like the first picture below that go from a symmetrical cluster to one for which a particular orientation has to be chosen. … At first one might think that one could set up some kind of analog of a cellular automaton and just replace all relevant clusters of nodes at once. … In each pair of pictures in the upper part of the page , the top one shows the form of the network before the replacement, and the bottom one shows the result after doing the replacement—with the cluster of nodes involved in the replacement being highlighted in both cases.

For in a sense any program—including one that is very simple and runs very quickly—can be thought of as implementing a method of perception or analysis. For if one gives a piece of data as the input to the program, then the output one gets—whatever it may be—can be viewed as corresponding to some kind of description of the data. … A description based on output from a cellular automaton rule that one has never seen before is thus for example not likely to be useful.

But the crucial point is that as one looks at cellular automata with progressively greater computational capabilities, one will eventually pass the threshold of universality. … One might assume that by using more and more sophisticated underlying rules, one would always be able to construct systems with ever greater computational capabilities. … And by not imposing this constraint, one might expect that one would be able to find universal cellular automata that have at least somewhat simpler underlying rules.

For if one knew only about practical computers and about systems like the universal cellular automaton discussed early in this chapter , then one would probably assume that universality would rarely if ever be seen outside of systems that were specifically constructed to exhibit it. … So what this means is that if one looks at a sequence of systems with progressively more complicated rules, one should expect that the overall behavior they produce will become more complex only until the threshold of universality is reached. And as soon as this threshold is passed, there should then be no further fundamental changes in what one sees.

Example (a) shows a network that is effectively one-dimensional. … But there is nothing intrinsically one-dimensional about the structure of network systems. … Examples of networks that correspond to arrays in one, two and three dimensions.

As a simple example, consider a line of cells in which each cell is colored black or white, and in which the arrangement of colors is subject to the constraint that every cell should have exactly one black and one white neighbor. … A system consisting of a line of black and white cells whose form is defined by the constraint that every cell should have exactly one black and one white neighbor. The pattern shown is the only possible one that satisfies this constraint.

But with almost any general classification scheme there are inevitably borderline cases which get assigned to one class by one definition and another class by another definition. … But such rules are quite unusual, and in most cases the behavior one sees instead falls squarely into one of the four classes described above. … But sometimes one can tell at least a certain amount simply from the form of the underlying rule.

But they still in a sense have a simple one-dimensional arrangement of states in time. … Starting with a single state consisting of one element, the picture then shows that applying these rules immediately gives two possible states: one with a single element, and the other with two. … In multiway systems, however, A very simple multiway system in which one element in each sequence is replaced at each step by either one or two elements.

On the third line one removes numbers divisible by 3, and so on. As one goes on, fewer and fewer numbers remain. … One starts on the top line with all numbers between 1 and 100.

Probably complexity is not in any fundamental sense rarer in continuous systems than in discrete ones. But the point is that discrete systems can typically be investigated in a much more direct way than continuous ones. … And in fact, in the end one typically has rather little idea which aspects of what one sees are actually genuine features of the system, and which are just artifacts of the particular methods and approximations that one is using to study it.