On two new types of fuzzy rough sets via overlap functions and corresponding applications to three-way approximations
Introduction
The study and applications of new rough set models and their relevant properties have been a significant research subject in the research of rough set theory. This article attempts to undertake further research behind this line of work. Precisely, we propose two new types of fuzzy rough sets (FRSs) via overlap functions and their residual implications. Then, we put forward a novel three-way decision (3WD) method on an information table by using the new type of FRSs. Next, we introduce the relevant studies on overlap functions, FRSs as well as 3WDs in a short way.
The procedure of combining multiple values into one representative value is referred to as aggregation, and the numerical operators that implement this process are known as aggregation functions. Overlap functions are peculiar aggregation functions introduced by Bustince et al. [4], which are increasing continuous commutative binary functions defined on the unit closed interval , satisfying appropriate boundary constraints. This notion originates from some real-life problems in classification [24] and image processing [2]. Over the past decade, overlap functions have yielded a wealth of results in terms of theory and practical applications.
In theory, there are a number of discussions involving various aspects of overlap functions, like the work associated with important properties [33], [29], the additive and multiplicative generator pairs [10], [32], the corresponding implications [9], the interval overlap functions [31], the evaluation functions induced by overlap functions in a 3WD space [19] and so forth. In applications, overlap functions play a crucial position in numerous aspects of real problems, for example, in image processing [2], fuzzy community detection problems [13], decision-making problems [12], etc.
At the beginning of the 1980s, Pawlak [28] developed the concept of rough sets, which offers a lower and upper approximations of a crisp concept with the help of an equivalence relation of objects in a universe of discourse. Being a novel mathematical tool for processing uncertainty, rough sets have received a lot of attention in both theoretical studies and actual applications. In theory, one research direction is the model of rough sets. For example, Żakowski [50] proposed the concept of covering-based rough sets by means of extending the partition to a covering. Hu et al. [17] defined the neighborhood rough sets by introducing the neighborhood similarity relations of objects. Another research direction is the discussion of the characteristics and topological structures of the rough approximation operators. For instance, Liu [22] studied one axiom to characterize the rough approximation operators generated by the reflexive, symmetric, and transitive relations, respectively. Han [14] explored the topological (interior and closure) operators of the M-rough set structure and the Marcus-Wyse topological rough set structure. Applications of rough set theory are widespread and are especially prominent in feature selection and more specific in classification. For example, Chen and Yang [5] designed an attribute reduction method for decision systems with symbolic and real-valued condition attributes by composing classical rough set and FRS models. Based on Bayesian decision theory, Yao [39] investigated a decision-theoretic rough set, which classifies a universal set into three pairwise disjoint parts: acceptance, non-commitment and rejection. In addition, it is vital to note that the study and application of novel models of rough sets and their associated properties have been uninterrupted for nearly four decades.
However, since the traditional rough set is designed to process qualitative (discrete) data, it faces important limitations when dealing with real-valued data sets. Fuzzy set theory proposed in 1965 by Zadeh [45] is very useful to overcome these limitations, as it can deal effectively with vague concepts and graded indiscernibility. To integrate the fuzzy set theory and rough set theory together, Dubois and Prade [11] expanded classical rough sets to FRSs in 1990, here both the concept and the indiscernibility relation can be fuzzy. Specifically, they used the greatest t-norm, the smallest t-conorm, as well as a fuzzy equivalence relation to define a pair of lower and upper fuzzy rough approximations with respect to a fuzzy concept. Using fuzzy relations and fuzzy logical connectives, one can construct a variety of FRS models. Radzikowska and Kerre [34] presented a general approach to FRSs with reference to a t-norm, a special fuzzy implication, and a fuzzy similarity relation. They defined and studied three classes of FRSs by taking into account three classes of well-known implications (S-implications, R-implications and QL-implications). Depending on the constructive approach and the axiomatic approach, Wu et al. [37] investigated an -FRS model on a general fuzzy relation by use of a t-norm T and a fuzzy implication I. In a similar way, based on a general fuzzy relation, Mi et al. [25] defined the -FRS model with a continuous t-conorm and a continuous t-norm. Furthermore, Mi and Zhang [26] studied a -FRS model by constructing a -lower and -upper fuzzy rough approximation operators, where denotes a fuzzy implication and stands for a fuzzy coimplication. Other fuzzifying rough set models via fuzzy relations and fuzzy logical operators were also studied in [27], [36], [49].
