scholarly journals AN OVERVIEW OF ROUGH SET SEMANTICS FOR MODAL AND QUANTIFIER LOGICS

2000 ◽  
Vol 08 (01) ◽  
pp. 93-118 ◽  
Author(s):  
CHURN-JUNG LIAU
Keyword(s):  
Set Theory ◽  
Rough Set ◽  
Possible Worlds ◽  
Rough Set Theory ◽  
Modal Logics ◽  
The Other ◽  
Approximation Space ◽  
The Universe

In this paper, we would like to present some logics with semantics based on rough set theory and related notions. These logics are mainly divided into two classes. One is the class of modal logics and the other is that of quantifier logics. For the former, the approximation space is based on a set of possible worlds, whereas in the latter, we consider the set of variable assignments as the universe of approximation. In addition to surveying some well-known results about the links between logics and rough set notions, we also develop some new applied logics inspired by rough set theory.

Measures of uncertainty for a fuzzy probabilistic approximation space

2021 ◽  
pp. 1-24
Author(s):  
Lijun Chen ◽  
Damei Luo ◽  
Pei Wang ◽  
Zhaowen Li ◽  
Ningxin Xie
Keyword(s):  
Set Theory ◽  
Rough Set ◽  
Rough Set Theory ◽  
Fuzzy Relation ◽  
Point Of View ◽  
Approximation Space ◽  
Fuzzy Approximation ◽  
Entropy Measurement ◽  
Fuzzy Relation Matrix ◽  
Relation Matrix

 An approximation space (A-space) is the base of rough set theory and a fuzzy approximation space (FA-space) can be seen as an A-space under the fuzzy environment. A fuzzy probability approximation space (FPA-space) is obtained by putting probability distribution into an FA-space. In this way, it combines three types of uncertainty (i.e., fuzziness, probability and roughness). This article is devoted to measuring the uncertainty for an FPA-space. A fuzzy relation matrix is first proposed by introducing the probability into a given fuzzy relation matrix, and on this basis, it is expanded to an FA-space. Then, granularity measurement for an FPA-space is investigated. Next, information entropy measurement and rough entropy measurement for an FPA-space are proposed. Moreover, information amount in an FPA-space is considered. Finally, a numerical example is given to verify the feasibility of the proposed measures, and the effectiveness analysis is carried out from the point of view of statistics. Since three types of important theories (i.e., fuzzy set theory, probability theory and rough set theory) are clustered in an FPA-space, the obtained results may be useful for dealing with practice problems with a sort of uncertainty.

scholarly journals A Detailed Study of the Distributed Rough Set Based Locality Sensitive Hashing Feature Selection Technique

2021 ◽  
Vol 182 (2) ◽  
pp. 111-179
Author(s):  
Zaineb Chelly Dagdia ◽  
Christine Zarges
Keyword(s):  
Feature Selection ◽  
Big Data ◽  
Set Theory ◽  
Rough Set ◽  
Rough Set Theory ◽  
Search Space ◽  
Locality Sensitive Hashing ◽  
Data Dependency ◽  
Distributed Environment ◽  
The Universe

In the context of big data, granular computing has recently been implemented by some mathematical tools, especially Rough Set Theory (RST). As a key topic of rough set theory, feature selection has been investigated to adapt the related granular concepts of RST to deal with large amounts of data, leading to the development of the distributed RST version. However, despite of its scalability, the distributed RST version faces a key challenge tied to the partitioning of the feature search space in the distributed environment while guaranteeing data dependency. Therefore, in this manuscript, we propose a new distributed RST version based on Locality Sensitive Hashing (LSH), named LSH-dRST, for big data feature selection. LSH-dRST uses LSH to match similar features into the same bucket and maps the generated buckets into partitions to enable the splitting of the universe in a more efficient way. More precisely, in this paper, we perform a detailed analysis of the performance of LSH-dRST by comparing it to the standard distributed RST version, which is based on a random partitioning of the universe. We demonstrate that our LSH-dRST is scalable when dealing with large amounts of data. We also demonstrate that LSH-dRST ensures the partitioning of the high dimensional feature search space in a more reliable way; hence better preserving data dependency in the distributed environment and ensuring a lower computational cost.

scholarly journals Some algorithms related to consistent decision table

2017 ◽  
Vol 33 (2) ◽  
pp. 131-142
Author(s):  
Quang Minh Hoang ◽  
Vu Duc Thi ◽  
Nguyen Ngoc San
Keyword(s):  
Set Theory ◽  
Rough Set ◽  
Polynomial Time ◽  
Time Complexity ◽  
Rough Set Theory ◽  
Decision Rules ◽  
The Other ◽  
Decision Table ◽  
Mathematical Tool ◽  
The Given

