Publications

Books

• An introduction to probability and statistics (in Japanese), with Prof. Watanabe, Morikita, March 2020
• A Japanese translation of “Philosophy and Probability” by Timothy Childers, Kyukasha, 31 Jan 2020.
• New approach to calculus, Volume 1 (in Japanese)， with Prof. Nagaoka, Prof. Watanabe, Prof. Yazaki, Kodansha, 6 Feb 2017.
• New approach to calculus, Volume 2 (in Japanese)， with Prof. Nagaoka, Prof. Watanabe, Prof. Yazaki, Kodansha, 6 Feb 2017.

Outreach

• Algorithmic Randomness and Computable Measure Theory, Suugaku Seminar, Feb 2021.
• Algorithmic Information Theory and AI (in Japanese), pp.59-60, Encyclopedia of Artificial Intelligence, 31 Dec 2019.
• A survey of game-theoretic probability (with A. Takemura) (in Japanese), Suugaku Seminar, Oct 2014.

In preparation/submitted

• Quantifier Variations in Solovay Reducibility (with Kumabe and Suzuki), in preparation
• Real closed fields via strong Solovay reducibility (with Kumabe and Suzuki), submitted to a Journal
• The rate of the convergence of computable predictions, submitted to a Journal
• Solovay reducibility via Lipschitz functions and signed-digit representation (with Kumabe and Suzuki), to appear in Computability.

Papers in algorithmic randomness

• Rational sequences converging to left-c.e. reals of positive effective Hausdorff dimension (with Imai, Kumabe, Mizusawa, and Suzuki), Post-proceedings of CTFM2019, pp. 97–121, June 2022.
• Solovay reducibility and continuity (with Kumabe, Mizusawa, and Suzuki), Journal of Logic and Analysis, vol 12:2, pp. 1-22, Dec. 2020.
• Uniform relativization, Lecture Notes in Computer Science, vol 11558, pp 50-61, June 2019.
• Muchnik degrees and Medvedev degrees of the randomness notions, Proc. of 14th-15th ALC, pp. 108–128, January 2019.
• Coherence of reducibilities with randomness notions, Theory of Computing Systems, Volume 62, Issue 7, pp 1599–1619, 2019.
• Randomness and Solovay degrees (with A. Nies and F. Stephan), Jouanl of Logic and Analysis, Vol 10, pp.1–13, 2018.
• Using Almost-Everywhere Theorems from Analysis to Study Randomness (with A. Nies and J. Zhang), The Bulletin of Symbolic Logic, Volume 22, Issue 3, pp. 305-331, 2016
• Reducibilities relating to Schnorr randomness, Theory of Computing Systems, 58(3), 441-462, 2016.
• Schnorr triviality and its equivalent notions, Theory of Computing Systems: Volume 56, Issue 3 , pp 465-486, 2015
• Unified Characterizations of Lowness Properties via Kolmogorov Complexity (with T. Kihara), Archive for Mathematical Logic: Volume 54, Issue 3, Page 329-358, 2015
• Uniform Kurtz randomness (with T. Kihara), Journal of Logic and Computation, 24 (4): 863-882, 2014
• Van Lambalgen’s Theorem for uniformly relative Schnorr and computable randomness (with J. Rute), Proceedings of the 12th Asian Logic Conference, 251-270, 2013.
• The difference between optimality and universality, Logic Journal of the IGPL, 20(1):222-234, 2012.
• Truth-table Schnorr randomness and truth-table reducible randomness, Mathematical Logic Quarterly 57(3):323-338, 2011
• An Extension of van Lambalgen’s Theorem to Infinitely Many Relative 1-random Reals, Notre Dame Journal of Formal Logic, 51(3):337-349, 2010.

Papers in randomness and analysis

• Algorithmic randomness over general spaces, Mathematical Logic Quarterly, 60(3), 184–204 (2014).
• $L^1$-computability, Layerwise Computability and Solovay Reducibility, Computability, 2:15-29, 2013.
• Characterization of Kurtz Randomness by a Differentiation Theorem, Theory of Computing Systems, 52(1):113-132, 2013.

