粗糙逻辑中公式的Borel型概率粗糙真度

以一种特殊的粗糙逻辑为研究对象,视全体赋值之集为通常乘积拓扑空间,通过利用赋值集上的Borel概率测度,提出了能融合粗糙逻辑与计量逻辑为一体的公式的Borel型概率粗糙真度理论,给出了公式概率粗糙真度的公理化定义,建立起了相应的概率真度表示定理.公式的概率粗糙真度理论可被看作粗糙逻辑中已有工作的计量化,也可看作计量逻辑学中真度理论的粗糙化.基于这一核心概念,进一步给出了粗糙逻辑中已有概念的程度化表示形式,如公式的粗糙度、精确度、公式之间的粗糙相似度等,并建立起了基于粗糙相似度的3种近似推理模式.该结果实现了粗糙逻辑与计量逻辑的和谐统一,为进一步基于粗糙真值的程度化推理搭建了一个可能的框架.     

粗糙逻辑的计量化研究

将计量化方法引入到粗糙逻辑的研究当中,在一种典型的粗糙逻辑[LR]中引入了公式的粗糙真度概念。在此基础上,提出了公式之间的粗糙相似度、粗糙伪距离等概念,得到了粗糙逻辑度量空间。在粗糙度量空间中提出了两种不同的粗糙近似推理模式。这一结果实现了粗糙集与计量逻辑学这两种不同的处理近似问题理论的融合,同时对进一步丰富基于粗糙集的近似推理有一定启示。     

The further investigation of covering-based rough sets: Uncertainty characterization, similarity measure and generalized models

The notion of rough sets was originally proposed by Pawlak. In Pawlak’s rough set theory, the equivalence relation or partition plays an important role. However, the equivalence relation or partition is restrictive for many applications because it can only deal with complete information systems. This limits the theory’s application to a certain extent. Therefore covering-based rough sets are derived by replacing the partitions of a universe with its coverings. This paper focuses on the further investigation of covering-based rough sets. Firstly, we discuss the uncertainty of covering in the covering approximation space, and show that it can be characterized by rough entropy and the granulation of covering. Secondly, since it is necessary to measure the similarity between covering rough sets in practical applications such as pattern recognition, image processing and fuzzy reasoning, we present an approach which measures these similarities using a triangular norm. We show that in a covering approximation space, a triangular norm can induce an inclusion degree, and that the similarity measure between covering rough sets can be given according to this triangular norm and inclusion degree. Thirdly, two generalized covering-based rough set models are proposed, and we employ practical examples to illustrate their applications. Finally, relationships between the proposed covering-based rough set models and the existing rough set models are also made.     

A new measure of uncertainty based on knowledge granulation for rough sets

In rough set theory, accuracy and roughness are used to characterize uncertainty of a set and approximation accuracy is employed to depict accuracy of a rough classification. Although these measures are effective, they have some limitations when the lower/upper approximation of a set under one knowledge is equal to that under another knowledge. To overcome these limitations, we address in this paper the issues of uncertainty of a set in an information system and approximation accuracy of a rough classification in a decision table. An axiomatic definition of knowledge granulation for an information system is given, under which these three measures are modified. Theoretical studies and experimental results show that the modified measures are effective and suitable for evaluating the roughness and accuracy of a set in an information system and the approximation accuracy of a rough classification in a decision table, respectively, and have a much simpler and more comprehensive form than the existing ones.     

A rough set approach to reasoning under uncertainty

Abstract

Reasoning with uncertain information is a problem of key importance when dealing with information about the real world. Obtaining the precise numbers required by many uncertainty handling formalisms can be a problem. The theory of rough sets makes it possible to handle uncertainty without the need for precise numbers, and so has some advantages in such situations. This paper presents an introduction to various forms of reasoning under uncertainty that are based on rough sets. In particular, a number of sets of numerical and symbolic truth values which may be used to augment propositional logic are developed, and a semantics for these values is provided based upon the notion of possible worlds. Methods of combining the truth values are developed so that they may be propagated when augmented logic formulae are combined, and their use is demonstrated in theorem proving.     

