7. Concluding Remarks

The CCM-based systems shown in this paper can be interpreted as self-organizing systems in a sense. However, they are still far from our goal of real self-organizing system, which is even hard to be defined. This paper contributes to this grand challenge in the following points.

The self-organization paradigm is explained, and the relation between this paradigm and computation from local information to global result via emergent behavior are explained.
A computation model, CCM, which will lead us to a better understanding of self-organizing systems, is proposed, and a ``programming'' style using local operations and local evaluation functions is shown.
An emergent property that a solution can be found without falling into local maxima is reported using the N queens system, for example.
Methods of controlling the locality of computation by adding catalysts to rules or composing rules in CCM are proposed, and their effects are explained.

The main focuses of future work are as follows. First, the N queens problem and other problems mentioned in this paper are very simple and basically closed problems, that means we can write specifications for these problems. We have to develop CCM-based open systems for problems that are not just constraint satisfaction nor optimization problems, and to observe and to analyze more complex emergent properties in those systems. Second, rules and order degrees are invariant in the model shown in this paper. However, not only the rules but also the goal or target of computation, which is related to the value of evaluation functions, may change during computation in real self-organizing systems. Thus, CCM should be extended to express self-organization or reflection of rules or LODs.

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Y. Kanada