Characteristic matrix of covering and its application to boolean matrix decomposition and axiomatization
This work provides incremental improvements for researchers in rough set theory and data analysis by applying boolean matrices to coverings.
The paper tackles the representation and axiomatization of covering approximation operators in rough set theory by defining characteristic matrices and using boolean matrix decomposition, resulting in an algorithm for decomposition and axiomatization of three operator types.
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.