The Word Problem for Finitary Automaton Groups

A finitary automaton group is a generated by an invertible, deterministic finite-state letter-to-letter transducer whose only cycles are self-loops at identity state. We show that, for this presentation of finite groups, the uniform word problem coNP-complete. Here, input consists together with state sequence and question whether acts trivially on all words. Additionally, we also that respective compressed problem, where given as straight-line program, PSpace-complete. In both cases, give direct reduction from satisfiablity (quantified) boolean formulae.