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As I discuss on page 1026 I suspect that in fact the universe as a whole probably had what were ultimately very simple initial conditions, and it is just that the effective rules for the evolution of matter led to rapid randomization, whereas those for gravity did not.
The same basic approach can be used to deduce the rule for an additive cellular automaton from vertical sequences.
With modus ponens as the rule of inference, the shortest single-axiom system that works is known to be {((a ∘ b) ∘ c) ∘ ((c ∘ a) ∘ (d ∘ a))} .
Given a network of the kind discussed in the main text of this section, one can assign a color to each node, and then update this color at each step according to a rule that depends on the colors of the nodes to which the connections from that node go.
As discussed on page 122 , the values can in fact be obtained by a simple arithmetic rule, without explicitly following each step in the evolution of the register machine.
And it is then a general result that any system of limited size that involves discrete elements and follows definite rules must always eventually exhibit repetitive behavior.
Averaging out small-scale randomness yields apparent uniformity, as shown here for a rule 30 pattern.
And all of this supports my strong belief that in the end it will turn out that every detail of our universe does indeed follow rules that can be represented by a very simple program—and that everything we see will ultimately emerge just from running this program.
The rules are the same as shown on pages 83 and 84 .
The Notion of Computation Computation as a Framework
In earlier parts of this book we saw many examples of the kinds of behavior that can be produced by cellular automata and other systems with simple underlying rules.