多粒化的模糊粗糙集代数

众所周知,一个粗糙集代数是由一个集合代数加上一对近似算子构成的。首先利用公理化的方法探讨经典的多粒化模糊粗糙集代数系统,可知经典的多粒化模糊粗糙集代数没有很好的性质;其次,引入 具有最小(大)元的等价关系的定义,并给出了基于具有最小(大)元等价关系的多粒化模糊近似算子的概念,在此基础上讨论了模糊粗糙集代数的性质,并得到了诸多结果。     

Similarity Relations and the Preservation of Solidity

The partitions of a given set stand in a well known one-to-onecorrespondence with the equivalence relations on that set. We askwhether anything analogous to partitions can be found which correspondin a like manner to the similarity relations (reflexive, symmetricrelations) on a set, and show that (what we call) decompositions – of acertain kind – play this role. A key ingredient in the discussion is akind of closure relation (analogous to the consequence relationsconsidered in formal logic) having nothing especially to do with thesimilarity issue, and we devote a final section to highlighting some ofits properties.     

I-Fuzzy equivalence relations and I-fuzzy partitions

A T-fuzzy equivalence relation is a fuzzy binary relation on a set X which is reflexive, symmetric and T-transitive for a t-norm T. Recently, Mesiar et al. [R. Mesiar, B. Reusch, H. Thiele, Fuzzy equivalence relations and fuzzy partitions, J. Multi-Valued Logic Soft Comput. 12 (2006) 167-181] have generalised the t-norm T to any general conjunctor C and investigated the minimal assumptions required on such operations, called duality fitting conjunctors, such that the fuzzification of the equivalence relation admits any value from the unit interval and also the one-one correspondence between the fuzzy equivalence relations and fuzzy partitions is preserved. In this work, we conduct a similar study by employing a related form of C-transitivity, viz., I-transitivity, where I is an implicator. We show that although every I-fuzzy equivalence relation can be shown to be a C-fuzzy equivalence relation, there exist C-fuzzy equivalence relations that are not I-fuzzy equivalence relations and hence these concepts are not equivalent. Most importantly, we show that the class of duality fitting implicators I is much richer than the residuals of the duality fitting conjunctors in the study of Mesiar et al. We also show that the I-fuzzy partitions have a “constant-wise” structure.     

多粒化的粗糙集代数

众所周知,一个粗糙集代数是由一个集合代数加上一对近似算子构成的。一方面 ,在公理化的方法下对经典的多粒化粗糙集代数系统进行了讨论,可知经典的粗糙集代数没有很好的性质;另一方面,给出了单调等价关系的定义,并给出了基于单调等价关系的多粒化近似算子的概念,在此基础上讨论了粗糙集代数的性质,并得到了诸多结果。     

基于等价关系的混合多粒度粗糙集

以基于等价关系诱导的划分为基础,提出了混合多粒度空间的概念,以便研究同时具有析取和合取关系的多 粒度空间。利用混合多粒度空间中的划分对目标概念进行近似逼近,提出了混合多粒度粗糙集模型。讨论了混合多 粒度粗糙集模型的基本性质,证明了混合多粒度粗糙集是乐观和悲观多粒度粗糙集的广义化表现形式。     

Fuzzy Rough Sets: The Forgotten Step

Traditional rough set theory uses equivalence relations to compute lower and upper approximations of sets. The corresponding equivalence classes either coincide or are disjoint. This behaviour is lost when moving on to a fuzzy T-equivalence relation. However, none of the existing studies on fuzzy rough set theory tries to exploit the fact that an element can belong to some degree to several "soft similarity classes" at the same time. In this paper we show that taking this truly fuzzy characteristic into account may lead to new and interesting definitions of lower and upper approximations. We explore two of them in detail and we investigate under which conditions they differ from the commonly used definitions. Finally we show the possible practical relevance of the newly introduced approximations for query refinement     

Vague相似关系与Vague划分

通过对单位区间[0,1]的闭子区间集中的元素——Vague值的定义,引入二元Vague关系、关系运算、逆关系、关系的合成、λ-截关系等新的定义。从基于可能度的区间数序关系的概念以及Vague集投票模型的实际意义两个方面,引入闭子区间集中Vague值的偏序关系的定义,并利用偏序关系证明了Vague关系的一些主要性质。给出了满足sup-min传递性的Vague相似关系以及Vague划分的定义,建立了Vague相似关系与Vague划分一一对应性,进而以[0,1]上连续t模作为广义的“与”逻辑算子,引入Vague相似关系的特例——满足T-传递性的Vague T-相似关系与Vague T-划分新的定义。     

