Fuzzy Negations Research Articles

An Algorithm for Producing Fuzzy Implications via Conical Sections

In the present study, a novel parametric family of fuzzy implications is introduced and its properties are examined. The parametric family of implications is produced only via a fuzzy negation. This in turn enables the effortless production of a wide range of implications from which to select the one that best fits a given problem, for example in fuzzy inference systems or fuzzy neural networks. The fuzzy negations that have been selected as a basis for the proposed methodology are strong, i.e., involutions, thus leading, in general, to the generated fuzzy implications possessing many desirable additional properties. We have examined which of these properties hold for the implications produced by our algorithm and under which conditions. Finally, it is demonstrated that the family of implications generated via the proposed methodology generalizes other well-established implications, including the Łukasiewicz implication.

Innovative Methods of Constructing Strict and Strong Fuzzy Negations, Fuzzy Implications and New Classes of Copulas

This paper presents new classes of strong fuzzy negations, fuzzy implications and Copulas. It begins by presenting two theorems with function classes involving the construction of strong fuzzy negations. These classes are based on a well-known equilibrium point theorem. After that, a construction of fuzzy implication is presented, which is not based on any negation. Finally, moving on to the area concerning copulas, we present proof about the third property of copulas. To conclude, we will present two original constructions of copulas. All the above constructions are motivated by a specific formula. For some specific conditions of the variables x, y and other conditions for the function f(x), the formula presented produces strict and strong fuzzy negations, fuzzy implications and copulas.

Lift and generalized ordinal sum of negations on bounded posets

In this paper, two construction methods for fuzzy negations on bounded structures are presented. The first method focuses on lifting a fuzzy negation from a more general subposet to the complete lattice, extending the existing research. It provides an effective approach to lift fuzzy negations, even when the top and bottom elements of the subposet differ from those of the complete lattice. The second method explores the generalized ordinal sum of negations on an abelian partially ordered group, achieved by complementing a given fuzzy negation. A sufficient and necessary condition for the generalized ordinal sum, with a familiar fuzzy negation as the complement, to be a fuzzy negation is given. Building upon this, the paper introduces a supplementary fuzzy negation based on the provided family of subintervals, which guarantees that the generalized ordinal sum is still a fuzzy negation.

Novel strict intuitionistic fuzzy similarity measures-based on fuzzy negation and their applications

The concept of similarity measure is fundamental for evaluating the consistency between intuitionistic fuzzy sets (IFSs). Several researchers have developed various intuitionistic fuzzy similarity measures (IFSimMs) utilizing intuitionistic fuzzy distance measures (IFDisMs) or combinations of membership and non-membership degrees of IFSs. Nevertheless, there is still ample room for the improvement, as some of these measures do not satisfy the axiomatic properties of IFSimMs and produce unreasonable results. In this paper, we first establish a dual relationship between IFDisMs and IFSimMs based on fuzzy negations, enabling the construction of an infinite number of IFSimMs from a given IFDisM. We then propose two novel IFSimMs by directly manipulating the membership and non-membership degrees of IFSs and prove that they satisfy the axiomatic properties of strict IFSimMs (SIFSimMs). Furthermore, a comparative analysis reveals that the proposed SIFSimMs do not yield any unreasonable results in various scenarios. Finally, we apply these SIFSimMs to pattern recognition, medical diagnosis, and clustering analysis problems, demonstrating their superior performance compared to some existing measures.

Interval-valued pre-(quasi-)grouping functions and its application in constructing interval-valued directional monotonic fuzzy implications

How to expand the variable domain and monotonicity of aggregation functions to generate new aggregation functions is an important research content in aggregation functions. In this work, the concept of interval-valued pre-(quasi-)grouping functions is given by relaxing the interval monotonicity of interval-valued (quasi-)grouping functions to interval directional monotonicity. Then, some basic properties of interval-valued pre-(quasi-)grouping functions and the relationship between interval-valued pre-(quasi-)grouping functions and pre-(quasi-)grouping functions are presented. Accordingly, several construction methods of interval-valued pre-(quasi-)grouping functions are proposed. Finally, the concept of ( I G , IN ) -interval-valued directional monotonic fuzzy implications and QL -interval-valued directional monotonic operations are introduced on the basis of interval-valued pre-(quasi-)grouping functions I G , interval-valued overlap functions IO and interval-valued fuzzy negations IN. In addition, related studies were conducted on the basic properties of ( I G , IN ) -interval-valued directional monotonic fuzzy implications and QL -interval-valued directional monotonic operations.

