Difference patterns [in cellular automata] The maximum rate at which a region of change can grow is determined by the range of the underlying cellular automaton rule.

And these amplitudes a i are assumed to be complex numbers with a continuous range of possible values, subject only to the conventional constraint of unit total probability Sum[Abs[a i ] 2 , {i, 2 n }]  1 . … But these correspond to periodicities in the list Mod[a^Range[m], m] . Given n spins one can imagine using their 2 n possible configurations to represent each element of Range[m] .

Presumably it is best to add axioms that allow the widest range of new statements to be proved.

But if one uses instead s = {1, 2} then starts with {1} and {2} one gets any of {{}, {1}, {2}, {1, 2}} and in general with s = Range[n] one gets any of the 2 n elements in the powerset Distribute[Map[{{}, {#}} &, s], List, List, List, Join] But applying Complement[s, Intersection[a, b]] to these elements still always produces the same equivalences as with a ⊼ b .

The first one on the bottom (with 63 comparisons) has a nested structure and uses the method invented by Kenneth Batcher in 1964: Flatten[Reverse[Flatten[With[{m = Ceiling[Log[2, n]] - 1}, Table[With[{d = If[i  m, 2 t , 2 i + 1 - 2 t ]}, Map[ {0, d} + # &, Select[Range[n - d], BitAnd[# - 1, 2 t ]  If[i  m, 0, 2 t ] &]]], {t, 0, m}, {i, t, m}]], 1]], 1] The second one on the bottom also uses 63 comparisons, while the last one is the smallest known for n = 16 : it uses 60 comparisons and was invented by Milton Green in 1969.

The invention of fluxions by Isaac Newton in the late 1600s, however, introduced the idea of continuous variables—numbers with a continuous range of possible sizes.

And with the constraint of reversibility, it turns out that it is impossible to get a non-trivial phase transition in any 1D system with the kind of short-range interactions that exist in a cellular automaton.

This result has turned out to be in respectable agreement with a range of experimental data, but its physical significance has remained somewhat unclear.

My classical English education—in elementary school (Dragon School) and high school (Eton)—emphasized such pursuits as writing, and exposed me to a certain range of subjects, a remarkable fraction of which have ended up being useful, especially in the historical research for this book.

The following generates explicit lists of n -input Boolean functions requiring successively larger numbers of Nand operations: Map[FromDigits[#, 2] &, NestWhile[Append[#, Complement[Flatten[Table[Outer[1 - Times[##] &, # 〚 i 〛 , # 〚 -i 〛 , 1], {i, Length[#]}], 2], Flatten[#, 1]]] &, {1 - Transpose[IntegerDigits[Range[2 n ] - 1, 2, n]]}, Length[Flatten[#, 1]] < 2 2 n &], {2}] The results for 2-step cellular automaton evolution in the main text were found by a recursive procedure.