Nor have we followed the probability computer or statistical decision theory approach exemplified by Braverman (7) because these usually require some sort of preassigned coordinate system (8). Neither will the reader find much indication of formal logic (9) or heuristic (10) programming. Instead, we view a self-organizing system more as a mirror whose appearance reflects the environment rather than its own intrinsic nature. With this viewpoint, a self-organizing system appears very flexible because it possesses few internal constraints which would tend to distort the reflection of the environment and hinder its ability to adapt. 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). [Illustration: Figure 2—Simulation of (a portion of) the environment] [Illustration: 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.