New Class Of Fuzzy Implications Research Articles

Modus Ponens property of T-power based implications

Among the great bunch of additional properties which fuzzy implications may satisfy, the so-called invariance with respect to T-powers highlights due to its applications in approximate reasoning. This property ensures the invariance of the truth value of the fuzzy implication when both the antecedent and the consequent are modified using the same quantifier modeled by using powers of t-norms. In order to find the fuzzy implications that satisfy this property, a new class of fuzzy implications named T-power based implications has been proposed. In this paper we study when this kind of fuzzy implication satisfies the Modus Ponens property with respect to a continuous t-norm T♯. Then a characterization theorem on this topic in terms of residual implication generated from a summand of T♯ is presented.

Distributivity of a class of ordinal sum implications over t-norms and t-conorms

Recently, Baczyński, Drygaś, Król and Mesiar have proposed two constructions of ordinal sums of fuzzy implications based on the construction of ordinal sum of overlap functions. In this paper, we study the class of ordinal sum implications constructed using the first method, and show that this new class of fuzzy implications is different from the known (S,N)-, R-, QL-, f- and g-implications. Furthermore, we explore this new class of ordinal sum implications with respect to distributivity.

Fuzzy Implications Generated by h-Generators

In this work, by using h-generators, we introduce a new class of fuzzy implications, which are called (h, c)-generated implications. Some properties of (h, c)-implications are studied, including left neutrality, order property, exchange principle, law of contraposition, law of importation and distributivity over t-norms or t-conorms. We also discuss the intersections of both (h; c)-generated implications with R-, (S,N)-, QL-, f-, g- and h-implications and prove that (h, c)-generated implications are different from the ones above. In addition, we define (h, a, b, c)-implications by generalizing (h, c)-generated implications and obtain their properties similarly.

Distributivity of the Ordinal Sum Implications Over t-Norms and t-Conorms

Recently, Su and Liu have introduced a new class of fuzzy implications, called ordinal sum implications, and discussed some of their desirable properties, such as neutrality property, consequent boundary, exchange principle, etc. In this paper, we explore the class of ordinal sum implications with respect to distributivity. Necessary and sufficient conditions, under which ordinal sum implications are distributive over t-norms and t-conorms are given.

On ordinal sum implications

A new class of fuzzy implications, called ordinal sum implications, is introduced by means of the ordinal sum of a family of given implications, which is similar to the ordinal sum of t-norms (or t-conorms). Basic properties of ordinal sum implications are discussed. It is shown that the ordinal sum implication is really a new class, which is different from the known (S,N)-, R-, QL- and Yager’s f- and g-implications.

A new class of fuzzy implications derived from generalized h-generators

A new class of fuzzy implications, called (h,min)-implications, is introduced by means of generalized h-generators. Basic properties of these implications are discussed. It is shown that the (h,min)-implications are really a new class different from the known (S,N)-, R-, QL- and Yager's f- and g-implications. Generalizations of three classical logic tautologies with implications, viz., law of importation, contraction law and distributivity over triangular norms (t-norms) and triangular conorms (t-conorms) are investigated. A series of necessary and sufficient conditions are established, under which the corresponding functional equations are satisfied.

A generalization of Yager’s f-generated implications

Since Yager introduced f-generated implications in 2004, this class of fuzzy implications has been extensively investigated. In this paper, we generalize f-generated implications and get a new class of fuzzy implications called (f,g)-implications, which is different from the usual known classes of fuzzy implications. We discuss the basic algebraic properties of (f,g)-implications and study some classical logic tautologies (i.e., law of importation, contrapositive symmetry and distributivity over t-norms or t-conorms) for (f,g)-implications. Characterization of solutions to the corresponding fuzzy functional equations is obtained.

ON A NEW CLASS OF IMPLICATIONS: (g,min)-IMPLICATIONS AND SEVERAL CLASSICAL TAUTOLOGIES

A new class of fuzzy implications, called (g, min )-implications, is introduced by means of the additive generators of continuous Archimedean t-conorms, called g-generators. Basic properties of these implications are discussed. It is shown that the (g, min )-implications are really a new class different from the known ( S , N )-, R -, QL - and Yager's f- and g-implications. Generalizations of three classical logic tautologies with implications, viz. law of importation, contraction law and distributivity over triangular norms ( t -norms) and triangular conorms ( t -conorms) are investigated. A series of necessary and sufficient conditions are proposed, under which the corresponding functional equations are satisfied.

On a new class of fuzzy implications: h-Implications and generalizations

A new class of fuzzy implications called the h-implications is introduced. They are implications generated from an additive generator of a representable uninorm in a similar way of Yager’s f- and g-implications which are generated from additive generators of continuous Archimedean t-norms and t-conorms. Basic properties of these implications are studied in detail. Modifications and generalizations of the initial definition are presented and their properties studied and compared between them. One of the modifications, called ( h, e)-implications, is another example of a fuzzy implication satisfying the exchange principle but not the law of importation for any t-norm, in fact for any function F : [0, 1] 2 → [0, 1].

Yager’s new class of implications Jf and some classical tautologies

Recently, Yager [R. Yager, On some new classes of implication operators and their role in approximate reasoning, Information Sciences 167 (2004) 193–216] has introduced a new class of fuzzy implications, denoted J f , called the f-generated implications and has discussed some of their desirable properties, such as neutrality, exchange principle, etc. In this work, we discuss the class of J f implications with respect to three classical logic tautologies, viz., distributivity, law of importation and contrapositive symmetry. Necessary and sufficient conditions under which J f implications are distributive over t-norms and t-conorms and satisfy the law of importation with respect to a t-norm have been presented. Since the natural negations of J f implications, given by N J f ( x ) = J f ( x , 0 ) , in general, are not strong, we give sufficient conditions under which they become strong and possess contrapositive symmetry with respect to their natural negations. When the natural negations of J f are not strong, we discuss the contrapositivisation of J f . Along the lines of J f implications, a new class of implications called h-generated implications, J h , has been proposed and the interplay between these two types of implications has been discussed. Notably, it is shown that while the natural negations of J f are non-filling those of J h are non-vanishing, properties which determine the compatibility of a contrapositivisation technique.

A new class of fuzzy implications. Axioms of fuzzy implication revisited

Many different fuzzy implication operators have been proposed; most of them fit into one of the two classes: implication operations that are based on an explicit representation of implication A → B in terms of &, ∨, and ¬ (e.g., S-implications that are based on the formula B ∨ ¬ A), and R-implications that are based on an implicit representation of implication A → B as the weakest C for which C&B implies A. However, some fuzzy implication operations (such as b a ) cannot be naturally represented in this form. To describe such operations, we propose a new (third) class of implication operations called A-implications whose relation to &, ∨, and ¬ is described by (implicit) axioms.