Wolfram class behavior

Fig. 12.16. Wolfram s behavioral classes cl, c2, c3 and c4 overlayed on top of a plot of Complexity versus Langton s A-parameter see text. Patterned after figures 22 and 23 in [lang92b].

The dependence of the different phase space regimes on A is illustrated in Fig. 9.12, in which the four Wolfram class of behavior are indicated. 

That neither complex nor random-like, behavior, need have a complex dynamical origin is well exemplified by the elementary one-dimensional class c3 rule R30, studied extensively by Wolfram in [wolf84c]. It is defined explicitly by... 

Wolframl discovered analogous behavior from simulation studies with cellular automata. His work shows that, notwithstanding well-defined short-range interaction rules between components on a microscopic level, macroscopic dynamic behavior can become unpredictable. This implies that external disturbances can have an important outcome on both temporal and structural events. This is consistent with a condition of life, where there is change due to evolutionary response. Examples of the four basic classes of behavior Wolfram discovered are shown in Fig. 9.11. 

Hence rules have been set up in which the color of the cell in a row n depends on the coloring of the three neighboring cells in row n — 1. Each system is determined by eight different rules. In total, 256 different choises can be made, generating an amazing variety in pattern behavior. Wolfram distinguishes four classes of behavior. 

At intermediate values of A, a phase transition can occur between periodic and chaotic dynamics. At either end of the A spectrum, behavior seems simple and predictable. It is the behavior in the vicinity of this phase transition that is complex and unpredictable. There are long transients, intermediate and sometimes periodic structures. They are generated and later collapse, very similar to the class 4 system of Wolfram. 

Such models can be developed for the computational design of catalytic sterns that self organize and adapt themselves for optimum catalytic performance. Adaptation occurs in the reproduction process with mutation of the self-replicating molecules coupled to the metabolic system. The metabolic system acts as the bio-immune molecular recognition and response system. The conditions for the emergence of such a stem are far from equilibrium in the complex regime. The behavior as a function of time is unpredictable, similar to the class 4 system proposed by Wolfram. 

The membership requirements for class 4 are not rigorous. Li and Packard (1990) claimed that ECA 110 (Figure 4.2), for example, has typical class 4 behavior. Wolfram (1984) speculated that all class 4 CAs (except the EC As) are capable of universal computation, and that it could be implemented in a way similar to that described above for Life—by building circuits out of propagating localized structures such as gliders. However, since there can be no general method for proving... 

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