Mod[Binomial[t, n], k] is given for prime k by With[{d = Ceiling[Log[k, Max[t, n] + 1]]}, Mod[Apply[Times, Apply[Binomial, Transpose[ {IntegerDigits[t, k, d] , IntegerDigits[n, k, d] }], {1}]], k]] The patterns obtained for any k are nested.

With this setup, the evolution of any register machine can be implemented using the functions (a typical initial condition is {1, {0, 0}} ) RMStep[prog_, {n_Integer, list_List}] := If[n > Length[prog], {n, list}, RMExecute[prog 〚 n 〛 , {n, list}]] RMExecute[i[r_], {n_, list_}] := {n + 1, MapAt[(# + 1)&, list, r]} RMExecute[d[r_, m_], {n_, list_}] := If[list 〚 r 〛 > 0, {m, MapAt[(# - 1)&, list, r]}, {n + 1, list}] RMEvolveList[prog_, init:{_Integer, _List}, t_Integer] := NestList[RMStep[prog, #]&, init, t] The total number of possible programs of length n using k registers is (k (1 + n)) n .

The rules for the multiway system can then be given for example as {"AAB"  "BB", "BA"  "ABB"} The evolution of the system is given by the functions MWStep[rule_List, slist_List] := Union[Flatten[ Map[Function[s, Map[MWStep1[#, s] &, rule]], slist]]] MWStep1[p_String  q_String, s_String] := Map[StringReplacePart[s, q, #] &, StringPosition[s, p]] MWEvolveList[rule_, init_List, t_Integer] := NestList[MWStep[rule, #] &, init, t] An alternative approach uses lists instead of strings, and in effect works by tracing the internal steps that Mathematica goes through in trying out possible matchings.

And this immediately implies that the pattern must always have a nested form.

All generalized additive rules ultimately yield nested patterns.

means that the network of connections between stems necessarily has a very simple nested form.

what happens if one divides an image into a collection of nested squares, but imposes a lower limit on the size of these squares.

As I will discuss in the next chapter , my general expectation is that more or less any system whose behavior is not somehow fundamentally repetitive or nested will in the end turn out to be universal.

We know from Chapter 11 that systems whose behavior is purely repetitive or purely nested cannot be universal.

Nevertheless, if one looks carefully at examples (a) through (h) each of them shows large regions of either repetitive or nested behavior.