SINCE Zadeh s compositional rule of fuzzy inference

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1 IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 14, NO. 6, DECEMBER Error Estimation of Perturbations Under CRI Guosheng Cheng Yuxi Fu Abstrat The analysis of stability robustness of fuzzy reasoning is an important issue in areas like intelligent systems fuzzy ontrol. An interesting aspet is to what extent the perturbation of input in a fuzzy reasoning sheme auses the osillation of the output. In partiular, when the error limits (restritions) of the input values are given, what the error limits of the output values are. In this orrespondene, we estimate the upper lower bounds of the output error affeted by the perturbation parameters of the input, obtain the limits of the output values when the input values range over some interval in many fuzzy reasoning shemes under ompositional rule of fuzzy inferene (CRI). Index Terms Error estimation, fuzzy reasoning, fuzzy set, interval perturbation, simple perturbation. I. INTRODUCTION SINCE Zadeh s ompositional rule of fuzzy inferene [1] (CRI) was proposed, many other methods of fuzzy reasoning have been known [2] [10]. Appliations of these fuzzy reasoning tehniques have been suessful in various areas, espeially in fuzzy ontrol [11]. When fuzzy reasoning is applied, the stability robustness of the fuzzy reasoning beomes one of the prominent problems. The fuzzy ontrollers are available to transform human expertise subjetivity to quantitative terms, so the deviation of human expertise from its orresponding quantitative representations gives rise to the problem of the stability of fuzzy ontrollers [11]. The orresponding problem in fuzzy reasoning is the variane of output aused by perturbations of input. Here, the analysis of the stability of a fuzzy reasoning sheme onsists of two aspets: One is how the output values of the sheme are hanged by the perturbation parameters of input values. The other is how to estimate orresponding limits of output values of the sheme when osillation limits of input values are given. For the first question, many researhers have provided their answers [12] [15]. Their approahes to perturbations of input are based on some proximity of fuzzy sets, using proximity measure in [12], or -similarity measure in [13], or maximum perturbation in [14], or -equality in [15]. In a sense, the problem of stability of fuzzy reasoning has been well studied. However, the effets of the perturbation parameters of the input in fuzzy reasoning shemes still all for investigation. For one thing, when a Manusript reeivedaugust 9, 2003; revised April 13, 2005 April 19, The work was supported by the National Distinguished Young Sientist Fund of NNSFC under Grant , by the National 973 Projet under 2003CB317005, by the National Nature Siene Foundation of China under Grant , by the BoShiDian Researh Fund under , by the Jiangsu Eduation Offie Researh Fund under Grant 05KJD110123, by the Nanjing University of Information Siene Tehnology Researh Fund under Grant QD39. G. Cheng is with the Department of Mathematis, Nanjing University of Information Siene Tehnology, Nanjing , China ( gshheng@sohu.om). Y. Fu is with the Department of Computer Siene, Shanghai Jiaotong University, Shanghai , China ( fu-yx@s.sjtu.edu.n). Digital Objet Identifier /TFUZZ sequene of the perturbations of the input for a fuzzy reasoning sheme has an asymptoti limit, it is obvious that the orresponding sequene of the output annot be preisely demonstrated in the above approahes. The orrespondene is strutured as follows: After introduing the onepts of the simple perturbation the interval perturbation of the fuzzy sets, we obtain an estimation of the upper the lower bounds of the output error affeted by the simple perturbation of the input under CRI. The stability of fuzzy reasoning shemes is haraterized. Also the asymptoti performane in fuzzy reasoning shemes is exhibited. Next we investigate interval perturbation of input in fuzzy reasoning get the interval estimation of fuzzy sets inferred from CRI with some abstrat impliation operators onjuntion operators. Some final remarks are made in Setion V. II. PRELIMINARIES The basi form of the CRI methods is as follows: Anteedent If is then is Fat is Conlusion where, are fuzzy sets on are fuzzy sets on usually defined as In the previous definition is some onjuntion operator, is an impliation operator. The pair is alled a sheme of fuzzy reasoning. The ordered triad is alled an input an output of fuzzy reasoning. Usually is either,orprodut,oralukasiewiz onjuntion operator (i.e.,, ), or a -norm or a -onorm. A. Some Notions The -norms, -onorms, the fuzzy omplementary operations in will be fundamental for the present orrespondene. A funtion is alled t-norm if only if 1) is nondereasing in eah argument; 2) is ommutative; 3) is assoiative; 4) has 1 as unit. A funtion is alled -onorm if only if satisfies 1) 3), as well as 4 ) given as follows: 4') has 0 as unit. is /$20.00 IEEE

