As I will discuss in the next chapter, my general expectation is that more or less any system whose behavior is not somehow fundamentally repetitive or nested will in the end turn out to be universal. But I suspect that this fact will be very much easier to establish for some systems than for others—with rule 110 being one of the easiest cases.

In general what one needs to do in order to prove universality is to find a procedure for setting up initial conditions in one system so as to make it emulate some general class of other systems. And at some level the main challenge is that our experience from programming and engineering tends to provide us with only a limited set of methods for coming up with such a procedure. Typically what we are used to doing is constructing things in stages. Usually we start by building components, and then we progressively assemble these into larger and larger structures. And the point is that at each stage, we need think directly only about the scale of structures that we are currently handling—and not for example about all the pieces that make up these structures.

In proving the universality of rule 110, we were able to follow essentially the same basic approach. We started by identifying various localized structures, and then we used these structures as components in building up the progressively larger structures that we needed.

What was in a sense crucial to our approach was therefore that we could readily control the transmission of information in the system. For this is what allowed us to treat different localized structures as being separate and independent objects.

And indeed in any system with class 4 behavior, things will typically always work in more or less the same way. But in class 3 systems they will not. For what usually happens in such systems is that a change made even to a single cell will eventually spread to affect all other cells. And this kind of uncontrolled transmission of information makes it very difficult to identify pieces that could be used as definite components in a construction.

So what can be done in such cases? The most obvious possibility is that one might be able to find special classes of initial conditions in which transmission of information could be controlled. And an example where this can be potentially done is rule 73.

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