Abstract: Granular computing is an important research hotspot in artificial intelligence field. As one of the most important granular computing models, the quotient space theory mainly focuses on constructing the quotient space granularity by the false preserving principle, which is granulating from fine to coarse granularity. However, the granulation is an information decreasing and irreversible process. Therefore, this paper focuses on the reversibility of quotient space granularity. Firstly, in order to get the mathematical formal method of reversibly converting granularity from coarse to fine one, it proposes the concept and constructing method of inverse quotient space.Secondly, by analyzing the consistency of quotient space and original space, it demonstrates and analyzes two reversible conditions of the quotient space granularity—defining a bijective function or ensuring the saturability of all open-sets on the original space. This paper aims to further enrich and improve the quotient space granularity transformation theory and granular computing method.

Key words: granular computing, quotient space theory, inverse quotient space, saturability, reversibility

摘要： 粒计算是近年来人工智能领域的重要研究热点，而商空间理论是最重要的粒计算模型之一，其主要思想是通过保假原理实现求解问题从细粒度到粗粒度的商空间构造过程。这个粒化过程是一个信息有损过程，是不可逆的，于是研究商空间粒度的可逆性。首先，提出逆商空间的概念并定义其构造方法，为商空间（粗）粒度到原空间（细）粒度的可逆转换提供形式化的数学方法；其次，通过分析逆商空间与原空间的一致性，论证并实例分析商空间粒度的两个可逆性条件——定义原空间上的双射函数或保证原空间上所有开集的饱和性。旨在进一步丰富和完善商空间粒度转换理论和粒计算方法。

关键词: 粒计算, 商空间理论, 逆商空间, 饱和性, 可逆性