A study of rough set approximation operators based on modal logic

Abstract

In the 1980s, Pawlak proposed a mathematical approach, known as rough set theory (RST), for dealing with vague, uncertain, and incomplete data [73]. In his approach, the knowledge base about a non-empty set W of objects is given by an equivalence relation R. The pair (W, R) is called an approximation space. A set A ⊆ W is approximated by its lower approximation AR = {x ∈ W | R(x) ⊆ A} and upper approximation AR = {x ∈ W | R(x) ∩ A 6= ∅}, where R(x) = {y ∈ W | (x, y) ∈ R}. The elements occurring in the set AR \ AR are called boundary elements of A. The set of all the boundary elements of A is denoted by BdR(A). Applications of rough set theory are mostly based on an attribute-value representation model, called (deterministic) information system. These are essentially tables giving values taken for some attributes by objects of the domain – such as ‘blue’ for attribute ‘colour’, or ‘round’ for attribute ‘shape’. The domain then gets partitioned into blocks of indiscernible objects, indiscernible as they match on all information available about them. Formally, the following is the mathematical representation of a deterministic information system and the induced indiscernibility among objects of the domain. Definition 1.1. A deterministic information system (in brief, DIS) S := (W, A, S a∈A Va, f) comprises a non-empty set W of objects, a non-empty finite set A of attributes, a non empty finite set Va of attribute-values for each a ∈ A, and an assignment f : W × A → S a∈A Va such that f(x, a) ∈ Va. Based on the information given by S, each subset B of A induces an equivalence relation IndS B on the domain W, termed the indiscernibility relation induced by B, as follows: (x, y) ∈ IndS B ⇐⇒ f(x, a) = f(y, a) for all a ∈ B. Thus, given a DIS S and a set B of attributes, we obtain an approximation space (W,IndS B), and this, in turn, determines approximation operators. Note that IndS ∅ = W × W.