Conceptual identification of fuzzy knowledge is one of the important knowledge-processing methods, which can be used for such tasks as concept matching, computation of concept similarity, re-engineering of conceptual hierarchies, etc. Since wildly used approaches to conceptual identification, which are based on the formal concept analysis and fuzzy formal concept analysis, do not consider the internal semantic dependencies among the attributes, it may lead to the construction of semantically inconsistent concepts. Therefore, in this paper, we propose a new approach to the conceptual identification of fuzzy knowledge within the decomposition of nodes of fuzzy object-oriented dynamic networks. The decomposition of fuzzy homogeneous classes of objects is considered the space for the identifying their fuzzy subconcepts within the corresponding identification lattice. To implement the proposed approach, we developed the algorithm for identifying semantically consistent subclasses of fuzzy homogeneous classes of objects. The algorithm constructs a semantically consistent lattice of fuzzy class subclasses and discovers all subclasses and superclasses for a selected fuzzy class subclass, creating a corresponding identification lattice. In addition, we introduce a notion of a subclass neighborhood within its identification lattice, which allows the consideration of a conceptual locus of the subclass instead of the subclass itself. It makes it possible to operate with subclasses of a fuzzy class in a broader sense, calculating their similarities and differences. To explain the proposed approach, we have provided a detailed example of the conceptual identification of a particular fuzzy homogeneous class of objects, demonstrating the application of the developed algorithm.
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, 0000-0003-7393-1426 (D. O. Terletskyi); 0000-0002-9895-777X (S. V. Yershov) © 2024 Copyright for this paper by its authors.
Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0). 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.
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.
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 [
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 P → Q , 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 [
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.
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.
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 [
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.
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].
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
Ai (T / M (T )) = T.xi / (T.xi ) , where T.xi / (T.xi ) P (T ) / M ( P (T )) F (T ) / M ( F (T )) is a crisp or fuzzy property or a method defined without using any other properties and (or) methods of the fuzzy class T / M (T ) , where P (T ) / M ( P (T )) and F (T ) / M ( F (T )) are collections of its properties and methods, respectively. collection Definition 2. A fuzzy molecule of a fuzzy homogeneous class of objects T / M (T ) is a
Mi (T / M (T )) = T.xi / (T.xi ) ,T.y j1 / (T.y j1 ),...,T.y jn / (T.y jn ) , where T.xi / (T.xi ) 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 T.y j1 / (T.y j1 ) ,...,T.y jn / (T.y jn ) P (T ) / M ( P (T )) F (T ) / M ( F (T )) which are crisp or fuzzy atoms and (or) parts of smaller fuzzy molecules of the fuzzy class T / M (T ) , where 1 j1 ... jn T / M (T ) , where P (T ) / M ( P (T )) and F (T ) / M ( F (T )) are collections of properties and methods of the fuzzy class T / M (T ) , respectively.
Definition 3. Internal semantic dependencies of a fuzzy homogeneous class of objects T / M (T ) is a set of its atoms and molecules, i.e.
ISD (T / M (T )) = A1 (T / M (T )) ,..., An (T / M (T )),
M1 (T / M (T )),..., M m (T / M (T )) where Ai (T / M (T )) , i = 1, n are fuzzy atoms of the fuzzy class T / M (T ) , while M j (T / M (T )) , j = 1, m are its fuzzy molecules, respectively.
defines the concept of a geographic place and has the following structure: Gp ( p1 = (latitude, ( , )) / 0.91, p2 = (longitude, ( , )) / 0.91, p3 = (name, (n Vn , str )) / 0.71, p4 = (region, (r Vr , str )) / 0.78, f1 = get _ latitude ( gp, ) / 0.95, f2 = get _ longitude ( gp, ) / 0.95, f3 = get _ name ( gp, str ) / 1, f4 = get _ region ( gp, str ) / 1) / 0.9 where Gp.p1 / 0.91 and Gp.p2 / 0.91 are fuzzy quantitative properties, which mean the latitude and longitude of the geographic place gp / ( gp) defined in degrees, i.e.
gp.latitude = / ( ), gp.longitude = / ( ); Gp.p3 / 0.71 is a fuzzy quantitative property which means the name of the geographic place gp / ( gp) , and is defined by the following fuzzy set
Vn = official _ name / 0.95, historical _ name / 0.74, regional _ name / 0.62, local _ name / 0.55, where elements of the fuzzy set Vn are corresponding names of the geographic place gp / ( gp) , i.e.
gp.name = n / (n) Vn ; Gp.p4 / 0.78 is a fuzzy quantitative property, which means the region name where the geographic place gp / ( gp) is located, and is defined by the following fuzzy set
Vr = official _ name / 0.95, historical _ name / 0.73, local _ name / 0.56 , where elements of the fuzzy set Vr are corresponding names of the region where the geographic place gp / ( gp) is located, i.e.
gp.region = r / (r ) Vr ; Gp. f1 / 0.95 , Gp. f2 / 0.95 , Gp. f3 /1 , and Gp. f4 /1 are fuzzy methods, which return values of corresponding properties of the geographic place gp / ( gp) , i.e.
gp.get _ latitude() → round ( gp.latitude, 1) , gp.get _ longitude() → round ( gp.longitude, 1) , gp.get _ name() → gp.name, gp.get _ region() → gp.region.
homogeneous class of objects Tr / 0.95 , which determines the concept of transfer from one geographic place to another and has the following structure:
Tr ( p1 = ( place1, ( gp1, Gp)) / 1, p2 = ( place2 , ( gp2 , Gp)) / 1, p3 = (distance, (dst, km)) / 0.94, p4 = (transport, (t Vt , str )) / 0.81, p5 = (duration, (dr, h)) / 0.87, p6 = ( price, ( p Vp , UAH )) / 0.91, f1 = get _ place1 (tr, GP) / 1, f2 = get _ place2 (tr, GP) / 1, f3 = get _ distance (tr, km) / 0.98, f4 = get _ transport (tr, str ) / 1, f5 = get _ duration (tr, h) / 0.94, f6 = get _ price (tr, UAH ) / 0.96) / 0.95 where Tr.p1 /1 and Tr.p2 /1 are fuzzy quantitative properties, which define two different geographic places for the transfer tr / (tr ) between them, i.e.
