Chapter 6: Starting from Randomness

Section 3: Sensitivity to Initial Conditions

Difference patterns [in cellular automata]

The maximum rate at which a region of change can grow is determined by the range of the underlying cellular automaton rule. If the rule involves up to r nearest neighbors, then at each step a change in the color of a given cell can affect cells up to r away—so that the edge of the region of change can move by r cells.

For most class 3 rules, once one is inside the region of change, the colors of cells usually become essentially uncorrelated. However, for additive rules the pattern of differences is just exactly the pattern that would be obtained by evolution from an initial condition consisting only of the changes made. In general the pattern of probabilities for changes can be thought of as being somewhat like a Green's function in mathematical physics—though the nonadditivity of most cellular automata makes this analogy less useful. (Note that the pattern of differences between two initial conditions in a rule with k possible colors can always be reproduced by looking at the evolution from a single initial condition of a suitable rule with 2k colors.) In 2D class 3 cellular automata, the region of change usually ends up having a roughly circular shape—a result presumably related to the Central Limit Theorem (see page 976).

For any additive or partially additive class 3 cellular automaton (such as rule 90 or rule 30) any change in initial conditions will always lead to expanding differences. But in other rules it sometimes may not. And thus, for example, in rule 22, changing the color of a single cell has no effect after even one step if the cell has a block on either side. But while there are a few other initial conditions for which differences can die out after several steps most forms of averaging will say that the majority of initial conditions lead to growing patterns of differences.

From Stephen Wolfram: A New Kind of Science [citation]