robustness of fuzzy reasoning and δ-equalities of fuzzy sets

738 IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 9, NO. 5, OCTOBER 2001

Robustness of Fuzzy Reasoning and�-Equalities of Fuzzy SetsKai-Yuan Cai

Abstract—Fuzzy reasoning methods or approximate reasoningmethods are extensively used in intelligent systems and fuzzy con-trol. In this paper we discuss how errors in premises affect con-clusions in fuzzy reasoning, that is, we discuss robustness of fuzzyreasoning. After reviewing the previous work, we present robust-ness results for various implication operators and inference rules.All the robustness results are formulated in terms of -equalitiesof fuzzy sets. Two fuzzy sets are said to be-equal if they are equalto an extent of .

Index Terms—Approximate reasoning, -equality, fuzzy rea-soning, fuzzy relation, fuzzy rule, generalized (fuzzy) hypotheticalsyllogism, generalized (fuzzy) modus ponens, generalized (fuzzy)modus tollens, inference rule, robustness of fuzzy reasoning.

I. INTRODUCTION

Fuzzy reasoning, or approximate reasoning, has been an ac-tive topic in the fuzzy community since the inception of Zadeh’spioneering work [1]. Various methods have been proposedfor fuzzy reasoning, including compositional rule of infer-ence (CRI) methods [1], possibilistic methods [2], evidentialmethods [3], [4], interpolative methods [5], truth-value methods[6], interval-value methods [7], and triple implication methods[8], among others [9]. For each method, certain implicationoperator and connectives are used to build a link between fuzzyantecedents and fuzzy consequences. Consequently, numerousimplication operators and connectives are practiced in fuzzyreasoning literature [10], [11]. Fuzzy reasoning techniqueshave been extensively applied in various areas and especiallyin fuzzy control [12], [13], where systematic formalisms areavailable to transform human expertise and subjectivity toquantitative terms.

Ideally, there should be one-to-one correspondence betweenhuman expertise and quantitative terms. In this way, fuzzy rea-soning in mathematically quantitative terms will exactly rep-resent or capture the essence of human fuzzy reasoning. Un-fortunately, the one-to-one correspondence can seldom be ob-served. For example, the fuzzy observation “about 5” in real-world sense can be represented in terms of a triangular fuzzynumber or of a Gaussian fuzzy number in a mathematicallyquantitative formalism, but we are really not sure which is betterto capture the essence of “about 5.” Then, a question arises: Isthe difference between the two fuzzy numbers in mathemati-cally quantitative formalism important for fuzzy reasoning? or:Is a mathematical fuzzy reasoning scheme robust or perturba-

Manuscript received March 27, 2000; revised September 10, 2000 and De-cember 4, 2000. This work was supported by the National Outstanding YouthFoundation of China and the Key Project of China.

The author is with the Department of Automatic Control, Beijing Uni-versity of Aeronautics and Astronautics, Beijing 100083, China (e-mail:[email protected]).

Publisher Item Identifier S 1063-6706(01)02824-7.

tion-resistant against the deviation of human expertise from itscorresponding mathematically quantitative representations?

In this paper we focus on the CRI methods and study robust-ness of fuzzy reasoning in terms of-equalities of fuzzy sets.A -equality of two fuzzy sets means that the two fuzzy setsare equal to each other to an extent of[14, Definition 2.3 ].First, we review some previous work. It is interesting to find thatalthough different authors approached this topic from differentperspectives, their work overlaps to a certain extent. This will beexplained in Section II. In Section III, we present a number oflemmas to be used in the subsequent sections. In Section IV, wediscuss -equalities for various implication operators. In Sec-tion V, we discuss -equalities for -norm and fuzzy relations,whereas in Section VI we discuss-equalities for generalizedmodus pollens and generalized modus tollens. We devote Sec-tion VII to -equalities in generalized hypothetical syllogism.Finally, we treat -equalities in inference with multiple rules inSection VIII and contain concluding remarks in Section IX.

