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The smallest product of these numbers is 24 (compare note below), and the rule he gave in this case is:
Note that these results concern Turing machines which can halt (see page 1137 ); the Turing machines that I consider do not typically have this feature.
The forms j[j] and j[j[j]] appear to be the simplest that can be used for s and k ; j and j[j] , for example, do not work.
If one now ignores machines that do not allow the head to move more than one step in one of the two directions, that always yield the same color when moving in a particular direction, or that always leave the tape unchanged, one is finally left with just 25 distinct cases.
(Some of the 20 standard amino acids do however occur more frequently than others.)
Typically the idea of these models is to approximate those elements of a system about which one does not know much by random variables.
The structure does not correspond to the way that chemical bonds are arranged in any common crystalline materials, probably because it would be likely to be mechanically unstable.
But when in general does complexity occur?
The new kind of science in this book, however, explores much more general classes of programs—and in doing so begins to shed new light on various longstanding questions in computational complexity theory.
A more practical alternative is to build up patterns iteratively, starting with a small region, and then adding new cells in essentially all possible ways, at each stage backtracking if the constraint for the system does not end up being satisfied.
But what about initial conditions that do not just consist of a single block repeated forever?