The repetitive structure of picture (a) implies that to reproduce this picture all we need do is to specify the colors in a 49×2 block, and then say that this block should be repeated an appropriate number of times. Similarly, the nested structure of picture (b) implies that to reproduce this picture, all we need do is to specify the colors in a 3×3 block, and then say that as in a two-dimensional substitution system each black cell should repeatedly be replaced by this block.

Sensitive dependence on initial conditions thus does not in and of itself imply that a system will behave in a random way. … And indeed if this were true, then it would imply that typical initial conditions would inevitably involve random digit sequences.

Almost inevitably, however, having such a maximal period implies a certain regularity. And in fact, as we shall see later in this book, the very possibility of any detailed mathematical analysis tends to imply the presence of at least some deviations from perfect randomness.

more formal level, it also implies that everything we can observe can be captured by a causal network. … As I discussed in the last two sections [ 14 , 15 ], causal invariance of the underlying rules implies that such structures should be able to move at a range of uniform speeds through the background.

For what the result implies is that in many kinds of systems particular rules can be found that achieve universality and thus show the same level of computational sophistication. … So far from universality being some rare and special property that exists only in systems that have carefully been built to exhibit it, the Principle of Computational Equivalence implies that instead this property should be extremely common.

And from the arguments above this in turn implies that there must be specific integer equations that have no solutions but where this fact cannot be proved from the normal axioms of arithmetic. … And in particular if an infinite process is computationally irreducible then there cannot in general be any useful finite summary of what it does—since the existence of such a summary would imply computational reducibility.

The presence of universality implies that there must at some level be computational irreducibility—and thus that there must be theorems that cannot be reached by any short procedure. … And indeed the Principle of Computational Equivalence implies that such phenomena should be close at hand in almost every direction: it is merely that—despite its reputation for generality—mathematics has in the past implicitly tended to define itself to avoid them.

But the basic point is that the phenomenon of entropy increase implied by the Second Law is a more or less direct consequence of the phenomenon discovered in this book that even with simple initial conditions many systems can produce complex and seemingly random behavior. … The reversibility of the underlying rules implies that at some level it must be possible to recognize outcomes from different kinds of initial conditions.

And this implies that if one knew all of the underlying details of the network that makes up our universe, it should always be possible to work out the result of any measurement. … And indeed this fact has often been taken to imply that quantum phenomena can ultimately never be the result of any definite underlying process of evolution.

Yet traditional mathematical models often in effect imply that systems in nature can do things like this. … And this has in the past often been taken to imply that systems in nature are able to do computations that are somehow fundamentally more sophisticated than standard computational systems.