Multidimensional fuzzy sets: Negations and an algorithm for multi-attribute group decision making

Multidimensional fuzzy sets (MFS) is a new extension of fuzzy sets on which the membership values of an element in the universe of discourse are increasingly ordered vectors on the set of real numbers in the interval [0,1]. This paper aims to investigate fuzzy negations on the set of increasingly ordered vectors on [0,1], i.e. on L∞([0,1]), MFN in short, with respect to some partial order. In this paper we study partial orders, giving special attention to admissible orders on Ln([0,1]) and L∞([0,1]). In addition, we study the possibility of existence of strong multidimensional fuzzy negations and some properties and methods to construct such operators. In particular, we define the ordinal sums of n-dimensional negations and ordinal sums of multidimensional fuzzy negations on a multidimensional product order. A multi-attribute group decision making algorithm is presented.

New results of (U,N)-implications satisfying I(r,I(s,t))=I(I(r,s),I(r,t))

Generalized Frege's law has been extensively explored by numerous scholars in the field of fuzzy mathematics, particularly within the framework of fuzzy logic. This study aims to further investigate the (U,N)-implications that satisfy this law and presents a multitude of novel findings. First, to efficiently determine the satisfiability of the generalized Frege's law for any (U,N)-implication, two new necessary conditions have been introduced that are simple and practical: for the fuzzy negation N, it must be noncontinuous, and its values in the interval [0,e] should remain the constant 1. Next, the necessary and sufficient conditions for any (U,N)-implication to satisfy the generalized Frege's law are provided. Several complete characterizations are described depending on the position of α in [e,1]. To be more specific, the full characterization is achieved when α=e (α=1) and a disjunctive uninorm with a continuous underlying t-norm (t-conorm). The necessary and sufficient conditions are presented when α∈]e,1[ and U is a locally internal and disjunctive uninorm.

Two Extensions of the Sugeno Class and a Novel Constructed Method of Strong Fuzzy Negation for the Generation of Non-Symmetric Fuzzy Implications

In this paper, we present two new classes of fuzzy negations. They are an extension of a well-known class of fuzzy negations, the Sugeno Class. We use it as a base for our work for the first two construction methods. The first method generates rational fuzzy negations, where we use a second-degree polynomial with two parameters. We investigate which of these two conditions must be satisfied to be a fuzzy negation. In the second method, we use an increasing function instead of the parameter δ of the Sugeno class. In this method, using an arbitrary increasing function with specific conditions, fuzzy negations are produced, not just rational ones. Moreover, we compare the equilibrium points of the produced fuzzy negation of the first method and the Sugeno class. We use the equilibrium point to present a novel method which produces strong fuzzy negations by using two decreasing functions which satisfy specific conditions. We also investigate the convexity of the new fuzzy negation. We give some conditions that coefficients of fuzzy negation of the first method must satisfy in order to be convex. We present some examples of the new fuzzy negations, and we use them to generate new non-symmetric fuzzy implications by using well-known production methods of non-symmetric fuzzy implications. We use convex fuzzy negations as decreasing functions to construct an Archimedean copula. Finally, we investigate the quadratic form of the copula and the conditions that the coefficients of the first method and the increasing function of the second method must satisfy in order to generate new copulas of this form.

On Equivalence Operators Derived from Overlap and Grouping Functions

This paper introduces the concept of equivalence operators based on overlap and grouping functions where the associativity property is not strongly required. Overlap functions and grouping functions are weaker than positive and continuous t-norms and t-conorms, respectively. Therefore, these equivalence operators do not necessarily satisfy certain properties, such as associativity and the neutrality principle. In this paper, two models of fuzzy equivalence operators are obtained by the composition of overlap functions, grouping functions and fuzzy negations. Their main properties are also studied.

Fuzzy difference operators derived from overlap functions

This paper introduces the concept of (O, N)-difference, for an overlap function O and a fuzzy negation N. (O, N)-differences are weaker than fuzzy difference constructed from positive and continuous t-norms and fuzzy negations, in the sense that (O, N)-differences do not necessarily satisfy certain properties, as the right neutrality principle, but only weaker versions of these properties. This paper analyzes the main properties satisfied by (O, N)-differences, and provides a characterization of (O, N)-difference.

