Approaches to the representations and logic operations of fuzzy concepts in the framework of axiomatic fuzzy set theory I☆
Introduction
In today’s massive storage era, knowledge acquisitions and representations constitute a major knowledge engineering bottleneck. There are various approaches aimed at alleviating this problem. The incorporation of fuzzy sets into the representations of fuzzy concepts makes it possible for us to combine the uncertainty handling and approximate reasoning capabilities with comprehensibility. In many fuzzy theories [22], the membership functions of the fuzzy sets are often given manually by human intuition and their logic operators are implemented by a t-norm, a t-conorm and a negation operator. These norms and negation are chosen from infinite kinds of options in advance and the choices are independent of the distribution of the original data. The large-scale intelligence systems in engineering applications are usually very complicated and contain a large number of concepts. It is impossible or difficult to define the membership functions in such way and to choose suitable logic operators from infinite kinds of options. In addition, different logic operator choices may also lead to different results for the same data set.
This paper is organized as follows: In Section 2, a brief review is given on AFS theory. In Section 3, following the EII, EIII algebra representations of fuzzy concepts proposed in [7], [11], we propose E#I algebras, which are lattices finer than EII, EIII algebras. Furthermore, we give two kinds of E#I algebra representations of fuzzy concepts. In Section 4, the norms of AFS algebras (including EII, EIII, E#I algebras) are defined and studied. This leads to a new algorithmic framework for determining fuzzy sets (membership functions) and their logic operations. The membership functions of fuzzy sets and their logic operations are decided by a consistent algorithm based upon the distribution of the raw data. We also study a number of representations in the framework of AFS theory and their relationships. 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.
Section snippets
A review of the AFS theory
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. AFS theory is based on AFS structures, which is a special kind of combinatorial objects [3], and the AFS algebras, which is a family of completely
E#In algebra representations of fuzzy concepts
In this section, we propose the E#In algebras as lattices which are finer than the EIn algebras and give two kinds of E#I algebra representations of the fuzzy concepts in EM. Definition 10 Let X1, X2, … , Xn be n non-empty sets. A binary relation R# on the setis defined as follows. For any , , ∀a1ia2i … ani (i ∈ I), ∃b1hb2h … bnh (h ∈ J) such that ari ⊆ brh, r = 1, 2, … n; ∀b1jb2j … bnj (j ∈ J), ∃a1ka
The norms for the AFS algebras
In this section, we propose a special family of measures. With the measures, the EIn, (n > 1), E#I algebras will become lattices with norms based on which we can convert the AFS algebra represented membership degrees to values in the interval [0, 1] and with great extent to preserve the information contained in the AFS algebra representations. Definition 11 Continuous case: Let X be a set, X ⊆ Rn. ρ : X → R+ = [0, ∞) is integrable on X under Lebesgue measure with . S (S ⊆ 2X) is the set of Borel sets in X. For all
Conclusion
In this paper, we propose the E#I algebras which are finer than the EII, EIII algebras. We also give two kinds of the E#I algebra representations of the fuzzy concepts in EM. The norms of the AFS algebras, which have similar properties to the t-norms and the s-conorms, are proposed and studied. Furthermore, it is shown that the EIn, E#I algebra representations of fuzzy concepts can be converted to the interval [0, 1] representations. This preserves the order relationships in the AFS algebras
Acknowledgement
The authors would like to thank the anonymous referees whose comments and suggestions have helped greatly in improving of this paper.
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This work is supported in part by the National Natural Science Foundation of China under Grant 60534010 60575039 and in part by the National Key Basic Research and Development Program of China under Grant 2002CB312201-06.