tr.place1 = gp1 / ( gp1 ), tr.place2 = gp2 / ( gp2 ) ; Tr.p3 / 0.94 is a fuzzy quantitative property, which means a distance between two geographic places Tr.p1 /1 and Tr. p2 / 1, and is defined in the following way tr.distance = d1 + d2 , d1 = (tr.place1.get _ latitude() − tr.place2.get _ latitude())2 , d1 = (tr.place1.get _ longitude() − tr. place2.get _ longitude())2 ; Tr.p4 / 0.81 is a fuzzy quantitative property, which means a kind of transport for a transfer between two geographic places and is defined as the following fuzzy set
Vt = bus / 0.82, train / 0.67, plane / 0.93, where elements of the fuzzy set Vt are possible kinds of transport for a transfer, i.e.
tr.transport = t / (t ) Vt ; Tr. p / 0.87 is a fuzzy quantitative property, which means duration of the transfer between 5 two geographic places and is defined in the following way
tr.duration = D , if tr.transport = train / 0.67,
t D , if tr.transport = bus / 0.82,
b Dp , if tr.transport = plane / 0.93, where D = xi− / ( x− ) , duration / 1, xi+ / ( x+ ) , i = 1,... is a fuzzy set, such that b i b i , x+ = duration + 4 i, durationb duration + 4 i i b b tr.distance 80
, − + , + = 1− ( x+ ) − ( x+ ) , ( x+ ) = 1− ( x+ ) ,
i i i i i i Dt = xi− / ( x− ) , durationt / 1, xi+ / ( x+ ) , i = 1,... is a fuzzy set, such that i i x+ = durationt + 3 i, durationt durationt + 3 i i ( x+ ) = i tr.distance − x+ 50
i tr.distance − durationt 50 − + , + = 1− ( x+ ) − ( x+ ) , ( x+ ) = 1− ( x+ ) ,
i i i i i i and D = xi− / ( x− ) , duration / 1, xi+ / ( x+ ) , i = 1,... is a fuzzy set, such that p i p i duration − 5 i duration , p p priceb − tr.distance 20 tr.distance 60 − x+
i tr.distance 60 − priceb x− − tr.distance 15 i pricet − tr.distance 15 tr.distance 35 − x + i tr.distance 35 − pricet x− 500 − tr.distance i duration 500 − tr.distance
p − + , + = 1− ( x+ ) − ( x+ ) , ( x+ ) = 1− ( x+ );
i i i i i i Tr. p / 0.91 is a fuzzy quantitative property, which means a price of a transfer between two 6 geographic places and is defined as follows
tr. price = Pt , if tr.transport = train / 0.67, P , if tr.transport = bus / 0.82,
b
P , if tr.transport = plane / 0.93; p where P = xi− / ( x− ) , priceb / 1, xi+ / ( x+ ) , i = 1,... is a fuzzy set, such that b i i
tr.distance 20 priceb tr.distance 60, x− = priceb − 4 i, tr.distance 20 priceb − 4 i priceb ,
i x+ = priceb + 4 i, priceb priceb + 4 i tr.distance 60, i − − , − = 1− ( x− ) − ( x− ) , ( x− ) = 1− ( x− ) ,
i i i i i i − + , + = 1− ( x+ ) − ( x+ ) , ( x+ ) = 1− ( x+ ) ,
i i i i i i Pt = xi− / ( x− ) , pricet / 1, xi+ / ( x+ ) , i = 1,... is a fuzzy set, such that i i
tr.distance 15 pricet tr.distance 35, x− = pricet − 2 i, tr.distance 15 pricet − 2 i pricet ,
i x+ = pricet + 2 i, pricet pricet + 2 i tr.distance 35, i − − , − = 1− ( x− ) − ( x− ) , ( x− ) = 1− ( x− ) ,
i i i i i i − + , + = 1− ( x+ ) − ( x+ ) , ( x+ ) = 1− ( x+ ) ,
i i i i i i and P = xi− / ( x− ) , price / 1, xi+ / ( x+ ) , i = 1,... is a fuzzy set, such that p i p i
tr.distance 55 pricep tr.distance 85, xi− = pricep − 3 i, tr.distance 55 pricep − 3 i pricep , ( xi− ) = ( xi+ ) =
xi− − tr.distance 55 pricep − tr.distance 55
tr.distance 85 − xi+ tr.distance 85 − pricep xi+ = pricep + 3 i, pricep pricep + 3 i tr.distance 85, − i− , i− = 1− ( xi− ) − ( xi− ) , ( xi− ) = 1− ( xi− ) , − i+ , i+ = 1− ( xi+ ) − ( xi+ ) , ( xi+ ) = 1− ( xi+ ); Tr. f1 / 1 , Tr. f2 / 1 , Tr. f3 / 0.98 , Tr. f4 / 1 , Tr. f5 / 0.94 , and Tr. f6 / 0.96 are fuzzy methods, which return values of corresponding properties of the transfer tr / (tr ) , i.e. tr.get _ place1() → tr. place1, tr.get _ place2 () → tr. place2 , tr.get _ distance() → round (tr.distance, 0) , tr.get _ transport() → tr.transport, tr.get _ duration() → tr.duration,
tr.get _ price() → tr. price.
which determines the concept of a journey through the sequence of geographic places and has the following structure:
Jrn ( p1 = (transfers, ((tr1, Tr ) ,..., (trn , Tr ))) / 0.93, p2 = (distannce, (dst, km)) / 0.88 p3 = ( duration, ( dr, h)) / 0.79, p4 = ( price, ( p, UAH )) / 0.84, f1 = get _ transfer ( jrn, i, Tr ) / 1, f2 = get_distance ( jrn, km) / 0.91, f3 = get _ duration ( jrn, h) / 0.89, f4 = get _ price ( jrn, UAH ) / 0.82, f5 = compute _ discount ( jrn, UAH ) / 0.78) / 0.87 where Jrn. p1 / 0.93 is a fuzzy quantitative property, defining a sequence of transfers between different geographic places in the scope of the journey jrn / ( jrn) , i.e.
