Families Of Fuzzy Implications Research Articles

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.

Two types of ordinal sums of fuzzy implications on bounded lattices

In this paper, two kinds of ordinal sums of fuzzy implications on bounded lattices are provided. The first one is complementing a specific fuzzy implication to a given family of fuzzy implications defined on pairwise disjoint closed subintervals of a bounded lattice. The second way is defining specific values outside the given family of countably many subintervals, where the endpoints of the subintervals constitute a chain. For the first way, necessary and sufficient conditions for a fuzzy implication to be a complement of a given family of fuzzy implications such that the ordinal sum is a fuzzy implication are provided, which generalize some results of fuzzy implications on the real unit interval. Based on this, we deal with those elements that are incomparable with the endpoints of the given subintervals and provide a fuzzy implication as a complement such that for arbitrary family of fuzzy implications on the given subintervals, the ordinal sum is always a fuzzy implication. For the second ordinal sum, we provide two methods for defining values outside the given subintervals that are shown to be always fuzzy implications.

New results on ordinal sum implications based on ordinal sum of overlap functions

Recently, Baczyński et al. have introduced two new methods of constructing ordinal sums of fuzzy implications. These methods are based on a construction of ordinal sum of overlap functions and extend some type of algebraic structures. In this paper, we explore the family of ordinal sum implications constructed by one of the method presented in (M. Baczyński, P. Drygaś, A. Król, R. Mesiar (2017), [7]) and examine intersections of this new family of fuzzy implications with (S, N)-, (T, N)-, QL-, D-, R-, Yager's f- and g-implications, respectively. Concretely, this new family of fuzzy implications is shown to be different from the above mentioned implications. Moreover, the necessary and sufficient conditions under which these new ordinal sum implications are distributive over t-norms and t-conorms are given.

Analyzing Differentiable Fuzzy Logic Operators

The AI community is increasingly putting its attention towards combining symbolic and neural approaches, as it is often argued that the strengths and weaknesses of these approaches are complementary. One recent trend in the literature is weakly supervised learning techniques that employ operators from fuzzy logics. In particular, these use prior background knowledge described in such logics to help the training of a neural network from unlabeled and noisy data. By interpreting logical symbols using neural networks, this background knowledge can be added to regular loss functions, hence making reasoning a part of learning.We study, both formally and empirically, how a large collection of logical operators from the fuzzy logic literature behave in a differentiable learning setting. We find that many of these operators, including some of the most well-known, are highly unsuitable in this setting. A further finding concerns the treatment of implication in these fuzzy logics, and shows a strong imbalance between gradients driven by the antecedent and the consequent of the implication. Furthermore, we introduce a new family of fuzzy implications (called sigmoidal implications) to tackle this phenomenon. Finally, we empirically show that it is possible to use Differentiable Fuzzy Logics for semi-supervised learning, and compare how different operators behave in practice. We find that, to achieve the largest performance improvement over a supervised baseline, we have to resort to non-standard combinations of logical operators which perform well in learning, but no longer satisfy the usual logical laws.

A Study of (T, N)– and (N ′, T, N)– Implications

In this paper, we revisit and study the basic properties of two families of fuzzy implications, the so-called and implications. More specific, we study when these fuzzy implications satisfy, or not, the neutrality property , the exchange principle , the identity principle and the ordering property . Moreover, a study is presented for the law of importation with respect to a t- norm. Also, we study the relation of conjugation in implications.

An Application of Classical Logic’s Laws in Formulas of Fuzzy Implications

The crucial role that fuzzy implications play in many applicable areas was our motivation to revisit the topic of them. In this paper, we apply classical logic’s laws such as De Morgan’s laws and the classical law of double negation in known formulas of fuzzy implications. These applications lead to new families of fuzzy implications. Although a duality in properties of the preliminary and induced families is expected, we will prove that this does not hold, in general. Moreover, we will prove that it is not ensured that these applications lead us to fuzzy implications, in general, without restrictions. We generate and study three induced families, the so-called D ′ -implications, QL ′ -implications, and R ′ -implications. Each family is the “closest” to its preliminary-“creator” family, and they both are simulating the same (or a similar) way of classical thinking.

Importation lattices

In recent years, many works have appeared that propose order from basic fuzzy logic connectives. However, all of them assume the connectives to possess some kind of monotonicity, which succinctly implies that the underlying set is already endowed with an order. In this work, given a set P≠∅, we define an algebra based on an implicative-type function I without assuming any order-theoretic properties, either on P or I. Terming it the importation algebra, since the law of importation becomes one of the main axioms in this algebra, we show that such algebras can impose an order on the underlying set P to obtain importation posets. In the case P=[0,1], it is well-known that only R-implications obtained from left-continuous t-norms lead us to richer order-theoretic structures. However, we show here that we can obtain new order-theoretic structures on it even from other families of fuzzy implications, in fact, even when I is not a fuzzy implication, and that one can recover the usual order on [0,1] even from fuzzy implications that do not have the classical ordering property. We also give some sufficient conditions under which such importation posets become lattices. Finally, we present some further possible explorations.

