Quite often part of the reason for this, as illustrated in the pictures on the facing page , is that even with a single very simple initial condition the actual evolution of a system will generate blocks that correspond to essentially all possible initial conditions.

All the statements in the top block above can be proved true from the axiom system.

Logic circuits [from cellular automata] The rules for the cellular automaton shown here are {{0, 1, 1 | 3}  1, {0, 3, 3}  3, {1, 0, 0 | 1 | 3}  1, {1, 1, 3}  4, {1, 3, 0}  3, {1, 3, 3}  2, {2, 1, 3}  3, {2, 3, 0}  2, {2, 0, _}  4, {3, 3, 0}  3, {4, 0, 0 | 1 | 2 | 4}  2, {4, 3, 3}  3, {4, 1, 3}  1, {4, 3, 0}  4, {_, _, _}  0} The initial conditions are given by Flatten[Block[{And, Or}, Map[{0, 2 (# + 1)} &, expr, {-1}] //. {!

The issue is essentially no different from the one that we encountered in previous sections [ 9 , 10 , 11 ] for blocks of elements in substitution systems on strings.

And one can then assume that in the visual cortex there is a corresponding array of cells, with each cell receiving input from, say, a 2×2 block of squares, and following the rule that it responds whenever the colors of these squares form some particular pattern.

With two colors of elements additional conditions can be constructed involving counting elements of each color, or various blocks of elements. … When a sequence of blocks leads to upper and lower strings that disagree, the rectangle is left white. … Note that in case (c) the presence of only one color in either block means that strings will always agree so far.

It implies that every block of length m that occurs at a particular step has exactly 4 immediate predecessor blocks of length m + 2 (see page 960 ).

Another approach was to look at actual possible transformations between partitionings, and this led from the late 1950s to various studies of so-called shift-commuting block maps (or sliding-block codes)—which turn out to be exactly 1D cellular automata (see page 878 ).

The pattern can be viewed as a tessellation of 5×5 blocks of cells.

In the case of the 19-color universal cellular automaton on page 645 it turns out that encodings in which individual black and white cells are represented by particular 20-cell blocks are sufficient to allow the universal cellular automaton to emulate all 256 possible elementary cellular automata—with one step in the evolution of each of these corresponding to 53 steps in the evolution of the original system.