Jason Cawley
Wolfram Science Group
Phoenix, AZ USA
Registered: Aug 2003
Posts: 712 |
Sufficient can be doubted by pointing to relatively simple systems which count through possibilities, but don't actually do anything further with them. A binary counter may serve as an example. Yes it runs through all possible strings of length 1, then length 2, then length 3, etc. But these local patterns do not give rise to any further intricate behavior, they just disappear to be replaced with the next string in counting order. The overall pattern this produces looks nested, which is a kind of simplicity - if a bit more intricate than say repetition.
Necessary can be doubted by noticing that some CAs may disallow blocks of various types, because the form of the rule can't allow certain combinations at the lowest level. Alternating black and white for instance. Yet such a rule might still do perfectly complicated things with larger structures.
Consider rule 54, which rapidly resolves to only a few allowed local patterns and a modest variety of stable structures moving about on a repeating background of 1-1-1-0 repeated, alternating with 0-0-0-1 repeated. Yet the boundaries of regions "tiled" this way may move and interact to produce new structures. Elaborate "glider guns" have been found for rule 54, which emit new offsets between periodic regions every few steps, while remaining stable themselves.
As for "sufficiently infinite", page 726 notes that "it may be quite impractical to find particular computations that one wants". Infinities can make lots of unusual things seem plausible, and sometimes analysis in terms of them helps us see general relationships we might miss otherwise. But they can also mislead. Overall, the analysis serves as a good reason to expect rule 30 to never "settle down", and indirectly for its evolution to be irreducible. More than that I would not read into it.
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01-23-2006 06:55 AM
