And it turns out to be rather straightforward to generalize cellular automaton fluids to handle these.

And by late 1985, cellular automaton fluids were generating considerable interest throughout the fluid mechanics community. … But in practice over the years since 1985, cellular automaton methods have grown steadily in popularity, and are now widely used in physics and engineering. Yet despite all the work that has been done, the fundamental issues about the origins of turbulence that I had originally planned to investigate in cellular automaton fluids have remained largely untouched.

For we just used a simple cellular automaton rule, and just started from a simple initial condition containing a single black cell. … The next two pages [ 29 , 30 ] show progressively more steps in the evolution of the rule 30 cellular automaton from the previous page .

At some level the situation will no doubt be a little like in the evolution of a typical class 4 cellular automaton, as illustrated below. … But as a first approximation, one can consider A collision between localized structures in the rule 110 class 4 cellular automaton.

And indeed one can already see very much the same kind of thing going on in a simple system like the cellular automaton below. For A cellular automaton whose behavior seems to show an analog of free will.

So as an example the picture on the facing page shows how one type of problem about a so-called non-deterministic Turing machine can be translated to a different type of problem about a cellular automaton. … So what about the cellular automaton below in the picture?

[Examples of] reducible systems The color of a cell at step t and position x can be found by starting with initial condition Flatten[With[{w = Max[Ceiling[Log[2, {t, x}]]]}, {2 Reverse[IntegerDigits[t, 2, w]] + 1, 5, 2 IntegerDigits[x, 2, w] + 2}]] then for rule 188 running the cellular automaton with rule {{a : (1 | 3), 1 | 3, _}  a, {_, 2 | 4, a : (2 | 4)}  a, {3, 5 | 10, 2}  6, {1, 5 | 7, 4}  0, {3, 5, 4}  7, {1, 6, 2}  10, {1, 6 | 11, 4}  8, {3, 6 | 8 | 10 | 11, 4}  9, {3, 7 | 9, 2}  11, {1, 8 | 11, 2}  9, {3, 11, 2}  8, {1, 9 | 10, 4}  11, {_, a_ /; a > 4, _}  a, {_, _, _}  0} and for rule 60 running the cellular automaton with rule {{a : (1 | 3), 1 | 3, _}  a, {_, 2 | 4, a : (2 | 4)}  a, {1, 5, 4}  0, {_, 5, _}  5, {_, _, _}  0}

Implementation [of TM cellular automaton] Given a non-deterministic Turing machine with rules in the form above, the rules for a cellular automaton which emulates it can be obtained from NDTMToCA[tm_] := Flatten[{{_, h, _}  h, {s, _c, _}  e, {s, _, _}  s, {_, s, c[i_]}  s[i], {_, s, x_}  x, {a[_, _], _s, _}  s, {_, a[x_, y_], s[i_]}  a[x, y, i], {x_, _s, _}  x, {_, _, s[i_]}  s[i], Map[Table[With[{b = (# 〚 Min[Length[#], z] 〛 &)[ {x, #} /. tm]}, If[Last[b]  -1, {{a[_], a[x, #, z], e}  h, {a[ _], a[x, #, z], s}  a[x, #, z], {a[_], a[x, #, z], _}  a[b 〚 2 〛 ], {a[x, #, z], a[w_], _}  a[b 〚 1 〛 , w], {_, a[w_], a[x, #, z]}  a[w]}, {{a[_], a[x, #, z], _}  a[b 〚 2 〛 ], {a[x, #, z], a[w_], _}  a[w], {_, a[w_], a[x, #, z]}  a[b 〚 1 〛 , w]}]], {x, Max[Map[# 〚 1, 1 〛 &, tm]]}, {z, Max[Map[Length[# 〚 2 〛 ] &, tm]]}] &, Union[Map[# 〚 1, 2 〛 &, tm]]], {_, x_, _}  x}]

Cellular Automata…Cellular Automata The cellular automata that we have discussed so far in this book are all purely one-dimensional, so that at each step, they involve only a single line of cells. But one can also consider two-dimensional cellular automata that involve a whole grid of cells, with the color of each cell being updated according to a rule that depends on its neighbors in all four directions on the grid, as in the picture below. The form of the rule for a typical two-dimensional cellular automaton.

Such rules can be constructed by taking ordinary cellular automata and adding dependence on colors two steps back. … The next two pages [ 438 , 439 ] show examples of the behavior of such cellular automata with both random and simple initial conditions. An example of a cellular automaton that is explicitly set up to be reversible.