Analysis of relevance, similarity, and the diference between extracted or acquired new knowledge items and previously obtained ones are important stages of the knowledge integration process for modern knowledge-based systems. These stages can be performed via the application of special operations deifned over the knowledge representation structures provided by chosen representation model. Within the object-oriented approach, classes are one of the main knowledge representation structures, consequently, the analysis of the relevance, similarity, and diference between them requires the development of corresponding operations over classes. Therefore the concept of universal diference exploiter of fuzzy homogeneous classes of objects, within such a knowledge representation model as fuzzy objectoriented dynamic networks, was introduced in the paper. To implement the proposed exploiter, which allows computing of the diference of one fuzzy homogeneous class of objects from another one, we developed the corresponding algorithm and provided an example of its application. The proposed approach provides an opportunity to evaluate the relevance and novelty level for extracted or acquired fuzzy knowledge items, compared with previously obtained ones, by computing the diference between them.
The analysis of knowledge structures for their proper integration into the knowledge base and organizing future reasoning processes is a crucial and important task for modern knowledgebased systems (KBSs). To manage such a challenge, a KBS should be able to perform the comparative analysis of extracted or acquired new knowledge items with previously obtained ones. It allows a system to estimate the level of novelty for extracted knowledge items as well as to conclude about their similarity and diferences with other knowledge items, which are already integrated within a knowledge base. Taking into account such parameters, it is possible to integrate new extracted or acquired knowledge items into the knowledge base avoiding their representation redundancy.
According to the concept of knowledge integration proposed by Murray and Porter in [
3. Adaptation: Resolving of previously detected abnormalities within the knowledge base. The model was implemented within the knowledge acquisition tool KI for management of a large Botany Knowledge Base, which incorporates knowledge about plant anatomy, physiology, and development.
The recognition stage of the knowledge integration process requires performing the verification of such relations between new and previously obtained knowledge items as equivalence, inclusion, similarity, diference , etc. However, each knowledge representation model provides a suite of particular representation structures and corresponding tools for their management. Therefore knowledge analysis approaches are oriented toward certain knowledge representation paradigms or models. Within the object-oriented knowledge representation approach such structures as objects, classes, metaclasses, and relations among them, form a basic representation suite. Consequently, knowledge analysis within such an approach can be interpreted as a comparison of corresponding representation structures. Since a class is a widely-used representation structure, thus the comparative analysis of classes is one of the important and priority knowledge analysis tasks within the object-oriented approach. One of the known approaches to solving this task is to define the special operations over classes, in particular set-theoretical ones. According to the approach, a class is considered as a collection of properties and methods, consequently, the application of basic set-theoretical operations such as union, intersection, diference , and symmetric diference , provides a corresponding framework for the analysis of knowledge items, which are represented in terms of classes.
However, diferent object-oriented knowledge representation models use distinct concepts of a class, consequently, all mentioned set-theoretical operations should be adapted for a particular knowledge representation model and appropriate interpretation of a class. In addition, the classical (crisp) object-oriented paradigm has some representation restrictions, which limit its application for modeling vague, imprecise as well as uncertain entities and domains. Therefore in this paper, we adapted the set-theoretical diference for such knowledge representation model as fuzzy object-oriented dynamic networks in the form of the corresponding universal exploiter of fuzzy homogeneous classes of objects. To implement the proposed universal diference exploiter, we developed an appropriate algorithm and provided a representative example of its application for the knowledge analysis within fuzzy object-oriented dynamic networks.
A class is one of the main representation structures in class-based object-oriented programming, in a variety of object-oriented knowledge representation models as well as in object-oriented databases. According to the classical (crisp) definition, a class can be interpreted as the collection of properties (specification) and collection of methods (signature), which define a common structure and typical behavior for all instances of the class, respectively.
However, as it was noted in [
Many of the introduced interpretations of fuzzy types and fuzzy classes were proposed as fuzzy extensions of the object-oriented entity-relationship model, used in databases, where attributes of a class are considered separately from each other. However, such interpretation of a class or a type is distinguished from corresponding interpretation within the class-based object-oriented programming and many object-oriented knowledge representation models, where properties and methods of a class can have internal dependencies from each other. As it was shown in [21], the internal structure of a class consists of structural and functional atoms as well as structural and functional molecules, created by properties and methods of a class. It specifies the main diference between the concept of a class within object-oriented programming as well as knowledge representation and similar concept within object-oriented databases.
