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So if such processes can correspond to the evolution of systems like cellular automata, then it follows at least formally that differential equations should be able to do in finite time computations that would take a discrete system like a cellular automaton an infinite time to do. … Does it also follow the Principle of Computational Equivalence? Or does it somehow manage to do computations that are more sophisticated than the Principle of Computational Equivalence implies?
But others do not—and thus in effect do not obey the Second Law of Thermodynamics.
So the fact that it is possible does not immediately establish universality in any ordinary sense. But it does once again support the idea that almost any cellular automaton whose behavior seems to us complex can be made to do computations that are in a sense as sophisticated as one wants. … Summaries of how various underlying cellular automata do in emulating a single step in the evolution of each of the 256 possible elementary cellular automata using the scheme from the facing page with blocks of successively greater widths.
Needless to say, we do not know what a truly optimal organism would be like. … So why then do higher organisms exist at all? … But after a while it becomes clear what makes sense and what does not.
So how does universality actually work in the case of arithmetic? … At first it does not seem obvious that such a statement could ever be set up as a statement in arithmetic. … And in fact the main technical difficulty in the original proof of Gödel's Theorem had to do with showing—by doing what amounted to establishing the universality of arithmetic—that the statement could indeed meaningfully be encoded as a statement purely in arithmetic.
But as a practical matter we often end up describing what systems do in terms of purpose when this seems to us simpler than describing it in terms of mechanism. … For while rule 30 does generate such a pattern, it also does a lot else that appears irrelevant to this purpose. … But an immediate issue is that in traditional engineering we normally do not come even close to getting systems that are minimal.
At first biology seems to do better by repeatedly making random modifications to genetic programs, and then applying natural selection. But while this process does quite often yield programs with complex behavior, I argued earlier in this book that it does not usually manage to mold anything but fairly simple aspects of this behavior. So what then can one do?
But even if one does all sorts of parallel processing this approach presumably in the end becomes quite impractical. So what can one then do? … It turns out that one does not.
And only after 1017 steps does it finally become clear that the pattern in fact dies out. … And this means that the general question of what the system will ultimately do can be considered formally undecidable, in the sense there can be no finite computation that will guarantee to decide it. For any particular initial condition it may be that if one just runs the system for a certain number of steps then one will be able to tell what it will do.
In this chapter what I will do is to take what we have learned, and look at a sequence of fairly specific kinds of systems in nature and elsewhere, and in each case discuss how the most obvious features of their behavior arise. … And in fact, to do this for even just one kind of system would most likely take at least another whole book, if not much more. But what I do want to do is to identify the basic mechanisms that are responsible for the most obvious features of the behavior of each kind of system.
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