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The bottom row shows how a cellular automaton can be made to emulate this behavior when given a succession of different initial conditions. The cellular automaton is set up to produce a vertical black stripe if the head of the Turing machine ever goes further to the right than it starts—as it does in cases 6 and 8. … The cellular automaton takes 2t 2 +t steps to emulate t steps of evolution in the Turing machine.

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Five hundred steps in the evolution of the rule 30 cellular automaton from page 27 . … The asymmetry between the left and right-hand sides is a direct consequence of asymmetry that exists in the particular underlying cellular automaton rule used.

A mollusc shell, like a one-dimensional cellular automaton, in effect grows one line at a time, with new shell material being produced by a lip of soft tissue at the edge of the animal inside the shell. … And given this, the simplest hypothesis in a sense is that the new state of the element is determined from the previous state of its neighbors—just as in a one-dimensional cellular automaton.
Examples of patterns produced by the evolution of each of the simplest possible symmetrical one-dimensional cellular automaton rules, starting from a random initial condition.

Note (b) for The Rule 110 Cellular Automaton

A cellular automaton with a simple rule that generates a pattern which seems in many respects random. The rule used is of the same type as in the previous examples, and the cellular automaton is again started from a single black cell. … In the numbering scheme of Chapter 3 , the cellular automaton shown here is rule 30.

The picture below shows a simple example based on the rule 30 cellular automaton that I have discussed several times before in this book. The idea is to generate an encrypting sequence by sampling the evolution of the cellular automaton, starting from initial conditions that are defined by a key.
… But as the width of the cellular automaton increases, the total number of possible initial conditions
Encryption using a column of rule 30 as the encrypting sequence.

So just how complex can the behavior of a cellular automaton that starts from random initial conditions be? … The facing page and the one that follows show as an example the cellular automaton that we first discussed on page 32 . … But the cellular automaton quickly organizes itself into a set of definite localized structures.

There are several details of the cellular automaton used above that differ from actual physical systems of the kind usually studied in thermodynamics. … The pictures on the next two pages [ 446 , 447 ] show a particular two-dimensional cellular automaton in which black squares representing particles move around and collide with each other, essentially like particles in an ideal gas. This cellular automaton shares with the cellular automaton at the beginning of the section the property of being reversible.

The underlying rules for the cellular automaton used in the picture below are precisely reversible. … For as the picture on the facing page demonstrates, if the
A reversible cellular automaton that exhibits seemingly irreversible behavior. … The specific cellular automaton used here is rule 122R.

In the examples on page 248 , however, such behavior always seems to occur superimposed on some kind of repetitive background—much as in the case of the rule 110 one-dimensional cellular automaton on page 229 .
… And so as one example page 249 shows a two-dimensional cellular automaton often called the Game of Life in which all sorts of localized structures occur even on a white background. If one watches a movie of the behavior of this cellular automaton its correspondence to a one-dimensional class 4 system is not particularly obvious.