## Abstract

We investigate the solution and the complexity of algorithmic problems on finitely generated subgroups of free groups. Margolis and Meakin showed how a finite inverse monoid Synt(H) can be canonically and effectively associated to such a subgroup H. We show that H is pure (closed under radical) if and only if Synt(H) is aperiodic. We also show that testing for this property for 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 — whereas they are known to be in NC for permutation automata.

Original language | English |
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Title of host publication | Automata, Languages and Programming - 21st International Colloquium, ICALP 1994, Proceedings |

Editors | Serge Abiteboul, Eli Shamir |

Publisher | Springer Verlag |

Pages | 274-285 |

Number of pages | 12 |

ISBN (Print) | 9783540582014 |

DOIs | |

State | Published - 1994 |

Externally published | Yes |

Event | Proceedings of the 1994 21st International Colloquium on Automata, Languages and Programming, ICALP'94 - Jerusalem, Isr Duration: 1 Jul 1994 → 1 Jul 1994 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 820 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Conference

Conference | Proceedings of the 1994 21st International Colloquium on Automata, Languages and Programming, ICALP'94 |
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City | Jerusalem, Isr |

Period | 1/07/94 → 1/07/94 |

### Bibliographical note

Funding Information:Margolis and Meakin \[8\]f irst introduced the automata-theoretic point of view on this correspondance by showing that the finite Z-labeled graph -~H associated with * The first three authors were supported by NSF grant DMS 9203"81. The last author was partly supported by PRC Math~matiques et Informatique and by ESPRIT-BRA WG 6317 ASMICS-2. F_,-mai\]:( birget,mvxgolis)@cse.unl.edu, meakinQhoss.unl.edu,

Funding Information:

The first three authors were supported by NSF grant DMS 920381. The last author was partly supported by PRC Math?matiques et Informatique and by ESPRIT-BRAWG 6317 ASMICS-2.

Publisher Copyright:

© 1994, Springer Verlag. All rights reserved.