PSPACE-complete problems for subgroups of free groups and inverse finite automata

We investigate the complexity of algorithmic problems on finitely generated subgroups of free groups. Margolis and Meakin showed how a finite monoid Synt(H) can be canonically and effectively associated with such a subgroup H. We show that H is pure (that is, closed under radical) if and only if Synt(H) is aperiodic. We also show that testing for this property of H is PSPACE-complete. In the process, we show that certain problems about finite automata which are PSPACE-complete in general remain PSPACE-complete when restricted to injective and inverse automata (with single accept state), whereas they are known to be in NC for permutation automata (with single accept state).

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