Fuzzy Negations Research Articles

Correlation in Interval-Valued Atanassov’s Intuitionistic Fuzzy Sets — Conjugate and Negation Operators

This paper studies the conjugate functions related to main connectives of the Intervalvalued Atanassov’s Intuitionistic Fuzzy Logic. The relationships among automorphism classes are formalized by the ϕ-representability theorem, passing from automorphisms to interval-valued intuitionistic automorphisms, also visiting other two ones, intuitionistic automorphisms and interval-valued automorphisms. Additionally, the ϕ-conjugate of an interval-valued Atanassov’s intuitionistic fuzzy negation can be obtained either from an interval-valued fuzzy negation or from an Atanassov’s intuitionistic fuzzy negation, including a discussion presenting such reverse constructions. The ϕ-conjugate of an interval-valued Atanassov’s intuitionistic fuzzy negation not only preserves the main properties of its corresponding fuzzy negation but also of two other ones, the intuitionistic fuzzy negation and interval-valued fuzzy negation. Moreover, an extension of the intuitionistic fuzzy index as well as the correlation coefficient is discussed in terms of fuzzy negations, by considering the Atanassov’s Intuitionistic Fuzzy Logic.

Representation of fuzzy subsets by Galois connections

There is a great deal of fuzziness in our everyday natural language, and thus fuzzy subsets have come to represent a direct generalisation of the indicator function of a classical subset. On the other hand, a Galois connection is given by two opposite order-inverting maps whose composition yields two closure operations between ordered sets. We present the one-to-one correspondence between a set of all fuzzy subsets and a set of all Galois connections. The essential correspondences are built with the help of α-cuts, which represent fuzzy subsets by means of classical sets. Moreover, we present a relationship between strong fuzzy negations in the lattices and Galois connections. The various extensions of fuzzy subsets from the point of view of nestedness and negations are recalled. Other fruitful properties and connections with related studies are included.

QL-operations and QL-implication functions constructed from tuples (O,G,N) and the generation of fuzzy subsethood and entropy measures

Considering the important role played by overlap and grouping functions in several applications in which associativity is not demanded, in this paper we introduce the notion of QL-operations constructed from tuples (O,G,N), where overlap functions O, grouping functions G and fuzzy negations N are used for the generalization of the implication p→q≡¬p∨(p∧q), which is defined in quantum logic (QL). We also study under which conditions QL-operations constructed from tuples (O,G,N) are fuzzy implication functions, presenting a general form for obtaining QL-implication functions, and particular forms of such fuzzy implication functions according to specific properties of O and G. We analyze the main properties satisfied by QL-operations and QL-implication functions, establishing under which conditions of O, G and N, the derived QL-operations (implication functions) satisfy the different known properties for fuzzy implication functions. We show that QL-implication functions constructed from tuples (O,G,N) are richer than QL-implication functions constructed from t-norms and positive t-conorms. We provide a comparative study of QL-implication functions and other classes of fuzzy implication functions constructed from fuzzy negations, overlap and grouping functions, analyzing the intersections among such classes. Finally, we present the application of both QL-operations and QL-implication functions constructed from tuples (O,G,N) to the generation of fuzzy subsethood and derived entropy measures, which are useful for several practical applications.

Robustness of f- and g-generated Fuzzy (Co)Implications: The Yager's (Co)Implication Case Study

This paper studies the robustness of intuitionistic fuzzy implications in fuzzy reasoning based on Atanassov's intuitionistic fuzzy logic. Starting with an evaluation of the sensitivity in representable fuzzy negations, we apply the results in the Yager's classes of fuzzy implications called the f- and g-generated fuzzy implications. The paper formally states that the robustness preserves the projection functions in such class and also discusses their corresponding dual operators.

Some results on extension of lattice-valued QL-implications

A very important issue in lattice theory is how to extend a given operator preserving its algebraic properties. For lattice-valued fuzzy operators framework, in 2008 Saminger-Platz presented a way to extend t-norms which was generalized by Palmeira et al. (2011) for t-norms, t-conorms, fuzzy negations and implications, considering the scenery provided by the (r,s)-sublattice. In this paper we investigated how to extend QL-implications and which properties of it are preserved by the extension method via retractions (EMR). As results, we proved that properties (LB), (RB), (CC1), (CC2), (CC3), (CC4), (L-NP), (EP) and (IP) are preserved by EMR. However, the extension method via retractions fails in preserving the important properties (NP), (OP), (IBL), (CP), (P) and (LEM).

