=Paper=
{{Paper
|id=Vol-3013/20210024
|storemode=property
|title=Difference of Fuzzy Homogeneous Classes of Objects
|pdfUrl=https://ceur-ws.org/Vol-3013/20210024.pdf
|volume=Vol-3013
|authors=Dmytro Terletskyi,Sergey Yershov
|dblpUrl=https://dblp.org/rec/conf/icteri/TerletskyiY21
}}
==Difference of Fuzzy Homogeneous Classes of Objects==
https://ceur-ws.org/Vol-3013/20210024.pdf
Difference of Fuzzy Homogeneous Classes of Objects
Dmytro O. Terletskyi1 , Sergey V. Yershov2
1
V. M. Glushkov Institute of Cybernetics of NAS of Ukraine, Academician Glushkov Avenue, 40, Kyiv, 03187, Ukraine
2
V. M. Glushkov Institute of Cybernetics of NAS of Ukraine, Academician Glushkov Avenue, 40, Kyiv, 03187, Ukraine
Abstract
Analysis of relevance, similarity, and the difference 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 de-
fined over the knowledge representation structures provided by chosen representation model. Within
the object-oriented approach, classes are one of the main knowledge representation structures, conse-
quently, the analysis of the relevance, similarity, and difference between them requires the development
of corresponding operations over classes. Therefore the concept of universal difference exploiter of
fuzzy homogeneous classes of objects, within such a knowledge representation model as fuzzy object-
oriented dynamic networks, was introduced in the paper. To implement the proposed exploiter, which
allows computing of the difference of one fuzzy homogeneous class of objects from another one, we
developed the corresponding algorithm and provided an example of its application. The proposed ap-
proach 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 difference between
them.
Keywords
Fuzzy class, Fuzzy type, Universal difference exploiter, Difference of fuzzy classes
1. Introduction
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 knowledge-
based 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 differences 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 [1, 2, 3,
4, 5], it can be defined as a task of incorporating new information into a knowledge base, which
requires elaborating new information and resolving inconsistencies with existing knowledge.
17th International Conference on ICT in Education, Research and Industrial Applications. Integration, Harmonization
and Knowledge Transfer, September 28 β October 2, 2021, Kherson, Ukraine
dmytro.terletskyi@gmail.com (D. O. Terletskyi); ErshovSV@nas.gov.ua (S. V. Yershov)
0000-0002-7393-1426 (D. O. Terletskyi); 0000-0002-9895-777X (S. V. Yershov)
Β© 2021 Copyright for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
CEUR
Workshop
Proceedings
http://ceur-ws.org
ISSN 1613-0073 CEUR Workshop Proceedings (CEUR-WS.org)
The corresponding computational model of knowledge integration incorporates the following
main stages:
1. Recognition: Identification of relevance between new and previously obtained knowledge.
2. Elaboration: Determination of interactions of obtained new knowledge items on previ-
ously obtained ones, and how they can affect them.
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 verifica-
tion of such relations between new and previously obtained knowledge items as equivalence,
inclusion, similarity, difference, 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 represen-
tation 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,
difference, and symmetric difference, provides a corresponding framework for the analysis of
knowledge items, which are represented in terms of classes.
However, different 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 difference 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 difference
exploiter, we developed an appropriate algorithm and provided a representative example of its
application for the knowledge analysis within fuzzy object-oriented dynamic networks.
2. Fuzzy Classes and Types
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 [6, 7], despite all benefits of crisp classes and objects, they
can be inefficient for the construction of realistic representation models of vague, imprecise,
or uncertain entities or domains, because of their descriptive restrictions. Therefore a crisp
class-based object-oriented model was extended using notions of fuzzy sets and linguistic
variables, which were introduced by Zade in [8, 9, 10, 11]. The main idea of such extension is to
define values of class properties as fuzzy sets or as linguistic variables, which are more complex
structures defined as the term-set interpreted using appropriate fuzzy variables defined by fuzzy
sets. It led to the appearance of concepts of fuzzy attributes and fuzzy objects, described in [6].
The next step in fuzzification of the object-oriented paradigm was to introduce a measure of
truth for each property of the class, which is defined on the interval of real numbers [0, 1]. Then,
concepts of fuzzy methods and fuzzy types were introduced in [12, 13, 14, 15], as well as the
concept of fuzzy classes and fuzzy class hierarchies were proposed in [7]. One more step was to
define a membership degree for objects of a class as well as membership degree for subclasses
of a class, which led to the appearance of classification of fuzziness levels, introduced in [16],
and the notion of fuzzy classes of fuzzy objects, which were introduced in [17, 18, 19, 20].
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 difference 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].
Definition 1. A fuzzy homogeneous class of objects is a collection
π /π (π ) = (π (π‘), πΉ (π‘))/π (π ) = ((π1 (π‘)/π(π1 (π‘)), . . . ,
ππ (π‘)/π(ππ (π‘))), (π1 (π‘)/π(π1 (π‘)), . . . , ππ (π‘)/π(ππ (π‘))))/π (π ),
where π‘ is a fuzzy type which is defined by the class π , ππ (π‘)/π(ππ (π‘)) β π (π‘) is a crisp or fuzzy
property of the class π , ππ (π‘)/π(ππ (π‘)) β πΉ (π‘) is its crisp or fuzzy method, π(ππ (π‘)) : ππ (π‘) β
(0, 1] and π(ππ (π‘)) : ππ (π‘) β (0, 1] are measures of fuzziness of a property ππ (π‘) and a method
ππ (π‘), and π (π ) is a measure of fuzziness of the class π , i.e.
