an approach to measure the robustness of fuzzy reasoning

21
An Approach to Measure the Robustness of Fuzzy Reasoning Yongming Li, 1,2,† Dechao Li, 1,‡ Witold Pedrycz, 3,4, * Jingjie Wu 1,§ 1 Institute of Fuzzy Systems, College of Mathematics and Information Sciences, Shaanxi Normal University, Xi’an, 710062, China 2 Department of Computer Science and Technology, Tianjing University, Tianjing, 300072, China 3 Department of Electrical and Computer Engineering, University of Alberta, Edmonton, T6G2V4, Canada 4 Systems Research Institute, Polish Academy of Sciences, Warsaw, Poland Fuzzy reasoning is intensively used in intelligent systems including fuzzy control, classifica- tion, expert systems, and networks to name a few dominant categories of such architectures. As being a fundamental construct permeating so many diverse areas, fuzzy reasoning was studied with respect to its fundamental properties such as robustness. The notion of robustness or sensitivity becomes of paramount importance by leading to a more comprehensive under- standing of the way in which reasoning processes are developed. In this study, we introduce and study properties of some measures of robustness (or sensitivity) of fuzzy connectives and implication operators and discuss their relationships with perturbation properties of fuzzy sets. The results produced here are compared and contrasted with the previous findings available in the literature. © 2005 Wiley Periodicals, Inc. 1. INTRODUCTION Fuzzy reasoning, or approximate reasoning, has been an active research pur- suit since the very inception of fuzzy sets. 1 Various methods of fuzzy reasoning (refer, e.g., to Refs. 2 and 3) have been proposed along with numerous alternative realizations of implication operators and logic connectives (that is, and and or operators). 4,5 The results of fuzzy reasoning are dependent on the choice of fuzzy sets of fuzzy antecedent and fuzzy consequences as well as fuzzy connectives and fuzzy implication operators that link fuzzy antecedents and fuzzy consequences. *Author to whom all correspondence should be addressed: e-mail: [email protected]. e-mail: [email protected]. e-mail: [email protected]. § e-mail: [email protected]. INTERNATIONAL JOURNAL OF INTELLIGENT SYSTEMS, VOL. 20, 393–413 (2005) © 2005 Wiley Periodicals, Inc. Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/ int.20072

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An Approach to Measure the Robustnessof Fuzzy ReasoningYongming Li,1,2,† Dechao Li,1,‡ Witold Pedrycz,3,4,* Jingjie Wu1,§

1Institute of Fuzzy Systems, College of Mathematics and InformationSciences, Shaanxi Normal University, Xi’an, 710062, China2Department of Computer Science and Technology, Tianjing University,Tianjing, 300072, China3Department of Electrical and Computer Engineering, University of Alberta,Edmonton, T6G2V4, Canada4Systems Research Institute, Polish Academy of Sciences, Warsaw, Poland

Fuzzy reasoning is intensively used in intelligent systems including fuzzy control, classifica-tion, expert systems, and networks to name a few dominant categories of such architectures.As being a fundamental construct permeating so many diverse areas, fuzzy reasoning wasstudied with respect to its fundamental properties such as robustness. The notion of robustnessor sensitivity becomes of paramount importance by leading to a more comprehensive under-standing of the way in which reasoning processes are developed. In this study, we introduceand study properties of some measures of robustness (or sensitivity) of fuzzy connectives andimplication operators and discuss their relationships with perturbation properties of fuzzy sets.The results produced here are compared and contrasted with the previous findings available inthe literature. © 2005 Wiley Periodicals, Inc.

1. INTRODUCTION

Fuzzy reasoning, or approximate reasoning, has been an active research pur-suit since the very inception of fuzzy sets.1 Various methods of fuzzy reasoning(refer, e.g., to Refs. 2 and 3) have been proposed along with numerous alternativerealizations of implication operators and logic connectives (that is, and and oroperators).4,5 The results of fuzzy reasoning are dependent on the choice of fuzzysets of fuzzy antecedent and fuzzy consequences as well as fuzzy connectives andfuzzy implication operators that link fuzzy antecedents and fuzzy consequences.

*Author to whom all correspondence should be addressed: e-mail: [email protected].†e-mail: [email protected].‡e-mail: [email protected].§e-mail: [email protected].

INTERNATIONAL JOURNAL OF INTELLIGENT SYSTEMS, VOL. 20, 393–413 (2005)© 2005 Wiley Periodicals, Inc. Published online in Wiley InterScience(www.interscience.wiley.com). • DOI 10.1002/int.20072

An interesting question arises: How much do the differences reported betweentwo fuzzy sets, fuzzy connectives, and fuzzy implication operators translate intovisible and meaningful differences in the schemes of fuzzy reasoning? Put itdifferently:

Is a mathematical fuzzy reasoning scheme robust or perturbation resistant againstthe deviation of human expertise from its corresponding mathematically quantita-tive representations? 6

To address this problem, Ying7 proposed the concepts of maximum and aver-age perturbations of fuzzy sets. Subsequently perturbation parameters of variousmethods of fuzzy reasoning could be estimated. Those perturbation parameterscould help formulate certain criteria useful in selecting a specific method of fuzzyreasoning. Cai8 used the notion of d equalities of fuzzy sets (being dual to themaximum perturbation of fuzzy sets of Ying’s) to study robustness of fuzzy rea-soning that led to some general results in case of fuzzy connectives, fuzzy impli-cation operators, and generalized modus ponens and generalized modus tollens.Some others approached this concept from a different perspective as presented inRefs. 9 and 10; refer also to some comments made in Ref. 6.

In the previous work, the authors directly used the changes of membershipfunctions of fuzzy sets to study the robustness of fuzzy reasoning. However, thebehavior of a fuzzy logic system is mainly determined by its internal structure—the fuzzy connectives and fuzzy implication operators. Having this in mind, weinvestigate the issues of the robustness of fuzzy logic connectives and implicationoperators. We develop some estimates of robustness of the main schemes of fuzzyreasoning (as will be shown, those are less conservative than those proposed byYing and Cai).

The material is organized in the following way. In Section 2, we study therobustness of fuzzy logic connectives and formulate some general results concern-ing robustness of t- and s-norms, and various types of implication operators (i.e.,R, S, and QL implication). In Section 3, we study the robustness of the key catego-ries of fuzzy inference systems. Conclusions are presented in Section 4. Through-out the study we use a standard notation encountered in fuzzy sets. In particularwe use sup (∨) to represent the supremum operation and inf (∧) is used to repre-sent the infimum operation.

