Relating Domain Concepts Intensionally by Ordering Connections - Formal Methods and Hybrid Real-Time Systems - page 191

Concept relations are an important part of domain and requirements model.
Some concept relations define concept specialization and generalization. We shall
say that such relations stand between concepts of the same kind as they are kind-
of relations. Other concept relations hold between what we shall call concepts of
distinct kinds . That is, concept pairs which are not naturally related by kind-of
relations. The distinction has been treated further in [2] and [3].
Several domain and requirements acquisition methods provide notations for
expressing both kinds of concept relations. Basically, we can classify concept
relation approaches in four levels concerning rigidity and formality. On the first
level, we have graphical and other informal notations like UML. Such approaches
define implicit concept relations . Hence, it is not possible (without further for-
malization) to check nor justify that two arbitrarily chosen objects satisfy the
criteria of the concept relation. On the second level, we have modelling ap-
proaches which relate concepts by expressing simple criteria for distinct concept
objects to relate. When modelling databases, such criteria are expressed in join
structures. However, the criteria are merely that objects (here tuples) have iden-
tical values on certain fields. The concept relation is here explicit, but still very
simple as the objects are related based on having properties in common. Not all
domain concepts have that. Hence, there would be concepts that could not be
related this way. On the third level, we have approaches which offer a full math-
ematical range of expressions for formalizing the criteria. The approaches (as in
this paper) may include facilities for implicitly expressing the change effects on
objects. Relations on this level are explicit concept relations . On the fourth level
we have approaches which — in addition to what is on the third level — also
facilitate expressing how objects of one kind can be calculated explicitly from
objects of other concepts. However, this is certainly not be possible for objects
of all kinds of domain concepts. The approach taken in this paper belongs to the
third level; including the mentioned facilities.

3 Intensional Modelling: A Proposed Method

This section outlines a proposed method for relating domain concepts formally.
The method is motivated by the analysis from Sect. 1 and draws on the defi-
nitions from Sect. 2; namely (i) a Galois connection between sets of objects of
the two concepts considered, and (ii) an order-preserving connection between
classifications of objects of the two concepts. We shall refer to these as the char-
acteristics connection and the information flow connection , defined as follows:
Definition 4 (Characteristics Connection). By a characteristics connec-
tion , we understand a relation between objects and their common properties.
The connection is founded by a Galois connection. In Formal Concept Analysis,
a context relates objects to properties. The Galois connection is then utilized for
defining general concepts by clustering objects and properties [1].
In this paper, we shall consider the characteristics connection between objects
and the common properties these have in virtue of standing in a certain relation
to objects of another concept.