π (π ) = ((π(π1 (π‘)) + Β· Β· Β· + π(ππ (π‘))) + (π(π1 (π‘)) + Β· Β· Β· + π(ππ (π‘))))/(π + π).
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 [12, 13, 14, 15] as well as to the concept of the fuzzy class described in [17, 18, 19, 20],
while the measure of fuzziness of the class itself is defined as the arithmetic mean of measures of
the truth of all its properties and methods. Similar to homogeneous classes, fuzzy homogeneous
classes of objects define only a single fuzzy type of objects, therefore as in class-based object-
oriented programming, a fuzzy type of objects and a fuzzy homogeneous class of objects can be
considered as equivalent concepts.
3. Difference of Fuzzy Homogeneous Classes
As it was mentioned above, to perform knowledge integration in the proper way, a KBS should
be able to analyze the relevance, similarity, and differences 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 difference adapted for fuzzy classes as a tool
for dynamic creation of new fuzzy classes of objects which represent of difference 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 [27, 28, 19, 29, 17, 30], relational
uncertain databases [31], the blurry classes within the fuzzy object-oriented databases [32],
supporting fuzzy XML queries [33, 19] and handling uncertain spatiotemporal data [34, 35],
biomedical fuzzy HBase databases [36], etc.
All mentioned algebras provide different signatures, however, all of them contain an adapted
operation of set-theoretical difference, 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 difference 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 difference 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 difference exploiter for fuzzy
homogeneous classes of objects via generalizing its versions for the classes of fuzzy objects.
Definition 2. Difference π1 β π2 of two fuzzy homogeneous classes of objects π1 /π (π1 ) and
π2 /π (π2 ), which define fuzzy types of objects π‘1 and π‘2 correspondingly, is a fuzzy homogeneous
class of objects π1β2 /π (π1β2 ), which define fuzzy type of objects π‘1β2 , such that
(οΈ )οΈ (οΈ * (οΈ
*
)οΈ (οΈ
*
)οΈ (οΈ
*
)οΈ)οΈ
π‘1β2 β π‘1 | β π‘1β2 β© π‘2 β§ βπ‘1β2 | π‘1β2 β π‘1β2 β§ π‘1β2 β π‘1 β§ β π‘1β2 β© π‘2 .
The fuzzy class π1β2 /π (π1β2 ) exists if and only if βππ1 (π‘1 ), βππ2 (π‘2 ), such that ππ1 (π‘1 ) ΜΈβ‘ ππ2 (π‘2 ),
or βππ1 (π‘1 ), βππ2 (π‘2 ), such that ππ1 (π‘1 ) ΜΈβ‘ ππ2 (π‘2 ), where ππ1 (π‘1 ) is an π1 -th property of the fuzzy
type π‘1 , π1 = 1, |π (π‘1 )|, ππ2 (π‘2 ) is an π2 -th property of the fuzzy type π‘2 , π2 = 1, |π (π‘2 )|, ππ1 (π‘1 )
is an π1 -th method of the fuzzy type π‘1 , π1 = 1, |πΉ (π‘1 )|, ππ2 (π‘2 ) is an π2 -th method of the fuzzy
type π‘2 , π2 = 1, |πΉ (π‘2 )|.
The universal difference exploiter creates a fuzzy homogeneous class of objects π1β2 /π (π1β2 ),
which represents the difference of the fuzzy class of objects π1 /π (π1 ) from the fuzzy class
of objects π2 /π (π2 ) if such difference exists. The class π1β2 /π (π1β2 ) defines a fuzzy type of
objects π‘1β2 β π‘1 , which consists of crisp and (or) fuzzy properties and (or) methods, which are
typical only for the fuzzy type of objects π‘1 .
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
π1 /π (π1 ), π2 /π (π2 ) and find properties and methods of the fuzzy class π1 /π (π1 ), which
are not typical for the fuzzy class of objects π2 /π (π2 ). 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 difference exploiter for fuzzy homogeneous classes of objects.
Analyzing Algorithm 1, we can see that it uses fuzzy homogeneous classes of objects
π1 /π (π1 ), π2 /π (π2 ) as the input data and computes the difference of the class π1 /π (π1 ) from
the class π2 /π (π2 ) in a form of a new fuzzy homogeneous class of objects π1β2 /π (π1β2 ) =
π1 /π (π1 ) β π2 /π (π2 ) if such difference exists. The algorithm successively constructs the
specification and signature of class π1β2 /π (π1β2 ) 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 ππ (π1 )/π(ππ (π1 )),
ππ (π2 )/π(ππ (π2 )) or methods ππ (π1 )/π(ππ (π1 )), ππ (π2 )/π(ππ (π2 )) and if they are equivalent
ones it returns 1, otherwise it returns 0. It can be implemented in various ways using corre-
sponding 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 π1 /π (π1 ) and π2 /π (π2 ) the algorithm
checks the equivalence of |π (π1 )| Γ |π (π2 )| = π Γ π properties and |πΉ (π1 )| Γ |πΉ (π2 )| = π Γ π
methods. In addition, to construct the specification π (π1β2 ) and signature πΉ (π1β2 ) of the fuzzy
class π1β2 /π (π1β2 ), it performs copying of π€1 properties and π€2 methods of the fuzzy class
π1 /π (π1 ), which are not typical for the fuzzy class π2 /π (π2 ), where 0 β€ π€1 β€ |π (π1 )| and
0 β€ π€2 β€ |πΉ (π1 )|. Therefore, the time complexity of Algorithm 1 is equal to
π(π Γ π) + π(π Γ π) + π(π€1 ) + π(π€2 ) β π(π2 + π 2 + π€1 + π€2 ),
Algorithm 1 Difference of fuzzy homogeneous classes.