2. THE ROBUSTNESS OF FUZZY LOGIC CONNECTIVES

Fuzzy reasoning relies heavily on the fuzzy logic connectives. Having this inmind, our study of robustness of the reasoning schemes is linked directly to theproperties of the logic connectives and their robustness. In what follows, the mem-bership grades of fuzzy sets (say A, B, etc.) will be denoted by small letters; say xassuming values in [0,1] will be used instead of the explicit notation x � A~u! with

394 LI ET AL.

u being treated as a certain element of the universe of discourse. Fuzzy logic con-nectives involve fuzzy conjunction, fuzzy disjunction, fuzzy complement, and fuzzyimplication. There are numerous ways for their implementation. In general, weuse t-norm to represent fuzzy conjunction, whereas s-norms are models of the fuzzydisjunction. Negation is used to represent fuzzy complement. Fuzzy implicationcomes with its realization known as R implications, S implications, and QL impli-cations, where all of those are the extensions of some well-known classical impli-cation operators.11 Let us recall that a binary operation f : @0,1#� @0,1#r @0,1# iscalled an R implication if there exists a t-norm t such that t~a, c! � b ? c �f ~a, b!, that is,

f ~a, b! � ∨$c � @0,1#6 t~a, c!� b%

for all a, b � @0,1# . Subsequently a binary operation f is called an S implication ifthere exists a certain s-norm s and negation n such that f ~a, b!� s~n~a!, b! for alla, b � @0,1# . A binary operation f is called a QL implication if there exists a t-normt, an s-norm s, and a negation n for which we have f ~a, b!� s~n~a!, t~a, b!! for alla, b � @0,1# . For more details refer to Refs. 2, 4, 5, and 11. A binary fuzzy connec-tive can be considered as a binary operator from [0,1] � [0,1] to [0,1], that is, afuzzy connective is a function f from [0,1] � [0,1] to [0,1]. If f is continuous, wecan consider the modulus of continuity of f as defined in Ref. 12. We extend thedefinition of modulus of continuity of f to any n-order (argument) fuzzy connec-tives in the following manner.

Definition 1. Let f : @0,1# n r @0,1# be an n-order function and d � @0,1# ,x � @0,1# n, where x � ~x1, x2, . . . , xn ! .

(1) The d sensitivity of f at point x, or a pointwise sensitivity, denoted by Df ~x,d! , isdefined in the form

Df ~x,d! � sup $6 f ~x!� f ~y!6 : y � @0,1# n and 6xi � yi 6� d, i � 1,2, . . . , n%

(2) The maximum d sensitivity of f (in the sequel we will be referring to it as sensitivity),denoted as Df ~d! , is defined as follows:

Df ~d! � ∨x�@0,1# n

Df ~x,d!

(3) The average d sensitivity of f (average sensitivity for brief), denoted as Sf ~d! , is de-fined as follows:

Sf ~d! � �0

1�0

1

. . .�0

1

Df ~x,d! dx1 dx2 . . . dxn

Remark 1. The notion of the maximum d sensitivity of f was introduced byNguyen et al.13 in the context of the robustness of fuzzy logic; their study wasfocused on fuzzy conjunction and disjunction.

ROBUSTNESS OF FUZZY REASONING 395

Definition 2. Let f and g be two n-order connectives. We say that f is at least asrobust as g at point x if ∀d � 0,Df ~x,d! � Dg~x,d! . Furthermore, if there existsd � 0 such that Df ~x,d! � Dg~x,d! , then f is called more robust than g at point x.

We say that f is at least as robust as g if ∀d� 0,Df ~d!� Dg~d! . Furthermoreif there exists d � 0 such that Df ~d! � Dg~d! , then f is called more robust than g.

f is called more average robust than or the same as g if ∀d � 0, Sf ~d! �Sg~d! . Moreover if there exists d� 0 such that Sf ~d!� Sg~d! then f is called moreaverage robust than g.

In what follows, we study these notions in more detail.

2.1. The Pointwise Sensitivity of Fuzzy Connectives

Theorem 1. For a binary connective f : @0,1# 2 r @0,1# , the following holds:

(1) If f is increasing with respect to its variables, x � x ', y � y ' ] f ~x, y!� f ~x ', y ' ! , then

Df ~~x, y!,d! � @ f ~x, y!� f ~~x � d! ∨ 0, ~ y � d! ∨ 0!# ∨

@ f ~~x � d! ∧ 1, ~ y � d! ∧ 1!� f ~x, y!#

In particular, if f is any t-norm or an s-norm, the above formula holds.(2) If f is decreasing with respect to its first variable and increasing with respect to its

second variable, x ' � x, y � y ' ] f ~x, y!� f ~x ', y ' ! , then

Df ~~x, y!,d! � @ f ~x, y!� f ~~x � d! ∧ 1, ~ y � d! ∨ 0!# ∨

@ f ~~x � d! ∨ 0, ~ y � d! ∧ 1!� f ~x, y!#

In particular, if f is an R implication or S implication, the above formula is satisfied.(3) If f is increasing with respective to its second variable, ∀x � @0,1# , y � y ' ] f ~x, y!�

f ~x, y ' !, then

Df ~~x, y!,d! � ∨x�d�x '�x�d

6 f ~x ', ~ y � d! ∨ 0!� f ~x, y!6 ∨

∨x�d�x '�x�d

6 f ~x ', ~ y � d! ∧ 1!� f ~x, y!6

In particular, if f is a QL implication, the above formula holds.

Proof.

(1) Suppose 6x � x ' 6 � d,6y � y ' 6 � d, then x � d � x ' � x � d, y � d �y ' � y � d. Let

b � 6 f ~x ', y ' !� f ~x, y!6, c � @ f ~x, y!� f ~~x � d! ∨ 0, ~ y � d! ∨ 0!# ∨

@ f ~~x � d! ∧ 1, ~ y � d! ∧ 1!� f ~x, y!#

396 LI ET AL.

We discuss two cases:

(i) If f ~x ', y ' ! � f ~x, y!, then b � f ~x ', y ' ! � f ~x, y! � f ~~x � d! ∧ 1, ~ y � d! ∧ 1! �f ~x, y!� c.

(ii) If f ~x ', y ' ! � f ~x, y!, then b � f ~x, y!� f ~x ', y ' !� f ~x, y!� f ~~x � d! ∨ 0, ~ y � d! ∨0!� c.

Hence the relation b � c always holds. From the definition of Df ~~x, y!,d!, it fol-lows that Df ~~x, y!,d!� c.

If we now choose x '� ~x � d! ∧ 1, y '� ~ y � d! ∧ 1 or x '� ~x � d! ∨ 0, y '�~ y � d! ∨ 0, then 6x � x ' 6 � d and 6y � y ' 6 � d, and in this case the relationshipb � c holds. Therefore, Df ~~x, y!,d!� c.

The proofs of statements (2) and (3) are similar to that considered abovefor (1). �

Theorem 2. Let f : @0,1# r @0,1# be a reverse order function, that is, x � y ]f ~x!� f ~ y!; then ∀x � @0,1# ,∀y � @0,1# , and the following relationship holds:

Df ~x,d! � @ f ~x!� f ~~x � d! ∧ 1!# ∨ @ f ~~x � d! ∨ 0!� f ~x!#

In particular, the above formula holds for the negation operation.

