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(Operators exist for any k .)
But in this chapter, I study actual systems that exist in nature, and as a result, most of what I say cannot be said with any absolute certainty, but instead must involve a significant component of hypothesis.
What the picture then shows is that every region of white that exists in the initial conditions progressively gets filled in with black, so that in the end all that remains is a uniform state with every cell black.
And in fact it turns out to be quite common for there to exist special initial conditions for one cellular automaton that make it behave just like some other cellular automaton.
And for all repetition periods up to 10—with the exception of 7—at least one fixed or moving structure ultimately turns out to exist.
the past taken as one of the key pieces of evidence for the idea that simple laws of nature could exist.
So this strongly suggests that no simple formula exists—at least of the type used here—that can describe patterns generated by any significant number of steps of evolution in a system like rule 30.
Obvious methods involve for example showing that no solutions exist for real variables, or for variables reduced modulo some n . … For d > n , on the other hand, it suggests that there will often be no solutions, and that any solutions that exist will usually be small. … When solutions do exist, most are fairly small.
But in capturing the most obvious aspects of human will what seems to be most important is at least short-term coherence and consistency of action—as often exists in class 4, but not class 3, systems.
Using the results of this section it then follows that in the evolution of all 2D cellular automata of the type discussed on page 170 there exist purely repetitive configurations that remain unchanged.