=Paper=
{{Paper
|id=Vol-2919/paper17
|storemode=property
|title=Formalization of Conceptual Classification Models of Knowledge
|pdfUrl=https://ceur-ws.org/Vol-2919/paper17.pdf
|volume=Vol-2919
|authors=Sergey Matorin,Vladimir Mikhelev
}}
==Formalization of Conceptual Classification Models of Knowledge==
https://ceur-ws.org/Vol-2919/paper17.pdf
Formalization of Conceptual Classification Models of
Knowledge
Sergey Matorin 1[0000-0003-1800-4424], Vladimir Mikhelev 1[0000-0001-9163-8269]
1
National Research University "Belgorod State University",
85 Pobeda St., Belgorod 308015, Russia
matorin@bsu.edu.ru,keeper121@ya.ru
Abstract. The paper explores the information (conceptual) systems in terms of
the hierarchies of conceptual and material systems. Such systems are an object-
oriented from the system-object approach side. The conceptual systems of
which are external systems (systems-classes) that determine the properties of
specific objects and objects are material or internal systems (facts systems) that
implement real interactions. The paper also considers the results of a formal de-
scription of the hierarchy of conceptual systems. It takes into account their sys-
temic relationships using a description logic for formalizing certain states of the
system-object approach. The syntax and semantics of the ALCOIQ descriptive
logic and its extension to the SHOIQ logic are described. Introduced and for-
mally described the concepts of the volume and content of a conceptual system.
It expands the system theory based on a system-object approach.
Keywords: information systems, system-object approach, conceptual systems,
systems-classes, material systems, systems-facts, functional query, external de-
terminant, description logic.
1 Introduction
Nowadays, information systems are widely used in various fields of human activity.
Such systems are used to store, search and process information. They include general
organizational resources (human, technical, financial, etc.) and information dissemi-
nation. In our study, the information system is understood as a material system (i.e.,
real, objectively existing), organizers, custodians and transforming data. Thus, infor-
mation is a resource.
For example, a set of data under certain conditions, classification, knowledge mod-
el, including ontology. An important example for our research is conceptual systems
as semantic networks associated with certain rules of concepts and concepts, or con-
ceptual systems that consist of non-physical objects.
In [1, 2] substantiated, the importance of studying such systems and developing
principles applicable to both material and conceptual systems for constructing general
systems theories as well as for overcoming the omissions that separate the natural
sciences and the humanities.
Copyright © 2021 for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0
International (CC BY 4.0).
Proceedings of the of the XXIII International Conference "Enterprise Engineering and Knowledge Management"
(EEKM 2020), Moscow, Russia, December 8-9, 2020.
The study and formalization of the structure of such systems as well as the con-
cepts of a system-object approach system-class and property-class are described with
the concepts of descriptive logic. Description logics are a family of languages for the
formal description of knowledge [3]. In this sense, the use of DL allows you to de-
scribe the elements of the system using some logical expressions. The structure of the
hierarchy of class systems is formally substantiated by describing the syntax and se-
mantics of the ALCOQ and SHIOQ DLs. The mandatory implementation of the
monocentrism principle for conceptual systems is shown. The concepts of the volume
and content of system-classes are introduced and described by means of DL. These
states expand the system theory based on the system-object approach. The results
obtained in the future will allow us to improve existing and create new classifiers.
From the point of view of the authors of this article the use of this approach is a prom-
ising direction for system-object approach.
2 Conceptual Systems in Terms of System-Object Approach
A feature of the system-object approach is the consideration of two fundamentally
different types of systems: internal systems (material systems according to Ackoff)
and external systems (conceptual systems according to Ackoff) [1, 4]. We adopted the
terms “systems-fact” and “systems-classes” in accordance with [5] using system-
object approach [6, 7]. Both ways of systems formation (internal and external) corre-
spond to the basic dialectical principles of the system approach: integrity, systemicity,
hierarchy and development presented in [8]. In addition, in [9] it was demonstrated
that the main well-known system-wide regularities are fulfilled both for systems-facts
and for systems-classes.
