fuzzy values in fuzzy logic

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Fuzzy Values in Fuzzy Logic* L. Godo C. €.A. B.IC.S.I.C.,Cami Sta. Barbara, 17300 Blanes, Girona, Spain J. Jacas E.T.S.A. Barcelona, Univ. Polit. Catalunya, Diagonal 649, 08028 Barcelona, Spain L. Valverde Dep. Cienc. Mat., Univ. llles Balears, Ctra. Valldemosa Km 7,5, 07071 Palma Mallorca, Spain One of the main features of Fuzzy Logic is its capability to deal with the concept of compatibility between two propositions, in such a way that the inference process mod- eled through the Compositional Rule of Inference is independent from the particular possibility distributions involved. It is in this context that the compatibility functions can be considered as fuzzy truth values, labels or qualifications, playing the same role as the values true and false play in the Classical Logic, where the meaning of propositions is nothing but its truth value. In this article we consider a restricted family of labels having the following desirable properties: (a) easy parametric representation, (b) easy semantic interpretation, (c) to allow a gradation in the family according to the modifica- tions performed by each label, and (d) to be closed under inference processes (FR- functions), and also under some suitable and meaningful operations between them. 1. INTRODUCTION AND PRELIMINARIES It is known that one of the main features of Fuzzy Logic is its capability to deal with the concept of compatibility between two propositions like “X is A” and “X is A”’, where A and A’ are linguistic terms represented by fuzzy subsets of a given universe of discourse U, inducing, respectively, possibility distributions on the values of the variable X. Such a compatibility is usually estimated by the so-called “inverse-truth functional modification” proposed by Baldwin’ in the following way Sup{A’(ir)lu E A-’({u})} if A-’({u}) # 0; otherwise. PAA~Q) = *Research partially supported by the CSIC-CAICYT. project No. 836. INTERNATIONAL JOURNAL OF INTELLIGENT SYSTEMS, VOL. 6, 199-212 (1991) 0 1991 John Wiley & Sons, Inc. CCC 0884-8173/91/020199- 14$04.00

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Page 1: Fuzzy values in fuzzy logic

Fuzzy Values in Fuzzy Logic* L. Godo C. €.A. B.IC.S.I.C., Cami Sta. Barbara, 17300 Blanes, Girona, Spain

J. Jacas E.T.S.A. Barcelona, Univ. Polit. Catalunya, Diagonal 649, 08028 Barcelona, Spain

L. Valverde Dep. Cienc. Mat., Univ. llles Balears, Ctra. Valldemosa Km 7,5, 07071 Palma Mallorca, Spain

One of the main features of Fuzzy Logic is its capability to deal with the concept of compatibility between two propositions, in such a way that the inference process mod- eled through the Compositional Rule of Inference is independent from the particular possibility distributions involved. It is in this context that the compatibility functions can be considered as fuzzy truth values, labels or qualifications, playing the same role as the values true and false play in the Classical Logic, where the meaning of propositions is nothing but its truth value. In this article we consider a restricted family of labels having the following desirable properties: (a) easy parametric representation, (b) easy semantic interpretation, (c) to allow a gradation in the family according to the modifica- tions performed by each label, and (d) to be closed under inference processes (FR- functions), and also under some suitable and meaningful operations between them.

1. INTRODUCTION AND PRELIMINARIES

It is known that one of the main features of Fuzzy Logic is its capability to deal with the concept of compatibility between two propositions like “X is A ” and “X is A”’ , where A and A’ are linguistic terms represented by fuzzy subsets of a given universe of discourse U , inducing, respectively, possibility distributions on the values of the variable X.

Such a compatibility is usually estimated by the so-called “inverse-truth functional modification” proposed by Baldwin’ in the following way

Sup{A’(ir)lu E A-’({u})} if A-’({u}) # 0;

otherwise. P A A ~ Q ) =

*Research partially supported by the CSIC-CAICYT. project No. 836.