On retractions and extension of quasi-overlap and quasi-grouping functions defined on bounded lattices

Saminger-Platz, Klement, and Mesiar (2008) extended t-norms from a complete sublattice to its respective lattice using the conventional definition of sublattice. In contrast, Palmeira and Bedregal (2012) introduced a more inclusive sublattice definition, via retractions. They expanded various important mathematical operators, including t-norms, t-conorms, fuzzy negations, and automorphisms. They also introduced De Morgan triples (semi-triples) for these operators and provided their extensions in their groundbreaking work. In this paper, we propose a method of extending quasi-overlap functions and quasi-grouping functions defined on bounded sublattices (in a broad sense) to a bounded superlattice. To achieve that, we use the technique proposed by Palmeira and Bedregal. We also define: quasi-overlap (resp. quasi-grouping) functions generated from quasi-grouping (resp. quasi-overlap) functions and frontier fuzzy negations, De Morgan (semi)triples for the classes of quasi-overlap functions, quasi-grouping functions and fuzzy negations, as well as its respective extensions. Finally we study properties of all extensions defined.

Generator of Fuzzy Implications

In this research paper, a generator of fuzzy methods based on theorems and axioms of fuzzy logic is derived, analyzed and applied. The family presented generates fuzzy implications according to the value of a selected parameter. The obtained fuzzy implications should satisfy a number of axioms, and the conditions of satisfying the maximum number of axioms are denoted. New theorems are stated and proven based on the rule that the fuzzy function of fuzzy implication, which is strong, leads to fuzzy negation. In this work, the data taken were fuzzified for the application of the new formulae. The fuzzification of the data was undertaken using four kinds of membership degree functions. The new fuzzy functions were compared based on the results obtained after a number of repetitions. The new proposed methodology presents a new family of fuzzy implications, and also an algorithm is shown that produces fuzzy implications so as to be able to select the optimal method of the generator according to the value of a free parameter.

Fuzzy correlational analysis for dynamic consolidation of virtual machines in cloud computing environment

This theoretical research in fuzzy correlation analysis integrates data uncertainty analysis by measuring the strength of the linear relationship restricted to two fuzzy sets. As the relevant contribution to this research area, this paper presents the axiomatic definition of the n-dimensional generalized fuzzy correlation coefficient (nGCC), assigning to n input fuzzy sets an output value in the interval [−1,1]. Thus, the properties of general overlap functions and fuzzy negations are studied, discussing binary non-normed restricted dissimilarity functions, and the n-dimensional non-normed conjunctive functions. This study provides new methods for the n-dimensional generalized fuzzy correlation coefficient analysis, regarding their applications in solving multi-criteria and decision-making problems founded on fuzzy logic extensions. The n-dimensional non-normed conjunctive aggregation functions are also introduced, as range domain extensions from [0,1] to [−1.1], covering the interpretation of negative to positive variable associations. The proposal correlation analysis promotes a better evaluation and data selection even when more than one algorithm is applied to evaluate the reduction methods in the defuzzification process based on Interval-valued Fuzzy Logic. We also investigate the relevance to determine a reliable result from the fuzzy inference system based on the nGCC methodology. This correlation methodology applied to the Interval Fuzzy Load Balancing for Cloud Computing (Int-FLBCC) model contributes as a flexible approach for virtual machines dynamic consolidation enabling improvements in resource usage and power efficiency, improving the computational system’s energy efficiency. So, the nGCC methodology extends the Int-FLBCC model by adding other degrees of reliability to the results obtained with diverse evaluations through n-dimensional generalized fuzzy correlation coefficient expressions, exploring average aggregations as arithmetic and exponential means and the median operator.

Novel construction methods of interval-valued fuzzy negations and aggregation functions based on admissible orders

In this work, we present a review of the existing admissible orders and propose a generalization of all of these orders in the discrete setting. Also, we propose some construction methods of interval-valued fuzzy negations w.r.t. the existing admissible orders on L([0,1]) via arbitrary pairs of fuzzy logic connectives that are defined over the unit interval [0,1] and show that these proposed methods generalize some of the existing construction methods proposed by Asiain et al. Further, we propose also some construction methods of interval-valued aggregation functions w.r.t. the existing admissible orders on L([0,1]) via standard fuzzy logic connectives on [0,1] and also, as a special case, interval-valued t-norms and t-conorms are constructed.

