intro_ca

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=Introduction to [|Cellular automata]=

(**Concepts**: localness (speed of light), mesoscale pattern, simple to complex, predictable to unpredictable, let life live its life)

This is a model formalism where a full model (grid) is constructed as a [|tesselation] of small FSMs, i.e. each grid point on a lattice is its own FSM (). It has in particular been used as a prototype for studying how simple local behaviour can lead to complex behaviour on a larger scale.

To illustrate some properties of CAs we consider the case of a 2-state 1-dimensional CA with a 4 neighbourhood (i.e. the input of each grid point depends on two neighbours to its left and two to its right). In such a CA there are 2 5 possible transition rules (for the grid points) and thus 2 2^5 possible transition rule tables. In contrast if one would take a full 1000 x 1000 grid as an FSM defined on the whole field there would be 2 10^6 rules, which is by far a much greater number of rules to specify (In fact more than the number of atoms in the universe!). Defining rules on a grid point level therefore makes a considerable short cut in the number of rules that need to be specified. This short cut results from the assumption that the transition rule is the same for all grid points (depending on input from its neighbours). Note that each small FSM (gridpoint) requires //input// and is therefore //non-autonomous// in that sense. However, when the whole grid is updated synchronously the full grid is an //autonomous// FSM.

Shifting from global rules to local interactions requires a contracting of possibilities by introducing **localness**, or in other words **[|the speed of light]** (the localness of the universe). This is the case because the introduction of localness ensures that perturbations need time to percolate through the field. In light of our own universe such a restriction makes sense and therefore may be very informative.

Binary rule tables can of course also be represented by a single general rule. Below we discuss two classical examples of CAs and their special properties.

//Modulo Prime//
The rules of modulo prime can be defined for any prime number //p//. The state of each gridpoint is one of the numbers {0, 1, 2, ..., (//p// - 1)} (binary CA if //p// = 2), and the next state function of a cell is defined as follows: - Compute the sum of all neighbours in the von Neumann-4 neighbourhood. - The next state is this sum%//p// (sum [|modulo] //p//).

When this rule is implemented and a specific initial configuration (e.g. a dog) is used, after a given amount of timesteps this doggy will be repeated and replicated at other places in the field. Actually this replication continues into infinity provided one uses an infinite field. Interestingly, because it moves in the grid the pattern (doggy) cannot be recognized on the level of the whole field (macroscale) and neither is it a result of the local or microscale rules (i.e. any initial pattern is replicated). Only on an intermediate scale (**mesoscale**), that was //not predefined// in our model, something special is happening and the pattern can be recognized. This mesoscale pattern (local configuration of cells), which is not an attractor, can be seen as a regularity on this non-predefined scale. Hence there is a clear predictability, but only at this intermediate scale.



//[|Game of Life]//
This rule was originally put forward by **[|John Conway]** **(1970)** and is a simple rule where - a cell remains 1 when it has 2 or 3 neighbours (1s), - or becomes 1 when it has 3 neighbours, - and otherwise becomes 0. This fully deterministic (arbitrary) rule however leads to totally //unpredictable// behaviour, meaning that the final outcome of simulations from different initial conditions cannot be predicted (undecidable), i.e. there is no procedure that can predict whether life will persevere or not, for all possible initial conditions.

In this rule, various mesoscale entities are formed, each with different dynamical properties in time and space (e.g. blinkers, glider gun). Furthermore, this system displays long range interactions that occur between traveling signals (e.g. traveling gliders). Therefore, although the system is locally fully specified, new levels of information processing emerge. However, even when all mesoscale patterns are defined, including their interactions, the system still remains unpredictable. Nonetheless, those mesoscale patterns which have explanatory value are worth identifying since they have (ir)relevance to the behaviour of the system as a whole. **Here we see how very simple local rules lead to (maximally) complex behaviour through the mesoscale patterns they generate.**

This example illustrates one very fundamental point: that fully deterministic systems at a local level can generate unpredictable behaviour at a global level through emergence of novel entities at an intermediate level. This means that in order to find out the particular behaviour of the system for a particular initial condition we must "**let life live its life**". In an answer to the question of why God would let mankind suffer if he is all powerful, **Fratkin** suggested that "**this universe is the fastest simulation for the universe in the eye of god.**"

Next: CA as paradigm system


 * General References**
 * Burks AW** (ed.) (1970) Essays on Cellular Automata. Univ. of Illinois Press 375pp
 * von Neumann J** (1966) Theory of Self reproducing automata. University of Illinois Press 388pp
 * Berlekamp E, Conway JH & Guy R** (1982) Winnings ways for your mathematical plays, Vol 2 Academic Press.

