Some New Implication Operations Emerging From Fuzzy Logic

We choose, from fuzzy set theory, t-norms, t-conorms and fuzzy compliments which forms dual triplet that is (i,u,c) that satisfy the DeMorgan's law, these dual triplet are used in the construction of fuzzy implications in fuzzy logic. In this work introduction of fuzzy implication is given, which included definition of fuzzy implications and their properties and also distinct classes of fuzzy implication (S, R and QL-implications). Further also described previous work on fuzzy implication and supporting literature of construction of fuzzy implication are given. Finally main contribution of work is to design new fuzzy implication and their graphical representations

Interval-valued algebras and fuzzy logics

In this chapter, we present a propositional calculus for several interval-valued fuzzy logics, i.e., logics having intervals as truth values. More precisely, the truth values are preferably subintervals of the unit interval. The idea behind it is that such an interval can model imprecise information. To compute the truth values of ‘p implies q’ and ‘p and q’, given the truth values of p and q, we use operations from residuated lattices. This truth-functional approach is similar to the methods developed for the well-studied fuzzy logics. Although the interpretation of the intervals as truth values expressing some kind of imprecision is a bit problematic, the purely mathematical study of the properties of interval-valued fuzzy logics and their algebraic semantics can be done without any problem. This study is the focus of this chapter

A practical inference method with several implicative gradual rules and a fuzzy input: one and two dimensions

International audienceA general approach to practical inference with gradual implicative rules and fuzzy inputs is presented. Gradual rules represent constraints restricting outputs of a fuzzy system for each input. They are tailored for interpolative reasoning. Our approach to inference relies on the use of inferential independence. It is based on fuzzy output computation under an interval-valued input. A double decomposition of fuzzy inputs is done in terms of alpha-cuts and in terms of a partitioning of these cuts according to areas where only a few rules apply. The case of one and two dimensional inputs is consideredCet article présente une méthode d'inférence avec des règles implicatives graduelles pour une entrée floue. Les règles graduelles représentent des contraintes qui restreignent l'univers de sortie pour chacune des entrées. Elles sont conçues pour réaliser des interpolations. L'algorithme que nous proposons s'appuie sur le principe de indépendance inférentielle. Il met en oeuvre une double décomposition de l'ensemble flou d'entrée, par alpha-coupes et suivant le partitionnement de l'univers des variables d'entrée. Les cas étudiés correspondent à des systèmes à une et deux dimension

Bandler–Kohout Subproduct With Yager’s Classes of Fuzzy Implications

The Bandler-Kohout subproduct (BKS) inference mechanism is one of the two established fuzzy relational inference (FRI) mechanisms; the other one being Zadeh's compositional rule of inference (CRI). Both these FRIs are known to possess many desirable properties. It can be seen that many of these desirable properties are due to the rich underlying structure, viz., the residuated algebra, from which the employed operations come. In this study, we discuss the BKS relational inference system, with the fuzzy implication interpreted as Yager's classes of implications, which do not form a residuated structure on [0,1] . We show that many of the desirable properties, viz., interpolativity, continuity, robustness, which are known for the BKS with residuated implications, are also available under this framework, thus expanding the choice of operations available to practitioners. Note that, to the best of the authors' knowledge, this is the first attempt at studying the suitability of an FRI where the operations come from a nonresiduated structure

Order algebraizable logics

AbstractThis paper develops an order-theoretic generalization of Blok and Pigozziʼs notion of an algebraizable logic. Unavoidably, the ordered model class of a logic, when it exists, is not unique. For uniqueness, the definition must be relativized, either syntactically or semantically. In sentential systems, for instance, the order algebraization process may be required to respect a given but arbitrary polarity on the signature. With every deductive filter of an algebra of the pertinent type, the polarity associates a reflexive and transitive relation called a Leibniz order, analogous to the Leibniz congruence of abstract algebraic logic (AAL). Some core results of AAL are extended here to sentential systems with a polarity. In particular, such a system is order algebraizable if the Leibniz order operator has the following four independent properties: (i) it is injective, (ii) it is isotonic, (iii) it commutes with the inverse image operator of any algebraic homomorphism, and (iv) it produces anti-symmetric orders when applied to filters that define reduced matrix models. Conversely, if a sentential system is order algebraizable in some way, then the order algebraization process naturally induces a polarity for which the Leibniz order operator has properties (i)–(iv)

