Data volume reduction in covering approximation spaces with respect to twenty-two types of covering based rough sets

In this paper, we investigate whether consistent mappings can be used as homomorphism mappings between a covering based approximation space and its image with respect to twenty-two pairs of covering upper and lower approximation operators. We also consider the problem of constructing such mappings and minimizing them. In addition, we investigate the problem of reducing the data volume using consistent mappings as well as the maximum amount of their compressibility. We also apply our algorithms against several datasets.     

Topology vs generalized rough sets

This paper investigates the relationship between topology and generalized rough sets induced by binary relations. Some known results regarding the relation based rough sets are reviewed, and some new results are given. Particularly, the relationship between different topologies corresponding to the same rough set model is examined. These generalized rough sets are induced by inverse serial relations, reflexive relations and pre-order relations, respectively. We point that inverse serial relations are weakest relations which can induce topological spaces, and that different relation based generalized rough set models will induce different topological spaces. We proved that two known topologies corresponding to reflexive relation based rough set model given recently are different, and gave a condition under which the both are the same topology.     

On some types of neighborhood-related covering rough sets

Covering rough sets are natural extensions of the classical rough sets by relaxing the partitions to coverings. Recently, the concept of neighborhood has been applied to define different types of covering rough sets. In this paper, by introducing a new notion of complementary neighborhood, we consider some types of neighborhood-related covering rough sets, two of which are firstly defined. We first show some basic properties of the complementary neighborhood. We then explore the relationships between the considered covering rough sets and investigate the properties of them. It is interesting that the set of all the lower and upper approximations belonging to the considered types of covering rough sets, equipped with the binary relation of inclusion ?, constructs a lattice. Finally, we also discuss the topological importance of the complementary neighborhood and investigate the topological properties of the lower and upper approximation operators.     

Reduction about approximation spaces of covering generalized rough sets

The introduction of covering generalized rough sets has made a substantial contribution to the traditional theory of rough sets. The notion of attribute reduction can be regarded as one of the strongest and most significant results in rough sets. However, the efforts made on attribute reduction of covering generalized rough sets are far from sufficient. In this work, covering reduction is examined and discussed. We initially construct a new reduction theory by redefining the approximation spaces and the reducts of covering generalized rough sets. This theory is applicable to all types of covering generalized rough sets, and generalizes some existing reduction theories. Moreover, the currently insufficient reducts of covering generalized rough sets are improved by the new reduction. We then investigate in detail the procedures to get reducts of a covering. The reduction of a covering also provides a technique for data reduction in data mining.     

Four matroidal structures of covering and their relationships with rough sets

Covering is a common form of data representation, and covering-based rough sets serve as an efficient technique to process this type of data. However, many important problems such as covering reduction in covering-based rough sets are NP-hard so that most algorithms to solve them are greedy. Matroids provide well-established platforms for greedy algorithm foundation and implementation. Therefore, it is necessary to integrate covering-based rough set with matroid. In this paper, we propose four matroidal structures of coverings and establish their relationships with rough sets. First, four different viewpoints are presented to construct these four matroidal structures of coverings, including 1-rank matroids, bigraphs, upper approximation numbers and transversals. The respective advantages of these four matroidal structures to rough sets are explored. Second, the connections among these four matroidal structures are studied. It is interesting to find that they coincide with each other. Third, a converse view is provided to induce a covering by a matroid. We study the relationship between this induction and the one from a covering to a matroid. Finally, some important concepts of covering-based rough sets, such as approximation operators, are equivalently formulated by these matroidal structures. These interesting results demonstrate the potential to combine covering-based rough sets with matroids.     

On uniqueness of （fuzzy） relation in some generalized rough set model

In rough set theory, crisp and/or fuzzy binary relations play an important role in both constructive and axiomatic considerations of various generalized rough sets. This paper considers the uniqueness problem of the （fuzzy） relation in some generalized rough set model. Our results show that by using the axiomatic approach, the （fuzzy） relation determined by （fuzzy） approximation operators is unique in some （fuzzy） double-universe model.     

单向奇异粗糙模糊集合的结构

用模糊集合与模糊等价关系对单向奇异粗集进行了研究,并给出了单向奇异粗糙模糊集合的数学结构及其并、交、补运算和性质.同时证明了单向奇异粗糙模糊集合对并、交、补运算构成完全可无限分配的软代数.     

On πgs-closed sets in topological spaces

Summary A new class of sets called πgs-closed sets is introduced and its properties are studied. Moreover the notions of πgs-T_1/2 spaces and πgs-continuity are introduced.     

