[Paper Review] Relation matroid and its relationship with generalized rough set based on relation

Yanfang Liu, William Zhu|arXiv (Cornell University)|Sep 24, 2012

Rough Sets and Fuzzy Logic28 references3 citations

TL;DR

This paper introduces predecessor and successor relation matroids derived from neighborhoods in generalized rough sets based on binary relations, establishing a bidirectional link between relations and matroids. It proves that a relation induces a matroid whose circuits generate an equivalence relation, and shows that the upper approximation operator in rough sets and the closure operator in matroids coincide when the relation is an equivalence relation.

ABSTRACT

Recently, the relationship between matroids and generalized rough sets based on relations has been studied from the viewpoint of linear independence of matrices. In this paper, we reveal more relationships by the predecessor and successor neighborhoods from relations. First, through these two neighborhoods, we propose a pair of matroids, namely predecessor relation matroid and successor relation matroid, respectively. Basic characteristics of this pair of matroids, such as dependent sets, circuits, the rank function and the closure operator, are described by the predecessor and successor neighborhoods from relations. Second, we induce a relation from a matroid through the circuits of the matroid. We prove that the induced relation is always an equivalence relation. With these two inductions, a relation induces a relation matroid, and the relation matroid induces an equivalence relation, then the connection between the original relation and the induced equivalence relation is studied. Moreover, the relationships between the upper approximation operator in generalized rough sets and the closure operator in matroids are investigated.

Motivation & Objective

• To establish a formal connection between generalized rough sets based on binary relations and matroid theory using neighborhood structures.
• To define and analyze predecessor and successor relation matroids based on predecessor and successor neighborhoods from a binary relation.
• To investigate how a matroid can induce an equivalence relation through its circuits, and how this relates back to the original relation.
• To compare the upper approximation operator in generalized rough sets with the closure operator in matroids, identifying conditions under which they are equal.

Proposed method

• Define predecessor and successor neighborhoods from a binary relation R on a universe U, using the relation's reachability structure.
• Construct two matroids—predecessor and successor relation matroids—based on the neighborhoods, with the successor matroid being the primary focus.
• Characterize the matroid using standard matroid concepts: dependent sets, circuits, rank function, and closure operator, all expressed in terms of the relation's neighborhoods.
• Induce a relation from a matroid by defining xR(M)y if {x,y} is a circuit or x=y, proving the induced relation is always an equivalence relation.
• Compare the closure operator of the matroid with the upper approximation operator of the induced relation, showing equality when the original relation is an equivalence relation.
• Establish bidirectional induction: a relation induces a matroid, and the matroid induces an equivalence relation, proving the induced equivalence relation equals the original relation if and only if the original is an equivalence relation.

Experimental results

Research questions

• RQ1 How can predecessor and successor neighborhoods from a binary relation be used to construct matroids?
• RQ2 What are the key matroid-theoretic properties (e.g., circuits, closure) of the relation matroid induced by a binary relation?
• RQ3 Can a matroid induce a relation, and if so, what type of relation results from this construction?
• RQ4 Under what conditions do the closure operator of a matroid and the upper approximation operator of the induced relation coincide?
• RQ5 What is the relationship between the original binary relation and the equivalence relation induced by its associated relation matroid?

Key findings

• The successor relation matroid induced by a binary relation R can be fully characterized using the successor neighborhood structure of R.
• The closure operator of the relation matroid and the upper approximation operator in generalized rough sets based on R are equal if and only if R is an equivalence relation.
• A matroid induces a relation R(M) via its circuits, and this induced relation is always an equivalence relation.
• The equivalence relation induced by the relation matroid M(R) is equal to the original relation R if and only if R is an equivalence relation.
• The induced equivalence relation R(M(R)) can be explicitly expressed as {(x,y) ∈ U×U : RS_R(x) = RS_R(y)}.
• The closure of a singleton {x} in the matroid equals the upper approximation of {x} in the induced equivalence relation, plus all singletons that are circuits.