CONCEPTUAL MODEL
Definition
A system is said to be self-organizing if, after observing the input and output of an unknown phenomenon (transfer relation), the system organizes itself into a simulation of the unknown phenomenon.
Implicit in this definition is the requirement that the self-organizing machine (SOM) not possess a preassigned coordinate system. In fact it is just this ability to acquire that coordinate system implicit in the input-output spaces which define the phenomenon that we designate as self-organization. Thus any a priori information programmed into the SOM by means of, for example, stored or wired programs, constrains the SOM and limits its ability to adapt. We do not mean to suggest that such preprogramming is not useful or desirable; merely that it is inconsistent with the requirement for self-organization. As shown in [Figure 2], it is the given portion of the environment which the SOM is to simulate, which via the defining end spaces, furnishes the SOM with all the data it needs to construct the coordinate system intrinsic to those spaces.
The motivation for requiring the ability to simulate as a feature of self-organization stems from the following examples.
Consider the operation of driving an automobile. [Figure 3] depicts the relation characterized by a set of inputs; steering, throttle, brakes, transmission, and a set of outputs; the trajectory. Operation of the automobile requires a device (SOM) which for a desired trajectory can furnish those inputs which realize the desired trajectory. In order to provide the proper inputs to the automobile, the SOM must contain a simulation of ⨍⁻¹(x).
Figure 2—Simulation of (a portion of) the environment
Figure 3—Simulation of a relation
Since ⨍(x) is completely defined in terms of the inputs and the resulting trajectories, exposure to them provide the SOM with all the information necessary to simulate ⨍⁻¹(x). And if the SOM possesses internal processes which cause rearrangement of the input-output relation of the SOM to correspond to ⨍⁻¹(x) in accordance with the observed data, the SOM can operate an automobile. It is this internal change which is implied by the term “self-organizing,” but note that the instructions which specify the desired organization have their source in the environment.
As a second example consider adaptation to the environment. Adapt (from Webster) means: “to change (oneself) so that one’s behavior, attitudes, etc., will conform to new or changed circumstances. Adaptation in biology means a change in structure, function or form that produces better adjustment to the environment.” These statements suggest a simulation because adjustment to the environment implies survival by exposing the organism to the beneficial rather than the inimical effects of the environment. If we represent the environment (or portion thereof) as a relation as shown in [Figure 2], we note that the ability to predict what effect a given disturbance will have is due to a simulation of the cause-effect relation which characterizes the environment.
It would be a mistake to infer from these examples that simulation preserves the appearance of the causes and effects which characterize a relation. We clarify this situation by examining a relation and its simulation.
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