As a reduced analog of algorithmic information theory one can for example ask what the simplest cellular automaton rule is that will generate a given sequence if started from a single black cell. Page 1186 gives some results, and suggests that sequences which require more complicated cellular automaton rules do tend to look to us more complicated and more random.

Note (e) for Cellular Automata…Ulam systems Having formulated the system around 1960, Stanislaw Ulam and collaborators (see page 877 ) in 1967 simulated 120 steps of the process shown below, with black cells after t steps occurring at positions Map[First, First[Nest[UStep[p[q[r[#1], #2]] &, {{1, 0}, {0, 1}, {-1, 0}, {0, -1}}, #] &, ({#, #} &)[{{{0, 0}, {0, 0}}}], t]]] UStep[f_, os_, {a_, b_}] := {Join[a, #], #} &[f[Flatten[ Outer[{#1 + #2, #1} &, Map[First, b], os, 1], 1], a]] r[c_]:= Map[First, Select[Split[Sort[c], First[#1]  First[#2] &], Length[#]  1 &]] q[c_, a_] := Select[c, Apply[And, Map[Function[u, qq[#1, u, a]], a]] &] p[c_]:= Select[c, Apply[And, Map[Function[u, pp[#1, u]], c]] &] pp[{x_, u_}, {y_, v_}] := Max[Abs[x - y]] > 1 || u  v qq[{x_, u_}, {y_, v_}, a_] := x  y || Max[Abs[x - y]] > 1 || u  y || First[Cases[a, {u, z_}  z]]  y These rules are fairly complicated, and involve more history than ordinary cellular automata. … Ulam also in 1967 considered the pure 2D cellular automaton with outer totalistic code 12 (though he stated its rule in a complicated way).

Properties of [initially random cellular automaton] patterns For a random initial condition, the average density of black cells is exactly 1/2.

Self-similarity of additive [cellular automaton] rules The fact that rule 90 can emulate itself can be seen fairly easily from a symbolic description of the rule.

[Structures in] the Game of Life The 2D cellular automaton described on page 949 supports a whole range of persistent structures, many of which have been given quaint names by its enthusiasts. … If one looks at the history of a single row of cells, it typically looks much like the complete histories we have seen in 1D class 4 cellular automata. … As in other class 4 cellular automata, there are structures in Life which take a very long time to settle down.

So as a simple idealization, one can consider having a large number of particles move around on a fixed discrete grid, and undergo collisions governed by simple cellular-automaton-like rules.

For the patterns we saw are in effect built according to very simple plans—that just tell us to start with a single black cell, and then repeatedly to apply a simple cellular automaton rule.

But if the behavior of the system is computationally irreducible—as I suspect is so for the cellular automaton on the facing page and for many other systems with simple underlying rules—then the point is that ultimately no such shortcut is possible.

And in fact one might consider this not all that unlikely for the kind of fairly straightforward exhaustive searches that I ended up using to find the cellular automaton rules in the pictures on the previous page .

Note (e) for Emulating Other Systems with Cellular Automata…Sequential substitution systems [from cellular automata] Given a sequential substitution system with rules in the form used on page 893 , the rules for a cellular automaton which emulates it can be obtained from SSSToCA[rules_] := Flatten[{{v[_, _, _], u, _}  u, {_, v[rn_, x_, _], u}  r[rn + 1, x], {_, v[_, x_, _], _}  x, MapIndexed[ With[{r n = #2 〚 1 〛 , rs = #1 〚 1 〛 , rr = #1 〚 2 〛 }, {If[Length[rs]  1, {u, r[rn, First[rs]], _}  q[0, rr], {u, r[rn, First[rs]], _}  v[rn, First[rs], Take[rs, 1]]], {u, r[rn, x_], _}  v[rn, x, {}], {v[rn, _, Drop[rs, -1]], Last[rs], _}  q[Length[rs] - 1, rr], Table[{v[rn, _, Flatten[{___, Take[rs, i - 1]}]], rs 〚 i 〛 , _}  v[ rn, rs 〚 i 〛 , Take[rs, i]], {i, Length[rs] - 1, 1, -1}], {v[rn, _, _], y_, _}  v[rn, y, {}]}] & , rules /. s  List], {_, q[0, {x__, _}], _}  q[0, {x}], {_, q[0, {x_}], _}  r[1, x], {_, q[0, {}], x_}  r[1, x], {_, q[_, {___, x_}], _}  x, {_, q[_, {}], x_}  x, {_, x_, q[0, _]}  x, {_, _, q[n_, {}]}  q[n - 1, {}], {_, _, q[n_, {x___, _}]}  q[n - 1, {x}], {q[_, {}], _, _}  w, {q[0, {__, x_}], p[y_, _], _}  p[x, y], {q[0, {__, x_}], y_, _}  p[x, y], {p[_, x_], p[y_, _], _}  p[x, y], {p[_, x_], u, _}  x, {p[_, x_], y_, _}  p[x, y], {_, p[x_, _], _}  x, {w, u, _}  u, {w, x_, _}  w, {_, w, x_}  x, {_, r[rn_, x_], _}  x, {_, u, r[_, _]}  u, {_, x_, r[rn_, _]}  r[rn, x], {_, x_, _}  x}] The initial condition is obtained by applying the rule s[x_, y__]  {r[1, x], y} and then padding with u 's.