Elsevier

Fuzzy Sets and Systems

Volume 456, 30 March 2023, Pages 144-172
Fuzzy Sets and Systems

On fuzzy Sheffer strokes: New results and the ordinal sums

https://doi.org/10.1016/j.fss.2022.09.006Get rights and content

Abstract

Recently, Baczyński et al. introduced the axiomatic definition of fuzzy Sheffer strokes, which is the generalization of the Sheffer stroke operators in classical logic. On the basis of this work, we point out that fuzzy Sheffer strokes can be transformed into common fuzzy logical operators through a fuzzy negation and fuzzy Sheffer strokes can be obtained from two increasing unary operators acting on a given fuzzy Sheffer stroke. We further discuss the relationship between fuzzy Sheffer strokes and overlap functions as well as grouping functions. The main contribution is to present the concept of ordinal sums of fuzzy Sheffer strokes which provides constructing methods of fuzzy Sheffer strokes and analyze some related properties. We argue under which condition a fuzzy Sheffer stroke can be represented as the ordinal sum of a family of fuzzy Sheffer strokes. We relate ordinal sums of fuzzy Sheffer strokes with its induced fuzzy conjunctions and fuzzy disjunctions as well as overlap functions and grouping functions under certain conditions. In the end, we give real-life application examples of fuzzy Sheffer strokes.

Introduction

Known as NAND operations or alternative denial, the so-called Sheffer strokes [34] play a special role unlike other fuzzy connectives in classical Boolean logic. In fact, this operator can be used to constitute a logical formal system by itself without usage of any other logical connective, while no other unary or binary connective fulfills this property. As we know, fuzzy set theory introduced by Zadeh [37] brought new applications of multivalued logic and new directions in examination of logical connectives. Despite the metalogical importance of this operator in Boolean logic, in the context of fuzzy logic [37], most of the theoretical efforts have been devoted to the study of fuzzy conjunctions, fuzzy disjunctions and fuzzy implications. However, there is still a dearth of research on Sheffer strokes in the frame of fuzzy logic.

To compensate for the absence of Sheffer strokes in fuzzy logic, in [29], basic properties of Sheffer stroke fuzzy implications were analyzed. Soon after, Helbin et al. [20] proposed the notion of fuzzy Sheffer strokes from classical logic to the fuzzy logic framework and discussed the characterization of such operators. On the fundamental of these preliminary results and the fact that all mathematically definable connectives in Boolean logic can be defined using only Sheffer Stroke, Baczyński et al. [1] gave the first extensive study on fuzzy Sheffer stroke operations recently, including defining different fuzzy logical operators by itself and constructing new fuzzy Sheffer strokes. Furthermore, the close connection of this operation with a pair of a fuzzy conjunction and a fuzzy negation was discussed.

As previously stated, fuzzy conjunctions and fuzzy disjunctions have received a great deal of attention, which are aggregation functions [19] satisfying different boundary conditions. As a special kind of aggregation functions, the concepts of overlap functions [7], [8] and grouping functions [9] were introduced by Bustince et al. initially in 2009-2010 and 2012, respectively. Since then, overlap and grouping functions have been used in various application areas, especially in classification [25] as well as image processing [22], decision making [9], [17] and forest fire detection [18]. From theoretical point of view, scholars studied different ways to construct overlap and grouping functions, including ordinal sums [12], [32], additive generators [13], [14] and multiplicative generators [31].

On the other hand, the ordinal sum construction is interesting and powerful, which provides a way to create, from a family of operators of a certain class, a new operator of the same class. The idea of ordinal sums has originated from the extension of algebraic structures, namely of posets and lattices [5] and semigroups [10]. In the context of fuzzy logic, the ordinal sums were first studied for t-norms and t-conorms [33] to provide a method to construct new t-norms and t-conorms from given ones preserving some common properties of the summands. Meanwhile, ordinal sums also provided a method for representing several classes of t-norms [23]. Researchers have also paid attention to ordinal sums of aggregation functions different from t-norms, including the most general aggregation operators [11] and some special ones, such as uninorms [27], overlap and grouping functions [12], [32]. Apart from aggregation functions, ordinal sums of other operators, for example, copulas [26], [28], hoops [6] and fuzzy implications [2], [16], [35], [38] have also been studied adequately. Specifically, the ordinal sums of fuzzy negations were introduced in [4] with the goal of studying the ordinal sums of families of t-norms, t-conorms and fuzzy negations when they form De Morgan's triples. Researchers [24] further explored some known classes of fuzzy negations and some properties of ordinal sums of fuzzy negations. As several types of construction methods and representation are known in the literature and the increasing interest of fuzzy logical operators, it seems quite natural to consider the related research of fuzzy Sheffer strokes.

As we have stated, the reason why the fuzzy Sheffer strokes are special is that it can be utilized by itself, without any other logical operators, to define all mathematically definable connectives in Boolean logic and thereby constitute a logical formal system. Besides, a real-life application was proposed in [1], where the fuzzy Sheffer stroke can represent the fuzzy modelization and model the alarm behavior of the refrigerator. The fuzzy Sheffer stroke operators provide us with a direct research way and perspective when we need the output value to be negatively correlated with the input value. Since there is still not much research on fuzzy Sheffer strokes, we attempt to investigate them and hope our theoretical results might inspire future research.

