Characteristic matrix of covering and its application to boolean matrix decomposition and axiomatization
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.
💡 Research Summary
This paper revisits covering‑based rough set theory from a matrix‑theoretic perspective and shows how Boolean (0‑1) matrices can replace the traditional set‑theoretic definitions of covering approximation operators. The authors first introduce two characteristic matrices for a covering C = {C₁,…,Cₘ}. The first matrix M₁ is the usual incidence matrix: entry (i, j) equals 1 iff element xᵢ belongs to covering set Cⱼ. The second matrix M₂ is defined as the Boolean product M₁·M₁ᵀ, i.e., M₂(i, j)=∨ₖ(M₁(i, k)∧M₁(j, k)). They prove a series of identities (M₁·M₁ᵀ = M₂, M₁ᵀ·M₁ = M₁ᵀ·M₁ᵀᵀ, etc.) and demonstrate that M₁ and M₂ together capture the entire covering structure.
Using these matrices, the three classical covering approximation operators are expressed compactly: the lower approximation L_C(x) = M₁·x, the upper approximation U_C(x) = M₂·x, and the boundary approximation B_C(x) = U_C(x) − L_C(x), where x is a Boolean vector indicating a target subset and “·” denotes Boolean matrix‑vector multiplication (logical AND followed by OR). The authors verify that these matrix formulas are exactly equivalent to the original set‑based definitions, thereby enabling all approximation calculations to be performed by simple Boolean algebra.
The second major contribution concerns Boolean matrix decomposition. The paper asks: for a given square Boolean matrix A, when does there exist a Boolean matrix B such that A = B·Bᵀ? The authors establish a necessary and sufficient condition: A must be symmetric (A = Aᵀ) and idempotent under Boolean multiplication (A·A = A). In other words, A must be a symmetric idempotent Boolean matrix. This result provides a clear diagnostic test for decomposability.
Based on the condition, an explicit O(n³) algorithm is proposed. The algorithm first checks symmetry and idempotence, then forces all diagonal entries to 1, and finally constructs B by examining inclusion relationships among rows (or columns). After constructing B, the algorithm verifies that B·Bᵀ reproduces A. The authors report that the procedure works efficiently for matrices of several thousand rows/columns, making it practical for real‑world covering data.
Finally, the paper leverages the matrix framework to axiomatize the three covering approximation operators. Instead of relying on set inclusion axioms, the authors present three Boolean matrix axioms that any characteristic matrix must satisfy: (1) idempotence (M·M = M), (2) symmetry (M = Mᵀ), and (3) closure under multiplication with Boolean vectors (M·x ∈ {0,1}ⁿ for any Boolean vector x). They prove that any 0‑1 matrix obeying these axioms can serve as a characteristic matrix of some covering, and consequently defines valid lower, upper, and boundary approximations. This axiomatization unifies the theory, simplifies verification, and opens the door to extending covering‑based rough sets to other combinatorial structures such as hypergraphs or multi‑sets.
In summary, the paper makes three substantive contributions: (1) the definition and analysis of two Boolean characteristic matrices that encode covering information; (2) a rigorous necessary‑and‑sufficient condition for Boolean matrix decomposition into a matrix and its transpose, together with an efficient construction algorithm; and (3) a matrix‑based axiomatization of covering approximation operators. By translating covering‑based rough set concepts into Boolean linear algebra, the work provides a computationally attractive and theoretically elegant framework that can be readily incorporated into data mining, knowledge discovery, and network analysis pipelines.