An approach to measure the robustness of fuzzy reasoning

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<ul><li><p>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, Xian, 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</p><p>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.</p><p>1. INTRODUCTION</p><p>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.</p><p>*Author to whom all correspondence should be addressed: e-mail:</p><p>INTERNATIONAL JOURNAL OF INTELLIGENT SYSTEMS, VOL. 20, 393413 (2005) 2005 Wiley Periodicals, Inc. Published online in Wiley InterScience( DOI 10.1002/int.20072</p></li><li><p>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:</p><p>Is a mathematical fuzzy reasoning scheme robust or perturbation resistant againstthe deviation of human expertise from its corresponding mathematically quantita-tive representations? 6</p><p>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 Yings) 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.</p><p>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 structurethe 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).</p><p>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.</p><p>2. THE ROBUSTNESS OF FUZZY LOGIC CONNECTIVES</p><p>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 xA~u! with</p><p>394 LI ET AL.</p></li><li><p>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,</p><p>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.</p><p>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 ! .</p><p>(1) The d sensitivity of f at point x, or a pointwise sensitivity, denoted by Df ~x,d! , isdefined in the form</p><p>Df ~x,d! sup $6 f ~x! f ~y!6 : y @0,1# n and 6xi yi 6 d, i1,2, . . . , n%</p><p>(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:</p><p>Df ~d! x@0,1# n</p><p>Df ~x,d!</p><p>(3) The average d sensitivity of f (average sensitivity for brief), denoted as Sf ~d! , is de-fined as follows:</p><p>Sf ~d! 0</p><p>10</p><p>1</p><p>. . . 0</p><p>1</p><p>Df ~x,d! dx1 dx2 . . . dxn</p><p>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.</p><p>ROBUSTNESS OF FUZZY REASONING 395</p></li><li><p>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.</p><p>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.</p><p>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.</p><p>In what follows, we study these notions in more detail.</p><p>2.1. The Pointwise Sensitivity of Fuzzy Connectives</p><p>Theorem 1. For a binary connective f : @0,1# 2 r @0,1# , the following holds:</p><p>(1) If f is increasing with respect to its variables, x x ', y y ' ] f ~x, y! f ~x ', y ' ! , thenDf ~~x, y!,d! @ f ~x, y! f ~~x d! 0, ~ y d! 0!# </p><p>@ f ~~x d! 1, ~ y d! 1! f ~x, y!#</p><p>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</p><p>second variable, x ' x, y y ' ] f ~x, y! f ~x ', y ' ! , thenDf ~~x, y!,d! @ f ~x, y! f ~~x d! 1, ~ y d! 0!# </p><p>@ f ~~x d! 0, ~ y d! 1! f ~x, y!#</p><p>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!</p><p>f ~x, y ' !, then</p><p>Df ~~x, y!,d! xdx 'xd</p><p>6 f ~x ', ~ y d! 0! f ~x, y!6 </p><p>xdx 'xd</p><p>6 f ~x ', ~ y d! 1! f ~x, y!6</p><p>In particular, if f is a QL implication, the above formula holds.</p><p>Proof.</p><p>(1) Suppose 6x x ' 6 d,6y y ' 6 d, then x d x ' x d, y d y ' y d. Let</p><p>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!#</p><p>396 LI ET AL.</p></li><li><p>We discuss two cases:</p><p>(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.</p><p>(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.</p><p>Hence the relation b c always holds. From the definition of Df ~~x, y!,d!, it fol-lows that Df ~~x, y!,d! c.</p><p>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.</p><p>The proofs of statements (2) and (3) are similar to that considered abovefor (1). </p><p>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:</p><p>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.</p><p>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).</p><p>Corollary 1.</p><p>(1) Let f : @0,1# 2 r @0,1# be a t-norm or an s-norm; thenDf ~~0,0!,d! f ~d,d!,Df ~~0,1!,d! Df ~~1,0!,d! dDf ~~1,1!,d! 1 f ~1 d,1 d!</p><p>(2) Let f : @0,1# 2 r @0,1# be an R implication; thenDf ~~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!</p><p>(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</p><p>Df ~~0,0!,d! 1 n~d! Df ~~0,1!,d!1 s~n~d!,1 d!</p><p>Df ~~1,0!,d! S~n~1 d!,d! Df ~~1,1!,d! d</p><p>ROBUSTNESS OF FUZZY REASONING 397</p></li><li><p>In particular, if n is the linear negation c, that is, c~x!1 x, thenDf ~~0,0!,d! d Df ~~0,1!,d!1 S~1 d,1 d!</p><p>Df ~~1,0!,d! S~d,d! Df ~~1,1!,d! d</p><p>(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</p><p>Df ~~0,0!,d! 1 n~d! Df ~~0,1!,d!1 n~d!</p><p>Df ~~1,0!,d! S~n~1 d!,d!</p><p>~1 S~n~1 d!,1 d!! d Df ~~1,1!,d! 1 t~1 d,1 d!</p><p>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</p><p>Df ~~0,0!,d! d Df ~~1,0!,d! d Df ~~1,0!,d! S~d,d!</p><p>~1 S~d,1 d!! d Df ~~1,1!,d! S~d,d!</p><p>Corollary 2.</p><p>(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-</p><p>ing relationships hold:</p><p>Df ~0,0! d Df ~~0,1!,d! 0 Df ~~1,0!, s! 2d 1</p><p>Df ~~1,1!,d! d</p><p>(3) If f ~a, b! ~1 a! b, the KleeneDienes implication, then we have:Df ~~x, y!,d! d, x 0,1, y 0,1</p><p>(4) If f ~a, b! ~1 a! ~a b! , (which is the Zadeh implication) then weobtain</p><p>Df ~~x, y!,d! d, x 0,1, y 0,1.</p><p>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 KleeneDienes implication, and QL implication and Zadeh implication leadto the similar results.</p><p>398 LI ET AL.</p></li><li><p>2.2. Maximum Sensitivity of Fuzzy Connectives</p><p>Lemma 1.</p><p>(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! .</p><p>(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! .</p><p>Proof. It can be inferred from the definition of the continuity of the function. </p><p>Lemma 2. If d1 d2 , then Df ~x,d1! Df ~x,d2 ! , and Df ~d1! Df ~d2 !, Sf ~d1!Sf ~d2 ! .</p><p>Proof. This follows from the definition of Df ~x,d!. </p><p>The following is a well-known result encountered in mathematical analysis;compare Ref. 6.</p><p>Lemma 3. Assume that f, g;Ur R is a bounded function; then</p><p>xU</p><p>f ~x! xU</p><p>g~x! xU6 f ~x! g~x!6</p><p>xU</p><p>f ~x! xU</p><p>g~x! xU6 f ~x! g~x!6</p><p>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!; thend @0,1# ,Df ~d! Dg~d! .</p><p>Theorem 3. The minimum , the maximum , and the linear negation are themost robust t-norm, s-norm, and negation, respectively.</p><p>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.</p><p>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!.</p><p>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.</p><p>ROBUSTNESS OF FUZZY REASONING 399</p></li><li><p>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 </p><p>is the most robust s-norm.Let f ~x! 1 x; it follows from Theorem 2 that Df ~d! d. Let g be any</p><p>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</p><p>Dg~d! d.Case 2. If g~d! 1 d, then 6g~1! gg~d!...</p></li></ul>


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