Introduction

Nowadays, conceptual (class) hierarchies are the most common complex knowledge representation structure within modern object-oriented knowledge-based systems and programming languages. It provides an opportunity to formalize a particular domain via constructing a corresponding hierarchy of concepts that encapsulates the representation of concepts themselves and the relations among them. The knowledge-based systems can use such hierarchies for conceptual knowledge processing, including representation, analysis, classification, integration, identification, retrieval, inferring, and transferring. Concept identification is one of the important tasks related to knowledge analysis, integration, retrieval, and inferring since it allows the system to detect a place of particular concepts and how they are connected with other concepts in the hierarchy. Consequently, a concept can be considered not only as a single node from the hierarchy but also as its neighborhood or sub-hierarchy, which includes some number of adjacent nodes and relations among them. Using such a representation, the system can operate by concepts in the broader meaning, for example, for computation of similarity or difference of certain concepts from the same hierarchy or a few different hierarchies. In addition, a hierarchical neighborhood of a concept can be used to reduce a hierarchy representation and consequently the search space during the concept retrieval or inferring, as well as for detecting the best place for integrating a new concept into a hierarchy.

Conceptual identification has a few interpretations depending on the specifics of a certain hierarchy, the nature of the concepts, and the relations among them. Since sometimes concepts themselves, as well as the relations among them, can be vague and imprecise, knowledge-based systems should be able to perform the identification of fuzzy concepts. Therefore, in this paper, we study the identification of fuzzy concepts in fuzzy object-oriented dynamic networks, considering the decomposition of fuzzy homogeneous classes of objects, which are nodes of the networks, as spaces for the identification of fuzzy sub-concepts. As a result, we propose a new approach to identifying semantically consistent fuzzy sub-concepts of fuzzy homogeneous classes of objects.

The paper has the following structure. Section 2 contains the analysis of the main approaches to the conceptual identification of fuzzy knowledge. Section 3 presents a morphological analysis of a particular fuzzy homogeneous class of objects. Section 4 provides an approach to reducing the space for identifying fuzzy subclasses via the semantically consistent decomposition of the fuzzy homogeneous class of objects. Section 5 presents the algorithm for identifying fuzzy sub-concepts and an example of its application to identifying semantically consistent subclasses of the fuzzy homogeneous class of objects. The conclusions and acknowledgments sections finish the paper.

Conceptual Identification

Nowadays, one of the most common approaches to formal representation of concept hierarchies is a formal concept analysis (FCA) proposed by Ganter and Wille in [5]. It is a powerful framework that focuses on representing concept hierarchies in terms of so-called, concept lattices. Such representation of the essences from a chosen domain consists of three main stages. Firstly, to define a formal context for the domain by constructing the crosstable with a set of attributes of domain entities as columns and the set of objects, which model their particular instances, as rows. The cells of such a table usually contain a Boolean value meaning that a particular object has or does not have a corresponding attribute. Secondly, to define formal concepts within the formal context by constructing a collection of pairs of appropriate extents and intents, where the extent is a set of common attributes for the particular set of objects, while the intent is a set of objects with common attributes. And finally, to build a corresponding concept lattice by constructing a complete lattice of objects and a complete lattice of attributes, which are isomorphic to each other. The lattice structure assumes that a set of formal concepts is a partially ordered set with defined subconcept-super-concept relations. Later, the FCA framework was extended for the representation of fuzzy domains (FFCA), consequently, notions of fuzzy formal context, fuzzy formal concepts, and fuzzy concept lattice were introduced [3,11]. In contrast to the crisp formal concepts, the fuzzy ones are defined using the confidence threshold, which allows modeling the membership measure of a particular attribute for a certain object, and a certain object for the set of objects. Constructing a concept lattice or fuzzy concept lattice for a particular domain creates its lattice-based formal model that can be used for conceptual identification.

