Property Oriented Concept Lattices Research Articles

Multi-adjoint concept lattices via quantaloid-enriched categories

With quantaloids carefully constructed from multi-adjoint frames, it is shown that multi-adjoint concept lattices, multi-adjoint property-oriented concept lattices and multi-adjoint object-oriented concept lattices are derivable from Isbell adjunctions, Kan adjunctions and dual Kan adjunctions between quantaloid-enriched categories, respectively.

Attribute Logic Formula Description of Granule and Its Application to Build Concept Lattice

This paper focus on the granule description based on formal concept analysis. First, we introduce the notion of attribute logic formula in a formal context, and then prove a general granule description theorem by using attribute logic formulas. Second, we prove some basic granule description theorems based on concept, property-oriented concept and object-oriented concept respectively. Third, we propose some methods that use attribute logic formulas to describe concept, property-oriented concept and object-oriented concept respectively, and prove that a property-oriented concept lattice and an object-oriented concept lattice are anti-isomorphic. Finally, we apply granule description methods to build concept lattice, property-oriented concept lattice and object-oriented concept lattice, propose some algorithms for building concept lattice, property-oriented concept lattice and object-oriented concept lattice, and give some examples and experiments to show the utility of algorithms.

A theoretical study on object-oriented and property-oriented multi-scale formal concept analysis

In traditional formal concept analysis, the attributes in the formal context are considered fixed. However, in the real world data set, attributes always have different levels of granularity, correspondingly, the derived concept lattice may reveal different information and patterns. Therefore, the capability to change the level of granularity of an attribute in formal concept analysis to capture relevant patterns in data is a natural requirement. In this paper, a theoretical study has been undertaken in multi-scale formal contexts, where attributes with different levels of granularity possess different attribute values. Two types of formal concepts, i.e., object-oriented and property-oriented multi-scale concepts, are introduced and studied in detail. The collection of object-oriented concept lattices and property-oriented concept lattices can be obtained at different granularity levels of attributes. It has been shown that the set of extents in the derived concept lattices increases when we choose to use a finer level of granularity. Moreover, a corresponding bidirectional approach to concept construction(i.e., from coarser to finer and from finer to coarser, respectively) is exhibited, and some characterization theorems have been obtained.

Three-way dual concept analysis

Three-way concept analysis is a mathematical theory, which combines formal concept analysis and three-way decision. The existing models, i.e., three-way concept lattices and three-way object oriented concept lattices, have been used successfully in many fields. However, these two models are established to cater some specific applications and cannot fulfill some special requirements such as the needs to seek potential collaborators in international import and export transactions. In this paper, a novel type of three-way concept lattices is presented based on dual concept analysis, which enables ones to characterize specific set by pointing out the attributes that are not possessed by at least one object in the complement of this specific set. And then, the connections between three-way dual concept lattices and classical dual concept lattices are explored. Besides, the relationships among four types of three-way concept analysis models, i.e., three-way concept lattices, three-way dual concept lattices, three-way object oriented concept lattices and three-way property oriented concept lattices, are investigated.

Object-oriented interval-set concept lattices

Formal concept analysis and rough set are two kinds of efficient mathematical tools for data analysis and knowledge discovery. By combining these two theories, object-oriented and property-oriented concept lattices are proposed. Interval set theory is proposed to describe a partially-known concept by a lower bound and a upper bound. In order to obtain the more accurate extension and intension for a partially-known object-oriented concept, we introduce the theory of interval sets into the object-oriented concept lattice, and propose an object-oriented interval-set concept lattice. Properties of them are investigated. Relationships among interval-set concept lattices, object-oriented interval-set concept lattices and property-oriented interval-set concept lattices are discussed. By discussing the relationships between the object-oriented concept lattice and the object-oriented interval-set concept lattice, an approach to construct object-oriented interval-set concept lattices are established.

A theoretical study on the object (property) oriented concept lattices based on three-way decisions

The three-way object oriented lattice and the three-way property oriented lattice are extend researches of rough concept analysis by combining three-way decisions. In this paper, we investigate them more comprehensively and detailedly. Firstly, the relationship between the object (property) oriented concept lattice and the three-way object (property) oriented concept lattice is studied, respectively. In addition, approaches to construct two types of three-way concept lattices based on apposition and subposition of formal contexts are given based on these relationships. Finally, since the methods for constructing three-way concept lattices and three-way object oriented and property oriented concept lattices are similar, the connections among them are discussed.