II. PREVIOUS WORK

A. Pappis’ Work

With an attempt to show that “precise membership valuesshould normally be of no practical significance,” Pappis intro-duced the following definition [15] (but reformulated here).

Definition 2.1: Let be a universe of discourse. Letandbe two fuzzy sets on , and and their mem-

bership functions, respectively. Then and are said to beapproximately equal, denoted by , if

The number is said to be a proximity measure ofand .Theorem 2.1:Let and be fuzzy sets on , and andfuzzy relations from to . Then

a) impliesb) implies

where is the max–min composition.We see that this theorem actually addresses the perturbation

or robustness problem of fuzzy reasoning. For example, in gen-eralized modus ponens, can represent the implication oper-ator. Of course, various implication operators and compositionoperators can be used in generalized modus pollens. See Sec-tion VI for more details.

B. Hong and Hwang’s Work

Hong and Hwang reformulated Pappis’ definition in terms ofsimilarity measure [16].

Definition 2.2: Let be a universe of discourse. Letandbe two fuzzy sets on , and and their membership

1063–6706/01$10.00 © 2001 IEEE

IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 9, NO. 5, OCTOBER 2001 739

functions, respectively. Then and are said to be -similar,denoted by , if ,

.Hong and Hwang then generalized Theorem 2.1 to be the

following.Theorem 2.2:Let and be fuzzy sets on , and andfuzzy relations from to . If and , then

.Evidently, the theorem is highly related to generalized modus

ponens, as explicitly stated in Hong and Hwang’s paper [16].

C. Cai’s Work

A problem with Pappis’ and Hong and Hwang’s work wasthat they employed a fixed numberor . This constraint wasremoved by Cai [14]. In order to comply with human intuitionmore directly, Cai used the term “-equal” as follows.

Definition 2.3: Let be a universe of discourse. Letandbe two fuzzy sets on , and and their membershipfunctions, respectively. Then and are said to be -equal,denoted by , if

In this way, we say and construct an -equality.In Cai’s paper [14], he stated:

“The notion of -equality is important both in fuzzy sta-tistics and fuzzy reasoning. In fuzzy statistics we need toestimate a membership function and errors are almost al-ways attached to the estimates. Although one may claimthat the estimation errors are not important in some cases,this does not imply that the estimation errors can be disre-garded in all circumstances. In fuzzy reasoning, we needto account for the effects of errors of a fuzzy premise onthe fuzzy consequence .”

He further introduced the notion of proximity amplificationfactor, proximity reduction factor, equality amplification factor,and equality reduction factor, for an algebraic operator, withthe understanding that

“In fuzzy reasoning we may be concerned with theerror propagation of a membership function. The processof fuzzy reasoning is essentially the process of operationwith membership functions. So the following definition[omitted here] may be useful when we consider whethererrors of membership functions are amplified or reducedby fuzzy reasoning.”

Cai then presented various results of-equalities with respectto algebraic operators (e.g., union, intersection, complement),fuzzy relations, extension principle and triangular norms. Someof them will be revisited in Section III. A particularly interestingresult is as follows.

Theorem 2.3:Let , , and be universes of discourse,the collection of all fuzzy sets defined on , andthe collection of all fuzzy sets defined on . Let

and , i.e., and

are fuzzy relations. Further, let be the composition ofand , and the composition of and

Suppose , . Then,.

We see that Theorem 2.3 generalizes Theorems 2.1 and 2.2in two dimensions. First, different numbers and are con-cerned. Second, the composition of a fuzzy set and a fuzzy re-lation is generalized to the composition of two fuzzy relations.Here we note that a fuzzy set on can be treated as a spe-cial fuzzy relation from to or from to with beinga singleton universe. Following the proof of Theorem 2.3, othercomposition forms can be easily treated.

A highly related work is due to Ying who considered the per-turbation problem of fuzzy reasoning [17]. On the other hand,Pappis and Karacapilidis compared a set of similarity measuresof fuzzy sets that are related to the notion of-equalities of fuzzyset [18].