According to [20], conceptual identification is the detection of the taxonomic position of a particular object within a certain classification. In the case of FCA/FFCA, the concept lattice is used as such classification, therefore identification of a concept transforms into the detection of sub-concepts and super-concepts within the lattice. One of the commonly used approaches to conceptual identification is rule-based identification. The main idea of the approach is to define a system of implication rules extracting them from the defined formal context and corresponding concept lattice. In general, an implication rule can be defined in the form PQ → , where P and Q are subsets of attributes of the set of all tributes used to determine a formal context or a fuzzy formal context. In the case of FCA, an implication rule can be interpreted in the following way -if an object has all attributes from the set P , it also has all attributes from the set Q . This approach was used to identify: a set of professional competencies that can help people successfully take a new position when professional retraining or changing jobs [15]; conservative access patterns, minimum behavior patterns, and canonical access patterns in two-mode social networks [13]. In the case of FFCA, an implication rule can be interpreted as follows -if a fuzzy object has all fuzzy attributes from the set P to the corresponding degree, then it also has all fuzzy attributes from the set Q to the corresponding degree. This version of the approach was used to identify: differential diagnoses for patients by a conversational recommender system [2]; causes and consequences of customer complaints within customer relationship management in financial services helping managers to accommodate the required dynamic changes according to customer expectations [14]; exceptional or suspicious cases specific to the event logs, NTFS file system, the Windows operating system, or a type of anomaly, to provide warnings for the security analysts [16][17]. However, the approach assumes the rules extraction from the formal context and corresponding concept lattice analyzing subconcept and super-concept relations. In the case of big formal contexts, this task becomes more complicated from the computational perspective. Moreover, to identify specific concepts within a concept lattice, the system must discover in the set of rules those rules that are associated with these concepts, including all transitive rules. Another approach to conceptual identification is the multi-stage intersection identification of formal concepts. The approach involves extracting new concepts via the sequence-based intersection of formal concepts within a constructed formal context. The discovered hidden concepts are identified and then integrated into the classification, where identification of the concepts means retrieval of their sub-concepts and super-concepts.

Such integration extends the initial formal context enriching it with previously non-obvious or hidden concepts. The approach was used to detect missing or hidden concepts and improve the completeness of concept coverage in biomedical terminologies NCI Thesaurus and SNOMED CT [23][24]. However, the larger the size of the formal context, the more difficult identification becomes due to the increasing number of intersections being calculated.

One more approach to conceptual identification is the criterion-based identification of a group of concepts. The main idea of the approach is to identify a group of concepts within a formal context or fuzzy formal context, which satisfies particular identification criteria. The FCA-version of the approach was used to identify: key nodes in massive networks using cross-face scalable centrality measure [9]; key nodes in a two-mode network, using bi-face bipartite centrality measure [10]; diversified topk maximal clique in a social Internet of things [8]; dynamic maximal clique in online social networks [21]; user-friendly communities in signed social networks and  -quasi-cliques for closely related users within them [22]; key structures from social networks [6]. The FFCA-version of the approach was used for the identification of location-based and content-based communities of users in social networks [4]; skyline ( ) , k  -cliques in a fuzzy attributed social network [7]; cloud services in collaborative filtering-based recommendation system [12].

Each of the considered approaches implements a specific strategy for solving the problem of conceptual identification based on FCA/FFCA. In all the mentioned FCA/FFCA applications, the formal context and the fuzzy formal context were constructed using a set of objects and a set of attributes. However, this approach has one major drawback: if we consider objects as instances of a particular class of objects, this means, that they are encapsulated containers for storing data, and we do not see how their attributes are defined relative to each other. It is known that the attributes of all class objects are defined at the class level, and, as was shown in [18][19], some attributes (properties and methods) of a class may have internal semantic dependencies on other attributes. It is crucial for the semantic consistency of formal concepts within the constructed concept lattice or fuzzy concept lattice because the construction of new formal concepts is based on the settheoretical intersection of extents ignoring internal semantic dependencies among the attributes [19]. Consequently, some constructed formal concepts may be semantically inconsistent and, therefore, physically impossible or unrealistic in a modeled domain. Another feature of FCA/FFCA is the fact that attributes are treated only as properties of objects, not as methods defined within an object class, and can be executed on all objects to change their state and attribute values. Therefore, we propose an alternative lattice-based approach to the conceptual identification of fuzzy knowledge, based on the analysis of internal semantic dependencies between the attributes (fuzzy properties and fuzzy methods) of fuzzy homogeneous classes of objects. In addition, we introduce the notion of concept' neighborhood, which allows the consideration of some subclass and superclass locus within a concept lattice instead of the single concept.

Fuzzy Concepts Morphology

To explain our approach to conceptual identification of fuzzy knowledge, we use the fuzzy homogeneous classes of objects which are nodes of fuzzy object-oriented dynamic networks. The formalization of internal semantic dependencies among the attributes (properties and methods) of fuzzy homogeneous classes of objects was introduced in [18]. For this purpose, the abstract model of chemical atoms and molecules was used, according to which, atoms are indivisible particles and molecules are the union of atoms and (or) smaller molecules. This model can be interpreted by attributes defined independently of other fuzzy class attributes (fuzzy atoms) and attributes defined based on them (fuzzy molecules).

Definition 1. A fuzzy atom of a fuzzy homogeneous class of objects( ) / T M T is a singleton collection ( ) ( ) ( )   / . / . i i i A T M T T x T x  = , where ( ) ( ) ( ) ( ) ( ) ( ) ( ) Definition 2.

