基于格值逻辑的模糊概念格

从逻辑的角度,将非经典逻辑之一的格值逻辑引入概念格,建立了格值模糊形式背景,通过格结构来刻画对象与属性之间的模糊关系,证明了由蕴涵算子诱导的算子对是伽罗瓦连接,并讨论了相关的一些性质,进而给出了格值模糊概念格的构造算法.格值模糊概念格的建立为模糊性与不可比较性信息的处理提供了可靠的数学工具.     

Power contexts and their concept lattices

We introduce a framework for the study of formal contexts and their lattices induced by the additional structure of self-relations on top of the traditional incidence relation. The induced contexts use subsets as objects and attributes, hence the name power context and power concept. Six types of new incidence relations are introduced by taking into account all possible combinations of universal and existential quantifiers as well as the order of the quantifications in constructing the lifted power contexts. The structure of the power concept lattice is investigated through projection mappings from the baseline objects and attributes to those of the power context, respectively. We introduce the notions of extensional consistency and intensional consistency, corresponding to the topological notions of continuity in the analogous setting when concepts are viewed as closed sets. We establish Galois connections for these notions of consistency. We further introduce the notion of faithfulness for the first type of lifted incidence relation based on the fact that it can be equivalently characterized by a concept-faithful morphism. We also present conditions under which the power concept lattice serves as a factor lattice of the base concept lattice.     

Axiomatic characterizations of dual concept lattices

Formal concept analysis is an algebraic model based on a Galois connection. It is used for symbolic knowledge exploration from an elementary form of a formal context. This paper mainly presents a general framework for concept lattice in which axiomatic approaches are used. The relationship between concept lattice and dual concept lattice is first studied. Based on set-theoretic operators, generalized concept systems are established. And properties of them are examined. By using axiomatic approaches, a pair of dual concept lattices is characterized by different sets of axioms. The connections between 0-1 binary relations and generalized concept systems are examined. And generalized dual concept systems can be constructed by a pair of dual set-theoretic operators. Axiomatic characterizations of the generalized concept systems guarantee the existence of a binary relation producing a formal context.     

模糊概念格

概念格是研究和处理概念内涵与外延确定性关系的数学方法 ,已成为一种有效的数据分析方法。本文进一步探讨概念内涵与外延的不确定关系的模糊映射 ,给出相应的隶属函数的一些基本数学性质 ,证明全体模糊概念构成一个完全格。     

Fuzzy Galois Connections

The concept of Galois connection between power sets is generalized from the point of view of fuzzy logic. Studied is the case where the structure of truth values forms a complete residuated lattice. It is proved that fuzzy Galois connections are in one-to-one correspondence with binary fuzzy relations. A representation of fuzzy Galois connections by (classical) Galois connections is provided.     

模糊概念格与模糊推理

基于伽罗瓦连接,分别在交换伴随对与对合剩余格条件下,讨论了模糊概念格的四种定义形式。并证明了在对合剩余格上,对偶性成立,四种模糊算子将具有与经典意义下一致的相互关系。最后我们提出了一种基于模糊概念格的模糊推理规则,并证明了其还原性。     

Study of a universal formal context

Studying a universal formal context, we obtain a number of properties of the context itself, its concepts, and the lattice formed by the set of these concepts. The most significant of these properties is represented by a theorem showing that there exists an embedding of the concept lattice of an arbitrary at most countable universal context into the concept lattice of a universal context under which the image of the embedding is an initial segment of the concept set of a universal formal context with infinite volumes, and the validity of the dual result. It is shown that the theorem also holds in the computable case. This theorem demonstrates the complexity of the structure of a universal formal context.     

基于粗集与概念格的属性简约讨论

概念格的属性简约是在形式背景下解决复杂问题的重要途径,通过对概念格、粗糙集的讨论,将两者有效结合,并借助粗糙集上(下)近似的方法,得出了一个对概念格属性简约的方法,方法将二维的概念格属性简约转化为一维的一种对象格的简约,避免了形式背景下的概念的计算和进一步的可辨识矩阵的计算,方法简便,算法简单易实现,是概念格属性简约有效的算法.     

A partition-based approach towards constructing Galois (concept) lattices

Galois lattices and formal concept analysis of binary relations have proved useful in the resolution of many problems of theoretical or practical interest. Recent studies of practical applications in data mining and software engineering have put the emphasis on the need for both efficient and flexible algorithms to construct the lattice. Our paper presents a novel approach for lattice construction based on the apposition of binary relation fragments. We extend the existing theory to a complete characterization of the global Galois (concept) lattice as a substructure of the direct product of the lattices related to fragments. The structural properties underlie a procedure for extracting the global lattice from the direct product, which is the basis for a full-scale lattice construction algorithm implementing a divide-and-conquer strategy. The paper provides a complexity analysis of the algorithm together with some results about its practical performance and describes a class of binary relations for which the algorithm outperforms the most efficient lattice-constructing methods.     

Possibility-theoretic extension of derivation operators in formal concept analysis over fuzzy lattices

Formal concept analysis (FCA) associates a binary relation between a set of objects and a set of properties to a lattice of formal concepts defined through a Galois connection. This relation is called a formal context, and a formal concept is then defined by a pair made of a subset of objects and a subset of properties that are put in mutual correspondence by the connection. Several fuzzy logic approaches have been proposed for inducing fuzzy formal concepts from L-contexts based on antitone L-Galois connections. Besides, a possibility-theoretic reading of FCA which has been recently proposed allows us to consider four derivation powerset operators, namely sufficiency, possibility, necessity and dual sufficiency (rather than one in standard FCA). Classically, fuzzy FCA uses a residuated algebra for maintaining the closure property of the composition of sufficiency operators. In this paper, we enlarge this framework and provide sound minimal requirements of a fuzzy algebra w.r.t. the closure and opening properties of antitone L-Galois connections as well as the closure and opening properties of isotone L-Galois connections. We apply these results to particular compositions of the four derivation operators. We also give some noticeable properties which may be useful for building the corresponding associated lattices.     