III. L EMMAS

From this section to the end of the paper, we use Defini-tion 2.3 and study the robustness problem of fuzzy reasoningin terms of -equalities. In our previous work [14],-equali-ties with respect to various algebraic operators are confined toa single universe. For example, let and be fuzzysets defined on and , , then

. Note , are also fuzzy setson . In this section, we assume that two universes are involvedand develop the corresponding-equalities of fuzzy relations. If

and are fuzzy sets defined on and , respectively, thendefines a fuzzy relation from to . We only discuss a

few algebraic operators that will be used in the reminder of thepaper, but obviously, other algebraic operators can be treated ina similar manner.

Lemma 3.1:Let

i.e., is the Lukasiewicz conjunction. Then

1)2)3)4)5)6)7)

Proof: Trivial. Q.E.D.


Lemma 3.2:Let be bounded, real valued function on aset . Then

Proof: See, e.g., [16]. Q.E.D.Lemma 3.3:Let be fuzzy sets defined on, and

fuzzy sets defined on . Suppose , . Letrepresent the relational union1 of and , and

the relational union of and

Then

Proof: From Lemma 3.2, we have

Then

Q.E.D.Lemma 3.4:Suppose . Let represent the com-

plement of , and the complement of

Then .Proof: Trivial. Q.E.D.

Lemma 3.5:Let and be fuzzy sets defined on, andand fuzzy sets defined on . Suppose ,

. Let represent the relational intersection2 ofand , and the relational intersection of and , i.e.,

Then .Proof: Similar to that of Lemma 3.3. Q.E.D.

Lemma 3.6:Let and be fuzzy sets defined on, andand fuzzy sets defined on . Suppose ,

. Let represent the relational product of and ,and the relational product of and , i.e.,

1We use the term “relational union” because it defines a fuzzy relation.2This is actually the Cartesian product. However, we use the term “relational

intersection” to emphasize that it is an extension of standard intersection of twofuzzy sets.

Then


Further

Q.E.D.Lemma 3.7:Let be a universe, and , , , , ,

and be fuzzy sets defined on. Suppose

and

(i.e., ; ). Then,.

Proof: See [14]. Q.E.D.We need another result for the proof of Proposition 4.7 in

Section IV.Lemma 3.8:Suppose is a real-valued function defined on

a set , and a constant independent of. Then there alwaysholds

Proof: See [19, p. 60]. Q.E.D.

IV. I MPLICATION OPERATORS

Fuzzy rules are extensively used in intelligent systems andfuzzy control. They are often in form ofIF is , THEN is

, where and are linguistic variables, and and arefuzzy sets defined on universesand , respectively. In orderto conduct fuzzy reasoning mathematically, the fuzzy rules needto be transformed into quantitative terms, and this often hap-pens by applying various implication operators to fuzzy rules togenerate fuzzy relations. However, different implication opera-tors may lead to different fuzzy relations. In this section we areconcerned with robustness of several typical implication oper-ators frequently used in fuzzy control [12], [13]. In general, let

denote the fuzzy relation from to defined by animplication operator in the fuzzy ruleIF is THEN is .


Proposition 4.1: Suppose Dienes–Rescher implication oper-ator is applied to the fuzzy ruleIF is THEN is , that is

or

Further, let and be two fuzzy sets with

Then

Proof: From Lemma 3.4 we have . Then,Lemma 3.3 leads to . Q.E.D.

Proposition 4.2: Suppose Lukasiewicz implication operatoris applied to the fuzzy ruleIF is , THEN is , that is3


Then

Proof: By using Lemma 3.2, we have

Further, we note

Therefore

Q.E.D.Proposition 4.3: Suppose Zadeh implication operator is ap-

plied to the fuzzy ruleIF is , THEN is , that is

3Lukasiewicz implication operator is often defined by� (x; y) =min(1; 1 � � (x) + � (y)) in the literature [20, Ch. 11]. The propositionis still valid under this definition. In order to coincide with the Lukasiewiczconjunction used in Lemma 3.1, we use the� operator throughout this paper.For the same reason, we do not use other notation such as “! .”