Relations between complex vague soft sets

Complex vague soft sets are essentially vague soft sets characterized by an additional parameter called the phase term which is defined over the set of complex numbers. In this study, we introduce and discuss the relations between complex vague soft sets. We present the definitions of the Cartesian product of complex vague soft sets and subsequently that of complex vague soft relations. The definition of the composition of complex vague soft sets is also provided. The notions of symmetric, transitive, reflexive and equivalence complex vague soft relations are then proposed and the algebraic properties of these concepts are verified. The relation between complex vague soft sets is then discussed in the context of a real-life problem: the relation between the financial indicators of the Chinese economy which are characterized by their degrees of influence on the financial indicators of the Malaysian economy, and the time required for the former to affect the latter. Interpretations of the results obtained from this example are then proposed by relating them to recent significant real-life events in the Chinese and Malaysian economies.     

概念格与粗糙集的关系研究

概念格理论和粗糙集理论是两种不同的知识表示与知识发现的工具,都已被成功应用于许多领域.粗糙集理论的数据表现方式是信息系统,其研究基础是对象之间的等价关系;概念格理论的数据表现方式是形式背景,其研究基础是一种有序的层次结构--概念格.本文讨论了概念格理论与粗糙集理论之间的基本联系;重点分析了形式概念与等价类之间、概念格与分划之间的关系;证明了概念格与分划之间可以相互转换,给出了转换算法,并用例子加以说明.     

Effectiveness of NAQ-tree as index structure for similarity search in high-dimensional metric space

Similarity search (e.g., k-nearest neighbor search) in high-dimensional metric space is the key operation in many applications, such as multimedia databases, image retrieval and object recognition, among others. The high dimensionality and the huge size of the data set require an index structure to facilitate the search. State-of-the-art index structures are built by partitioning the data set based on distances to certain reference point(s). Using the index, search is confined to a small number of partitions. However, these methods either ignore the property of the data distribution (e.g., VP-tree and its variants) or produce non-disjoint partitions (e.g., M-tree and its variants, DBM-tree); these greatly affect the search efficiency. In this paper, we study the effectiveness of a new index structure, called Nested-Approximate-eQuivalence-class tree (NAQ-tree), which overcomes the above disadvantages. NAQ-tree is constructed by recursively dividing the data set into nested approximate equivalence classes. The conducted analysis and the reported comparative test results demonstrate the effectiveness of NAQ-tree in significantly improving the search efficiency.     

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.     

CONSENSUS-BASED ENSEMBLES OF SOFT CLUSTERINGS

The problem of obtaining a single “consensus” clustering solution from a multitude or ensemble of clusterings of a set of objects, has attracted much interest recently because of its numerous practical applications. While a wide variety of approaches including graph partitioning, maximum likelihood, genetic algorithms, and voting-merging have been proposed so far to solve this problem, virtually all of them work on hard partitionings, i.e., where an object is a member of exactly one cluster in any individual solution. However, many clustering algorithms such as fuzzy c-means naturally output soft partitionings of data, and forcibly hardening these partitions before applying a consensus method potentially involves loss of valuable information. In this article we propose several consensus algorithms that can be applied directly to soft clusterings. Experimental results over a variety of real-life datasets are also provided to show that using soft clusterings as input does offer significant advantages, especially when dealing with vertically partitioned data.     

积模糊粗集模型及其模糊知识粒的表示和分解

为处理人工智能中不精确和不确定的数据和知识,Pawlak提出了粗集理论。之后粗集理论被推广,其方法主要有二:一是减弱对等价关系的依赖;二是把研究问题的论域从一个拓展到多个。结合这两种思想,研究基于两个模糊近似空间的积模糊粗集模型及其模糊粗糙集的表示和分解。根据这种思想,可以从论域分解的角度探索降低高维模糊粗糙集计算的复杂度问题。先对模糊近似空间的分层递阶结构———λ-截近似空间进行研究,得到不同层次知识粒的相互关系;然后定义模糊等价关系的积,并研究其性质及算法;最后构建基于积模糊等价关系的积模糊粗集模型,并讨论了该模型中模糊粗糙集的表示及分解问题,分别从λ-截近似空间和一维模糊近似空间的角度去处理,给出了可分解集的上(下)近似的一个刻画,及模糊可分解集的上(下)近似的λ-截集分解算法。     

粗集商空间的格论性质

设U是论域，尺是U上的等价关系，我们在U的幂集P(U)上研究上粗相等、下粗相等及粗相等这三个等价关系，证明这三个等价关系确定的商集是有界分配格，且上粗相等、下粗相等确定的商集是布尔代数，并把这些结论完整地推广到模糊集上。     