Because t-norm is a generalization of the conjunction connective, s-norm gen-eralizes the disjunction connective, and R implication, S implication, QL implica-tion are the extension of the classical implication, it is instructive to study thesensitivity of these types of connectives at the boundary points being used in thetwo valued logic, that is, (0,0), (0,1), (1,0), and (1,1).

Corollary 1.

(1) Let f : @0,1# 2 r @0,1# be a t-norm or an s-norm; then

Df ~~0,0!,d! � f ~d,d!,Df ~~0,1!,d!� Df ~~1,0!,d!� d

Df ~~1,1!,d! � 1 � f ~1 � d,1 � d!

(2) Let f : @0,1# 2 r @0,1# be an R implication; then

Df ~~0,0!,d! � 1 � f ~d,0! Df ~~0,1!,d!� 1 � f ~d,1 � d!

Df ~~1,0!,d! � f ~1 � d,d! Df ~~1,1!,d!� 1 � f ~1,1 � d!

(3) Let f : @0,1# 2 r @0,1# be an S implication, that is, f ~a, b! � S~n~a!, b! ,where S is an s-norm and n is a negation; then

Df ~~0,0!,d! � 1 � n~d! Df ~~0,1!,d!� 1 � s~n~d!,1 � d!

Df ~~1,0!,d! � S~n~1 � d!,d! Df ~~1,1!,d!� d

ROBUSTNESS OF FUZZY REASONING 397

In particular, if n is the linear negation c, that is, c~x!� 1 � x, then

Df ~~0,0!,d! � d Df ~~0,1!,d!� 1 � S~1 � d,1 � d!

Df ~~1,0!,d! � S~d,d! Df ~~1,1!,d!� d

(4) If f : @0,1# 2r @0,1# is a QL implication, that is, f ~a, b!� S~n~a!, t~a, b!! ,where S is an s-norm, n is a negation, and t is a t-norm, then we have

Df ~~0,0!,d! � 1 � n~d! Df ~~0,1!,d!� 1 � n~d!

Df ~~1,0!,d!� S~n~1 � d!,d!

~1 � S~n~1 � d!,1 � d!! ∨ d � Df ~~1,1!,d!� 1 � t~1 � d,1 � d!

In particular, if n is a linear negation, and ~S, t ! are dual norms, that is,S~a, b!� 1 � t~1 � a,1 � b! , then

Df ~~0,0!,d! � d Df ~~1,0!,d!� d Df ~~1,0!,d!� S~d,d!

~1 � S~d,1 � d!! ∨ d � Df ~~1,1!,d!� S~d,d!

Corollary 2.

(1) If f � ∨ or ∧, then Df ~~x, y!,d!� d holds for x, y � 0,1.(2) If f ~a, b!� ~1 � a � b! ∧ 1, the Lukasiewicz implication, then the follow-

ing relationships hold:

Df ~0,0! � d Df ~~0,1!,d!� 0 Df ~~1,0!, s!� 2d ∧ 1

Df ~~1,1!,d! � d

(3) If f ~a, b!� ~1 � a! ∨ b, the Kleene–Dienes implication, then we have:

Df ~~x, y!,d! � d, ∀x � 0,1, ∀y � 0,1

(4) If f ~a, b!� ~1 � a! ∨ ~a ∧ b! , (which is the Zadeh implication) then weobtain

Df ~~x, y!,d! � d, ∀x � 0,1, ∀y � 0,1.

Remark 2. If f is a t-norm, then from the results of Corollaries 1 and 2, it followsthat f is more robust than minimum connective ∧ at the point (0,0), and if f � ∧, fis more robust than ∧ at the point (0,0); f is the same robust as ∧ at the points (0,1)and (1,0); but ∧ is more robust than other t-norms at the point (1,1). The relationsbetween s-norm and ∨, R implication and Lukasiewicz implication, S implicationand Kleene–Dienes implication, and QL implication and Zadeh implication leadto the similar results.

398 LI ET AL.

2.2. Maximum Sensitivity of Fuzzy Connectives

Lemma 1.

(1) If f : @0,1# 2 r @0,1# is continuous at x � @0,1# 2, then ∀« � 0,there exists d � @0,1# such that Df ~x,d! � «; moreover, if f : @0,1# 2 r@0,1# is continuous, then ∀« � 0, there exists d � @0,1# such thatDf ~d!� «. In this case, Sf ~d!� «.

(2) If f : @0,1# 2 r @0,1# is discontinuous at x � @0,1# 2, then there exists« � 0, such that for any d � @0,1# , Df ~d!� Df ~x,d! � «, and sf ~d! � «.

Proof. It can be inferred from the definition of the continuity of the function. �

Lemma 2. If d1 � d2 , then Df ~x,d1!� Df ~x,d2 ! , and Df ~d1!� Df ~d2 !, Sf ~d1!�Sf ~d2 ! .

Proof. This follows from the definition of Df ~x,d!. �

The following is a well-known result encountered in mathematical analysis;compare Ref. 6.

Lemma 3. Assume that f, g;Ur R is a bounded function; then

� ∨x�U

f ~x!� ∨x�U

g~x!� � ∨x�U6 f ~x!� g~x!6

� ∧x�U

f ~x!� ∧x�U

g~x!� � ∨x�U6 f ~x!� g~x!6

Lemma 4. Assume that f, g : @0,1# 2 r @0,1# are two binary connectives, andg is the dual of f, that is, ∀x, y � @0,1# , g~x, y! � 1 � f ~1 � x, 1 � y!; then∀d � @0,1# ,Df ~d!� Dg~d! .

Theorem 3. The minimum ∧, the maximum ∨, and the linear negation are themost robust t-norm, s-norm, and negation, respectively.

Proof. Let f ~x, y!� x ∧ y, if 6x � x ' 6� d and 6y � y ' 6� d; then from Lemma 4,6 f ~x, y!� f ~x ', y ' !6� 6x ∧ y � x ' ∧ y ' 6� 6x � x ' 6 ∨ 6y � y ' 6� d. In particular, ifwe let x � y � d, x '� y '� 0, then 6 f ~x, y!� f ~x ', y ' !6� d, and thus Df ~d!� d.

On the other hand, let g be any t-norm; then ∀d � 0, Dg~d!� Dg~~1,1!,d!�1 � g~1 � d, 1 � d! � 1 � ~1 � d! ∧ ~1 � d! � d, that is, ∀d � 0, Dg~d! � d �Df ~d!.

Moreover, if g is another t-norm different from f, then there exist x, y � ~0,1!such that g~x, y! � f ~x, y! � x ∧ y, or to put it more precisely, g~x, y! � x ∧ y.Without any loss of generality we assume that x � y; then g~x, y! � x � y,and thus 6g~x, y! � g~1,1!6 � 1 � g~x, y! � 1 � x. This demonstrates thatDg ~1 � x! � 1 � x.