Then

Proof: This is obvious from Lemmas 3.4 and 3.5. Q.E.D.Proposition 4.4: Suppose Godel implication operator is ap-

plied to the fuzzy ruleIF is , THEN is , that is

if

otherwise.


Then

where

Proof: Note

For , we consider three cases:Case 1: : Then

.Case 2: . Then

If , we have.

If , we have.

If , we have.

Case 3: : Then

If , we have.

If , we have.

If , we have.

In summary

For , we still consider threecases.

Case 1: . Then.

Case 2: . Then

If , we have.


If , we have.

If , we have.

Case 3: . Then

If , we have.

If , we have.

If , we have.

In summary

Therefore

By the symmetry of and , we can also obtain

In this way

Q.E.D.From the above proposition, we see thatis not explicitly

related to , , , or , but is explicitly related toand . This suggests that the Godel implication op-

erator be very perturbation sensitive as a result of its disconti-nuity. We can imagine that this conclusion is also valid for otherimplication operators with discontinuity. Let us consider the fol-lowing example.

Example 4.1:Let

We see

Further

Actually, it is easy to verify

Thus

Proposition 4.5: Suppose Mamdani min implication oper-ator is applied to the fuzzy ruleIF is , THEN is , thatis


Then

Proof: This is just Lemma 3.5. Q.E.D.Proposition 4.6: Suppose Mamdani product implication op-

erator is applied to the fuzzy ruleIF is THEN is , that is


Then

Proof: This is just Lemma 3.6. Q.E.D.


Proposition 4.7: Suppose the following implication operatoris applied to the fuzzy ruleIF is , THEN is , that is

if

otherwise.


Then

where

Proof: Note

For , we consider three cases.Case 1: : Then

.Case 2: : Then

If , we have.

If , we have

If , we have

Case 3: : Then

If , we have.

If , we have

If , we have

In summary

For , we still consider threecases.

Case 1: . Then.

Case 2: . Then

If , we have

If , we have

If , we have.

Case 3: . Then

If , we have.

If , we have

If , we have

In summary

Therefore

Similarly, following


we have

In this way

From Lemma 3.9, we arrive at

Q.E.D.Example 4.2:Consider Example 4.1 again, but the implica-

tion operator is replaced by that used in Proposition 4.7. We have

Note

We see that the implication operator is less robust thanDienes–Rescher, Lukasiewicz, Zadeh, Mamdani min, andMamdani product implication operators.

Proposition 4.8: Suppose Reichenbach implication operatoris applied to the fuzzy ruleIF is , THEN is , that is


Then

Proof: This is because

Q.E.D.

V. -NORM AND FUZZY RELATIONS

-norm is a generalized union of two fuzzy sets or two mem-bership functions [13, Sec. 3.2].

Definition 5.1: Any function thatsatisfies Axioms s1–s4 is called an-norm.

Axiom s1: (boundarycondition).

Axiom s2: (commutative condition).Axiom s3: If and , then

(nondecreasing condition).Axiom s4: (associative

condition).Lemma 5.1:Suppose is an -norm. Then

where

ififotherwise.

Proof: Trivial. Q.E.D.Proposition 5.1: Let and be fuzzy sets defined on,

and and fuzzy sets defined on . Suppose


Let be -norm of and , -norm of and

Then

where

Proof: From Lemma 5.1 we have

From Lemma 3.3, we obtain

Further, we note

if

if

if

if

otherwise

if

if

if

if

otherwise.

Consider the case . We have. If , then

. If, then . Therefore

or

On the other hand,

if

if

otherwise

By using Lemma 3.7, we immediately complete theproof. Q.E.D.

Now let us consider a variant of Theorem 2.3.Proposition 5.2: If the sup-min composition in Theorem 2.3

is replaced by the sup-product composition, that is

Then


Since

we arrive at

Q.E.D.Since , from the above proposition,

we see that the sup-min composition of fuzzy relations is morerobust than the sup-product composition of fuzzy relations.

