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The picture below shows in sorted order the configurations obtained at each successive step in the evolution of all 256 elementary cellular automata starting from a single black cell.
Since numbers can be factored uniquely into products of powers of primes, a number can be specified by a list in which 1's appear at the positions of the appropriate Prime[m] n (which can be sorted by size) and 0's appear elsewhere, as shown below.
No doubt the methods of this book will in the future lead to all sorts of algorithms based on much more complex patterns of behavior.
And indeed with this thought in mind all sorts of elaborate combinations of linear congruential and other generators have been proposed.
And it is then assumed that these estimates are ultimately affected by all sorts of events that go on in the world, making random movements
Or is it in fact common in all sorts of simple programs?
In cases where all strings that appear both in rules and initial conditions are sorted—so that for example A 's appear before B 's—any string generated will also be sorted, so it can be specified just by giving a list of how many A 's and how many B 's appear in it.
In many places in the book—especially these notes—I discuss all sorts of specific problems and issues of direct relevance to current computer science.
But at least at first, I suspect that huge simplifications will be made, with the result that all sorts of misleading conclusions will probably be reached, perhaps in some cases even seemingly contradicting the principle.
Before the input can have a chance of specifying meaningful action there are often all sorts of issues about whether variables in it refer to entities that can be considered to exist. … For the kinds of systems like cellular automata that I have discussed in this book programs chosen at random do very often produce some sort of non-trivial behavior.
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