ROBUSTNESS OF FUZZY REASONING 399

Hence, f ~x, y!� x ∧ y is the most robust t-norm.From the dual principle of Lemma 3, it follows that the maximum operator ∨

is the most robust s-norm.Let f ~x! � 1 � x; it follows from Theorem 2 that Df ~d! � d. Let g be any

negation; there are three cases to discuss.Case 1. If g~d! � 1 � d, then 6g~0! � g~d!6 � 1 � g~d! � d, and thus

Dg~d! � d.Case 2. If g~d! � 1 � d, then 6g~1! � gg~d!6 � 60 � d6 � d, and thus

Dg~d! � d.Case 3. If g~d! � 1 � d, then 6g~1! � g~1 � d!6 � 60 � g~1 � d!6 � g~1 �

d! � d, and thus Dg~s! � d.For any case, we always have Dg~d! � d � Df ~x!, and if g � f, then there

exists x � ~0,1! such that g~x! � 1 � x. If g~x! � 1 � x, then Dg~x! � x as inCase 1; if g~x! � 1 � x, then Dg~x! � x as in Case 3.

Hence, the linear negation f ~x!� 1 � x is the most robust negation. �

Remark 3. The same results of Theorem 3 for t-norms and s-norms were obtainedin Refs. 13 and 14 but in a more sophisticated manner. Here our proof exploits thenotion of pointwise sensitivity.

Next we provide results dealing with sensitivity of some commonly usedt-norms and s-norms.

Example 1. Let d � @0,1# , then we have the following.(1) If f ~x, y! � xy, then Df ~d! � 2d � d2 . Dually if g~x, y! � x � y � xy

(probabilistic sum), then Dg~d!� 2d� d2 . The detailed derivations are carried outas follows.

From Lemma 4, we are concerned with f ~x, y!, first, Df ~~x, y!,d!� 2d� d2 ,which holds for all arguments in the unit interval. Now let a � @ f ~x, y! �f ~~x � d! ∨ 0, ~ y � d! ∨ 0!# , and b � @ f ~~x � d! ∧ 1, ~ y � d! ∧ 1!� f ~x, y!# ; thenDf ~~x, y!,d! � a ∨ b, and we get a � 2d � d2 and b � 2d � d2 . Note that (i) ifx � d or y � d, then a � f ~x, y!� xy � d� 2d� d2 ; (ii) if x � d and y � d, thena � xy � ~x � d!~ y � d!� ~x � y!d� d2 � 2d� d2 ; (iii) if x � d � 1 and y �d� 1, then xy � ~1 � d!2 , and thus b � 1 � xy � 1 � ~1 � d!2 � 2d� d2 ; (iv) ifx � d � 1 and y � d � 1, then b � ~x � d!~ y � d! � xy � ~x � y!d � d �2~1 � d!d � d2 � 2d � d2 , and the last case (v) if x � d � 1 and y � d � 1 orx � d � 1 and y � d � 1, it is no loss of generality to assume x � d � 1 and y �d � 1; then b � y � d � xy � ~1 � x!y � d � d~1 � d! � d � 2d � d2 . So wealways have Df ~~x, y!,d!� 2d� d2 , and thus Df ~d!� 2d� d2 .

If we choose x � y � 1, x ' � y ' � 1 � d, then xy � x 'y ' � 1 � ~1 � d!2 �2d� d2 . This demonstrates that Df ~d!� 2d� d2 .

(2) If f ~x, y! � 0 ∨ ~x � y � 1! (Lukasiewicz connective), then Df ~d! �2d ∧ 1. Alternatively, if g~x, y!� 1 ∧ ~x � y!, which is known as bounded sum orLukasiewicz t-norm, then Dg~d!� 2d ∧ 1.

Note that if 6x � x ' 6� d,6y � y ' 6� d, then 6 f ~x, y!� f ~x ', y ' !6� 60 ∨ ~x �y �1!� 0 ∨ ~x '� y '�1!6� 60 � 0 6 ∨ 6~x � y �1!� ~x '� y '�1!6� 6~x � x ' !�~ y � y ' !6� 6x � x ' 6� 6y � y ' 6� 2d, and therefore Df ~~x, y!,d!� 2d ∧ 1, that is,

400 LI ET AL.

Df ~d!� 2d ∧ 1. On the other hand, if we choose x � 0.5 � d, y � 0.5 � d, x '� y '�0.5, then 6 f ~x, y!� f ~x ', y ' !6� 2d ∧ 1, and this demonstrates that Df ~d!� 2d ∧ 1.

(3) Let f ~x, y! � x ∧ y; then Df ~d! � d. Dually, if g~x, y! � x ∨ y, thenDg~d! � d.

(4) If

f ~x, y! � �x ∧ y, if x ∨ y � 1

0, otherwise

then Df ~d!� 1. Subsequently, if

g~x, y! � �x ∨ y, if x ∧ y � 0

1, otherwise

then Dg~d!� 1.

Next we derive a formula for the sensitivity of some commonly used Rimplications.

Example 2. For d � @0,1# , we have the following.(1) If f ~x, y!� ~1 � x � y! ∧ 1, then Df ~d!� 2d ∧ 1.The calculations adhere to the following scheme. If 6x � x ' 6� d,6y � y ' 6� d,

then

6 f ~x, y!� f ~x ', y ' !6 � 6~1 � x � y! ∧ 1 � ~1 � x ' � y ' ! ∧ 16

� 6~1 � x � y!� ~1 � x ' � y ' !6 ∨ 61 � 16.

This reveals that Df ~d!� 2d ∧ 1. On the other hand, from the results of Corollary2, we know that Df ~~1,0!,d!� 2d ∧ 1; hence Df ~d!� 2d ∧ 1.

(2) If

f ~x, y! � �1, x � y

y, x � y

(which is a Gödel implication), then Df ~d!� 1.More specifically

Df ~~0,d!,d! � @ f ~0,d!� f ~d,0!# ∨ @ f ~0,2d ∧ 1!� f ~0,d!#

� ~1 � 0! ∨ ~1 � 1!� 1

As Df ~d!� 1, we get Df ~d!� 1.(3) If

f ~x, y! �y

x∧ 1 � �1, x � y

y

x, x � y

(Goguen implication), then Df ~d!� 1. We have Df ~~0,d!,d!� 1.

ROBUSTNESS OF FUZZY REASONING 401

Theorem 4. The Lukasiewicz implication is the most robust R implication.

