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And indeed—just like with the cellular automata in this section—most of the systems in Chapter 11 that exhibit universal computation can readily be made reversible with only slight overhead.
The basic origin of this phenomenon is the averaging effect of randomness discussed in Chapter 7 (technically, it is the survival only of leading operators at renormalization group fixed points).
The number of distinct sequences at step t in these three systems is respectively Ceiling[t/2] , t and Fibonacci[t+1] (which increases approximately like 1.618 t ).
… Note that in case (a), the total number of possible states at step t increases roughly like t 2 , while in case (b) it increases only like t .
Is it only cellular automata with very specific underlying rules that produce such behavior?
But in trying to explain our actual experience of the natural world, we need to consider not only how phenomena are produced in nature, but also how we perceive and analyze these phenomena.
The pictures on the right are obtained by keeping only the steps indicated by arrows on the left.
And its implications are both broad and deep, addressing a host of longstanding issues not only in science, but also in mathematics, philosophy and elsewhere.
alternates between 0 and 1, but the second register progressively grows. … It turns out that this program is one of only two (which differ just by interchange of the first and second registers) out of the 248,832 possible programs with five instructions that yield anything other than strictly repetitive behavior.
Feedback [in visual processing]
Most of the lowest levels of visual processing seem to involve only signals going successively from one layer in the eye or brain to the next.
And in all cases black cells appear only in blocks that are an odd number of cells wide.