Require: π1 /π (π1 ), π2 /π (π2 ) β fuzzy homogeneous classes
Ensure: π /π (π ) = π1 /π (π1 ) β π2 /π (π2 )
1: π := {};
2: π’ππππ’π := true;
3: for all ππ /π(ππ ) β π (π1 ) do
4: for all ππ /π(ππ ) β π (π2 ) do
5: if is_equivalent(ππ /π(ππ ), ππ /π(ππ )) then
6: π’ππππ’π := false;
7: break;
8: if π’ππππ’π then
9: if π (π ) ΜΈβ π then
10: π := {};
11: π.add(π );
12: π (π ).add(ππ /π(ππ ))
13: else
14: π’ππππ’π := true;
15: π’ππππ’π := true;
16: for all ππ /π(ππ ) β πΉ (π1 ) do
17: for all ππ /π(ππ ) β πΉ (π2 ) do
18: if is_equivalent(ππ /π(ππ ), ππ /π(ππ )) then
19: π’ππππ’π := false;
20: break;
21: if π’ππππ’π then
22: if πΉ (π ) ΜΈβ π then
23: πΉ := {};
24: π.add(πΉ );
25: πΉ (π ).add(ππ /π(ππ ))
26: else
27: π’ππππ’π := true;
28: return π .
where π2 is a number of properties equivalence checks, π 2 is a number of methods equivalence
checks, π€1 and π€2 is a number of copying operations of properties and methods of fuzzy class
π1 /π (π1 ). To perform main computations Algorithm 1 uses π units of memory for storing fuzzy
type π‘1β2 , therefore its space complexity is equal to π(π ), where 0 β€ π β€ |π (π1 )| + |πΉ (π1 )|.
4. Application Example
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 π»ππππΉ πππππ
defines the fuzzy type π‘βπ 1 , which describes a fuzzy concept of a home fridge, which has the
following representation:
π»ππππΉ πππππ(
π1 = (πππ πππππππ‘ππ_π£πππ’ππ, (π£ β π (π_π£πππ’ππ), π π‘π))/1,
π2 = (π ππππ§ππ_π£πππ’ππ, (π£ β π (π _π£πππ’ππ), π π‘π))/1,
π3 = (πππ, (π£ β ππππ , π π‘π))/1,
π4 = (πππ, (π£ β ππππ , ππ β))/1,
π5 = (π ππ§ππ , ((π£1 β πβπππβπ‘ , ππ), (π£2 β ππ€πππ‘β , ππ), (π£3 β πππππ‘β , ππ))/1,
π6 = (πππππππ‘πππ π , (π£π6 (π»ππππΉ πππππ.π ππ§ππ ), π£ β [0, 1]))/0.93,
π7 = (πππππ, (π£ β ππππππ , π π‘π))/1,
π8 = (π€πππβπ‘, (π£ β π (π€πππβπ‘), π π‘π))/1,
π9 = (ππππ ππππ π , (π£ β π (ππππ ππππ π ), π π‘π))/0.75,
π10 = (πππππ, (π£ β ππππππ , N+ ))/1,
π1 = πππ‘_ππππ π_π€πππβπ‘()/0.93,
π2 = πππ‘_π π’π§π§π¦_πππππ(π, π, π)/0.87
)/0.96,
refrigerator volume is a fuzzy quantitative property defined as a linguistic variable, which has
the following term-set
π (π_π£πππ’ππ) = {π£πππ¦ π ππππ, π ππππ, πππππ’π, πππ, π£πππ¦ πππ},
where fuzzy variables very small, small, medium, big, and very big are defined over the interval
of integer numbers π = [40, 425], which means the volume of the refrigerator in ππ3 , and
have the following interpretation:
π (π£πππ¦ π ππππ) = {40/1 + 50/0.95 + 60/0.85 + 70/0.7 + 80/0.65} ππ3 ,
π (π ππππ) = {95/0.92 + 110/0.78 + 125/0.63 + 140/0.55 + 150/0.45} ππ3 ,
π (πππππ’π) = {170/0.78 + 190/0.92 + 210/1 + 230/0.92 + 250/0.78} ππ3 ,
π (πππ) = {270/0.82 + 290/0.94 + 310/1 + 330/0.94 + 350/0.82} ππ3 ,
π (π£πππ¦ πππ) = {365/0.65 + 380/0.72 + 395/0.86 + 410/0.93 + 425/1} ππ3 ;
freezer volume is a fuzzy quantitative property defined as a linguistic variable, which has the
following term-set
π (π _π£πππ’ππ) = {π£πππ¦ π ππππ, π ππππ, πππππ’π, πππ, π£πππ¦ πππ},
where fuzzy variables very small, small, medium, big, and very big are defined over the interval
of integer numbers π = [10, 275], which means the volume of the freezer in ππ3 , and have the
following interpretation:
π (π£πππ¦ π ππππ) = {10/1 + 17/0.94 + 24/0.85 + 31/0.78 + 38/0.65} ππ3 ,
π (π ππππ) = {50/1 + 57/0.92 + 64/0.86 + 71/0.73 + 78/0.61} ππ3 ,
π (πππππ’π) = {90/0.85 + 100/0.93 + 110/1 + 120/0.93 + 130/0.85} ππ3 ,
π (πππ) = {140/0.82 + 155/0.93 + 170/1 + 185/0.93 + 200/0.82} ππ3 ,
π (π£πππ¦ πππ) = {215/0.67 + 230/0.79 + 245/0.88 + 260/0.95 + 275/1} ππ3 ;
cee is a crisp quantitative property, which means the class of energy efficiency to which the fridge
