)In (2) the symbols -, + and A are the extended ones. In the sequel we shall usethe extended operations for fuzzy sets without a comment. For calculation it isbetter to take a strong a-cut of (2). We have

P-,Q,=(l-p,+Q,bl (3)where 5 ={u 1 f+(u)> (Y} and h&u), IA E [O, 11, is the membership function of 2.When P+ Q and _P are given, (3 is easily obtained by solving (3).In many cases we can assume that P + Q is normal, its membership function isnon-decreasing in its domain [0, l] and also _P is normal and convex. DenoteR = P + Q. Then it fol lows from the above assumptions that

% = (rb), 11, (4)po = h(a), P*(d), (5)

where r(a)~[O, 11 s determined from h, and pi(a), p,(a) from he Solving (3),Q is obtained asP

Q, = ((PI(Q) + 40~) - 1) v (All.- (6)Here we should take a caution of the term -E in (3). For simplicity suppose an


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Multi-dimensional fuzzy reasoning 315equation R, = -Pm + Q,. If we directly solve Q, according to extended opera-tions, we do not have & = R,+P,. More simply it is not even true thatQ-p,)+p,=Q, . So we should transfer P, from the right-hand side to the leftin (3). This sort of problem will be also discussed in the Appendix.The solution in (6) is the same to that obtained by solving fust (1) and thenapplying the extension principle. Now ho is drawn since (6) implies that h, is a-non-decreasing function.Fuzzy reasoning based on modus ponens is carried out as follows, where modusponens is written

P, P+QQ' *(1) Given a premise P, set P = P is E. By converse of truth qualification, _P isfound as

_p &wL (7)where hp is the extension of hp for a fuzzy set P.(2) Calculate Q from c and P 4 Q which is assumed to be given.(3) Set Q = Q is Q. Then Q is the consequence of fuzzy reasoning. By truthqualification, Q is obtained as

Q' = h,(Q), (8)where h


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316 M. Sugeno, T. Takagiparticular when P + Q is u-true and Q is a singleton yO, we have

b(x) = (-h&J + I, + 1) A 1 (14)

3. ~Multi-dimensional reasoningAs stated in the introduction, we suggest in this section a method of multi-dimensional fuzzy reasoning. Unfortunately we have not had a general method inthe multi-dimensional case so far. Of course we can use the compositional rule Ofinference. However there are two difficulties: one is to identify or construct afuzzy relation not merely as, for example, a Cartesian product of some number offuzzy sets, and the other is to calculate a multi-dimensional fuzzy relationalcomposition, for example, in order to compute a fuzzy control algorithm in realtime.The idea suggested here is quite simple and reasonable, which is very similar to

linear interpolation.We should firs t consider that we cannot logically infer anything from a singleimplication, say, A + B unless the same premise to A is given. This is alsosomewhat true in fuzzy reasoning. This is the reason why there are some personswho translate A 4 B into a fuzzy relation R = A + B and not A 4 V whereV is the universe of discourse of B. Suppose x is big* y is small. Then no onecan infer the value of y from this single implication when, for example, x ismedium. We should have at least two different implications, Ai + B, andA24 B2 for one-dimensional reasoning. What we can do then is to infersomething when we are given a premise lying between A1 and AZ. This reasoningmust anyhow be carried out using some method like interpolation.Apart from computational technique, it is sufficient for the purpose to discusstwo-dimensional case. Thus we can start with 4 implications such that(x is Ai, y is B,) --, z is C1r,(x is Al, y is BJ+ t is Ci2,(x is A*, y is B,)+ z is Ctl,(x is Aa, y is B2) + z is C,,.

