So it is not that nesting is somehow fundamentally difficult to recognize; it is just that the particular processes that happen to occur in human visual perception do not in general manage to do it. … The two-dimensional pointer-based encoding scheme from page 571 does however manage to recognize the structure in all cases.

But it does not tell one what will happen if, for example, one starts from typical simple initial conditions. … So what this means is that even, say, in cellular automata that start from very simple initial conditions, one can expect that those aspects of their behavior that do not look obviously simple will usually correspond to computations of equivalent sophistication. According to the Principle of Computational Equivalence therefore it does not matter how simple or complicated either the rules or the initial conditions for a process are: so long as the process itself does not look obviously simple, then it will almost always correspond to a computation of equivalent sophistication.

Do they behave any differently? … Indeed, the only way to do it seems to be to lay the balls down carefully one after another. … In no case does the global behavior seen come even close to satisfying the simple constraints that determine the invariant state.

But in recent times it has usually been assumed that it has something to do with an ability to perform sophisticated computations. … And certainly some everyday turns of phrase might suggest that we do. … Certainly one can identify all sorts of specific features of human intelligence: the ability to understand language, to do mathematics, solve puzzles, and so on.

Note that the first replacement only removes elements and does not insert new ones. … At least with the initial condition used here, despite considerable early apparent randomness, the differences in number of elements do repeat (shifted by 1) every 1071 steps.

As in one dimension, knowing the constraint does not immediately provide a procedure for finding a pattern which satisfies it. But a little experimentation reveals that the simple repetitive pattern above satisfies the constraint, and in fact it is the only pattern to do so.

How the Discoveries in This Chapter Were Made This chapter —and the last —have described a series of surprising discoveries that I have made about what simple programs typically do. … But what allowed me to discover much more was that I used instead a methodology fundamentally based on doing computer experiments. … And the great advantage of such an experimental approach is that it does not require one to know in advance exactly what kinds of behavior can occur.

out that there is behavior which does not happen to fit in with these assumptions, then typically the experiment will fail to notice it. … It is usually much better, for example, to do a mindless search of a large number of possible cases than to do a carefully crafted search of a smaller number. … For in making a summary one inevitably has to pick out only certain features, and in doing this one can remove or obscure the most interesting effects.

However certain one might be that simple programs could never do more than produce simple behavior, the pictures on the past few pages [ 32 , 33 , 34 , 35 , 36 , 37 , 38 ] should forever disabuse one of that notion. … But instead, they seem to be doing quite different things. … But by doing the appropriate computer experiments one can easily find out what actually happens—and in effect begin the process of exploring a whole new world of remarkable phenomena associated with simple programs.

And indeed for me to do it certainly took carrying out quite large-scale computer experiments on a considerable number of different cellular automata. If one already has a clear idea about the basic features of a particular phenomenon, then one can often get more details by doing fairly specific experiments. But in my experience the only way to find phenomena that one does not already know exist is to do very systematic and general experiments, and then to look at the results with as few preconceptions as possible.