News
4 Nov 2024, New Page
Title
General Learning THeory
Type
Survey in Japanese
Journal
RIMS Kokyuroku 2228
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Solovay reducibility via Lipschitz functions and signed-digit representation
News
7 Nov 2023, submitted to a journal
Title
Solovay reducibility via Lipschitz functions and signed-digit representation
Masahiro Kumabe, Kenshi Miyabe, and Toshio Suzuki
Type
Submitted to a journal
Publication
TBA
Abstract
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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preprint (23 May, 2024)
Read the paper on Meiji University Institutional Repository
Rational sequences converging to left-c.e. reals of positive effective Hausdorff dimension
News
Feb 2021, accepted to publication
June 2022, Published
Title
Rational sequences converging to left-c.e. reals of positive effective Hausdorff dimension
Hiroyuki Imai, Masahiro Kumabe, Kenshi Miyabe, Yuki Mizusawa, Toshio Suzuki
Type
Post proceedings of a conference
Publication
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)
Abstract
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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Read the paper on Tokyo Metropolitan University Institutional Repository
Computable prediction
News
July 2019, Proof reading
Title
Computable prediction
Type
Conference paper
Publication
Miyabe K. (2019) Computable Prediction. In: Hammer P., Agrawal P., Goertzel B., Iklé M. (eds) Artificial General Intelligence. AGI 2019. Lecture Notes in Computer Science, vol 11654. Springer, Cham
DOI https://doi.org/10.1007/978-3-030-27005-6_14
Abstract
We try to predict the next bit from a given finite binary string
when the sequence is sampled from a computable probability measure on the Cantor space.
There exists the best betting strategy among a class of effective ones up to a multiplicative constant,
the induced prediction from which is called algorithmic probability or universal induction by Solomonoff.
The prediction converges to the true induced measure for sufficiently random sequences.
However, the prediction is not computable.
We propose a framework to study the properties of computable predictions.
We prove that all sufficiently general computable predictions also converge to the true induced measure.
The class of sequences along which the prediction converges is related to computable randomness.
We also discuss the speed of the convergence.
We prove that, even when a computable prediction predicts a computable sequence,
the speed of the convergence cannot be bounded by a computable function monotonically decreasing to $0$.
Uniform relativization
News
19 June 2019, Online
Title
Uniform relativization
Type
Conference survey paper
Publication
In: Manea F., Martin B., Paulusma D., Primiero G. (eds) Computing with Foresight and Industry. CiE 2019. Lecture Notes in Computer Science, vol 11558. Springer, Cham
Abstract
This paper is a tutorial on uniform relativization. The usual relativization considers computation using an oracle, and the computation may not work for other oracles, which is similar to Turing reduction. The uniform relativization also considers computation using oracles, however, the computation should work for all oracles, which is similar to truth-table reduction. The distinction between these relativizations is important when we relativize randomness notions in algorithmic randomness, especially Schnorr randomness. For Martin-Löf randomness, its usual relativization and uniform relativization are the same so we do not need to care about this uniform relativization.
We focus on two specific examples of uniform relativization: van Lambalgen’s theorem and lowness. Van Lambalgen’s theorem holds for Schnorr randomness with the uniform relativization, but not with the usual relativization. Schnorr triviality is equivalent to lowness for Schnorr randomness with the uniform relativization, but not with the usual relativization. We also discuss some related known results.