belongs, and is defined over the set of string values ππππ = {π΄+++ , π΄++ , π΄+ , π΄, π΅, πΆ, π·, πΉ };
aec is a crisp quantitative property, which means the annual energy consumption by the fridge
in ππ β, and is defined over the following interval of integer numbers ππππ = [100, 360]; sizes
is a crisp multiple-valued quantitative property, which means dimensions of the fridge in ππ,
and is defined over the intervals of integer numbers πβπππβπ‘ = [45, 205], ππ€πππ‘β = [35, 95],
πππππ‘β = [55, 85]; compactness is a fuzzy qualitative property defined by verification function
π£π6 (π»ππππΉ πππππ) : π»ππππΉ πππππ.π ππ§ππ β [0, 1],
where π£π6 (π»ππππΉ πππππ) is defined as follows
π β ππππ
π£π6 (π»ππππΉ πππππ) = ,
ππππ₯ β ππππ
where
π = π ππ§ππ .π£1 Β· π ππ§ππ .π£2 Β· π ππ§ππ .π£3 ,
ππππ = min(πβπππβπ‘ ) Β· min(ππ€πππ‘β ) Β· min(πππππ‘β ),
ππππ₯ = max(πβπππβπ‘ ) Β· max(ππ€πππ‘β ) Β· max(πππππ‘β );
color is a crisp quantitative property, which means the color of the fridge, and is defined over
the following set of string values
ππππππ = {πππππ, π€βππ‘π, ππππβππ‘π, ππππππ, ππππ€π, πππ, π π‘ππππππ π π π‘πππ,
π πππ£ππ, ππππ¦, π‘ππ‘ππππ’π, πππππ, πππππ§π, πππ’π, πππππ, ππππππ, ππππ, ππ£πππ¦, ππ’ππππ};
weight is a fuzzy quantitative property defined as a linguistic variable, which has the following
term-set
π (π€πππβπ‘) = {πππβπ‘π€πππβπ‘, πππππ’π, βπππ£π¦, π£πππ¦ βπππ£π¦},
where fuzzy variables lightweight, medium, heavy, and very heavy are defined over the interval
of integer numbers π = [10, 135], which means the weight of the fridge in ππ, and have the
following interpretation:
π (πππβπ‘π€πππβπ‘) = {10/1 + 20/0.95 + 30/0.88 + 40/0.79 + 50/0.68} ππ,
π (πππππ’π) = {55/0.83 + 60/0.94 + 65/1 + 70/0.94 + 75/0.83} ππ,
π (βπππ£π¦) = {80/0.86 + 85/0.95 + 90/1 + 95/0.95 + 100/0.86} ππ,
π (π£πππ¦ βπππ£π¦) = {115/0.71 + 120/0.79 + 125/0.88 + 130/0.96 + 135/1} ππ;
noisiness is a fuzzy quantitative property defined as a linguistic variable, which has the following
term-set
π (ππππ ππππ π ) = {πππ€, πππππ’π, βππβ},
where fuzzy variables low, medium, and high are defined over the interval of real numbers
π = [30, 45], which means the noisiness of the fridge in ππ΅, and have the following meaning:
π (πππ€) = {30/1 + 31/0.97 + 32/0.93 + 33/0.89 + 34/0.82} ππ΅,
π (πππππ’π) = {35/0.92 + 36/0.98 + 37/1 + 38/0.98 + 39/0.92} ππ΅,
π (βππβ) = {40/0.83 + 41/0.88 + 42/0.92 + 43/0.97 + 44/1} ππ΅;
price is a crisp quantitative property, which means the price of the fridge in UAH, and is
defined over the interval of integer numbers ππππππ = [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:
βοΈ|π€πππβπ‘.π£|
π=1 π(π€πππβπ‘.π£) Β· π€πππβπ‘.π£
πππ‘_ππππ π_π€πππβπ‘() = βοΈ|π€πππβπ‘.π£| ;
π=1 π(π€πππβπ‘.π£)
get_fuzzy_price is a fuzzy method that computes fuzzification representation of the crisp quan-
titative property price and defined in the following way:
πππ‘_π π’π§π§π¦_πππππ(π, π, π) = {π₯β β + +
π /π(π₯π ), πππππ.π£/1, π₯π /π(π₯π )},
where π < πππππ.π£ < π and π is the incremental for the generation of π₯β
π and π₯π , π = 1, . . .
+
π₯β
π = πππππ.π£ β π * π, π < πππππ.π£ β π * π < πππππ.π£,
π₯+
π = πππππ.π£ + π * π, πππππ.π£ < πππππ.π£ + π * π < π,
and where
π₯β
π βπ
π(π₯β
π )= β πΏπβ , πΏπβ = 1 β π(π₯β β β β
π ) β π(π₯π ), π(π₯π ) = 1 β π(π₯π ),
πππππ.π£ β π
π β π₯+
π(π₯+
π ) = π
β πΏπ+ , πΏπ+ = 1 β π(π₯+ + + +
π ) β π(π₯π ), π(π₯π ) = 1 β π(π₯π ).
π β πππππ.π£
As we can see, the fuzzy class of objects π»ππππΉ πππππ has a measure of its fuzziness, which
is equal to 0.96 according to Definition 1.