Attribute reduction and rule acquisition of formal decision context based on object (property) oriented concept lattices

The study of concept lattices, property oriented concept lattices and object oriented concept lattices provides complementary conceptual structures, which can be used to search, analyze and extract information from data sets. This paper is devoted to the study of rule acquisition and attribute reduction of formal decision context. Based on object oriented concepts and property oriented concepts, the notions of object oriented decision rules and property oriented decision rules are proposed. By using some equivalence relations on the set of extents of the related conditional concept lattices and decision concept lattices, the rule acquisition methods are presented. The attribute reduction approaches for formal decision context to preserve the object oriented decision rules and property oriented decision rules are put forward by using discernibility attributes.

Fuzzy Concept Lattices and Fuzzy Relation Equations in the Retrieval Processing of Images and Signals

This paper considers the introduced relations between fuzzy property-oriented concept lattices and fuzzy relation equations, on the one hand, and mathematical morphology, on the other hand, in the retrieval processing of images and signals. In the first part, it studies how the original images and signals can be retrieved using fuzzy property-oriented concept lattices and fuzzy relation equations. In the second one we analyze two of the most important tools in fuzzy mathematical morphology from the point of view of the fuzzy property-oriented concepts and the aforementioned study. Both parts are illustrated with practical examples.

Attribute Reduction in Property Oriented Concept Lattices of Formal Contexts and Formal Decision Contexts

This paper studies notions and approaches to attribute reduction in property oriented concept lattices of formal contexts and formal decision contexts based on congruence relations. Firstly, dependence space based on the property oriented concept lattice is researched to obtain the relationship among property oriented concept lattices and the corresponding congruence relations. Then notions of attribute reduction is defined for formal contexts and decision contexts to find minimal attribute subsets which can preserve all congruence classes determined by the original attribute set, and also keep all property oriented extents and their original hierarchy in the property oriented concept lattice. Finally, approaches of discernibility matrix are presented to calculate all attribute reducts.

Granular Transformation and Irreducible Element Judgment Theory Based on Pictorial Diagrams.

As an important tool for data analysis and knowledge processing, formal concept analysis has been applied to many fields. Currently, granular computing (GrC) provides a novel approach for data analysis. In this paper, pictorial diagrams of a formal context are first defined based on equivalence relations. And then, from the viewpoint of GrC, the transformation between equivalence classes and three kinds of concepts (formal concept, property oriented concept, and object oriented concept) in formal contexts are studied. Further, using the relationships between arrow relations proposed by Wille and Ganter and irreducible elements of a concept lattice, the new judgment theorems for join(meet)-irreducible elements of a concept lattice, join-irreducible elements of a property oriented concept lattice and an object oriented concept lattice are achieved.

Approximate concepts acquisition based on formal contexts

Formal concepts are abstraction and formalization of the concepts in philosophy. Acquisition of formal concepts is the base of formal concept analysis. Besides the concrete concepts, uncertain but meaningful concepts sometimes are more appropriate for real life. The paper begins with this practical problem and obtains some interesting knowledge based on formal contexts. First, k-grade relation on the object set of a formal context is defined, and the different values of k in different cases are discussed in detail. Then the algorithms to acquire approximate concepts associated with each object are proposed and these approximate concepts are explained. Second, the relationships between approximate concepts associated with objects and concepts of concept lattice (property oriented concept lattice) are studied. Parallel to the above idea, k-grade relation on the attribute set is proposed dually, and approximate concepts associated with each attribute are obtained and interpreted. Meanwhile, the relationships between approximate concepts associated with attributes and concepts of concept lattice (object oriented concept lattice) are also discussed.

Attribute reduction based on property pictorial diagram.

This paper mainly studies attribute reduction which keeps the lattice structure in formal contexts based on the property pictorial diagram. Firstly, the property pictorial diagram of a formal context is defined. Based on such diagram, an attribute reduction approach of concept lattice is achieved. Then, through the relation between an original formal context and its complementary context, an attribute reduct of complementary context concept lattice is obtained, which is also based on the property pictorial diagram of the original formal context. Finally, attribute reducts in property oriented concept lattice and object oriented concept lattice can be acquired by the relations of attribute reduction between these two lattices and concept lattice of complementary context. In addition, a detailed illustrative example is presented.