jrn.transfers = (tr1 / (tr1 ),..., trn / (trn )), such that tri. place2 = tri+1. place1 , i = 1, n −1 ; Jrn. p2 / 0.88 is a fuzzy quantitative property, meaning the total distance of the transfer, during the journey jrn / ( jrn) , and is defined in the following way
n jrn.distance = jrn.transfers[i].get _ distance();
i=1 Jrn. p3 / 0.79 is a fuzzy quantitative property, meaning the total duration of the transfer, during the journey jrn / ( jrn) , and is defined in the following way n a jrn.duration = i=1 b , m a = ( jrn.transfers[i].get _ duration()[ j]) jrn.transfers[i].get _ duration()[ j], j=1
m b = ( jrn.transfers[i].get _ durtion()[ j]) ,
j=1 where n = jrn.transfers ,
m = jrn.transfers[i].duration ; Jrn. p4 / 0.84 is a fuzzy quantitative property, meaning the total price of the transfer, during the journey jrn / ( jrn) , and is defined in the following way
n a jrn. price = − jrn.compute _ diccount(),
i=1 b m a = ( jrn.transfers[i].get _ price()[ j]) jrn.transfers[i].get _ price()[ j], j=1
m b = ( jrn.transfers[i].get _ price()[ j]) ,
j=1 where n = jrn.transfers , m = jrn.transfers[i]. price ;
Jrn. f1 / 1 ,
Jrn. f2 / 0.91, Jrn. f3 / 0.89 , and Jrn. f4 / 0.82 are fuzzy methods, returning values for corresponding properties of the journey object jrn / ( jrn) , i.e.
jrn.get _ transfer(i) → jrn.transfers[i],
jrn.get _ distance() → jrn.distance, jrn.get _ duration() → round ( jrn.duration, 0) ,
jrn.get _ price() → round ( jrn. price, 1); Jrn. f5 / 0.87 is a fuzzy method, calculating a discount on the price of a journey, depending on its distance, and that is defined as follows where d is defined by the following set of rules 0.15, if jrn.transfer[i].distance 1000, 0.12, if jrn.transfer[i].distance 800,
Jrn. p1 / 0.93 meaning transfer between two geographic places, is defined without using any other class members, therefore, it determines a corresponding fuzzy atom of the fuzzy class Jrn / 0.87 , i.e.
A1 ( Jrn / 0.87) = Jrn. p1 / 0.93.
Fuzzy property Jrn. p2 / 0.88 , meaning the total distance of the transfer during the journey, is defined using the property Jrn. p1 / 0.93 , consequently, it determines a fuzzy molecule of the fuzzy class Jrn / 0.87 , i.e.
M1 ( Jrn / 0.87) = ( Jrn. p2 / 0.88, Jrn. p1 / 0.93).
Fuzzy property Jrn. p3 / 0.79 , meaning the total duration of the transfer during the journey, is defined using the property Jrn. p1 / 0.93 , therefore, it determines a fuzzy molecule of the fuzzy class Jrn / 0.87 , i.e.
M2 ( Jrn / 0.87) = ( Jrn. p3 / 0.79, Jrn. p1 / 0.93).
Fuzzy method Jrn. f1 / 1 , returning transfers list during the journey, is defined using the property Jrn. p1 / 0.93 , therefore, it determines a fuzzy molecule of the fuzzy class Jrn / 0.87 , i.e.
M3 ( Jrn / 0.87) = ( Jrn. f1 / 1, Jrn. p1 / 0.93).
Fuzzy method Jrn. f5 / 0.78 , returning the total price of the transfer during the journey, is defined using the property Jrn. p1 / 0.93 , therefore, it determines a fuzzy molecule of the fuzzy class Jrn / 0.87 , i.e.
M4 ( Jrn / 0.87) = ( Jrn. f5 / 0.78, Jrn. p1 / 0.93).
Fuzzy property Jrn. p4 / 0.84 , meaning the total price of the transfer during the journey, is defined using the property Jrn. p1 / 0.93 and fuzzy method Jrn. f5 / 0.78 , consequently, it determines a fuzzy molecule of the fuzzy class Jrn / 0.87 , i.e.
M5 ( Jrn / 0.87) = ( Jrn. p4 / 0.84, Jrn. f5 / 0.78, Jrn. p1 / 0.93).
Fuzzy method Jrn. f2 / 0.91, returning the total distance of the transfer during the journey, is defined using the property Jrn. p2 / 0.88 , consequently, it determines a fuzzy molecule of the fuzzy class Jrn / 0.87 , i.e.
M6 ( Jrn / 0.87) = ( Jrn. f2 / 0.91, Jrn. p2 / 0.88, Jrn. p1 / 0.93).
Fuzzy method Jrn. f3 / 0.89 , returning the discount on the price of a journey, is defined using the property Jrn. p3 / 0.79 , therefore, it determines a fuzzy molecule of the fuzzy class Jrn / 0.87 , i.e.
M7 ( Jrn / 0.87) = ( Jrn. f3 / 0.89, Jrn. p3 / 0.79, Jrn. p1 / 0.93).
Fuzzy method Jrn. f4 / 0.82 , returning the total price of the transfer during the journey, is defined using the properties Jrn. p4 / 0.84 , Jrn. p1 / 0.93 and fuzzy method Jrn. f5 / 0.78 , consequently, it determines a fuzzy molecule of the fuzzy class Jrn / 0.87 , i.e.
M8 ( Jrn / 0.87) = ( Jrn. f4 / 0.8, Jrn. p4 / 0.84, Jrn. f5 / 0.78, Jrn. p1 / 0.93).
ISD ( Jrn / 0.87) = A1 ( Jrn / 0.87), M1 ( Jrn / 0.87),..., M8 ( Jrn / 0.87).
find some similarities and intersections among them. To observe the connections among different dependencies we visualized them in Fig. 1. The orange nodes depict corresponding internal semantic dependencies, while the violet ones mean the attributes of the fuzzy class. As we can see, all molecules of the fuzzy class Jrn / 0.87 contain the atom A1 ( Jrn / 0.87) , i.e. A1 ( Jrn / 0.87) Mi ( Jrn / 0.87), where i = 1,8 , while the bigger molecules contain some of the smaller ones, i.e.
M1 ( Jrn / 0.87) M 6 ( Jrn / 0.87) , M 2 ( Jrn / 0.87) M 7 ( Jrn / 0.87) , M 4 ( Jrn / 0.87) M5 ( Jrn / 0.87) , M 4 ( Jrn / 0.87) M8 ( Jrn / 0.87) ,
M5 ( Jrn / 0.87) M8 ( Jrn / 0.87).
