The most important point seems to be that it is mostly derived from experience with building things and doing engineering—where it so happens that one avoids encountering systems like the ones in the previous section . … And in fact one of the important conclusions of this book is that such systems are actually very common in nature. But because the only situations in which we are routinely aware both of underlying rules and overall behavior are ones in which we are building things or doing engineering, we never normally get any intuition about systems like the ones at the end of the previous section .

And certainly it seems not inconceivable that there could be a fundamental result that the only kinds of regularities that both occur frequently in actual systems and can be recognized quickly enough to provide a basis for practical methods of perception and analysis are ones like repetition and nesting. … Indeed, as one example, one could imagine just enumerating all possible simple descriptions of some particular type, and then testing in each case to see whether what one gets matches a piece of data that one has.

And one of the consequences of this is that it implies that most systems whose behavior seems complex should be universal. Yet as of now we only know for certain about fairly few systems that are universal, albeit including ones like rule 110 that have remarkably simple rules. … But when one examines the known examples of such systems—all of which have very intricate underlying rules—one finds that even though the particular part of their behavior that is identified as output is sufficiently restricted to avoid universality, almost every other part of their behavior nevertheless does exhibit universality—just as one would expect from the Principle of Computational Equivalence.

One can go for 1000 steps and still not know what is going to happen. … But even having traced the evolution this far, one still has no idea what its final outcome will be. … For any particular initial condition it may be that if one just runs the system for a certain number of steps then one will be able to tell what it will do.

When one does practical computing one tends to assess the difficulty of a computation by seeing how much time it takes and perhaps how big a program it involves and how much memory it needs. But normally one has no way to tell whether the scheme one has for doing a particular computation is the most efficient possible. … For with such programs it becomes realistic to enumerate every single one of a particular kind, and then just to see explicitly which is the most efficient at performing some specific computation.

But what if one allows Turing machines with more complicated rules? … Nevertheless, if one looks carefully at examples (a) through (h) each of them shows large regions of either repetitive or nested behavior. … Once one has a system that is universal it can in principle be made to do any computation.

So as a simple example one could imagine having a pattern laid out on a three-dimensional array with each successive vertical plane giving the evolution of some one-dimensional universal system from each of its successive possible initial conditions.

Meaning and regularity If one considers something to show regularity one may or may not consider it meaningful. But if one considers something random then usually one will also consider it meaningless. For to say that something is meaningful normally implies that one somehow comes to a conclusion from it.

There is no doubt that they do, and as one example I will briefly discuss here what is probably the most obvious feature of essentially all financial markets: the apparent randomness with which prices tend to fluctuate. Whether one looks at stocks, bonds, commodities, currencies, derivatives or essentially any other kind of financial instrument, the sequences of prices that one sees at successive times show some overall trends, but also exhibit varying amounts of apparent randomness.

One simple strategy for assigning codewords is to number all distinct blocks in order of decreasing frequency, and then just to use the resulting numbers—given, say, in one of the representations discussed above—as the codewords. But if one takes into account the actual frequencies of different blocks, as well as their ranking, then it turns out that there are better ways to assign codewords.