Publication – Kenshi Miyabe

要旨
We explore Solovay reducibility in the context of computably approximable reals, extending its natural characterization for left-c.e. reals via computable Lipschitz functions. Our paper offers two distinct characterizations: the first employs Lipschitz functions, while the second utilizes Turing reductions with bounded use with respect to signed-digit representation. Additionally, we examine multiple related reducibilities and establish separations among them. These results contribute to a refined perspective of the relationship between Solovay reducibility and computable Lipschitz functions.

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6月 19, 2022October 21, 2024

Rational sequences converging to left-c.e. reals of positive effective Hausdorff dimension

履歴
2021年2月　アクセプト
2022年6月出版

タイトル
Rational sequences converging to left-c.e. reals of positive effective Hausdorff dimension

Hiroyuki Imai, Masahiro Kumabe, Kenshi Miyabe, Yuki Mizusawa, Toshio Suzuki

種類
査読ありの事後会議録

国際会議と雑誌
Computability Theory and Foundations of Mathematics
Proceedings of the 9th International Conference on Computability Theory and Foundations of Mathematics
The 9th International Conference on Computability Theory and Foundations of Mathematics, Wuhan, China, 21 – 27 March 2019
https://doi.org/10.1142/12917 | June 2022
Pages: 196
Edited By: NingNing Peng (Wuhan University of Technology, China), Kazuyuki Tanaka (Tohoku University, Japan), Yue Yang (National University of Singapore, Singapore), Guohua Wu (Nanyang Technological University, Singapore) and Liang Yu (Nanjing University, China)

要約
In our previous work, we characterized Solovay reducibility using Lipschitz condition,
and introduced quasi Solovay reducibility (qS-reducibility, for short) as a H ̈older condition counterpart.
In this paper, we investigate effective dimensions and ideals closely related to quasi Solovay reducibility by means of the rate of convergence.
We show that the qS-completeness among left-c.e. reals is equivalent to having a positive effective Hausdorff dimension.
The Solovay degrees of qS-complete left-c.e. reals form a filter. On the other hand, the Solovay degrees of non-qS-complete left-c.e. reals do not form an ideal.
Based on observations on the relationships between rational sequences and reducibility, we introduce a stronger version of qS-reducibility.
Given a degree of this reducibility, the lower cone (including the given degree) forms an ideal.
By developing these investigations, we characterize the effective dimensions by means of the rate of convergence.
We give a variation of the first incompleteness theorem based on Solovay reducibility.

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