Among principles in science the Principle of Computational Equivalence is almost unprecedentedly broad—for it applies to essentially any process of any kind, either natural or artificial.

So what happens if one applies the same criterion in other settings?

Non-deterministic Turing machines Generalizing rules from page 888 by making each right-hand side a list of possible outcomes, the list of configurations that can be reached after t steps is given by NTMEvolve[rule_, inits_, t_Integer] := Nest[ Union[Flatten[Map[NTMStep[rule, #]&, #], 1]]&, inits, t] NTMStep[rule_List, {s_, a_, n_}] /; 1 ≤ n ≤ Length[a] := Apply[{#1, ReplacePart[a, #2, n], n + #3}&, Replace[{s, a 〚 n 〛 }, rule], {1}]

In the first 200 billion digits, the frequencies of 0 through 9 differ from 20 billion by {30841, -85289, 136978, 69393, -78309, -82947, -118485, -32406, 291044, -130820} An early approximation to π was 4 Sum[(-1) k /(2k + 1), {k, 0, m}] 30 digits were obtained with 2 Apply[Times, 2/Rest[NestList[Sqrt[2 + #]&, 0, m]]] An efficient way to compute π to n digits of precision is (# 〚 2 〛 2 /# 〚 3 〛 )& [NestWhile[Apply[Function[{a, b, c, d}, {(a + b)/2, Sqrt[a b], c - d (a - b) 2 , 2 d}], #]&, {1, 1/Sqrt[N[2, n]], 1/4, 1/4}, # 〚 2 〛 ≠ # 〚 2 〛 &]] This requires about Log[2, n] steps, or a total of roughly n Log[n] 2 operations (see page 1134 ).

The total number of commutative groups with k elements is just Apply[Times, Map[PartitionsP[Last[#]] &, FactorInteger[k]]] (Relabelling of elements makes the number of possible operator forms up to k!

In each block the second entry is the rule obtained by interchanging black and white, the third entry is the rule obtained by interchanging left and right, and the fourth entry the rule obtained by applying both operations.

Mathematical Functions The last section showed that individual numbers obtained by applying various simple mathematical functions can have features that are quite complex.

And does it really apply to all of the various kinds of systems that we see in nature?

So if two expressions are equivalent then by applying the rules of the appropriate axiom system it must be possible to get from one to the other—and in fact the picture on page 775 shows an example of how Values of expressions obtained by using operators of various forms.

[Generating] random networks One way to generate the connections for a "completely random" trivalent network with n nodes is just to apply a random permutation: RandomNetwork[n_?