Three-way decisions (3WDs), derived from Pawlak’s rough sets [28], decision-theoretic rough sets [39] and some practical decision-making problems, have received extensive attention in academia since their proposal by Yao in 2009 [40]. 3WDs, as a generalization of the classical two-way decisions, possess three kinds of decision rules, that is, acceptance rules, rejection rules and non-commitment rules. Furthermore, Yao [41] researched the interplay of 3WDs and cognitive computing, and discussed general applicability of 3WDs. Recently, Yao [42] introduced a framework of three-way granular computing based on a hexagon induced by a trisection of a set. Since the concept of 3WDs is very close to people’s actual decision-making, this decision theory has developed rapidly both in theory and applications in recent years.
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In theory, many scholars have proposed a variety of 3WD models on the basis of different decision scenarios. Considering that the objects and attributes will simultaneously change over time, Huang et al. [18] provided a dynamic 3WD model in incomplete hybrid information systems. Based on the classification precision difference between two adjacent granularity layers, Zhang et al. [48] proposed a sequential 3WD model based on penalty function to improve the classification accuracy. Combining the idea of multi-granulation and the Bayesian decision theory, Zhang et al. [47] introduced a multi-granulation 3WD model in a q-rung orthopair fuzzy information system. Based on the thought of three-way multi-granularity learning, Yang et al. [43] discussed the fuzzy interpretation and representation of 3WDs with FRSs. To improve the efficiency of searching for all optimal scale combinations in multi-scale decision tables, Cheng et al. [7] established a multi-scale 3WD model. By exploring the unified theory of 3WDs through an axiomatic method, Hu [16] extended 3WD model to 3WD space. Furthermore, Jia and Qiao [19] discussed the construction of the evaluation function in 3WD space based on the overlap functions.
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In terms of applications, many researchers have applied 3WDs to a variety of practical problems. Sun et al. [35] developed a 3WD approach to multi-attribute group decision-making with linguistic information and applied it to emergency decision-making problem of unconventional emergency events. Based on a three-way conflict analysis model, Lang [21] discussed the conflict analysis problems in making development plans for Gansu Province in China. In view of the multilevel characteristics of recommendation information and the interpretability of recommendation results, Ye and Liu [44] put forward a novel interpretable sequential three-way recommendation strategy in recommender system. In light of the intrinsic uncertainty of Focal Liver Lesions diagnosis, Chen et al. [6] integrated the objective judgments from classification algorithms and the subjective judgments from human expert experiences, and proposed a data-human-driven 3WD support for Focal Liver Lesions diagnosis.
As we have stated in SubSection 1.2, one can conclude that t-norms and t-conorms play an influential factor in the construction of FRSs. Indeed, t-norms (resp. t-conorms) are important binary aggregation operators, which fulfill associative, commutative, increasing in every term and having a unit element 1 (resp. 0). Nevertheless, there exist diverse applications [4], where the associativity property of t-norms (resp. t-conorms) is unnecessary, like classification and decision-making problems. Thus, if a logical connective operator does not fulfill associativity, how to define lower and upper fuzzy rough approximation operators is a topic worth studying. Notice that, overlap functions can be seen as a novel generalization of the classical logical operator distinguished from the general fuzzy logical operator t-norms. The following Venn diagram visualizes the connection between t-norms and overlap functions, see Fig. 1. Meanwhile, compared with t-norms, the requirement for the associativity of overlap functions is not strong due to the context of their birth, which makes overlap functions more complicated but more flexible than t-norms. Therefore, it is necessary to define new FRS models via overlap functions from a theoretical aspect. In 2021, Qiao [30] first utilized overlap functions and their residual implications separately to replace t-norms and their residual implications to define a novel model of -FRSs. In accordance with the idea of [30], Jiang and Hu [20] further introduced an -FRS by using overlap and grouping functions on a complete lattice. Recently, Han and Qiao [15] studied a -FRS by use of overlap and grouping functions. These successful efforts prompt us to continue this research. In fact, whether a new FRS model can be built using overlap functions is important for the development of both overlap functions and FRSs from the theoretical and application standpoint.
