One might have thought that in the literature of traditional science new models would be proposed all the time. … For once one has a program, one can always find out what its behavior will be just by running it. Yet with an equation one may need to do elaborate mathematical analysis in order to find out what behavior it can lead to.

As a simple example, one might have imagined that systems like cellular automata, being made up of discrete cells, would never be able to reproduce realistic natural shapes. … But from experience in practical computing one knows that it is usually very difficult to foresee what even a simple program will do. … For if one could just look at a program

Indeed, one might assume that there are in fact an infinite number of universes, each with a different rule, and that we simply live in a particular—and essentially arbitrary—one of them. … But one of its consequences is that it gives us no reason to think that the rule for our particular universe should be in any way simple. … And knowing this, one might be led to believe that for some reason the universe is set up to have the simplest rules throughout.

For example, we can move from one point in space to another in more or less any way we choose. … And thus for example in a cellular automaton moving from one point in space to another just corresponds to shifting from one cell to another. But moving from one point in time to another involves actually applying the cellular automaton rule.

So what does one find if one allows templates that include nodes out to distance two? … But in trying to understand the ultimate underlying mechanisms of the universe, I believe that one must inevitably distinguish between these views. And I strongly believe that the second view is the one most likely to provide a meaningful underlying model for our universe.

At first one might not think so. … And assuming that one is able to look at a multiway system from the outside, one will immediately see that different paths exist corresponding to different histories. … If one were able to look at the multiway system from the outside, this path would seem quite arbitrary.

But an important difference is that it would occur even if one could in effect perfectly isolate a system from the rest of the universe. If in the past one had seen apparent randomness in such a system there might have seemed to be no choice but to assume something like an underlying multiway system. … But if one looks not just at individual strings but rather at the sequences of strings that exist along paths in the multiway system, then one finds that these can no longer be so arbitrary.

But unlike in example (e), this Turing machine is not the only one that computes the function it computes. … And indeed if one looks at all 4096 Turing machines with 2 states and 2 colors it turns out that the only rates of growth that one ever sees are linear, square and exponential. … So what happens if one considers more complicated Turing machines?

But as a practical matter, if one can set up a model that is based on sufficiently simple rules then it becomes more likely that one will be able to make designs and build control devices that work even with some system in nature that shows complex behavior. … Traditional intuition suggests that to be able to do sophisticated computations one would inevitably need a system with complicated underlying rules. … And one thing this would mean is that doing

fundamentally larger range if one allowed, say, four or five connections rather than just three. … So how then can one get networks that correspond to ordinary space? … In what follows I consider only non-degenerate networks, in which there is at most one connection between any two nodes.