### Abstract

We study finite-state transducers and their power for transforming infinite words. Infinite sequences of symbols are of paramount importance in a wide range of fields, from formal languages to pure mathematics and physics. While finite automata for recognising and transforming languages are well-understood, very little is known about the power of automata to transform infinite words. The word transformation realised by finite-state transducers gives rise to a complexity comparison of words and thereby induces equivalence classes, called (transducer) degrees, and a partial order on these degrees. The ensuing hierarchy of degrees is analogous to the recursion-theoretic degrees of unsolvability, also known as Turing degrees, where the transformational devices are Turing machines. However, as a complexity measure, Turing machines are too strong: they trivialise the classification problem by identifying all computable words. Finite-state transducers give rise to a much more fine-grained, discriminating hierarchy. In contrast to Turing degrees, hardly anything is known about transducer degrees, in spite of their naturality. We use methods from linear algebra and analysis to show that there are infinitely many atoms in the transducer degrees, that is, minimal non-trivial degrees.

Original language | English |
---|---|

Pages (from-to) | 825-843 |

Number of pages | 19 |

Journal | International Journal of Foundations of Computer Science |

Volume | 29 |

Issue number | 5 |

DOIs | |

Publication status | Published - Aug 2018 |

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### Keywords

- degrees of infinite words
- finite-state transducers
- Infinite words

### Cite this

*International Journal of Foundations of Computer Science*,

*29*(5), 825-843. https://doi.org/10.1142/S0129054118420066

}

*International Journal of Foundations of Computer Science*, vol. 29, no. 5, pp. 825-843. https://doi.org/10.1142/S0129054118420066

**Degrees of Infinite Words, Polynomials and Atoms.** / Endrullis, Jörg; Karhumäki, Juhani; Klop, Jan Willem; Saarela, Aleksi.

Research output: Contribution to Journal › Article › Academic › peer-review

TY - JOUR

T1 - Degrees of Infinite Words, Polynomials and Atoms

AU - Endrullis, Jörg

AU - Karhumäki, Juhani

AU - Klop, Jan Willem

AU - Saarela, Aleksi

PY - 2018/8

Y1 - 2018/8

N2 - We study finite-state transducers and their power for transforming infinite words. Infinite sequences of symbols are of paramount importance in a wide range of fields, from formal languages to pure mathematics and physics. While finite automata for recognising and transforming languages are well-understood, very little is known about the power of automata to transform infinite words. The word transformation realised by finite-state transducers gives rise to a complexity comparison of words and thereby induces equivalence classes, called (transducer) degrees, and a partial order on these degrees. The ensuing hierarchy of degrees is analogous to the recursion-theoretic degrees of unsolvability, also known as Turing degrees, where the transformational devices are Turing machines. However, as a complexity measure, Turing machines are too strong: they trivialise the classification problem by identifying all computable words. Finite-state transducers give rise to a much more fine-grained, discriminating hierarchy. In contrast to Turing degrees, hardly anything is known about transducer degrees, in spite of their naturality. We use methods from linear algebra and analysis to show that there are infinitely many atoms in the transducer degrees, that is, minimal non-trivial degrees.

AB - We study finite-state transducers and their power for transforming infinite words. Infinite sequences of symbols are of paramount importance in a wide range of fields, from formal languages to pure mathematics and physics. While finite automata for recognising and transforming languages are well-understood, very little is known about the power of automata to transform infinite words. The word transformation realised by finite-state transducers gives rise to a complexity comparison of words and thereby induces equivalence classes, called (transducer) degrees, and a partial order on these degrees. The ensuing hierarchy of degrees is analogous to the recursion-theoretic degrees of unsolvability, also known as Turing degrees, where the transformational devices are Turing machines. However, as a complexity measure, Turing machines are too strong: they trivialise the classification problem by identifying all computable words. Finite-state transducers give rise to a much more fine-grained, discriminating hierarchy. In contrast to Turing degrees, hardly anything is known about transducer degrees, in spite of their naturality. We use methods from linear algebra and analysis to show that there are infinitely many atoms in the transducer degrees, that is, minimal non-trivial degrees.

KW - degrees of infinite words

KW - finite-state transducers

KW - Infinite words

UR - http://www.scopus.com/inward/record.url?scp=85051933899&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85051933899&partnerID=8YFLogxK

U2 - 10.1142/S0129054118420066

DO - 10.1142/S0129054118420066

M3 - Article

VL - 29

SP - 825

EP - 843

JO - International Journal of Foundations of Computer Science

JF - International Journal of Foundations of Computer Science

SN - 0129-0541

IS - 5

ER -