require as much effort as just tracing each step in the original process whose outcome the formula was supposed to give.

And the crucial point that turns out to be the basis for much of the success of traditional theoretical science is that in fact most standard mathematical functions can be evaluated in a number of steps that is far smaller than the numerical value of their input, and that instead normally grows only slowly with the length of the digit sequence of their input.

So the result of this is that if there is a traditional mathematical formula for the outcome of a process then almost always this means that the process must show great computational reducibility.

In practice, however, the vast majority of cases for which traditional mathematical formulas are known involve behavior that is ultimately either uniform or repetitive. And indeed, as we saw in Chapter 10, if one uses just standard mathematical functions then it is rather difficult even to reproduce many simple examples of nesting.

But as the pictures on the facing page and in Chapter 10 illustrate, if one allows more general kinds of underlying rules then it becomes quite straightforward to set up procedures that with very little computational effort can find the color of any element in any nested pattern.

So what about more complex patterns, like the rule 30 cellular automaton pattern at the bottom of the page?

When I first generated such patterns I spent a huge amount of time trying to analyze them and trying to find a procedure that would allow me to compute directly the color of each cell. And indeed it was the fact that I was never able to make much progress in doing this that first led me to consider the possibility that there could be a phenomenon like computational irreducibility.

And now, what the Principle of Computational Equivalence implies is that in fact almost any system whose behavior is not obviously simple will tend to exhibit computational irreducibility.

But particularly when the underlying rules are simple there is often still some superficial computational reducibility. And so, for example, in the rule 30 pattern on the right one can tell whether a cell at a given position has any chance of not being white just by doing a

## Captions on this page:

An example of a pattern where it is difficult to compute directly the color of a particular cell.