Application Of Logical Equivalence Operators In Fuzzy Reasoning

Fuzzy inference provides a theoretical foundation of fuzzy control. One possible method to evaluate fuzzy inference is to measure its robustness. As far as the robust-ness is concerned, it is crucial to choose an appropriate perturbation parameter. The most commonly used perturbation parameter is defined with respect to the usual met-ric on the unit interval [0,1]. However, the outcome of fuzzy reasoning relies heavily on the choices of the fuzzy connectives and fuzzy implication operators. Since the log-ical equivalence operator is induced by the implication operator, the correspondingly defmed perturbation parameter is consistent with logical inference. The objective of this thesis is twofold. Firstly, we aim to construct a series of perturbation param-eters by using the logical equivalence operators and study robustness of the triple I inference method. Secondly, we compare different perturbation parameters from the viewpoint of topology. Furthermore, the topological properties derived from some logical equivalence operators are investigated.The present thesis is divided into four chapters.Chapter 1 first introduces some definitions and methods of fuzzy inference. More-over, some preliminaries about residuated lattice and topology are reviewed, which are necessary for the remaining chapters.In chapter 2, the average logical similarity degree is constructed with respect to the logical equivalence operator. Taking the similarity degree as the perturbation pa-rameter, the robustness of the connectives and triple I inference method are discussed. In addition, minimum and average logical similarity degrees are compared from the perspective of topology. It is obtained that the metrics induced separately by these two similarity measures are equivalent. Different logical equivalence operators will induce different metrie spaces. The distributions of isolated points in different metric spaces as well as connectivity and dense subsets are studied. Based on the above results, comparative analysis of different perturbation parameters is presented.In chapter 3, firstly the minimum logical similarity degree between fuzzy sets is generalized to lattice valued fuzzy sets and the relevant topology spaces on FL(X) are induced by using different implication operators, where FL(X) denotes the set of all lattice valued fuzzy sets. In the topology spaces which are induced separately by using R0 and Godel implication operators, the condensation points are proved to be normal lattice valued fuzzy sets or their complements. The converse, however, is not true in general, and a counterexample is constructed. Furthermore, when L is residuated lattice, triangular norm and logical equivalence operator are used to construct two lat-tice valued similarity measures E1 and E2. It is pointed out that both E1 and E2 are L- equalities, and also, Ei- Cauchy sequence is Ei-convergence sequence respectively, i=1,2. In order to express the lattice valued similarity measure more intuitively, state m is taken as the tool to construct similarities Sm, Sm, Sm* and Sm*, whose value is taken from [0,1]. Moreover, these four similarity measures are proved satisfying the set of axioms of L-equality, and the robustness of the triple I method is discussed with respect to the measurement Sm. Logical equivalence operators in residuated lattice play very important roles during the construction process of aforementioned similarity measures. Based on these logical equivalence operators, uniform topologies and quo-tient residuated lattice are established in residuated lattice. Especially, compactness and convergence of sequence under the logical equivalence metrics on [0,1] residuated lattice are also presented.In chapter 4, firstly the composites of two pairs of residual operators in complete lattice are studied. The counterexamples show that there is no reversibility about the aforementioned composite. In addition, the conditions under which the bipolar t-norm and implication to be decomposed into two unipolar operators are given. And an application of bipolar information in decision-making is also presented. One of the tools for presenting bipolar information is bipolar fuzzy sets, which are also called intuitionistic fuzzy sets. As special lattice-valued fuzzy sets, intuitionistic fuzzy sets own special application background and properties. Four types of similarity measures of the intuitionistic fuzzy sets are constructed by using logical equivalence operators. Then, the equivalence of the metric spaces with respect to the four similarity measures is proved. Moreover, several similarity measures are compared from the view point of the perturbation parameters. Finally, an application example in pattern recognition is presented.