## Abstract

A regular language R of finite words induces three repetition languages of infinite words: the language lim(R), which contains words with infinitely many prefixes in R, the language ∞R, which contains words with infinitely many disjoint subwords in R, and the language R^{ω}, which contains infinite concatenations of words in R. Specifying behaviors, the three repetition languages provide three different ways of turning a specification of a finite behavior into an infinite one. We study the expressive power required for recognizing repetition languages, in particular whether they can always be recognized by a deterministic Büchi word automaton (DBW), the blow up in going from an automaton for R to automata for the repetition languages, and the complexity of related decision problems. For lim R and ∞R, most of these problems have already been studied or are easy. We focus on R^{ω}. Its study involves some new and interesting results about additional repetition languages, in particular R^{#}, which contains exactly all words with unboundedly many concatenations of words in R. We show that R^{ω} is DBW-recognizable iff R^{#} is ω-regular iff R^{#} = R^{ω}, and there are languages for which these criteria do not hold. Thus, R^{ω} need not be DBW-recognizable. In addition, when exists, the construction of a DBW for R^{ω} may involve a 2^{O}(n log n^{)} blow-up, and deciding whether R^{ω} is DBW-recognizable, for R given by a nondeterministic automaton, is PSPACE-complete. Finally, we lift the difference between R^{#} and R^{ω} to automata on finite words and study a variant of Büchi automata where a word is accepted if (possibly different) runs on it visit accepting states unboundedly many times.

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

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

## Keywords

- Büchi automata
- Expressive power
- Succinctness