The important point we want to stress is that we cannot translate (x is A, y is B)into, for example, (x, y) is A X B where A X B is Cartesian product, sincegenerally a pair of two propositions (x is A, y is B) could be any relation asmentioned also in the introduction. This means that we should deal with (x is A, yis B) directly without modification.The situation of 4 implications is shown in Fig. 1.Our problem is to infer z is C from a given premise (x is A, y is B) where Ais assumed to be between A1 and A2 and also B between Br and B2. When weapply fuzzy reasoning introduced in the previous section to this problem, we facea difficulty to find T such that (x is A, y is B) = (x is Al, y is B,) is 7, where r is alinguistic truth value. It is impossible to solve the above semantical equation


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Multi-dimensional fuzzy reasoning 317I Iy b-T

ITBIcc11I %-*I *2 x

B+-i-t;--s--r-T--4 i A2

Fig. 1. Four implications. Fig. 2. Outline of algorithm.unless the relation between x and y is given. That is, generally the truth value of apair of propositions is expressed using those of individual propositions as

(A,, &I =f(At, B,)-where a function f of arbitrary Ai and B, is unknown. Our method can do- -without f.The outline of the algorithm is shown in Fig. 2. First let us infer the value of zat the point (A,, B) from (A,, B,) as indicated by an arrow in Fig. 2. Modusponens is written

(x is Al, y is B), (x is Ai, y is BJ+ z is Cl12 is Cl1

Here Cl1 is easily obtained by one-dimensional reasoning shown in the previoussection since Al is fixed along the arrow. That is, set(x is Al, y is B)=(x is Ai, y is B,) is r (15)

then clearlyyisB=yisBrisr (16)

for the same r.Let us next infer z at the same point (A,, B) this time from (A,, B,). That is

(x is Al, B), (x is Ai, BJ +- z is Cl*z is Cl*

Now letDl=f(c;,+c;*). (17)

Then we obtain a new implication with respect to the point (A,, B) such that(x is Al, y is B)+ z is D1.

Here D1 may be obtained in another way: D1 = C;, tl Ci2. For the point (AZ, B)we can also obtain an implication such that(x is AZ, y is B) += z is D2.

Finally if we follow the same procedure along the dotted arrows in Fig. 2, wecan infer the value of z at (A, B). So the method is just like linear interpolation.In the following section we shall show how well the method works.


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318 M. Sugeno, T. Takagi

-Al -

Fig. 3. Outline of algorithm.

Let us consider the inverse problem such as finding x when y is B and z isC are given. The first half part of the algorithm is the same as in the previouscase. We start from the part where Q and D2 in Fig. 2 have been obtained. Thatis, we have two implications(x is A,, y is B)-+ z is D,,(x is A*, y is B) + z is I&.

The second half of the algorithm is illustrated this time in the (y, z) plane as inFig. 3.Now consider the following modus tollens:z is C, (x is Al, y is B) + z is Q

(x is A;, y is B)Since y is B is fixed, the problem is also reduced to one-dimensional modustollens. So x is A; can be inferred, which is shown by the lower arrow in Fig. 3.Along the upper arrow, x is A; is also obtained. By setting A = (A i + A;)/2, xis A is finally inferred from (y is B, z is C).

4. Illmlrative example4.1. Reasoning based on modus ponens

Let us consider the following linear functional relationz=x+ay (18)

to examine the validity of the method. This relation is understood as an underly-ing relation which we want to express by a set of fuzzy implications. For simplicityfuzzy variables used in implications are chosen asPI(X) =& x +;, x E [-z, a], (1%N,(x) = P1(-x>, (20)PAY)=& y+3, YE L-7, Yl, (21)WY) = M-y), (22)


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Multi-dimensional fuzzy reasoning 319

Fig. 4. Membership functions of P,, N,. Fig. 5. Membership functions of P2, N2.

where P and N imply positive and negative, respectively, and Pi(x) is themembership function of Pi. Those are shown in Figs. 4 and 5. In the sequel themembership function of a fuzzy set A will be written A(x).Let us further assume that the value of z, for example, at the point (x is Pi, y isPJ is precisely given according to the underlying relation shown in (18). Then wehave an implication(x is PI, y is PJ -+ z is PI + aPz.