Proof. If f ~x, y!� ~1 � x � y! ∧ 1, then Df ~d!� 2d ∧ 1.Let g be another R implication. From Theorem 1(2) we obtain Dg~~1 � d,

1 � d!,d!� @g~1 � d,1 � d!� g~1, ~1 � 2d! ∨ 0!# ∨ @g~~1 � 2d! ∨ 0,1!� g~1 � d,1 � d!#� @1 � g~1, ~1 � 2d! ∨ 0!# ∨ 61 � 16� 1 � g~1, ~1 � 2d! ∨ 0!. Note that forany R implication, we always have g~1, x!� x; it follows that Dg~~1�d,1�d!,d!�1 � ~1 � 2d! ∨ 0 � 2d ∧ 1, and thus Dg~d!� ∨x�@0,1# 2 Dg~x,d!� 2d ∧ 1 � Df ~d!,which demonstrates that the Lukasiewicz implication is the most robust Rimplication. �

Theorem 5.

(1) Let f be an S implication, f � S~n~a!, b! , then Df ~d!� DS ~Dn~d!! .(2) If n is a linear negation and f is an S implication, then Df ~d!� DS ~d! .

Proof.

(1) If Dn~d!� d and 6x � x ' 6� d,6y � y ' 6� d, then 6n~x!� n~x ' !6� Dn~d!and 6y � y ' 6 � d � Dn~d!. Hence Df ~~x, y!,d! � DS ~~n~x!, y!,Dn~d!!,from which we conclude that Df ~d!� Ds~Dn~d!!.

(2) For the linear negation n we have Dn~d!� d, and this combined with therelationship 6x � x ' 6� d? 6~1 � x!� ~1 � x ' !6� d and (1) produces theequality Df ~d!� DS ~d!. �

Example 3.

(1) If f ~x, y! � ~1 � x! ∨ y (Kleene–Dienes implication), then Df ~d! � d.From Theorem 5, it follows that the Kleene–Dienes implication is themost robust S implication (with the linear form of the negation operator).

(2) If f ~x, y!� 1 � a � ab (Reichenbach implication), then from Theorem 5we note that f ~x, y! has the same sensitivity as the product operation, andtherefore Df ~d!� 2d� d2 .

(3) If f ~x, y! � ~1 � a � b! ∧ 1, then based on Theorem 5 we observe thatf ~x, y! exhibits the same sensitivity as the bounded sum, and thereforeDf ~d!� 2d ∧ 1.

Theorem 6. If f is a QL implication, f ~x, y!� S~n~x!, t~x, y!! , then

Df ~d! � DS ~Dn~d! ∨ Dt ~d!!

Proof. If 6x � x ' 6� d, 6y � y ' 6� d, then 6n~x!� n~x ' !6� Dn~d! and 6 t~x, y!�t~x ', y ' !6 � Dt ~d!. We note that Df ~~x, y!,d! � DS ~~n~x!, t~x, y!!,Dn~d! ∨ Dt ~d!!and therefore Df ~d!� DS ~Dn~d! ∨ Dt ~d!!. �

Example 4. (1) If f ~x, y!� ~1� x! ∨ ~x ∧ y! (Zadeh implication), then Df ~d!� d.From Theorem 6, Df ~d!� d, and Corollary 2, we have Df ~~0,0!,d!� d, and hence

402 LI ET AL.

Df ~d!� d. From Theorem 6 and Theorem 3, it follows that the Zadeh implicationis the most robust QL implication.

Remark 4. Based on the above discussion, we observe that the minimum, maxi-mum, Lukasiewicz implication, Kleene–Dienes implication, and Zadeh implica-tion are the most robust operators among t-norms, s-norms, R implications, Simplications (in case of linear negation), and QL implications, respectively. Fromthe standpoint of robustness of fuzzy reasoning we see that these operators play aspecial role. As a matter of fact, one can observe that they are the most commonlyused fuzzy connectives. Of course, the maximum sensitivity can only reflect theextreme property of the connective. Being more conservative, the measures ofaverage sensitivity give us a better insight into the global robustness properties ofthe logic connectives.

2.3. The Average Sensitivity of a Fuzzy Connective

The average sensitivity of a fuzzy connective f can be regarded as a functiony � Sf ~t ! from [0,1] to [0,1] when t changes in [0,1]. By plotting these relation-ships, see Figures 1–5, we gain a better sense as to the behavior of the property.

These figures reveal some general tendency as to the average sensitivity ofthe logic operators. For example, the Godel implication is more average-robustthan the Lukasiewicz implication, whereas the Lukasiewicz implication is moremaximum-robust than other R implications. It seems that the fuzzy connectivesusing t-norm as minimum and using s-norm as maximum are more average-robustthan other fuzzy connectives.

There are some other methods to capture the average robustness of fuzzyconnectives (cf. Refs. 14 and 15) and those use the partial derivatives of the fuzzy

Figure 1. Plots of average sensitivity of selected t-norms: (a) Minimum; (b) Product;(c) Lukasiewicz t-norm.

ROBUSTNESS OF FUZZY REASONING 403

connectives. For a fuzzy connective f : @0,1# n r @0,1# , its average robustness isdefined as follows:

S~ f ! � �0

1�0

1

{{{�0

1

(i�1

n � ]f]xi�2

dx1 dx2 . . . dxn

This average robustness of f is easier to calculate than the one introduced in thisarticle, but obviously it does not reflect the influence of changes in the premise (asour notion of average robustness does).

Figure 2. Average sensitivity of selected s-norms: (a) Maximum; (b) Probabilistic sum;(c) Lukasiewicz.

Figure 3. Average sensitivity of R implications: (a) Godel; (b) Goguen; (c) Lukasiewicz.

404 LI ET AL.

3. THE ROBUSTNESS OF FUZZY INFERENCE SYSTEMS

Let F~X ! denote the set of all fuzzy subsets of X. For any A, B � F~X !, and0 � d� 1. If for any x � X,6A~x!� B~x!6� d, that is, 7A � B7`� supx�X 7A~x!�B~x!7 � d, where 7{7` denote the infinite norm of F~X !, then B is called a dperturbation of A, and we write A[ ~d!B. Using the d equality of Cai,8 A[ ~d!Bmeans that A is 1 � d equal to B, that is, 7A � B7`� 1 � ~1 � d!, so the notion of

Figure 4. Average sensitivity of selected S implications: (a) Kleene–Dienes; (b) Reichenbach;(c) Lukasiewicz.

Figure 5. Average sensitivity of selected QL implications: (a) Zadeh; (b) f ~a, b! � 1 � a �a2b; (c) Lukasiewicz.

ROBUSTNESS OF FUZZY REASONING 405

maximum perturbation of fuzzy sets is dual to that of d equality of fuzzy sets asdefined by Cai in Ref 6. We study the robustness of fuzzy inference systems bylinking it to the sensitivity of fuzzy connectives, which also constitutes the mainresults of our study.

