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Of the 274,499 axiom systems of the form {… a} where … involves ∘ up to 6 times, 32,004 allow only operators {6,9} , while 964 allow only {1,7} .
Long halting times [in symbolic systems]
Symbolic systems with rules of the form ℯ [x_][y_] Nest[x, y, r] always evolve to fixed points—though with initial conditions of size n this can take of order Nest[r # &, 0, n] steps (see above ).
Part (c) shows the instructions that are executed for the first 400 times that one of the registers is decreased to zero.
The picture below shows a simple example based on the rule 30 cellular automaton that I have discussed several times before in this book.
The cellular automaton effectively works by adding the original number n together n times.
The pictures below show what happens if the programs operate by applying elementary cellular automaton rules t times to 2t + 1 inputs.
Matrix memories
Many times since the 1950s it has been noted that methods from linear algebra suggest ways to construct associative memories in which data can potentially be retrieved on the basis of some form of similarity.
In recent times one reason for studying animals that live in cone shells is that they produce potent neurotoxins that show promise as pain-control drugs.
The pictures show the evolution of a 30×30 array, which is nearly 10 times the area of the array shown in the previous picture.
And indeed, as we have seen many times in this book, and as the pictures below illustrate, even rules that are extremely similar can produce quite different behavior.