And what this means is that in effect the same rules will apply regardless of how fast one is going. … And from this idea one expects for example that light should always travel at its usual speed with respect to whatever emitted it. But what if one happens to be moving with respect to this emitter?

But until there are interactions that change the structure of the cores, these common connections will then remain—and will continue to define a thread that goes directly from one particle to the other. … For earlier in this chapter I discussed measuring distance on a network just by counting the minimum number of successive individual connections that one has to follow in order to get from one point to another. Yet if one uses this measure of distance then the distance between two particles will always tend to remain fixed as the number of connections in the thread.

And in general perception and analysis can be viewed as equivalent to finding models that reproduce whatever aspects of data one considers relevant. … As one example, the pictures on the facing page were all generated by starting from a single black cell and then applying very simple two-dimensional cellular automaton rules. Yet if one looks just at these final pictures, there is no easy way to tell how they were made.

So even if one allows rather general structure, the evidence is that in the end there is no way to set up any simple formula that will describe the outcome of evolution for a system like rule 30. And even if one settles for complicated formulas, just finding the least complicated one in a particular case rapidly becomes extremely difficult. … Human Thinking When we are presented with new data one thing we can always do is just apply our general powers of human thinking to it.

So given a particular elementary cellular automaton one can then ask what other elementary cellular automata it can emulate using blocks up to a certain length. … And as an example the picture below shows how rule 30 can be made to emulate the basic action of one step in rule 90. The idea is to set up a configuration in rule 30 so that if one inserts input at particular positions the output from the underlying rule 30 evolution corresponds exactly to what one would get from a single step of rule 90 evolution.

One of the features of cellular automata and most of the other computational systems that I have discussed in this book is that they are in some fundamental sense discrete. … And indeed, as I have discussed several times in this book, it is in many cases clear that the whole notion of continuity is just an idealization—although one that happens to be almost required if one wants to make use of traditional mathematical methods. Fluids provide one obvious example.

It is not that one cannot find underlying rules for such behavior. … One can always in effect do an experiment, and just watch the actual behavior of whatever system one wants to study. But what one

And often one finds that even a short string can take a rather large number of steps to produce. … Certainly one can take the rules for any multiway system and add transformations that immediately generate particular short strings. … Just like in multiway systems, one can always add axioms to make it easier to prove particular theorems.

But if one removes just a single axiom from any of the axiom systems above then it turns out that they no longer work, and for example they cannot establish the equivalence result stated by whichever axiom one has removed. In general one can think of axioms for an operator system as giving constraints on the form of the operator. And if one is going to reproduce all the equivalences that hold for a particular form then these constraints must in effect be such as to force that form to occur.

But in practice if one just receives a signal one normally has no way to tell which of all possible rules for producing it were in fact used. … In the past, one might have thought that it would be enough for the production of the signal to involve sophisticated computation. … One might nevertheless imagine that any sufficiently advanced intelligence would somehow at least consider the primes significant.