3.1. The Robustness of Fuzzy MP Rule

We start with the characterization of robustness of fuzzy MP rules. The multi-dimensional fuzzy MP rule is of the following form:

Premise 1: If x1 is A11, x2 is A12, . . . , xn is A1n , then y is B1

Premise 2: x1 is A11' , x2 is A12

' , . . . , xn is A1n'

Conclusion: y is B1' (1)

where x1, x2, . . . , xn , y are linguistic variables, A1j and Bj are fuzzy sets defined inXj and Y ~ j � 1,2, . . . , n!. R � A r B denotes a fuzzy relation in X � Y, whereA~x! � A1 � A2 � {{{ � An � Xi�1

n Ai ~x! denote a Cartesian product definedin X � X1 � X2 � {{{� Xn that reads as A~x!� A1~x1! * A2~x2 ! * {{{ * An~xn !,x � ~x1, x2, . . . , xn ! � X. Furthermore * is a t-norm, andr denotes a fuzzy impli-cation. If we let A' � F~X ! be defined as A'~x!� A1

' ~x1! * A2' ~x2 ! * {{{ * An

' ~xn !,then the fuzzy modus ponens leads to the expression A' 8 ~Ar B!� B ' , namely,

B '~ y! � ∨x�X@A'~x! * ~A~x!r B~ y!!# , ∀y � Y, (2)

In what follows we discuss robustness of fuzzy modus ponens under some com-monly used t-norms and implication operators. First we start with the followingdefinition.

Definition 3. Let A1i , A2i , A1i' , A2i

' � F~Xi !, B1, B2, B1' , B2

' � F~Y !, ~i �1,2, . . . , n! Aj � Xi�1

n Aji Aj' � Xi�1

n Aji' ~ j � 1,2!, Rj~x, y! � Aj~x! r Bj~ y!,

Rj' ~x, y!� Aj

' ~x!r Bj' ~ y! , and Bj

'� Aj'8 Rj~ j �1,2! . For d� @0,1# , we define the

robustness index of fuzzy MP rule B '� A' 8 R, denoted as DB ' ~d! , as follows:

DB ' ~d! � sup $7B1' � B2

' 7` : A1i[ ~d!A2i , B1[ ~d!B2,

A1i' [ ~d!A2i

' , i � 1,2, . . . , n% (3)

Remark 5. Obviously, if A1i [ ~d!A2i , B1[ ~d!B2, A1i' [ ~d!A2i

' , i � 1,2, . . . , n,then B1

' [ ~DB ' ~d!!B2' , and DB ' ~d! is the least number among h such that B1

' [~h!B2

' , so DB ' ~d! is the best estimator or the least conservative bound for therobustness index of fuzzy MP rule.6

Definition 4. Suppose that f : @0,1#mr @0,1# is any n-order fuzzy connective. If∀d� @0,1# , there exist x, y � @0,1#m, 6xi � yi 6� d~1 � i � m! , such that Df ~d!�6 f ~x!� f ~ y!6; then f is called a sensitivity-attainable operator.

406 LI ET AL.

If f is a continuous function, then f is a sensitivity-attainable operator. Fur-thermore, all the fuzzy connectives we discussed in the previous section are in factsensitivity-attainable operators. More specifically, we write

f ~x1' , . . . , xn

' ; x1, . . . , xn ; y! � ~x1' *' {{{ *' xn

' ! * @~x1 *' {{{ *' xn !r y#

� x ' * ~xr y!

and treat it as a (2n � 1)-order fuzzy connective, if f is a sensitivity-attainableoperator; then from Lemma 3 and unlimited restriction on the shapes of fuzzy setsin Definition 3, we have the following interesting result.

Lemma 5. If f ~x1' , . . . , xn

' ; x1, . . . , xn ; y!� ~x1' *' {{{ *' xn

' ! * @~~x1 *' {{{ *' xn !ry!#� x ' * ~xr y! is a sensitivity-attainable operator, then the robustness of fuzzyMP rule is equal to the d sensitivity of f, that is, Df ~d!� DB ' ~d! .

Owing to this lemma, the robustness of many commonly encountered fuzzyMP rules is determined in the setting of their corresponding fuzzy connectives;obviously these are more convenient to handle. We first discuss the case wheren � 1 and then proceed with the general multivariable case.

Lemma 6. If f ~x, y, z!� x * ~ yr z!, then Df ~d!� D*~Dr~d!! .

Lemma 7. If the t-norm * is specified as the minimum, product, or Lukasiewiczt-norm, then it satisfies the following condition:

D*~d! � D*~~1,1!,d! (4)

Lemma 8. The min operator, Kleene–Dienes implication, and Lukasiewicz impli-cation satisfy the following condition:

∃y, z, y ', z ',6y � y ' 6 � d,6z � z ' 6� d

such that yr z � 1 and Dr~d!� 1 � y ' r z ' (5)

Theorem 7. Consider the one-dimensional fuzzy MP rule in the form 1; letDr~d!� l:

(i) If *� ∧, then DB ' ~d!� l.(ii) If * is the product operator then the following holds: (a) if the implication operatorr satisfies condition 5, then DB ' ~d! � d � l � dl; (b) if the implication r is theReichenbach implication, then DB ' ~d!� 2d� d2.

(iii) If * is the Lukasiewicz t-norm, this implies the following: (1) If the implication oper-atorr satisfies condition 5, then DB ' ~d!� ~d� l! ∧ 1; (2) ifr is the Reichenbachimplication, then DB ' ~d!� 2d� d2.

Proof. Let f ~x, y, z!� x * ~ yr z!; then DB ' ~d!�Df ~d! by Lemma 5. We need tocompute Df ~d!:

ROBUSTNESS OF FUZZY REASONING 407

(i) From Lemma 6, we have Df ~d!� D∧~Dr~d!!� Dr~d!� l. Moreover,if we choose x �1, then f ~x, y, z!� ~ yr z!, and then Df ~d!� Dr~d!�l. Hence Df ~d!� l.

(ii) Suppose now that f ~x, y, z!� x~ yr z! and let 6x � x ' 6� d,6y � y ' 6�d,6z � z ' 6 � d. Without any loss of generality, we can assume thatf ~x, y, z!� f ~x ', y ', z ' !. Then (1) 6 f ~x, y, z!� f ~x ', y ', z ' !6� f ~x, y, z!�f ~x ', y ', z ' !� x~ yr z!� x '~ y ' r z ' ! � x~ yr z!� ~x � d!~ yr z �Dr~d!! � xDr~d! � ~ y r z!d � dDr~d! � d � Dr~d! � dDr~d!.Because the implication operatorr satisfies condition 5, we can choosex � 1, yr z � 1, x ' � 1 � d, y ' r z ' � 1 Dr~d! to let the above equal-ity be satisfied. In the sequel, this leads to the relationship Df ~d!� d�l � dl. (2) Because y r z � 1 � y � yz � 1 � y~1 � z!, 6 f ~x, y, z! �f ~x ', y ', z ' !6� f ~x, y, z!� f ~x ', y ', z ' !� x@1 � y~1 � z!#� x ' @1 � y '~1 �z ' !#� x@1 � y~1 � z!#� ~x � d!@1 � y '~1 � z ' !#� x@ y '~1 � z ' !� y~1 �z!# � d@1 � y '~1 � z ' !# � x@ y '~1 � z ' ! � ~ y ' � d!~1 � z ' � d!# �d@1� y '~1� z ' !#� x@~ y '� ~1� z ' !!d� d2 #� d@1� y '~1� z ' !#� ~ y '�~1 � z ' !!d� d2 � d@1 � y '~1 � z ' !#� ~2 � y 'z '� z ' !d� d2 � 2d� d2 .Now if x � 1, y � 1 � d, z � d; x ' � 1 � d, y ' � 1, z ' � 0, the equalityholds, and thus Df ~d!� 2d� d2 .

