Fuzzy Formal Context Research Articles

INCOMPLETE DATA ANALYSIS FOR BUILDING FORMAL ONTOLOGIES

The article considers the problem of automating the formation of ontological specifications subject domains on the basis of measurements. This problem is the pivotal issue of ontological analysis. The article presents models and methods, aimed at identifying conceptual structure and, ultimately, detecting the formal ontology of the considered subject domain. Fundamental realities of accumulation of empirical data (multiple independent measurements for each training sample properties of the object; congruence of the measurement procedures; differentiation of trust to different data sources) are reflected in the "objects-properties" summary table model. Imperfection (inaccuracy, inconsistency, uncertainty) of this information implies the need to use many-valued logic models for its primary processing. The result of this treatment - fuzzy formal context - should be approximated as a unique context, from which the possible conclusion of formal concepts in the framework of applied branch of the lattice theory, known as the "formal concept analysis". The genesis of the "properties’ limits of existence" that affect the correctness of the approximation of fuzzy formal context is studied. The models and the method of accounting for this additional information are proposed. Guidelines for conversion of lattice formal concepts into a formal ontology are formulated. A model example of the developed models and ontological data analysis methods is presented.

Lattice-theoretic contexts and their concept lattices via Galois ideals

This paper introduces a concept of lattice-theoretic contexts as well as their concept lattices. A lattice-theoretic context is a triple (G, M, I) with two complete lattices G, M and their Galois ideal I. A lattice-theoretic context and its concept lattice are a common generalization of classical FCA, Pócs’s formal fuzzy context, one-sided concept lattices, generalized concept lattices and L-fuzzy concept lattices (with hedges). When the lattices G, M are completely distributive, a reduction of the relation I in the lattice-theoretic context (G, M, I) can be obtained. Related algorithms to construct concept lattices of L-fuzzy contexts considered as lattice-theoretic contexts are presented. In the case of L being a completely distributive lattice, we can reduce the number of elements (objects or/and attributes) before computing the whole concept lattice. Then the related algorithm has lower complexity.

Knowledge Reduction in Crisply Generated Fuzzy Concept Lattices

Knowledge reduction is a basic issue in knowledge representation and data mining. Al- though various methods have been developed to reduce the size of classical formal contexts, the reduction of formal fuzzy contexts based on fuzzy lattices remains a difficult problem owing to its complicated derivation operators. To address this problem, this paper proposes a method of knowl- edge reduction by reducing attributes in a formal fuzzy context based on the crisply generated fuzzy concept lattice. Employing the proposed approach, attributes which are non-essential to the struc- ture of the crisply generated fuzzy concept lattice are removed. Discernibility matrix and Boolean function are employed to compute the attribute reducts of the formal fuzzy contexts, by which all the attribute reducts of the formal fuzzy contexts are deter mined without changing the structure of the lattice. Further, all the attributes are classified into three types by their significance in construct- ing the crisply generated fuzzy concept lattice. The characteristics of these types of attributes are also analyzed. Finally, the proposed method is used to conduct knowledge reduction in the variable threshold concept lattices, which is a complement to the existing knowledge reduction methods.

Knowledge representation using interval-valued fuzzy formal concept lattice

Formal concept analysis (FCA) is a mathematical framework for data analysis and processing tasks. Based on the lattice and order theory, FCA derives the conceptual hierarchies from the relational information systems. From the crisp setting, FCA has been extended to fuzzy environment. This extension is aimed at handling the uncertain and vague information represented in the form of a formal context whose entries are the degrees from the scale [0, 1]. The present study analyzes the fuzziness in a given many-valued context which is transformed into a fuzzy formal context, to provide an insight into generating the fuzzy formal concepts from the fuzzy formal context. Furthermore, considering that a major problem in FCA with fuzzy setting is to reduce the number of fuzzy formal concepts thereby simplifying the corresponding fuzzy concept lattice structure, the current paper solves the problem by linking an interval-valued fuzzy graph to the fuzzy concept lattice. For this purpose, we propose an algorithm for generating the interval-valued fuzzy formal concepts. To measure the weight of fuzzy formal concepts, an algorithm is proposed using Shannon entropy. The knowledge represented by formal concepts using interval-valued fuzzy graph is compared with entropy-based-weighted fuzzy concepts at chosen threshold.

