Granular fuzzy rough sets based on fuzzy implicators and coimplicators
Introduction
Fuzzy implicator (coimplicator can be regarded as a negation dual operator of implicator) is one of the main operations in fuzzy logic. It generalizes the classical implicator, which ranges in , in the context of fuzzy logic, where the truth values belong to . There are different models for fuzzy implicators. Some of them are derived from t-norms and t-conorms, such as -implicator [2], R-implicators [14], QL-implicators [5] and D-implicators. Moreover, these types of implicators have been extended to replace t-norm and t-conorm with some other binary operators, such as copulas [17], uninorms [4], [13], [21], overlap and grouping functions [16], and representable aggregation functions [23].
Fuzzy rough set theory [35] is a general mathematical tool for data analyzing. It can deal with the fuzziness of concepts and vagueness of information in real life, as an extension of rough set theory [25]. Granular structure of fuzzy rough sets was first introduced by Chen et al. [12]. They proposed the fuzzy granules based on t-norms and t-conorms on fuzzy similarity relations. Yao et al. [33] generalized the granular fuzzy rough sets into granular variable precision fuzzy rough sets. Considering the limitation of fuzzy similarity relations, Wang and Hu [31] proposed granular variable precision fuzzy rough sets with general fuzzy relations. Moreover, Qiao and Hu [26] studied granular variable precision L-fuzzy rough sets based on residuated lattices. On the other hand, Dubois and Prade [15] firstly proposed fuzzy rough sets based on Max and Min. Then Radzikowska and Kerre [27] generalized fuzzy rough sets from Max and Min to a border implicator and a t-norm. Another upper approximation operator based on coimplicator was proposed by Mi and Zhang [22] to obtain the dual approximation operator. Wu et al. [32] generalized fuzzy rough approximation operators determined by fuzzy implicators. However, the granular structure of fuzzy rough sets based on implicators and coimplicators has not been studied. The main aim of this paper is to fill this gap.
In this paper, we present a general framework for studying the granular structure of fuzzy rough sets based on fuzzy implicators and coimplicators. Moreover, granular fuzzy rough sets and granular variable precision fuzzy rough sets determined by implicators and coimplicators are discussed on arbitrary fuzzy relations from the mathematical point of view. There has been no comparative analysis of the properties that hold for the granular structure of fuzzy rough sets based on fuzzy implicators and coimplicators when the fuzzy relations are arbitrary. This suggests that the fuzzy granules based on fuzzy implicators and coimplicators in arbitrary fuzzy relations should be studied more thoroughly in this context.
The remainder of this paper is organized as follows: Section 2 reviews briefly related basic concepts and proposes our motivations. Section 3 introduces the constructive and axiomatic approach to fuzzy granules based on fuzzy implicators and coimplicators. Moreover, we consider the connection between fuzzy granules and fuzzy relations and give some properties with proofs. Section 4 introduces granular structures of lower and upper approximations of fuzzy sets based on implicators and coimplicators, and studies the relations between granular structures of lower and upper approximations and four lower and upper approximation operators. Section 5 proposes granular variable precision fuzzy rough sets based on fuzzy implicators and coimplicators. Finally, Section 6 concludes the paper.
Section snippets
Related work and motivation
Let U and W be two non-empty universes of discourse and R be a fuzzy relation from U to W. The triple is called a fuzzy approximation space. When and R is a fuzzy relation on U, we call a fuzzy approximation space.
We denote for simplicity and denote as a family of fuzzy sets on U and , , .
Fuzzy granules based on fuzzy implicators and coimplicators
In this section we define four kinds of fuzzy granules based on fuzzy implicators and coimplicators, employ constructive and axiomatic approaches to investigate the properties of these fuzzy granules, compare fuzzy granules introduced in this paper with fuzzy granules proposed in [12], and examine the connection between fuzzy granules and fuzzy relations.
Granular structures of fuzzy sets based on fuzzy implicators and coimplicators
In this section, we introduce granular structures of lower and upper approximations of fuzzy sets based on implicators and coimplicators, and study the relations between granular structures of lower and upper approximations and four lower and upper approximation operators , , and . Definition 4.1 Suppose U is a universe of discourse, R is a fuzzy relation on . For , define Then ,
Granular variable precision fuzzy rough sets based on fuzzy implicators and coimplicators
The granular structures of lower and upper approximations of fuzzy sets described in Section 4, are consistent with the approach of defining the lower and upper approximations of crisp sets in classical rough sets. However, this kind of fuzzy rough set model can not tolerate even small errors and it is not very suitable to process uncertain information. To remedy these defects, a new granular variable precision fuzzy rough set model is proposed in this section. Then we present four propositions
Conclusion
In the paper, the granular structures of fuzzy rough sets based on the fuzzy implicators and coimplicators have been discussed. Both of the constructive and axiomatic approach to fuzzy granules based on fuzzy implicators and coimplicators have been studied. For different types of fuzzy implicators and coimplicators, we have approximated the novel granular variable precision fuzzy rough set model efficiently. This paper provides a theoretical background for the granular variable precision fuzzy
Acknowledgements
This research was supported by the National Natural Science Foundation of China (Grant nos. 11571010 and 61179038).
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2022, Fuzzy Sets and SystemsCitation Excerpt :Fuzzy implication operators play an important role in fuzzy logic and approximate reasoning, as well as in many research areas where these theories apply, including composition of fuzzy relations [10,16], fuzzy relational equations [18,24], fuzzy mathematical morphology [33], fuzzy neural networks [42], fuzzy rough sets [11,17], data mining [45], and so on.