Yet with undecidability one believes that there is absolutely no construct that can explicitly exist in our universe that allows the problem to be solved in any finite way.

Extended instruction sets [for register machines] One can consider also including instructions such as RMExecute[eq[r1_, r2_, m_], {n_, list_}] := If[list 〚 r1 〛  list 〚 r2 〛 , {m, list}, {n + 1, list}] RMExecute[add[r1_, r2_], {n_, list_}] := {n + 1, ReplacePart[list, list 〚 r1 〛 + list 〚 r2 〛 , r1]} RMExecute[jmp[r1_], {n_, list_}] := {list 〚 r1 〛 , list} Note that by being able to add and subtract only 1 at each step, the register machines shown in the main text necessarily operate quite slowly: they always take at least n steps to build up a number of size n .

With suitable distinct integers a[n] one can represent any number by Sum[1/a[n], {n, ∞ }] .

Most such groups come in families that are easy to characterize; a handful of so-called sporadic ones are much more difficult to find.

Fitting the result to 2 h n 2 one finds h ≃ 0.589 , but no exact formula for h has ever been found.

One possibility is to allow dependence on next-nearest as well as nearest neighbors.

One issue however is that in the simplest cellular automaton fluids molecules are in effect counted in unary: each molecule is traced separately, rather than just being included as part of a total number that can be manipulated using standard arithmetic operations.

In the phyllotaxis process discussed in the main text one new element is produced at a time.

Implementation [of generalized substitution systems] Sequential substitution systems in which only one replacement is ever done at each step can just be implemented using /. as described on page 893 .

And although one might not expect it on the basis of traditional mathematical intuition, there is an analog of this even for topological equivalence of ordinary continuous manifolds.