Proposition 5.3: If the min composition in Theorem 2.3 isreplaced by the Lukasiewicz composition, that is

Then

Proof: Similar to that of Proposition 5.2. Refer to Propo-sition 4.2. Q.E.D.

VI. GENERALIZED MODUS PONENS AND GENERALIZED

MODUS TOLLENS

A. Generalized Modus Ponens

Generalized modus ponens is a basic inference rule in fuzzyreasoning. It states

Premise 1: isPremise 2: is isConclusion: is .


where and are linguistic variables, and are fuzzy setsdefined on , and are fuzzy sets defined on. More specif-ically, generalized modus ponens concludes

where stands for an -norm, for a fuzzy relationfrom to defined by imposing an implication operator on

and . In this subsection we discuss robustness of general-ized modus ponens under various specific-conjunctions andimplication operators.

Proposition 6.1: Suppose the min conjunction and theDines–Rescher implication operator are used in generalizedmodus ponens; that is

Further

Then


Thus

Actually, we can follow a simpler manner to draw the conclu-sion. From Proposition 4.1, we have

By using Theorem 2.3, we immediately arrive at

Q.E.D.Proposition 6.2: Suppose the min conjunction and the Luk-

siewicz implication operator are used in generalized modus po-nens; that is

Further

Then

Proof: Trivial from Proposition 4.1 and Theorem2.3. Q.E.D.

Proposition 6.3: Suppose the min conjunction and the Zadehimplication operator are used in generalized modus ponens; thatis

Further

Then


Proposition 6.4: Suppose the min conjunction and the Mam-dani min implication operator are used in generalized modusponens; that is

Further

Then


Proposition 6.5: Suppose the min conjunction and the Mam-dani product implication operator are used in generalized modusponens; that is

Further

Then


Proposition 6.6: Suppose the min conjunction and the Re-ichenbach implication operator are used in generalized modusponens; that is


TABLE IROBUSTNESSRESULTS OFGENERALIZED MODUS PONENSUNDER VARIOUS CONJUNCTIONS ANDIMPLICATION OPERATORS

Further

Then


We can also discuss the robustness of generalized modus po-nens if the product conjunction or the Lukasiewicz conjunctionis used in place of the min conjunction. Similar results can beobtained for the -equalities of as long as we pay attention toPropositions 5.2 and 5.3. In general, let

Table I summarizes the relationships among, , , and .

B. Generalized Modus Tollens

Generalized modus tollens is another basic inference rule infuzzy reasoning. It states

Premise 1: isPremise 2: is isConclusion: is .

Or

Obviously, the -equalities in generalized modus ponens canapply to generalized modus tollens, as long as we note thatand here are in place of and in generalized modus po-nens, respectively. Let

Then Table I is still valid for generalized modus tollens.

VII. GENERALIZED HYPOTHETICAL SYLLOGISM

As generalized modus ponens and generalized modus tollens,generalized hypothetical syllogism is also a basic inference rulein fuzzy reasoning. It states

Premise 1: is isPremise 2: is isConclusion: is is

where , , and are linguistic variables, is fuzzy sets de-fined on , , and are fuzzy sets defined on, and and

are fuzzy sets defined on . Generalized hypothetical syl-logism constructs a fuzzy relation from to . In the contextof membership functions, the resulting fuzzy relation fromto

, denoted by , is determined by

wheredenotes an-norm;implication relation from to defined byand ;implication relation from to defined byand .

We see that is just a composition of two fuzzy relations andthus we can follow Theorem 2.3, Propositions 5.2 and 5.3 to dis-cuss robustness of. Of course, the robustness results dependon the choices of-norms and implication operators.

As an example, let us consider the min conjunction and theDienes–Rescher implication operator. Then

Suppose

Let denotes the fuzzy relation from to correspondingto and . From Proposition 4.1 we have

By using Theorem 2.3, we immediately arrive at

In general, let


TABLE IIROBUSTNESSRESULTS OFGENERALIZED HYPOTHETICAL SYLLOGISM UNDER VARIOUS CONJUNCTIONS ANDIMPLICATION OPERATORS

varies with the choices of conjunctions and implication oper-ators. Table II summarizes the corresponding results.

