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.

Starting from satisfiability it is possible to show that all sorts of well-known computational problems in discrete mathematics are NP-complete.

Yet all sorts of non-living systems—from crystals to flames—also do this.

No doubt there will be all sorts of specific applications of particular results and ideas.

So this means that there is in the end no difference between the level of computational sophistication that is achieved by humans and by all sorts of other systems in nature and elsewhere.

But with the new kinds of models based on simple programs that I explore in this book it becomes possible to capture all sorts of much more complex features that can only really be seen in explicit images of behavior.

No doubt there will quite quickly be all sorts of claims about applications of my ideas to the social sciences.

And so similarly, one would be unlikely to think that generating the center column from rule 30 could represent any sort of meaningful purpose—unless one was operating within the framework that I have developed in this book.

. • Different programs for doing all sorts of different things can be set up. • Any given program can be implemented in many ways. • Programs can behave in complicated and seemingly random ways—particularly when they are not working properly. • Debugging a program can be difficult. • It is often difficult to foresee what a program can do by reading its code. • The lower the level of representation of the code for a program the more difficult it tends to be to understand. • Some computational problems are easy to state but hard to solve. • Programs that simulate natural systems are among the most computationally expensive. • It is possible for people to create large programs—at least in pieces. • It is almost always possible to optimize a program more, but the optimized version may be more difficult to understand. • Shorter programs are sometimes more efficient, but optimizations often require many cases to be treated separately, making programs longer. • If programs are patched too much, they typically stop working at all.

But the problem is that it is hard to be sure that the system really is in the same state—and that there are not all sorts of large differences that do not happen to have been observed.)