 Author
 Yun Shi (UGent)
 Promoter
 Etienne Kerre (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 QLimplications satisfy the thirteen fuzzy implication axioms. In Chapter 4 we develop the complete interrelationships between the eight supplementary axioms FI6FI13 for fuzzy implications satisfying the five basic axioms FI1FI15. We prove all the dependencies between the eight fuzzy implication axioms, and provide for each independent case a counterexample. The counterexamples 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 QLimplications for an iterative booleanlike 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 QLimplications 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 tnorms, tconorms and Simplications their robustness against different perturbations in a fuzzy rulebased 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, tconorms, tnorms, fuzzy implication axioms, fuzzy logic, fuzzy implications, fuzzy morphology operations
Downloads

PhDYunShi.pdf
 full text
 
 open access
 
 
 1.39 MB
Citation
Please use this url to cite or link to this publication: http://hdl.handle.net/1854/LU758878
 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 QLimplications satisfy the thirteen fuzzy implication axioms. In Chapter 4 we develop the complete interrelationships between the eight supplementary axioms FI6FI13 for fuzzy implications satisfying the five basic axioms FI1FI15. We prove all the dependencies between the eight fuzzy implication axioms, and provide for each independent case a counterexample. The counterexamples 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 QLimplications for an iterative booleanlike 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 QLimplications 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 tnorms, tconorms and Simplications their robustness against different perturbations in a fuzzy rulebased 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,tconorms,tnorms,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/RUG01001354713\_2010\_0001\_AC.pdf}, year = {2009}, }