Rough set theory is useful mathematical tool developed to deal with vagueness and uncertainty. As an important concept of rough set theory, an attribute reduct is a subset of attributes that are jointly sufficient and individually necessary for preserving a particular property of the given information table. Rough set theory is also the most popular for generating decision rules from decision table. In this paper, we propose an algorithm finding object reduct of consistent decsion table. On the other hand, we also show an algorithm to find some attribute reducts and the correctness of our algorithms is proof-theoretically. These our algorithms have polynomial time complexity. Our finding object reduct helps other algorithms of finding attribute reducts become more effectively, especially as working with huge consistent decision table.

scholarly journals Rough Set Theory and its Applications: A recent Survey

2020 ◽  
pp. 548-552
Keyword(s):  
Set Theory ◽  
Rough Set ◽  
Intelligent Agents ◽  
Rough Set Theory ◽  
Boundary Region ◽  
Process Industry ◽  
Decision Making Process ◽  
Upper Approximation ◽  
Analysis Process ◽  
The Universe

Rough set theory is a mathematical method proposed by Pawlak . Rough set theory has been developed to manage uncertainties in information that presents missing and noises. Rough set theory is an expansion of the conventional set theory that supports approximations in decision making process. Fundamental of assumption of rough set theory is that with every object of the universe has some information associated it. Rough set theory is correlate two crisp sets, called lower and upper approximation. The lower approximation of a set consists of all elements that surely belong to the set, and the upper approximation of the set constitutes of all elements that possibly belong to the set. The boundary region of the set consists of all elements that cannot be classified uniquely as belonging to the set or as belonging to its complement, with respect to the available knowledge Rough sets are applied in several domains, such as, pattern recognition, medicine, finance, intelligent agents, telecommunication, control theory ,vibration analysis, conflict resolution, image analysis, process industry, marketing, banking risk assessment etc. This paper gives detail survey of rough set theory and its properties and various applications of rough set theory.

scholarly journals Rough Set Approximation as Formal Concept

2006 ◽  
Vol 10 (5) ◽  
pp. 606-611 ◽  
Author(s):  
Nozomi Ytow ◽  
◽  
David R. Morse ◽  
David McL. Roberts ◽  
◽  
...  
Keyword(s):  
Set Theory ◽  
Rough Set ◽  
Rough Sets ◽  
Formal Concept Analysis ◽  
Rough Set Theory ◽  
The Other ◽  
Formal Concept ◽  
Object Based ◽  
Formal Concepts ◽  
Set Approximation

Formal Concept Analysis (FCA) defines a formal concept as a pair of sets: objects and attributes, called extent and intent respectively. A rough set, on the other hand, approximates a concept using sets of objects only (in terms of FCA). We show that 1) a formal concept can be composed using a set of objects and its complement, 2) such object-based formal concepts are isomorphic to formal concepts based on objects and attributes, 3) upper and lower approximations of rough sets give generalization of formal concept, and 4) the pair of positive and negative sets (sensu rough set theory) are isomorphic to complemental formal concepts when the equivalence of the rough set gives positive and negative sets unique to each of the formal concepts. Implications of this are discussed.

Characteristics of Rough SetC-Means Clustering

2018 ◽  
Vol 22 (4) ◽  
pp. 551-564 ◽  
Author(s):  
Seiki Ubukata ◽  
◽  
Keisuke Umado ◽  
Akira Notsu ◽  
Katsuhiro Honda
Keyword(s):  
Set Theory ◽  
Rough Set ◽  
Malignant Disease ◽  
Rough Set Theory ◽  
Fuzzy Theory ◽  
Classification Performance ◽  
Approximation Space ◽  
Clustering Model ◽  
Positive Region ◽  
Medical Dataset

HardC-means (HCM), which is one of the most popular clustering techniques, has been extended by using soft computing approaches such as fuzzy theory and rough set theory. FuzzyC-means (FCM) and roughC-means (RCM) are respectively fuzzy and rough set extensions of HCM. RCM can detect the positive and the possible regions of clusters by using the lower and the upper areas which are respectively analogous to the lower and the upper approximations in rough set theory. RCM-type methods have the problem that the original definitions of the lower and the upper approximations are not actually used. In this paper, rough setC-means (RSCM), which is an extension of HCM based on the original rough set definition, is proposed as a rough set-based counterpart of RCM. Specifically, RSCM is proposed as a clustering model on an approximation space considering a space granulated by a binary relation and uses the lower and the upper approximations of temporal clusters. For this study, we investigated the characteristics of the proposed RSCM through basic considerations, visual demonstrations, and comparative experiments. We observed the geometric characteristics of the examined methods by using visualizations and numerical experiments conducted for the problem of classifying patients as having benign or malignant disease based on a medical dataset. We compared the classification performance by viewing the trade-off between the classification accuracy in the positive region and the fraction of objects classified as being in the positive region.