G ( Jrn / 0.87) = ( A( Jrn / 0.87), DL ( Jrn / 0.87)), where A( Jrn / 0.87) is a set of attributes of the fuzzy class Jrn / 0.87 , i.e.
A( Jrn / 0.87) = Jrn. p1 / 0.93, Jrn. p2 / 0.88, Jrn.p3 / 0.79, Jrn.p4 / 0.84, Jrn. f1 / 1,
Jrn. f2 / 0.91, Jrn. f3 / 0.89, Jrn. f4 / 0.82, Jrn. f5 / 078, and DL ( Jrn / 0.87) is a set of dependency links among the attributes of the fuzzy class Jrn / 0.87 , i.e.
DL ( Jrn / 0.87) = Jrn. p1 / 0.93 ⊥, Jrn. p2 / 0.88 Jrn. p1 / 0.93,
Jrn. p3 / 0.79 Jrn. p1 / 0.93, Jrn. f1 / 1 Jrn. p1 / 0.93, Jrn. f5 / 0.78 Jrn. p1 / 0.93, Jrn. p4 / 0.84 Jrn. f5 / 0.78, Jrn. f2 / 0.91 Jrn. p2 / 0.88, Jrn. f3 / 0.89 Jrn. p3 / 0.79,
Jrn. f4 / 0.82 Jrn. p4 / 0.84.
The graph of internal semantic dependencies G ( Jrn / 0.87) is represented in Fig. 2. Violet nodes represent attributes of a fuzzy class Jrn / 0.87 , 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.
to use graph-based interpretation of its subclasses. Therefore, each subgraph of graph G ( Jrn / 0.87) defines an appropriate subclass of the fuzzy class Jrn / 0.87 . However, as we mentioned above, not all subclasses are semantically consistent, i.e. do not contradict the internal semantic dependencies of a fuzzy class. Therefore, let us define the satisfiability of internal semantic dependencies for the fuzzy homogeneous class of objects. Definition 4. Any subclass SC (T ) / M (SC ) of a fuzzy homogeneous class of objects T / M (T ) is semantically consistent if only if its graph of internal semantic dependencies
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.
Let us compute the complete decomposition D ( Jrn / 0.87) of the fuzzy class Jrn / 0.87 ,
using an algorithm for decomposing fuzzy homogeneous classes of objects via
constraintbased filtering, proposed in [
(T / M (T ) = Jrn / 0.87, C = ISD ( Jrn / 0.87) , N = 1,...,8, M = [
SC11 ( Jrn) / 0.93 = ( p1 / 0.93) , SC12 ( Jrn) / 0.91 = ( p1 / 0.93, p2 / 0.88) , SC22 ( Jrn) / 0.86 = ( p1 / 0.93, p3 / 0.79) , SC72 ( Jrn) / 0.97 = ( p1 / 0.93, f1 / 1) , SC229 ( Jrn) / 0.86 = ( p1 / 0.93, f5 / 0.78) ,
SC13 ( Jrn) / 0.87 = ( p1 / 0.93, p2 / 0.88, p3 / 0.79) , SC53 ( Jrn) / 0.94 = ( p1 / 0.93, p2 / 0.88, f1 / 1) , SC63 ( Jrn) / 0.91 = ( p1 / 0.93, p3 / 0.79, f1 / 1) , SC131 ( Jrn) / 0.91 = ( p1 / 0.93, p2 / 0.88, f2 / 0.91) , SC232 ( Jrn) / 0.87 = ( p1 / 0.93, p3 / 0.79, f3 / 0.89) , SC537 ( Jrn) / 0.86 = ( p1 / 0.93, p2 / 0.88, f5 / 0.78) , SC538 ( Jrn) / 0.83 = ( p1 / 0.93, p3 / 0.79, f5 / 0.78) , SC630 ( Jrn) / 0.85 = ( p1 / 0.93, p4 / 0.84, f5 / 0.78) ,
SC633 ( Jrn) / 0.9 = ( p1 / 0.93, f1 / 1, f5 / 0.78) , SC24 (Jrn)/ 0.9 = ( p1 / 0.93, p2 / 0.88, p3 / 0.79, f1 /1), SC64 (Jrn)/ 0.88 = ( p1 / 0.93, p2 / 0.88, p3 / 0.79, f2 / 0.91), SC140 (Jrn)/ 0.93 = ( p1 / 0.93, p2 / 0.88, f1 /1, f2 / 0.91), SC146 (Jrn)/ 0.87 = ( p1 / 0.93, p2 / 0.88, p3 / 0.79, f3 / 0.89), SC241(Jrn)/ 0.9 = ( p1 / 0.93, p3 / 0.79, f1 /1, f3 / 0.89), SC741(Jrn)/ 0.85 = ( p1 / 0.93, p2 / 0.88, p3 / 0.79, f5 / 0.78), SC742 (Jrn)/ 0.86 = ( p1 / 0.93, p2 / 0.88, p4 / 0.84, f5 / 0.78), SC743 (Jrn)/ 0.84 = ( p1 / 0.93, p3 / 0.79, p4 / 0.84, f5 / 0.78), SC745 (Jrn)/ 0.9 = ( p1 / 0.93, p2 / 0.88, f1 /1, f5 / 0.78), SC746 (Jrn)/ 0.88 = ( p1 / 0.93, p3 / 0.79, f1 /1, f5 / 0.78), SC748 (Jrn)/ 0.89 = ( p1 / 0.93, p4 / 0.84, f1 /1, f5 / 0.78), SC841(Jrn)/ 0.88 = ( p1 / 0.93, p2 / 0.88, f2 / 0.91, f5 / 0.78), SC942 (Jrn)/ 0.85 = ( p1 / 0.93, p3 / 0.79, f3 / 0.89, f5 / 0.78),
SC1409 (Jrn)/ 0.84 = ( p1 / 0.93, p4 / 0.84, f4 / 0.82, f5 / 0.78), 16 subclasses of cardinality 5. i.e.
