But now that the results of this chapter are known, one can go back and see quite a number of times in the past when they came at least somewhat close to being discovered. … The question of whether complex behavior could occur in cellular automata was occasionally raised, but on the basis of intuition from engineering it was generally assumed that to get any substantial complexity, one would have to have very complicated underlying rules. … One example known for well over two thousand years concerns the distribution of prime numbers (see page 132 ).

And in fact one can often do something similar for networks. … But the standard formalism of quantum theory says that this is not correct, and that in fact one has to look at so-called probability amplitudes, not ordinary probabilities. At a mathematical level, such amplitudes are analogous to ones for things like waves, and are in effect just numbers with directions.

And what we will discover in this section is that to come up with an appropriate definition one has no choice but to consider issues of perception and analysis. One might have thought that from traditional mathematics and statistics there would long ago have emerged some standard definition of randomness. … And indeed I believe that it is only with the discoveries in this book that one is finally now in a position to develop a real understanding of what randomness is.

specific process one can apply it to a piece of raw data, and then see how the results compare with those obtained from all possible sequences. If the process is sufficiently simple then by using traditional mathematics one can sometimes work out fairly completely what will happen with all possible sequences. But in the vast majority of cases this cannot be done, and so in practice one has no choice but just to compare with results obtained by sampling some fairly limited collection of possible sequences.

But typically these regularities are ones that can also be found quite easily by many of the standard methods of perception and analysis discussed earlier in this chapter . … But there is at least one important difference between the way typical databases operate, and the way human memory operates. For in a standard database one tends to be able to find only data that meets some precise specification, such as containing an exact match to a particular string of text.

But what is somewhat special about the setup above is that inputs which yield the same output tend to be ones that might reasonably be considered similar, while inputs that yield different outputs tend to be significantly different. … In a scheme like the one above the output cells would presumably be connected to cells that actually perform actions of some kind—perhaps causing muscles to move, or perhaps just providing inputs to further nerve cells. … But if one looks, say, at nerve cells involved in the early stages of the visual system, then once the brain has matured past some point these never seem to change their properties

This is a very remarkable result, and one which will turn out to be crucial to the new kind of science that I develop in this book. … For one might have thought that every different kind of system that we discussed for example in Chapter 3 would be able to perform completely different kinds of computations. … And instead it has turned out that essentially every single one of these systems is ultimately capable of exactly the same kinds of computations.

Region (c) shows what happens when the information corresponding to one element in a block passes through the kind of object produced in region (a). … Region (g) shows the analog of region (a), but now for a white element instead of a black one. … Starting around the middle of the region, however, the behavior becomes quite different from region (a): while region (a) yields an object that allows information to pass through, region (g) yields one that stops all information, as shown in regions (h) and (i).

One slightly subtle issue in thinking about computational irreducibility is that given absolutely any system one can always at least nominally imagine speeding up its evolution by setting up a rule that for example just executes several steps of evolution at once. … For the point of such formulas is usually to allow one to give a number as

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. … And so, for example, in the rule 30 pattern below one can tell whether a cell at a given position has any chance of not being white just by doing a An example of a pattern where it is difficult to compute directly the color of a particular cell.