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Abstract: Two basic inference models of fuzzy reasoning are fuzzy modus ponens (FMP) and fuzzy modus tollens (FMT). The Triple I method is a very important method to solve the problems of FMP and FMT. The aim of this paper is to extend the Triple I method of approximate reasoning on Atanassov's intuitionistic fuzzy sets. In the paper, we first investigate the algebra operators' properties on the lattice structure of intuitionistic fuzzy information and provide the unified form of residual implications which indicates the relationship between intuitionistic fuzzy implications and fuzzy implications. Then we present the intuitionistic fuzzy reasoning version of the Triple I principles based on the models of intuitionistic fuzzy modus ponens (IFMP) and intuitionistic fuzzy modus tollens (IFMT) and give the Triple I method of intuitionistic fuzzy reasoning for residual implications. Moreover, we discuss the reductivity of the Triple I methods for IFMP and IFMT. Finally, we propose α-Triple I method of intuitionistic fuzzy reasoning. [Copyright &y& Elsevier]

Abstract: In this paper, we propose a granularity-based framework of deduction, induction, and abduction using variable precision rough set models proposed by Ziarko and measure-based semantics for modal logic proposed by Murai et al. The proposed framework is based on -level fuzzy measure models on the basis of background knowledge, as described in the paper. In the proposed framework, deduction, induction, and abduction are characterized as reasoning processes based on typical situations about the facts and rules used in these processes. Using variable precision rough set models, we consider -lower approximation of truth sets of nonmodal sentences as typical situations of the given facts and rules, instead of the truth sets of the sentences as correct representations of the facts and rules. Moreover, we represent deduction, induction, and abduction as relationships between typical situations. [Copyright &y& Elsevier]

Rough set theory is an important approach to granular computing. Type-1 fuzzy set theory permits the gradual assessment of the memberships of elements in a set. Hybridization of these assessments results in a fuzzy rough set theory. Type-2 fuzzy sets possess many advantages over type-1 fuzzy sets because their membership functions are themselves fuzzy, which makes it possible to model and minimize the effects of uncertainty in type-1 fuzzy logic systems. Existing definitions of type-2 fuzzy rough sets are based on vertical-slice or α -plane representations of type-2 fuzzy sets, and the granular structure of type-2 fuzzy rough sets has not been discussed. In this paper, a definition of type-2 fuzzy rough sets based on a wavy-slice representation of type-2 fuzzy sets is given. Then the concepts of granular type-2 fuzzy sets are proposed, and their properties are investigated. Finally, granular type-2 fuzzy sets are used to describe the granular structures of the lower and upper approximations of a type-2 fuzzy set, and an example of attribute reduction is given. [ABSTRACT FROM AUTHOR]

Abstract: This paper focuses on modelling fuzzy numbers with meaningful membership functions. More precisely, it proposes a method to construct trapezoidal fuzzy number approximations from raw discrete data. In many applications, input information is numerical, and therefore, particular fuzzy sets, such as fuzzy numbers, hold great interest and relevance in managing data imprecision and vagueness. The proposed technique provides an efficient way to obtain trapezoidal numbers using linear regression. The technique is simple, fast, and effective. Preliminary tests are performed using different types of input data: a Gaussian function, a Sigmoidal function, three datasets of synthetic discrete data, and an histogram obtained from a colour satellite image. [Copyright &y& Elsevier]

Abstract: This paper aims to give a new perspective for definability in rough set theory. First, a counterpart of definability is introduced in textural approximation spaces. Then a complete field of sets for texture spaces is defined and using textural arguments, some new results are obtained for rough sets. It is shown that definability can be also discussed in terms of a complete field of fuzzy sets on a fuzzy lattice for the various fuzzy approximation spaces. It is also given a partial affirmative answer to an open problem posed by Wei-Zhi Wu in On some mathematical structures of T-fuzzy rough set algebras in infinite universes of discourse in Fundamenta Informaticae 108 (3–4) 2011 337–369. [Copyright &y& Elsevier]

Abstract: To discover knowledge from hybrid data using rough sets, researchers have developed several fuzzy rough set models and a neighborhood rough set model. These models have been applied to many hybrid data processing applications for a particular purpose, thus neglecting the issue of selecting an appropriate model. To address this issue, this paper mainly concerns the relationships among these rough set models. Investigating fuzzy and neighborhood hybrid granules reveals an important relationship between these two granules. Analyzing the relationships among rough approximations of these models shows that Hu’s fuzzy rough approximations are special cases of neighborhood and Wang’s fuzzy rough approximations, respectively. Furthermore, one-to-one correspondence relationships exist between Wang’s fuzzy and neighborhood rough approximations. This study also finds that Wang’s fuzzy and neighborhood rough approximations are cut sets of Dubois’ fuzzy rough approximations and Radzikowska and Kerre’s fuzzy rough approximations, respectively. [Copyright &y& Elsevier]

Abstract: The approximation space model was originally proposed by Pawlak (1982) . It was Orłowska who first observed that approximation spaces serves as a formal counterpart of perception, or observation [16, §2, p. 8], in which approximations provide a means of approximating one set of objects with another set of objects using the indiscernibility relation. Topology has been used to enrich the original model of an approximation space as well as more recent models of generalized approximation spaces. In this paper, an extension of th e topology neighborhood based on AFS (Axiomatic Fuzzy Sets) theory is introduced, and some interesting properties are given. Furthermore, a new generalized approximation space model is established with two application examples, which can be used to deal with information tables with many category features and viewed as a multi-granulations form of nearness approximation space models. [Copyright &y& Elsevier]

Abstract: Similarity based rough set theory (RST) involving choice in the formation of approximations was recently introduced by the present author. Though the theory can be used to develop improved semantics and models of knowledge and belief with ontology, application requires a priori concepts of granules and granulation as opposed to the more common a posteriori or not a priori concepts of the same prevalent in the literature. In this research, we clarify the desirable semantic features of a context for seamless application of the theory to more general situations, formalise them and refine the semantics. A new axiomatic theory of granules in general RST (including hybrid versions involving fuzzy set theories) is also developed in the process. Interesting new applications to human learning are also illustrated in this paper. [ABSTRACT FROM AUTHOR]

Abstract: Rough set theory provides a methodology for data analysis based on the approximation of concepts in information systems. It revolves around the notion of discernibility: the ability to distinguish between objects, based on their attribute values. It allows to infer data dependencies that are useful in the fields of feature selection and decision model construction. In many cases, however, it is more natural, and more effective, to consider a gradual notion of discernibility. Therefore, within the context of fuzzy rough set theory, we present a generalization of the classical rough set framework for data-based attribute selection and reduction using fuzzy tolerance relations. The paper unifies existing work in this direction, and introduces the concept of fuzzy decision reducts, dependent on an increasing attribute subset measure. Experimental results demonstrate the potential of fuzzy decision reducts to discover shorter attribute subsets, leading to decision models with a better coverage and with comparable, or even higher accuracy. [Copyright &y& Elsevier]