But despite all sorts of discussion over the years, no clear understanding has ever emerged of just why such processes should in the end actually lead to much complexity at all. … So from this one might think that the complexity we see in biological organisms must all just be a reflection of complexity in their underlying rules—making discoveries about simple programs not really relevant.

If one managed to find the positions and properties of all the particles in the system, then no information about the system would remain unknown, and the entropy of the system would just be zero. … There are then a large number of possible detailed arrangements of particles that are all consistent with the results of such measurements. … But here we can imagine numbering all the possible arrangements of particles that are consistent with the results of our

But how can we ever expect to find any kind of precise general characterization of what all our various standard methods of perception and analysis do? The key point that will emerge in this chapter is that in the end essentially all these methods can be viewed as being based on rather simple programs. … So does one really need to try essentially all sufficiently simple programs in order to determine this?

For other systems will tend to perform computations that are just as sophisticated as those we can do, even with all our mathematics and computers. … For it implies that when it comes to computation—or intelligence—we are in the end no more sophisticated than all sorts of simple programs, and all sorts of systems in nature.

And indeed it is almost inevitable that the scheme will have to be at least somewhat complicated: for if a system is to be universal then it must be able to emulate any of the huge range of other systems that are universal—with the result that specifying which particular such system it is going to emulate for the purposes of a proof will typically require giving a fair amount of information, all of which must somehow be part of the encoding scheme. … And usually the only way to be sure of this is to have a more or less complete analysis of all possible behavior that the system can exhibit. … But as we saw in Chapter 10 , in almost no other case do standard methods of perception and analysis allow one to make much progress at all.

And this means that to ask whether there is any sequence in the satisfiability problem that obeys all the constraints is equivalent to finding the answer to the Turing machine problem described above. Starting from satisfiability it is possible to show that all sorts of well-known computational problems in discrete mathematics are NP-complete. … But almost always what is actually constructed is quite complicated—and certainly not something one would expect to occur at all often.

So in as much as intelligence is associated with the ability to do sophisticated computations it should in no way require billions of years of biological evolution to produce—and indeed we should see it all over the place, in all sorts of systems, whether biological or otherwise. … 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.

But if one in effect explicitly searches all 8 trillion or so rules that involve less than four colors, it turns out that one can find 4277 three-color rules that work. … Each uses at least a slightly different scheme, but all achieve the same purpose of doubling their input. … After all, Examples of cellular automata that can be viewed as achieving the purpose of doubling the width of the pattern given in their input.

No doubt there will be all sorts of specific applications of particular results and ideas. … And indeed one of the things that emerges from this book is that traditional engineering has actually considered only a tiny and quite unrepresentative fraction of all the kinds of systems and processes that are in principle possible. … Other examples include all

In each case, all cells are initially white, and one of the rules given on the left is applied for the specified number of steps.