Bisimulation and simulation algorithms on probabilistic transition systems by abstract interpretation

We show how bisimulation equivalence and simulation preorder on probabilistic LTSs (PLTSs), namely the main behavioural relations on probabilistic nondeterministic processes, can be characterized by abstract interpretation. Both bisimulation and simulation can be obtained as completions of partitions and preorders, viewed as abstract domains, w.r.t. a pair of concrete functions that encode a PLTS. This approach provides a general framework for designing algorithms that compute bisimulation and simulation on PLTSs. Notably, (i)?we show that the standard bisimulation algorithm by Baier et al. (2000) can be viewed as an instance of such a framework and (ii)?we design a new efficient simulation algorithm that improves the state of the art.     

Equivalence of queries that are sensitive to multiplicities

The query equivalence problem has been studied extensively for set-semantics and, more recently, for bag and bag-set semantics. However, SQL queries often combine set, bag and bag-set semantics. For example, an SQL query that returns a multiset of elements may call a subquery or view that returns a set of elements. Queries may access both relations that do not contain duplicates, as well as relations with duplicates. As another example, in SQL one can compute a multiset-union of queries, each of which returns a set of answers. This paper presents combined semantics, which formally models query evaluation combining set, bag and bag-set semantics. The equivalence problem for queries evaluated under combined semantics is studied. A sufficient condition for equivalence is presented. For several important common classes of queries necessary and sufficient conditions for equivalence are presented. An early version of this article appeared in [7]. This paper extends [7] to include bag semantics, in addition to set and bag-set semantics. This work was partially supported by the Israel Science Foundation (Grant 1032/05).     

Fuzzy relations and fuzzy relational databases

This paper deals with the connections existing between fuzzy set theory and fuzzy relational databases. Our new result dealing with fuzzy relations is how to calculate the greatest lower bound (glb) of two similarity relations. Our main contributions in fuzzy relational databases are establishing from fuzzy set theory what a fuzzy relational database should be (the result is both surprising and elegant), and making fuzzy relational databases even more robust.Our work in fuzzy relations and in fuzzy databases had led us into other interesting problems—two of which we mention in this paper. The first is primarily mathematical, and the second provides yet another connection between fuzzy set theory and artificial intelligence. In understanding similarity relations in terms of other fuzzy relations and in making fuzzy databases more robust, we work with closure and interior operators; we present some important properties of these operators. In establishing the connection between fuzzy set theory and artificial intelligence, we show that an abstraction on a set is in fact a partition on the set; that is, an abstraction defines an equivalence relation on the underlying set.     

On the expressive power of behavioral profiles

Behavioral profiles have been proposed as a behavioral abstraction of dynamic systems, specifically in the context of business process modeling. A behavioral profile can be seen as a complete graph over a set of task labels, where each edge is annotated with one relation from a given set of binary behavioral relations. Since their introduction, behavioral profiles were argued to provide a convenient way for comparing pairs of process models with respect to their behavior or computing behavioral similarity between process models. Still, as of today, there is little understanding of the expressive power of behavioral profiles. Via counter-examples, several authors have shown that behavioral profiles over various sets of behavioral relations cannot distinguish certain systems up to trace equivalence, even for restricted classes of systems represented as safe workflow nets. This paper studies the expressive power of behavioral profiles from two angles. Firstly, the paper investigates the expressive power of behavioral profiles and systems captured as acyclic workflow nets. It is shown that for unlabeled acyclic workflow net systems, behavioral profiles over a simple set of behavioral relations are expressive up to configuration equivalence. When systems are labeled, this result does not hold for any of several previously proposed sets of behavioral relations. Secondly, the paper compares the expressive power of behavioral profiles and regular languages. It is shown that for any set of behavioral relations, behavioral profiles are strictly less expressive than regular languages, entailing that behavioral profiles cannot be used to decide trace equivalence of finite automata and thus Petri nets.     

一种基于信息熵的模糊粗糙集知识获取方法

模糊粗糙集融合了模糊集和粗糙集的思想,是一种新的处理模糊和不确定性知识的软计算工具。针对属性为模糊值的信息系统,提出了一种基于熵的模糊粗糙集知识获取方法:首先通过模糊相似度量计算出各属性下对象的模糊相似值,再根据模糊相似关系构造模糊等价关系,然后根据模糊等价关系建立属性集的信息熵表示,继而使用基于信息熵的决策表属性约简算法获取规则。最后,通过一个实例,分析说明了这种算法的合理有效性。