In connection with this feature and system-object approach, a system is considered
as a functional object or class, a function or role of which is determined by the func-
tion or role of an object or class of a higher tier (a supersystem), which clarifies the
definition of the system from [11].
A system’s function mentioned in this definition is considered a function of a
supersystem as a functional request of a supersystem to a system with a specific func-
tion. It is called the external determinant of the system. The external determinant is
the cause of the creating of the system the purpose of its existence and the main de-
terminant of its structural, functional and substantial properties. Thus, the external
determinant of the system is considered as a universal system-forming factor.
The process of functioning of a system is its internal determinant, since it directly
determines its internal properties (properties of subsystems). Correspondence of the
internal determinant of the system to its external determinant establishes between the
system and the supersystem the relation of maintaining the functional ability of the
whole.
In addition to the above-mentioned correspondence of conceptual systems repre-
senting systems-classes to systems-facts from the point of view of a system-object
approach mentioned in [6, 7, 9]. Next, it's necessary to consider some features of con-
ceptual systems as classes, which are systems (i.e., systems-classes).
The systemicity of such systems-classes is due to the fact that each class depends
on the functional ability of a class of a higher tier. For clarity, we give an example:
the classes “car” and “truck” functionally support the class “vehicle” as types of road
transport i.e. are systems (subsystems of the “vehicle” system). For example, classes
“green car” and “blue car” are formally also species of the same class, but functional-
ly does not depend from class “vehicle”, therefore, they are not systems (subsystems).
Similarly, for systems-facts: the engine as part of the car functionally depends on the
car and is its subsystem. At the same time, a piece cut from a car is also a part of the
car, but does not support it functionally and, therefore, is not a subsystem of it.
From the point of view of this study, it is important to emphasize that systems-
classes’ hierarchical structure has some differences from the hierarchy of systems-
facts. This feature consists in the fact that the hierarchy of system-facts formed by the
part-whole relation. It does not have an upper boundary in accordance with the well-
known principle of infinity. The hierarchy of system-classes formed by the genus-
species relation has an upper boundary in accordance with a known logical law the
inverse relationship between the volume and content of concepts (classes) [5, 15]. The
fact is that the said law requires a reduction in content, i.e. reducing the amount of
information, which corresponds to the number of features describing the content of
the class, with an increase in the volume of the class, i.e. the number of subclasses
that make up the class. Moreover, the content, of course, can only decrease to zero.
This determines the upper boundary of the hierarchy of systems-classes (conceptual
systems).
These features are essential for our study for the reason that the basic properties of
any system (including a system-class) are determined by a supersystem (in this case, a
supersystem-class). The reason for the existence of a system in accordance with the
system-object approach is functional request of the supersystem. Those the reason for
the presence of certain properties of the system is determined by the hierarchy. More-
over, the analysis of the hierarchy of systems-facts, due to its infinity, does not allow
us to determine the final reason for the presence of system properties, which contra-
dicts the principle of determinism. An analysis of the hierarchy of systems-classes
allows you to determine the final reason for the presence of system properties due to
the finiteness of this hierarchy. Thus, the hierarchy of systems-classes, which does not
contradict the provision on the infinity of the world (in terms of the volume of clas-
ses), does not contradict the principle of determinism.
3 Application of Description Logic
3.1 Description logic basis
Descriptive logic (DL) is a knowledge representation language for describing the
concepts of a subject area in an unambiguous, formalized form. Any DL has syntax
and semantics. The basic syntactic elements of the language of descriptive logic are
the atomic concept and role, corresponding to predicates of the language of mathe-
matical logic. Concepts are used to describe classes, roles – to describe the relation-
ship between concepts. Concepts and roles allow you to describe classes and their
properties [21]. One of the basic descriptive logics is ALC [3, 22]. The syntax of the
ALC logic is presented below in short form:
The symbols and are concepts (called true and false). A – atomic concept, C, D
– arbitrary concepts. R is the atomic role.