基于条件概率的粗糙集不确定性度量

通过语义分析,提出一种修正的粗糙集不确定性度量公理化定义。首先,对该定义的数学特征进行分析,提出两种基于条件概率的粗糙集不确定性度量方法;然后,证明它们满足所提出的公理化定义,并导出相应的知识不确定性度量,发现其中一个是现有条件信息熵,另一个与确定性度量形成互补关系。设计算例对各种不确定性度量进行比较分析,验证了所提出的度量公式与不确定性语义保持一致。     

Generalized fuzzy rough sets determined by a triangular norm

The theory of rough sets has become well established as an approach for uncertainty management in a wide variety of applications. Various fuzzy generalizations of rough approximations have been made over the years. This paper presents a general framework for the study of T-fuzzy rough approximation operators in which both the constructive and axiomatic approaches are used. By using a pair of dual triangular norms in the constructive approach, some definitions of the upper and lower approximation operators of fuzzy sets are proposed and analyzed by means of arbitrary fuzzy relations. The connections between special fuzzy relations and the T-upper and T-lower approximation operators of fuzzy sets are also examined. In the axiomatic approach, an operator-oriented characterization of rough sets is proposed, that is, T-fuzzy approximation operators are defined by axioms. Different axiom sets of T-upper and T-lower fuzzy set-theoretic operators guarantee the existence of different types of fuzzy relations producing the same operators. The independence of axioms characterizing the T-fuzzy rough approximation operators is examined. Then the minimal sets of axioms for the characterization of the T-fuzzy approximation operators are presented. Based on information theory, the entropy of the generalized fuzzy approximation space, which is similar to Shannon’s entropy, is formulated. To measure uncertainty in T-generalized fuzzy rough sets, a notion of fuzziness is introduced. Some basic properties of this measure are examined. For a special triangular norm T = min, it is proved that the measure of fuzziness of the generalized fuzzy rough set is equal to zero if and only if the set is crisp and definable.     

一类特殊的粗糙逻辑代数的不确定性度量

基于已有的包含度理论,在一类特殊的粗糙逻辑代数中首次引入了元素的粗糙真度,粗糙度等概念。进一步,引入了针对两个元素的粗糙相似度及粗糙包含度的概念,详细研究了它们的性质。这些概念可用于展开带有粗糙信息特征的近似推理。     

On the generalization of fuzzy rough sets

Rough sets and fuzzy sets have been proved to be powerful mathematical tools to deal with uncertainty, it soon raises a natural question of whether it is possible to connect rough sets and fuzzy sets. The existing generalizations of fuzzy rough sets are all based on special fuzzy relations (fuzzy similarity relations, T-similarity relations), it is advantageous to generalize the fuzzy rough sets by means of arbitrary fuzzy relations and present a general framework for the study of fuzzy rough sets by using both constructive and axiomatic approaches. In this paper, from the viewpoint of constructive approach, we first propose some definitions of upper and lower approximation operators of fuzzy sets by means of arbitrary fuzzy relations and study the relations among them, the connections between special fuzzy relations and upper and lower approximation operators of fuzzy sets are also examined. In axiomatic approach, we characterize different classes of generalized upper and lower approximation operators of fuzzy sets by different sets of axioms. The lattice and topological structures of fuzzy rough sets are also proposed. In order to demonstrate that our proposed generalization of fuzzy rough sets have wider range of applications than the existing fuzzy rough sets, a special lower approximation operator is applied to a fuzzy reasoning system, which coincides with the Mamdani algorithm.     

覆盖粗糙直觉Fuzzy集模型的一点注记

通过对一类覆盖粗糙直觉模糊集模型中粗糙度定义的分析,对其所存在疏漏进行了改进;再将粗糙熵的概念引入到该模型,研究直觉模糊集的不确定度量;通过例子说明该度量的有效性。     

A first-order logic for reasoning under uncertainty using rough sets

Reasoning with uncertain information is a problem of key importance when dealing with knowledge from real situations. Obtaining the precise numbers required by many uncertainty-handling formalisms can be a problem when building real systems. The theory of rough sets allows us to handle uncertainty without the need for precise numbers, and so has some advantages in such situations. The authors develop a set of symbolic truth values based upon rough sets which may be used to augment predicate logic, and provide methods for combining these truth values so that they may be propagated when augmented logic formulae are used in automated reasoning.     

