But if one allows progressively more molecules computational irreducibility can make it take progressively more computational work to see what will happen. And much as in determining whether constraints like those on page 213 can be satisfied for an infinite region, it can take an infinite amount of computational work to determine bulk properties of an infinite collection of molecules.

The rule shown here takes the new gray level of each cell to be the average of its own gray level and those of its immediate neighbors.

And what one sees is that it takes only a small change in the initial speed to make the ball come to rest in a completely different orientation.

Their role in effect is to take large volumes of raw data and extract from it summaries that we can use.

Each connection in each network has a certain probability associated with it, and the model takes sequences of black and white squares to be generated by tracing paths through the networks according to these probabilities.

But most will probably take decades to emerge.

Whenever a number x is known to satisfy Sum[a[i] f[i][x], {i, n}]  0 with fixed f[i] one can take the early digits of x and use LatticeReduce to find integer solutions for the a[i] .

Machine 1447 (example (e)) computes the function which takes the digit sequence of x and replaces its first 3 + IntegerExponent[x + 1, 2] 0's by 1's. … The number of distinct functions that can be computed is about 36,392 (or 75,726 for {f[x], t[x]} pairs). 8934 machines compute x + 1 (by 25 different methods, including ones like machine 164850 that take exponential steps), 14 compute x + 2 , and none compute x + 3 . Those machines that take times that grow precisely like 2 n all tend to compute very straightforward functions which can be computed much faster by other machines.

And indeed in the course of this chapter we have seen that in every single one of the general kinds of systems that we have discussed, it ultimately takes only very simple rules to produce behavior of great complexity.

But I have little doubt that within a matter of a few decades what I have done will have led to some dramatic changes in the foundations of technology—and in our basic ability to take what the universe provides and apply it for our own human purposes.