and especially a simple one—then it can be extremely difficult to determine whether or not the rule is universal.

As we discussed in the previous chapter, the usual way to demonstrate that a rule is universal is to find a scheme for setting up initial conditions and for decoding output that makes the rule emulate some other rule that is already known to be universal.

But the problem is that in any particular case there is almost no limit on how complicated such a scheme might need to be. In fact, about the only restriction is that the scheme itself should not exhibit universality just in setting up initial conditions and decoding output.

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.

It is often even more difficult to prove that a system is not universal than to prove that it is. For what one needs to show is that no possible scheme can be devised that will allow the system to emulate any other universal system. 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.

If this behavior always has an obvious repetitive or nested form then it will often be quite straightforward to analyze. 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.

As mentioned in Chapter 10, however, I do know of a few systems based on numbers for which a fairly complete analysis can be given even though the overall behavior is not repetitive or nested or otherwise obviously simple. And no doubt some other examples like this do exist. But it is my strong belief—as embodied in the Principle of Computational Equivalence—that in the end the vast majority of systems whose behavior is not obviously simple will turn out to be universal.

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From Stephen Wolfram: A New Kind of Science [citation]