2 710 IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 14, NO. 6, DECEMBER 2006 A funtion is alled fuzzy omplement if only if 1), ; 2) is noninreasing. In this orrespondene the impliation operator will st for one of the following, where : Mamdani: ; Kleene Dienes: ; Lukasiewiz: ; Reihenbah: ; Zadeh: ; in [16]: ; -impliations:, where is a fuzzy omplement, is a -onorm; -impliations:, where is a t-norm; -impliations:, where is required to be a de Morgan triplet; -norm impliations:. Now, we introdue the onepts of simple perturbation, interval perturbation of a fuzzy set stability of a fuzzy reasoning sheme. Definition 1: Let be a universe of disourse, two fuzzy sets defined on. If there exists a mapping :, suh that for all,, then is alled a simple perturbation of is a fator of the perturbation of. The following definition will use the onept of fuzzy interval. If, are two fuzzy sets on, for all,, then is alled a fuzzy interval on. Definition 2: Let be a fuzzy interval on, a fuzzy set on. If for all,, then has an interval perturbation on, written. Definition 3: Let be fuzzy sets on, a fuzzy set on. Suppose, are perturbations of, with fator,, respetively. A fuzzy reasoning sheme is said to be stable if, given, there exists suh that on whenever,. A perturbation sequene of an input with a orresponding sequene of fators in some fuzzy reasoning sheme is said to be (asymptoti) stable if, given, there exist some natural number, suh that on for all,,,. Definition 4: Let, be fuzzy sets on, a fuzzy set on. A fuzzy reasoning sheme is said to be stable if, given, there exist a fuzzy interval on, suh that for eah,, whenever,,. B. Related Work 1) Pappis Work: Pappis introdued the approximately equal of two fuzzy sets on in [12], i.e., let be fuzzy sets on,if then are said to be approximately equal, denoted by. is said to be a proximity measure of. The Pappis result is as follows. Let be fuzzy sets on, fuzzy relations from to. Then implies ; implies ; where is the max min omposition. We see that this result atually addresses the stability of some fuzzy reasoning shemes when,, of the input are perturbed, respetively. 2) Hong Hwang s Work: Hong Hwang defined the -similarity of two fuzzy sets on in [13], i.e., let be fuzzy sets on,if then are said to be -similar, denoted by. Hong Hwang generalized the Pappis result to be the following. Let be fuzzy sets on, fuzzy relations from to. If, then. By means of this result, the stability of some fuzzy reasoning shemes is obtained when,, of the input are all perturbed. 3) Ying s Work: Ying introdued the onept of maximum perturbation of fuzzy set in [14] as follows. Let be fuzzy sets on. If for eah,, then is alled a maximum perturbation of. One of the main results in [14] was to obtain the maximum perturbation of the output when all of, of the input have the maximum perturbations. See [14] for more details. Evidently, the stability of some fuzzy reasoning shemes may be preisely haraterized by the Ying s result. 4) Cai s Work: Cai used the term -equality in [15] as follows. Let be fuzzy sets on. Then, are said to be -equal, if,. Cai investigated -equalities for some impliation operators, -onorm, fuzzy relations generalized modus pollens in [15]. The stability instability of some fuzzy reasoning shemes are easily addressed by the Cai s results. C. Two Lemmas Lemma 1 an be easily established. Lemma 1: Let be bounded, real-valued funtions defined on (or ), fuzzy sets on. Then, the following properties hold: 1) ; 2) ; 3) ; 4) ; 5) ; 6) ; 7), ;

3 CHENG AND FU: ERROR ESTIMATION OF PERTURBATIONS UNDER CRI 711 8),. If some impliation operators are perturbed, one gets the following. Lemma 2: Let be a fuzzy set on, a fuzzy set on., are fators of perturbation of, respetively. Let st for Then, the following inequalities hold. 1) If is Mamdani impliation, then 2) If is Kleene Dienes impliation, then Again by 3) of Lemma 1, one has that 3) If is Lukasiewiz impliation, then 4) If is Reihenbah impliation, then that 5) If is Zadeh impliation, then 6) If is in [16], then Therefore by 1) 3) of Lemma 1, one has the equation shown at the bottom of the page, that Proof: We only prove 5). The proofs of the other ases are similar. For,, one has, by 2) of Lemma 1, that It follows that III. SIMPLE PERTURBATION In this setion, when the input values are simply perturbed, we estimate the upper lower bounds of the output values in

