Will all the structures that are produced eventually annihilate each other, leaving only a very regular pattern? … The only sure way to answer these questions, it seems, is just to run the cellular automaton for as many steps as are needed, and to

And indeed throughout most of history it has been taken almost for granted that such complexity—being so vastly greater than in the works of humans—could only be the work of a supernatural being. … And the reason that such complexity is not usually seen in human artifacts is just that in building these we tend in effect to use programs that are specially chosen to give only behavior simple enough for us to be able to see that it will achieve the purposes we want.

And in fact my guess is that the only way to show this with any certainty would be actually to find a specific set of multiway system rules with the property that regardless of the path that gets followed these rules would always yield behavior that agrees with the various observed features of our universe. … But as we saw there, certain underlying rules have the A multiway system in which strings of any length can be generated—but in which only specific sequences of lengths actually occur on any path.

But if one drops some of the later forms—thereby reducing the number of weights that have to be specified—one gets only an approximation to the image. … Images that are sufficiently simple can already be recognized even when only a very small fraction of the forms are included—corresponding to a very high level of compression.

But the fact that we have only three kinds of color-sensitive cells means that our eyes essentially sample only three features of this spectrum.

But how can one arrange that only the intended recipient of the message knows the encrypting sequence (b)? … But much more common is to be able to transmit only some short key in a secure way, and then to have to generate the encrypting sequence from this key.

And at some level the main challenge is that our experience from programming and engineering tends to provide us with only a limited set of methods for coming up with such a procedure. … And the point is that at each stage, we need think directly only about the scale of structures that we are currently handling—and not for example about all the pieces that make up these structures.

Given these models the only way to find out what they do will usually be just to run them. … For if the evolution of a system corresponds to an irreducible computation then this means that the only way to work out how the system will behave is essentially to perform this computation—with the result that there can fundamentally be no laws that allow one to work out the behavior more directly.

And only after 1017 steps does it finally become clear that the pattern in fact dies out. … And in any system the only way to be able to guarantee to know this in general is to have some way to shortcut the evolution of the system, and to be able to reduce to a finite computation what takes the system an infinite number of steps to do.

But unlike in example (e), this Turing machine is not the only one that computes the function it computes. … And indeed if one looks at all 4096 Turing machines with 2 states and 2 colors it turns out that the only rates of growth that one ever sees are linear, square and exponential. … With 3 states and 2 colors there are a total of 2,985,984 possible machines.