Given any finite group $G$ and some set of generators $S$, are there some heuristic algorithm to find presentation of $x \in G$ using products of elements in $S$ and their inverse elements? If we define the word length of a presentation to be the number of elements used in it, how to find the presentation with minimal word length?
It may be difficult to find exact solution, so we should try a heuristic algorithm. Could we first generate lots of formulas that simplifies the presentation, and try to apply it? We may complexize the presentation to avoid local minimum.
The minimal step problem for Rubik's cube is a special type of this problem. Does Korf's algorithm generalizes to arbitrary finite group?