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 homomorphism is provided to relate two covering approximation spaces. We also examine the properties of approximations preserved by the operations and homomorphisms, respectively.     

Evidence-theory-based numerical algorithms of attribute reduction with neighborhood-covering rough sets

Covering rough sets generalize traditional rough sets by considering coverings of the universe instead of partitions, and neighborhood-covering rough sets have been demonstrated to be a reasonable selection for attribute reduction with covering rough sets. In this paper, numerical algorithms of attribute reduction with neighborhood-covering rough sets are developed by using evidence theory. We firstly employ belief and plausibility functions to measure lower and upper approximations in neighborhood-covering rough sets, and then, the attribute reductions of covering information systems and decision systems are characterized by these respective functions. The concepts of the significance and the relative significance of coverings are also developed to design algorithms for finding reducts. Based on these discussions, connections between neighborhood-covering rough sets and evidence theory are set up to establish a basic framework of numerical characterizations of attribute reduction with these sets.     

Weighted covering numbers of convex sets

In this paper we define the notions of weighted covering number and weighted separation number for convex sets, and compare them to the classical covering and separation numbers. This sheds new light on the equivalence of classical covering and separation. We also provide a formula for computing these numbers via a limit of classical covering numbers in higher dimensions.     

On topological properties of ultraproducts of finite sets

In [3] a certain family of topological spaces was introduced on ultraproducts. These spaces have been called ultratopologies and their definition was motivated by model theory of higher order logics. Ultratopologies provide a natural extra topological structure for ultraproducts. Using this extra structure in [3] some preservation and characterization theorems were obtained for higher order logics. The purely topological properties of ultratopologies seem interesting on their own right. We started to study these properties in [2], where some questions remained open. Here we present the solutions of two such problems. More concretely we show 1. that there are sequences of finite sets of pairwise different cardinalities such that in their certain ultraproducts there are homeomorphic ultratopologies and 2. if A is an infinite ultraproduct of finite sets, then every ultratopology on A contains a dense subset D such that |D| < |A|. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On a topological closeness of perturbed Julia sets

In the present work we expand our previous work in [1] by introducing the Julia Deviation Distance and the Julia Deviation Plot in order to study the stability of the Julia sets of noise-perturbed Mandelbrot maps. We observe a power-law behaviour of the Julia Deviation Distance of the Julia sets of a family of additive dynamic noise Mandelbrot maps from the Julia set of the Mandelbrot map as a function of the noise level. Additionally, using the above tools, we support the invariance of the Julia set of a noise-perturbed Mandelbrot map under different noise realizations.     

On Λ-generalized closed sets in topological spaces

We introduce new classes of sets called Λ _g -closed sets and Λ _g -open sets in topological spaces. We also investigate several properties of such sets. It turns out that Λ _g -closed sets and Λ _g -open sets are weaker forms of closed sets and open sets, respectively and stronger forms of g-closed sets and g-open sets, respectively. Dedicated to Professor Maximilian Ganster on the occasion of his 50th birthday     

On a topological closeness of perturbed Mandelbrot sets

In the present work we propose a numerical and visual tool for the study of the deformation of the Mandelbrot sets of perturbed Mandelbrot maps by noise in comparison with the original Mandelbrot set. Further, by employing these numerical tools, we support the invariance of the Mandelbrot set of a noise-perturbed Mandelbrot map under different noise realizations. Finally, we provide evidence for the non-fractal structure of the Mandelbrot set of a noise-perturbed Mandelbrot map.     

On generalized perfect difference sets constructed from Sidon sets

Let $\mathbb{N}$ be the set of positive integers. For a nonempty set A of integers and every integer u, denote by $d_A(u)$ the number of $(a,a^{\prime })$ with $a,a^{\prime }\in A$ such that $u=a-a^{\prime }$. For a sequence S of positive integers, let $S(x)$ be the counting function of S. The set $A\subseteq \mathbb{N}$ is called a perfect difference set if $d_A(u)=1$ for every positive integer u. In 2008, Cilleruelo and Nathanson (2008) [4] constructed dense perfect difference sets from dense Sidon sets. In this paper, as a main result, we prove that: let $f:\mathbb{N}\to \mathbb{N}$ be an increasing function satisfying $f(n)\ge 2$ for any positive integer n, then for every Sidon set B and every function $\omega (x)\to \infty$, there exists a set $A\subseteq \mathbb{N}$ such that $d_A(u)=f(u)$ for every positive integer u and $B(x/3)-\omega (x)\le A(x)\le B(x/3)+\omega (x)$ for all $x\ge C_{f,B,\omega }$.