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But at least with blocks up to length 25, rule 30 for example is not able to emulate any non-trivial rules at all.
My guess is that for all practical purposes one cannot.
For it implies that even if in principle one has all the information one needs to work out how some particular system will behave, it can still take an irreducible amount of computational work actually to do this.
But what my discoveries about computational irreducibility now imply is that this is not in fact the case, and that instead there are many common systems whose behavior cannot in the end be determined at all except by something like an explicit simulation.
So the result is that computational irreducibility can in the end be expected to be common, so that it should indeed be effectively impossible to outrun the evolution of all sorts of systems.
And it turns out that the kinds of compressed descriptions that can be obtained by the methods of perception and analysis that we use in practice and that we discussed in Chapter 10 all essentially have this property.
In other cases the proof may be more difficult—say being based on establishing some large maximum size for a solution, then checking all integers up to that size.
For typically our experience is that if we are able to get a particular kind of system to generate a particular outcome at all, then normally the behavior involved in doing so is quite simple.
And if one identifies a feature—such as repetition or nesting—that is common to many possible systems, then it becomes inevitable that this feature will appear not only when intelligence or mathematics is involved, but also in all sorts of systems that just occur in nature.
So what does all this mean about extraterrestrial intelligence?