effect on overall patterns of flow, one cannot realistically attribute the large-scale randomness that one sees in a turbulent fluid to randomness that exists at the level of individual particles. … And although many details are different from what one sees in real fluids, the overall mixture of regularity and randomness is strikingly similar. … But while some of the simpler ones have been captured quite completely by methods based on traditional mathematical equations, the more complex ones have not.

And this implies that if one knew all of the underlying details of the network that makes up our universe, it should always be possible to work out the result of any measurement. … One might however imagine that as a first approximation one could take account of underlying apparent randomness just by saying that there are certain probabilities for particles to behave in particular ways. … For in particular, if one takes two particles that have come from a single source, then the result of a measurement on one of them is found in a sense to depend too much on what measurement gets done on the other—even if there is not enough time for information travelling at the speed of light to get from one to the other.

But what if a sequence one actually observes has 9 black squares out of 10? Even though this is not the most likely thing to see, one certainly cannot conclude from seeing it that the model is wrong. … For in such a case one can always just make up an extreme model in which only one very long block is allowed, with this block being precisely the sequence that is observed in the data.

One practical example where this is done is a simple procedure often used for looking up names by sound rather than spelling. … So how can one achieve this in general? … And this means that if one looks only at the output from these nerve cells, then one gets a representation of visual images in which two images that differ only in certain kinds of details will be assigned the same representation.

Once one has an explicit system that successfully emulates human thinking, however, one can imagine progressively removing some of this complexity, and seeing just which features of human thinking end up being preserved. … When one learns a language—at least as a young child—one implicitly tends to deduce simple grammatical rules that are in effect specific generalizations of examples one has encountered. And I suspect that in doing this the types of generalizations that one makes are essentially those that correspond to the types of similarities that one readily recognizes in retrieving data from memory.

For knowing that a particular rule is universal just tells one that it is possible to set up initial conditions that will cause a sophisticated computation to occur. But it does not tell one what will happen if, for example, one starts from typical simple initial conditions. Yet the Principle of Computational Equivalence asserts that even in such a case, whenever the behavior one sees is not obviously simple, it will almost always correspond to a computation of equivalent sophistication.

But can one go still further? And what happens for example if one just tries to search simple axiom systems for ones that work? … But if one now looks at operators involving 3 possible values then it turns out that this axiom system allows ones not equivalent to Nand Axiom systems for basic logic (propositional calculus) formulated in terms of Nand ( ⊼ ).

And then, depending on the colors of these elements, one of several possible blocks is tagged onto the end of the sequence. The pictures below show examples of tag systems in which just one element is removed at each step. And already in these systems one sometimes sees behavior that looks somewhat complicated.

But if one extends again the type of constraints one considers, it turns out to become possible to construct examples that force more complex behavior. … In this particular system, only the 33 templates shown above (out of the 512 possible ones) are allowed to occur. … The system shown was specifically constructed in correspondence with the rule 60 elementary one-dimensional cellular automaton.

If one just looks at a rule in its raw form, it is usually almost impossible to tell much about the overall behavior it will produce. But in cases where this behavior ends up being simple, one can often recognize in it specific mechanisms that seem to be at work. … A rather straightforward one, illustrated in the first set of pictures A sequence of elementary cellular automata whose rules differ from one to the next only at one position (a Gray code sequence).