SC35 (Jrn)/ 0.9 = ( p1 / 0.93, p2 / 0.88, p3 / 0.79, f1 /1, f2 / 0.91), SC85 (Jrn)/ 0.9 = ( p1 / 0.93, p2 / 0.88, p3 / 0.79, f1 /1, f3 / 0.89), SC152 (Jrn)/ 0.88 = ( p1 / 0.93, p2 / 0.88, p3 / 0.79, f2 / 0.91, f3 / 0.89), SC557 (Jrn)/ 0.84 = ( p1 / 0.93, p2 / 0.88, p3 / 0.79, p4 / 0.84, f5 / 0.78), SC558 (Jrn)/ 0.88 = ( p1 / 0.93, p2 / 0.88, p3 / 0.79, f1 /1, f5 / 0.78), SC559 (Jrn)/ 0.89 = ( p1 / 0.93, p2 / 0.88, p4 / 0.84, f1 /1, f5 / 0.78), SC650 (Jrn)/ 0.87 = ( p1 / 0.93, p3 / 0.79, p4 / 0.84, f1 /1, f5 / 0.78), SC652 (Jrn)/ 0.86 = ( p1 / 0.93, p2 / 0.88, p3 / 0.79, f2 / 0.91, f5 / 0.78), SC653 (Jrn)/ 0.87 = ( p1 / 0.93, p2 / 0.88, p4 / 0.84, f2 / 0.91, f5 / 0.78), SC656 (Jrn)/ 0.9 = ( p1 / 0.93, p2 / 0.88, f1 /1, f2 / 0.91, f5 / 0.78), SC752 (Jrn)/ 0.85 = ( p1 / 0.93, p2 / 0.88, p3 / 0.79, f3 / 0.89, f5 / 0.78), SC754 (Jrn)/ 0.85 = ( p1 / 0.93, p3 / 0.79, p4 / 0.84, f3 / 0.89, f5 / 0.78), SC757 (Jrn)/ 0.88 = ( p1 / 0.93, p3 / 0.79, f1 /1, f3 / 0.89, f5 / 0.78), SC953 (Jrn)/ 0.85 = ( p1 / 0.93, p2 / 0.88, p4 / 0.84, f4 / 0.82, f5 / 0.78), SC954 (Jrn)/ 0.83 = ( p1 / 0.93, p3 / 0.79, p4 / 0.84, f4 / 0.82, f5 / 0.78),
SC959 (Jrn)/0.87 =( p1 /0.93, p4 /0.84, f1 /1f4 /0.82, f5 /0.78), 14 subclasses of cardinality 6. i.e.
SC46 (Jrn)/ 0.9 = ( p1 / 0.93, p2 / 0.88, p3 / 0.79, f1 /1, f2 / 0.91, f3 / 0.89), SC269 (Jrn)/ 0.87 = ( p1 / 0.93, p2 / 0.88, p3 / 0.79, p4 / 0.84, f1 /1, f5 / 0.78), SC360 (Jrn)/ 0.86 = ( p1 / 0.93, p2 / 0.88, p3 / 0.79, p4 / 0.84, f2 / 0.91, f5 / 0.78), SC361(Jrn)/ 0.88 = ( p1 / 0.93, p2 / 0.88, p3 / 0.79, f1 /1, f2 / 0.91, f5 / 0.78), SC362 (Jrn)/ 0.9 = ( p1 / 0.93, p2 / 0.88, p4 / 0.84, f1 /1, f2 / 0.91, f5 / 0.78), SC365 (Jrn)/ 0.85 = ( p1 / 0.93, p2 / 0.88, p3 / 0.79, p4 / 0.84, f3 / 0.89, f5 / 0.78), SC366 (Jrn)/ 0.88 = ( p1 / 0.93, p2 / 0.88, p3 / 0.79, f1 /1, f3 / 0.89, f5 / 0.87), SC368 (Jrn)/ 0.87 = ( p1 / 0.93, p3 / 0.79, p4 / 0.84, f1 /1, f3 / 0.89, f5 / 0.78), SC460 (Jrn)/ 0.86 = ( p1 / 0.93, p2 / 0.88, p3 / 0.79, f2 / 0.91, f3 / 0.89, f5 / 0.78), SC560 (Jrn)/ 0.84 = ( p1 / 0.93, p2 / 0.88, p3 / 0.79, p4 / 0.84, f4 / 0.82, f5 / 0.78), SC562 (Jrn)/ 0.88 = ( p1 / 0.93, p2 / 0.88, p4 / 0.84, f1 /1, f4 / 0.82, f5 / 0.78), SC563 (Jrn)/ 0.86 = ( p1 / 0.93, p3 / 0.79, p4 / 0.84, f1 /1, f4 / 0.82, f5 / 0.78), SC566 (Jrn)/ 0.86 = ( p1 / 0.93, p2 / 0.88, p4 / 0.84, f2 / 0.91, f4 / 0.82, f5 / 0.78), SC667 (Jrn)/ 0.84 = ( p1 / 0.93, p3 / 0.79, p4 / 0.84, f3 / 0.89, f4 / 0.82, f5 / 0.78),
SC97 (Jrn)/ 0.88 = ( p1 / 0.93, p2 / 0.88, p3 / 0.79, p4 / 0.84, f1 /1, f2 / 0.91, f5 / 0.78), SC170 (Jrn)/ 0.87 = ( p1 / 0.93, p2 / 0.88, p3 / 0.79, p4 / 0.84, f1 /1, f3 / 0.89, f5 / 0.78), SC171(Jrn)/ 0.86 = ( p1 / 0.93, p2 / 0.88, p3 / 0.79, p4 / 0.84, f2 / 0.91, f3 / 0.89, f5 / 0.78), SC172 (Jrn)/ 0.88 = ( p1 / 0.93, p2 / 0.88, p3 / 0.79, f1 /1, f2 / 0.91, f3 / 0.89, f5 / 0.78), SC176 (Jrn)/ 0.86 = ( p1 / 0.93, p2 / 0.88, p3 / 0.79, p4 / 0.84, f1 /1, f4 / 0.82, f5 / 0.78), SC177 (Jrn)/ 0.85 = ( p1 / 0.93, p2 / 0.88, p3 / 0.79, p4 / 0.84, f2 / 0.91, f4 / 0.82, f5 / 0.78), SC179 (Jrn)/ 0.88 = ( p1 / 0.93, p2 / 0.88, p4 / 0.84, f1 /1, f2 / 0.91, f4 / 0.82, f5 / 0.78), SC272 (Jrn)/ 0.85 = ( p1 / 0.93, p2 / 0.88, p3 / 0.79, p4 / 0.84, f3 / 0.89, f4 / 0.82, f5 / 0.78), SC275 (Jrn)/ 0.86 = ( p1 / 0.93, p3 / 0.79, p4 / 0.84, f1 /1, f3 / 0.89, f4 / 0.82, f5 / 0.78), and 4 subclasses of cardinality 8. i.e.
