[Paper Review] Characteristic matrix of covering and its application to boolean matrix decomposition and axiomatization

Shiping Wang, Qingxin Zhu+2 more|arXiv (Cornell University)|Jul 2, 2012

Rough Sets and Fuzzy Logic27 references15 citations

TL;DR

This paper introduces two types of characteristic matrices for coverings to represent three classes of covering approximation operators via Boolean matrices. It establishes a sufficient and necessary condition for decomposing a Boolean matrix into the Boolean product of another matrix and its transpose, enabling a novel matrix-based axiomatization of covering-based rough sets with algorithmic implementation.

ABSTRACT

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.

Motivation & Objective

• To bridge covering-based rough set theory with Boolean matrix decomposition by introducing characteristic matrices.
• To provide a matrix-based representation of three types of covering approximation operators (second, fifth, sixth) using Boolean matrices.
• To derive a sufficient and necessary condition for decomposing a square Boolean matrix into the Boolean product of a matrix and its transpose.
• To develop an algorithm for Boolean matrix decomposition based on the derived condition.
• To axiomatize the three types of covering approximation operators using Boolean matrix properties.

Proposed method

• Define two types of characteristic matrices (M_C and Gamma(C)) from a covering C to represent covering approximation operators.
• Represent the second, fifth, and sixth upper and lower approximation operators as Boolean matrix operations using the characteristic matrices.
• Establish a sufficient and necessary condition for a square Boolean matrix A to be decomposed as B · B^T, requiring A^T = A and A_ii = 1 for all i.
• Design an algorithm to compute such a decomposition by finding a minimal Boolean matrix B such that A = B · B^T.
• Use the matrix decomposition condition to axiomatize the three types of covering approximation operators via Boolean matrix properties.
• Leverage relational matrix correspondence: reflexive and symmetric relations for second operator, reflexive and transitive for fifth and sixth operators.

Experimental results

Research questions

• RQ1 Can covering approximation operators be equivalently represented using Boolean matrices through characteristic matrices?
• RQ2 What is the sufficient and necessary condition for a square Boolean matrix to be decomposed into the Boolean product of a matrix and its transpose?
• RQ3 How can an algorithm be designed to compute such a decomposition?
• RQ4 Can the three types of covering approximation operators be axiomatized using Boolean matrix decomposition?
• RQ5 What is the relationship between covering-based rough sets and generalized rough sets based on relations through matrix representations?

Key findings

• The second upper approximation operator SH_C is equivalent to a Boolean matrix A satisfying A^T = A and A_ii = 1 for all i.
• The fifth upper approximation operator IH_C corresponds to a Boolean matrix A satisfying A^2 = A and A_ii = 1 for all i.
• The sixth upper approximation operator XH_C is characterized by a Boolean matrix A = B · B^T where B^2 = B and B_ii = 1 for all i.
• A sufficient and necessary condition for Boolean matrix decomposition A = B · B^T is that A is symmetric and has all diagonal entries equal to 1.
• The algorithm for decomposition is based on finding a minimal Boolean matrix B such that A = B · B^T, ensuring correctness and completeness.
• The axiomatization of the three approximation operators is achieved via Boolean matrix properties, linking them to reflexive and symmetric or reflexive and transitive relations.