Chapter 12: The Principle of Computational Equivalence

Section 9: Implications for Mathematics and Its Foundations

Properties [of example multiway systems]

The second rule shown has the property that black elements always appear before white, so that strings can be specified just by the number of elements of each color that they contain--making the rule one of the sorted type discussed on page 937, based on the difference vector {{2,-1}, {-1,3}, {-4,-1}}. The question of whether a given string can be generated is then analogous to finding whether there is a solution with certain positivity properties to a set of linear Diophantine equations.

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