## Abstract

Each language L ? S induces an infinite sequence {Pr(L, n)}8n=1, where for all n = 1, the value Pr(L, n) ? [0, 1] is the probability of a word of length n to be in L, assuming a uniform distribution on the letters in S. Previous studies of {Pr(L, n)}8n=1 for a regular language L, concerned zero-one laws, density, and accumulation points. We study monotonicity of {Pr(L, n)}8n=1, possibly in the limit. We show that monotonicity may depend on the distribution of letters, study how operations on languages affect monotonicity, and characterize classes of languages for which the sequence is monotonic. We extend the study to languages L of infinite words, where we study the probability of lasso-shaped words to be in L and consider two definitions for Pr(L, n). The first refers to the probability of prefixes of length n to be extended to words in L, and the second to the probability of word w of length n to be such that w? is in L. Thus, in the second definition, monotonicity depends not only on the length of w, but also on the words being periodic.

Original language | American English |
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Title of host publication | 43rd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science, FSTTCS 2023 |

Editors | Patricia Bouyer, Srikanth Srinivasan, Srikanth Srinivasan |

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

Pages | 26:1-26:19 |

Number of pages | 19 |

ISBN (Electronic) | 9783959773041 |

DOIs | |

State | Published - Dec 2023 |

Event | 43rd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science, FSTTCS 2023 - Hyderabad, India Duration: 18 Dec 2023 → 20 Dec 2023 |

### Publication series

Name | Leibniz International Proceedings in Informatics, LIPIcs |
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Volume | 284 |

ISSN (Print) | 1868-8969 |

### Conference

Conference | 43rd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science, FSTTCS 2023 |
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Country/Territory | India |

City | Hyderabad |

Period | 18/12/23 → 20/12/23 |

### Bibliographical note

Publisher Copyright:© 2023 Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing. All rights reserved.

## Keywords

- Automata
- Monotonicity
- Probability
- Regular Languages