(iii) Suppose f ~x, y, z!� ~x � yr z � 1! ∨ 0. If 6x � x ' 6 � d,6y � y ' 6 � d,6z � z ' 6� d, then (1) f ~x, y, z!� f ~x ', y ', z ' !6� 6~x � yr z � 1! ∨ 0 �~x '� y 'r z '�1! ∨ 0 6� 6x � x '� �y 'r z ' 6 ∨ 0 � 6x � x ' 6� 6yr z �y ' r z ' 6� d� Dr~d!.

Because the implication operator r satisfies condition 5, we can choosex � 1, yr z � 1, x '� 1 � d, y 'r z '� 1 � Dr~d! and Df ~d!� ~d� l! ∧ 1. (2) Ify r z � 1 � y � yz, we can assume (not losing any generality of our consider-ations) that f ~x, y, z! � f ~x ', y ', z ' !. Then 6 f ~x, y, z! � f ~x ', y ', z ' !6 � f ~x, y, z! �f ~x ', y ', z ' !� ~x �1 � y � yz! ∨ 0 � ~x '�1 � y '� y 'z ' ! ∨ 0 � ~x �1 � y � yz!�~x ' � 1 � y ' � y 'z ' ! � x � x ' � @ y~1 � z! � y '~1 � z ' !# � d � @~ y ' � d! �~1 � z ' � d!� y '~1 � z ' !#� d� d@1 � z '� d#� 2d� d2 � z 'd� 2d� d2 . If weselect x � 1, y � 1 � d, z � d; x '� 1 � d, y '� 1, z � d, then above equality holds,and therefore Df ~d!� 2d� d2 . �

Now we can move on to the multidimensional case.

Lemma 9. Suppose h~x1, . . . , xn ; y!� x1 * {{{ * xnr y � xr y, x � x1 * {{{ *xn, and let l� D*n ~d! , if *n satisfies the condition l� D*n ~d!� D*n ~~1, . . . ,1!,d!;then the expressions for Dh~d! considering various t-norms and implication oper-ators are contained in Table I. More specifically, if we choose the t-norm * asminimum, product, or Lukasiewicz t-norm, the above results hold.

The explicit formula for l � D*n ~d! is given in the form of the followingproposition.

408 LI ET AL.

Lemma 10. Let g~x1, . . . , xn !� x1 * {{{ * xn;

(1) if *� ∧ or ∨, then Dg~d!� d;

(2) if * is chosen as the product operation or x * y � x � y � xy, then Dg~d!�l, where l�

(i�1n ~�1!i�1 �n

i �d i;

(3) if x * y � ~x � y � 1! ∨ 0 or x * y � ~x � y! ∧ 1, then Dg~d!� nd ∧ 1.

These results can be derived by induction.

Theorem 8. For the multidimensional fuzzy MP rule coming in the form of 2,suppose that the corresponding t-norm *' satisfies condition 4, and D*'n~d! � h,Dh~d!� l, where h~x1, . . . , xn , y!� ~x1 *' {{{ *' xn !r y.

(i) If *� ∧, then DB ' ~d!� l.(ii) If * is specified as the product, then

(1) if the implication operatorr satisfies condition 6, then DB ' ~d!� h� l� hl;(2) ifr is chosen as the Reichenbach implication, then DB ' ~d!� 2h� h2.

(iii) If * is taken as the Lukasiewicz t-norm, then(1) if the implication operatorr satisfies condition 5, then DB ' ~d!� ~h� l! ∧ 1;(2) ifr is equal to Reichenbach implication, then DB ' ~d!� 2h� h2.

Proof. Let f ~x1' , . . . , xn

' ; x1, . . . , xn ; y!� ~x1' *' {{{ *' xn

' ! * @~~x1 *' {{{ *' xn ! ry!# � x ' * ~x r y!, and then DB ' ~d! � Df ~d! by Lemma 5, so we only show theequation about Df ~d! instead of DB ' ~d!. The left proofs are similar to those ofTheorem 7. �

The best estimates of the robustness indexes are given in Theorems 7 and 8(cf. Ref 6.). The results lead to explicit comparisons of the robustness indexesdepending on the different logic connectives being used in the reasoning scheme.

3.2. The Robustness of Fuzzy MT Rule

Fuzzy modus tollens (MT) comes in the following format.

Premise 1: If x is A, then y is B

Premise 2: y is B '

Conclusion: x is A' (6)

Table I. Robustness results of multidimensional fuzzy MP rule under various t-normand implications.

r

Mamdani minimplication

Kleene–Dienesimplication

Lukasiewiczimplication

Reichenbachimplication

Zadehimplication

Dh~d! d ∨ l� l d ∨ l� l ~d� l! ∧ 1 d� l� dl d ∨ l� l

ROBUSTNESS OF FUZZY REASONING 409

Or

A'~x! � ∨y�Y@B '~ y! * ~A~x!r B~ y!!# (7)

Obviously, the discussion of the robustness of the fuzzy MP rule in case n �1can apply to that of fuzzy MT rule, as long as we note that A' and B ' are to bepositioned in place of B ' and A' in the fuzzy MP rule. Let

DA'~d! � ∨$7A1' � A2

' 7: A1[ ~d!A2, B1[ ~d!B2, B1' [ ~d!B2

' % (8)

denote the robustness measure of the fuzzy MT rule. Then Theorem 7 is still validfor fuzzy MT. We have the following theorem.

Theorem 9. For the fuzzy MT rule 7, let Dr~d!� l.

(i) If *� ∧, then DA~d!� l.(ii) If * is the product operator, then

(1) if the implication operatorr satisfies condition 5, then DA'~d! � d � l � dl;(2) if ther is the Reichenbach implication, then DA'~d! � 2d � d2.

(iii) If * is the Lukasiewicz t-norm, then(1) if the implication operatorr satisfies condition 5, then DA'~d! � ~d � l! ∧ 1;(2) ifr is the Reichenbach implication, then DA'~d! � 2d � d2.

3.3. The Robustness of Fuzzy Inference Schemes

A fuzzy rule base normally comprises several rules and the inference scheme16

states that

Premise 1: If x1 is A11, x2 is A12, . . . , xn is A1n , then y is B1

. . .