Summarization of information systems based on rough set theory

2021 ◽  
Vol 40 (1) ◽  
pp. 1001-1015
Author(s):  
Yen-Liang Chen ◽  
Fang-Chi Chi
Keyword(s):  
Set Theory ◽  
Rough Set ◽  
Rough Set Theory ◽  
Summary Table ◽  
Information Distortion ◽  
Reduced Representation ◽  
The Past ◽  
Upper Limit ◽  
Original Table ◽  
The Universe

In the rough set theory proposed by Pawlak, the concept of reduct is very important. The reduct is the minimum attribute set that preserves the partition of the universe. A great deal of research in the past has attempted to reduce the representation of the original table. The advantage of using a reduced representation table is that it can summarize the original table so that it retains the original knowledge without distortion. However, using reduct to summarize tables may encounter the problem of the table still being too large, so users will be overwhelmed by too much information. To solve this problem, this article considers how to further reduce the size of the table without causing too much distortion to the original knowledge. Therefore, we set an upper limit for information distortion, which represents the maximum degree of information distortion we allow. Under this upper limit of distortion, we seek to find the summary table with the highest compression. This paper proposes two algorithms. The first is to find all summary tables that satisfy the maximum distortion constraint, while the second is to further select the summary table with the greatest degree of compression from these tables.

Reduction in a fuzzy probabilistic information system

2021 ◽  
pp. 1-17
Author(s):  
Damei Luo ◽  
Zhaowen Li ◽  
Liangdong Qu
Keyword(s):  
Information System ◽  
Probability Distribution ◽  
Set Theory ◽  
Rough Set ◽  
Rough Set Theory ◽  
Mathematical Tool ◽  
Fuzzy Relations ◽  
Probabilistic Information ◽  
Entropy Measurement ◽  
The Universe

An information system (IS) is an important mathematical tool for artificial intelligence. A fuzzy probabilistic information system (FPIS), the combination of some fuzzy relations in the same universe which satisfies the probability distribution, can be seen as an IS with fuzzy relations. A FPIS overcomes the shortcoming that rough set theory assumes elements in the universe with equal probability and leads to lose some useful information. This paper integrates the probability distribution into the fuzzy relations in a FPIS and studies its reduction. Firstly, the concept of a FPIS is introduced and its reduction is proposed. Then, the fuzzy relations in a FPIS are divided into three categories (P-necessary, P-relatively necessary and P-unnecessary fuzzy relations) according to their importance. Next, entropy measurement for a FPIS is investigated. Moreover, some reduction algorithms are constructed. Finally, an example is given to verify the effectiveness of these proposed algorithms.

An Improved Covering Fuzzy Rough Set Model

2012 ◽  
Vol 490-495 ◽  
pp. 1397-1401
Author(s):  
Qing Hai Wang
Keyword(s):  
Set Theory ◽  
Rough Set ◽  
Fuzzy Set ◽  
Rough Set Theory ◽  
Real Life ◽  
Fuzzy Rough Set ◽  
Application Range ◽  
Model Based ◽  
Lower And Upper Approximations ◽  
The Universe

In this paper, we proposed the covering fuzzy rough set model based on multi-granulations and discussed some interesting properties about the model. The research may enlarge the application range of the rough set theory in real life. The lower and upper approximations of fuzzy set are defined by multi-covering relations on the universe, and some basic properties are introduced. It is shown that the fuzzy rough set model based on multi-covering relations is an extension of the rough set model based on multi-granulations.

scholarly journals Bi-covering rough sets

Filomat ◽  
2021 ◽  
Vol 35 (7) ◽  
pp. 2361-2369
Author(s):  
Mohamed Abo-Elhamayel
Keyword(s):  
Data Mining ◽  
Set Theory ◽  
Rough Set ◽  
Rough Sets ◽  
Rough Set Theory ◽  
Inclusion Relation ◽  
Different Types ◽  
Paper Construct ◽  
Lower And Upper Approximations ◽  
The Universe

Rough set theory is a useful tool for knowledge discovery and data mining. Covering-based rough sets are important generalizations of the classical rough sets. Recently, the concept of the neighborhood has been applied to define different types of covering rough sets. In this paper, based on the notion of bi-neighborhood, four types of bi-neighborhoods related bi-covering rough sets were defined with their properties being discussed. We first show some basic properties of the introduced bi-neighborhoods. We then explore the relationships between the considered bi-covering rough sets and investigate the properties of them. Also, we show that new notions may be viewed as a generalization of the previous studies covering rough sets. Finally, figures are presented to show that the collection of all lower and upper approximations (bi-neighborhoods of all elements in the universe) introduced in this paper construct a lattice in terms of the inclusion relation ?.

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