Papers in algorithmic probability

• Computable prediction, Lecture Notes in Computer Science, vol 11654, 137-147, 2019.
• An optimal superfarthingale and its convergence over a computable topological space, Lecture Notes in Computer Science 7070, 273-284, 2013.

Papers in game-theoretic probability

• Erdos-Feller-Kolmogorov-Petrowsky law of the iterated logarithm for self-normalized martingales: a game-theoretic approach (with T. Sasai and A. Takemura), Annals of Probability, Vol. 47, No. 2, 1136-1161, March 2019.
• Relation between the rate of convergence of strong law of large numbers and the rate of concentration of Bayesian prior in game-theoretic probability (with R. Sato and A. Takemura), Stochastic Processes and their Applications. Volume 128, Issue 5, May 2018, Pages 1466-1484.
• Derandomization in Game-Theoretic Probability (with A. Takemura), Stochastic Processes and their Applications 125, 39-59, 2015.
• The law of the iterated logarithm in game-theoretic probability with quadratic and stronger hedges (with A. Takemura), Stochastic Processes and their Application, 123, 3132-3152, 2013.
• Convergence of random series and the rate of convergence of the strong law of large numbers in game-theoretic probability (with A. Takemura), Stochastic Processes and their Applications, 122:1-30, 2012.

Publication without review

• Computable Measure Theory, RIMS kokyuroku, 25-27 Dec, 2017.
• Lowness for uniform Kurtz randomness (with T. Kihara), in informal proceedings in CiE 2013.
• Computation with advice of degree of randomness, RIMS kokyuroku, No. 1849, 1-7, 2013.
• Analytical approach to algorithmic randomness, RIMS kokyuroku, No. 1832, 114-126, 2013.
• Schnorr and Kurtz randomness versions of Merkle’s criterion, IEICE Technical Report Vol. 112 No. 498, COMP 2012-60, pp55-59
• Generalizing randomness, RIMS kokyuroku, No.1729, 84-94, 2011.

Other writings

• Van Lambalgen’s Theorem for uniform Kurtz randomness., Annual meeting of Mathematical Society of Japan 2013, (21 Mar 2013), Kyoto University
• The other history of probability theory., Annual meeting of Mathematical Society of Japan 2013, (20 Mar 2013), Kyoto University
• Computability of conditional probability, preliminary report on LA symposium 2012, (28 Jan 2013), Kyoto University
• Natural properties that a randomness notion should have, The Mathematical Society of Japan, (18-21 Sep 2012), Kyushu University
• Schnorr triviality and being a basis for tt-Schnorr randomness (Japanese), preliminary report on Proof theory and complexity, (12-14 Sep 2012), RIMS
• On the notion of randomness, Summer school of foundations of mathematics, (4-7 Sep 2012), Tokyo Institute of Technology
• On the effectivization of Lusin’s theorem (Japanese), preliminary report on LA symposium 2012, (17 July, 2012), Kyoto
• The convergence rate of SLLN in the case of non-i.i.d., Annual meeting of Mathematical Society of Japan 2012, (26-29 Mar 2012), Tokyo University of Science
• At which points are functions differentiable?, Annual meeting of Mathematical Society of Japan 2012, (26-29 Mar 2012), Tokyo University of Science

Unpublished

• Computable measure theory and algorithmic randomness (with J. Rute), in preparation.
• Differentiability of Polynomial-time computable functions (with A. Kawamura), in preparation.
• Null-additivity in the theory of algorithmic randomness (with T. Kihara), unpublished.
• Weak L^1-computability and Limit L^1-computability, unpublished.
• An integral test for Schnorr randomness and its application, unpublished.
• A computable topological space of measures, unpublished.
• A Random Sequence of Reals, unpublished.