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# A Deep Study of Fuzzy Implications

Yun Shi (UGent)
(2009)
Author
Promoter
(UGent) and Da Ruan
Organization
Abstract
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.
Keywords
fuzzy negations, fuzzy adjunctions, t-conorms, t-norms, fuzzy implication axioms, fuzzy logic, fuzzy implications, fuzzy morphology operations

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## Citation

Chicago
Shi, Yun. 2009. “A Deep Study of Fuzzy Implications”. Ghent, Belgium: Ghent University. Faculty of Sciences.
APA
Shi, Yun. (2009). A Deep Study of Fuzzy Implications. Ghent University. Faculty of Sciences, Ghent, Belgium.
Vancouver
1.
Shi Y. A Deep Study of Fuzzy Implications. [Ghent, Belgium]: Ghent University. Faculty of Sciences; 2009.
MLA
Shi, Yun. “A Deep Study of Fuzzy Implications.” 2009 : n. pag. Print.
@phdthesis{758878,
abstract     = {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{\textbackslash}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 \${\textbackslash}mathcal\{F\}({\textbackslash}mathbb\{R\}\^{ }\{n\})\$. And then we study the conditions under which a fuzzy dilation which is defined from a conjunction \${\textbackslash}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 \${\textbackslash}mathcal\{C\}\$ on the unit interval and the implication \$I\$ or the implication \$I\^{ }\{'\}\$ play important roles in such conditions.},
author       = {Shi, Yun},
keyword      = {fuzzy negations,fuzzy adjunctions,t-conorms,t-norms,fuzzy implication axioms,fuzzy logic,fuzzy implications,fuzzy morphology operations},
language     = {eng},
pages        = {V, 147},
publisher    = {Ghent University. Faculty of Sciences},
school       = {Ghent University},
title        = {A Deep Study of Fuzzy Implications},
url          = {http://lib.ugent.be/fulltxt/RUG01/001/354/713/RUG01-001354713\_2010\_0001\_AC.pdf},
year         = {2009},
}