One might at first imagine that if p and q are both short strings then one could also drop any very long strings that are produced. … But as soon as one knows that there is a path from a string u to a string v , one can also imagine applying the rule u  v to any string—in effect like a lemma. … Nevertheless, one can for example always in effect work forwards from p , and backwards from q , seeing whether there is any overlap in the sets of strings one gets.

When one looks at systems in nature, one of the striking things one notices is that even when systems have quite different underlying physical, biological or other components their overall patterns of behavior can often seem remarkably similar. … And if this is the case, then it means that one can indeed expect to get insight into the behavior of natural systems by studying the behavior of simple programs.

If one looks at individual particles, then changing the position of even one particle will typically have an effect that spreads rapidly. But if one looks instead at the average behavior of many particles, such effects get completely washed out. … And in the vast majority of cases, small changes instead seem to damp out rapidly—just as one might expect from everyday experience with viscosity in fluids.

In addition, occasionally one can identify situations of seemingly pure speculation in which trading occurs without the possibility of any significant external input—and in such situations prices tend to show more, rather than less, seemingly random fluctuations. And knowing this, one might then think that perhaps random fluctuations are just an inevitable feature of the way that prices adjust to their correct values. … And certainly one can construct algorithms that operate between larger numbers of parties that would also lead to fairly smooth behavior.

So from this one can use essentially standard arguments to derive all the various phenomena familiar from ordinary relativity theory. … But more generally one can think of following criss-crossing connections that exist in some fixed fragment of a causal network. … The internal mechanism of the clock is shown as a zig-zag black line—with each sweep of this line corresponding to the passage of one unit of time.

cone and asks how the number of events that lie within the cone increases as one goes to successive slices. … But I suspect that if one has replacement rules that are causal invariant and that in effect successfully maintain a fixed number of dimensions they will almost inevitably lead to behavior that follows something close to the Einstein equations. … As a first step, one can ask what effect the structure of space has on something like a particle—assuming that one can ignore the effect of the particle back on space.

advance how many steps of evolution one will need to look at in order to be sure that any particular piece of behavior will not occur. … In fact, if one followed such a definition most of the pictures in this book that I have said look random—including for example picture (c) on page 553 —would be considered not random. … But in fact they typically correspond just to seeing whether some particular process—and usually a rather simple one—succeeds in recognizing regularities and thus in generating a shorter description.

The pictures on the facing page show some examples of the results one gets. … One might have thought that to capture all these kinds of regularities would require a whole collection of complicated procedures. … Indeed, as it happens, every single one of the pictures on the facing page can for example be generated from very short cellular automaton programs.

In the one specific case shown at the top of the facing page it turns out to be fairly easy. … And as the table below illustrates, the entries in Pascal's triangle are simply the binomial coefficients that appear when one expands out the powers of 1 + x . So to determine whether a particular square in the pattern is black or white, all one need do is to compute the corresponding binomial coefficient, and see whether or not it is an odd number.

In the first instance, each term can be set up to correspond directly to one of the cases in the original rule. … The picture at the top of the facing page shows what happens if one considers two steps of cellular automaton evolution. … They are set up so that only at most one term in each formula is ever relevant for any particular configuration of colors.