Fuzzy Truth Values Research Articles

Migrativity of extended binary operations on fuzzy truth values

Migrativity of extended binary operations on fuzzy truth values

Analysis of power asymmetry conflict based on fuzzy options graph models

Asymmetric power conflicts occur frequently. Because of the complexity of the conflict as well as the vagueness of the decision makers’ cognition, it becomes urgent and highly motivated to propose an appropriate method to solve power asymmetry conflict. In this study, we consider that decision makers provide option choices quantified by some degrees of membership. The choice of an option is determined by the thresholds of selection degree. At the same time, due to the influence of the power, the follower adjusts its degree of option choice to reach consensus with the leader. The computational rules determining fuzzy truth value are given, and a fuzzy truth value option prioritization method is proposed to calculate the ranking of the states, where the states ordering is related to the fuzzy degree of option selection. Different from the previous studies, this paper is the first one to study the asymmetric power conflict from the perspective of options, considering the psychological threshold of decision maker for option selection, and pointing out that the option choice is described with the fuzzy values rather than being treated as two-valued (Boolean). Furthermore, the introduced stability analysis also reflects the interaction of the options of different decision makers, which makes the proposal being more in rapport with real-world scenarios. Finally, a case study of carbon emission reduction power asymmetry conflict in supply chain is studied to demonstrate the performance of the proposed method.

Distributivity equations of extended aggregation functions in type-2 fuzzy sets

In this paper, we completely characterize the distributivity between the extended conjunctive aggregation functions and the extended disjunctive aggregation functions. We give two necessary and sufficient conditions for the distributivity of the extended aggregation functions and the convexity of fuzzy truth values. Most of the current research focuses on the distributivity of the extended t-norm (resp. t-conorm), overlap function (resp. grouping function) and uninorm (resp. nullnorm), and so on. It is well-known that a conjunctive (resp. disjunctive) aggregation function is a more general aggregation function than a t-norm (resp. t-conorm). Therefore, the obtained results show that the distributivity equation is more general than the previous ones.

Revisiting type-2 triangular norms on normal convex fuzzy truth values

This paper studies t-norms on the space L of all normal and convex fuzzy truth values. We first prove that the only non-convolution form type-2 t-norm constructed by Wu et al. satisfies the distributivity law for meet-convolution and show that t-norm in the sense of Walker and Walker is strictly stronger than tr-norm on L, which is strictly stronger than t-norm on L. Furthermore, we characterize some restrictive axioms of tr-norms for convolution operations on L and obtain some necessary conditions for tr-(co)norm convolution operations on L.

Extension operators for type-2 fuzzy sets derived from overlap functions

Overlap and grouping functions, as two novel continuous binary aggregation functions, have been discussed in the literature for applications in image processing, decision making, classification and so on. Since Walker et al. proved that the algebraic structure of fuzzy truth values has the same property as the algebraic structure of type-2 fuzzy sets, the study of extension operators for fuzzy truth values has become a hot research topic. This article considers this research topic and discusses mainly the so-called O-extension operators based on overlap functions. Firstly, the O-extension operators of general binary operators for fuzzy truth values on a linearly ordered set are introduced, and some of their basic properties are discussed. Secondly, the properties of O-extended maximum operators, O-extended minimum operators, O-extended overlap functions and O-extended grouping functions are studied. Thirdly, O-extension operators on normal fuzzy sets and O-convex fuzzy sets are investigated. Fourthly, the O-extension operators for fuzzy truth values are extended to type-2 fuzzy sets. Finally, we provide a brief comparison of O-extension operators with other common operators for fuzzy truth values.

A Complementary Study on General Interval Type-2 Fuzzy Sets

Liang <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">et al.</i> in 2000 defined interval type-2 fuzzy sets (IT2FSs), which constitute a subset of type-2 fuzzy sets. While the membership degrees in the former are functions from [0, 1] to [0, 1] (fuzzy truth values), the membership degrees in IT2FSs only take their values in <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\lbrace {\text{0}},{\text{1}}\rbrace$</tex-math></inline-formula> . Although all the initial work on IT2FSs involved convex membership degrees only, in 2015, Bustince <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">et al.</i> began the study on IT2FSs in general, including certain sets with nonconvex membership degrees. However, these are obviously early stages, with a lot of open problems regarding the theoretical structure of IT2FSs. For example, as far as we know, no negation operator has been obtained in this context. Therefore, it seems appropriate to continue with the study started in previous papers, delving deeper into the properties and operations of IT2FSs. Consequently, this work studies the structure of the set of functions from [0, 1] to <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\lbrace {\text{0}},{\text{1}}\rbrace$</tex-math></inline-formula> (expanding the set considered by Bustince <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">et al.</i> ), from which we have removed the constant function <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathbf{0}$</tex-math></inline-formula> , to offer a different study to the one carried out by Walker and Walker. More specifically, we consider join and meet operations, partial order derived from each one, and the negation operators in that set. Among other results, we provide new characterizations of join and meet operations and of partial orders on the set of functions from [0, 1] to <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$ \lbrace {\text{0}},{\text{1}} \rbrace $</tex-math></inline-formula> ; we also present the first negation operators on this set.

