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Both ideas rely on the local nature of cellular automaton rules.
If v is known then such equations in essence provide explicit rules for computing u .
In organisms with a total of just a few thousand cells, the final position and type of every cell seems to be determined directly by the genetic program of the organism; most likely what happens is that each cell division leads to some modification in genetic material, perhaps through rules like those in a multiway system.
Iterated Maps and the Chaos Phenomenon
The basic idea of an iterated map is to take a number between 0 and 1, and then in a sequence of steps to update this number according to a fixed rule or "map".
And indeed it is one of the central ideas of this book to go beyond mathematical equations, and to consider models that are based on programs which can effectively involve rules of any kind.
So as a simple idealization, one can consider having a large number of particles move around on a fixed discrete grid, and undergo collisions governed by simple cellular-automaton-like rules.
Many times in this book we have seen examples where different systems can yield very much the same overall behavior, even though the details of their underlying rules are quite different.
Similarly, I discovered the randomness of rule 30 (page 27 ) when I was in the process of setting up large simulations for an early parallel-processing computer.
But moving from one point in time to another involves actually applying the cellular automaton rule.
But such systems work not by being required to satisfy constraints, but instead by just repeatedly applying explicit rules.