By effectively enumerating all such sequences, it is easy to see that such a model predicts that in any particular sequence the fraction of black squares is most likely to be 1/2 .

(Note that these substitution systems are the simplest ones that yield equal frequencies of all blocks up to lengths 1, 2 and 3 respectively.)

These pictures can be thought of as matrices with 1's at the position of each black dot, and 0's elsewhere.

And then in the specific case shown one compares corresponding digits in these two sequences, and if these digits are ever respectively 0 and 1, then the square is white; otherwise it is black.

And these amplitudes a i are assumed to be complex numbers with a continuous range of possible values, subject only to the conventional constraint of unit total probability Sum[Abs[a i ] 2 , {i, 2 n }]  1 . … In a classical system like a cellular automaton with n cells a probabilistic ensemble of states can similarly be described by a vector of 2 n probabilities p i —now satisfying Sum[p i , {i, 2 n }]  1 , and evolving by multiplication with 2 n × 2 n matrices having a single 1 in each row. … It was found to be sufficient to do operations on just one and two spins at a time, and in fact it was shown that any 2 n × 2 n unitary matrix can be approximated arbitrarily closely by a suitable sequence of for example underlying 2-spin {x, y}  {x, Mod[x + y, 2]} operations (assuming values 0 and 1), together with 1-spin arbitrary phase change operations.

The maximum length DNF for elementary rules after 1 step is 4, and this is achieved by rules 105, 107, 109, 121, 150, 151, 158, 182, 214 and 233.

In such cases, combinator expressions can be viewed as binary trees without labels, equivalent to balanced strings of parentheses (see page 989 ) or sequences of 0's and 1's.

If only a single symbol ever appears, then all that matters is the overall structure of an expression, which can be captured as in the main text by the sequence of opening and closing brackets, given by Flatten[Characters[ToString[expr]]/.{"["  1,"]"  0, " ℯ "  {}}]

Properties [of multiway system example] The total number of strings grows approximately quadratically; its differences repeat (offset by 1) with period 1071.

Given a flat interface, the layer of cells immediately on either side of this interface behaves like the rule 150 1D cellular automaton.