4 712 IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 14, NO. 6, DECEMBER 2006 fuzzy reasoning aording to some hoie of onjuntion impliation operators. Let be, where are the perturbations of fuzzy sets on, is the perturbation of the fuzzy set on. For simpliity, we denote,,, by,,, respetively. The next three theorems desribe the main results. Theorem 1: Let be fuzzy sets on, a fuzzy set on. Suppose, are perturbations of, with fators,, respetively. If is then the following properties hold. 1) If is Mamdani impliation, then By 3) of Lemma 2, one has 2) If is Kleene Dienes impliation, then 3) If is Lukasiewiz impliation, then Obviously,. We are done by applying (7) (8) of Lemma. When is produt, we have the following results. Theorem 2: Let be produt. The other onditions are the same as in Theorem 1. Then, the following inequalities hold. 1) If is Mamdani impliation, then 4) If is Reihenbah impliation, then 2) If is Kleene Dienes impliation, then 5) If is Zadeh impliation, then 3) If is Lukasiewiz impliation, then 6) If is in [16], then 4) If is Reihenbah impliation, then 7) If is Zadeh impliation, then 5) If is Zadeh impliation, then Proof: We only provide the proof of 3). The proofs of the other ases are similar. By 4) 6) of Lemma 1, one obtains that 6) If is as defined in [16], then Proof: We only give the proof of 2). To start with, one has by 2) of Lemma 2 that

5 CHENG AND FU: ERROR ESTIMATION OF PERTURBATIONS UNDER CRI 713 6) If is as defined in [16], then Then Proof: Now, we only verify that 4) holds. Sine the equation shown at the bottom of the page holds, one has. Therefore The rest of the proof of is similar. If is the Lukasiewiz onjuntion, then we have the following result. Theorem 3: Let be Lukasiewiz onjuntion. The other onditions are the same as in Theorem 1. Then, the following properties hold. 1) If is Mamdani impliation, then 2) If is Kleene Dienes impliation, then It is lear that. Thus We omit the details of the rest of the proof of 4) beause of similarity. The proofs of the other assertions are similar. Let, be fuzzy sets on, a fuzzy set on. Suppose that there exists a perturbation sequene of an input with a orresponding sequene of fators in some fuzzy reasoning sheme. If,, satisfy where or 3) If is Lukasiewiz impliation, then Then, by previous theorems, we have 4) If is Reihenbah impliation, then 5) If is Zadeh impliation, then On the other h, if there are a small positive real number, mappings for or suh that, where,, then

6 714 IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 14, NO. 6, DECEMBER 2006 Now it is lear that the aformentioned fuzzy reasoning shemes using Zadeh s CRI methods are stable. Their asymptoti performane has been demonstrated. are gradually ontrated, i.e., there exists a fuzzy interval perturbation sequene of the input suh that IV. INTERVAL PERTURBATION In this setion,, where is some -norm or -onorm, is one of -impliations, -impliations, -impliations, or -norm impliations. We give the estimations of the limits of the output with the interval perturbation of the input in CRI methods. Let be fuzzy intervals on, a fuzzy interval on, st for,,,, respetively. The main results are as follows. Theorem 4: Suppose that, are fuzzy sets on, is a fuzzy set on, is either a -norm or a -onorm. 1) If is one of -impliations, -impliations, then 2) If is an -impliation, then where, are, respetively, a -norm, a -onorm a fuzzy omplement. 3) If is a -norm impliation, then Proof: 1) Using the properties of -impliations, -impliations, -norms, or -onorms, we have The inequalities of 2) 3) are verified diretly from the monotoniity of the -norm, -onorm, the fuzzy omplement. We omit the details. When the -norm, -onorm, the fuzzy omplement are ontinuous in Theorem 4, the fuzzy reasoning sheme using the Zadeh s CRI method with (some -norm or -onorm) an -impliation or an -impliation, or a -norm impliation is stable. When the osillation limits of an input in some fuzzy reasoning sheme using Zadeh s CRI methods hold, then the output values onverge stably to some value. On the other h, the fuzzy reasoning shemes using Zadeh s CRI methods with ( -norm or -onorm) some -impliations are not stable. V. CONCLUSION We know that a fuzzy reasoning sheme applied to pratie is probably perturbed by noises in various ways. In addition to many proximity measures of fuzzy sets, the simple perturbation interval perturbation may effetively simulate suh noises as well. In fat, Ying s maximum perturbation of a fuzzy set [14] is a simple perturbation, some error estimation of the output values in fuzzy reasoning is more preise using the approah of this orrespondene. Pappis proximity measure of two fuzzy sets [12], Hong Hwang s -similarity of two fuzzy sets [13], Cai s -equality of two fuzzy sets [15] Ying s maximum perturbation of a fuzzy set are all formulated by the interval perturbation of the fuzzy sets. Therefore, in ertain sense this orrespondene is a further development of the previous work. On the other h, we take into aount the effets of realisti noise aurately evaluate the output errors of fuzzy reasoning. Therefore, we may hoose a fuzzy reasoning sheme aording to the requirement of the output errors in appliations. ACKNOWLEDGMENT The authors would like to thank the referees for the invaluable omments suggestions. The proof of 1) in Theorem 4 is due to the anonymous referee. REFERENCES [1] L. A. Zadeh, The onept of a linguisti variable its appliations to approximate reasoning, I,II,III, Inform. Si., vol. 8, pp , [2] D. Dubois H. Prade, Fuzzy sets in approximate reasoning, part 1: Inferene with possibility distributions, Fuzzy Sets Syst., vol. 40, pp , [3] H. Nakanishi, I. B. Turksen, M. Sugeno, A review omparison of six reasoning methods, Fuzzy Sets Syst., vol. 57, pp , [4] E. S. Lee Q. Zhu, Fuzzy Evidene Reasoning. Hiedelberg, Germany: Physia-Verlag, [5] J. W. Guan D. A. Bell, Approximate reasoning evidene theory, Inform. Si., vol. 96, pp , [6] W. H. Hsiao, S. M. Chen, C. H. Lee, A new interpolative reasoning methods in sparse rule-based systems, Fuzzy Sets Syst., vol. 93, pp , [7] J. L. Castro, E. Trillas, J. M. Zurita, Non-monoti fuzzy reasoning, Fuzzy Sets Syst., vol. 94, pp , [8] Y. Liu E. E. Kerre, An overview of fuzzy quantifiers(ii): Reasoning appliations, Fuzzy Sets Syst., vol. 95, pp , [9] M. S. Ying, A logi for approximate reasoning, J. Symb. Logi, vol. 59, pp , 1994.

