Approaches to the representations and logic operations of fuzzy concepts in the framework of axiomatic fuzzy set theory II

Abstract

In part II of this paper, firstly, we study the relationship between the AFS (Axiomatic Fuzzy Zet) and FCA (Formal Concept Analysis, which has become a powerful theory for data analysis, information retrieval, and Knowledge discovery) and some algebraic homomorphisms between the AFS algebras and the concept lattices are established. Then, the numerical approaches to determining membership functions proposed in part I of this paper are used to study the fuzzy description and data clustering problems by mimicking human reasoning process. Finally, illustrative examples show that the framework of AFS theory offers a far more flexible and effective approach to artificial intelligence system analysis and design with applications to knowledge acquisition and representations in practice.

A Study of Notations and Illustrations of Axiomatic Fuzzy Set Theory

“…It is known that the AFS theory provides an effective tool to convert the information into the training examples and databases into the membership functions and their fuzzy logic operations by taking both the subjective imprecision and objective uncertainty into consideration [14][15][16][17][18][19][20][21]. AFS theory is based on AFS structurea kind of mathematical description of the data structure, which is a special kind of combinatorics systems and AFS algebra -a kind of semantic methodology which is a family of completely distributive lattices.…”

Approaches to the representations and logic operations of fuzzy concepts in the framework of axiomatic fuzzy set theory I

“…In this section, we recall some basic ideas, notations and results given by [7][8][9][10][11][12][13][14][15][16][17][18][19]. It is known that the AFS framework provides an effective tool to convert the information in the training examples and databases into the membership functions and their fuzzy logic operations.…”

“…The results obtained in this paper can be used in selecting appropriate representations based on the data size of the original information, the quantity of original information to be preserved and the number of objects in the universe of discourse to be compared. The examples in part II of this paper [13] will help us to reveal some advantages of the proposed framework.…”

Building multi-adjoint concept lattices

“…Formal concept analysis (FCA) has been extended with ideas from fuzzy set theory [1,2,3], fuzzy logic reasoning [4,5,6], rough set theory [7,8,9], some integrated approaches such as fuzzy and rough [10], or rough and domain theory [11].…”