SC28 (Jrn)/0.88 = ( p1 /0.93, p2 /0.88, p3 /0.79, p4 /0.84, f1 /1, f2 /0.91, f3 /0.89, f5 /0.78), SC38 ( Jrn) / 0.87 = ( p1 / 0.93, p2 / 0.88, p3 / 0.79, p4 / 0.84, f1 / 1, f2 / 0.91, SC48 ( Jrn) / 0.87 = ( p1 / 0.93, p2 / 0.88, p3 / 0.79, p4 / 0.84, f1 / 1, f3 / 0.89,
of its semantically consistent decomposition D ( Jrn / 0.87) . Analyzing graphs of internal semantic dependencies for each constructed subclass of fuzzy class Jrn / 0.87 , we can see, that all of them are semantically consistent, according to Def. 4.
complete bounded lattice [21]. Therefore, the power set of the set of properties and methods of the fuzzy homogeneous class of objects Jrn / 0.87 is a poset ( PS ( Jrn / 0.87), ) , which defines a complete bounded lattice of subclasses of the fuzzy class Jrn / 0.87 . i.e.
L ( Jrn / 0.87) = ( PS ( Jrn / 0.87), , , , 0, 1), where and are the least upper bound (join) and the greatest lower bound (meet) operations, defined on the set PS ( Jrn / 0.87) i.e.
SC1 / M ( SC1 ) PS ( Jrn / 0.87) , SC2 / M ( SC2 ) PS ( Jrn / 0.87) → → SC1 / M ( SC1 ) SC2 / M ( SC2 ) PS ( Jrn / 0.87) ,
SC1 / M ( SC1 ) SC2 / M ( SC2 ) PS ( Jrn / 0.87) , and where 0 is the least element SC10 ( Jrn) / 0.0 , and 1 is the greatest elements SC19 ( Jrn) / 0.87 of the lattice, i.e.
SC / M ( SC ) PS ( Jrn / 0.87) → SC10 ( Jrn) / 0.0 SC / M ( SC ) = SC10 ( Jrn) / 0.0,
SC19 ( Jrn) / 0.87 SC / M ( SC ) = SC19 ( Jrn) / 0.87.
As we know, the cardinality of PS ( Jrn / 0.87) is equal to 2n = 29 = 512 , where n = Jrn / 0.87 , therefore, illustrating the Hasse diagram of such a lattice is a non-trivial task, because of its size. Therefore, let us construct and illustrate the tower of subclass lattice L ( Jrn / 0.87) , using the corresponding approach proposed in [20] (see Fig. 3). As we can see, Fig. 3 represents three objects similar to tower buildings, which consist of sections and floors of a certain capacity. The tower sections are vertical columns of floors represented by grey and lime circles. The capacity of all floors, within the particular section, is represented by yellow circles with a corresponding number. The tower of the subclass lattice L ( Jrn / 0.87) is depicted on the left side of Fig. 3. The circles colored in gray can be interpreted as the unlighted tower floors because they mean semantically inconsistent subclasses of the fuzzy homogeneous class of objects Jrn / 0.87 . The circles colored in lime have an opposite interpretation since they mean semantically consistent subclasses, detected by the decomposition algorithm.
The second and third towers depicted in the middle and on the right in Fig. 3 are towers of sublattices of the subclass lattice L ( Jrn / 0.87) , which contain only semantically inconsistent and only semantically consistent subclasses of the fuzzy homogeneous class of objects Jrn / 0.87 , respectively. Comparing the number of subclasses of both kinds, we can see, that there are only 71 semantically consistent proper non-empty subclasses among the 510 formally possible. In more detail, this comparison can be represented by Tab. 1. The first row of the table means the cardinality of subclasses, while the second and the third rows contain the number of all formally possible and all semantically consistent subclasses of the fuzzy homogeneous class of objects Jrn / 0.87 of certain cardinality. The fourth row represents the ratio third row to the second row in percent. According to [20], the decomposition consistency of the fuzzy homogeneous class of objects Jrn / 0.87 is approximately equal to 13.9 % , i.e. DC ( Jrn / 0.87) = It means that only 13.9 % of all possible proper non-empty subclasses of the fuzzy homogeneous class of objects Jrn / 0.87 are semantically consistent ones. Consequently, we can reduce the search space for the conceptual identification within the semantically consistent decomposition of a fuzzy homogeneous class of objects. To do this, let us construct the sublattice of the subclass lattice L ( Jrn / 0.87) , which contains only semantically consistent subclasses of the fuzzy class Jrn / 0.87 . According to the definition provided in [21], the sublattice of the lattice L is a subset X of L , such that a X , b X → a b X , a b X . It is obvious that the empty subclass SC10 ( Jrn) / 0.0 Jrn / 0.87 , as well as the subclass SC19 ( Jrn) / 0.87 Jrn / 0.87 , are semantically consistent subclasses of the fuzzy homogeneous class of objects Jrn / 0.87 , therefore the set of all its semantically consistent subclasses is defined as follows
CS ( Jrn / 0.87) = D ( Jrn / 0.87) SC10 ( Jrn) / 0.0, SC19 ( Jrn) / 0.87.