Aggregating Fuzzy QL- and (S,N)-Subimplications: Conjugate and Dual Constructions

<p>Fuzzy (S,N)- and <span style="text-decoration: underline;">QL</span>-<span style="text-decoration: underline;">subimplication</span> classes can be obtained by a distributive n-<span style="text-decoration: underline;">ry</span> aggregation performed over the families of t-<span style="text-decoration: underline;">subnorms</span> and t-<span style="text-decoration: underline;">subconorms</span> along with a fuzzy negation. Since these classes of <span style="text-decoration: underline;">subimplications</span> are explicitly represented by t-<span style="text-decoration: underline;">subconorms</span> and t-<span style="text-decoration: underline;">subnorms</span> verifying the generalized associativity, the corresponding (S,N)- and <span style="text-decoration: underline;">QL</span>-<span style="text-decoration: underline;">subimplications</span>, referred as I(S,N) and I_(S,T,N), are characterized as distributive n-<span style="text-decoration: underline;">ary</span> aggregation together with related generalizations as the exchange and neutrality principles. Based on these results, the both subclasses I_(S,n) and I_QL of (S,N)- and <span style="text-decoration: underline;">QL</span>-<span style="text-decoration: underline;">subimplications</span> which are obtained by the median aggregation operation performed over the standard negation N_S together with the families of t-<span style="text-decoration: underline;">subnorms</span> and t-<span style="text-decoration: underline;">subconorms</span> S_P and T_P, respectively. In particular, the subclass T_P extends the product t-norm T_P as well as S_P extends the algebraic sum S_P. As the main results, the family of <span style="text-decoration: underline;">subimplications</span> I_(S_P,N) and I_(S_P,T_P,N) extends the implication class by preserving the corresponding properties. We also present an extension from (S,N)- and <span style="text-decoration: underline;">QL</span>-<span style="text-decoration: underline;">subimplications</span> to (S,N)- and <span style="text-decoration: underline;">QL</span>-implications and discuss dual and conjugate constructions.</p><pre><!--EndFragment--></pre>

RU and (U,N)-implications satisfying Modus Ponens

In this paper it is investigated when some kinds of fuzzy implication functions derived from uninorms satisfy the Modus Ponens with respect to a continuous t-norm T, or equivalently, when they are T-conditionals. The study is done for RU-implications and (U,N)-implications with N a continuous fuzzy negation leading to a lot of solutions in both cases. For RU-implications T-conditionality only depends on the underlying t-norm of the uninorm used to derive the residual implication. On the contrary, for (U,N)-implications the underlying t-norm is never relevant and only the region out of the t-norm is so. Even the t-conorm can be not relevant also in some cases.

New types of contrapositivisation of fuzzy implications with respect to fuzzy negations

Among the most studied properties of fuzzy implication functions, the so-called contrapositive symmetry or law of contraposition with respect to a fuzzy negation is of special interest because it is a key property in approximate reasoning, deductive systems and formal methods of proof. Even when contraposition is not satisfied, there exist some methods for modifying an implication function with the aim that the new implication satisfies this property, these methods are known as contrapositivisations. In this paper some new methods of contrapositivisation with respect to any fuzzy negation are presented and the properties preserved by these new methods are studied. Along this study we will see that these new methods not only preserve the usual properties preserved by the already known methods, but they also have some additional properties.

Natural negation of interval-valued t-(co) norms and implications

In this paper, we investigate interval-valued fuzzy negations induced by interval-valued $$t$$t-norms, $$t$$t-conorms or implications. Some properties of interval-valued fuzzy negations induced by interval-valued sup-morphism $$t$$t-norms, inf-morphism $$t$$t-conorms or $$R$$R-implications are firstly obtained. We also show interval-valued automorphisms acting on the interval-valued fuzzy negations induced by interval-valued $$t$$t-norms, $$t$$t-conorms or implications. Finally, the relations among the interval-valued fuzzy negations induced by interval-valued $$t$$t-norms, $$t$$t-conorms or implications are explored.

Aggregation distance measure and its induced similarity measure between intuitionistic fuzzy sets

Various distance and similarity measures in Atanassov’s intuitionistic fuzzy sets (IFSs) are applied to practical problems. However, it is difficult to decide which one is the most suitable for measuring the degree of distance or similarity, especially in the case of the occurrence of conflict results in practice. In this paper, we propose the concept of aggregation distance measure for IFSs based on aggregation functions without zero divisors to help decision makers to make final decisions. Furthermore, a novel technique is introduced to construct similarity measures from a given distance measure with respect to non-filling fuzzy negations, which gives a new direction for the construction method for similarity measures of IFSs. Some illustrative examples in applications such as pattern recognition, fuzzy clustering and decision making are used to investigate the effectiveness of the proposed measures.

On a New Class of Fuzzy Coimplications Derived from Generalizedh-Generators and Fuzzy Negations

In this paper, a new class of fuzzy coimplications, called (h, N)-coimplications, is introduced. Basic properties of this class of coimplications are discussed. Furthermore, the classical tautologies that involve coimplications, such as the importation law, the contraction law and the distributive law, are investigated.

A fuzzy morphological hit-or-miss transform for grey-level images: A new approach

The translation of the hit-or-miss transform (HMT) for grey-level images from the HMT for binary images is not simple. Initially established as a powerful tool for morphological binary image processing, some generalizations for grey-level images have been proposed in the literature. The difficulty lies in the definition of the complement of a given grey-level image. In this paper, after performing a detailed review of the different approaches proposed in the literature, including those based on fuzzy logic, we propose the definition of the hit-or-miss transform for grey-level images in the framework of the fuzzy mathematical morphology based on fuzzy conjunctions and implications under the duality with respect to a fuzzy negation approach. Some theoretical properties of this operator are studied when a general fuzzy conjunction is considered. In particular, we prove that this fuzzy morphological HMT reduces to the binary one when it is applied to binary images. After that, we focus on the fuzzy morphological HMT derived from t-norms and we introduce the concept of “part of an image” which will guide the study of the desirable properties of our operator and the interpretability of the obtained results. Some preliminary experimental results provide evidence of the potential of this tool to be feasible in design algorithms for detecting patterns. Moreover, some comparisons with other hit-or-miss transforms proposed in the literature are performed.

