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And in a 10×10 array the chance of finding a pattern where the fraction of squares that violate the constraints is even less than 50% is only one in a thousand, while the chance of finding a pattern where the fraction is less than 25% is one in four trillion.
In the first picture for each rule, replacements are made at randomly chosen steps, while in the second picture, they are in a sense always made at the earliest possible step.
But while in a substitution system one always carries out just a single set of replacements at each step, getting a single new string, in a multiway system one instead carries out every possible replacement, thereby typically generating many new strings.
The next page shows a few examples in detail, while the following two pages [ 55 , 56 ] show what happens in all 256 possible cases.
While properties such as the density of black cells are typically set up to agree between the data and the model, the pictures make it clear that more detailed features do not.
Once again, lighter colors in the cellular automaton represent ordinary cells in the Turing machine, while darker colors represent the cell under the head, with a specific darker color corresponding to each possible state of the head.
And while these particular types of equations have never seriously been proposed as idealizations of actual processes in nature or elsewhere, it turns out that a related phenomenon can presumably occur in differential equations—which represent the most common basis for mathematical models in most areas of traditional science.
But while this means that for a given input each of them yields the same output, the pictures demonstrate that they usually take a different number of steps to do so.
For the completeness of the system guarantees that one must always reach one or the other, while its consistency implies that reaching one allows one to conclude that one will never reach the other.
So while we might have imagined that science would eventually show us how to rise above all our human details what we now see is that in fact these details are in effect the only important thing about us.
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