Implementation [of multiway systems] It is convenient to represent the state of a multiway system at each step by a list of strings, where an individual string is for example "ABBAAB" . 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. … Note that the rules are set up so that a string for which there are no applicable replacements at a given step is simply dropped.

Substitution systems in which all replacements are done that are found to fit in a left-to-right scan can be implemented as follows GSSEvolveList[rule_, s_, n_] := NestList[GSSStep[rule, #] &, s, n] GSSStep[rule_, s_] := g[rule, s, f[StringPosition[s, Map[First, rule]]]] f[{ }] = { }; f[s_] := Fold[If[Last[Last[#1]] ≥ First[#2], #1, Append[#1, #2]]&, {First[s]}, Rest[s]] g[rule_, s_, { }] := s; g[rule_, s_, pos_] := StringReplacePart[ s, Map[StringTake[s, #] &, pos] /. rule, pos] with rules given as {"ABA"  "BAAB", "BBBB"  "AA"} .

Instead of looking separately at strings produced at each step, one instead maintains just a single copy of each distinct string ever produced, and makes that correspond to a node in the network. … If each node in a network is associated with a point in spacetime, then one slightly peculiar feature is that every such point would have an associated string—something like an encoded position coordinate.

And it is common in spacetime physics to draw "light cones" of the kind shown below to indicate the region that will be reached by a light signal emitted from a particular position in space at a particular time. … For the different elements in the system are always just laid out in a one-dimensional string, with the result that local replacement rules can only ever propagate effects to nearby elements in the string—much like in a one-dimensional cellular automaton.

But the crucial point is that if the complete state of our universe is in effect like a single string in a multiway system, then there is no way for us ever to look at the multiway system from the outside. … Up until a few centuries ago, it was widely believed that the Earth had some kind of fundamentally unique position in space.

And by using rules such as s[x___, 1, 0, y___]  {s[x, 0, 1, 0, y], Length[s[x]]} one can keep track of the positions at which substitutions are made. ( StringReplace replaces all occurrences of a given substring, not just the first one, so cannot be used directly as an alternative to having a flat function.)

With k colors each giving a string of the same length s the recurrence relation is Thread[Map[ ϕ [#][t + 1, ω ] &, Range[k] - 1]  Apply[Plus, MapIndexed[Exp[  ω (Last[#2] - 1) s t ] ϕ [#1][t, ω ] &, Range[k] - 1 /. rules, {-1}], {1}]/ √ s ] Some specific properties of the examples shown include: (a) (Thue–Morse sequence) The spectrum is essentially Nest[Range[2 Length[#]] Join[#, Reverse[#]] &, {1}, t] . The main peak is at position 1/3, and in the power spectrum this peak contains half of the total.