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In all cases the evolution starts from the same random initial condition, and is continued until it stabilizes.
Apart from exceptional cases where no randomization occurs, the behavior obtained with different initial conditions is eventually quite indistinguishable in its overall properties.
But there is still an important constraint on the behavior: even though black and gray cells may in effect move around randomly, their total number must always be conserved.
Some level of checking can be done by tracing the emulation of random initial conditions for the universal system.
Note also that as discussed in the main text having maximal entropy does not by any means imply perfect randomness.
The quantity FoldList[Plus, 0, Table[MoebiusMu[i], {i, n}]] behaves very much like a random walk.
Part (b) of the picture shows a version of the evolution compressed to include only
A register machine whose behavior seems in some ways random.
If the curve were continued further, it would spend more time above the axis, and no aspect of what is seen provides any evidence that the digit sequence is anything but perfectly random.
Sequences whose spectra contain no dominant peaks typically sound like random noise, although sometimes explicit time variation can be heard, and indeed sequence (c) just sounds like a succession of idealized frog ribbets.
But with rule 30 most sequences that are generated—even from simple initial conditions—appear completely random with respect to all of the methods of perception and analysis discussed so far.