Denote P3 =Pl+aP,, then follows by a simple calculation.PJz)=(P1+aP*)(z)= -I2(x + ujq z+$. (23)

The method of calculation such as P, + aP, is discussed in the Appendix.In a similar manner, we have implications for the other three points as is shownin Fig. 6. They are(x is Ni, y is NJ + z is NJ,(x is Ni, y is P2) + z is P4,(x is PI, y is NJ + z is N4,(x is P,, y is PZ) + z is P3,

wherePJz)= ?2(x + aj3 z+;,N(z) = PA-z),P,$(z) = 12(aY - 3) z+$,N,(z) = PA-z),

(24)(25)(26)(27)

and a7 + ff # 0, ajj -2 # 0 are assumed of course.Now let us consider a special premise such as (x is x0, y is yo) where x0 and y,are singletons. What we expect by this is to obtain z. = x0+ aye from our fuzzyreasoning. According to the algorithm described in the previous section, we havetwo m.odus ponens along the lines reaching (x is P,, y is y,) in Fig. 6 such that

(PI, Yo), (PI, Pd --, P3 (PI, YO), (p,, NJ --* N46 , W .


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M. Sugeno, T. TakagiI I

pz- 4 -I----p3-Ye-y--t-N-N2 I3 x0 y-Nl Pl

Fig. 6. Situation of implications.

Assuming that the truth values of all the implications are u-true, that is, r(a)=&in (4), we obtain from the procedure of fuzzy reasoning shown in Section 2p;(z) = P3(z) + 1 - P*(Yo)

1 1= 2(@ + f) z-- ycJ+ 1,27 (28)

N(z) = N$(z) + 1 - ~z(y,)1=- 2(@-x) f +2gyo+l.

They are shown in Fig. 7.(2%

0 1 I Z(a-il$yo (a-W Y0

Fig. 7. Membership functions of Pi and N;.Let

K = $(P; + NA),then

K(2)=&(z+2f-ay,)which is shown in Fig. 8.

] T-m- ---_K(Z)[ !0 z=y0

Fig. 8. Membership function of K. Fig. 9. Membership function of H.

(30)

(31)

0 =y Z0


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Multi-dimensional fuzzy reasoningWe have also two modus ponens for the point (N,, yO) such that

WI, Yo), WI, PA + p4 UC YCJ, w,, Nz) + N3p; 9 N;

321

LetH = $(P; + N;),

then we also obtainH(z)= -&(z-2f-ay,)

which is shown in Fig. 9.Now the new implications with respect to (P,, yO) and (N,, yO) are(x is P,, y is yO) + z is K,(x is Nr, y is yJ-+ z is H.

(32)

(33)

Finally we have the following modus ponens along the dotted lines in Fig. 6.(x0,Yoh (PI, Yo) + K (xc,, YO), WI, YO)+ HK , H

where H and K are calculated asK(z)=K(z)-Pr(xJ+l,H(Z) = H(z) - N,(x,) + 1.

They are shown in Fig. 10. Let

(34)(35)

L = ;(K+ H), (36)then L is the final consequence of fuzzy reasoning, which is just found to be asingleton as is shown in Fig. 11. That is

L=x,+ay,, (37)which gives the precise value of z at (x,, y,,).Thus we can state that our method of multi-dimensional fuzzy reasoningrepresents precisely a linear functional relation. We should recall that we haveused only 4 implications at all.

-;-g------T@-0 I I z1 ----I;;-- z

aat% -x ayo+xo+Ti x0 +ayoFig. 10. Membership functions of K and H. Fig. 11. Membership function of L.


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322 M. Sugeno, T. Takagi4.2. Reasoning based on modus tollens

Now let us consider the inverse problem of that discussed in the previoussection. Our problem is to And x when (y is yo, z is zo) is given. According to thealgorithm, we can start from Fig. 12.

4 Y I IyQ-H-K-

I INl Pl x

Fig. 12. Situation of two implications.

We have already obtained two implications on the line y = y, in Fig. 12 such as(x is IV,, y is yo) ---, z is H,(x is PI, y is yo) + 2 is K,

where H and K are given in (32) and (30).When (yo, zo) is given, we can consider the following two modus tollens.20, (PI, Yo) + H 20, WI, yo) + K

UT Yo) WY> Yo) .

Fig. 13. Two modus tol lens.They are illustrated on the (y, z) plane in Fig. 13. We have from (14)

P;(x) = -PJx) + K(Z0) + 1, (3%N;(x)=-N,(x)+H(z,)+l. (39)

Lets =f(P:+N;), (40)

then it follows thatS=zo-ay, (41)

which also gives the precise value of x, i.e. x0 = zo- aye.