Now let us suppose that the second fuzzy homogeneous class of objects π»ππ‘πππΉ πππππ defines
the fuzzy type π‘βπ 2 , which describes a fuzzy concept of a hotel fridge, which has the following
representation:
π»ππ‘πππΉ πππππ(
π1 = (πππ πππππππ‘ππ_π£πππ’ππ, (π£ β π (π_π£πππ’ππ), π π‘π))/1,
π2 = (π ππππ§ππ_π£πππ’ππ, (π£ β π (π _π£πππ’ππ), π π‘π))/0.78,
π3 = (πππ, (π£ β ππππ , π π‘π))/1,
π4 = (πππ, (π£ β ππππ , ππ β))/1,
π5 = (π ππ§ππ , ((π£1 β πβπππβπ‘ , ππ), (π£2 β ππ€πππ‘β , ππ), (π£3 β πππππ‘β , ππ))/1,
π6 = (πππππππ‘πππ π , (π£π6 (π»ππ‘πππΉ πππππ.π ππ§ππ ), π£ β [0, 1]))/0.93,
π7 = (πππππ, (π£ β ππππππ , π π‘π))/1,
π8 = (π€πππβπ‘, (π£ β π (π€πππβπ‘), π π‘π))/1,
π9 = (ππππ ππππ π , (π£ β π (ππππ ππππ π ), π π‘π))/0.82,
π10 = (πππππ, (π£ β ππππππ , N+ ))/1,
π1 = πππ‘_ππππ π_π€πππβπ‘()/0.93,
π2 = πππ‘_π π’π§π§π¦_πππππ(π, π, π)/0.87
)/0.94,
refrigerator volume is a fuzzy quantitative property defined as a linguistic variable, which has
the following term-set
π (π_π£πππ’ππ) = {π ππππ, πππππ’π, πππ},
where fuzzy variables small, medium, and big are defined over the interval of integer numbers
π = [30, 45], which means the volume of the refrigerator in ππ3 , and have the following
interpretation:
π (π ππππ) = {30/1 + 31/0.95 + 32/0.91 + 33/0.87 + 34/0.82} ππ3 ,
π (πππππ’π) = {35/0.9 + 36/0.96 + 37/1 + 38/0.96 + 39/0.9} ππ3 ,
π (πππ) = {40/0.78 + 41/0.83 + 42/0.89 + 43/0.95 + 44/1} ππ3 ;
freezer volume is a fuzzy quantitative property defined as a linguistic variable, which has the
following term-set
π (π _π£πππ’ππ) = {π ππππ, πππππ’π, πππ},
where fuzzy variables small, medium, and big are defined over the interval of real numbers π =
[5, 6.5], which means the volume of the freezer in ππ3 , and have the following interpretation:
π (π ππππ) = {5.0/1 + 5.1/0.92 + 5.2/0.87 + 5.3/0.82 + 5.4/0.78} ππ3 ,
π (πππππ’π) = {5.5/0.8 + 5.6/0.93 + 5.7/1 + 5.8/0.93 + 5.9/0.8} ππ3 ,
π (πππ) = {6.0/0.82 + 6.1/0.88 + 6.2/0.93 + 6.3/0.97 + 6.4/1} ππ3 ;
cee is a crisp quantitative property, which means the class of energy efficiency to which the fridge
belongs, and is defined over the set of string values ππππ = {π΄+++ , π΄++ , π΄+ , π΄, π΅, πΆ, π·, πΉ };
aec is a crisp quantitative property, which means the annual energy consumption by the fridge
in ππ β, and is defined over the following interval of integer numbers ππππ = [95, 110]; sizes
is a crisp multiple-valued quantitative property, which means dimensions of the fridge in ππ,
and is defined over the intervals of integer numbers πβπππβπ‘ = [45, 55], ππ€πππ‘β = [40, 60],
πππππ‘β = [40, 50]; compactness is a fuzzy qualitative property defined by verification function
π£π6 (π»ππ‘πππΉ πππππ) : π»ππ‘πππΉ πππππ.π ππ§ππ β [0, 1],
where π£π6 (π»ππ‘πππΉ πππππ) is defined as follows
π β ππππ
π£π6 (π»ππ‘πππΉ πππππ) = ,
ππππ₯ β ππππ
where
π = π ππ§ππ .π£1 Β· π ππ§ππ .π£2 Β· π ππ§ππ .π£3 ,
ππππ = min(πβπππβπ‘ ) Β· min(ππ€πππ‘β ) Β· min(πππππ‘β ),
ππππ₯ = max(πβπππβπ‘ ) Β· max(ππ€πππ‘β ) Β· max(πππππ‘β );
color is a crisp quantitative property, which means the color of the fridge, and is defined over
the following set of string values
ππππππ = {πππππ, π€βππ‘π, ππππβππ‘π, ππππππ, ππππ€π, πππ, π π‘ππππππ π π π‘πππ,
π πππ£ππ, ππππ¦, π‘ππ‘ππππ’π, πππππ, πππππ§π, πππ’π, πππππ, ππππππ, ππππ, ππ£πππ¦, ππ’ππππ};
weight is a fuzzy quantitative property defined as a linguistic variable, which has the following
term-set
π (π€πππβπ‘) = {πππβπ‘π€πππβπ‘, πππππ’π},
where fuzzy variables lightweight, and medium are defined over the interval of integer numbers
π = [14, 23], which means the weight of the fridge in ππ, and have the following interpretation:
π (πππβπ‘π€πππβπ‘) = {14/1 + 15/0.95 + 16/0.91 + 17/0.87 + 18/0.81} ππ,
π (πππππ’π) = {19/0.96 + 20/1 + 21/0.95 + 22/0.91 + 23/0.87} ππ;
noisiness is a fuzzy quantitative property defined as a linguistic variable, which has the following
term-set
π (ππππ ππππ π ) = {πππ€, πππππ’π},
where fuzzy variables low, and medium are defined over the interval of real numbers π =
[35, 42], which means the noisiness of the fridge in ππ΅, and have the following meaning:
π (πππ€) = {35/1 + 36/0.94 + 37/0.89 + 38/0.83} ππ΅,
π (πππππ’π) = {39/0.95 + 40/1 + 41/0.95 + 42/0.93} ππ΅;
price is a crisp quantitative property, which means the price of the fridge in UAH, and is defined
over the interval of integer numbers ππππππ = [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:
βοΈ|π€πππβπ‘.π£|
π(π€πππβπ‘.π£) Β· π€πππβπ‘.π£
πππ‘_ππππ π_π€πππβπ‘() = π=1βοΈ|π€πππβπ‘.π£| ;
π=1 π(π€πππβπ‘.π£)
get_fuzzy_price is a fuzzy method that computes fuzzification representation of the crisp quan-
titative property price and defined in the following way:
πππ‘_π π’π§π§π¦_πππππ(π, π, π) = {π₯β β + +
π /π(π₯π ), πππππ.π£/1, π₯π /π(π₯π )},
where π < πππππ.π£ < π and π is the incremental for the generation of π₯β
π and π₯π , π = 1, . . .