INTERNATIONAL JOURNAL OF INTELLIGENT SYSTEMS, VOL. 6, 199-212 (1991) 0 1991 John Wiley & Sons, Inc. CCC 0884-8173/91/020199- 14$04.00

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200 GODO, JACAS, AND VALVERDE

What is most important is the fact that from the Compositional Rule of Inference point of view, “X is A”’ is assumed to be equivalent to “X is pM, 0 A,” in the sense that the traditional inference schema

If X is A then Y is B

Y is B’ X is A’

with B’(v) = Sup ml(A’(u), f (A(u) , B(v ) ) ) , can be reformulated by means of

the so-called Forward Reasoning Functions (FR-Functions)* M, as I (

where

I stands for an implication function of some multiple-valued system of logic underlying Fuzzy Set Theory and ml is its associate Modus Ponens generating function. 2-3

In this way, it is clear that the inference process is independent from the particular possibility distribution involved. It only employs two compatibility functions: the first one between the facts which appear in the premise of the rule and the observed facts and, the second one between the fact that appears in the conclusion of the rule and the inference fact.

It is in this context that the compatibility functions can be considered as fuzzy truth values, labels or qualifications, playing the same role as the values true and false play in the Classical Logic, where the meaning of propositions is nothing but its truth value,

In order to work with a system using these kind of labels, it would be convenient and necessary to consider a restricted family having the following desirable properties:

(a) Easy parametric representation. (b) Easy semantic interpretation. (c) To allow a gradation in the family according to the modifications per-

(d) To be closed under inference processes (FR-functions), and also under formed by each label.

some suitable and meaningful operations between them.

Furthermore, looking closely at the FR-functions associated to R-irnplica- tions defined in this introduction, it is worth noting that they are actually “consequence operators” in the sense of Tarski, that is, they fulfil the follow- ing conditions2

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FUZZY VALUES 20 1

(a) Ml(h) L h for any h E [0, 1]1°+11 (b) If h 2 h’ then M l ( h ) 2 M I ( h ’ ) (c) M I 0 MI = M / .

Taking into account this observation (mainly condition “c”), it becomes natural to study the closure system associated with a given M I in order to characterize its elements that, in this context, can be understood as the fuzzy truth labels that are invariant under inference.

The next section is devoted to this subject.

11. FR-FUNCTIONS CLOSURE SYSTEMS

In the sequel only R-implications will be considered, that is, those implica- tion functions defined by

R d x . y) = Sup{aJT(a , x) 5 y}.

T being a continuous t -no rn~ ,~ so that the Modus Ponens generating function is the t-norm T, and therefore, the FR-functions considered looks like

where h#(y) =Sup { h ( x ) } and h*(y) = Sup T ( h ( x ) , l ( x , y)). t 5 Y r>i

For T ( x , y ) = Min(x, y)

1. i f x s y

y, otherwise, RdX, Y ) =

MI(h)(y) = h#(y) v (y A H?), where H,. = Sup{h(x), x E ( y , I]} and, if h( I ) = I then MI(h)(y) = hx(y) A y.

For T ( x , y ) = x - y

1, i f x s y

otherwise, Rf(X,Y) =

M~(h)(y) = h#(y) v ( y . P,), where P,. = Sup{h(x)/x, x E (y. I]}.

M~(h)(y) = h#(y) v ( y + L,), where L,. = Sup{h(x) - x, x E (y . I]} . For T ( x , y ) = Max(x + y - l ,O) , R A x , y ) = Min(1 - x + y, I ) and,

The following definition plays a basic role in the characterization of the

Definition 2.1. A fuzzy label h is regular with respect to a R-implication I, labels invariant under a given FR-function.

if

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202 GODO, JACAS, AND VALVERDE

I ( x , y) 5 1 0 ( h x h) (x , y ) for any (x, y ) E [O, I ] x [0, 11.

It is easy to show that if h is a regular label with respect to the R-implica-

The following theorem characterizes the invariant labels Theorem 2. A fuzzy label h is invariant under a FR-function M I if, and

only if it is a regular element of I. In order to select a set of invariant labels suitable for applications, through-

out the reset of this article only labels that preserve the boolean values, that is, h(0) = 0 and h(1) = 1, will be considered, with the exception of the label h,, defined by h,,(x) = 1 for any x E X. This set will be denoted by 'If.

The structure of the closure system relative to the set 'If associated to a FR-function M I , is given by the next theorem.

Theorem 2.2 The closure set of a given FR-function M I is closed under arbitrary suprema and arbitrary infima and has a maximum element h,,, and a minimum element j.

tion I, then h is an increasing function.