Generalized rough sets over fuzzy lattices

This paper studies generalized rough sets over fuzzy lattices through both the constructive and axiomatic approaches. From the viewpoint of the constructive approach, the basic properties of generalized rough sets over fuzzy lattices are obtained. The matrix representation of the lower and upper approximations is given. According to this matrix view, a simple algorithm is obtained for computing the lower and upper approximations. As for the axiomatic approach, a set of axioms is constructed to characterize the upper approximation of generalized rough sets over fuzzy lattices.     

Fuzzy rough DEA model: A possibility and expected value approaches

Uncertainty is certain in the world of uncertainty. Measuring the performance of any entity in such an uncertain environment is unavoidable. Fuzzy rough data envelopment analysis (FRDEA) provides a room to evaluate the relative efficiency of homogenous entities, widely known as decision making units (DMUs) in the data envelopment analysis (DEA) literature. This paper attempts to create a fuzzy rough DEA model by integrating the classical DEA, fuzzy set theory, and rough set theory, which apparently provide a way to accommodate the uncertainty. Moreover, in contrast to the probability approach, this paper provides a pavement to measure the relative efficiency of any given DMUs in line with the possibility approach along with the fuzzy rough expected value operator.     

命题逻辑系统Ln中公式集上的真度函数

在n值Lukasiewicz命题逻辑系统中引入了公式集F（S）上真度函数的公理化定义,给出了真度函数的若干重要性质,利用真度函数从形式上定义了相似度和伪距离,建立了逻辑度量空间,为从语构的角度展开近似推理提供了一种可能的框架。     

粗糙集的不确定性度量准则

针对现有粗糙集不确定性度量中有些定义在某种情况下并不合理,给出粗糙集不确定性度量的基本准则,证明除二次模糊度外其它几种不确定性度量都是满足基本准则的不确定性度量。由于满足基本准则的不确定性度量仍然可能存在不足,文中对基本准则中的单调性进行进一步限制,提出不确定性度量的扩展准则,并证明模糊熵和修正模糊度是满足扩展准则的不确定性度量,而粗糙度、粗糙熵和线性模糊度都不满足扩展准则。这些结论为已有的不确定性度量的合理性(或不合理性)提供理论说明,也为设计新的不确定性度量方法提供依据。     

MEASURING UNCERTAINTY IN ROUGH SET THEORY

A well justified measure of uncertainty for rough set theory is presented along with an axiomatic derivation. The connection between this measure and classical measures of uncertainty is provided.     

Measures of general fuzzy rough sets on a probabilistic space

Fuzzy rough set is a generalization of crisp rough set, which deals with both fuzziness and vagueness in data. The measures of fuzzy rough sets aim to dig its numeral characters in order to analyze data effectively. In this paper we first develop a method to compute the cardinality of fuzzy set on a probabilistic space, and then propose a real number valued function for each approximation operator of the general fuzzy rough sets on a probabilistic space to measure its approximate accuracy. The functions of lower and upper approximation operators are natural generalizations of the belief function and plausibility function in Dempster-Shafer theory of evidence, respectively. By using these functions, accuracy measure, roughness degree, dependency function, entropy and conditional entropy of general fuzzy rough set are proposed, and the relative reduction of fuzzy decision system is also developed by using the dependency function and characterized by the conditional entropy. At last, these measure functions for approximation operators are characterized by axiomatic approaches.     

关于近似空间中粗糙包含可能度的研究

粗糙包含关系是粗糙集理论中一个重要概念,粗糙包含关系也是近似空间中的一个拟序关系。文章给出了在同一近似空间中比较两个不同粗糙集的包含可能度的计算公式,研究了粗糙包含可能度的性质,并讨论了粗糙包含度、粗糙度和划分加细关系之间的联系。