Multi-adjoint fuzzy rough sets: Definition, properties and attribute selection

This paper introduces a flexible extension of rough set theory: multi-adjoint fuzzy rough sets, in which a family of adjoint pairs are considered to compute the lower and upper approximations. This new setting increases the number of applications in which rough set theory can be used. An important feature of the presented framework is that the user may represent explicit preferences among the objects in a decision system, by associating a particular adjoint triple with any pair of objects.Moreover, we verify mathematical properties of the model, study its relationships to multi-adjoint property-oriented concept lattices and discuss attribute selection in this framework.

Multi-adjoint relation equations: Definition, properties and solutions using concept lattices

This paper generalizes fuzzy relation equations following the multi-adjoint philosophy. Moreover, the solutions of these general fuzzy relation equations and the concepts of a multi-adjoint property-oriented concept lattice are related, and several results are obtained from the theory of concept lattices.As a consequence of this relevant relation, more properties about these general equations can be proven from the theory of concept lattices and the algorithms developed to compute concept lattices can be used to obtain solutions. Furthermore, an interesting application to fuzzy logic programming has been introduced, in which an important problem in this topic has been interpreted in terms of solving a multi-adjoint relation equation.

Solving systems of fuzzy relation equations by fuzzy property-oriented concepts

This paper relates systems of fuzzy relation equations with the theory of fuzzy property-oriented concept lattices. Consequently, the results obtained from this theory of concept lattices can be transformed in order to obtain properties about the former. Moreover, the algorithms developed to obtain concept lattices can be applied to obtain solutions for systems of fuzzy relation equations.

The concept lattice functors

This paper is concerned with the relationship between contexts, closure spaces, and complete lattices. It is shown that, for a unital quantale L, both formal concept lattices and property oriented concept lattices are functorial from the category L-Ctx of L-contexts and infomorphisms to the category L-Sup of complete L-lattices and suprema-preserving maps. Moreover, the formal concept lattice functor can be written as the composition of a right adjoint functor from L-Ctx to the category L-Cls of L-closure spaces and continuous functions and a left adjoint functor from L-Cls to L-Sup.

Relating attribute reduction in formal, object-oriented and property-oriented concept lattices

Attribute reduction is an important step in reducing computational complexity in order to extract information from relational systems.Three of these systems are the formal, object-oriented and property oriented concept lattices. Attribute reduction in the last two concept lattices has recently been studied. The relation with the first concept lattice is very important since two important, independent tools to extract information from databases–the formal concept analysis and rough set theory–will be related. This paper studies attribute reduction in these three frameworks. The main results are that the classification of each attribute into absolutely necessary, relatively necessary and absolutely unnecessary attributes is independent of the framework considered and that an attribute reduct in one of these relational systems is also an attribute reduct in the others.

Relations of attribute reduction between object and property oriented concept lattices

As one of the basic problems of knowledge discovery and data analysis, knowledge reduction can make the discovery of implicit knowledge in data easier and the representation simpler. In this paper, relations of attribute reduction between object and property oriented formal concept lattices are discussed. And beautiful results are obtained that attribute reducts and attribute characteristics in the two concept lattices are the same based on new approaches to attribute reduction by means of irreducible elements. It turns out to be meaningful and effective in dealing with knowledge reduction, as attribute reducts and attribute characteristics in the object and property oriented formal concept lattices can be acquainted by only investigating one of the two concept lattices.

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Formal concept analysis and rough set are two kinds of efficient mathematical tools for data analysis and knowledge discovery. By combining these two theories, object-oriented and property-oriented concept lattices are proposed. Interval set theory is proposed to describe a partially-known concept by a lower bound and a upper bound. In order to obtain the more accurate extension and intension for a partially-known object-oriented concept, we introduce the theory of interval sets into the object-oriented concept lattice, and propose an object-oriented interval-set concept lattice. Properties of them are investigated. Relationships among interval-set concept lattices, object-oriented interval-set concept lattices and property-oriented interval-set concept lattices are discussed. By discussing the relationships between the object-oriented concept lattice and the object-oriented interval-set concept lattice, an approach to construct object-oriented interval-set concept lattices are established.