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In the detailed behavior of the various localized structures that separate these domains of repetitive behavior there is all sorts of complexity.
that they ultimately tend to be equivalent in their computational sophistication—and thus show all sorts of similar phenomena.
Such rule numbers can be converted to general form using
FromDigits[Map[Last, Sort[Flatten[Map[Thread, Thread[{s, IntegerDigits[n, 2, 12]}]], 1]]], 2]
If one allows more rapid growth, however, then there presumably start to be all sorts of multiway systems that never show any such regularity.
Note that rule (a) effectively sorts the elements in its initial conditions, always placing black before white.
The first rule shown has the effect of sorting the elements in the string.
In most kinds of mathematics there are all sorts of additional details, particularly about how to determine which parts of one or more previous expressions actually get used at each step in a proof.
The statements are sorted so that the simplest are first.
Ever since the 1960s all sorts of schemes for nonlinear processing of images have been discussed and used in particular communities.
But the Principle of Computational Equivalence suggests at some level a remarkable uniformity among systems, that allows all sorts of general scientific statements to be made without dependence on context.