+
π₯β
π = πππππ.π£ β π * π, π < πππππ.π£ β π * π < πππππ.π£,
π₯+
π = πππππ.π£ + π * π, πππππ.π£ < πππππ.π£ + π * π < π,
and where
π₯β
π βπ
π(π₯β
π )= β πΏπβ , πΏπβ = 1 β π(π₯β β β β
π ) β π(π₯π ), π(π₯π ) = 1 β π(π₯π ),
πππππ.π£ β π
π β π₯+
π(π₯+
π ) = π
β πΏπ+ , πΏπ+ = 1 β π(π₯+ + + +
π ) β π(π₯π ), π(π₯π ) = 1 β π(π₯π ).
π β πππππ.π£
As we can see, the fuzzy class of objects π»ππ‘πππΉ πππππ has a measure of its fuzziness, which
is equal to 0.94 according to Definition 1.
Now let us compute the difference between the fuzzy homogeneous classes of objects
π»ππππΉ πππππ and π»ππ‘πππΉ πππππ and then vice versa, i.e. π»ππππΉ πππππ β π»ππ‘πππΉ πππππ and
π»ππ‘πππΉ πππππ β π»ππππΉ πππππ, using Algorithm 1. Analyzing the specifications and signatures
of the fuzzy homogeneous classes of objects π»ππππΉ πππππ and π»ππ‘πππΉ πππππ, we can conclude
that
π»ππππΉ πππππ.πππ β‘ π»ππ‘πππΉ πππππ.πππ,
π»ππππΉ πππππ.πππππππ‘πππ π β‘ π»ππ‘πππΉ πππππ.πππππππ‘πππ π ,
π»ππππΉ πππππ.πππππ β‘ π»ππ‘πππΉ πππππ.πππππ,
π»ππππΉ πππππ.πππ‘_ππππ π_π€πππβπ‘() β‘ π»ππ‘πππΉ πππππ.πππ‘_ππππ π_π€πππβπ‘(),
π»ππππΉ πππππ.πππ‘_π π’π§π§π¦_πππππ(π, π, π) β‘ π»ππ‘πππΉ πππππ.πππ‘_π π’π§π§π¦_πππππ(π, π, π).
Therefore, using Algorithm 1 we have constructed two fuzzy homogeneous classes of objects,
which have the following representations:
π»ππππΉ πππππ β π»ππ‘πππΉ πππππ(
π1 = (πππ πππππππ‘ππ_π£πππ’ππ, (π£ β π (π_π£πππ’ππ), π π‘π))/1,
π2 = (π ππππ§ππ_π£πππ’ππ, (π£ β π (π _π£πππ’ππ), π π‘π))/1,
π3 = (πππ, (π£ β ππππ , ππ β))/1,
π4 = (π ππ§ππ , ((π£1 β πβπππβπ‘ , ππ), (π£2 β ππ€πππ‘β , ππ), (π£3 β πππππ‘β , ππ))/1,
π5 = (π€πππβπ‘, (π£ β π (π€πππβπ‘), π π‘π))/1,
π6 = (ππππ ππππ π , (π£ β π (ππππ ππππ π ), π π‘π))/0.75,
π7 = (πππππ, (π£ β ππππππ , N+ ))/1,
)/0.96,
π»ππ‘πππΉ πππππ β π»ππππΉ πππππ(
π1 = (πππ πππππππ‘ππ_π£πππ’ππ, (π£ β π (π_π£πππ’ππ), π π‘π))/1,
π2 = (π ππππ§ππ_π£πππ’ππ, (π£ β π (π _π£πππ’ππ), π π‘π))/0.78,
π3 = (πππ, (π£ β ππππ , ππ β))/1,
π4 = (π ππ§ππ , ((π£1 β πβπππβπ‘ , ππ), (π£2 β ππ€πππ‘β , ππ), (π£3 β πππππ‘β , ππ))/1,
π5 = (π€πππβπ‘, (π£ β π (π€πππβπ‘), π π‘π))/1,
π6 = (ππππ ππππ π , (π£ β π (ππππ ππππ π ), π π‘π))/0.82,
π7 = (πππππ, (π£ β ππππππ , N+ ))/1,
)/0.94,
Created classes represent unique parts of fuzzy homogeneous classes of objects π»ππππΉ πππππ
and π»ππ‘πππΉ πππππ. Class π»ππππΉ πππππ β π»ππ‘πππΉ πππππ has a measure of fuzziness, which is
equal to 0.96 according to Definition 1, and defines the fuzzy type π‘βπ 1ββπ 2 β π‘βπ 1 , which
consists of properties and methods which are typical only for the fuzzy homogeneous class of
objects π»ππππΉ πππππ/0.96. Class π»ππ‘πππΉ πππππ β π»ππππΉ πππππ has a measure of fuzziness,
which is equal to 0.94 according to Definition 1, and defines the fuzzy type π‘βπ 2ββπ 1 β π‘βπ 2 ,
which consists of properties and methods which are typical only for the fuzzy homogeneous
class of objects π»ππ‘πππΉ πππππ/0.94.
