### 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 - 2003 |

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### ASJC Scopus subject areas

- Hardware and Architecture
- Software
- Computational Theory and Mathematics
- Theoretical Computer Science

### Cite this

*Acta Cybernetica*,

*16*(1), 1-28.

**Temporal logic with cyclic counting and the degree of aperiodicity of finite automata.** / Ésik, Z.; Ito, M.

Research output: Contribution to journal › Article

*Acta Cybernetica*, vol. 16, no. 1, pp. 1-28.

}

TY - JOUR

T1 - Temporal logic with cyclic counting and the degree of aperiodicity of finite automata

AU - Ésik, Z.

AU - Ito, M.

PY - 2003

Y1 - 2003

N2 - We define the degree of aperiodicity of finite automata and show that for every set M of positive integers, the class QAM 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 QAM 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.

AB - We define the degree of aperiodicity of finite automata and show that for every set M of positive integers, the class QAM 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 QAM 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.

UR - http://www.scopus.com/inward/record.url?scp=18744436808&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=18744436808&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:18744436808

VL - 16

SP - 1

EP - 28

JO - Acta Cybernetica

JF - Acta Cybernetica

SN - 0324-721X

IS - 1

ER -