覆盖的特征矩阵及其在布尔矩阵分解和公理化中的应用
Characteristic matrix of covering and its application to boolean matrix decomposition and axiomatization
摘要：
覆盖是一种重要的数据结构，基于覆盖的粗糙集理论为覆盖数据的处理提供了一种有效而系统的理论。本文用布尔矩阵来表示和公理化三类覆盖逼近算子。首先，我们定义了覆盖的两类特征矩阵，并研究了它们的性质。通过特征矩阵，简洁地等价表示了三种重要的覆盖逼近算子。其次，覆盖近似算子的矩阵表示用于布尔矩阵分解。给出了一个布尔方阵分解为另一个布尔方阵与其转置的布尔乘积的充要条件。我们开发了一个布尔矩阵分解的算法。最后，在上述结果的基础上，利用布尔矩阵将这三类覆盖近似算子公理化。总之，本文的工作广泛借鉴了布尔矩阵，为研究覆盖粗糙集提供了一个新的视角。
英文摘要：
Covering is an important type of data structure while covering-based rough sets provide an efficient and systematic theory to deal with covering data. In this paper, we use boolean matrices to represent and axiomatize three types of covering approximation operators. First, we define two types of characteristic matrices of a covering which are essentially square boolean ones, and their properties are studied. Through the characteristic matrices, three important types of covering approximation operators are concisely equivalently represented. Second, matrix representations of covering approximation operators are used in boolean matrix decomposition. We provide a sufficient and necessary condition for a square boolean matrix to decompose into the boolean product of another one and its transpose. And we develop an algorithm for this boolean matrix decomposition. Finally, based on the above results, these three types of covering approximation operators are axiomatized using boolean matrices. In a word, this work borrows extensively from boolean matrices and present a new view to study covering-based rough sets.
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