Another extension of the class-based object-oriented model was implemented within such knowledge representation model, as fuzzy object-oriented dynamic networks (FOODNs), which was proposed in [22, 23] and later extended in [24, 25]. Similar to other object-oriented models, such concepts as fuzzy objects, fuzzy classes, and fuzzy relations are also used within the FOODNs. The structure of a fuzzy class also defined by a collection of crisp and (or) fuzzy properties, while its behavior is determined by a collection of crisp and (or) fuzzy methods. However, in contrast to other object-oriented models, the specification of a fuzzy class of objects consists of quantitative and qualitative properties. Quantitative properties represent evident numerical or symbolic single-valued or multi-valued characteristics, while qualitative properties represent more complex, not obvious features defined based on other properties and methods of the class [24, 25]. The signature of a fuzzy class of objects consists of methods, which define a common behavior and opportunities to check and (or) to modify the structure of a particular instance of the class. Similar to other object-oriented models, each fuzzy class within FOODNs defines the particular fuzzy type, which identifies a common structure and behavior for all fuzzy objects of the class. Therefore such classes can be called homogeneous ones. Let us consider the definition of the fuzzy homogeneous class of objects within FOODNs introduced in [24, 25].
A fuzzy homogeneous class of objects is a collection
/ ( ) = ( (), ())/ ( ) = ((1()/ (1()), . . . , ()/ (())), (1()/ (1()), . . . , ()/ (())))/ ( ), where is a fuzzy type which is defined by the class , ()/ (()) ∈ () is a crisp or fuzzy property of the class , ()/ ( ()) ∈ () is its crisp or fuzzy method, (()) : () → (0, 1] and ( ()) : () → (0, 1] are measures of fuzziness of a property () and a method (), and ( ) is a measure of fuzziness of the class , i.e.
( ) = (( (1()) + · · · + (())) + ( (1()) + · · · + (())))/( + ).
Analyzing Definition 1, we can see that in the context of fuzziness of attributes and methods,
a concept of a fuzzy homogeneous class of objects is similar to the concept of the fuzzy type
described in [
As it was mentioned above, to perform knowledge integration in the proper way, a KBS should
be able to analyze the relevance, similarity, and diferences between extracted or acquired new
knowledge items and previously obtained ones, which are represented in terms of fuzzy classes.
Therefore we propose to consider set-theoretical diference adapted for fuzzy classes as a tool
for dynamic creation of new fuzzy classes of objects which represent of diference of one fuzzy
class from another one. The idea to apply set-theoretical operations to fuzzy classes, fuzzy
objects as well as fuzzy relations is widely used in many interpretations. As the result, it was
used to implement algebraic operations for fuzzy object-oriented database language [26], query
processing within the fuzzy relational object-oriented databases [
All mentioned algebras provide diferent signatures, however, all of them contain an adapted operation of set-theoretical diference, defined over fuzzy objects, or fuzzy classes, or fuzzy relations. However, in most cases, it is used for database querying on the level of objects, classes, and relations, when the data or knowledge are already integrated within the database or knowledge base. Therefore we propose to consider the application of diference operation, defined over the fuzzy homogeneous classes of objects, in the context of knowledge analysis as a part of the knowledge integration process.
The concept of diference of classes of objects was introduced in [ 22, 23] in a form of a corresponding universal exploiter for classes of fuzzy objects. Later the concept of a fuzzy class of objects was proposed in [24, 25]. Let us define the notion of diference exploiter for fuzzy homogeneous classes of objects via generalizing its versions for the classes of fuzzy objects.
