## Abstract

We construct group codes over two letters (i.e., bases of subgroups of a two-generated free group) with special properties. Such group codes can be used for reducing algorithmic problems over large alphabets to algorithmic problems over a two-letter alphabet. Our group codes preserve aperiodicity of inverse finite automata. As an application we show that the following problems are PSpace-complete for two-letter alphabets (this was previously known for large enough finite alphabets): The intersection-emptiness problem for inverse finite automata, the aperiodicity problem for inverse finite automata, and the closure-under-radical problem for finitely generated subgroups of a free group. The membership problem for 3-generated inverse monoids is PSpace-complete.

Original language | English |
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Pages (from-to) | 159-168 |

Number of pages | 10 |

Journal | Semigroup Forum |

Volume | 76 |

Issue number | 1 |

DOIs | |

State | Published - Jan 2008 |

### Bibliographical note

Funding Information:Both authors were supported in part by NSF grant DMS-9970471. The first author was also supported in part by NSF grant CCR-0310793. The second author acknowledges the support of the Excellency Center, “Group Theoretic Methods for the Study of Algebraic Varieties” of the Israeli Science Foundation.

## Keywords

- Free groups
- Inverse automata
- Inverse semigroups