* Part of this research was done at Central Research Laboratory, Hitachi Ltd.

[*1.1] There may be no final state if the problem solving is an infinite process, such as that in human society. Even the state-space may be impossible to be predefined in such cases.

[*2.1] In this context, natural systems include social systems which are developed naturally but not artificially. Artificial systems based on system theories, such as automatic controlling systems with negative feedback (process controllers and so on), are self-stable. However, computational systems are not self-stable. Computational systems may crash due to a very small bug or unexpected input, which are types of noise.

[*2.2] However, systems with no such law, such as computational systems, may be closed systems.

[*2.3] ``Self-organization'' does not mean growth of rules in this simplified model. However, growth of rules is considered to be important in open systems and thus it should be included in our future theory.

[*3.1] The word ``program'' is not used in this model because a program is a whole and complete plan. We use ``cast'' and ``caster'' instead.

[*3.2] Although molecules possess a hierarchical structure, which is important in a complex system, molecules currently play no important role in CCM, so we make no further mention of them in this paper.

[*3.3] Instances are not elements of a conflict set. The system interpreter for CCM does not generate conflict sets.

[*3.4] LODs may be analogical to chemical binding energy of atoms.

[*3.5] This condition, IODbefore <= IODafter, may be replaced by the following condition: IODbefore < IODafter.

[*3.6] LODs must be defined to satisfy this condition.

[*4.1] The LHS and RHS of this rule can be reversed and this rule can also be a reversible rule, because the LHS and RHS are symmetric, i.e., they are identical if variables r2 and r3 are exchanged.

[*4.2] Shimizu and Hayashi [Shi 90], and Sosic [Sos 91] independently developed methods of solving the N queens problem by swapping columns. However, a global evaluation function that is equal to the GOD, explained in Section 5.1, is used in these methods.

[*4.3] It can be proved that this caster (program) is correct, that means, a solution is obtained every time the system terminates. However, it is much easier to write a caster that tests the correctness of the result.

[*4.4] We do not call SOOC (Self-Organization-Oriented Computing) a programming language because it is not a language to describe a complete procedure or a complete specification for computation. SOOC is a textual language. The current SOOC (SOOC-92) compiler and interpreter are written in Common Lisp.

2 GOD is not computed nor used in CCM, but it may be used in macroscopic models.

3 GOD may be analogical to total energy or entropy of chemical systems, and may be analogical to energy of Hopfield networks.

[*5.1] CCM-based systems maximize the GOD when their target is to solve constraint satisfaction or optimization problems. However, they may work in other way if their target is different.

1 Sosic [Sos 91] states that there is no local maxima when N >= 1000.

2 A conflicting system may have cooperative local maxima, where no reaction decreases the GOD. Thus, the condition that the system is a conflicting system is not a sufficient condition for this characteristic.

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(C) Copyright 1994 by Yasusi Kanada and IEEE
Y. Kanada