Because the rule is probabilistic a different detailed pattern of evolution will in general be obtained each time the cellular automaton is run—as in the top row of pictures above. Despite this, however, any particular probabilistic cellular automaton will typically exhibit some characteristic overall pattern of behavior, as illustrated in the array of pictures above. … Probabilistic cellular automata can be viewed as generalizations of so-called directed percolation models.

[No text on this page] A cellular automaton that never settles down to a stable state, but instead continues to show behavior that seems in many respects random.

The basic idea is to look at the evolution of an additive cellular automaton in a register of limited width. The key then gives the initial condition for the cellular automaton, and the encrypting sequence is extracted, for example, by sampling a particular cell on successive steps. Encryption using the rule 60 additive cellular automaton.

[No text on this page] A typical example of the behavior of the rule 110 cellular automaton with random initial conditions.

More Cellular Automata…More Cellular Automata The first picture below shows the rules used in the four cellular automata on the facing page . … In the very simplest cases, all the cells in the cellular automaton end up just having the same color after one step. … The sequence of 256 possible cellular automaton rules of the kind shown above.

And thus for example in a cellular automaton moving from one point in space to another just corresponds to shifting from one cell to another. But moving from one point in time to another involves actually applying the cellular automaton rule. When we make a picture of the behavior of a cellular automaton, however, we do nevertheless tend to represent space and time in the same visual kind of way—with space going across the page and time going down.

So given the underlying rule for a particular cellular automaton, can one tell what class of behavior the cellular automaton will produce? In most cases there is no easy way to do this, and in fact there is little choice but just to run the cellular automaton and see what it does. … Different definitions based on different specific properties will place these cellular automata into different classes.

[No text on this page] The effect of small changes in initial conditions in the rule 110 class 4 cellular automaton.

As an example of what can happen when simple processes are applied to data, the pictures on the facing page show the results of evolution according to various cellular automaton rules, with initial conditions given by the sequences from page 594 . On each row the first picture illustrates the typical behavior of each cellular automaton. … With these sequences none of the simple cellular automaton rules shown here yield behavior that can readily be distinguished from what is typical.

Cellular Automata…[No text on this page] Three-dimensional objects formed by stacking successive two-dimensional patterns produced in the evolution of the cellular automaton from the previous page .