When multiple granulated knowledge in multigranulation spaces are involved in decision making, protocol principles are adopted to arrive at the final consensus. Multigranulation rough set theory utilizes a voting principle to combine the decision options derived from individual granulated knowledge. Note that those knowledge may provide different degrees of support to the final results, some are key, some are of less importance and some are even of no use. Selecting valuable knowledge and reducing worthless one are thus necessary for data processing, which can alleviate the storage occupancy and facilitate the logical and statistical analysis. However, the basic reduction foundation of multigranulation spaces has been rarely touched by researchers, which brings in many difficulties in algorithmic and real applications. This work aims to disclose the principles of multiple knowledge reduction in multigranulation spaces from the viewpoint of discernibility. First, the notions of knowledge reduction of multigranulation spaces are defined based on multigranulation rough set theory. Second, a decision function mapping each object into the decision options of its neighborhood granule is introduced. Third, several pairs of discernibility matrices and discernibility functions are successively developed using the decision function. We claim that the valuable and worthless knowledge in multigranulation spaces can be explicitly chose and eliminated respectively by using the proposed discernibility matrices and discernibility functions. That is to say, these discernibility tools provide a precise criterion for the knowledge reduction of multigranulation spaces. As a theoretical extension, a multigranulation information entropy is proposed and an approximate algorithm is constructed to compute a suboptimal reduct of a multigranulation space based on this entropy. In the end, numerical experiments are performed on public data sets to verify the effectiveness of the proposed reduction methods. This study can get us a grasp of the foundational principle of knowledge reduction and may bring a new insight for the designation of substantial reduction algorithms of multigranulation knowledge.

中文翻译：

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使用可辨识性的多粒度粗糙集约简基础

当多粒度空间中的多粒度知识参与决策时，采用协议原则达成最终共识。多粒度粗糙集理论利用投票原则来组合从单个粒度知识派生的决策选项。请注意，这些知识可能会为最终结果提供不同程度的支持，有些是关键，有些不太重要，有些甚至毫无用处。因此，选择有价值的知识，减少无价值的知识是数据处理的必要条件，可以减少存储占用，便于逻辑和统计分析。然而，多粒度空间的基本约简基础很少被研究人员触及，这给算法和实际应用带来了许多困难。这项工作旨在从可辨别性的角度揭示多粒度空间中多知识约简的原理。首先，基于多粒度粗糙集理论定义了多粒度空间的知识约简概念。其次，引入了将每个对象映射到其邻域粒度的决策选项的决策函数。第三，利用决策函数先后开发了几对可辨别矩阵和可辨别函数。我们声称，通过使用所提出的辨别矩阵和辨别函数，可以分别明确地选择和消除多粒度空间中有价值和无价值的知识。也就是说，这些可辨别性工具为多粒度空间的知识减少提供了精确的标准。作为理论扩展，提出了多粒度信息熵，并构造了一种近似算法，以基于该熵计算多粒度空间的次优约简。最后，在公共数据集上进行数值实验，以验证所提出的约简方法的有效性。本研究可以让我们掌握知识约简的基本原理，并可能为多粒度知识的实质性约简算法的设计带来新的见解。