Rough matroids based on coverings

The introduction of covering-based rough sets has made a substantial contribution to the classical rough sets. However, many vital problems in rough sets, including attribution reduction, are NP-hard and therefore the algorithms for solving them are usually greedy. Matroid, as a generalization of linear independence in vector spaces, it has a variety of applications in many fields such as algorithm design and combinatorial optimization. An excellent introduction to the topic of rough matroids is due to Zhu and Wang. On the basis of their work, we study the rough matroids based on coverings in this paper. First, we investigate some properties of the definable sets with respect to a covering. Specifically, it is interesting that the set of all definable sets with respect to a covering, equipped with the binary relation of inclusion $\subseteq$, constructs a lattice. Second, we propose the rough matroids based on coverings, which are a generalization of the rough matroids based on relations. Finally, some properties of rough matroids based on coverings are explored. Moreover, an equivalent formulation of rough matroids based on coverings is presented. These interesting and important results exhibit many potential connections between rough sets and matroids.Comment: 15page

Topological and algebraic characterization of coverings sets obtained in rough sets discretization and attribute reduction algorithms

Abstract. A systematic study on approximation operators in covering based rough sets and some relations with relation based rough sets are presented. Two different frameworks of approximation operators in covering based rough sets were unified in a general framework of dual pairs. This work establishes some relationships between the most important generalization of rough set theory: Covering based and relation based rough sets. A structured genetic algorithm to discretize, to find reducts and to select approximation operators for classification problems is presented.Se presenta un estudio sistemático de los diferentes operadores de aproximación en conjuntos aproximados basados en cubrimientos y operadores de aproximación basados en relaciones binarias. Se unifican dos marcos de referencia sobre operadores de aproximación basados en cubrimientos en un único marco de referencia con pares duales. Se establecen algunas relaciones entre operadores de aproximación de dos de las más importantes generalizaciones de la teoría de conjuntos aproximados. Finalmente, se presenta un algoritmo genético estructurado, para discretizar, reducir atributos y seleccionar operadores de aproximación, en problemas de clasificación.Doctorad

Study on Rough Sets and Fuzzy Sets in Constructing Intelligent Information System

Since human being is not an omniscient and omnipotent being, we are actually living in an uncertain world. Uncertainty was involved and connected to every aspect of human life as a quotation from Albert Einstein said: �As far as the laws of mathematics refer to reality, they are not certain. And as far as they are certain, they do not refer to reality.� The most fundamental aspect of this connection is obviously shown in human communication. Naturally, human communication is built on the perception1-based information instead of measurement-based information in which perceptions play a central role in human cognition [Zadeh, 2000]. For example, it is naturally said in our communication that �My house is far from here.� rather than let say �My house is 12,355 m from here�. Perception-based information is a generalization of measurement-based information, where perception-based information such as �John is excellent.� is hard to represent by measurement-based version. Perceptions express human subjective view. Consequently, they tend to lead up to misunderstanding. Measurements then are needed such as defining units of length, time, etc., to provide objectivity as a means to overcome misunderstanding. Many measurers were invented along with their methods and theories of measurement. Hence, human cannot communicate with measurers including computer as a product of measurement era unless he uses measurement-based information. Perceptions are intrinsic aspect in uncertainty-based information. In this case, information may be incomplete, imprecise, fragmentary, not fully reliable, vague, contradictory, or deficient in some other way. 1In psychology, perception is understood as a process of translating sensory stimulation into an organized experience Generally, these various information deficiencies may express different types of uncertainty. It is necessary to construct a computer-based information system called intelligent information system that can process uncertainty-based information. In the future, computers are expected to be able to make communication with human in the level of perception. Many theories were proposed to express and process the types of uncertainty such as probability, possibility, fuzzy sets, rough sets, chaos theory and so on. This book extends and generalizes existing theory of rough set, fuzzy sets and granular computing for the purpose of constructing intelligent information system. The structure of this book is the following: In Chapter 2, types of uncertainty in the relation to fuzziness, probability and evidence theory (belief and plausibility measures) are briefly discussed. Rough set regarded as another generalization of crisp set is considered to represent rough event in the connection to the probability theory. Special attention will be given to formulation of fuzzy conditional probability relation generated by property of conditional probability of fuzzy event. Fuzzy conditional probability relation then is used to represent similarity degree of two fuzzy labels. Generalization of rough set induced by fuzzy conditional probability relation in terms of covering of the universe is given in Chapter 3. In the relation to fuzzy conditional probability relation, it is necessary to consider an interesting mathematical relation called weak fuzzy similarity relation as a generalization of fuzzy similarity relation proposed by Zadeh [1995]. Fuzzy rough set and generalized fuzzy rough set are proposed along with the generalization of rough membership function. Their properties are examined. Some applications of these methods in information system such as α-redundancy of object and dependency of domain attributes are discussed. In addition, multi rough sets based on multi-context of attributes in the presence of multi-contexts information system is defined and proposed in Chapter 4. In the real application, depending on the context, a given object may have different values of attributes. In other words, set of attributes might be represented based on different context, where they may provide different values for a given object. Context can be viewed as background or situation in which somehow it is necessary to group some attributes as a subset of attributes and consider the subset as a context. Finally, Chapter 5 summarizes all discussed in this book and puts forward some future topics of research

Covering rough sets based on neighborhoods: An approach without using neighborhoods

Rough set theory, a mathematical tool to deal with inexact or uncertain knowledge in information systems, has originally described the indiscernibility of elements by equivalence relations. Covering rough sets are a natural extension of classical rough sets by relaxing the partitions arising from equivalence relations to coverings. Recently, some topological concepts such as neighborhood have been applied to covering rough sets. In this paper, we further investigate the covering rough sets based on neighborhoods by approximation operations. We show that the upper approximation based on neighborhoods can be defined equivalently without using neighborhoods. To analyze the coverings themselves, we introduce unary and composition operations on coverings. A notion of homomorphismis provided to relate two covering approximation spaces. We also examine the properties of approximations preserved by the operations and homomorphisms, respectively.Comment: 13 pages; to appear in International Journal of Approximate Reasonin

Covering matroid

In this paper, we propose a new type of matroids, namely covering matroids, and investigate the connections with the second type of covering-based rough sets and some existing special matroids. Firstly, as an extension of partitions, coverings are more natural combinatorial objects and can sometimes be more efficient to deal with problems in the real world. Through extending partitions to coverings, we propose a new type of matroids called covering matroids and prove them to be an extension of partition matroids. Secondly, since some researchers have successfully applied partition matroids to classical rough sets, we study the relationships between covering matroids and covering-based rough sets which are an extension of classical rough sets. Thirdly, in matroid theory, there are many special matroids, such as transversal matroids, partition matroids, 2-circuit matroid and partition-circuit matroids. The relationships among several special matroids and covering matroids are studied.Comment: 15 page