Classical Fuzzy Theory Research Articles

Neutrosophic C-means clustering with local information and noise distance-based kernel metric image segmentation

The traditional FCM algorithm is developed on the basis of classical fuzzy theory, though the classical fuzzy theory has its own limitations. The lack of expressive ability of uncertain information makes it hard for FCM algorithm to handle clustered boundary pixels and outliers. This paper proposes a Neutrosophic C-means Clustering with local information and noise distance-based kernel metric for image segmentation (NKWNLICM). At first, noisy distance and fuzzy spatial information are introduced to NCM model to improve the robustness of noise image segmentation. Then, the kernel function is used to measure the distance between pixels. By mapping low-dimensional data into high-dimensional data, the classification performance is further improved. At last, the fuzzy factor is redefined based on the distance between the center pixel and its neighborhood. The new fuzzy factor can excellently reflect the influence of neighborhood pixels on central pixels and improve the classification accuracy much better. The experimental results on Berkeley Segmentation Database demonstrates the excellent performance of the proposed method for noisy image segmentation.

A Logical Study of Local and Global Graded Similarities

In this work we study the relationship between global and local similarities in the graded framework of fuzzy class theory (FCT), in which there already exists a graded notion of similarity. In FCT we can express the fact that a fuzzy relation is reflexive, symmetric, or transitive up to a certain degree, and similarity is defined as a first-order sentence, which is the fusion of three sentences corresponding to the graded notions of reflexivity, symmetry, and transitivity. This allows us to speak in a natural way of the degree of similarity of a relation. We consider global similarities defined from local similarities using t-norms as aggregation operators, and we obtain some results in the framework of FCT that, adequately interpreted, allow us to say that when we take a t-norm as an aggregation operator, the properties of reflexivity, symmetry, and transitivity of fuzzy binary relations are inherited from the local to the global level, and that the global similarity is a congruence if some of the local similarities are congruences.

Graded dominance and related graded properties of fuzzy connectives

Graded properties of binary and unary fuzzy connectives (valued in MTLΔ-algebras) are studied, including graded monotony, a generalized Lipschitz property, commutativity, associativity, unit and null elements, and the dominance relation between fuzzy connectives. The apparatus of Fuzzy Class Theory (or higher-order fuzzy logic) is employed as a tool for easy derivation of graded theorems on the connectives.

Graded properties of unary and binary fuzzy connectives

The paper studies basic graded properties of unary and binary fuzzy connectives, i.e., unary and binary operations on the set of truth degrees of a background fuzzy logic extending the logic MTL of left-continuous t-norms. The properties studied in this paper are graded generalizations of monotony, Lipschitz continuity, null and unit elements, idempotence, commutativity, and associativity. The paper elaborates the initial study presented in previous papers and focuses mainly on parameterization of graded properties by conjunction-multiplicities of subformulae in the defining formulae, preservation of graded properties under compositions and slight variations of fuzzy connectives, the values of graded properties for basic connectives of the ground logic, and the dependence of the values on the ground logic. The results are proved in the formal framework of higher-order fuzzy logic MTL, also known as Fuzzy Class Theory (FCT). General theorems provable in FCT are illustrated on several semantic examples.

Relational compositions in Fuzzy Class Theory

We present a method for mass proofs of theorems of certain forms in a formal theory of fuzzy relations and classes. The method is based on formal identification of fuzzy classes and inner truth values with certain fuzzy relations, which allows transferring basic properties of sup-T and inf-R compositions to a family of more than 30 composition-related operations, including sup-T and inf-R images, pre-images, Cartesian products, domains, ranges, resizes, inclusion, height, plinth, etc. Besides yielding a large number of theorems on fuzzy relations as simple corollaries of a few basic principles, the method provides a systematization of the family of relational notions and generates a simple equational calculus for proving elementary identities between them, thus trivializing a large part of the theory of fuzzy relations.

Filters in fuzzy class theory

A concept of filter is introduced in fuzzy class theory. Graded properties of filters, prime filters, and related constructions are investigated. In particular, the properties of filters relativized with respect to the three basic t-norms (Gödel, product, and Łukasiewicz) are studied. A naturally arising example of a graded filter is constructed and it is shown that the models of filters include well-known families of set functions.

Relations in Fuzzy Class Theory:: Initial steps

This paper studies fuzzy relations in the graded framework of Fuzzy Class Theory (FCT). This includes (i) rephrasing existing work on graded properties of binary fuzzy relations in the framework of Fuzzy Class Theory and (ii) generalizing existing crisp results on fuzzy relations to the graded framework. Our particular aim is to demonstrate that Fuzzy Class Theory is a powerful and easy-to-use instrument for handling fuzzified properties of fuzzy relations. This paper does not rephrase the whole theory of (fuzzy) relations; instead, it provides an illustrative introduction showing some representative results, with a strong emphasis on fuzzy preorders and fuzzy equivalence relations.

Fuzzy class theory

The paper introduces a simple, yet powerful axiomatization of Zadeh's notion of fuzzy set, based on formal fuzzy logic. The presented formalism is strong enough to serve as foundations of a large part of fuzzy mathematics. Its essence is elementary fuzzy set theory, cast as two-sorted first-order theory over fuzzy logic, which is generalized to simple type theory. We show a reduction of the elementary fuzzy set theory to fuzzy propositional calculus and a general method of fuzzification of classical mathematical theories within this formalism. In this paper we restrict ourselves to set relations and operations that are definable without any structure on the universe of objects presupposed; however, we also demonstrate how to add structure to the universe of discourse within our framework.

About the axiomatizations of fuzzy set theory

Starting from the results of French [9] we discuss the notions of Fuzzy Set and of fuzziness: we show some characteristic properties that should be fulfilled by an axiomatic theory of fuzzy classes. We discuss two axiomatizations of fuzzy classes already proposed [5, 18] and we show the first axioms of a new axiomatic theory of fuzzy classes satisfying the properties previously showed.