A Deep Study of Fuzzy Implications

This thesis contributes a deep study on the extensions of the IMPLY operator in classical binary logic to fuzzy logic, which are called fuzzy implications. After the introduction in Chapter 1 and basic notations about the fuzzy logic operators In Chapter 2 we first characterize In Chapter 3 S- and R- implications and then extensively investigate under which conditions QL-implications satisfy the thirteen fuzzy implication axioms. In Chapter 4 we develop the complete interrelationships between the eight supplementary axioms FI6-FI13 for fuzzy implications satisfying the five basic axioms FI1-FI15. We prove all the dependencies between the eight fuzzy implication axioms, and provide for each independent case a counter-example. The counter-examples provided in this chapter can be used in the applications that need different fuzzy implications satisfying different fuzzy implication axioms. In Chapter 5 we study proper S-, R- and QL-implications for an iterative boolean-like scheme of reasoning from classical binary logic in the frame of fuzzy logic. Namely, repeating antecedents $n$ times, the reasoning result will remain the same. To determine the proper S-, R- and QL-implications we get a full solution of the functional equation $I(x,y)=I(x,I(x,y))$, for all $x$, $y\in[0,1]$. In Chapter 6 we study for the most important t-norms, t-conorms and S-implications their robustness against different perturbations in a fuzzy rule-based system. We define and compare for these fuzzy logical operators the robustness measures against bounded unknown and uniform distributed perturbations respectively. In Chapter 7 we use a fuzzy implication $I$ to define a fuzzy $I$-adjunction in $\mathcal{F}(\mathbb{R}^{n})$. And then we study the conditions under which a fuzzy dilation which is defined from a conjunction $\mathcal{C}$ on the unit interval and a fuzzy erosion which is defined from a fuzzy implication $I^{'}$ to form a fuzzy $I$-adjunction. These conditions are essential in order that the fuzzification of the morphological operations of dilation, erosion, opening and closing obey similar properties as their algebraic counterparts. We find out that the adjointness between the conjunction $\mathcal{C}$ on the unit interval and the implication $I$ or the implication $I^{'}$ play important roles in such conditions

Extending possibilistic logic over Gödel logic

In this paper we present several fuzzy logics trying to capture different notions of necessity (in the sense of possibility theory) for Gödel logic formulas. Based on different characterizations of necessity measures on fuzzy sets, a group of logics with Kripke style semantics is built over a restricted language, namely, a two-level language composed of non-modal and modal formulas, the latter, moreover, not allowing for nested applications of the modal operator N. Completeness and some computational complexity results are shown

A short note on fuzzy relational inference systems

This paper is a short note contribution to the topic of fuzzy relational inference systems and the preservation of their desirable properties. It addresses the two main fuzzy relational inferences – compositional rule of inference (CRI) and the Bandler–Kohout subproduct (BK-subproduct) – and their combination with two fundamental fuzzy relational models of fuzzy rule bases, namely, the Mamdani–Assilian and the implicative models. The goal of this short note article is twofold. Firstly, we show that the robustness related to the combination of BK-subproduct and implicative fuzzy rule base model was not proven correctly in [24]. However, we will show that the result itself is still valid and a valid proof will be provided. Secondly, we shortly discuss the preservation of desirable properties of fuzzy inference systems and conclude that neither the above mentioned robustness nor any other computational advantages should automatically lead to a preference of the combinations of CRI with Mamdani–Assilian models or of the BK-subproduct with the implicative models