Chapter 9: Fundamental Physics

Section 11: Uniqueness and Branching in Time

Conditions for convergence [in string rewriting]

One way to guarantee that there is convergence after one step is to require as in the previous section that blocks to be replaced cannot overlap with themselves or each other. And of the 196 possible rules involving two colors and blocks of length at most three, 112 have this property. But there are also an additional 20 rules which allow some overlap but which nevertheless yield convergence after one step. Examples are "AAA"->"A" and "AA"->"ABA". In these rules some of the elements essentially just supply context, but are not affected by the replacement. These elements can then overlap while not affecting the result. Note that unless one excludes the context elements from events, paths in the multiway system will converge, but the causal networks on these paths will be locally slightly different.

Much as in the previous section, even if paths do not converge for every possible string, it can still be true that paths converge for all strings that are actually generated from a particular initial string.

In general, one can consider convergence after any number of steps, requiring that any two strings which have a common ancestor must at some point also have a common successor. Note that a rule such as {"A"->"B", "A"->"C", "B"->"A", "B"->"D"} exhibits convergence for all paths that have diverged for only one step, but not for all those that have diverged for longer. In general it is formally undecidable whether a particular multiway system will eventually exhibit convergence of all paths.

From Stephen Wolfram: A New Kind of Science [citation]