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Problem 9. What is the correspondence between cellular automata and continuous systems? Cellular automatat are discrete in several respects. First, t… - Stephen Wolfram
" "Problem 9. What is the correspondence between cellular automata and continuous systems?
Cellular automatat are discrete in several respects. First, they consist of a discrete spatial lattice of sites. Second, they evolve in discrete steps. And finally, each site has only a finite discrete set of possible values.
The first two forms of discreteness are addressed in the numerical analysis of approximate solutions to, say, differential equations. ...
The third form of discreteness in cellular automata is not so familiar from numerical analysis. It is an extreme form of round-off, in which each "number" can have only a few possible values (rather than the usual 2<sup>16</sup> or 2<sup>32</sup>).
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About Stephen Wolfram
Stephen Wolfram (born 29 August 1959) is a British scientist known for his work in theoretical particle physics, cellular automata, complexity theory, and computer algebra. He is the creator of the computer program Mathematica.
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This is what you... learn from this principle of computational equivalence. ...[I]t's both a message of ...hope, and ...[that] you're not as special as you think you are... We're just doing computations like things in nature do computations, like those gas molecules do computations, like the weather does computations. The only thing about the computations that we do that's very special is that we understand what they are... because they're connected to our purposes, our ways of thinking...
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Cellular automata are discrete dynamical systems with simple construction but complex self-organizing behaviour. Evidence is presented that all one-dimensional cellular automata fall into four distinct universality classes. Characterizations of the structures generated in these classes are discussed. Three classes exhibit behaviour analogous to limit points, limit cycles and chaotic attractors. The fourth class is probably capable of universal computation, so that properties of its infinite time behaviour are undecidable.
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