And what we see immediately from these pictures is that while some systems exhibit exactly the kind of randomization implied by the Second Law, others do not. … There is, I believe, no choice but to conclude that for practical purposes rule 37R simply does not obey the Second Law. … How do such systems work?

Ones that do not last long can be very different from ones that would persist forever. … One immediate difference, however, is that in traditional particle physics one does not imagine a pattern of behavior as definite and determined as in the picture above. … And in effect what this background does is to introduce a kind of random environment that can make many different detailed patterns of behavior occur with certain probabilities even with the same initial configuration of particles.

But what the picture on the facing page demonstrates is that if one just does statistical analysis by computing frequencies of blocks one will see no evidence of any such underlying simplicity. … So what kinds of quantities can one in the end use in doing statistical analysis? … For in each case all one has to do is to compute the value of a quantity from a particular sequence of data, and then compare this value with what would be obtained by averaging over all possible sequences.

But for none of the other 3-state 2-color Turing machines shown do 4-state rules offer any speedup. … Yet now—much as for the 4-state Turing machines in Chapter 3 —the actual behavior seen does not show any obvious computational reducibility. … Once one has a system that is universal it can in principle be made to do any computation.

And I believe the basic answer to this has to do with the fact that when we as humans set up artifacts we usually need to be able to foresee what they will do—for otherwise we have no way to tell whether they will achieve the purposes we want. … And in fact I have argued that among systems that appear in nature a great many exhibit computational irreducibility—so that in a sense it becomes irreducibly difficult to foresee what they will do.

do not also show sensitive dependence on initial conditions. … Case (c) does not show such sensitivity to initial conditions, but instead always evolves to 0, independent of its initial conditions.

So what does it take to get patterns with more complicated structure? … Even though this basic rule does not involve overlapping squares, the pattern obtained even by step 3 already has squares that overlap.

But where does this randomness ultimately come from? … The presence of randomness in initial conditions—together with sensitive dependence on initial conditions—does imply at least some degree of randomness in the behavior of any class 3 system.

below, is that some process can start at one point in space and then progressively spread, doing the same thing at every point it reaches. … Here different elements in the system do interact, but the result is still that all of them evolve to the same state.

Does this also fail to find regularities, or does it provide some special way—at least within the context of a setup like the one shown below—to recognize whatever regularities are necessary for one to be able to deduce the initial condition and thus determine the key?