# Notes

## Section 11: Uniqueness and Branching in Time

Confluence [in string rewriting]

As mentioned on page 938, multiway systems have been studied in mathematical logic, typically under names such as rewrite systems, since the early 1900s. The property of path convergence discussed in the main text has been considered since the 1930s, usually under the name of confluence, or sometimes the Church–Rosser property. (Also considered is strong confluence—that paths can always converge in at most one step, and local confluence—that paths can converge after diverging for one step but not necessarily more. Early in its history confluence was most often studied for symbolic systems and lambda calculus rather than ordinary multiway systems.)

Confluence is important in defining a notion of equivalence for strings. One can say that two strings are equivalent if they can both be transformed to the same string by using the rules of the multiway system. And with such a definition, confluence is what is needed to obtain transitivity for equality, so that p==q and q==r implies p==r.

Most often confluence is studied in the context of terminating multiway systems—multiway systems in which eventually strings are produced to which no further replacements apply. If a terminating multiway system has the confluence property, then this implies that regardless of the path taken, a given string will always evolve to a unique string that can be thought of as giving a canonical or normal form for the original string. Examples (a) through (c) below have this property; (d) does not. In example (a), the canonical form is all elements black; in (b) it is a single black element, and in (c) all elements are black, except the last one, which is white if there were any initial white elements. Note that the first example on page 507 has a canonical form consisting of a sorted string.

The process of evaluation in mathematics or in a computer language such as Mathematica can be thought of as involving the application of a sequence of replacement rules. Only if these rules have the confluence property will the results always be unique, and independent of the order of rule application.

The evaluation of functions with attribute Flat in Mathematica provides an example of confluence. If f is Flat, then in evaluating f[a,b,c] one can equally well start with f[f[a,b],c] or f[a,f[b,c]]. Showing only the arguments to f, the pictures below illustrate how the flat functions Xor and And are confluent, while the non-flat function Implies is not.

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