If at every stage the tissue in each region produced grows at the same rate, and all that differs is what final type of cells will exist in each region, then inevitably a simple and highly regular overall structure will emerge, as in the idealized picture below. … Here the subdivisions are taken to occur in two directions, always giving three simple rectangles which all grow at the same rate.

The answer that I will develop in considerable detail later in this book is to view all such processes as computations. … The consequence of this is that no reasonable experiment can ever involve setting up the kind of initial conditions that will lead to decreases in randomness, and that therefore all practical experiments will tend to show only increases in randomness.

And could it even be that underneath all the complex phenomena we see in physics there lies some simple program which, if run for long enough, would reproduce our universe in every detail? … Instead, it would be a complete and precise representation of the actual operation of the universe—but all reduced to readily stated rules.

All eight pictures on the facing page were generated from the two-dimensional substitution systems shown, and thus correspond to purely nested patterns. … But at the end of that section I showed that the fairly simple procedure of two-dimensional pointer Examples of all the distinct repetitive patterns that can be formed from arrays of 2×2 and 3×3 blocks.

But by now the vast majority of practical electronic devices, despite all their apparent differences, are based on computers that are universal. … Indeed, essentially all of them are based on the same kinds of logic circuits, the same basic layout of data paths, and so on.

Indeed, most often they have in effect been engineered out of all sorts of components that are direct idealizations of various elaborate structures that exist in practical digital electronic computers. … And among other things this means that universality can be expected to occur not only in many kinds of abstract systems but also in all sorts of systems in nature.

But as I discussed in the previous section , beyond asserting that there is an upper limit to computational sophistication, the Principle of Computational Equivalence also makes the much stronger statement that almost all processes except those that are obviously simple actually achieve this limit. And this is related to what I believe is a very fundamental abstract fact: that among all possible systems with behavior that is not obviously simple an overwhelming fraction are universal.

So what this means is that systems one uses to make predictions cannot be expected to do computations that are any more sophisticated than the computations that occur in all sorts of systems whose behavior we might try to predict. … But it follows from the Principle of Computational Equivalence that in practically all other cases it will be computationally irreducible.

Substitution Systems One of the features that cellular automata, mobile automata and Turing machines all have in common is that at the lowest level they consist of a fixed array of cells. … In the cases on the facing page , I start from a single element represented by a long box going all the way across the picture.

Indeed, away from the right-hand edge, all the elements effectively behave as if they were lying on a regular grid, with the color of each element depending only on the previous color of that element and the element immediately to its right. … The substitution systems that we discussed in the previous section work by replacing each element in such a string by a new sequence of elements—so that in a sense these systems operate in parallel on all the elements that exist in the string at each step.