My work on cellular automata I began serious work on cellular automata in the middle of 1981. … And in June 1982, I finished my first paper on cellular automata, entitled "Statistical Mechanics of Cellular Automata". … In the early summer of 1985 I was doing consulting at a startup company called Thinking Machines Corporation, which had developed a massively parallel computer called the Connection Machine that was fairly well suited to cellular automaton simulation.

For the different elements in the system are always just laid out in a one-dimensional string, with the result that local replacement rules can only ever propagate effects to nearby elements in the string—much like in a one-dimensional cellular automaton.

Fractal dimensions [of additive cellular automata] The total number of nonzero cells in the first t rows of the pattern generated by the evolution of an additive cellular automaton with k colors and weights w (see page 952 ) from a single initial 1 can be found using g[w_, k_, t_] := Apply[Plus, Sign[NestList[Mod[ ListCorrelate[w, #, {-1, 1}, 0], k] &, {1}, t - 1]], {0, 1}] The fractal dimension of this pattern is then given by the large m limit of Log[k,g[w, k,k m + 1 ]/g[w, k, k m ]] When k is prime it turns out that this can be computed as d[w_, k_:2] := Log[k,Max[Abs[Eigenvalues[With[ {s = Length[w] - 1}, Map[Function[u, Map[Count[u, #] &, #1]], Map[Flatten[Map[Partition[Take[#, k + s - 1], s, 1] &, NestList[Mod[ListConvolve[w, #], k] &, #, k - 1]], 1] &, Map[Flatten[Map[{Table[0, {k - 1}], #} &, Append[#, 0]]] &, #]]] &[Array[IntegerDigits[#, k, s] &, k s - 1]]]]]]] For rule 90 one gets d[{1, 0, 1}] = Log[2, 3] ≃ 1.58 .

Note that if one uses base 6 rather than base 2, then as shown on page 614 powers of 3 still yield a complicated pattern, but all operations are strictly local, and the system corresponds to a cellular automaton with 6 possible colors for each cell and rule {a_, b_, c_}  3 Mod[b, 2] + Floor[c/2] (see page 1093 ).

It is this characteristic length that presumably determines the size of an individual cell in the cellular automaton model.

Inverse [cellular automaton] rules Some reversible rules are self-inverse, so that applying the same rule twice yields the identity.

For example, with three inputs and one output, w = {{-1, +1, -1}} yields essentially the rule for the rule 178 elementary cellular automaton. … Particularly in early investigations of neural networks, it was common to consider systems more like very simple cellular automata, in which the s[i] corresponded not to states of successive layers of neurons, but rather to states of the same set of neurons at successive times.

Implementation [of cellular automaton state networks] One can represent a network by a list such as {{1  2}, {0  3, 1  2}, {0  3, 1  1}} where each element represents a node whose number corresponds to the position of the element, and for each node there are rules that specify to which nodes arcs with different values lead. … = {} Given a set of sequences of values represented by a particular network, the set obtained after one step of cellular automaton evolution is given by NetCAStep[{k_, r_, rtab_}, net_] := Flatten[ Map[Table[# /.

In the typical case of the finite difference method one sets up a system with discrete cells in space and time that is much like a continuous cellular automaton, and then hopes that when the cells in this system are made small enough its behavior will be close to that of the continuous PDE.

Derivation of the diffusion equation With some appropriate assumptions, it is fairly straightforward to derive the usual diffusion equation from a cellular automaton.