A Note on Computing the Crisp Order Context of a Fuzzy Formal Context for Knowledge Reduction

Fuzzy Formal Concept Analysis (FCA) is a mathematical tool for the effective representation of imprecise and vague knowledge. However, with a large number of formal concepts from a fuzzy context, the task of knowledge representation becomes complex. Hence, knowledge reduction is an important issue in FCA with a fuzzy setting. The purpose of this current study is to address this issue by proposing a method that computes the corresponding crisp order for the fuzzy relation in a given fuzzy formal context. The obtained formal context using the proposed method provides a fewer number of concepts when compared to original fuzzy context. The resultant lattice structure is a reduced form of its corresponding fuzzy concept lattice and preserves the specialized and generalized concepts, as well as stability. This study also shows a step-by-step demonstration of the proposed method and its application. Keywords—Crisp Context, Concept Lattice, Formal Concept Analysis, Fuzzy Formal Concept, Fuzzy Relation, Knowledge Reduction

Knowledge reduction in formal fuzzy contexts

Knowledge reduction is a basic issue in knowledge representation and data mining. Although various methods have been developed to reduce the size of classical formal contexts, the reduction of formal fuzzy contexts based on fuzzy lattices remains a difficult problem owing to its complicated derivation operators. To address this problem, we propose a general method of knowledge reduction by reducing attributes and objects in formal fuzzy contexts based on the variable threshold concept lattices. Employing the proposed approaches, we remove attributes and objects which are non-essential to the structure of a variable threshold concept lattice, i.e., with a given threshold level, the concept lattice constructed from a reduced formal context is made identical to that constructed from the original formal context. Discernibility matrices and Boolean functions are, respectively, employed to compute the attribute reducts and object reducts of the formal fuzzy contexts, by which all the attribute reducts and object reducts of the formal fuzzy contexts are determined without changing the structure of the lattice.

Bipolar fuzzy graph representation of concept lattice

Formal Concept Analysis (FCA) is a mathematical framework for knowledge processing tasks. FCA has been successfully incorporated into fuzzy setting and its extension (interval-valued fuzzy set) for handling vagueness and impreciseness in data. However, the analysis in such settings is restricted to unipolar space. Recently, some applications of bipolar information are shown in bipolar fuzzy graph, lattice theory as well as in FCA. The adequate analysis of bipolar information using FCA requires incorporation of bipolar fuzzy set and an appropriate lattice structure. For this purpose, we propose an algorithm for generating the bipolar fuzzy formal concepts, a method for (α,β)-cut of bipolar fuzzy formal context and its implications with illustrative examples.

Construction Method of λ-Concept Lattice in Fuzzy Formal Context

From the view of fuzzy cut set, the fuzzy concept lattice was analyzed and researched and a new kind of concept lattice was proposed. The concept lattice with every different confidence can be constructed through selecting different level of . From two different angles of context and concept, two corresponding construction methods, context cut method and concept cut method, were discussed, and some results were given. Furthermore, an example was provided to illustrate the method correct.

Representation of Fuzzy Concept Lattices in the Framework of Classical FCA

We describe a representation of the fuzzy concept lattices, defined via antitone Galois connections, within the framework of classical Formal Concept Analysis. As it is shown, all needed information is explicitly contained in a given formal fuzzy context and the proposed representation can be obtained without a creation of the corresponding fuzzy concept lattice.

Parallel Construction of Variable Precision Concept Lattice in Fuzzy Formal Context

Based on the fuzzy cut method, the fuzzy concept lattice was analyzed and researched and a new kind of concept lattice which called variable precision concept lattice was proposed. This kind of concept lattice can be constructed with every different confidence through selecting different level. Two corresponding construction methods, context cut method and concept cut method, were discussed in the two different angles, and some results were given. Furthermore, a case study was carried on.

Two kinds of multi-level formal concepts and its application for sets approximations

In this paper, we introduce two pairs of operators in fuzzy formal contexts. Based on the proposed operators, we present two kinds of multi-level formal concepts. We also propose two pairs of rough approximation operators by employing the two kinds of multi-level formal concepts. By the proposed rough set approximation operators, we not only approximate a crisp set, but also approximate a fuzzy set. Finally, we discuss the properties of the proposed approximation operators in details.

