Generating causal networks If every element generated in the evolution of a generalized substitution system is assigned a unique number, then events can be represented for example by {4, 5}  {11, 12, 13} —and from a list of such events a causal network can be built up using With[{u = Map[First, list]}, MapIndexed[Function[ {e, i}, First[i]  Map[(If[# === {}, ∞ , # 〚 1, 1 〛 ] &)[ Position[u, #]]) &, Last[e]]], list]]

Each element in each string in a multiway system corresponds to a connection in a causal network.

If the rules for a one-element-dependence tag system are given in the form {2, {{0, 1}, {0, 1, 1}}} (compare page 1114 ), the initial conditions for the Turing machine are TagToMTM[{2, rule_}, init_] := With[{b = FoldList[Plus, 1, Map[Length, rule] + 1]}, Drop[Flatten[{Reverse[Flatten[{1, Map[{Map[ {1, 0, Table[0, {b 〚 # + 1 〛 }]} &, #], 1} &, rule], 1}]], 0, 0, Map[{Table[2, {b 〚 # + 1 〛 }], 3} &, init]}], -1]] surrounded by 0 's, with the head on the leftmost 2 , in state 1 . An element -1 in the tag system corresponds to halting of the Turing machine.

By inserting k = 6 Ceiling[Length[subs]/6] in the definition of TS1ToCT from page 1113 one can construct a cyclic tag system of this kind to emulate any one-element-dependence tag system.

Mathematical properties [of branching model] If an element c of the list b is real, so that there is a stem that goes straight up, then the limiting height of the center of the pattern is obtained by summing a geometric series, and is given by 1/(1 - c) .

Substitution Systems and Fractals One-dimensional substitution systems of the kind we discussed on page 82 can be thought of as working by progressively subdividing each element they contain into several smaller elements.

Earlier in this section we discussed cases in which negation simply reverses the color of each element in a string.

An element m in a message is encoded as c = PowerMod[m, d, n] .

The first stage, illustrated in the top picture on the next page , is to get a cyclic tag system to emulate an ordinary tag system with the property that its rules depend only on the very first element that appears at each step.

Generators and relations [and axiom systems] In the axiom systems of page 773 , a single variable can stand for any element—much like a Mathematica pattern object such as x_ .