Organization
Undergraduate School School of Science and Technology Associate Professor
Title
Associate Professor
Homepage
External link
Degree
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学士（理学） ( 京都大学 )
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Master of science ( Kyoto University )
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Doctor of science ( Kyoto University )
Research Areas
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Natural sciences / Applied mathematics and statistics
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Natural sciences / Basic mathematics
Education
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Kyoto University Graduate School, Division of Natural Science
2007.4 - 2010.3
Country/Region： Japan
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Kyoto University Graduate School, Division of Natural Science
2005.4 - 2007.3
Country/Region： Japan
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Kyoto University Faculty of Science Department of Science
2001.4 - 2005.3
Country/Region： Japan
Research History
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Meiji University School of Science and Technology Associate Professor Associate Professor
2017.10
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Meiji University Department of Mathematics, School of Science and Technology Lecturer Senior Assistant Professor
2014.4 - 2017.9
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JSPS Special researcher of the Japan Society for the Promotion of Science Research Fellow at Tokyo University
2013.4 - 2014.3
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Kyoto University Researher Postdoc Global COE Research fellow
2011.4 - 2013.3
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Kyoto University Researher Postdoc Global COE Research fellow
2010.4 - 2011.3
Papers
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Randomness with respect to c.e. semimeasures Reviewed
Kenshi Miyabe
Information and Computation 307 105384 2025.11
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Solovay reducibility via Lipschitz functions and signed-digit representation
Masahiro Kumabe, Kenshi Miyabe, Toshio Suzuki
Computability 14 ( 1 ) 38 - 62 2025.2
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Rational Sequences Converging to Left-c.e. Reals of Positive Effective Hausdorff Dimension Reviewed International journal
Hiroyuki Imai, Masahiro Kumabe, <U>Kenshi Miyabe</U>, Yuki Mizusawa and Toshio Suzuki
Proceedings of the 9th International Conference on Computability Theory and Foundations of Mathemati 97 - 121 2022
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Solovay reducibility and continuity.
Masahiro Kumabe, Kenshi Miyabe, Yuki Mizusawa, Toshio Suzuki
J. Log. Anal. 12 2020
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Computable Prediction. Reviewed
Kenshi Miyabe
Lecture Notes in Computer Science 11654 137 - 147 2019.7
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Uniform Relativization. Invited Reviewed
Kenshi Miyabe
Lecture Notes in Computer Science 11558 50 - 61 2019.6
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Erdos-Feller-Kolmogorov-Petrowsky law of the iterated logarithm for self-normalized martingales: a game-theoretic approach Reviewed
Takeyuki Sasai, Kenshi Miyabe, and Akimichi Takemura
Annals of Probability 47 ( 2 ) 1136 - 1161 2019.3
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Muchnik degrees and Medvedev degrees of the randomness notions Reviewed
Kenshi Miyabe
Proceedings of the 14th and 15th Asian Logic Conferences 108 - 128 2019.1
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Relation between the rate of convergence of strong law of large numbers and the rate of concentration of Bayesian prior in game-theoretic probability Reviewed
Ryosuke Sato, Kenshi Miyabe, Akimichi Takemura
Stochastic Processes and their Applications 128 ( 5 ) 1466 - 1484 2018.5
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Coherence of Reducibilities with Randomness Notions. Reviewed
Kenshi Miyabe
Theory Comput. Syst. 62 ( 7 ) 1599 - 1619 2018
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Randomness and Solovay degrees. Reviewed
Kenshi Miyabe, André Nies, Frank Stephan 0001
J. Log. Anal. 10 1 - 13 2018
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Using Almost-everywhere theorems from Analysis to Study Randomness. Reviewed
Kenshi Miyabe, André Nies, Jing Zhang
Bulletin of Symbolic Logic 22 ( 3 ) 305 - 331 2016
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Reducibilities Relating to Schnorr Randomness. Reviewed
Kenshi Miyabe
Theory Comput. Syst. 58 ( 3 ) 441 - 462 2016
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Derandomization in game-theoretic probability Reviewed
Kenshi Miyabe, Akimichi Takemura
STOCHASTIC PROCESSES AND THEIR APPLICATIONS 125 ( 1 ) 39 - 59 2015.1
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Schnorr Triviality and Its Equivalent Notions. Reviewed
