R₀-algebras And Theory Of Similarity Measures Between Vague Sets

In the filed of artificial intelligence, fuzzy control and fuzzy information processing have been drawing deep interest of many researchers in the world. However, its mathematical foundation is not so perfect. To solve the problem, many formal deduction systems based on different practical backgrounds are built up, and the algebraic sematics with respect to those fuzzy logic also reach many significant results. Among them, BL-algebras and MTL-algebras based on continuous t-norms and left-continuous t-norms are more meaningful. NM-algebras, as an important extension of MTL-algebras, is proved to be equivalent to R₀-algebras, which was proposed by Professor Wang Guo-jun in 1998 based on L^*-Lindenbaum algebras. This conclusion makes it possible that the relatively mature theories and methods of R₀-algebras can be transplanted into NM-algebras in a paralleling way. It is a significant step towards the perfection and maturity of the theory of MTL-algebras. Spired by it, this work devotes to extract the fundamental characteristics, it concerned with simplifying the definition, classifying and constructing of .R₀-algebras. By working on the relationship between R₀-algebras and the corresponding logic system L^*, some valuable results have been developed, it includes the reducing of the complexity of the formalized reasoning in system L^*; the axiomatizing of all extensions of system L^* and the discussing of its E-completeness.On the other side, as one of the hotspots in the field of information amalgamation technology, fuzzy information processing needs more powerful tools to represent and handle information with uncertainty. Vague sets was proposed by Gau and Buehrer in 1993, and has been applied in many fields of AI, such as, machine learning, decision analysis, knowledge representation and mode matching, etc. Among those applications, the interesting spot is how to measure the similarity between Vague sets. It is such an essential key technology that it gained close attention of many researcheres. Many methods were presented for the measure of similarly degree. Based on the analysis of known results, the work described in this thesis provides the normal definition of the similarity degree between vague sets. Moreover, a new method was purposed based on Hausdorff distance.This work is described in three chapters.In Chapter 1, the simplifying of the definition of R₀-algebras is studied sys-...