procedure. And as a result, the center column of rule 30 cannot be considered truly random according to such definitions.

But while definitions of this type have a certain conceptual appeal, they are not likely to be useful in discussions of randomness in nature. For as we will see later in this book, it is almost certainly impossible for any natural process ever to generate a sequence which is guaranteed to be truly random according to such definitions.

For our purposes more useful definitions tend to concentrate not so much on whether there exists in principle a simple way to generate a particular sequence, but rather on whether such a way can realistically be recognized by applying various kinds of analysis to the sequence. And as discussed above, there is good evidence that the center column of rule 30 is indeed random according to all reasonable definitions of this kind.

So whether or not one chooses to say that the sequence is truly random, it is, as far as one can tell, at least random for all practical purposes. And in fact sequences closely related to it have been used very successfully as sources of randomness in practical computing.

For many years, most kinds of computer systems and languages have had facilities for generating what they usually call random numbers. And in Mathematica—ever since it was first released—Random[Integer] has generated 0's and 1's using exactly the rule 30 cellular automaton.

The way this works is that every time Random[Integer] is called, another step in the cellular automaton evolution is performed, and the value of the cell in the center is returned. But one difference from the picture two pages ago is that for practical reasons the pattern is not allowed to grow wider and wider forever. Instead, it is wrapped around in a region that is a few hundred cells wide.

One consequence of this, as discussed on page 259, is that the sequence of 0's and 1's that is generated must then eventually repeat. But even with the fastest foreseeable computers, the actual period of repetition will typically be more than a billion billion times the age of the universe.

Another issue is that if one always ran the cellular automaton from page 315 with the particular initial condition shown there, then one would always get exactly the same sequence of 0's and 1's. But by using different initial conditions one can get completely different