So how can one understand what is going on? … And so as a minimal idealization one can for example try viewing a market as being like a simple one-dimensional cellular automaton. … One can imagine all sorts of schemes by which such colors could be updated.

But how good are the results one then gets? If one looks at quantities such as the overall density of black cells that were in effect used in finding the model in the first place then inevitably the results one gets seem quite good. But as soon as one looks at explicit pictures like the ones below, one immediately sees dramatic differences between the original data and what one gets from the model.

At first one sees only simple repetitive behavior. … And as one continues one sees various other repetitive and nested forms. … It could be that if one went just a little further in looking at initial conditions one would see more complicated behavior.

If one looks at axiom systems of the form {…  a, a ∘ b  b ∘ a} the first one that one finds that allows only Nand and Nor with 2-value operators is {(a ∘ a) ∘ (a ∘ a)  a, a ∘ b  b ∘ a} . … So what this means is that if one were just to go through a list of the simplest few thousand axiom systems one would already be quite likely to find one that represents logic. … But if one looks just at axiom systems is there anything obviously special about the ones for logic?

But why do only one replacement at each step? The pictures on the next page show what happens if one again scans from left to right, but now one performs all replacements that fit, rather than just the first one. … So what happens with replacements that involve more than just one element?

One issue—beyond the obvious fact that sounds cannot be included directly in a printed book—is that while one can study the details of a picture at whatever pace one wants, a sound is in a sense gone as soon as it has finished playing. But everyday experience makes it quite clear that one can still learn a lot by listening to sounds. … But if the sequence is random then what one hears is just an amorphous hiss.

For as we discussed, once one has a universal system such a system can emulate any of the kinds of systems that we considered—even ones whose construction is more complicated than its own. … For as soon as one identifies any such class of computations, one can imagine setting up a system which includes an infinite table of their results. … So what might make one think that this is true?

But one feature is that normally the resulting axiom system is in a sense more general than the objects one started from. … One can think of an axiom system—say one of those listed on pages 773 and 774 —as giving a set of constraints that any object it describes must satisfy. … One might imagine that if one were to add more axioms to an axiom system one could always in the end force there to be only one kind of object that would satisfy the constraints of the system.

The pictures below show what happens if one adds a number other than 1 at each step. … If instead of addition one uses multiplication, however, then the results one gets can be very different. … It turns out that if one represents numbers as digit sequences in base 2, then the operation of multiplying by 2 has a very simple effect: it just shifts the digit sequence one place to the left, adding a 0 digit on the right.

Of the 65,536 possible mobile automata with rules of the kind discussed so far it turns out that not a single one shows more complex behavior. … One can extend the set of rules one considers by allowing not only the color of the active cell itself but also the colors of its immediate neighbors to be updated at each step. … If one samples these rules at random, one finds that more than 99% of them just yield simple repetitive behavior.