Convex combinations of fuzzy logical operations

One may ask when (non-trivial) convex combinations of fuzzy logical operations result in operations of the same type. This question has been intensively studied for triangular norms and it still remains open for continuous ones. An equivalent problem is obtained for triangular conorms. We show equivalence with the analogous problem for S-implications. Negative answers are obtained for strong fuzzy negations and for R-implications corresponding to continuous triangular norms. The situation for Q-implications and related problems are discussed.

On [formula omitted]-implications derived from grouping functions

Overlap and grouping functions are special kinds of non-necessarily associative aggregation operators recently proposed for applications in classification problems involving the overlap problem and/or when the associativity property is not strongly required, as in image processing and decision making based on fuzzy preference relations, respectively. The concepts of indifference and incomparability defined in terms of overlap and grouping functions may allow their application in several different contexts. This paper introduces the concept of (G,N)-implication, for a grouping function G and a fuzzy negation N. (G,N)-implications are weaker then (S,N)-implications for positive and continuous t-conorms S, in the sense that (G,N)-implications do not necessarily satisfy certain properties, as the exchange and the left neutrality principles, which are not demanded for applications in decision making based on fuzzy preference relations. We analyze several related important properties, providing a characterization of (G,N)-implications.

Typical Hesitant Fuzzy Negations

Since the seminal paper of fuzzy set theory by Zadeh in 1965, many extensions have been proposed to overcome the difficulty for assigning the membership degrees. In recent years, a new extension, the hesitant fuzzy sets, has attracted a lot of interest due to its usefulness to handle those problems in which it is difficult to provide accurately a single membership value; since for hesitant sets, membership values are given by a whole set of values. On the other hand, since fuzzy negations have an important role in applications as well as in the theoretical approach to of fuzzy logics, it is important to study an extension of the concept of fuzzy negation for hesitant fuzzy degrees (elements). In this paper, we propose such a definition and we study some of the main properties of this new concept.

REVISITING Xor-IMPLICATIONS: CLASSES OF FUZZY (CO)IMPLICATIONS BASED ON f-Xor (f-XNor) CONNECTIVES

The main contribution of this paper is the introduction of an intrinsic definition of the connective “fuzzy exclusive or” E (f-Xor E), based only on the properties of boundary conditions, commutativity and partial isotonicity-antitonicity on the the end-points of the unit interval U = [0,1], in a way that the classical definition of the boolean Xor is preserved. We show three classes of the f-Xor E that can be also obtained from the composition of fuzzy connectives, namely, triangular norms, triangular conorms and fuzzy negations. A discussion about extra properties satisfied by the f-Xor E is presented. Additionally, the paper introduces a class of fuzzy equivalences that generalizes the Fodor and Roubens's fuzzy equivalence, and four classes of fuzzy implications induced by the f-Xor E, discussing their main properties. The relationships between those classes of fuzzy implications and automorphisms are explored. The action of automorphisms on f-Xor E is analyzed.

A class of implications related to Yager’s f-implications

In the paper we introduce IU,f,g implications generated by two fuzzy negations and a uninorm. These implications are generalizations of the formula ¬(A∧¬B) from the classical logic and further we generalize the f-generated fuzzy implications introduced by Yager. We study basic properties of the newly proposed implications as well as properties of their φ-transformations.

A new way to extend t-norms, t-conorms and negations

This work presents a method of extending t-norms, t-conorms and fuzzy negations to a lattice-valued setting by preserving the largest possible number of properties of these fuzzy connectives which are invariants under homomorphisms. Further, we also apply this method to extend De Morgan triples, automorphisms and n-dimensional t-norms.

Some new approaches to constructing similarity measures

The purpose of this paper is threefold. Firstly, based on the concept of fuzzy equivalence given by Fodor and Roubens, we propose different methods for the construction of fuzzy equivalencies from biresiduations, automorphisms, fuzzy negations, t-norms and t-conorms. Secondly, we give two manners of generating similarity measures of P.Z. Wang from fuzzy equivalencies. Finally, some properties of the two kinds of similarity measures are discussed and compared.

A construction method of semicopulas from fuzzy negations

In this paper a method of defining commutative semicopulas from fuzzy negations is introduced. Some properties are investigated that lead to understand these semicopulas as non-associative generalizations of the Łukasiewicz t-norm. In particular, it is proved that some well known examples of copulas and t-norms can be obtained by this method. Moreover, any commutative semicopula constructed by this method can be always obtained from a negation N which is symmetric with respect to the diagonal. Then, those symmetric fuzzy negations N for which the corresponding semicopula is a copula are characterized. Also, several examples of symmetric negations N are given such that the corresponding semicopula is a t-norm.