The semantics of DL is described using the concept of interpretation. Interpretation
is a pair , consisting of a nonempty set, called the domain of this interpreta-
tion, and an interpreting function I that compares: for each atomic concept, is
an arbitrary subset of is the set of all concepts; each atomic role
is an arbitrary subset of is the set of all roles.
Theories that describe knowledge bases distinguish common knowledge about
concepts and their relationships, which are expressed using general statements – ter-
minologies, or axioms, as well as knowledge about individual objects, their properties
and relationships with other objects – statements about individuals. The DL distin-
guishes a set of terminological axioms called TBox, and a set of statements about the
relationships and properties of individuals – ABox. Together they form a knowledge
base, or ontology .
In [29], an extension of the ALC logic to ALCOIQ is presented. The following ex-
tensions are introduced in this logic: nominals (O) – representation of the individual
in the form of a concept. If a is an individual, then is a concept. Thus, individual
names enclosed in braces become full-fledged concepts. inverse roles (I). If R is an
atomic role, then R – is the inverse role; numerical limitations (Q).
Each new symbol of the designation of logic means some expansion of it. Using
these extensions individually, they say that we get the family of
logics. L – Logic lying in the interval, belonging to this family.
3.2 Formalization of system-class hierarchy
You can define concepts for objects-classes using the DL. Roles in the DL will corre-
spond to the properties-classes. However, expressing the logic of ALCOIQ will not be
enough to describe the hierarchy of roles. To solve the problem of constructing a hier-
archy of conceptual systems, we use the SHOIQ DL [3]. This DL extends ALCOIQ
and has axioms for RBox roles (similar to TBox and ABox), which allows us to de-
scribe the hierarchy of roles as class systems. ALCOIQ logic expands with the follow-
ing points:
hierarchy of roles (H) – axioms of the form are allowed, where R, S are
arbitrary roles;
transitive roles (S): axioms of the form or are allowed, where R is an
arbitrary role, is a transitive role.
In SHOIQ DL, axioms for RBox R roles are added to TBox and ABox, i.e.
knowledge base . Moreover, it is said that R is a sub-role
of S, and S is a super-role of R.
However, it is necessary to expand the SHOIQ logic to justify the structure of the
hierarchy of conceptual systems. Will do it by formally introducing into it the con-
cepts of “volume” and “content” of a class system.
The volume of the system-class (Vol) is the totality of the species class systems in-
cluded in this system-class, which is generic to them.
The content of the system-class (Cont) includes the supersystem-class (generic
class) as well as a set of distinctive features (roles in the supersystem) of this class
system.
We describe these concepts in terms of DL. The content of the system-class is ex-
pressed through the role that supports the functional ability of the class super-system,
as well as through the supersystem-class itself.
,
where , is the number of the tier of the hierarchy; – numbers inside
the same tier of the hierarchy.
The concept of the volume of a class system can be described using the operation
of combining concepts:
,
where, , , – the number of nodes of the i-level hierarchy.
Roles are also class systems. Therefore, they also have content (properties-
properties) and volume.
Next, will see the possibility of creating a formal model of the hierarchy of sys-
tems-classes (conceptual systems) using descriptive logic that describes the system
relationships between classes. In accordance with the system (system-object) ap-
proach a system is considered both as a fact (material object) and as a class (concep-
tual system). The function or role of this class is determined by the function of the
fact or the role of the class of a higher tier (i.e., the supersystem-fact or supersystem-
class). A formalized description of this understanding of the system using the notation
adopted in the DL is as follows:
. (1)
In expression (1), a formal description of the system is presented in accordance
with the Abadi-Kardeli calculus [19]. There are and are a system-class
to indicate a system-class (node) of a higher tier of the hierarchy. Thus,
– a method corresponding to the role (function) of the system in the super-
system ; is a functional role (property-class) that supports the functional abil-
ity of a supersystem-class (concept).
