But with eight instructions, 126 out of the 11,019,960,576 possible programs finally do show more complicated behavior.

In particular, after many years of using Mathematica, I have now got to the point where I can effectively think directly in Mathematica, so that I can start entering a Mathematica program even though I may be a long way from being able to explain in English what I want to do.

But even from this I do not know, for example, how to prove that the density of universal rules cannot decrease when rules become more complicated.

But in all other cases, the sequence does not repeat, and in fact it is known that a uniform distribution of values is obtained.

Neighbor-dependent [2D] substitution systems Given a list of individual replacement rules such as {{_, 1}, {0, 1}}  {{1, 0}, {1, 1}} , each step in the evolution shown corresponds to Flatten2D[Partition[list, {2, 2}, 1, -1] /. rule] One can consider rules in which some replacements lead to subdivision of elements but others do not.

As we will see on page 225 some cellular automata evolve to invariant configurations from any initial conditions, but most do not.

Indeed, questions like "does there exist a way to transmit information faster than light?"

In fact, as mentioned on page 1155 , if one does not allow lemmas some proofs perhaps have to become exponentially longer. … The longest of these are respectively {57, 94, 42, 57, 55, 53, 179, 157} and occur for theorems {(((a ⊼ a) ⊼ b) ⊼ b)  (((a ⊼ b) ⊼ a) ⊼ a), (a ⊼ (a ⊼ (a ⊼ a)))  (a ⊼ ((a ⊼ b) ⊼ b)), (((a ⊼ a) ⊼ a) ⊼ a)  (((a ⊼ a) ⊼ b) ⊼ a), (((a ⊼ a) ⊼ b) ⊼ b)  (((a ⊼ b) ⊼ a) ⊼ a), (a ⊼ ((b ⊼ b) ⊼ a))  (b ⊼ ((a ⊼ a) ⊼ b)), ((a ⊼ a) ⊼ a)  ((b ⊼ b) ⊼ b), ((a ⊼ a) ⊼ a)  ((b ⊼ b) ⊼ b), ((a ⊼ a) ⊼ a)  ((b ⊼ b) ⊼ b)} Note that for systems that do not already have it as an axiom, most theorems use the lemma (a ⊼ b)  (b ⊼ a) which takes respectively {6, 1, 8, 49, 8, 1, 119, 118} steps to prove.

Network constraint systems Cases (a), (f) and (p) allow all networks that do not contain respectively cycles of length 1 (self-loops), cycles of length 3 or less, and cycles of length 5 or less. … If one does allow dangling connections to be joined within a single template, the results are similar to those discussed so far.

Studying simple systems Over the years, I have watched with disappointment the continuing failure of most scientists and mathematicians to grasp the idea of doing computer experiments on the simplest possible systems.