In that case, the evolution is effectively 1D, and turns out to follow elementary rule 22, thus producing the infinitely growing nested pattern shown on page 263 .

For not knowing about the phenomenon of intrinsic randomness generation, it has normally been assumed that with the kinds of discrete elements and fairly simple rules common in such models, realistically complicated behavior can only ever be obtained if explicit randomness is continually introduced.

For I essentially always consider systems that are based on explicit evolution rules rather than implicit constraints.

For as suggested by the bottom row of pictures on page 732 one can imagine having localized structures whose interactions emulate the rules of the cellular automaton.

Given the combinator rules crules = {s[x_][y_][z_]  x[z][y[z]], k[x_][y_]  x} the setup was that any function f would be written as some combination of s and k —which Schönfinkel referred to respectively as "fusion" and "constancy"—and then the result of applying the function to an argument x would be given by f[x] //. crules . … The rule 110 combinator on page 713 provides however a much more direct proof of this.

Given an integer a for which IntegerDigits[a, 2] gives the cell values for a cellular automaton, a single step of evolution according say to rule 30 is given by BitXor[a, 2 BitOr[a, 2a]] where (see page 871 ) BitXor[x, y]  BitOr[x, y] - BitAnd[x, y] and a is assumed to be padded with 0's at each end. The corresponding form for rule 110 is BitXor[BitAnd[a, 2a, 4a], BitOr[2a, 4a]] The final equation is then obtained from {1 + x 4 + x 12  2 (1 + x 3 ) (x 1 + 2 x 3 ) , x 2 + x 13  2 x 1 , 1 + x 5 + x 14  2 x 1 , 2 x 3 x 5 + 2 x 1 + 2 x 3 x 6 + 2 x 1 + x 3 x 15 + x 16  x 4 , 1 + x 15 + x 17  2 x 3 , 1 + x 16 + x 18  2 x 3 , 2 1 + x 3 (1 + x 1 + 2 x 3 ) (-1 + x 2 ) - x 10 + x 11  2 x 4 , x 7  BitAnd[x 6 , 2 x 6 ] ∧ x 8  BitOr[x 6 , 2 x 6 ], x 9  BitAnd[x 6 , 2 x 7 ] ∧ x 19  BitOr[x 6 , 2 x 7 ], x 10  BitAnd[x 9 , 2 x 8 ] ∧ x 11  BitOr[x 9 , 2 x 8 ]} where x 1 through x 4 have the meanings indicated in the main text, and satisfy x i ≥ 0 .

For it is almost certain that experiments on, say, some specific cellular automaton whose rule has been picked at random from a large set will never have been done before.

For one step in rule 30, for example, this yields {{1, 0, 0}, {0, 1, 1}, {0, 1, 0}, {0, 0, 1}} , as shown on page 616 .

But the behavior in the course of the evolution can depend on how the combinator rules are applied; here expr /. crules is used at each step, as in the symbolic systems of page 896 .

In general, the probability distribution for the displacement of a particle that executes a random walk is With[{ σ = 1}, (d/(2 π σ t)) d/2 Exp[-d r 2 /(2 σ t)]] The same results are obtained, with a different value of σ , for other random microscopic rules, so long as the variance of the distribution of step lengths is bounded (as in the Central Limit Theorem).