Investigations of fuzzy implications satisfying generalized hypothetical syllogism

Generalized hypothetical syllogism (GHS) is one of the fuzzy inference rules that plays a central role in the selection of suitable fuzzy implications for a specific task. Because of the important role played by the classical hypothetical syllogism, in this work, we attempt to investigate the fuzzy implications that do satisfy (GHS). Due to the variety of fuzzy implications and the complexity of the functional equation, we restrict our investigations of fuzzy implications only for those which come from some well known families of fuzzy implications, viz., (S,N)-, R-, QL-, and Yager's families of f-, g-implications. Since there do not exist many fuzzy implications satisfying (GHS) from these families of fuzzy implications, we propose some classes of fuzzy implications and show that every element from these classes satisfies (GHS). Finally, we examine the preservation of (GHS) by some important generating methods of fuzzy implications that exist in the literature.

Bijective transformations of fuzzy implications – An algebraic perspective

Bijective transformations play an important role in generating fuzzy implications from fuzzy implications. In [Representations through a Monoid on the set of Fuzzy Implications, Fuzzy Sets and Systems, 247, 51–67], Vemuri and Jayaram proposed a monoid structure on the set of fuzzy implications, which is denoted by I, and using the largest subgroup S of this monoid discussed some group actions on the set I. In this context, they obtained a bijective transformation which ultimately led to hitherto unknown representations of the Yager's families of fuzzy implications, viz., f-, g-implications. This motivates us to consider whether the bijective transformations proposed by Baczyński & Drewniak and Jayaram & Mesiar, in different but purely analytic contexts, also possess any algebraic connotations. In this work, we show that these two bijective transformations can also be seen as being obtained from some group actions of S on I. Further, we consider the most general bijective transformation that generates fuzzy implications from fuzzy implications and show that it can also be obtained as a composition of group actions of S on I. Thus this work tries to position such bijective transformations from an algebraic perspective.

Homomorphisms on the monoid of fuzzy implications and the iterative functional equation [formula omitted

Recently, Vemuri and Jayaram proposed a novel method of generating fuzzy implications, called the ⊛-composition, from a given pair of fuzzy implications (Vemuri and Jayaram, 2014). However, as with any generation process, the ⊛-composition does not always generate new fuzzy implications. In this work, we study the generative power of the ⊛-composition. Towards this end, we study some specific functional equations all of which lead to the solutions of the iterative functional equation I(x,I(x,y))=I(x,y) involving fuzzy implications which has been studied extensively for different families of fuzzy implications in this very journal, see Balasubramaniam (2007), Xie and Qin (2010) and Xie et al. (2012). In this work, unlike in other existing works, we do not restrict the solutions to a particular family of fuzzy implications. Thus we take an algebraic approach towards solving these functional equations. Viewing the ⊛-composition as a binary operation ⊛ on the set I of all fuzzy implications one obtains a monoid structure (I,⊛) on the set I. From the Cayley’s theorem for monoids, we know that any monoid is isomorphic to the set of all right translations. We determine the complete set K of fuzzy implications w.r.t. which the right translations also become semigroup homomorphisms on the monoid (I,⊛) and show that K not only answers our questions regarding the generative power of the ⊛-composition but also contains many as yet unknown solutions of the iterative functional equation I(x,I(x,y))=I(x,y).

Contrapositive symmetry of fuzzy implications

Contrapositive symmetry of R- and QL-implications defined from t-norms, t-conorms and strong negations is studied. For R-implications, characterizations of contrapositive symmetry are proved when the underlying t-norm satisfies a residuation condition. Contrapositive symmetrization of R-implications not having this property makes it possible to define a conjunction so that the residuation principle is preserved. Cases when this associated conjunction is a t-norm are characterized. As a consequence, a new family of t-norms (called nilpotent minimum) owing several attractive properties is discovered. Concerning QL-implications, contrapositive symmetry is characterized by solving a functional equation. When the underlying t-conorm is continuous and the t-norm is Archimedean, the t-conorm must be isomorphic to the Lukasiewicz one, while the t-norm must be isomorphic to a member from the well-known Frank family of t-norms. Finally, contrapositive symmetry for some new families of fuzzy implications is investigated.