Upper and Lower Bounds of the α-Universal Triple I Method for Unified Interval Implications

Theα-universal triple I(α-UTI)method is a recognized scheme in the field of fuzzy reasoning,whichwas proposed by our research group previously.The robustness of fuzzy reasoning determines the quality of reasoning algorithms to a large extent,which is quantified by calculating the disparity between the output of fuzzy reasoning with interference and the output without interference.Therefore,in this study,the interval robustness(embodied as the interval stability)of theα-UTI method is explored in the interval-valued fuzzy environment.To begin with,the stability of theα-UTI method is explored for the case of an individual rule,and the upper and lower bounds of its results are estimated,using four kinds of unified interval implications(including the R-interval implication,the S-interval implication,the QL-interval implication and the interval t-norm implication).Through analysis,it is found that theα-UTI method exhibits good interval stability for an individual rule.Moreover,the stability of theα-UTI method is revealed in the case of multiple rules,and the upper and lower bounds of its outcomes are estimated.The results show that theα-UTI method is stable for multiple rules when four kinds of unified interval implications are used,respectively.Lastly,theα-UTI reasoning chain method is presented,which contains a chain structure with multiple layers.The corresponding solutions and their interval perturbations are investigated.It is found that theα-UTI reasoning chain method is stable in the case of chain reasoning.Two application examples in affective computing are given to verify the stability of theα-UTImethod.In summary,through theoretical proof and example verification,it is found that theα-UTImethod has good interval robustness with four kinds of unified interval implications aiming at the situations of an individual rule,multi-rule and reasoning chain.

Implicative and Fuzzy Implicative Ideals of Lattice Implication Algebras

In this paper, we defined the concept of implicative and fuzzy implicative ideals of lattice implication algebras, and discussed the properties of them. And then, we pointed out the relations between implicative ideal and LI _ideal, implicative iedal and implicative filter, implicative ideal and fuzzy implicative ideal, fuzzy implicative ideal and fuzzy implicative filter, and fuzzy implicative ideal and fuzzy LI _ideal.

Prime MP-filter Spaces of Fuzzy Implication Algebras

In this paper,the topological space(PF_(MP)(X),T) based on prime MP-filters of a lattice FI-algebra X is constructed firstly and we proved that it is a compact T_0-space if X with condition(P).Secondly,we restricted T to the set of all maximal MP-filters MF_(MP)(X) of X and concluded that(PF_(MP)(X),T |_(PF_(MP)(X)) )is a compact T_2 space if X with conditions(P) and(S).

Fully implicational methods for approximate reasoning based on interval-valued fuzzy sets

The aim of this paper is to discuss the approximate rea- soning problems with interval-valued fuzzy environments based on the fully implicational idea. First, this paper constructs a class of interval-valued fuzzy implications by means of a type of impli- cations and a parameter on the unit interval, then uses them to establish fully implicational reasoning methods for interval-valued fuzzy modus ponens （IFMP） and interval-valued fuzzy modus tel- lens （IFMT） problems. At the same time the reversibility properties of these methods are analyzed and the reversible conditions are given. It is shown that the existing unified forms of α-triple I （the abbreviation of triple implications） methods for FMP and FMT can be seen as the particular cases of our methods for IFMP and IFMT.

Some kinds of falling fuzzy filters of lattice implication algebras

In this paper, the concepts of falling fuzzy(implicative, associative) filters of lattice implication algebras based on the theory of falling shadows and fuzzy sets are presented at first. And then the relations between fuzzy(implicative, associative) filters and falling fuzzy(implicative, associative) filters are provided. In particular, we put forward an open question on a kind of falling fuzzy filters of lattice implication algebras. Finally, we apply falling fuzzy inference relations to lattice implication algebras and obtain some related results.

Some properties of fuzzy positive implicative ideals in a BCK-algebras

In this paper, we use the notion of fuzzy point to study ideal and positive implicative ideal in BCK algebras, then we clarify the links between the fuzzy point approach, the classical fuzzy approach and the ordinary case.

Probability representations of fuzzy systems

In this paper, the probability significance of fuzzy systems is revealed. It is pointed out that COG method, a defuzzification technique used commonly in fuzzy systems, is reasonable and is the optimal method in the sense of mean square. Based on different fuzzy implication operators, several typical probability distributions such as Zadeh distribution, Mamdani distribution, Lukasiewicz distribution, etc, are given. Those distributions act as ＂inner kernels＂ of fuzzy systems. Furthermore, by some properties of probability distributions of fuzzy systems, it is also demonstrated that CRI method, proposed by Zadeh, for constructing fuzzy systems is basically reasonable and effective. Besides, the special action of uniform probability distributions in fuzzy systems is characterized. Finally, the relationship between CRI method and triple I method is discussed. In the sense of construction of fuzzy systems, when restricting three fuzzy implication operators in triple I method to the same operator, CRI method and triple I method may be related in the following three basic ways： 1） Two methods are equivalent; 2） the latter is a degeneration of the former; 3） the latter is trivial whereas the former is not. When three fuzzy implication operators in triple I method are not restricted to the same operator, CRI method is a special case of triple I method; that is, triple I method is a more comprehensive algorithm. Since triple I method has a good logical foundation and comprises an idea of optimization of reasoning, triple I method will possess a beautiful vista of application.

Pattern classification using fuzzy relation and genetic algorithm

Purpose–This paper aims to consider a soft computing approach to pattern classification using the basic tools of fuzzy relational calculus(FRC)and genetic algorithm(GA).Design/methodology/approach–The paper introduces a new interpretation of multidimensional fuzzy implication(MFI)to represent the author’s knowledge about the training data set.It also considers the notion of a fuzzy pattern vector(FPV)to handle the fuzzy information granules of the quantized pattern space and to represent a population of training patterns in the quantized pattern space.The construction of the pattern classifier is essentially based on the estimate of a fuzzy relation Ri between the antecedent clause and consequent clause of each one-dimensional fuzzy implication.For the estimation of Ri floating point representation of GA is used.Thus,a set of fuzzy relations is formed from the new interpretation of MFI.This set of fuzzy relations is termed as the core of the pattern classifier.Once the classifier is constructed the non-fuzzy features of a test pattern can be classified.Findings–The performance of the proposed scheme is tested on synthetic data.Subsequently,the paper uses the proposed scheme for the vowel classification problem of an Indian language.In all these case studies the recognition score of the proposed method is very good.Finally,a benchmark of performance is established by considering Multilayer Perceptron(MLP),Support Vector Machine(SVM)and the proposed method.The Abalone,Hosse colic and Pima Indians data sets,obtained from UCL database repository are used for the said benchmark study.The benchmark study also establishes the superiority of the proposed method.Originality/value–This new soft computing approach to pattern classification is based on a new interpretation of MFI and a novel notion of FPV.A set of fuzzy relations which is the core of the pattern classifier,is estimated using floating point GA and very effective classification of patterns under vague and imprecise environment is performed.This new approach to pattern classification avoids the curse of high dimensionality of feature vector.It can provide multiple classifications under overlapped classes.