That is, the closure set is a complete lattice under max-min operations. It is immediate to prove that for any increasing fuzzy label h

and therefore, if h is invariant

Under these assumptions, from Theorem 2.2 it follows Proposition 2.1 If I is the Lukasiewicz implication I ( x , y ) = Min(1 -

x + y, 1)) then M , V ) n 'V = ci, h,}. Proposition 2.2. If I is the R-implication associated to the 1-norm product

( I ( x , y ) = I if x 5 y, I ( x , y) = y / x otherwise), then h E M1(V) if and only if h ( x ) = xg(x) where g(x) is any decreasing function such that g(1) = 1 and I 5

Proposition 2.3. If I is the R-implication associated to the 1-norm Min then h E M1W) if and only if h ( x ) 2 x.

So, if we only consider the R-implications associated to the f-norms prod- uct and minimum, the parametrized set of labels % = 'Vll U {h,} where 'Ifll, {hll, p E [0, 11) (Fig. 1) defined by:

0 i f x = O

1 otherwise - i f x s p i f p # 0, ho(x) =

1 otherwise

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FUZZY VALUES 203

Figure 1. Representation of parametrized labels h,.

fulfils the conditions a, b, c considered in the introduction. Its elements are invariant under inference. In the case of the r-norm minimum, they are invari- ant since h, 2 j and, in the product’s case, the labels h, are the images by the associated FR-function M I of the “boolean labels” b,, defined by

b,(x) = 0 if x < p .

b,(x) = I otherwise.

In the next section, it is shown that the family 93 is closed under a suitable conjunction. These labels when applied over a fuzzy set A , perform the trans- formation shown in Figure 2, where A” is less restrictive than A.

Table I gives an example of the finite set of labels suitable for applications. The MILORD System’ also uses this set of labels but with different associated linguistic terms.

111. CONJUNCTION BETWEEN LABELS

In this section we introduce operations between fuzzy labels in order to apply the FR-functions when the conditional statement has compound prem- ises. The basic idea we have followed in order to define such connectives is that they must be coherent with the inference process.

A’’ = h

0 1n Figure 2. Transformation of a fuzzy set by a label h,.

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204 GODO, JACAS, AND VALVERDE

Table I. A possible set of labels.

~~

hi : TRUE h.65: ALMOST TRUE h.1: QUITE TRUE h.05: SLIGHTLY TRUE h,,: UNDECIDED

Let us consider a inference schema like:

conditional statement with compound premises in an

If (X is A ) and ( Y is B ) then ( Z is C ) ( X is A ) is T I

( Y is B ) is T~

( Z is C) is ~3

We are looking for a label T for “ ( X is A ) and (Y is B)” from “(X is A ) is T ~ ”

and “( Y is B) is T ~ , ” in such a way that schema ( I ) would be equivalent to

If (X is A) and ( Y is B) then ( Z is C )

( Z is C) is ~3

[(X is A ) and ( Y is B ) is 7 (1)

Considering that the possibility distributions of the compound premise and the one associated to the observed facts are represented by

respectively, the Compositional Rule of Inference (CRI) leads to

So, defining a conjunction operation between T~ and 72 as

we have 73 = M/(T, AT Q) , that guarantees a correct behavior of the conjunc- tion operator with respect to the inference process.

This definition generalizes the definition of the conjunction operator intro- duced by Baldwin in Ref. I arising from a different point of view.

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FUZZY VALUES 205

The behavior of this operation can be summarized in the following prop- erties:

0 The operation is commutative and associative. 0 T I AT 72 2 Max(.rl, T?) , that is, the label obtained in a conjunction is

always less restrictive than any of its components. In particular, if T = Min, T I AT 7 2 = Max(q, 72).

j A T j = j .

Finally, let us observe that the family of fuzzy labels considered in the previous section is closed under this operation for the t-norms T = Prod and T = Min and, h, AT h b = hT((,.b).

IV. NEGATION

Three kinds of negations are considered. Case 1. If we have a rule like “If (X is A ) then ( Y is B),” and we get

information about the complementary of A , that is, “(Xis A ) is 7,’’ we can infer a qualification of the consequence as usual, by means of the CRI, that is,

So, we can define the negation of 7, n1(7) as T 0 n (see Fig. 3), where n is a

This negation clearly satisfies strong negation in [ O , 11 used to define the complementation.

Note. From this negation and the conjunction operation defined in Section 111, a disjunction can be defined by duality:

being S the dual 1-conorm of T by n.

1 - / 1

Figure 3. Modification of a label h, by the negation n l .

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206 GODO, JACAS. AND VALVERDE

P 1 - P

Figure 4. Modification of a label h, by the negation n2.