Automorphisms on normal and convex fuzzy truth values revisited

The present paper extends some previous works studying automorphisms in type-2 fuzzy sets. The framework for the analysis is the set of convex and normal functions from [0,1] to [0,1] (fuzzy truth values). The paper concentrates on those automorphisms that, in this framework, leave the constant function 1 fixed. This function is quite important since it defines the boundary between the functions that represent “TRUE” (increasing functions) and those that represent “FALSE” (decreasing functions), being at the same time the only normal function that is simultaneously increasing and decreasing. While C.L. Walker, E.A. Walker and J. Harding introduced in 2008 a family of functions leaving the constant function 1 fixed, the main goal of this paper is to prove that the functions of that family are in fact automorphisms, and moreover, that they are the only automorphisms (in the mentioned set of convex and normal functions from [0,1] to [0,1]) that preserve the function 1.

Practical aspects of equivalence of Baldwin’s and Zadeh’s fuzzy inference

The article presents a thorough analysis of fuzzy inference introduced by Baldwin and compares this approach to Zaheh’s compositional rule of inference. The comparison is performed in order to analyze the equivalence of the two methods and describe practical aspects of this fact for simple and compound premises, indicating advantages and disadvantages of both approaches. The main aim of the analysis is focus on the computational complexity of the methods. The most important feature of Baldwin’s inference is transfer of the inference process into a truth space, unified for all input variables. Such environment allows to obtain one fuzzy truth value describing a compound premise in a sequence of low dimensional computations. The article proves equality of such approach with the compositional rule of inference. Therefore, this solution is much more computationally efficient in case of compound cases, for which compositional rule of inference is multidimensional.

The Idempotency of Convolution Operations on Fuzzy Truth Values

Recently, convolution operations on fuzzy truth values (FTVs, that is, functions from [0, 1] to [0, 1]) and their basic properties have become a hot research area, such as type-2 t-norms, type-2 aggregation operations, type-2 implications, etc., and the closure, monotonicity, distributivity, etc. However, the idempotency of convolution operations has not been studied so far. In this article, we give the equivalent characterization of the idempotency of convolution operations on FTVs.

The Distributive Laws of Convolution Operations Over Meet-Convolution and Join-Convolution on Fuzzy Truth Values

Type-2 fuzzy sets (T2FSs) were introduced by Zadeh in 1975 as an extension of type-1 fuzzy sets (T1FSs). The membership grades of T2FSs are T1FSs in interval [0, 1], which are referred to as fuzzy truth values (that is, functions from [0, 1] to [0, 1]). Recently, (convolution) operations on T2FSs have become a hot research topic, such as type-2 t-norms, type-2 aggregation operations, type-2 implications, and so on. So, the distributive laws between these convolution operations have become an interesting and natural research area. In this article, we investigate the distributive laws of convolution operations over meet-convolution and join-convolution (two basic operations on fuzzy truth values), respectively. At first, we consider whether the distributive laws hold in various subsets of the set of all fuzzy truth values, such as set-valued T2FSs, singletons, interval-valued T2FSs, and the set of all normal convex fuzzy truth values. Then, we discuss whether the distributive laws hold for some specific convolution operations. Finally, we give a necessary and sufficient condition under which join-convolution is distributive over meet-convolution, and similarly give a necessary and sufficient condition under which meet-convolution is distributive over join-convolution.

Distributivity between extended nullnorms and uninorms on fuzzy truth values

This paper mainly investigates the distributive laws between extended nullnorms and uninorms on fuzzy truth values under the condition that the nullnorm is conditionally distributive over the uninorm. It presents the distributive laws between the extended nullnorm and t-conorm, and the left and right distributive laws between the extended generalized nullnorm and uninorm, where a generalized nullnorm is an operator from the class of aggregation operators with absorbing element that generalizes a nullnorm.

Composite Decision Makers in the Graph Model for Conflict Resolution: Hesitant Fuzzy Preference Modeling

Hesitant fuzzy preference relations (HFPRs) are formally proposed to model the conflict situation in which each decision maker (DM) consists of multiple individuals and each individual has its own fuzzy preferences over the feasible states within the framework of the graph model for conflict resolution (GMCR). Based on HFPRs, new definitions for hesitant fuzzy Nash stability, hesitant fuzzy general metarationality, hesitant fuzzy symmetric metarationality, and hesitant fuzzy sequential stability permit stability analyses to be carried out. Moreover, a new option prioritization technique, called hesitant fuzzy option prioritization, is developed for modeling a DM’s HFPRs based on the DM’s priority sequence of preference statements, the DM’s fuzzy truth values and levels of confidence. The groundwater contamination conflict of Elmira, Ontario, Canada, is utilized as a case study to illustrate the usefulness and applicability of the hesitant fuzzy option prioritization technique and GMCR with HFPRs.