As the result, Algorithm 1 provides an opportunity to verify the difference 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 difference exists. Fuzzy homogeneous classes of
objects, which are dynamically created by the algorithm, allow a KBS to estimate both the
similarity and the difference between extracted or acquired new knowledge items and those
ones, which already integrated within the knowledge base since the difference and similarity
are inverse concepts. Results of such analysis can be used for the efficient integration of new
knowledge into the knowledge base.
5. Conclusions
To perform the integration of new knowledge into the knowledge base efficiently, 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 difference 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
difference 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 difference 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.
Similar to other universal exploiters, difference exploiter can be adapted to compute the
difference of fuzzy inhomogeneous classes of objects as well as for the difference of fuzzy
homogeneous and inhomogeneous classes of objects. However, such extensions require the
development of the appropriate algorithms for their implementation.
References
[1] R. Bareiss, B. W. Porter, K. S. Murray, Supporting Start-to-Finish Development of Knowl-
edge Bases, Mach. Learn. 4 (1989) 259β283. doi:10.1007/BF00130714.
[2] K. S. Murray, B. W. Porter, Controlling Search for the Consequences of New Information
during Knowledge Integration, in: Proceedings of the 6th International Workshop on
Machine Learning, New York, USA, 1989, pp. 290β295.
[3] K. S. Murray, B. W. Porter, Developing a tool for knowledge integration: initial results, Int.
J. Man-Machine Studies 33 (1990) 373β383.
[4] K. S. Murray, Learning as Knowledge Integration, Ph.D. thesis, Faculty of the Graduate
School, University of Texas at Austin, Austin, Texas, USA, 1995.
[5] K. S. Murray, KI: A Tool for Knowledge Integration, in: Proceedings of the 13th National
Conference on Artificial Intelligence, AAAI 1996, Portland, Oregon, USA, 1996, pp. 835β842.
[6] T. D. Ndousse, Intelligent Systems Modeling with Reusable Fuzzy Objects, Int. J. Intell. Syst.
12 (1997) 137β152. doi:10.1002/(SICI)1098-111X(199702)12:2<137::AID-INT2>
3.0.CO;2-R.
[7] G. Bordogna, G. Pasi, D. Lucarella, A Fuzzy Object-Oriented Data Model for Managing
Vague and Uncertain Information, Int. J. Intell. Syst. 14 (1999) 623β651. doi:10.1002/
(SICI)1098-111X(199907)14:7<623::AID-INT1>3.0.CO;2-G.
[8] L. A. Zadeh, Fuzzy sets, Inform. Control 8 (1965) 338β353. doi:10.1016/S0019-9958(65)
90241-X.
[9] L. A. Zadeh, The concept of a linguistic variable and its application to approximate
reasoning β I, Inform. Sci. 8 (1975) 199β249. doi:10.1016/0020-0255(75)90036-5.
[10] L. A. Zadeh, The concept of a linguistic variable and its application to approximate
reasoning β II, Inform. Sci. 8 (1975) 301β357. doi:10.1016/0020-0255(75)90046-8.
[11] L. A. Zadeh, The concept of a linguistic variable and its application to approximate
reasoning β III, Inform. Sci. 9 (1975) 43β80. doi:10.1016/0020-0255(75)90017-1.
[12] N. MarΓn, O. Pons, M. A. Vila, Fuzzy Types: A New Concept of Type for Managing Vague
Structures, Int. J. Intell. Syst. 15 (2000) 1061β1085. doi:10.1002/1098-111X(200011)15:
11<1061::AID-INT5>3.0.CO;2-A.
[13] N. MarΓn, O. Pons, M. A. Vila, A Strategy for Adding Fuzzy Types to an Object-Oriented
Database System, Int. J. Intell. Syst. 16 (2001) 863β880. doi:10.1002/int.1039.
[14] F. Berzal, N. MarΓn, O. Pons, M. A. Vila, Managing Fuzziness on Conventional Object-
Oriented Platforms, Int. J. Intell. Syst. 22 (2007) 781β803. doi:10.1002/int.20228.
[15] F. Berzal, N. MarΓn, O. Pons, M. A. Vila, Using Classical Object-Oriented Features to Build
a Fuzzy O-O Database System, in: J. Lee (Ed.), Software Engineering with Computational
Intelligence, volume 121 of Studies in Fuzziness and Soft Computing, Springer, 2003, pp.
131β155. doi:10.1007/978-3-540-36423-8_6.
[16] A. Zvieli, P. P. Chen, Entity β Relationship modeling and fuzzy databases, in: Proc. IEEE
2nd Int. Conf. Data Eng., Los Angeles, CA, USA, 1986, pp. 320β327. doi:10.1109/ICDE.
1986.7266236.
[17] Z. M. Ma, W. J. Zhang, W. Y. Ma, Extending object-oriented databases for fuzzy informa-
tion modeling, Information Systems 29 (2004) 421β435. doi:10.1016/S0306-4379(03)
00038-3.