Diference 1 ∖ 2 of two fuzzy homogeneous classes of objects 1/ (1) and 2/ (2), which define fuzzy types of objects 1 and 2 correspondingly, is a fuzzy homogeneous class of objects 1∖2/ (1∖2), which define fuzzy type of objects 1∖2, such that 1∖2 ⊆ 1 | ∄ (︀ 1∖2 ∩ 2)︀ ∧ ︁(
The fuzzy class 1∖2/ (1∖2) exists if and only if ∃1 (1), ∃2 (2), such that 1 (1) ̸≡ 2 (2), or ∃1 (1), ∃2 (2), such that 1 (1) ̸≡ 2 (2), where 1 (1) is an 1-th property of the fuzzy type 1, 1 = 1, | (1)|, 2 (2) is an 2-th property of the fuzzy type 2, 2 = 1, | (2)|, 1 (1) is an 1-th method of the fuzzy type 1, 1 = 1, | (1)|, 2 (2) is an 2-th method of the fuzzy type 2, 2 = 1, | (2)|. typical only for the fuzzy type of objects 1.
The universal diference exploiter creates a fuzzy homogeneous class of objects 1∖2/ (1∖2), which represents the diference of the fuzzy class of objects 1/ (1) from the fuzzy class of objects 2/ (2) if such diference exists. The class 1∖2/ (1∖2 objects 1∖2 ⊆ 1, which consists of crisp and (or) fuzzy properties and (or) methods, which are ) defines a fuzzy type of
To implement a universal exploiter of fuzzy homogeneous classes of objects, the corresponding algorithm should analyze specifications and signatures of fuzzy homogeneous classes of objects 1/ (1), 2/ (2) and find properties and methods of the fuzzy class 1/ (1), which are not typical for the fuzzy class of objects 2/ (2). For this purpose, we used the [25, Algorithm 1] for checking the equivalence of fuzzy quantitative properties, [25, Algorithm 2] for checking the equivalence of fuzzy qualitative properties, and [25, Algorithm 3] for checking the equivalence of fuzzy methods. As the result, we developed Algorithm 1, which implements the idea of universal diference exploiter for fuzzy homogeneous classes of objects.
Analyzing Algorithm 1, we can see that it uses fuzzy homogeneous classes of objects 1/ (1), 2/ (2) as the input data and computes the diference of the class 1/ (1) from the class 2/ (2) in a form of a new fuzzy homogeneous class of objects 1∖2/ (1∖2) = 1/ (1) ∖ 2/ (2) if such diference exists. The algorithm successively constructs the specification and signature of class
1∖2/ (1∖2) and these stages are independent ones, consequently, such computations also can be performed in parallel mode. The polymorphic function is_equivalent checks the equivalence of two fuzzy properties (1)/ ((1)), (2)/ ( (2)) or methods (1)/ ((1)), (2)/ ( (2)) and if they are equivalent ones it returns 1, otherwise it returns 0. It can be implemented in various ways using corresponding algorithms for checking the equivalence of fuzzy properties and methods, which were proposed in [25].
Let us estimate the time and space complexity of Algorithm 1. As we can see, during the analysis of specifications and signatures of classes 1/ (1) and 2/ (2) the algorithm checks the equivalence of | (1)| × | (2)| = × properties and | (1)| × | (2)| = × methods. In addition, to construct the specification (1∖2) and signature (1∖2) of the fuzzy class 1∖2/ (1∖2), it performs copying of 1 properties and 2 methods of the fuzzy class 1/ (1), which are not typical for the fuzzy class 2/ (2), where 0 ≤ 1 ≤ | (1)| and 0 ≤ 2 ≤ | (1)|. Therefore, the time complexity of Algorithm 1 is equal to ( × ) + ( × ) + (1) + (2) ≈ (2 + 2 + 1 + 2), where 2 is a number of properties equivalence checks, 2 is a number of methods equivalence checks, 1 and 2 is a number of copying operations of properties and methods of fuzzy class 1/ (1). To perform main computations Algorithm 1 uses units of memory for storing fuzzy type 1∖2, therefore its space complexity is equal to (), where 0 ≤ ≤ | (1)| + | (1)|.