A fuzzy molecule of a fuzzy homogeneous class of objects

) ( / T M T is a collection ( ) ( ) ( ) ( ) ( )   11 / . / .

M T M T T x T x T y T y T y T y   = , where ( ) ( ) ( ) ( ) ( ) ( ) ( ) . / . / / ii T x T x P T M P T F T M F T   , and

( )

1/ i T M T  is a

crisp or fuzzy property or a method defined based on the other methods and (or) properties

( )( ) ( ) ( ) ( ) ( ) ( ) ( ) 11 .( )( ) ( ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) (

)

1 1 1 2 2 2 34 56 1 1 2 2 3 , , /1, , ,/1,== = =  = =  == = ( ) ( )( ) ) 4 56

.98, _ , /

       − − − − − − − − − +  = −   −   − = − = − − = − − = +  ( ) ( ) ( ) ( ) ( ) 4,80 80 , 1 , 1 , 80 bbi i i i i i i i b tr.distance tion duration i tr.distance x x x x x x tr.distance duration        + + + + + + + +  +   − = − = − − = − − ( ) ( )   / , /1, / t i i t i i D x x duration x x  − − + + = , 1,... i = is a        − − − − − − − − − + + = −   −   − = − = − − = − − = +   +   = ( ) ( ) ( ) ( ) 50 , 1 , 1 , .50i i i i i i i t distance x x x x x tr distance duration       + + + + + + + − − = − − = − − and ( ) ( )   / , /1           − − − − − − − − + + + + + − = − = − − = − − = +   +   − = − = − − ( ) ( ) ( ) ( ), 1

;

i i i i x x x x    + + + + − = − 6 . / 0. =  ==   =  where ( ) ( )   / , /1, / b i i b i i P x x price x x  − − + + = , 1,... i = is a fuzzy set, such that ( ) ( ) ( ) ( ) ( ) . 20 . 60, 4 , .20 4 , . 20 , 1 , 1 , . 20 4 , 4 .       − − − − − − − − − +     = −    −   − = − = − − = − − = +   +   ( ) ( ) ( ) ( ) ( )60, . 60 , 1 , 1 ,

. 60

i i i i i i i i b stance tr distance x x x x x x tr distance price        + + + + + + + +  − = − = − − = − − ( ) ( )   / , /1   ( ) ( ) ( ) ( ) ( ) ( ) 2 , .15 2 , . 15 , 1 , 1 , . 15 2 , 2 . 35,        − − − − − − − − − + + + = −    −   − = − = − − = − − = +   +    − = ( ) ( ) ( ) ( ) , 1 , 1 , 35 i i i i i i t x x x x istance price       + + + + + + − = − − = − − and ( ) ( )   / , /1        − − − − − − − − − + + + = −    −   − = − = − − = − − = +   +    − = ( ) ( ) ( ) ( ) , 1 , 1 ; 85 i i i i i i p x x x x istance price       + + + + + + − = − − = − − 1 . /( )( ) ( ) ( ) ( ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) (

)

1M Jrn M Jrn M Jrn M Jrn M Jrn M Jrn M Jrn M Jrn M Jrn M Jrn   

The directed arrows mean the dependencies between a pair of attributes. Analyzing the structure of each internal semantic dependency, illustrated in Fig. 1, we can construct a dependency graph combining all of them, i.e.

f = ⊥         4 . / 0.84 . Jrn p 

The graph of internal semantic dependencies ( ) / 0.87 G Jrn is represented in Fig. 2. Violet nodes represent attributes of a fuzzy class / 0.87 Jrn , edges depict dependency relations among the attributes, and edge titles mean the numbers of the molecules, which contain corresponding dependencies. To simplify the graph, we denote its nodes using only attribute identifiers.

/ / , / , G SC T M SC A SC T M SC DL A SC T M SC = where ( ) ( ) ( ) ( ) ( ) // A SC T M SC A T M T  , () ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )// DL SC T M SC DL T M T  , satisfies the following conditions: ( )) ( ) ( ) ( ) ( ), / | , / / / , / | / / , / , / . u v DL T M T u v DL SC T M SC u A SC T M SC v A SC T M SC v A T M T v A SC T M SC u A T M T u v DL T M T u v DL SC T M SC    → →             → ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

Those subclasses whose internal semantic dependency graphs do not satisfy these conditions are semantically inconsistent.

Conceptual Identification Space Reducing

Let us compute the complete decomposition ( )

/= = = = = = ( ) ( ) ( ) ( ) ( ) ( ) ( )25= = = = ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )13= = = = ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )4= = = = ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )= = = = ( ) ( ) ( ) ( ) ( ) ( ) ( )4