Behavioral Classes

Class cl All sites eventually attain the same value. 

Class c2 Simple stable states or periodic and separated structures emerge. Class c3 Chaotic nonperiodic patterns are generated. 

Class c4 Complex, localized, propagating structures are formed. 

All CA within a given class, regardless of the precise specification of the evolution rule, yield qualitatively similar behavior. Except for rules in the c4 class, the 

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Fig. 3.14 Totalistic d=l, fc = 3, r = 2 rules starting with random initial conditions. 3.1.2 Behavioral Classes...

Table 3.1 (taken from reference [wolfSfib]) gives the approximate fractions of a variety of legal totalistic rules belonging to each of the four behavioral classes. We see that rules generally become more likely to exhibit c3 or c4 behavior for larger A and V, and that c4 behavior does not appear at all in any of the elementary r = 1, k = 2 rules. 

Table 3.1 Approximate fraction of legal totalistic rules belonging to each of the four behavioral classes.

Kaneko [kaneko86a] has, in fact, pointed out that, despite being strictly defined only for infinitely large systems, the four generic behavioral classes appear to be well characterized by the number and length of their attractors on finite lattices ... 

What is the underlying basis for the observed behavioral classes ... 

Table 3.5 Behavioral classes for range-r one-ditnensional systems obeying a variety of T and OT rules. Rules whose behavioral classification changes with V are given at right.

The qualitative nature of the phase boundaries, however, is not simply related to the behavioral class of the deterministic rule corners. For example, while figure 7.7-c shows a phase transition to a class-3 rule, figures 7.7-a and 7.7-b show that the boundaries end at the class-3 rule. Similarly, while in figure 7.7-a the phase transition ends at a class-2 rule, there is only the absorbing stationary state close to the class-2 rule in figure 7.7-c. 

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 set of behaviors that defines the spectrum of a species reactivity is determined by its electronic state, which is dictated by internal and external influences. These influences may be established by the ground state, the excited state, or the effect of an external environment on the state. The chemical-behavior class uses a set of operations that define the chemical behavior of a chemical structure and proceed as follows ... 

Intraperitoneal and Intravenous toxicity of phenols to the mouse depend on small positive slopes of HI with no observable optimum. This simple behavior (Class 1) cannot be causally defined but suggests absorption-desorption from a lipid pool as the rate-limiting step. All other toxicities explored (oral, dermal, sc) to mouse, rat, guinea pig and chicken correlate with positive slopes of Zo and/ or 2,6-effects (Class 2). Diarylamines, anilines and pyrldines also appear to behave as Class 2 toxicants against mice. These reactivity factors Indicate target site expression, consistent with death by irreversible inhibition. 

Class 2 Almost all initial configurations relax after a transient period to some fixed point or some temporally periodic cycle of configurations, but which one depends on the initial configuration. (It should be pointed out that on finite lattices , there is only a finite number k ) of possible configurations, so all rules ultimately lead to periodic behavior. Class 2 refers not to this type of periodic behavior but rather to cycles with periods much shorter than. )... 

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