A Method for Decomposition of Fuzzy Formal Context

Recently, fuzzy FCA becomes popular in researchers for representing the knowledge precisely. In this study we present a method for decomposition of given fuzzy formal context into several crisp formal contexts with help of distinct granules of fuzzy relation. The presented method reduces the lattice of given fuzzy formal context into several lattices for refining the interested patterns. From the obtained crisp formal contexts, we can generate the formal concepts in order to visualize them in lattice structure for describing the knowledge of given research field. Proposed method can be helpful for the researchers for reducing number of concepts, visualization of concepts for distinct fuzzy relations, and reduction of fuzzy lattice into several lattices in fuzzy context related fields

Afuzzy formal concept analysis based approach for business component identification

Identifying business components is the basis of component-based software engineering. Many approaches, including cluster analysis and concept analysis, have been proposed to identify components from business models. These approaches classify business elements into a set of components by analyzing their properties. However, most of them do not consider the difference in their properties for the business elements, which may decrease the accuracy of the identification results. Furthermore, component identification by partitioning business elements cannot reflect which features are responsible for the generation of certain results. This paper deals with a new approach for component identification from business models using fuzzy formal concept analysis. First, the membership between business elements and their properties is quantified and transformed into a fuzzy formal context, from which the concept lattice is built using a refined incremental algorithm. Then the components are selected from the concepts according to the concept dispersion and distance. Finally, the effectiveness and efficiency are validated by applying our approach in the real-life cases and experiments.

A mathematical model for concept granular computing systems

Extent-intent and intent-extent operators are introduced between two complete lattices in this paper and a mathematical model for concept granular computing system is established. We proved that the set of all concepts in this system is a lattice with the greatest element and the least element. This framework includes formal concept lattices from formal contexts, L fuzzy concept lattices from L fuzzy formal contexts and three kinds of variable threshold concept lattices, i.e. the extension and the intension of a concept are a crisp set and a crisp set, a crisp set and a fuzzy set, a fuzzy set and a crisp set, respectively. Finally, some iterative algorithms for constructing concepts are proposed and they are proved to be optimal concepts under some conditions in this system.

Attribute reduction in fuzzy concept lattices based on the T implication

Attribute reduction in fuzzy concept lattices based on the kind of transitive regular implication operator ( T implication in short) is introduced and investigated. We first propose a kind of fuzzy concept lattices by using T implication in a fuzzy formal context at a level δ ∈ I T ⊆ [ 0 , 1 ] with respect to T implication. Then we introduce the notion of δ -reducts in a fuzzy formal context and give some equivalent characterizations of δ -consistent sets to determine δ -reducts. Based on δ -reducts, we divide attributes into three types (at a level δ ) and establish some characterization theorems to determine the type of an attribute.

Set approximations in fuzzy formal concept analysis

Formal concept analysis and rough set theory are two important tools in knowledge representation and knowledge discovery in relational information systems. The purpose of this paper is to study rough set approximations within formal concept analysis in fuzzy environment. Properties of existent fuzzy concept lattices derived from an adjoint pair of operations are first reviewed and examined. Based on both lattice-theoretic and fuzzy set-theoretic operators, two new pairs of rough fuzzy set approximations within fuzzy formal contexts are then defined. Finally, properties of the rough fuzzy set approximation operators are presented in detail.

Concept lattices and order in fuzzy logic

The theory of concept lattices (i.e. hierarchical structures of concepts in the sense of Port-Royal school) is approached from the point of view of fuzzy logic. The notions of partial order, lattice order, and formal concept are generalized for fuzzy setting. Presented is a theorem characterizing the hierarchical structure of formal fuzzy concepts arising in a given formal fuzzy context. Also, as an application of the present approach, Dedekind–MacNeille completion of a partial fuzzy order is described. The approach and results provide foundations for formal concept analysis of vague data—the propositions “object x has attribute y ”, which form the input data to formal concept analysis, are now allowed to have also intermediate truth values, meeting reality better.

Concept lattices and order in fuzzy logic

The theory of concept lattices (i.e. hierarchical structures of concepts in the sense of Port-Royal school) is approached from the point of view of fuzzy logic. The notions of partial order, lattice order, and formal concept are generalized for fuzzy setting. Presented is a theorem characterizing the hierarchical structure of formal fuzzy concepts arising in a given formal fuzzy context. Also, as an application of the present approach, Dedekind–MacNeille completion of a partial fuzzy order is described. The approach and results provide foundations for formal concept analysis of vague data—the propositions “object x has attribute y”, which form the input data to formal concept analysis, are now allowed to have also intermediate truth values, meeting reality better.