Distributivity between extended t-norms and t-conorms on fuzzy truth values

This paper concentrates on the distributive laws between extended t-norms and t-conorms on fuzzy truth values under the condition that the t-norm is conditionally distributive over the t-conorm or the t-conorm is conditionally distributive over the t-norm. It presents the distributive laws between extended t-norms and t-conorms, which are then used to discuss the distributive laws between extended uninorms and t-norms, and the distributive laws between extended uninorms and t-conorms, respectively.

On the Extensions of Overlap Functions and Grouping Functions to Fuzzy Truth Values

This article mainly investigates Z-extended overlap functions and grouping functions on fuzzy truth values. First, we introduce Z-extended overlap functions and grouping functions. Then, we present some properties of the Z-extended overlap functions and grouping functions, and use these properties to show the representations of the Z-extended overlap functions and grouping functions. Finally, we give the conditions under which the distributive laws between the Z-extended overlap functions and grouping functions are true.

Note on “On the extension of nullnorms and uninorms to fuzzy truth values” [Fuzzy Sets Syst. 352 (2018) 92-118

Xie recently extended nullnorms and uninorms to fuzzy truth values and investigated whether the extended nullnorms and extended uninorms form respectively type-2 nullnorms and type-2 uninorms on the algebra of fuzzy truth values. In this note, we show by counterexamples that there exist some flaws in a previous paper by Xie [Fuzzy Sets Syst. 352 (2018) 92-118], such as Corollaries 2.1, 3.1, 4.1 and 4.2, Lemmas 3.3, 3.4, 3.5, 3.6, 3.7 and 3.8, Propositions 3.4, 3.5, 3.10, 3.11, 4.1 and 4.4, Theorems 3.1, 3.3, 4.1 and 4.3, and then we provide the correct versions. In addition, Proposition 3.7 in that paper is correct, but its proof contains a flaw. We point out the flaw and give a more general result than Proposition 3.7 in that paper.

Structures of fuzzy truth values based on a type-2 fuzzy set

Structures of fuzzy truth values based on a type-2 fuzzy set

Type-2 fuzzy implications and fuzzy-valued approximation reasoning

In this paper, the quasi-distributivity laws of type-2 fuzzy implications with respect to extended supremum and extended infimum are investigated on different subalgebras of fuzzy truth values. Moreover, the properties of fuzzy-valued fuzzy implications are further discussed. Especially, new fuzzy-valued operations are obtained from a fuzzy-valued fuzzy implication induced by a fuzzy implication, which is right-continuous with respect to the second argument. As an application of fuzzy-valued fuzzy implications, fuzzy-valued approximate reasoning is further studied and a numerical example is given.

On the extension of nullnorms and uninorms to fuzzy truth values

In this work, by applying Zadeh's extension principle, we extend type-1 proper nullnorms and proper uninorms to fuzzy truth values. The definitions of type-2 nullnorms and uninorms are introduced. We mainly discuss in which subalgebras of fuzzy truth values extended proper nullnorms and proper uninorms are type-2 nullnorms and uninorms, respectively.

Method of fuzzy inference for one class of MISO-structure systems with non-singleton inputs

In fuzzy modeling, the inputs of the simulated systems can receive both crisp values and non-Singleton. Computational complexity of fuzzy inference with fuzzy non-Singleton inputs corresponds to an exponential. This paper describes a new method of inference, based on the theorem of decomposition of a multidimensional fuzzy implication and a fuzzy truth value. This method is considered for fuzzy inputs and has a polynomial complexity, which makes it possible to use it for modeling large-dimensional MISO-structure systems.

Comment on: “RETRACTED: Rift-related active fault-system and a direction of maximum horizontal stress in the Cairo-Suez district, northeastern Egypt: A new approach from EMR-Technique and Cerescope data” by

Xie recently extended nullnorms and uninorms to fuzzy truth values and investigated whether the extended nullnorms and extended uninorms form respectively type-2 nullnorms and type-2 uninorms on the algebra of fuzzy truth values. In this note, we show by counterexamples that there exist some flaws in a previous paper by Xie [Fuzzy Sets Syst. 352 (2018) 92-118], such as Corollaries 2.1, 3.1, 4.1 and 4.2, Lemmas 3.3, 3.4, 3.5, 3.6, 3.7 and 3.8, Propositions 3.4, 3.5, 3.10, 3.11, 4.1 and 4.4, Theorems 3.1, 3.3, 4.1 and 4.3, and then we provide the correct versions. In addition, Proposition 3.7 in that paper is correct, but its proof contains a flaw. We point out the flaw and give a more general result than Proposition 3.7 in that paper.