On the other hand, classical rough sets utilize equivalence relations to calculate the lower and upper approximations of a crisp set, where the equivalence classes derived from equivalence relations either coincide or do not intersect. However, in [8], De Cock et al. stated that “this behaviour disappears when switching to fuzzy equivalence relations. Nevertheless, none of the existing researches on FRS theory have attempted to make use of the truth that an object can belong to a few “soft similarity classes” simultaneously to some extent.”. On the basis of this fact, using t-norms and their residual implications, De Cock et al. [8] defined the tight and loose FRSs w.r.t. a reflexive and symmetric fuzzy relation by utilizing the truly fuzzy characteristic that an object can be in diverse fuzzy similarity classes simultaneously to some extent. Therefore, in terms of the standpoint of theoretical research, it is very necessary and meaningful to develop two new FRS models on a general fuzzy relation by using overlap functions and -implications to replace the t-norms and their residual implications in the two FRSs in [8]. However, to our knowledge, there are no discussions or studies on this work. So, as a supplement of this area, we propose two new FRS models via overlap functions and -implications in this paper. Some important properties and topological structures of the novel proposed FRS models are explored. Furthermore, we study the connection between these two new types of FRSs based on the order relation of fuzzy sets. We find that in a fuzzy approximation space, when the overlap function has 1 as the identity element, the type of lower approximation is contained in the type of lower approximation, and the type of upper approximation is contained in the type of upper approximation. In particular, if the overlap function fulfills the exchange principle and R is an O-similarity fuzzy relation, the type of lower approximation is equal to the type of lower approximation, and the type of upper approximation is equal to the type of upper approximation.
In addition, exploring the application of FRSs has been a topic of interest to scholars. As a complement to the theory of this paper, this paper further tries to investigate the application of the proposed new FRSs in decision-making problems. Note that, the approximation learning of fuzzy concepts associated with FRSs and 3WDs is a useful technology for the representation, learning, and transformation of uncertain knowledge. Therefore, from an application standpoint, we develop a novel 3WD method in an information table by using the proposed new FRSs, which provides an efficient strategy to the approximation learning of fuzzy concepts. The practicality and effectiveness of the constructed 3WD method are verified by a numerical example. Furthermore, through a series of experimental analyses, we observe that the constructed 3WD method based on the proposed overlap functions-based FRSs is superior to t-norms (or t-conorms)-based FRS models in classification ability.
The content of this paper is organized as follows. Section 2 revisits several essential knowledge along with related results based on a brief way, which are made use of in the whole article. Section 3 shows the definition of two new types of FRSs via overlap functions and some vital conclusions of these two FRS models are also studied. At the same time, the relationships among different FRS models derived from overlap functions are investigated. In Section 4, we display an application of the new type of FRSs to 3WDs on an information table. Section 5 summarizes our work and describes future research efforts.
Section snippets
Basic concepts
This part sorts out some basic definitions and related conclusions in a short way. It should be noted that the symbol always represents a non-empty index set in the sequel.
Two new types of FRSs via overlap functions
The primary purpose of this section is to introduce two new types of FRSs via overlap functions and study some critical properties of them.
Application to 3WDs on an information table
The goal of this section is to display an application of the new types of FRSs to 3WDs on an information table. For the sake of clarity and easiness, we only consider the type of FRSs in this section.
Conclusion
In this paper, we mainly study two new types of FRSs via overlap functions and their residual implications from a theoretical perspective, which can be considered as a further investigation of the generalization models of rough sets. Some significant properties and topological structures of the proposed two new types of fuzzy rough approximation operators are explored. The connections among different FRS models generated from overlap functions and -implications are also described. In
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgments
The authors would like to express their sincere thanks to the Editors and anonymous reviewers for their most valuable comments and suggestions in improving this paper greatly. The work described in this paper was supported by grants from the National Natural Science Foundation of China (Grant Nos. 11971365 and 11571010).
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