7 CHENG AND FU: ERROR ESTIMATION OF PERTURBATIONS UNDER CRI 715 [10] G. J. Wang, On the logi foundation of fuzzy reasoning, Inform. Si., vol. 117, pp , [11] L. X. Wang, A Course in Fuzzy Systems Control. Englewood Cliffs, NJ: Prentie-Hall, [12] C. P. Pappis, Value approximation of fuzzy systems variables, Fuzzy Sets Syst., vol. 39, pp , [13] D. H. Hong S. Y. Hwang, A note on the value similarity of fuzzy systems variables, Fuzzy Sets Syst., vol. 66, pp , [14] M. S. Ying, Perturbation of fuzzy reasoning, IEEE Trans. Fuzzy Syst., vol. 7, no. 5, pp , Ot [15] K. Y. Cai, Robustness of fuzzy reasoning -equalities of fuzzy sets, IEEE Trans. Fuzzy Syst., vol. 9, no. 5, pp , Ot [16] M. S. Ying, Impliation operators in fuzzy logi, IEEE Trans. Fuzzy Syst., vol. 10, no. 1, pp , Feb [17] S. Weber, A general onept of fuzzy onnetives,negations impliation based on t-norms t-onorm, Fuzzy Sets Syst., vol. 11, pp , [18] D. Boixader L. Godo,, E. H. Ruspini, P. P. Bonissone, W. Pedryz, Eds., Fuzzy inferene, in Hbook of Fuzzy Computation. Philadelphia, PA: Inst. Phys., [19] R. Fullér,, C. Carlsson, Ed., On fuzzy reasoning shemes, in The State of the Art of Information Systems Appliations in Turku, Finl: TUCS General Publiations, 1999, vol. 16, pp Guosheng Cheng reeived the B.S. degree from Huaibei Coal Industry Teaher s College, Huaibei, China, in 1986, the M.S. Ph.D. degrees from Xi an Jiaotong University, Xi an, China, in , respetively. He is a Professor in the Department of Mathematis, Nanjing University of Information Siene Tehnology, Nanjing, China. His researh interests inlude intelligent omputation omputability of the measures. Yuxi Fu reeived the B.S. degree from Tongji University, Shangai, China, the Ph.D. degree from the University of Manhester, Manhester, U.K., in , respetively. He is a Full Professor with the Department of Computer Siene, Shanghai Jiaotong University, Shangai, China. His researh interest lies in theoretial omputer siene, espeially in type theory onurreny theory. His reent work has foused on the studies of proess aluli, suh as pi-alulus hi-alulus.