Therefore, a set of all semantically consistent subclasses CS ( Jrn / 0.87) PS ( Jrn / 0.87) of the fuzzy class of objects Jrn / 0.87 defines a carrier (CS ( Jrn / 0.87), ) of the consistent subclass lattice CSL ( Jrn / 0.87) , which is a sublattice of the subclass lattice L ( Jrn / 0.87) , since SC1 / M ( SC1 ) CS ( Jrn / 0.87) , SC2 / M ( SC2 ) CS ( Jrn / 0.87) → → SC1 / M ( SC1 ) SC2 / M ( SC2 ) CS ( Jrn / 0.87) ,
SC1 / M ( SC1 ) SC2 / M ( SC2 ) CS ( Jrn / 0.87) , where and are the least upper bound (join) and the greatest lower bound (meet) operations, defined on the set CS ( Jrn / 0.87) . In addition, the carrier of the sublattice CSL ( Jrn / 0.87) contains the least element SC10 ( Jrn) / 0.0 , and the greatest element SC19 ( Jrn) / 0.87 , which makes it the a bounded one, since
SC / M ( SC ) CS ( Jrn / 0.87) → → SC01 ( Jrn) / 0.0 SC / M ( SC ) = SC01 ( Jrn) / 0.0,
SC19 ( Jrn) / 0.87 SC / M ( SC ) = SC19 ( Jrn) / 0.87.
homogeneous class of objects Jrn / 0.87 is defined as follows
CSL ( Jrn / 0.87) = (CS ( Jrn / 0.87), , , , 0, 1), where 0 means the least element, while 1 means the greatest element of the lattice. The Hasse diagram of the lattice CSL ( Jrn / 0.87) is depicted in the Fig. 4. The superscript of each lattice node means the cardinality of the corresponding subclass of the fuzzy class
subclasses of such cardinality.
Constructing the sublattice CSL ( Jrn / 0.87) allows us to reduce the subclass identification space by 512 / 73 7 times and to perform the subclass identification, analyzing only semantically consistent subclasses of the fuzzy class Jrn / 0.87 .
fuzzy homogeneous classes of objects via constraint-based filtering, we modified the corresponding decomposition algorithm, proposed in [19], adding subclasses and superclasses detection procedures (see Algorithm 1). The main idea of the algorithm is to detect sets of all semantically consistent subclasses S (SC / M ( SC )) and superclasses S (SC / M ( SC )) for the selected semantically consistent subclass SC / M ( SC ) of the fuzzy homogeneous class of objects T / M (T ) . As the input parameters, the algorithm uses the following: a fuzzy homogeneous class of objects T / M (T ) as a space for the fuzzy knowledge identification; a semantically consistent subclass SC / M ( SC ) of the fuzzy class T / M (T ) as an identification target; a set of constraints C = ISD (T / M (T )) defined by molecules of the fuzzy class T / M (T ) , to detect its semantically consistent subclasses; a required accuracy for computation of the measure of fuzziness of subclasses and superclasses of the class SC / M ( SC ) .
into a few successive stages. In the first stage, the algorithm constructs all formally possible subclasses of the fuzzy class T / M (T ) , which have a cardinality greater or less than the cardinality of the subclass SC / M ( SC ) . This reduces the identification space again, avoiding subclasses of the same cardinality as the SC / M ( SC ) subclass, since such subclasses definitely are not superclass or subclass of the subclass SC / M ( SC ) .
Require: T / M (T ) , SC / M ( SC ) , C , Ensure: S , S
fuzzy homogeneous class of objects T / M (T ) , among previously generated, performing the constraint-based filtering, using Procedure 1. It verifies the satisfiability of each constraint c C , defined by molecules of the fuzzy class T / M (T ) , for the subclass SC / M ( SC ) . In general, the subclass SC / M ( SC ) can satisfy or not satisfy the constraint c C as well as the constraint can be inapplicable to the subclass, therefore, the procedure can return as a value true, false, or none, respectively. In the third stage, the algorithm computes the fuzziness for each semantically consistent potential subclass or superclass of the class SC / M ( SC ) , using Procedure 2. After that, the algorithm verifies the subclass and (or) superclass relation between the detected semantically consistent subclasses of the fuzzy class T / M (T ) and subclass SC / M ( SC ) , using Procedure 3 and Procedure 4, respectively.
Procedure 1. is_satisfy (t, c) Input: t , c Output: satisfy → { true, false, none} 1: satisfy := none; 2: if c0 t then 3: 4: 5: 6: 3: 4: satisfy := false; for i 1,..., c do for all a / (a) ci do
if a / (a) t then 7: satisfy := true; 8: else 9: satisfy := false; 10: break; 11: if satisfy then 12: return satisfy; 13: return satisfy.
Input: t , Output: M (t ) → 0, 1 1: sum := 0; 2: for all a / (a) t do
sum := sum + (a); 5: return M (t ).
Procedure 2. compute_fuzziness (t, ) M (t ) := round ( sum / max ( t , 1), );
Procedure 3. is_subclass (SC / M (SC ), t / M (t )) Input: SC / M ( SC ) , t / M (t ) Output: is_subclass →{true, false} 1: if SC / M ( SC ) t / M (t ) then 2: return false; 3: for all a / (a) SC / M (SC ) do 4: if a / (a) t / M (t ) then 5: return false; 6: return true.
Procedure 4. is_superclass (SC / M (SC ), t / M (t )) Input: SC / M ( SC ) , t / M (t ) Output: is_superclass →{ true, false} 1: if SC / M ( SC ) t / M (t ) then 2: return false; 3: for all a / (a) t / M (t ) do 4: if a / (a) SC / M (SC ) then 5: return false; 6: return true.
subclasses S (SC / M ( SC )) and superclasses S (SC / M ( SC )) for the subclass SC / M ( SC ) .
