Solutions of Twisted Word Equations, EDT0L Languages, and Context-Free Groups
Volker Diekert, Murray J. Elder · arXiv (Cornell University) · 2017
It is shown that the subgroup membership problem for a virtually free group can be decided in polynomial time where all group elements are represented by so-called power words, i.e., words of the form p_1^{z_1} p_2^{z_2} ⋯ p_k^{z_k}. Here the p_i are explicit words over the generating set of the group and all z_i are binary encoded integers. As a corollary, it follows that the subgroup membership problem for the matrix group GL(2,ℤ) can be decided in polynomial time when all matrix entries are given in binary notation.