If x1 is Am1, x2 is Am2, . . . , xn is Amn , then y is Bm

Premise 2: x1 is A1, x2 is A2, . . . , xn is An

Conclusion: y is B ' (9)

where x1, x2, . . . , xn , y are linguistic variables, Aij and Bj are fuzzy sets defined inXj and Y ~i � 1,2, . . . , m; j � 1,2, . . . , n!. We can treat fuzzy inference machine as ageneralized form of fuzzy MP rule. However, because many rules are involved,different inference procedure can be utilized in this setting (see Refs. 2 and 11 formore detailed discussion). In general, the following schemes are of interest:

method 1: B '1 � �i�1

m

@A' 8 ~Ai r Bi !# method 2: B '2 � A' 8�i�1

m

~Ai r Bi !

method 3: B '3 � �i�1

m

@A' 8 ~Ai r Bi !# method 4: B '4 � A' 8�i�1

m

~Ai r Bi !

(10)

where �, � denote s-norm (in particular, the maximum operator) and t-norm (theminimum operator in some specific case).

410 LI ET AL.

We first develop a robustness index of the fuzzy inference mechanism (usingmethod 1).

Definition 5. Let Aij , Aij' , Ai

' , Ai'' � F~X !, Bi , B ', B '' � F~Y ! , ~i � 1, . . . , n;

j � 1,2, . . . m! , and write Ri � Air Bi , Ri'� Ai

' r Bi' , Ci � Xi�1

n Aij Ci'� Xi�1

n Aij' ,

C ' � Xi�1n Ai

'C '' � Xi�1n Ai

'' , B1' ~ y! � �i�1

m @C ''~x! * Ri ~x, y!# , B2''~ y! �

�i�1m @C ''~x! * Ri ~x, y!# . For d � @0,1# , the robustness index of the fuzzy in-

ference machine, denoted here as DB1' ~d! , is defined in the form

DB '1 ~d! � sup $7B ' � B '' 7` : Aij[ ~d!Aij' , Ai

' [ ~d!Ai'' , Bi[ ~d!Bi

' % (11)

Similarly, we can construct definitions of DB '2 ~d!, DB '3 ~d! , and DB '4 ~d! .

Just as it has been formulated above, we can express the robustness index ofthe fuzzy inference machine through the following theorem. We use DBi

' ~d! torepresent the robustness index of a single rule Air Bi :

A'

Ai r Bi

Bi' (12)

It is obvious that each single rule has the same robustness index, that is, DB1' ~d!�

DB2' ~d!� DB3

' ~d!� DB4' ~d!� DB ' ~d!� DB1

' ~d!, where DB ' ~d! is just the DB ' ~d! asgiven by Definition 3.

Theorem 10. Let l�DB1' ~d!�DB2

' ~d!�DB3' ~d!�DB4

' ~d!�DB ' ~d!�DB1' ~d! for

i � 1,2, . . . ,m, where DB'~d! is expressed by Definition 3.

(1) If � � ∨ and � � ∧, then DB1' ~d!� DB2

' ~d!� DB3' ~d!� DB4

' ~d!� l.(2) If � � probabilistic sum, and � � product, then DB1

' ~d! � DB2' ~d! � DB3

' ~d! �

DB4' ~d!�( i�1

m ~�1!i�1 �m

i �li.

(3) If � � Lukasiewicz s-norm and � � Lukasiewicz t-norm, then DB1' ~d! � DB2

' ~d! �DB3

' ~d!� DB4' ~d!� ml ∧ 1.

Proof. Owing to the distributivity of continuous t-norms, the first and secondinference methods are equivalent; the same holds for the third and fourth methods(see the detailed discussion in Ref. 11). �

From the above theorem, we see that if we take � � ∨ and � � ∧, then thefour inference methods come with the same robustness measure, and the robust-ness of multiple rule fuzzy reasoning is the same as that based on a single rulefuzzy reasoning. This implies that the number of fuzzy rules does not play any rolein the specification of the robustness of fuzzy reasoning. Furthermore, if we taket-norm as the minimum operator, then the robustness measure of the resulting fuzzyinference machine is the least. Being more specific, it is equal to the maximumperturbation of the fuzzy sets, that is, DB1

' ~d!� d. Of course, if we take � as someother s-norm different from the maximum operator and � is different from the

ROBUSTNESS OF FUZZY REASONING 411

minimum operation, then the four inference methods will not exhibit the samerobustness measure. Likewise the robustness indexes of multiple rule fuzzy rea-soning will depend on this number. This again emphasizes that the robustness offuzzy reasoning significantly relies on the choice of the fuzzy connectives beingused there.

4. CONCLUSIONS

We have discussed robustness of fuzzy reasoning in terms of the sensitivityof fuzzy connectives and maximum perturbation of fuzzy sets. Some general andsystematic results have been derived. In comparison with the previous work, ourformulation of the problem is more systematic and practical. We transformed theproblem of robustness of fuzzy reasoning to the task of robustness (or sensitivity)of the related fuzzy connectives and fuzzy implication operators. This equivalentproblem is more suitable for a comprehensive analysis. The results reveal that therobustness of fuzzy reasoning is directly linked to the selection of fuzzy connec-tives and implication operators.

There has been a substantial body of research concerning the problem ofrobustness of fuzzy reasoning. For example, the perturbation of fuzzy sets in thisarticle is expressed based on the notion of the maximum perturbation or d equali-ties of fuzzy sets. We can define another form of perturbation based on similarityof fuzzy sets:

A [ B � ~Ar B! * ~Br A!

where r is an implication operator, and * is a t-norm. Following this, we candevelop some results concerning fuzzy proposition calculus as expressed in Ref. 17for approximation solutions of fuzzy relational equations. Another question con-cerns linkages between the robustness property and random errors of membershipfunction determination. A place where robustness may be useful arises in the fol-lowing scenario (given a one input system): (a) the values of a system S are knownat points x1, x2, . . . , xn: ~x1, y1!,~x2, y2!, . . . , ~xn, yn!; (b) two models M and M ' areconstructed in such a way that they agree with S (or are good approximations of S!for some given data. Under these conditions, the following selection criterion maybe worth postulating. “Given two equally accurate models, the model that is morerobust over the domain is the one to be preferred.” It is worth stressing howeverthat the robustness of fuzzy reasoning is only one among its fundamental featuresto be considered along with many others. For example, the min and max operatorsassure a high degree of robustness, but they contribute to significant losses ofinformation that occur during the inference procedure. This stipulates that the stud-ies of robustness should also look at potential trade-offs between other propertiesof fuzzy inference schemes.

Acknowledgments

This work is supported by the National Science Foundation of China (Grant No. 60174016)and “TRAPOYT” of China and the National 973 Foundation Research Program (Grant No.

412 LI ET AL.

2002CB312200) as well as the Natural Sciences and Engineering Research Council of Canada(NSERC) and the Canada Research Chair (CRC) Program.

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