Starting from Randomness

An old kind of concept...?

Wolfram describes a CA classification with four classes:

*Class 1 Starting from random initial conditions, these CA converge to a uniform final state. This class also exhibits a resistance to perturbation (not surprising as the same fixed point is reached regardless of the initial conditions).

*Class 2 Starting from random initial conditions, CA from this class converge to one of many fixed points. Pertubations persist, but remain local and are not divergent. In effect, cells that are far away from the perturbation remain unaffected.

*Class 3 These are the pseudo-random CA, that produce random looking output. These CA display divergent behaviour when perturbed. i.e. small changes eventually have widespread effects on most of the system - chaos.

*Class 4 These CA exhibit random like behaviour, with some underlying regularity or nesting. These are the 'complex' CA. They are neither ordered, nor fully random, but seem poised on the boundary of Class 2 and 3 CA. In this sense, they remind me of Stuart Kauffman's K=3 Random Boolean Networks, that exhibit behaviour with both orderly and chaotic characteristics. Perturbations may have either transient effects, or may diverge and have more widespread effects - 'sporadic divergence'.

It is interesting that some class 3 CA can produce random like output even when started from simple, ordered initial conditions. (e.g. rule 30 started from a single black cell eventually produces a disordered, noisy looking output). This suggests that random looking output is not simply a 'translation and rotation' of random initial conditions. Rule 30 is an example of a system that takes ordered input, chews it up and spits out disorder. Or is it? Are we wrongly accusing a single black cell of being a simple initial condition? A single black cell is really a black cell surrounded by many white cells, and is therefore a rather precise position in state space. Should we really have any reason to think that this position in state space would not eventually map to some more complicated position, given that we have partly chaotic dynamics? Its like initialising inverted coupled pendulums with anlges 0 rad and 1 rad. These seem like orderly initial conditioins but the system treats them no differently from 0.000000... and 1.00000...

We can consider the attractors that may account for the behaviour of each class.

• Class 1 CA contain a single fixed point attractor, and a single basin that drains all state space.
• Class 2 CA contain many fixed point attractors, each with a basin draining some portion of state space. Given that class 2 CA exhibit some resistance to perturbation I would expect that these basins are mainly non-overlapping.
• Class 3 CA might be analagous to unstable, chaotic, nonlinear dynamical systems. e.g. inverted coupled pendulums. Or are these just systems with no easily observable attractors?
• Class 4 CA might contain strange attractors. i.e. some underlying regularity, with some superimposed random-like variation.

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