The Robustness Of Fuzzy Reasoning And Solutions Of Fuzzy Relation Equations

In this article, the robust analysis of fuzzy reasoning and solutions of fuzzy relation equations is discussed. As is well known, robustness theoryis always an important issue in the field of control theory. With the developmentof society, the application of fuzzy mathematics has been becoming broad, day byday, which relates to many aspects as follows: automatic control, the image andcharacters identification, artificial intelligence, geology earthquake, the malftmctiondiagnose, the analysis of meteorological phenomena, aerospace aviation, train automobile driving, traffic control, decision-making appraise, business administrationand economy, and so on. By the same token, the reasearch of perturbation theoryof fuzzy reasoning is imperative. That is, will a small variance of input cause a bigvariance of output of fuzzy reasoning? Now the different fuzzy reasoning, method hasbeen proposed. If perturbation parameters of various methods of fuzzy reasoningare estimated, then these perturbation parameters might serve as a certain criteriafor choices of methods of fuzzy reasoning in practical processes. In recent years, there have a number of progress in doing research in perturbation parameters offuzzy system.Mingsheng Ying [34] proposed the concepts of maximum and average perturations of fuzzy sets. Subsequently perturbation parameters of various methodsof fuzzy reasoning could be estimated. Cai[2] used the notion ofδ—equalities offuzzy sets(being dual to the maximum perturbation of fuzzy sets of Ying's) to studyrobustness of fuzzy reasoning that Ied to some general results in case of fuzzy con-nectives, fuzzy implication opexators, and generalized modus ponens and generalizedmodus tollens. Some others approached this concept from a different perspectiveas presented in references[26, 29]; reference also to some comments made in reference[11]. Considering that the results of fuzzy reasoning are heavily dependent onthe choice of fuzzy sets of fuzzy antecedent and fuzzy consequences as well as fuzzyconnectives and fuzzy implication operators that link fuzzy antecedents and fuzzyconsequences, Yongming Li, etc.[16, 17] investigated the issues of the robustness offuzzy logic connectives and implication operators and develop some estimates ofrobustness of the main schemes of fuzzy reasoning (as will be shown, those are lessconservative than those proposed by Ying and Cai). In the previous work, the perturbation of fuzzy sets was expressed based on thenotion of the maximum perturbation orδequalities of fuzzy sets via the distancemeasure on the unit interval [0, 1]. However, the behavior of a fuzzy logic system ismainly determined by its internal logic structure—the fuzzy connectives and fuzzyimplication operators. That doing research in the robustness of fuzzy reasoningbased on logically equivalent measure is rather theoretical and more practical. Having this in mind, the robustness results of the main schemes of fuzzy reasoning arederived and the problem posed in[2, 17] are solved too. Based on logically equivalentmeasure, the definition of maximum perturbation of fuzzy sets is proposed, firstly.Secondly, robustness results for various implication operators, inference rules andfuzzy reasoning machines are presented, and the relations between the robustnessof fuzzy reasoning and that of both fuzzy conjunction and implication operators arediscussed. Where, all the robustness results are presented in terms ofδ-equalitiesof fuzzy sets, and the maximum ofδthat ensures the correspondingδ—equalityholds is derived.In other aspect, the majority of fuzzy inference systems can be implemented byusing fuzzy relation equations. And there are many different types of fuzzy relationequations corresponding to different methods of fuzzy inference. Now the projectthat the analysis of the perturbation of fuzzy relation equations is the hot topic caredby research of theory and application of project, and there have many progress inthese fields studied by many authors. However, these conclusions and properties arenot derived from the aspect of how the solutions of fuzzy relational equation areimpacted by all perturbable elements of fuzzy matrix. Consequently, in this paper, the perturbation parameters of solutions of fuzzy relation equations are presentedbased on the definition of all perturbable elements of fuzzy matrix. Some usefulresults are obtained and the problem posed in[27] is solved. At first, the definitionof the perturbation of fuzzy relational equation based on max-* composition by theuse of the index of measure are established. Secondly the changing condition of thesolutions of fuzzy relational equation and its perturbable equation are discussed.The last a formula to estimate the perturbation of fuzzy relational equation, thechange of the number of its maximal solution and minimal solutions are derived.Noting that the exprensive application of logically equivalence measure. Basedon some special impliation operator and triangler norm, and the definition of maximum perturbation of fuzzy sets based on logically equivalent measure, the changingof the existence of solutions of fuzzy relation equation and its perturbable equation are discussed. The last but not the least, the perturbation parameters of the biggestand the minimal solutions are estimated.