2. The Basic Paradigm of Self-organization

Laszlo [Las 72] itemize four properties of natural systems. We believe artificial self-organizing systems should also have these properties. Thus, they are explained below from our point of view.

• Holisticity: Natural systems are holistic and have irreducible properties, such as mentioned in the previous section. This property may be undesirable for artificial systems, but is inevitable.
• Self-stability: Natural systems preserve themselves in changing environments. This property is also important for artificial systems. Self-stable systems work well even when there is a little noise (some errors) in the system or in the input. Although this property is important in natural systems or automatic control theory, it is usually ignored in computational systems or computation theories.[*2.1]
• Self-organizability: ``Natural systems create themselves in response to the challenge of the environment.'' No further explanation is given here because self-organization has been mentioned earlier.
• Hierarchy: Both natural and artificial systems are hierarchical. However, the hierarchical structure of natural systems is not as simple as tree structures that are common in artificial systems, nor they are static. Such a complex and dynamic (partial) hierarchical structure is called heterarchy [McC65, Foe 84].

A generic macroscopic model of self-organizing systems [Kan 92a], which CCM is based on, is explained here. The model is shown in Figure 1. This model can be applied to a wide range of self-organizing systems, but not all of them. It models our target self-organizing computational system, a thermodynamic system that generates a dissipative structure, and other varieties of self-organizing systems.

Figure 1. Generic model of self-organizing systems

Self-organizing systems generate order in a macroscopic level. The degree of order should be measurable because if it is not, we cannot claim objectively that the system is self-organizing. Thus, entropy or (global) order degree, which is a measure of the order or degree of organization, should be defined. If entropy is defined, it decreases with time. If order degree is defined, it increases with time. If the system is a thermodynamic system, entropy must be ``thrown away'' from the system, because it must satisfy the second law of thermodynamics which states that entropy increases in a closed system. Thus, it must be an open system in this case.[*2.2]

There must be a set of laws or rules that (partially) control how the system changes, though they probably do not control the system in a deterministic way for reasons described later. Probably, the rules work at a microscopic level and their emergent behavior generates complex macroscopic behavior of the system. Thus, the system is hierarchical (or heterarchical). For simplicity in this paper, rules, such as natural laws, are asserted as given and unchanging.[*2.3] There will be various ways of describing the laws or rules. The laws can be written as differential equations in some systems, and may be written as sequences of operations in some others, and so on.

The growth of a self-organizing system is autonomous, and, thus, its behavior is unpredictable, or it is observed as nondeterministic or driven by noise that comes from the outside of the system. However, these properties are not sufficient conditions for self-organization, of course. If the behavior is predictable, i.e., observed as deterministic, it is not a self-organizing system, but the organization is fully controlled by external laws or rules. In particular, in the case of computational systems, deterministic systems are (indirectly) organized by humans, because the rules, i.e., the programs, are written by humans. This does not constitute self-organizing computation. In thermodynamic or other physical systems, nondeterministic system development is called bifurcation or symmetry breaking [Pri 77, Pri 84].