Earlier, one of the basic system-wide principles was mentioned, the principle of
monocentrism, which directly determines that a stable system “will be characterized
by a single center and if it is a complex then it has one higher, common center” [16].
This principle is a consequence of the hierarchical ordering of systems, in our case,
the hierarchical structure of generic relations between systems-classes (conceptual
systems).
The following are statements that substantiate this principle and the relationship
structure of conceptual systems.
Statement 1. If the system-class is depending on of a system-class of a higher tier
and the properties (properties-classes) of a class system are depending on a type of
properties (properties-classes) of a system-class of a higher tier, then this hierarchy
has one root.
Let there exist systems-classes and ; i is the number of the tier of the hier-
archy; j is the ordinal number of the node in the tier and is the serial number of the
supersystem in the tier. In terms of the SHOIQ: – concept, – role (functional
role). Suppose that there are systems-classes (descendants) in , i.e.
. Let there exist systems-classes (properties-classes)
occurring in , . We describe the fragments TBox and
RBox corresponding to the above in the form of expression (2) (see Figure 2):
; . (2)
From (1) it is known that properties-classes (functional roles) support the function-
al ability of the super-system . Therefore, each class system must have supporting,
functional roles that determine the purpose of this system. In turn, is also a
system-class and is a form of . is a supersystem-class i.e. it must have sup-
porting features (properties-properties). We describe the system-class in terms of
the logic SHOIQ. We get a concept that can be described using the intersection opera-
tion.
,
where is the serial number of the supersystem class (property-class), with respect to
; is the serial number of the class system in the tier, with respect to .
We refine the above expression (2) in accordance with the definition of the system
in the form of expression (1) (see Figure 2):
; . (3)
As mentioned earlier, systems-classes must have species characteristics (proper-
ties- classes) that are different from the generic ones. This is necessary for construct-
ing subsequent tiers of the hierarchy and correlates with the logical law of the inverse
relationship of volume and content [19]. In accordance with this law a system-class
inherited from the current system-class must have a large number of species charac-
teristics, i.e. a large content , but a smaller volume
. Applying this law in its entirety to the entire hierarchy of
systems, the following relationships should be satisfied:
(4)
(5)
From (4) it follows that if you move along the tiers then each parent system-class
should have fewer content than the current one, therefore, have a larger volume. It
also follows from (5) that the number of features decreases to the limit state. This
state will be the case when the content is the most complete, in this case
. Moreover, we can talk about the root system-class . This confirms the unity of
the top of the classification scheme and statement 1 (see Figure 1).
S0
S110 RS110
...
Fig. 1. The root of system-class’s hierarchy
Statement 2. The root of the hierarchy of systems-classes is divided into systems-
classes, which are objects-classes and properties-class.
Let there be a root system-class , which has no parents. Suppose that it has two
descendants (systems-classes) and . The characteristics will be next
; .
In [18] – an extremely wide role, corresponding to the class “property”. In addi-
tion, this is consistent with the work of Melnikov [11], which describes the separation
of properties into boundary and qualitative, which can be correlated with our reason-
ing. It is fair to say that in this case relations (4) and (5) will also take a place. This
confirms the structure of the hierarchy of conceptual systems and statement 2.
Conclusion
The initial reason for the existence of systems and the presence of certain properties
in them is due to the hierarchy of conceptual or external systems (systems-classes).
Thus, reality is an object-oriented system. The classes of the system represent concep-
tual (external) systems-classes that determine the properties of objects. Objects –
material (internal) systems-facts that carry out real interactions.
The concepts of the system-object approach “system-class” and “property-class”
are unambiguously compared with the concepts of descriptive logic. The syntax and
semantics of the descriptive logic ALCOIQ and its original extension SHOIQ allow us
to justify the structure of the hierarchy of class systems and the mandatory implemen-
tation of the principle of monocentrism for conceptual systems. The introduction of
the concepts of “volume” and “content” of class systems and their description by
means of descriptive logic extends a system theory based on a system-object ap-
proach.
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