us apply the proposed approach to the subclass SC269 ( Jrn) / 0.87 of the fuzzy homogeneous class of objects Jrn / 0.87 , described in the previous section. Consequently, we obtain the following sets of proper non-empty subclasses
S ( SC269 ( Jrn) / 0.87) = SC557 ( Jrn) / 0.84, SC558 ( Jrn) / 0.88, SC559 ( Jrn) / 0.89, SC650 ( Jrn) / 0.87, SC24 ( Jrn) / 0.9, SC741 ( Jrn) / 0.85, SC742 ( Jrn) / 0.86, SC743 ( Jrn) / 0.84, SC745 ( Jrn) / 0.9, SC746 ( Jrn) / 0.88, SC748 ( Jrn) / 0.89, SC13 ( Jrn) / 0.87, SC53 ( Jrn) / 0.94, SC63 ( Jrn) / 0.91, SC537 ( Jrn) / 0.86, SC538 ( Jrn) / 0.83, SC630 ( Jrn) / 0.85, SC633 ( Jrn) / 0.9, SC12 ( Jrn) / 0.91, SC22 ( Jrn) / 0.86, SC72 ( Jrn) / 0.97, SC229 ( Jrn) / 0.86, SC11 ( Jrn) / 0.93, and superclasses
S ( SC269 ( Jrn) / 0.87) = SC7 ( Jrn) / 0.88, SC170 ( Jrn) / 0.87, SC176 ( Jrn) / 0.86, 9 As was noted above, subclasses SC0 ( Jrn) / 0.0 , and SC9 ( Jrn) / 0.87 are semantically 1 1 consistent subclasses of the fuzzy class Jrn / 0.87 , therefore the identification space for the subclass SC269 ( Jrn) / 0.87 is defined as follows
IS269 ( Jrn / 0.87) = S ( SC269 ( Jrn) / 0.87) S ( SC269 ( Jrn) / 0.87) Thus, the identification space IS269 ( Jrn / 0.87) of the subclass SC269 ( Jrn) / 0.87 defines the a sublattice of the consistent subclass lattice CSL ( Jrn / 0.87) and subclass lattice L ( Jrn / 0.87) , since IS269 ( Jrn / 0.87) CS ( Jrn / 0.87) PS ( Jrn / 0.87) and SC / M ( SC ) IS269 ( Jrn / 0.87) , SC / M ( SC ) IS269 ( Jrn / 0.87) → 1 1 2 2 where and are the least upper bound (join) and the greatest lower bound (meet) operations, defined on the set IS ( SC269 ( Jrn) / 0.87) . Since the carrier of the lattice ISL6 ( Jrn / 0.87) contains the least element SC0 ( Jrn) / 0.0 , and the greatest element 29 1 SC9 ( Jrn) / 0.87 , it makes the identification sublattice ISL6 ( Jrn / 0.87) a bounded one, 1 29 since
SC / M ( SC ) IS269 ( Jrn / 0.87) → Thus, the identification lattice for the subclass SC269 ( Jrn) / 0.87 of the fuzzy homogeneous class of objects Jrn / 0.87 is defined as follows
ISL629 ( Jrn / 0.87) = ( IS269 ( Jrn / 0.87), , , , 0, 1), Hasse diagram of the lattice ISL629 ( Jrn / 0.87) is depicted in the Fig. 5. The yellow node means the subclass SC269 ( Jrn) / 0.87 , while the lime nodes represent its subclasses and superclasses.
Definition 5. The neighborhood of the subclass SC / M ( SC ) of a fuzzy homogeneous class of objects T / M (T ) is a pair N (SC / M ( SC )) = (Sn, Sn ), where
Sn = SC1 / M ( SC1 ),..., SCk / M (SCk ),
Sn IS ( SC / M ( SC )) , S = SC1 / M ( SC ),..., SC / M ( SC ), n 1 w w are sets of subclasses and superclasses of the subclass SC / M ( SC ) such that SCi / M ( SC ) S , i = 1, k → SCi / M (SC ) SC / M (SC ) ,
i n i SC / M ( SC ) S , j = 1, w → SC / M ( SC ) SC / M (SC ) ,
j j n j j and where IS (SC / M (SC )) is a carrier of the subclass identification lattice ISL (SC / M (SC )) . measure.
Definition 6. The neighborhood measure of the subclass SC / M ( SC ) of a fuzzy and n are subclass and superclass neighborhood, respectively, and defined in the following way n = n =
S ( SC / M ( SC ))
n S ( SC / M ( SC )) S ( SC / M ( SC ))
n S ( SC / M ( SC )) , .
subclass SC269 ( Jrn) / 0.87 . Suppose that neighborhood of the subclass SC269 ( Jrn) / 0.87 is defined in the following way S = SC559 ( Jrn) / 0.89, SC650 ( Jrn) / 0.87, SC748 ( Jrn) / 0.89, n S = SC7 ( Jrn) / 0.88, SC176 ( Jrn) / 0.86, SC8 ( Jrn) / 0.87,
n 9 3 then its neighborhood measure is N (SC269 ( Jrn) / 0.87) = (0.13, 0.43 ) , where n =
S ( SC269 ( Jrn) / 0.87)
n S ( SC269 ( Jrn) / 0.87)
0.13, n =
Sn ( SC269 ( Jrn) / 0.87) S ( SC269 ( Jrn) / 0.87) It is clear that n 0,1 and n 0,1 , therefore if their values are closer to 0 , this means that the subclass and superclass neighborhood is low, but if it is closer to1 then their neighborhood is high. Using a neighborhood N (SC269 ( Jrn) / 0.87) of the subclass SC269 ( Jrn) / 0.87 we can consider a subclass locus defined by its subclasses and superclasses instead of the subclass itself.
In this paper, we analyzed known approaches to the conceptual identification of fuzzy knowledge using a formal concept analysis and its fuzzy extension. Since the FCA/FFCAbased computation of formal concepts can construct semantically inconsistent concepts, which are impossible or unreal within a modeled domain, we proposed another new latticebased approach to the conceptual identification of fuzzy knowledge. We study the conceptual identification of subclasses within the decomposition of fuzzy homogeneous classes of objects. The proposed approach allows us to identify semantically consistent subclasses within the decomposition of a fuzzy homogeneous class of objects constructing corresponding sub-class lattice. Such lattice is considered as a space for conceptual identification of any of its elements via detection of its subclasses and superclasses. Identification of a specific subclass involves the construction of a corresponding identification lattice, which is a sub-lattice of the subclass lattice. To implement the approach, we developed the corresponding identification algorithm, extending the algorithm for the decomposition of fuzzy homogeneous classes of objects via constraintbased filtering, proposed in [18]. As a result, the algorithm constructs all semantically consistent subclasses of a fuzzy homogeneous class of objects and then, verifies the subclass-superclass relation between each of them and a subclass, which needs to be identified.
To demonstrate the conceptual identification of fuzzy knowledge using the developed algorithm, we provided an example of conceptual identification of a semantically consistent subclass of a fuzzy homogeneous class of objects, which defines a fuzzy concept of a journey through the sequence of geographic places. To visualize the identification process, the corresponding identification lattice was constructed. Using this lattice, we introduced the notions of subclass neighborhood and its measure. The proposed approach can be extended for the conceptual identification of fuzzy knowledge within the fuzzy conceptual hierarchies and scaling of big concept lattices.
0123U103273 Development of Algorithms and Software Tools for the Analysis of Object