(CHANGELOG 2014-2015) - Clarified definition of a CA - Corrected rule description modulo prime

__(NOTES FROM COURSE 2006-2007)__

__A) Cellular Automata (CA)__ This is a model formalism where each grid is a FSM ( IS = O ) and has been used as a prototype for studying how simple local behaviour can lead to complex behaviour on a larger scale.

//Case study: 2 state 1-D CA with 4 neighbours.// 2 to the 5 transition rules, with 2 to the 2 to the 5 transition rule tables. These are non-automatous (need input). However is 1000 by 1000 grid (autonomous system without localness) there would be 2 to the 10 to the 6 rules (which is more molecules than in the universe). So short cut is needed. How do local FSM create short cut? By contracting possibilities by introducing "localness", or "speed of light" (the localness of the universe). This means that local pertubations take time to percolate. In that sense the restriction of localness makes sense and may be informative with respect to our own universe. Note that is the CA is updated synchronously this generates the global FSM.

Simple local rules to complex Binary rule table can also be represented by a single general rule.

i) Modulo Prime (CountMoore%Numberstates) Observed is a replication of a pattern on an infinite field (e.g. elephants). On the scale of the whole field one cannot recognise the elephant and one doens't see much happening. Neither on the level of the rules. But on an intermediate scale something special is happening (mesoscale). In this case there is a local configuration, which is not an attractor, which can be seen as a regularity on a non-predefined scale. I.e. there is a clear type of predictability, but that only works at some intermediate scale.

ii) Game of Life (Conway). Simple arbitrary rule which is fully deterministic which leads to a totally unpredictable behaviour. Various entities are formed at the mesoscale level, e.g. the glider gun, which is some pattern with a certain type of behaviour. From this rule the survival of the 1s in undecidable (unpredictable), i.e. there is no procedure that can predict for all possible initial states the outcome. Instead one must let Life live its life for each particular initial condition in order to find out. Therefore to explain the dilemna of why an all powerful god would let us suffer: "This universe is the fastest simulation for the universe in the eye of god." (Fratkin).

So why is Life unpredictable? Mainly because of long range interactions that occur between travelling signals. Although fully specified at a local level, information processing on a different level emerges. Even when all mesoscale patterns are defined including their interactions one can still not predict behaviour in the long run. However, those mesoscale patterns which have explanatory value are worth identifying. They are recognized as being (ir)relevant.

Review Q0 (non-advanced) (should this be here or in the ROCK OF GIBRALTAR?) GoL and Prime are particular instances of universes which only show properties of those universes. Spirals on the other hand are very common, but say nothing about the mechanism producing them.

On the other extreme one can have some kind of pattern: if GoL gives that pattern then GoL is THE correct model for that pattern. But if it is so sensitive to disturbance it reduces the confidence on the model relation.

However, SENSITIVE models are still worthwhile: in their role as PARADIGM systems to study WHAT CAN HAPPEN.

In contrast, VOTING shows robust patterns (similar to spirals). The pattern then runs at a certain time scale and shows critically slowing down behaviour, i.e. nearly stable state, which is mostly good enough biologically speaking.

Structural stability is an important feature when we know that a model is wrong: i.e. severly simplified. Example is the classical Lotka-Volterra. This is structurally unstable in a dynamical systems sense because the type of equilibrium changes with model structure. However a similar type of wrong model is used to decribe planet orbits, where for the time scale it is used it is good enough. What we require in a model stability depends on object-model relation (i.e what are the mistakes one accepts to make). Similarly for the mosquite-spider system, changing the model (and its equilibrium) does not really change the ecological consequences of the populations, i.e. these are fairly robust.

When verifying a model: generally CA parameters are more easily measured since they describe observable individuals, while population parameters need to be derived (i.e. for ODE models).