### Abstract

We define the degree of aperiodicity of finite automata and show that for every set M of positive integers, the class QA_{M} of finite automata whose degree of aperiodicity belongs to the division ideal generated by M is closed with respect to direct products, disjoint unions, subautomata, homomorphic images and renamings. These closure conditions define q-varieties of finite automata. We show that q-varieties are in a one-to-one correspondence with literal varieties of regular languages. We also characterize QA_{M} as the cascade product of a variety of counters with the variety of aperiodic (or counter-free) automata. We then use the notion of degree of aperiodicity to characterize the expressive power of first-order logic and temporal logic with cyclic counting with respect to any given set M of moduli. It follows that when M is finite, then it is decidable whether a regular language is definable in first-order or temporal logic with cyclic counting with respect to moduli in M.

Original language | English |
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Pages (from-to) | 1-28 |

Number of pages | 28 |

Journal | Acta Cybernetica |

Volume | 16 |

Issue number | 1 |

Publication status | Published - Jan 1 2003 |

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### ASJC Scopus subject areas

- Software
- Computer Science (miscellaneous)
- Computer Vision and Pattern Recognition
- Management Science and Operations Research
- Information Systems and Management
- Electrical and Electronic Engineering

### Cite this

*Acta Cybernetica*,

*16*(1), 1-28.