[18] Z. M. Ma, L. Yan, F. Zhang, Modeling fuzzy information in UML class diagrams and
object-oriented database models, Fuzzy Sets Syst. 186 (2012) 26β46. doi:10.1016/j.fss.
2011.06.015.
[19] L. Yan, Z. Ma, F. Zhang, Fuzzy XML Data Management, volume 311 of Studies in Fuzziness
and Soft Computing, Springer, Berlin, 2014. doi:10.1007/978-3-642-44899-7.
[20] Z. Ma, F. Zhang, L. Yan, J. Cheng, Fuzzy Knowledge Management for the Semantic Web,
volume 306 of Studies in Fuzziness and Soft Computing, Springer, Berlin, 2014. doi:10.
1007/978-3-642-39283-2.
[21] D. O. Terletskyi, Run-Time Class Generation: Algorithms for Decomposition of Homoge-
neous Classes, in: A. Lopata, R. ButkienΜe, D. GudonienΜe, V. SukackΜe (Eds.), Information
and Software Technologies. ICIST 2020, volume 1283 of CCIS, Springer, 2020, pp. 243β254.
doi:10.1007/978-3-030-59506-7_20.
[22] D. A. Terletskyi, A. I. Provotar, Fuzzy Object-Oriented Dynamic Networks. I, Cybern. Syst.
Anal. 51 (2015) 34β40. doi:10.1007/s10559-015-9694-0.
[23] D. A. Terletskyi, A. I. Provotar, Fuzzy Object-Oriented Dynamic Networks. II, Cybern.
Syst. Anal. 52 (2016) 38β45. doi:10.1007/s10559-016-9797-2.
[24] D. O. Terletskyi, O. I. Provotar, Algorithm for Intersection of Fuzzy Homogeneous Classes
of Objects, in: Proc. IEEE 2020 15th Int. Sci. Tech. Conf. Comput. Sci. Inform. Technol.
(CSIT), volume 2, Zbarazh, Ukraine, 2020, pp. 314β317. doi:10.1109/CSIT49958.2020.
9321914.
[25] D. O. Terletskyi, O. I. Provotar, Intersection of Fuzzy Homogeneous Classes of Ob-
jects, in: N. Shakhovska, M. O. Medykovskyy (Eds.), Advances in Intelligent Systems
and Computing V, volume 1293 of AISC, Springer, 2020, pp. 306β323. doi:10.1007/
978-3-030-63270-0_21.
[26] P. K. Panigrahi, A. Goswami, Algebra for Fuzzy Object Oriented Database Language, Int.
J. Comput. Appl. 26 (2004) 1β9. doi:10.1080/1206212X.2004.11441721.
[27] L. Yan, Z. M. Ma, Operations in Fuzzy Object-Oriented Databases, in: Proc. 2009 Int.
Conf. Comput. Intell. Softw. Eng., Wuhan, China, 2009, pp. 1β4. doi:10.1109/CISE.2009.
5364610.
[28] L. Yan, Z. M. Ma, F. Zhang, Algebraic operations in fuzzy object-oriented databases, Inf.
Syst. Front. 16 (2014) 543β556. doi:10.1007/s10796-012-9359-8.
[29] Z. M. Ma, F. Mili, Handling Fuzzy Information in Extended Possibility-Based Fuzzy
Relational Databases, Int. J. Intell. Syst. 17 (2002) 925β942. doi:10.1002/int.10057.
[30] T. T. Nguyen, B. V. Doan, C. N. Truong, T. T. T. Tran, A New Approach for Query
Processing and Optimization in Fuzzy Object-Oriented Database, in: V. Bhateja, B. L.
Nguyen, N. G. Nguyen, S. C. Satapathy, D.-N. Le (Eds.), Information Systems Design and
Intelligent Applications, volume 672 of AISC, Springer, 2018, pp. 49β63. doi:10.1007/
978-981-10-7512-4_6.
[31] O. Pivert, H. Prade, A Certainty-Based Model for Uncertain Databases, IEEE Trans. Fuzzy
Syst. 23 (2015) 1181β1196. doi:10.1109/TFUZZ.2014.2347994.
[32] D. V. Thang, Algebraic Operations in Fuzzy Object-Oriented Databases Based on Hedge
Algebras, in: V. E. Cong Vinh P., Ha Huy Cuong N. (Ed.), Context-Aware Systems and
Applications, and Nature of Computation and Communication. ICTCC 2017, ICCASA 2017,
volume 217 of LNICST, Springer, 2018, pp. 124β134. doi:10.1007/978-3-319-77818-1_
12.
[33] Z. M. Ma, J. Liu, L. Yan, Fuzzy Data Modeling and Algebraic Operations in XML, Int. J.
Intell. Syst. 25 (2010) 925β947. doi:10.1002/int.20424.
[34] L. Bai, L. Zhu, An Algebra for Fuzzy Spatiotemporal Data in XML, IEEE Access 7 (2019)
22914β22926. doi:10.1109/ACCESS.2019.2898228.
[35] L. Bai, X. Cao, W. Jia, Uncertain spatiotemporal data modeling and algebraic operations
based on XML, Earth. Sci. Inform. 11 (2018) 109β127. doi:10.1007/s12145-017-0322-6.
[36] L. Zhang, J. Sun, S. Su, Q. Liu, J. Liu, Uncertainty Modeling of Object-Oriented Biomedical
Information in HBase, IEEE Access 8 (2020) 51219β51229. doi:10.1109/ACCESS.2020.
2980553.