VIII. I NFERENCEWITH MULTIPLE RULES

A fuzzy rule base normally contains multiple or numerousrules, as in the case of fuzzy control. Inference with multiplerules states

Premise 1:

is is is

Premise 2:

(r1): is is is

is

(r2): is is is

is...

(rm): is is is

is

Conclusion: is

where and are linguistic variables,and are fuzzy sets defined on ,

, and are fuzzy sets definedon . We can treat inference with multiple rules as a general-ized form of generalized modus ponens. However, since mul-tiple rules are involved, different inference procedures can befollowed.

A. Combination Based Inference

In combination based inference, three steps are followed.

1) Each rule is transformed into a fuzzy relation fromto .

2) All the resulting fuzzy relations are combined into asingle (overall) fuzzy relation.

3) The resulting overall fuzzy relation is fired to generate anoutput (conclusion).

More specifically, let

where stands for an-norm, an implication re-lation defined by and . Then can becombined in Mamdani (union) manner or in Godel (intersec-tion) manner.

In Mamdani (union) combination based inference, we have

where stands for an -norm.Suppose the min, Mamdani product and max are used for the

-norm, implication operator and-norm, respectively. Then

Let

From Lemma 3.5, Proposition 4.6, Lemma 3.3, and Theorem2.3, we have


If , then

Following the line of the above formulation, we can easily ob-tain the corresponding robustness results of Mamdani (union)combination based inference if other norms are implication op-erators are used.

In Godel (intersection) combination based inference, we have

Specifically, suppose the min and Mamdani product are used forthe -norm and implication, respectively. Then

Let

we obtain

If , then

We see that in this particular case Godel (intersection) combina-tion based inference is as robust as Mamdani (union) combina-tion based inference. This observation should be generally true

as long as the intersection is taken as min and the union as max,since the min operator is as robust as the max operator.

B. Individual-Rule Based Inference

Individual-rule based inference also follows three steps but ina different manner.

1) Each rule is transformed into a fuzzy relation fromto .

2) Each resulting fuzzy relation is fired to generate an outputindividually.

3) All the individual outputs are combined into an overalloutput.

In other words, generalized modus ponens is applied to Premise1 and each rule (say, ) and draw a conclusion (say, is )one by one. We have

where is the same as that in combination based inference.Then are combined into an overall conclu-sion . In union individual-rule based inference

Specifically, suppose the min, Mamdani product and max areused for the -norm, implication operator and-norm, respec-tively. Then

Let

Then

If , then


In intersection individual-rule based inference

Specifically, suppose the min and Mamdani product are used forthe -norm and implication operator, respectively. Then

Let

we have

If , then

We see that the union individual-rule based inference is as robustas the intersection individual-rule based inference, as long as theunion operator is taken as max and the intersection operator istaken as min.

IX. CONCLUSION

Up to this point we have discussed robustness of various oper-ators and inference rules in fuzzy reasoning in terms of-equal-ities of fuzzy sets. In comparison with previous work, our for-mulation is rather general and systematic. We discuss the ef-fect of the errors in premises on the errors in conclusion. Sup-pose are fuzzy sets describing premises,is the fuzzy set describing the corresponding conclusion, and

. In general, the errors in premises areamplified to the error in conclusion in the procedure of fuzzyreasoning. In other words, if

it is generally true that . This is an undesiredproperty and may be a drawback of existing fuzzy reasoningmethods.

Of course, we should note that the robustness results pre-sented in this paper are conservative in certain sense. We onlyobtain a value of that ensures , but we did not ob-tain the maximum of that ensures the corresponding-equalityholds. This is a problem left for further investigation. Anotherproblem we need to investigate in the future is that, given therequired -equality , how to determine the largestvalues of such that

This problem is somewhat related to the continuity problem offuzzy reasoning [21].

ACKNOWLEDGMENT

The comments of the anonymous reviewers have helped theauthor to improve the readability of the paper.

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robustness of fuzzy reasoning and δ-equalities of fuzzy sets

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