The paper explores the information (conceptual) systems in terms of the hierarchies of conceptual and material systems. Such systems are an objectoriented 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 description of the hierarchy of conceptual systems. It takes into account their systemic 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 formally described the concepts of the volume and content of a conceptual system. It expands the system theory based on a system-object approach.
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 dissemination. In our study, the information system is understood as a material system (i.e., real, objectively existing), organizers, custodians and transforming data. Thus, information is a resource.
For example, a set of data under certain conditions, classification, knowledge model, including ontology. An important example for our research is conceptual systems as semantic networks associated with certain rules of concepts and concepts, or conceptual systems that consist of non-physical objects.
In [
The study and formalization of the structure of such systems as well as the
concepts 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 [
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) [
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
function or role of an object or class of a higher tier (a supersystem), which clarifies the
definition of the system from [
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 function. 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 determinant 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
representing systems-classes to systems-facts from the point of view of a system-object
approach mentioned in [
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 functionally 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
systemsclasses’ hierarchical structure has some differences from the hierarchy of
systemsfacts. 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
wellknown principle of infinity. The hierarchy of system-classes formed by the
genusspecies relation has an upper boundary in accordance with a known logical law the
inverse relationship between the volume and content of concepts (classes) [
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. Moreover, 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 contradicts 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 classes), does not contradict the principle of determinism.
3
3.1
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
mathematical logic. Concepts are used to describe classes, roles – to describe the
relationship between concepts. Concepts and roles allow you to describe classes and their
properties [
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 interpretation, 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 – terminologies, or axioms, as well as knowledge about individual objects, their properties and relationships with other objects – statements about individuals. The DL distinguishes 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 extensions 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
You can define concepts for objects-classes using the DL. Roles in the DL will
correspond 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
hierarchy of conceptual systems, we use the SHOIQ DL [
arbitrary roles; transitive roles (S): axioms of the form arbitrary role, is a transitive role.
are allowed, where R, S are or are allowed, where R is an
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 concepts of “volume” and “content” of a class system.
The volume of the system-class (Vol) is the totality of the species class systems included 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 expressed 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 (propertiesproperties) and volume.
Next, will see the possibility of creating a formal model of the hierarchy of systems-classes (conceptual systems) using descriptive logic that describes the system relationships between classes. In accordance with the system (system-object) approach a system is considered both as a fact (material object) and as a class (conceptual 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 supersystemclass). A formalized description of this understanding of the system using the notation adopted in the DL is as follows:
In expression (1), a formal description of the system is presented in accordance
with the Abadi-Kardeli calculus [
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” [
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 hierarchy; 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
. 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):
From (1) it is known that properties-classes (functional roles) support the functional 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 supporting 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 operation. , 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
(properties- classes) that are different from the generic ones. This is necessary for
constructing subsequent tiers of the hierarchy and correlates with the logical law of the inverse
relationship of volume and content [
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 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 conceptual (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 implementation 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 approach.