Case 2. Given a proposition “ ( X is A ) is 7,” a second kind of negation of the label T, n2 can be defined by ~ z ( T ) = n 0 T 0 n (see Fig. 4), that is, n2(7) is the label that applied on A, gives T 0 denotes the standard complementation of the fuzzy sets A and 7 0 A respectively).

as result (A and T 0

This definition makes proposition

“not [(X is A ) is 71’’ and proposition “(X is A) is n2(7)” equivalent.

This negation satisfies the properties:

As in Case 1, a disjunction operation can be defined by duality from the negation:

being S the t-conorm dual of T by n.

complementation n3(7) = n 0 7 (see Fig. 5) .

n3(7)” are equivalent.

Case 3. The third kind of negation (n3) we consider is defined by the usual

Under this negation the propositions “not [ ( X is A ) is 71’’ and “ ( X is A ) is

Figure 5. Modification of a label h, by the negation n3.

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FUZZY VALUES 207

The following properties are fulfilled by n3:

As in the previous cases, a disjunction operator can be defined by duality from the negation and the conjunction operator:

where, as usual, S stands for the dual t-conorm of T by n.

V. COMPOSITION AND COMPATIBILITY

The notion of compatibility between two possibility distributions, can be easily extended to fuzzy labels:

Definition 5.1. The compatibility 712 of “(X is A ) is 71” given “(X is A ) is T ~ ” is the compatibility of “X is A l ” given “X is A2,” where A l = T~ 0 A and A2 = 7 2 0 A , that is:

On the other hand, the qualification of an already qualified proposition is defined by the composition of the qualifying labels in the usual functional sense:

Definition 5.2. The composition of T~ and 7 2 is 7 = T~ 0 7 2 , and the proposi- tions “((X is A) is T ~ ) is 72” and “(X is A ) is 7’’ are equivalent.

These two operations, compatibility and compositions, are strongly re- lated. If 73 = 7 2 0 71. then the compatibility of T~ given 7 3 is 7 2 . Conversely, if the compatibility of T~ given 73 is r2, then 73 = 7 2 0 T~ when q and T? are bijections and, in general, 73 2 7 2 0 TI.

VI. QUALIFIED INFERENCES AND QUALIFIED RULES

As we noticed using FR-functions associated with a R-implication, from a softened hypothesis, a softened thesis is obtained, but this last output (never more restrictive thanj2) remains stable and a chaining of inferences and there- fore, the unrestricted level is never reached. However, in some cases like the Sorites’ paradox, it seems desirable that the iteration of the inference process, with the same or different rules, should lead to an unrestricted output.

In this section, we show that this behavior can be reached by qualifying the inference process itself, or qualifying the conditional statement.

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208 GODO, JACAS, AND VALVERDE

Let us notice that the stability of inference output obtained in the preced- ing section formulation is coherent with this new approach. In the former case, the fact that the inference process or the conditional statement were not quali- fied, could be interpreted now as being qualified by the identity function j , which usually stands for the fuzzy truth value true.

Let us consider the well known schema of inference

I fXisA then YisB

( Y is B ) is T

X is A’

where T = M / ( p A A , ) .

To qualify the inference process with a label p, means to infer T * = p 0

(PAA,) . It can be checked that, if p > j then the compatibilities obtained by iterating the application of this rule lead to the undecided level. On the con- trary, i f j > p, T * is more restrictive than T .

On the other hand, when a conditional rule is qualified we have a schema like:

[If (X is A) then (Y is B ) ] is p (X is A ) is T

( Y is B) is T * *

considering that p qualifies the possibility distribution associated to the condi- tional statement.

In this case, applying the CRI, we obtain

In particular, when the compatibility function T is increasing then,

that leads us to recover the property in [0, 1 1 for which a conjunction operator (a t-norm) is also a Modus Ponens generating function when the implication function is a R-implication generated by the t-norm used as conjunction opera- tor. As a consequence QM/(T, p ) will be always less restrictive that T .

Finally, it is interesting to note that when the family of labels 93 of Section I1 is considered, and the r-norm product is chosen, to qualify the inference process is equivalent to qualify the conditional statement. In this case, it is easy to show that the following property holds:

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FUZZY VALUES 209

ha 0 hb = ha A h b = hk, where k = a.b, for all a,b E [0, I].

Example. The Sorites paradox arises from the consideration of a rule like: “If a man with n hairs is bald, then another man with n + I hairs is also bald.” Assuming as true that a man with no hairs is certainly bald, it can be concluded by induction that every man is bald.

