Consider the relation between two mothers and their sons as pictured in [Figure 4]. Observe that if symbols (points) are substituted for the actual physical objects (mothers and sons), the relation is not altered in any way. This is what we mean by simulation and this is how a SOM simulates. It is not even necessary that the objects, used to display the relation, be defined; i.e., these objects may be primitive. (If this were not so, no mathematical or physical theory could model the environment.) The main prerequisite is sufficient resolution to distinguish the objects from each other.
Figure 4—A relation of objects—displayed and simulated
MATHEMATICAL MODEL
The mathematical model must represent both the environment and the SOM and for reasons given in the companion paper each is represented as a metrizable topology. For uniqueness we factor each space into equal parts and represent the environment as the channel
W ⟶ X. (Ref. 10a)
Consider now the SOM to be represented by the cascaded channels
X ⟶ Y ⟶ Z
where X ⟶ Y is a variable which represents the reorganization of the SOM existing input-output relation represented by Y ⟶ Z.
The solution of the three channels-in-cascade problem