On characterizations of a pair of covering-based approximation operators

Rough set approximations based on a matroidal structure over three sets

Pawlak's classical model of rough set approximations provides an efficient tool for extracting information exactly by employing available knowledge (i.e., known knowledge) in an information system, since many problems in rough set theory are NP-hard and their solution process is therefore greedy and approximate. Many extensions of Pawlak's classical model have been proposed in recent years. Most of them are considered over one or two sets, that is, one- or two-dimensional space or one- or two-dimensional data. Aided by relation-based rough set models, a few of these extensions are considered over three sets. However, the real world is in three-dimensional space. Therefore, it is necessary to solve these problems with other models, such as covering rough set models. For this purpose, we propose the TP-matroid-a matroidal structure over three sets. Employing the family of feasible sets of a TP-matroid as the available knowledge, a pair of rough set approximations-lower and upper approximations-is provided. In addition, for an information system defined over three sets, assisted by formal concept analysis, we establish a pair of rough set approximations. Furthermore, two TP-matroids are established based on the above pair of rough set approximations. The integration between the two pairs of rough set approximations presented here is discussed. The results show that for an information system in three-dimensional space, the rough set approximations provided here can effectively explore unknown knowledge by using available knowledge based on the family of feasible sets of a TP-matroid.

Approximation operators via TD-matroids on two sets

Rough set theory is an extension of set theory with two additional unary set-theoretic operators known as approximation in order to extract information and knowledge. It needs the basic, or say definable, knowledge to approximate the undefinable knowledge in a knowledge space using the pair of approximation operators. Many existed approximation operators are expressed with unary form. How to mine the knowledge which is asked by binary form with rough set has received less research attention, though there are strong needs to reveal the answer for this challenging problem. There exist many information with matroid constraints since matroid provides a platform for combinatorial algorithms especially greedy algorithm. Hence, it is necessary to consider a matroidal structure on two sets no matter the two sets are the same or not. In this paper, we investigate the construction of approximation operators expressed by binary form with matroid theory, and the constructions of matroidal structure aided by a pair of approximation operators expressed by binary form.

First, we provide a kind of matroidal structure—TD-matroid defined on two sets as a generalization of Whitney classical matroid.

Second, we introduce this new matroidal construction to rough set and construct a pair of approximation operators expressed with binary form.

Third, using the existed pair of approximation operators expressed with binary form, we build up two concrete TD-matroids.

Fourth, for TD-matroid and the approximation operators expressed by binary form on two sets, we seek out their properties with aspect of posets, respectively.

Through the paper, we use some biological examples to explain and test the correct of obtained results. In summary, this paper provides a new approach to research rough set theory and matroid theory on two sets, and to study on their applications each other.

Interpretations of belief functions in approximation operators by covering

The rough set theory and the evidence theory are two important methods used to deal with uncertainty. The relationships between the rough set theory and the evidence theory have been discussed. In covering rough set theory, several pairs of covering approximation operators are characterized by belief and plausibility functions. The purpose of this paper is to review and examine interpretations of belief functions in covering approximation operators. Firstly, properties of the belief structures induced by two pairs of covering approximation operators are presented. Then, for a belief structure with the properties, there exists a probability space with a covering such that the belief and plausibility functions defined by the given belief structure are, respectively, the belief and plausibility functions induced by one of the two pairs of covering approximation operators. Moreover, two necessary and sufficient conditions for a belief structure to be the belief structure induced by one of the two pairs of covering approximation operators are presented.

Interval-valued pythagorean fuzzy rough approximation operators and its application

By combining interval-valued Pythagorean fuzzy sets with rough sets, the interval-valued Pythagorean fuzzy rough set model is first constructed in this paper. The connections between special interval-valued Pythagorean fuzzy relations and interval-valued Pythagorean fuzzy approximation operators are established subsequently. Then, we study the axiomatic characterizations of interval-valued Pythagorean fuzzy lower and upper approximation operators. Different axiom sets of interval-valued Pythagorean fuzzy set-theoretic operators ensure the existence of different types of interval-valued Pythagorean fuzzy relations producing the same operators. Finally, we give an example to illustrate the practical application of the newly proposed model.