214
V. Babenko et al.
where: C
U —hierarchy of interconnected ODR objects by relations R; D
U —universal
set of value properties; F
U —a universal set of properties-objects, and each propertyobject F
U is the following logical structure < f, t : c
∈ C
U
>, where f—property,
the value type (t) of which the object is a class c
∈ C
U .
Let us consider in more detail the ontology component C
U of the ontological
knowledge base.
This component is the following hierarchical object-oriented structure:
C
U
=< C, R >,
(3)
where C—the set of ODR objects entering R.
To specify the mathematical implementation of the ontology of the ontological
knowledge base, we define the set of relations R as follows:
R = {sup, sub},
(4)
where: sup—the relation “to be an object-superclass”; sub—the relation “to be an
object-subclass”.
We modify the set of R ontologies of the ontological knowledge base by adding the
relationship “be an instance” between objects to organize communication between
class objects and specific instances of these class objects, as follows:
R = {sup, sub, inst}
(5)
where inst—the relation “to be an instance of a class”.
Then the ODR object hierarchy is formally defined as follows:
C
U
=< C, sup, sub, inst >,
(6)
where C—the set of ODR objects entering into relations sup, sub, inst.
Each ODR object represents a formal structure of an ODR entity, defined as
follows:
∀c ∈ C : c =< S, A, B, D, F, E, , >, ,
(7)
where: c—the ODR object; C—the set of ODR objects; S—the symbolic name of
the ODR object; A = {c
∈ C : c
supc}—the set of superclass objects of the
object c; B = {c
∈ C : c
sub c}—the set of subclasses of c; D = {< d 0 , v 0 >
, . . . , < d m , v m >}—the set of properties-values of the object c, where D ⊆ D
U ,
where D
U —the finite set of all properties-values of the ontology; d 0 , d m —symbolic
names of property values; ν 0 , ν m —values of property-values; F = {< f 0 , t 0 : c
∈
C, v 0 : c
inst
∈ E
c
>, . . . , < f q , t q : c
∈ C, v q : c
inst
∈ E
c
>}—the set of
object properties of the object c, where F ⊆ F
U , where F
U —the finite set of all
OKB object properties; f 0 , f q , t 0 , t q , ν 0 , ν q —symbolic names, types and values of
V. Babenko et al.
where: C
U —hierarchy of interconnected ODR objects by relations R; D
U —universal
set of value properties; F
U —a universal set of properties-objects, and each propertyobject F
U is the following logical structure < f, t : c
∈ C
U
>, where f—property,
the value type (t) of which the object is a class c
∈ C
U .
Let us consider in more detail the ontology component C
U of the ontological
knowledge base.
This component is the following hierarchical object-oriented structure:
C
U
=< C, R >,
(3)
where C—the set of ODR objects entering R.
To specify the mathematical implementation of the ontology of the ontological
knowledge base, we define the set of relations R as follows:
R = {sup, sub},
(4)
where: sup—the relation “to be an object-superclass”; sub—the relation “to be an
object-subclass”.
We modify the set of R ontologies of the ontological knowledge base by adding the
relationship “be an instance” between objects to organize communication between
class objects and specific instances of these class objects, as follows:
R = {sup, sub, inst}
(5)
where inst—the relation “to be an instance of a class”.
Then the ODR object hierarchy is formally defined as follows:
C
U
=< C, sup, sub, inst >,
(6)
where C—the set of ODR objects entering into relations sup, sub, inst.
Each ODR object represents a formal structure of an ODR entity, defined as
follows:
∀c ∈ C : c =< S, A, B, D, F, E, , >, ,
(7)
where: c—the ODR object; C—the set of ODR objects; S—the symbolic name of
the ODR object; A = {c
∈ C : c
supc}—the set of superclass objects of the
object c; B = {c
∈ C : c
sub c}—the set of subclasses of c; D = {< d 0 , v 0 >
, . . . , < d m , v m >}—the set of properties-values of the object c, where D ⊆ D
U ,
where D
U —the finite set of all properties-values of the ontology; d 0 , d m —symbolic
names of property values; ν 0 , ν m —values of property-values; F = {< f 0 , t 0 : c
∈
C, v 0 : c
inst
∈ E
c
>, . . . , < f q , t q : c
∈ C, v q : c
inst
∈ E
c
>}—the set of
object properties of the object c, where F ⊆ F
U , where F
U —the finite set of all
OKB object properties; f 0 , f q , t 0 , t q , ν 0 , ν q —symbolic names, types and values of