Let us consider a few fuzzy homogeneous classes of objects, represented in terms of fuzzy
object-oriented dynamic networks, which simultaneously have equivalent and nonequivalent
subclasses. Let us suppose that the first fuzzy homogeneous class of objects
defines the fuzzy type ℎ1, which describes a fuzzy concept of a home fridge, which has the
following representation:
(
1 = ( _, ( ∈ (_), ))/1,
2 = ( _, ( ∈ ( _), ))/1,
5 = (, ((1 ∈ ℎℎ, ), (2 ∈ ℎ, ), (3 ∈ ℎ, ))/1,
6 = (, (6( .), ∈ [
(_) = { , , , , }, where fuzzy variables very small, small, medium, big, and very big are defined over the interval of integer numbers = [40, 425], which means the volume of the refrigerator in 3, and have the following interpretation:
( ) = {40/1 + 50/0.95 + 60/0.85 + 70/0.7 + 80/0.65} 3, () = {95/0.92 + 110/0.78 + 125/0.63 + 140/0.55 + 150/0.45} 3, () = {170/0.78 + 190/0.92 + 210/1 + 230/0.92 + 250/0.78} 3, () = {270/0.82 + 290/0.94 + 310/1 + 330/0.94 + 350/0.82} 3, ( ) = {365/0.65 + 380/0.72 + 395/0.86 + 410/0.93 + 425/1} 3; freezer volume is a fuzzy quantitative property defined as a linguistic variable, which has the following term-set
( _) = { , , , , },
where fuzzy variables very small, small, medium, big, and very big are defined over the interval
of integer numbers = [
= {, ℎ, ℎ, , , , , , , , , , , , , , , }; weight is a fuzzy quantitative property defined as a linguistic variable, which has the following term-set
(ℎ) = {ℎℎ, , ℎ, ℎ},
where fuzzy variables lightweight, medium, heavy, and very heavy are defined over the interval
of integer numbers = [
() = {, , ℎℎ}, where fuzzy variables low, medium, and high are defined over the interval of real numbers = [30, 45], which means the noisiness of the fridge in , and have the following meaning: () = {30/1 + 31/0.97 + 32/0.93 + 33/0.89 + 34/0.82} , () = {35/0.92 + 36/0.98 + 37/1 + 38/0.98 + 39/0.92} ,
(ℎℎ) = {40/0.83 + 41/0.88 + 42/0.92 + 43/0.97 + 44/1} ; price is a crisp quantitative property, which means the price of the fridge in UAH, and is defined over the interval of integer numbers = [2200, 255000]; get_crisp_weight is a fuzzy method that computes defuzzification representation of the fuzzy quantitative property weight and defined in the following way: get_fuzzy_price is a fuzzy method that computes fuzzification representation of the crisp quantitative property price and defined in the following way:
_ _(, , ) = {− / (− ), ./1, +/ (+)},
where < . < and is the incremental for the generation of − and +, = 1, . . .
− = . − * , < . − * < .,
+ = . + * , . < . + * < ,
and where
1 = ( _, ( ∈ (_), ))/1,
2 = ( _, ( ∈ ( _), ))/0.78,
5 = (, ((1 ∈ ℎℎ, ), (2 ∈ ℎ, ), (3 ∈ ℎ, ))/1,
6 = (, (6( .), ∈ [
(_) = {, , }, where fuzzy variables small, medium, and big are defined over the interval of integer numbers = [30, 45], which means the volume of the refrigerator in 3, and have the following interpretation: () = {30/1 + 31/0.95 + 32/0.91 + 33/0.87 + 34/0.82} 3, () = {35/0.9 + 36/0.96 + 37/1 + 38/0.96 + 39/0.9} 3,
() = {40/0.78 + 41/0.83 + 42/0.89 + 43/0.95 + 44/1} 3; freezer volume is a fuzzy quantitative property defined as a linguistic variable, which has the following term-set
( _) = {, , }, where fuzzy variables small, medium, and big are defined over the interval of real numbers = [5, 6.5], which means the volume of the freezer in 3, and have the following interpretation: () = {5.0/1 + 5.1/0.92 + 5.2/0.87 + 5.3/0.82 + 5.4/0.78} 3, () = {5.5/0.8 + 5.6/0.93 + 5.7/1 + 5.8/0.93 + 5.9/0.8} 3,
() = {6.0/0.82 + 6.1/0.88 + 6.2/0.93 + 6.3/0.97 + 6.4/1} 3;
cee is a crisp quantitative property, which means the class of energy eficiency to which the fridge
belongs, and is defined over the set of string values = {+++, ++, +, , , , , };
aec is a crisp quantitative property, which means the annual energy consumption by the fridge
