Note that to generate the pictures that follow requires applying the underlying cellular automaton rule for individual cells a total of about 12 million times.

Note that to generate the pictures that follow requires applying the underlying cellular automaton rule for individual cells a total of about 12 million times.

Note that to generate the pictures that follow requires applying the underlying cellular automaton rule for individual cells a total of about 12 million times.

of the causal networks for rules (e) and (f) requires following the underlying mobile automaton evolution for 2447 and 731 steps respectively. … In most cases there will still be small differences; but the causal network on the right below is specifically constructed to be exactly reversible—much like the cellular automata we discussed near the beginning of this chapter . … Even to fill in the first few rows of such causal networks, many steps of underlying mobile automaton evolution must be traced.

Then in the early 1970s, considerable recreational computing interest developed in a specific two-dimensional cellular automaton known as the Game of Life, whose behavior is in some respects similar to the rule 110 cellular automaton discussed in this chapter . … In a sense it is surprising that so much could be done on the Game of Life without the much simpler one-dimensional cellular automata in this chapter ever being investigated.

Note (f) for Emulating Cellular Automata with Other Systems…Mobile automata [emulating cellular automata] Given the rules for an elementary cellular automaton in the form used on page 867 , the following will construct a mobile automaton which emulates it: vals = {x, p[0], q[0, 0], q[0, 1], q[1, 0], q[1, 1], p[1]} CAToMA[rules_] := Table[(#  Replace[#, {{q[a_, b_], p[c_], p[d_]}  {q[c, {a, c, d} /. rules], 1}, {q[a_, b_], p[c_], x}  {q[c, {a, c, 0} /. rules], 1}, {q[_, _], x, x}  {p[0], -1}, {q[_, _], q[_, a_], p[_]}  {p[a], -1}, {x, q[_, a_], p[_]}  {p[a], -1}, {x, x, p[_]}  {q[0, 0], 1}, {_, _, _}  {x, 0}}]) &[vals 〚 IntegerDigits[i, 7, 3] + 1 〛 ], {i, 0, 7 3 - 1}] The ordering in vals defines a mapping of symbolic cell values onto colors. Given a list of initial cell colors for the cellular automaton, the initial conditions for the mobile automaton are given by Flatten[{p[0], Map[p, list], p[0]}] surrounded by x 's, with the active cell being placed initially just before the first p[0] .

[No text on this page] The behavior of the code 20 cellular automaton from the top of the facing page for all initial conditions with black cells in a region of size less than nine.

The Threshold of Universality in Cellular Automata…The Threshold of Universality in Cellular Automata By showing that rule 110 is universal, we have established that universality is possible even among cellular automata with the very simplest kinds of underlying rules. But there remains the question of what is ultimately needed for a cellular automaton—or any other kind of system—to be able to achieve universality. … These cellular automata are necessarily all class 1 or class 2 systems.

By step 3, it becomes difficult to see any change if one just looks at an explicit picture of the cellular automaton evolution. … Networks representing possible sequences of black and white cells that can occur at successive steps in the evolution of several class 1 and 2 cellular automata.

Note (c) for Emulating Other Systems with Cellular Automata…Turing machines [from cellular automata] Given any Turing machine with rules in the form used on page 888 and k possible colors for each cell, a cellular automaton which emulates it can be constructed using TMToCA[rules_, k_:2] := Flatten[{Map[g[#, k]&, rules], {_, x_, _}  x}] g[{s_, a_}  {sp_, ap_, d_}, k_] := {If[d  1, Identity, Reverse][{k s + a, x_, _}]  k sp + x, {_, k s + a, _}  ap} If the Turing machine has s states for its head, then the cellular automaton has k (s+1) colors for each cell.