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Multi-dimensional fuzzy reasoning 323Conclusions

We have discussed some problems concerned with multi-dimensional fuzzyreasoning and proposed a method. The method enables us to precisely representany linear functional relation. On the other hand it is very difficult to find a fuzzyrelation in a compositional rule of inference which can represent a linearfunctional relation under the condition that four implications, for example, aregiven as discussed in the paper.The purpose of our research is to give a new mathematical tool to describesystems, or in other words to give a way to fuzzy modelling of systems. It is ofcrucial importance for general purpose to examine the ability of the description ofsuch and such a mathematical tool. So we have discussed our method in detail.

Appendix. Addition and subtraction in a class of fuzzy numbers with linearmonotone membership functionsHere w e deal with extended operations in a special class of fuzzy numbers.Suppose a class of fuzzy sets as are shown in Fig. 14.

(4Fig. 14.

b)

where fuzzy sets P and N have linear monotone membership functions P(x) andN(x), respectively.P(x)=-&- (x - 4, x Era,, 4192 1

N(Y) = -A (Y - bJ> Y E ib,, &I.2 1Their inverse functions are denoted by P- and N-.

P-(a) = (a* - a& + a,,N-(a) = -(b, - bl)a + bZ, a E [O, 1).

There are two types of fuzzy numbers, P and N, in the class. A singleton isconsidered as a special case of P or N.It is well known that a set of fuzzy numbers with an operation addition doesnot form a group. For example, letX=P+N.


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324 M.Sugeno, T. TakagiAccording to the extension principle, taking o-cut, we have

&=P,+N,= (P-(a), aJ+[b,, N-(d)= (P-'(a)+ bl, N-(a) + UJ.If we subtract P from X, then

x, -P, = (P-(o) + bl, N-(o) + ClJ -(P-(cr>, a,1= (P-(a) + br - a2, N-(a) - P-(a) + ~2)#Na.

That is(P+N)-Pf N.

Next let us take another more simple example. Let Y = 2P-P, then it is naturalto expect Y = P. However we haveY,=2P,-P,

= (2P-(o), 2a,]-(P-(a), a,]= (2P-(cr) - Q2. 2a2 - P-(a)]z PCS

That isYf P.

As for addition, it is true that 3P = 2P + P, for example. Notice that P + N is anoperation of the kind 2P-P since N is a fuzzy number of the type -P. Ingeneral we have to take caution of subtraction defined by the extension principle.We should indirectly calculate A -B by finding X such that A = X+B.So let us try to find Y such that2P= Y+P.

Setting Y, = (y,(a), y,(a)], we haveW-(a), 2a,l= (y,(a), Y2b)3+(~-1b), a21

= (YlM + p-v43 Y2b) + a,].This implies that Y = P since yl(a) = P-*(a) and y2(a) = u2 are obtained.Now we present new extended operations in a class of fuzzy numbers withlinear monotone membership functions which give a group structure.Addition :

Inverse :x = -P t-, X-'(a) = -P-'(a).


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Multi-dimensional fuzzy reasoningScalar multiplication:

x = UP f;, x-(a) = d-(a).

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The results of these operations belong to the same class; the operations areclosed. It is valid that (P + N) - P = N, 2P-P = P and 3P = 2P + P, etc. It is easilyseen that this class of fuzzy numbers with addition yields a commutative group.

References[l] J.F. Baldwin, A model of fuzzy reasoning through multi-valued logic and set theory, Int. J.Man-Machine Studies 11 (1979) 351-380.[2] J.F. Baldwin, Fuzzy logic and approximate reasoning for mixed input arguments, Int. J. Man-Machine Studies 11 (1979) 381-396.[3] S. Fukami, M. Mizumoto and K. Tanaka, Some considerations on fuzzy conditional inference,Fuzzy Sets and Systems 4 (1980) 243-273.[4] Y. Tsukamoto, An approach to fuzzy rsasoning method, in: M.M. Gupta, Ed. Advances in Fuzzy

Set Theory and Applications (North-Holland), Amsterdam, 1979) 137-149.[5] L.A. Zadeh, Fuzzy logic and approximate reasoning, Synthese 30 (1975) 407-428.

Multi_dimensional Fuzzy Reasoning

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