## Abstract

A regular language L of finite words is composite if there are regular languages L1, L2, . . ., Lt such that L = ^{Tt}_{i}_{=1} Li and the index (number of states in a minimal DFA) of every language Li is strictly smaller than the index of L. Otherwise, L is prime. Primality of regular languages was introduced and studied in [9], where the complexity of deciding the primality of the language of a given DFA was left open, with a doubly-exponential gap between the upper and lower bounds. We study primality for unary regular languages, namely regular languages with a singleton alphabet. A unary language corresponds to a subset of N, making the study of unary prime languages closer to that of primality in number theory. We show that the setting of languages is richer. In particular, while every composite number is the product of two smaller numbers, the number t of languages necessary to decompose a composite unary language induces a strict hierarchy. In addition, a primality witness for a unary language L, namely a word that is not in L but is in all products of languages that contain L and have an index smaller than L's, may be of exponential length. Still, we are able to characterize compositionality by structural properties of a DFA for L, leading to a LogSpace algorithm for primality checking of unary DFAs.

Original language | American English |
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Title of host publication | 45th International Symposium on Mathematical Foundations of Computer Science, MFCS 2020 |

Editors | Javier Esparza, Daniel Kral�, Daniel Kral� |

Publisher | Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing |

ISBN (Electronic) | 9783959771597 |

DOIs | |

State | Published - 1 Aug 2020 |

Event | 45th International Symposium on Mathematical Foundations of Computer Science, MFCS 2020 - Prague, Czech Republic Duration: 25 Aug 2020 → 26 Aug 2020 |

### Publication series

Name | Leibniz International Proceedings in Informatics, LIPIcs |
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Volume | 170 |

ISSN (Print) | 1868-8969 |

### Conference

Conference | 45th International Symposium on Mathematical Foundations of Computer Science, MFCS 2020 |
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Country/Territory | Czech Republic |

City | Prague |

Period | 25/08/20 → 26/08/20 |

### Bibliographical note

Funding Information:Funding Ismaël Jecker: This project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie Grant Agreement No. 754411. Nicolas Mazzocchi: PhD fellowship FRIA from the F.R.S.-FNRS.

Funding Information:

Ismaël Jecker: This project has received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie Grant Agreement No. 754411. Nicolas Mazzocchi: PhD fellowship FRIA from the F.R.S.-FNRS.

Publisher Copyright:

© Nathalie Bertrand; licensed under Creative Commons License CC-BY 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020).

## Keywords

- Deterministic Finite Automata (DFA)
- Primality
- Regular Languages