Generic+CAs

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TODO
 * CLARIFY: last paragraph, what is meant??

=Generic Behaviour of [|Cellular Automata]=

(**Concepts**: classification of CA models (classes), order parameter, almost all cases (highlighted by exceptions), vanishingly small parameter regimes)

In ODEs/MAPS we can generally distinguish three types of dynamical behaviour, that all systems eventually will result in: - Fixed points (equilibria) - Limit cycles - (deterministic) Chaos In a similar way one can ask: **What does an arbitrary CA do for arbitrary initial conditions?** [|Wolfram] ([|1984]) conducted such a study on 1D CAs using random transition rule tables with varying neighbourhoods and found the following classes of CA behaviour (with non-spatial analog in brackets):
 * Class I: to uniform state, all black or all white (fixed state)
 * Class IIa: localized domains (limit cycle)
 * Class IIb: non-localized domains (limit cycle)
 * Class III: non-periodic, non-localized (high dimensional chaos)
 * Class IV: long transient, unpredictable (universal computation)

These classes were defined according to the effect that perturbations have on the behaviour of the system, i.e. what happens to disturbances? In Class I the system returns to the fixed point. In Class II one can change from one attractor to another but the disturbance is limited to the attractor one is in. In Class III disturbances have a non-local impact and [|percolate] throughout the field. This class shows **high-dimensional** [|chaos], but its statistical properties have a short [|transient]. In Class IV disturbances can spread or not, die out or not, can be non-local and have a long transient. In other words they are highly unpredictable. Moreover, new entities with their own behaviour and interactions arise which lead to a new level of description and dynamics. Such entities have been suggested to have the capability of [|universal computation].

//Predictions about CAs//
In strict sense, it is impossible to predict from the rules in which class a CA will fall. However, for //almost all cases//, we can predict the class by using [|Langton] ([|1991])'s ordering parameter //lambda//: //the number of rules which lead to the quiescent state// (which is one of the CA states). He defined //lambda =// (K^N - Nq) / K^N, where K is the number of states, N is the number of neighbours and Nq is the number of rules leading to the quiescent state (i.e. the number of times the quiescent state is found in the transistion vector: the vector of all possible outcomes of the next state function). The ordering parameter //lambda// varies between 0 and (1 - 1/K). Hence, for binary CAs the maximal value of //lambda// is 0.5.

Using this ordering parameter it becomes clear that as //lambda// increases one transverses from Class I -- IIa -- IIb- IV - III, where Class IV is b etween the periodic and chaotic phase and occurs in a vanishingly small parameter region (i.e. a vanishlingly small region of //lambda//-values). This parameter regime clearly shows very interesting and unpredictable behaviour, but if it is so "rare", **is it important to consider?**

Let us consider some examples of the use of the ordering parameter //lambda//:

In Modulo Prime (with //p=2),// K=2, N=4 and Nq=8 (since half of the 16 possible combinations of the four neighbour states will lead to a next state of 0 (quiescent state). Hence, //lambda// = (16 - 8) / 16 = 0.5. Hence, according to such ordering Modulo Prime is class III, and is therefore highly disordered and chaotic. In fact this can be illustrated by tracking the percolation of a single pixel change in a given random initial condition. If such a change is tracked in time it displays a [|fractal pattern] which spreads throughout the field.

The **Voting rule** also has //lambda//=0.5, because again half of the possible neighbour combinations will lead to a next state of 0. However, this case is not chaotic, although it is maximally disordered. It is therefore Class I in most cases (no noise), and with the addition of noise it becomes Class II. Hence, this is one example of a CA which is an **exception** to the classification by ordering parameter //lambda//.

An important point shown here is that generalizations, or model outcomes, are based on **almost all cases** (i.e. there are exceptions to the rule). These should be seen as special cases which can exist, and an **all cases** model should be a different model.

Next: Mean Field Assumption