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One might at the outset have thought that leaves would get their shapes through some mechanism quite unrelated to other aspects of plant growth.
And as a result, my guess is that the only realistic way to find the rule in the first place will be to start from some very straightforward representation, and then just to search through large numbers of possible rules in this representation.
For one can think of successive slices through a causal network as corresponding to states at successive moments in time.
And second, that with causal invariance different slices through a causal network can be produced by the same underlying rules.
And the crucial point is that this happens just through the intrinsic evolution of the system—without the need for any additional input from outside or from any sort of explicit source of randomness.
And in fact I strongly suspect that even with just three pairs there is already computational irreducibility, so that in effect the only way to answer the question of whether the constraints can be satisfied is explicitly to trace through some fraction of all arbitrarily long sequences—making this question in general undecidable.
Yet now essentially this idea—viewed in computational terms through the discoveries in this book—emerges as crucial.
This feature of decrement-jump instructions may seem like a detail, but in fact it is crucial—for it is what makes it possible for our register machines to take different paths depending on values in registers through the programs they are given.
Typical running times for FactorInteger[n] in Mathematica 4 are shown below for the first 1000 numbers with each of 15 through 30 digits.
The pictures below show the smallest trivalent networks with girths 3 through 8 (so-called cages).