On Fuzzy Negations and Laws of Contraposition. Lattice of Fuzzy Negations

Summary This the next article in the series formalizing the book of Baczyński and Jayaram “Fuzzy Implications”. We define the laws of contraposition connected with various fuzzy negations, and in order to make the cluster registration mechanism fully working, we construct some more non-classical examples of fuzzy implications. Finally, as the testbed of the reuse of lattice-theoretical approach, we introduce the lattice of fuzzy negations and show its basic properties.

On a new contrapositivisation technique for fuzzy implications constructed from quasi-overlap and quasi-grouping functions

In this paper we propose a new class of contrapositivisation operators for fuzzy implications called (QO,QG,N)-contrapositivisators and whose generators are quasi-overlaps, quasi-groupings and fuzzy negations. We show that (QO,QG,N)-contrapositivisators generalize medium contrapositivisators and (G,N)-contrapositivisators. We present a brief study on the notion of N-compatibility for (QO,QG,N)-contrapositivisations and show that the class of (QO,QG,N)-contrapositivisators is invariant by the action of automorphisms. We extensively study the characterizations of (QO,QG,N)-contrapositivisators with respect to the main properties that are generally satisfied by fuzzy implications. This study was done from two points of view: first, considering that the input implication is arbitrary and second, assuming that the universe of the input implications consists of four specific classes of fuzzy implications generated from overlaps and/or groupings. In addition, we present new methods to construct quasi-overlap (overlap) and quasi-grouping (grouping) functions from (QO,QG,N)-contrapositivisators.

Determination of the continuous completions of conditionally cancellative pre-t-norms associated with the characterization of (S,N)-implications: Part II

The continuous completions of conditionally cancellative pre-t-norms defined in the regions linked to the characterization of (S,N)-implications when S is a continuous t-conorm and N is a fuzzy negation with one point of discontinuity are determined. The results have been divided in two parts. In this second part, we use the results of the first part for providing the continuous completions of a conditionally cancellative pre-t-norm defined in the four remaining regions. The most significant contribution in this second part is the study of the case that corresponds to a region of the type B=([0,1]2∖(a,1)2)∪({c}×[0,1])∪([0,1]×{c}) with 0<a<c<1, which has to be separated in six different cases.

Fuzzy implications based on quasi-copula and fuzzy negations

Fuzzy implications based on quasi-copula and fuzzy negations

Some Construction Methods for Pseudo-Overlaps and Pseudo-Groupings and Their Application in Group Decision Making

In many real-world scenarios, the importance of different factors may vary, making commutativity an unreasonable assumption for aggregation functions, such as overlaps or groupings. To address this issue, researchers have introduced pseudo-overlaps and pseudo-groupings as their corresponding non-commutative generalizations. In this paper, we explore various construction methods for obtaining pseudo-overlaps and pseudo-groupings using overlaps, groupings, fuzzy negations, convex sums, and Riemannian integration. We then show the applicability of these construction methods in a multi-criteria group decision-making problem, where the importance of both the considered criteria and the experts vary. Our results highlight the usefulness of pseudo-overlaps and pseudo-groupings as a non-commutative alternative to overlaps and groupings.

Determination of the continuous completions of conditionally cancellative pre-t-norms associated with the characterization of (S,N)-implications: Part I

The continuous completions of conditionally cancellative pre-t-norms defined in the regions linked to the characterization of (S,N)-implications when S is a continuous t-conorm and N is a fuzzy negation with one point of discontinuity are determined. The results have been divided in two parts. In this first part we provide three main results: we determine the existence and expression of the additive generator of a cancellative pre-t-norm known above a level curve and, using this result as a basis, we provide the continuous completions of pre-t-norms determined in regions of the type [0,1]2∖(0,a)2 or [0,1]2∖(a,1)2 with a∈(0,1). It is highlighted that, differently from the cancellative case where all the results provide unique continuous completions, in the analogous situations in the conditionally cancellative case the corresponding continuous completions might not be unique. This fact causes more complex results for the remaining regions, which are discussed in the second paper of this study.