### 計算可能な予測の収束速度

2020年9月16日　スライドアップロード

タイトル

ドイツ数学会　ミニシンポジウム”The impact of randomness on computation”

ダウンロード
DMV

### 逆ランダム

2019年11月4日　スライドアップロード

タイトル

2019年11月8日　京都大学数理情報論分野談話会

ダウンロード
kyoto

### 計算可能な予測

2019年7月　校正

タイトル
Computable prediction

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 $0$.

ダウンロード
preprint
slide

### 一様相対化

2019年6月19日　オンライン

タイトル
Uniform relativization

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
DOI: https://doi.org/10.1007/978-3-030-22996-2_5

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.

ダウンロード
preprint
slide

### Schnorr triviality via decidable machines

2019年3月29日　スライドアップロード

タイトル
Schnorr triviality via decidable machines

CTFM2019での招待講演
2019年3月25日，武漢理工大学@武漢(中国)

ダウンロード
slide