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2008 | Buch

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Fuzzy Implications

verfasst von: Michał Baczyński, Balasubramaniam Jayaram

Verlag: Springer Berlin Heidelberg

Buchreihe : Studies in Fuzziness and Soft Computing

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Inhaltsverzeichnis

Frontmatter

An Introduction to Fuzzy Implications

Abstract

The implication operator (→) plays a significant role in the classical two-valued logic. Firstly, from the classical implication one can obtain all other basic logical connectives of the binary logic, viz., the binary operators - and (\(\land\)), or (\(\lor\)) - and the unary negation operator (¬). Secondly, the implication operator holds the center stage in the inference mechanisms of any logic, like modus ponens, modus tollens, hypothetical syllogism in classical logic.

Michał Baczyński, Balasubramaniam Jayaram

Part I: Analytical Study of Fuzzy Implications

Frontmatter

Fuzzy Implications from Fuzzy Logic Operations

Abstract

In this first chapter of Part I, we discuss families of fuzzy implications obtained from binary operations on the unit interval [0, 1], i.e., from basic fuzzy logic connectives, viz., t-norms, t-conorms and negations. After giving the necessary background on the above connectives, we investigate three main ways of generating fuzzy implications from these connectives. We also study the properties possessed by the families of fuzzy implications thus generated, present their characterizations and their representations, in the cases where available.

Michał Baczyński, Balasubramaniam Jayaram

Fuzzy Implications from Generator Functions

Abstract

Recently, Yager [261] has introduced two new families of fuzzy implications, called the f- and g-generated implications, respectively, and discussed their desirable properties. In the next two sections we give the definitions of these newly proposed families of f- and g-generated implications and explore their algebraic properties.

Michał Baczyński, Balasubramaniam Jayaram

Intersections between Families of Fuzzy Implications

Abstract

In the previous chapters, Chaps. 2 and 3, we have dealt with five main families of fuzzy implication operations, viz., (S,N)-, R-, QL-, f- and g-implications. This chapter presents results regarding the intersections that exist among the above families of fuzzy implications. Firstly, we discuss the pair-wise intersections of the families from Chap. 2 and the intersections that exist among the Yager’s family of fuzzy implications in Chap. 3. Following this, we discuss the intersections that exist among the families of fuzzy implications across these two chapters, i.e., the intersections that exist among the Yager’s family of fuzzy implications, viz., f- and g-implications, with (S,N)-, R- and QL-implications.

Michał Baczyński, Balasubramaniam Jayaram

Fuzzy Implications from Uninorms

Abstract

In Chap. 2, we had seen the different ways of defining fuzzy implications based on the basic fuzzy logic connectives, viz., fuzzy negations, t-norms and t-conorms. Uninorms were introduced recently by Yager and Rybalov [263] (see also Fodor et al. [106]) as generalizations of t-norms and t-conorms and are thus another fertile source based on which one can define fuzzy implications. In this chapter, after giving the necessary introduction to uninorms, taking a similar approach as was done in the previous chapter we define fuzzy implications from uninorms and discuss their basic properties.

Michał Baczyński, Balasubramaniam Jayaram

Part II: Algebraic Study of Fuzzy Implications

Frontmatter

Algebraic Structures of Fuzzy Implications

Lattice of Fuzzy Implications

As noted in Sect. 1.1, in the family \({\mathcal FI}\) of all fuzzy implications we can consider the partial order induced from the unit interval [0, 1]. It is interesting and important to note that incomparable pairs of fuzzy implications generate new fuzzy implications by using the standard min (inf) and max (sup) operations. This is another method of generating new fuzzy implications from the given ones.

Michał Baczyński, Balasubramaniam Jayaram

Fuzzy Implications and Some Functional Equations

Abstract

Generally speaking functional equations are equations in which the unknowns are functions. In the previous chapters we have seen some functional equations, viz., the exchange property (EP), the contrapositive symmetry (CP) and the like. But as Prof. Aczél writes in his book [1] “merely stating properties (functional equations) satisfied by a function is different from solving and determining all functions that satisfy a given functional equation”.

In this chapter we deal with a few functional equations involving fuzzy implications. These equations, once again, arise as the generalizations of the corresponding tautologies in classical logic involving boolean implications.

Michał Baczyński, Balasubramaniam Jayaram

Part III: Applicational Study of Fuzzy Implications

Frontmatter

Fuzzy Implications in Approximate Reasoning

Abstract

Boolean implications are employed in inference schemas like modus ponens, modus tollens, etc., where the reasoning is done with statements or propositions whose truth-values are two-valued. Fuzzy implications play a similar role in the generalizations of the above inference schemas, where reasoning is done with fuzzy statements whose truth-values lie in [0, 1] instead of 0, 1.

One of the best known application areas of fuzzy logic is approximate reasoning (AR) ( Driankov et al. [84]), wherein from imprecise inputs and fuzzy premises or rules we obtain, often, imprecise conclusions. Approximate reasoning with fuzzy sets encompasses a wide variety of inference schemes and have been readily applied in many fields, among others, decision making, expert systems and fuzzy control.

Michał Baczyński, Balasubramaniam Jayaram

Backmatter

Titel: Fuzzy Implications
verfasst von: Michał Baczyński
Balasubramaniam Jayaram
Verlag: Springer Berlin Heidelberg
Electronic ISBN: 978-3-540-69082-5
Print ISBN: 978-3-540-69080-1
DOI: https://doi.org/10.1007/978-3-540-69082-5

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