On interval - and -implications derived from interval overlap and grouping functions
Introduction
As we all know, fuzzy implications [4], [6], [59] are the natural extension of classical one in fuzzy logic, which consider true values that change in the unit interval rather than in the set . The significance of fuzzy implications is not only that they are used for formalizing “If…then” rules in fuzzy systems, but also that many of their different meanings are applied to perform reasoning [10]. In fact, it is well known that there exist four basic fuzzy implications generated by fundamental fuzzy logic operators (e.g., t-norms, t-conorms and fuzzy negations), viz., R-implications [5], S-implications [3], [35], QL-implications [40], [46], [58] and D-implications [58]. In addition, fuzzy implications play a key role in various applications such as fuzzy mathematical morphology [26], data mining [75], image processing [22], [23], decision making [13] and so on.
Moreover, overlap functions [19], [20] and grouping functions [24] are two special unnecessarily associative binary aggregation operators [14], [43], which have rapidly developed both in theory and applications. In applications, they show great potential in dealing with real problems such as image processing [20], [47], decision making [24], [39] and classification [37], [38], [52], [53], [54], [55], [56], [57], [64]. In theory, compared with t-norms and t-conorms, the classes of overlap and grouping functions are richer than their classes, so that overlap and grouping functions have a wider range of properties (see, e.g., [9], [27], [29], [65], [66], [67], [68], [69], [70]), like idempotency, homogeneity, and, mainly, the self-closedness feature regarding the convex sum and the aggregation by generalized composition of overlap and grouping functions, as discussed in [28], [31], [32]. Specially for constructing fuzzy implications, the classes of fuzzy implications derived from overlap and grouping functions are still reacher than that derived from t-norms and t-conorms (see, e.g., [28], [30], [34]), which also shows the significance of studying fuzzy implications induced by overlap and grouping functions.
In fact, as a result of the extensive studies on overlap and grouping functions, many researchers attempt to use them to substitute, respectively, t-norms and t-conorms, when defining fuzzy implications. More specially, in 2014, the notion of -implications was put forward by Dimuro et al. [32], which are weaker than -implications constructed from positive and continuous t-conorms. In 2015, by means of the existent conclusions on residual implications derived from fuzzy conjunctions, -implications induced by overlap functions were studied by Dimuro and Bedregal [28]. In 2017, Dimuro et al. [30] investigated QL-implications induced by tuples , as the extension of the implication defined in quantum logic. And then, D-implications induced by grouping functions were introduced in [31], [34], which generalize from the Dishkant implication .
Additionally, interval mathematics as another form of information theory was investigated by Sunaga [73] and Moore [60]. Interval membership degrees can be expressed the uncertainty and the difficulty of an specialist, which exactly decide the fairest membership degree of an element regarding a linguistic term [10]. Furthermore, in order to deal with the cases of lacking sharp class boundaries and information [36], [51], interval-valued fuzzy sets have been studied in several works (see, e.g., [15], [16], [44], [45], [72], [74], [76], [78]) as the combination of interval mathematics and fuzzy set theory.
In recent years, many researchers have extensively investigated the different models of interval-valued fuzzy sets and made great progress (see, e.g., [2], [7], [10], [11], [12], [33], [49], [71]). In particular, in 2017, Bedregal et al. [8] introduced OWA operators with interval-valued weights based on interval-valued overlap functions. Almost simultaneously, the notions of interval overlap and grouping functions were presented by Qiao and Hu [66], including their interval additive generators. In 2018, on the basis of interval overlap and grouping functions, Cao et al. [25] proposed the interval -implications and interval -implications. Moreover, there are also many interesting works on interval-valued approach such as interval-valued fuzzy implications [77], interval-valued fuzzy inference using aggregation functions [50] and interval-valued fuzzy games [1].
In terms of applications, interval-valued fuzzy sets are a suitable tool for representing uncertain or incomplete information, which have been widely used in classification, image processing, decision making, medicine and many other aspects, as discussed in [17]. Specifically, the usefulness of implication functions in the interval-valued setting has been mentioned by many authors. For example, Lodwick [51] wrote, “the interval-valued fuzzy implications address intuitively not only vagueness (lack of sharp class boundaries) but also a feature of uncertainty (lack of information).”
However, as far as we know, except for interval -implications and interval -implications, no other researchers have investigated the interval generalizations of other fuzzy implications induced by interval overlap and grouping functions nowadays. Particularly, we observe that interval -implications are not the contraposition of interval -implications. Since an interval -operation is an interval fuzzy implication iff and is not strong, the study of interval -implications does not follow directly from related conclusions for interval -implications, which should be focused as an independent analysis. Consequently, as a supplement to the theoretical view of this topic, in this paper, we introduce interval -implications and interval -implications derived from interval overlap and grouping functions.
Through the previous narrative, we noticed that fuzzy implications, overlap and grouping functions, and interval-valued fuzzy sets are all useful in problems of image processing and decision making. Therefore, it is necessary to carry on theoretical study for interval fuzzy implications derive from interval overlap and grouping functions in order to provide theoretical bases for dealing with practical problems. Specifically, we point out that our extensive approach should have a precise application prospect, for instance, the situations that when the expert thinks over the belief degree of an imprecision by the interval degree of membership functions in actual problems.
The contents of this paper are listed in what follows. Section 2 introduces the fundamental concepts of the paper. In Section 3, the concept of interval -implications is proposed, showing related properties. Section 4 presents the concepts and some results of interval -operations (Subsection 4.1) and interval -implications (Subsection 4.2) in turn. Section 5 discusses the intersections between families of interval fuzzy implications, including interval -, -, - and -implications. Concluding remarks are given in Section 6.
Section snippets
Basic notions on interval mathematics
This subsection reviews certain fundamental notions on interval mathematics, coming mostly from [10], [11], [33], [62].
Let be the real unit interval and be the family of subintervals of U. For , define and . For , we use notations and . For convenience, denotes .
Correspondingly, regarding a function , we use notations and .
Furthermore, for , the
Interval -implications
In this section, we give the concept of interval -implications preserving the residuation property (IRP), as the interval extension of residual implications induced by overlap functions introduced in [28]. Meanwhile, we discuss various related properties of such interval -implications and show the actions of interval automorphisms on them.
Here, we firstly recall the definition of residual implications derived from overlap functions (-implications, shortly) as follows.
Let be an
Interval -operations and interval -implications
In this section, we propose the interval -operations and interval -implications, which are respectively the interval extensions of D-operations and D-implications derived from tuples introduced in [31]. Furthermore, we discuss related conclusions regarding them in turn, including their main properties and the actions of interval automorphisms on them.
Intersections between families of interval fuzzy implications
This section discusses the intersections between families of interval fuzzy implications, including interval -, -, - and -implications.
At first, the intersection between interval - and -implications is discussed.
Proposition 5.1 The interval -implication is not an interval -implication for any interval grouping function , any interval overlap function and any interval fuzzy negation .
Proof According to Proposition 4.7, we know that is an interval
Concluding remarks
This paper mainly shows a study of two interval fuzzy implications derived from interval overlap and grouping functions, namely interval - and -implications. The main results of this paper are listed in what follows.
- •
We propose the concept of interval -implications derived from inclusion monotonic interval overlap functions, which fulfill the residuation property (IRP). Meanwhile, we discuss various related properties of interval -implications and analyze the actions of interval
Declaration of Competing Interest
We wish to confirm that there are no known conflicts of interest associated with this publication and there has been no significant financial support for this work that could have influenced its outcome.
Acknowledgements
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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