in ℎ, and is defined over the following interval of integer numbers = [95, 110]; sizes
is a crisp multiple-valued quantitative property, which means dimensions of the fridge in ,
and is defined over the intervals of integer numbers ℎℎ = [45, 55], ℎ = [40, 60],
ℎ = [40, 50]; compactness is a fuzzy qualitative property defined by verification function
6( ) : . → [
(ℎ) = {ℎℎ, },
where fuzzy variables lightweight, and medium are defined over the interval of integer numbers
= [
() = {, }, where fuzzy variables low, and medium are defined over the interval of real numbers = [35, 42], which means the noisiness of the fridge in , and have the following meaning: () = {35/1 + 36/0.94 + 37/0.89 + 38/0.83} , () = {39/0.95 + 40/1 + 41/0.95 + 42/0.93} ; price is a crisp quantitative property, which means the price of the fridge in UAH, and is defined over the interval of integer numbers = [2800, 4700]; get_crisp_weight is a fuzzy method that computes defuzzification representation of the fuzzy quantitative property weight and defined in the following way: get_fuzzy_price is a fuzzy method that computes fuzzification representation of the crisp quantitative property price and defined in the following way: Therefore, using Algorithm 1 we have constructed two fuzzy homogeneous classes of objects, which have the following representations: ∖ ( 1 = ( _, ( ∈ (_), ))/1, 2 = ( _, ( ∈ ( _), ))/1, 3 = (, ( ∈ , ℎ))/1, 4 = (, ((1 ∈ ℎℎ, ), (2 ∈ ℎ, ), (3 ∈ ℎ, ))/1, 5 = (ℎ, ( ∈ (ℎ), ))/1, 6 = (, ( ∈ (), ))/0.75, 7 = (, ( ∈ , N+))/1, )/0.96, ∖ ( 1 = ( _, ( ∈ (_), ))/1, 2 = ( _, ( ∈ ( _), ))/0.78, 4 = (, ((1 ∈ ℎℎ, ), (2 ∈ ℎ, ), (3 ∈ ℎ, ))/1, 5 = (ℎ, ( ∈ (ℎ), ))/1, 6 = (, ( ∈ (), ))/0.82, Created classes represent unique parts of fuzzy homogeneous classes of objects and . Class ∖ has a measure of fuzziness, which is equal to 0.96 according to Definition 1, and defines the fuzzy type ℎ1∖ℎ2 ⊆ ℎ1, which consists of properties and methods which are typical only for the fuzzy homogeneous class of objects /0.96. Class ∖ has a measure of fuzziness, which is equal to 0.94 according to Definition 1, and defines the fuzzy type ℎ2∖ℎ1 ⊆ ℎ2, which consists of properties and methods which are typical only for the fuzzy homogeneous class of objects /0.94.
As the result, Algorithm 1 provides an opportunity to verify the diference of one fuzzy homogeneous class of objects from another one, as well as to compute it in the form of a new fuzzy homogeneous class of objects if such diference exists. Fuzzy homogeneous classes of objects, which are dynamically created by the algorithm, allow a KBS to estimate both the similarity and the diference between extracted or acquired new knowledge items and those ones, which already integrated within the knowledge base since the diference and similarity are inverse concepts. Results of such analysis can be used for the eficient integration of new knowledge into the knowledge base.
To perform the integration of new knowledge into the knowledge base eficiently, a KBS should be able to analyze and to compare extracted or acquired new knowledge items with those ones, which were integrated previously. A system should verify the diference and similarity between new knowledge items and previously obtained ones to perform the recognition stage of the knowledge integration process. For this purpose, we defined the concept of the universal diference exploiter for fuzzy homogeneous classes of objects and developed a corresponding algorithm for its implementation. The developed algorithm provides an opportunity to verify as well as to compute the diference and the similarity between extracted or acquired new knowledge items and previously obtained ones, in terms of fuzzy homogeneous classes of objects, within such knowledge representation model as fuzzy object-oriented dynamic networks.
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