Definability and Other Properties of Approximations for Generalized Indiscernibility Relations

Abstract

Abstract. In this paper we consider a generalization of the indiscernibility relation, i.e., a relation R that is not necessarily reflexive, symmetric, or transitive. On the basis of granules, defined by R, we introduce the idea of definability. We study 28 basic definitions of approximations, two approximations are introduced for the first time. Furthermore, we introduce additional 8 new approximations. Our main objective is to study definability and coalescence of approximations. We study definability of all 28 basic approximations for reflexive, symmetric, and transitive relations. In particular, for reflexive relations, the set of 28 approximations is reduced, in general, to the set of 16 approximations.

Rough Sets and Rule Induction in Imperfect Information Systems

“…This is a method of handling missing values summarized by Grzymala-Busse [9,10], in which, missing values are replaced by the most common value of the attribute. The method of handling missing attribute values is implemented, e.g., in well-known machine learning algorithm CN2 [3].…”

“…Now, it is able to define a relation R P (x, y) between objects x and y by controlling the degree of tolerance(similarity) using threshold α, such that R P (x, y) ⇔ φ P (x, y) ≥ α. Based on that, a neighbourhood, which consists of successor and predecessor sets, of an object [10,27] is determined. The successor set of x is the set of objects to which x is similar:…”

“…In the case of non-equivalence relations, which may not be reflexive, symmetric nor transitive, approximation spaces defined may lead to variant results [10]. Formally, approximations based on singleton, subset and concept approaches with a valued tolerance/similarity relation can be defined as follows:…”

Extended Tolerance Relation to Define a New Rough Set Model in Incomplete Information Systems

“…This is defined as is more tolerant to than based on extended tolerance relation. Now, with a relation, we can derive a neighbourhood, which consists of successor and predecessor sets, of an object [19,20]. Due to symmetric property of extended tolerance relation, successor is the same set as predecessor.…”

“…In the case of nonequivalence relations, which may not need to be reflexive, symmetric nor transitive, approximation spaces defined in such different ways may lead to variant results [14]. This section shall introduce lower and upper approximation definitions based on singleton, subset, and concept methods which are first studied and generalized by Grzymala-Busse [14,19].…”