In order to avoid this paradox, let us reformulate the conditional statement in the following way: “If a man X is bald, and another one Y has more or less the same number of hairs as X, then Y is also bald.” However, this rule is not absolutely true, since “baldness,” as an external characteristic of a man, is not mainly a matter of number of hairs but of the percentage of head surface covered by hair.

If we take the following notation for the linguistic values and variables:

H ( X ) = percentage of surface covered by hair of X, 6 ( X , Y ) = H ( Y ) - H ( X ) ,

A = Bald,

B = more or less null

the preceding statement can be represented by:

R: [IF “ X is A ” and “6(X , Y ) is B” THEN “ Y is A”] is

where the fuzzy subsets A and B can be represented by the possibility distribu- tions shown in Figure 6, and 71 is a fuzzy label “hs” which denotes a degree of truthfulness for the rule.

0 40 ”,

’1

-50 -20 0 20 50 ’Z

Figure 6. Possibility distributions for fuzzy sets A and B .

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210 GODO, JACAS, AND VALVERDE

1 0

Figure 7. Possibility distribution for fuzzy set C.

Suppose that there is evidence for the fact:

(1) John is bald. (2) Peter has a little bit more hair than John. (3) James has approximately as many surface covered by hair as Peter.

All these sentences can be reformulated as follows:

F I : John is A. Fz: 6 (John, Peter) is C. F3: 6 (Peter, James) is B.

where C can be represented by the fuzzy subset shown in Figure 7. From R, F I , F2, and F3 the following inference processes can be done:

( 1 ) R: [IF “X is A” and “ 6 ( X , Y ) is B” THEN “ Y is A”] is rl = hx FI: John is A F2: 6 (John, Peter) is C

CI: (Peter is A ) is 7 2

(2) R: [IF “X is A” and “ 6 ( X , Y ) is B” THEN “ Y is A”] is 71 = hl

CI: (Peter is a ) is 7 2

F2: 6 (Peter, James) is B

c2: (James is A ) is 7 3

where 7 3 = Q M , ( q A PEE, 71).

With the above representations, it can be verified that

PAA = hl, PEE = h l , and if PBC = h,? (Fig. 8) then 72 = h,, where c = T(&, k) , (7, = hk) 7 3 = hd, where d = T(c , k ) = T(T(P2 , k ) , k ) (Fig. 9)

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FUZZY VALUES 21 1

0 P,=2/3 1

Figure 8. Compatibility function of fuzzy set B, given C.

So that the truth qualifications of James’ baldness is less restrictive than the one obtained with unqualified rules, In this way the paradox is avoided, because after some iteration of the inference, as unrestricted output is ob- tained.

VII. CONCLUDING REMARKS

Throughout this article we have been concerned with the selection of a set of labels for Fuzzy Logic according to the following features:

0 It is closed under inferences (using t-norms min and Prod). 0 Its elements are the images of the boolean labels (i.e., h[0,1] C (0,l) and

h increasing) by the product inference operator. 0 Its regular behavior when dealing with qualified rules and chaining.

Those labels are obtained from the functions h, in the unit interval, defined as:

1, otherwise,

and the changes they perform on the possibility distributions affect specially then support and zero zone.

0 d c P , 1

Figure 9. Fuzzy truth-labels hd, h,, and hB2.

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212 GODO, JACAS, AND VALVERDE

References

1. J.F. Baldwin, “A new approach to approximate reasoning using fuzzy logic,” Fuzzy Sets and Systems, 1, 309-325 (1979).

2. E. Trillas and L. Valverde, “On inference in fuzzy logic,” Proc. Second I .F.S.A. Congress, Tokio, 1987, pp. 294-297.

3. L. Valverde and E. Trillas, “On modus ponens in fuzzy logic.” Proc. fnt. Symp, on Multiple-Valued Logic, Kingston, Ontario, 1985, pp. 294-301.

4. C. A h a , E. Trillas, and L. Valverde, “On some logical connectives for fuzzy set theory,” J. Math. Anal. and Appl. 93, 15-26 (1983).

5. L. Godo, R. L6pez de MBntaras, C. Sierra, and A. Verdaguer, Managing Linguis- tically Expressed Uncertainty in Milord. Application to Medical Diagnosis, Re- search Report 87/2, C.E.A.B./C.S.I.C., 1987.