Solving permutation problems using rewriting systems
A new approach is described for finding short expressions for arbitrary elements of a permutation group in terms of the original generators which uses rewriting methods. This forms an important component in a long term plan to find short solutions for “large” permutation problems (such as Rubik's cube), which are difficult to solve by existing search techniques. In order for this methodology to be successful, it is important to start with a short presentation for a finite permutation group. A new method is described for giving a presentation for an arbitrary permutation group acting on n letters. This can be used to show that any such permutation group has a presentation with at most n – 1 generators and (n – 1)2 relations. As an application of this method, an O(n4) algorithm is described for determining if a set of generators for a permutation group on n letters is a strong generating set (in the sense of Sims). The “back end” includes a novel implementation of the Knuth-Bendix technique on symmetrization classes for groups.
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