Elsevier

Information Sciences

Volume 537, October 2020, Pages 18-29
Information Sciences

On the Negation of discrete Z-numbers

https://doi.org/10.1016/j.ins.2020.05.106Get rights and content

Abstract

The negation of a problem provides a new perspective for information representation. However, existing negation method has limitations since it can only be applied to the accurately expressed knowledge. Real-world information is imperfect and imprecise. It usually describes in natural language. In view of this, Prof. Zadeh suggested the concept of Z-number as a more adequate way for description of real world information. As Z-number involves both fuzzy and probabilistic uncertainty, a novel method for the negation of Z-number in combination of probability and fuzziness is proposed from the reliability of probability transmission in this paper. Moreover, several examples are used to describe the negation process and result. As far as our latest knowledge is concerned, the negation of Z-number has not been covered by researchers, so this may be another door for us to process Z-number-based information.

Introduction

Knowledge and information representation [1], [2] problem is a crucial and open issue in the construction of Artificial Intelligence systems [3], [4]. In this regard, it is one of the hardest challenges especially in terms of representing the knowledge contained in the source of uncertain information. A large amount of literature has been developed to address this problem [5], [6], [7].

Negation is a new perspective and direction to represent knowledge. Zadeh raised the issue of determining the negation of probability distribution in his blog. Since then, discussion related to negation aroused widespread concern and attention among relevant researchers [8], mainly due to following advantages: Everything has two sides, people usually focus only on the front of the problem but ignore the opposite side of the problem. Since the introduction of negation, both angle of tackling questions and the representation of knowledge is more comprehensive, and more information that could not be obtained directly can be extracted more easily. Recently, Yager [9] proposed an excellent model to find the negation of probability distribution, and the model has been proved to satisfy the maximum entropy allocation. Thus, entropy [10], [11], [12], [13] and negation have an inseparable relationship. What’s more, the related work and model about negation has emerged, such as the negation of BPA (basic probability assignment) [14], [15], [16], generalization negation of probability distribution(PD) [17]. Z-number theory is proposed by Zadeh [18] in 2011 to describe uncertain information, which is a new concept in fuzzy logic. Inspired by the idea of Z-numbers, the Two Dimension Belief Function (TDBF) [19] and the Intuitionistic Evidence Sets (IES) [20] are presented. They can be applied to real-world problems in decision analysis [21], [22], [23], [24], economics, optimization and other fields [25], [26] are characterized by fuzzy and partially reliable information [27], [24]. A Z-number [28], an ordered pair Z=A,B of fuzzy number A and B, is often represented as linguistic terms, which always contains the imprecise, uncertain and incomplete information (A expresses imprecise estimation of a value X and B describes imprecise estimation of reliability of A). How to handle the Z-number-based information [29], [30], [31] in the construction of mathematical models [32], [33], [34] is a new challenge and open issue to researchers [35]. It is worth noting that there is a bridge that cannot be ignored between A and B: the underlying probability distribution. Inspired by Yager’s negation of a probability distribution on the maximum entropy [9], we proposed a negation model which associate the hidden probability distribution [36] in Z-numbers with Yager’s negation of a probability distribution. However, to the best of our knowledge, investigation results in this field are very scarce. The negation of Z-number has not been covered by researchers, so this may be another way for us to process Z-number-based information. In this paper, we proposed a novel method in the construction of the negation of Z-number, which uses a series of optimization models to find the hidden probability distributions in the Z-number based on the Maximum Shannon Entropy and Genetic Algorithm. Finally, the paper analyzes and discusses the effects and expressions of the negation of Z-number.

The paper is structured as follows: In Section 2, some related backgrounds and some basic knowledge associated with Z-number and the method of the negation of probability distribution are reviewed. In Section 3, the negation method for Z-number is proposed. Then, some numerical examples and further discussion are presented in Section 4. Future research directions are provided in Section 5. Finally, findings are summarized in Section 6.

Section snippets

Preliminaries

Some related concepts and definitions are reviewed in this section, including discrete fuzzy numbers, Z-numbers, discrete Z-numbers, some concepts that are easily confused with the negation of Z-number and the negation of probability distribution.

The negation of Z-number

In this section, we mainly introduce how to find the negation of Z-number from the perspective of hidden probability distribution of a Z-number instead of getting B for 1-B.

Examples and discussion

In this section, a calculation example is given to illustrate the effect of the negation. Secondly, we will discuss the reasons for the negation effect.

Suppose there is a Z-number in which the membership functions of the fuzzy number A and the fuzzy number B are μA,μB, and Z=A,B is as follows in Fig. 2:A=μA=01+0.52+0.73+14+0.75+0.56+07B=μB=00.1+0.30.2+0.90.3+10.4+0.90.5+0.30.6+00.7

Future research directions

  • 1.

    Z-number, is proposed by Zadeh in 2011 to model uncertain information. However, the opposite aspect of Z-number has never been considered and negation is an another effective way to indicate uncertainty information. This work proposes a way of finding the negation of Z-number and provides an idea for finding information behind the Z-number. This paper demonstrates the use of the negation of PDs to find the Z-number’s negation. It also reveals that the effect of this operation will make things

Conclusion

In this paper, we have established a reasonable model based on Maximum Entropy and Genetic Algorithm to obtain the hidden probability distribution, and a novel negation method is proposed to obtain the negation of Z-number from the perspective of hidden probability distribution. Examples and discussion are used to illustrate the meaning of negation. Through the increase of the number of iterations, the negation of underlying PD the results of this method are proved to be reasonable by the

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

The work is partially supported by the Fund of the National Natural Science Foundation of China (Grant No. 61903307), the Startup Fund from Northwest A&F University (Grant No. 2452018066), and the National College Students Innovation and Entrepreneurship Training Program (Grant No. 201910712081).

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