基于集值映射的形式概念分析

Formal concept analysis (FCA) is a discipline that studied the hierarchical structares induced by a binary relation between a pair of sets,and applies in data analysis,information retrieval,knowledge discovery,etc.In this paper,it is shown that a formal context T is equivalent to a set-valued mapping S :G →P(M),and formal concepts could be defined in the set-valued mapping S.It is known that the topology and set-valued mapping are linked.Hence,the advantage of this paper is that the conclusion make us to construct formal concept lattice based on the topology.     

模糊形式背景下的类近似算子

分别在形式背景和模糊形式背景下定义类下近似算子和类模糊下近似算子,并研究它们的性质.证明这两种算子分剐等价于形式背景和模糊形式背景下的*算子和模糊*算子.进一步给出类下近似算子与类模糊下近似算子的公理刺画.最后,对偶地讨论类上近似算子和类模糊上近似算子的定义和性质.     

基于对象导出三支概念格的决策背景规则获取

规则获取是当前形式概念分析领域的研究热点.首先给出了基于对象导出三支概念格间的细于关系,定义了基于对象导出三支概念格的三支弱协调性,并研究了其与经典概念格下的二支弱协调性之间的关系.然后,研究了基于对象导出三支概念格的规则获取,并与经典概念格的规则获取进行了比较.最后,定义了对象导出三支概念的弱闭标记,研究了基于弱闭标记的三支弱协调决策形式背景的规则获取,剔除了冗余规则,并且得到一些新的更为精简的三支规则.     

格蕴涵代数的区间值模糊子代数

将区间值模糊集的概念应用于格蕴涵代数,引入区间值模糊格蕴涵子代数的概念并研究它们的性质.讨论了区间值模糊格蕴涵子代数与（模糊）格蕴涵子代数之间的关系;定义了区间值模糊集的象和原象,获得了区间值模糊格蕴涵子代数的象和原象成为区间值模糊格蕴涵子代数的条件.     

On computable formal concepts in computable formal contexts

We introduce and study the notions of computable formal context and computable formal concept. We give some examples of computable formal contexts in which the computable formal concepts fail to form a lattice and study the complexity aspects of formal concepts in computable contexts. In particular, we give some sufficient conditions under which the computability or noncomputability of a formal concept could be recognized from its lattice-theoretic properties. We prove the density theorem showing that in a Cantor-like topology every formal concept can be approximated by computable ones. We also show that not all formal concepts have lattice-theoretic approximations as suprema or infima of families of computable formal concepts.     

概念格的粗糙近似比较研究

形式背景产生了概念格,每个节点由外延和内涵组成.对形式背景论域中的任何一个子集,可用外延来近似,在这方面已有了4种方法.对这些方法进行了比较研究,利用粗糙集理论证明了用这些方法所求出的概念的上近似外延是相同的,并利用粗糙集理论研究了概念格属性约简后,原来方法对结果的一致性.     

Locally convex quotient lattice cones

Walter Roth has investigated certain equivalence relations on locally convex cones in [W. Roth, Locally convex quotient cones, J. Convex Anal. 18, No. 4, 903–913 (2011)] which give rise to the definition of a locally convex quotient cone. In this paper, we investigate some special equivalence relations on a locally convex lattice cone by which the locally convex quotient cone becomes a lattice. In the case of a locally convex solid Riesz space, this reduces to the known concept of locally convex solid quotient Riesz space. We prove that the strict inductive limit of locally convex lattice cones is a locally convex lattice cone. We also study the concept of locally convex complete quotient lattice cones.     

Conjugate families of mappings in pointwise metric fuzzy lattices

In this paper some relations among the axioms of pointwise metric for fuzzy lattices introduced by Shi are studied. By means of the conjugate families of mappings, some representation results are presented concerning metric, pseudometric and pseudo-quasi-metric fuzzy lattices. These results reconcile the metric lattice theory generated via concept of neighborhoodwith the one generated via concept of remote-neighborhood.     

A method of inference in approximate reasoning based on interval-valued fuzzy sets

This paper introduces and discusses a method of approximate inference which operates on the extension of the concept of a fuzzy set by the concept of an interval-valued fuzzy set. This method allows a formal, fuzzy representation to be built for verbal decision algorithms. Furthermore, it can have an effective computer representation. An example showing how this method operates is provided.     

Discrete optimization algorithms and problems of decision making in a fuzzy environment

An approach to solving optimization problems with fuzzy coefficients is described. It consists in formulating and analyzing one and the same problem within the framework of mutually related models by constructing equivalent analogs with fuzzy coefficients in objective functions alone. Since the approach is applied within the context of fuzzy discrete optimization problems, modified algorithms of discrete optimization are discussed. These algorithms are based on a combination of formal and heuristic procedures and allow one to obtain quasi-optimal solutions after a small number of steps, thus overcoming the computational complexity posed by the NP-completeness of discrete optimization problems. The subsequent contraction of the decision uncertainty regions is associated with reduction of the problem to multiobjective decision making in a fuzzy environment using techniques based on fuzzy preference relations. The results of the paper are of a universal character and are already being used to solve practical problems in several fields.