In this paper, for one thing, we use fuzzy Sheffer strokes and fuzzy negations combined to construct new fuzzy conjunctions and fuzzy disjunctions, to be more specific, overlap and grouping functions, the boundary conditions of which are different from that of t-norms and t-conorms, respectively. This can bring us new perspectives of studying overlap and grouping functions and offer a convenient way for the selection of appropriate overlap and grouping functions in concrete problems. For another, we investigate two new construction methods of fuzzy Sheffer strokes from given ones and the ordinal sum construction, including the relationship between properties of fuzzy Sheffer strokes and the ordinal sum. In addition, we establish conditions under which a fuzzy Sheffer stroke can be written in the form of an ordinal sum and conditions under which ordinal sums of fuzzy Sheffer strokes and its induced functions (or together with ordinal sums of fuzzy negations) coincide.

The rest of this paper is structured as follows. In Section 2, we review some basic concepts needed in the sequel. In Section 3, we use fuzzy Sheffer strokes and fuzzy negations to generate overlap and grouping functions. Section 4 is devoted to notion and some properties of ordinal sums of fuzzy Sheffer strokes. Section 5 is devoted to the relation between ordinal sums of fuzzy Sheffer strokes and their induced functions. Section 6 is devoted to real-life applications of (ordinal sums of) fuzzy Sheffer strokes. Finally, we present our conclusion.

Section snippets

Preliminaries

In this section, we briefly review several basic definitions and properties on fuzzy logical operators, including fuzzy negations, fuzzy conjunctions, fuzzy disjunctions, fuzzy implications and fuzzy Sheffer strokes, which shall be used throughout the paper. Notice that in this section, we always assume that A is a non-empty countable index set and {]aα,bα[}αA be a family of nonempty, pairwise disjoint open subintervals of [0,1].

Fuzzy Sheffer strokes based on unary operators

At the beginning of this section, we provide a way to obtain a new fuzzy Sheffer stroke from a given one and unary operators on [0,1].

Proposition 3.1

Consider a[0,1] and b]a,1]. Let H:[0,1]2[0,1] be a fuzzy Sheffer stroke, ϑ:[0,1][a,b] and θ:[1b,1a][0,1] be two increasing functions such that ϑ(0)=a,ϑ(1)=b,θ(1b)=0 and θ(1a)=1. If H satisfies the following conditions:

  • (1)

    H(a,b)=H(b,a)=1a;

  • (1)

    H(b,b)=1b,

then the binary operator Hθ,ϑ:[0,1]2[0,1] given, for all x,y[0,1], byHθ,ϑ(x,y)=θ(H(ϑ(x),ϑ(y))), is a fuzzy

Ordinal sums of fuzzy Sheffer strokes and related properties

In this section, we focus on ordinal sums of fuzzy Sheffer strokes. As we know, for a subinterval [a,b][0,1], an aggregation operator F is first linearly transformed into some other binary operator F[a,b] defined on [a,b] by means of the transformation F[a,b]=φ1F(φ×φ), with φ:[a,b][0,1] defined by φ(x)=xaba. Then the newly constructed aggregation operator coincides on the square [a,b]2 with the operator F[a,b]. However, since a fuzzy Sheffer stroke H is decreasing, we consider H[a,b]=ψH

Ordinal sums of fuzzy Sheffer strokes and the induced fuzzy conjunctions and fuzzy disjunctions

In this subsection, we turn to relationship between ordinal sums of fuzzy Sheffer strokes and the induced fuzzy conjunctions as well as fuzzy disjunctions.

Theorem 5.1

Let {Hα}αA be a family of fuzzy Sheffer strokes and H={aα,bα,Hα}αA. If for each αA, there exists βA such that [aβ,bβ]=[1bα,1aα] and for each pair (α,β), the following condition holds:Nαd(Hα(x,y))=Nβd(Hα(x,y))for anyx,y[0,1], then it holds that {aα,bα,Cα}αA=CH, where the ordinal sum {aα,bα,Cα}αA is formed by Eqs. (2) and (3),

Application examples of the fuzzy Sheffer strokes

In this section, potential applications of fuzzy Sheffer strokes are presented in order to show the importance of disposing such operators in the fuzzy logic framework.

Concluding remarks

In this paper, we mainly studied fuzzy Sheffer strokes based on the work in [1], including its relationship with overlap and grouping functions as well as its ordinal sums. The major results of our work can be listed as follows:

  • (1)

    From the monotonicity and boundary conditions for fuzzy Sheffer strokes, we argue that a fuzzy Sheffer stroke can be converted into a fuzzy implication and a fuzzy disjunction by means of a fuzzy negation, respectively. Meanwhile, by applying a fuzzy negation to a fuzzy

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.

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 article was supported by grants from the National Natural Science Foundation of China (grant nos. 11971365 and 11571010).

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