Kenshi Miyabe
Theory Comput. Syst. 56 ( 3 ) 465 - 486 2015
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Unified characterizations of lowness properties via Kolmogorov complexity. Reviewed
Takayuki Kihara, Kenshi Miyabe
Arch. Math. Log. 54 ( 3-4 ) 329 - 358 2015
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Algorithmic randomness over general spaces. Reviewed
Kenshi Miyabe
Math. Log. Q. 60 ( 3 ) 184 - 204 2014
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Uniform Kurtz randomness. Reviewed
Takayuki Kihara, Kenshi Miyabe
J. Log. Comput. 24 ( 4 ) 863 - 882 2014
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The law of the iterated logarithm in game-theoretic probability with quadratic and stronger hedges Reviewed
Kenshi Miyabe, Akimichi Takemura
STOCHASTIC PROCESSES AND THEIR APPLICATIONS 123 ( 8 ) 3132 - 3152 2013.8
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Van Lambalgen's Theorem for uniformly relative Schnorr and computable randomness
<U>Kenshi Miyabe</U>, Jason Rute
Proceedings of the Twelfth Asian Logic Conference 251 - 270 2013.7
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L1-Computability, Layerwise Computability and Solovay Reducibility. Reviewed
Kenshi Miyabe
Computability 2 ( 1 ) 15 - 29 2013
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Characterization of Kurtz Randomness by a Differentiation Theorem. Reviewed
Kenshi Miyabe
Theory Comput. Syst. 52 ( 1 ) 113 - 132 2013
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Convergence of random series and the rate of convergence of the strong law of large numbers in game-theoretic probability Reviewed
Kenshi Miyabe, Akimichi Takemura
STOCHASTIC PROCESSES AND THEIR APPLICATIONS 122 ( 1 ) 1 - 30 2012.1
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The difference between optimality and universality. Reviewed
Kenshi Miyabe
Log. J. IGPL 20 ( 1 ) 222 - 234 2012
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An Optimal Superfarthingale and Its Convergence over a Computable Topological Space. Reviewed
Kenshi Miyabe
Lecture Notes in Computer Science 7070 273 - 284 2011
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Truth-table Schnorr randomness and truth-table reducible randomness. Reviewed
Kenshi Miyabe
Math. Log. Q. 57 ( 3 ) 323 - 338 2011
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An Extension of van Lambalgen's Theorem to Infinitely Many Relative 1-Random Reals. Reviewed
Kenshi Miyabe
Notre Dame Journal of Formal Logic 51 ( 3 ) 337 - 349 2010
Books
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確率統計入門:モデル化からその解析へ
渡辺浩, 宮部賢志（ Role： Joint author）
森北出版 2020.3
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確率と哲学
Childers,Timothy, 宮部, 賢志, 芦屋, 雄高
九夏社 2020 （ ISBN:9784909240033 ）
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新しい微積分＜上＞
長岡亮介/渡辺 浩/矢崎成俊/宮部賢志（ Role： Joint author）
講談社 2017.2
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新しい微積分＜下＞
長岡亮介/渡辺 浩/矢崎成俊/宮部賢志（ Role： Joint author）
講談社 2017.2
MISC
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ゲームによって予測不可能性を捉える (特集 ゲーム理論の数理)
宮部 賢志, 竹村 彰通
数学セミナー 53 ( 10 ) 15 - 19 2014.10
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Computation with advice of degree of randomness (New Trends in Theoretical Computer Science)
MIYABE KENSHI
RIMS Kokyuroku 1849 1 - 7 2013.8
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Analytical approach to algorithmic randomness (Proof theory and complexity)
MIYABE KENSHI
RIMS Kokyuroku 1832 114 - 126 2013.4
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MIYABE KENSHI
RIMS Kokyuroku 1729 84 - 94 2011.2
Presentations
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Randomness with respect to c.e. semimeasures
Kenshi Miyabe
Twenty-Second International Conference on Computability and Complexity in Analysis
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Variation of weakly computable reals in Solovay reducibility
Kenshi Miyabe
18th Asian Logic Conference
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Solovay reducibility for computably approximable reals
Kenshi Miyabe
Workshop on Logic and Computation @ BIMSA
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Solovay reducibility for computably approximable reals
Kenshi Miyabe
CCR 2025 : Computability, Complexity and Randomness & Journées Calculabilités
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アルゴリズム的ランダムネスと学習可能性 Invited
宮部賢志
数理論理学の最近の進展 2024.7
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Quantifier Variations in Solovay Reducibility
Kenshi Miyabe
American Mathematical Society 2025 Spring Western Sectional Meeting 2025.5
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Strong Solovay reducibility
Kenshi Miyabe
17th International Conference on Computability, Complexity and Randomness 2024.3
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Solovay reducibility and signed-digit representation
Kenshi Miyabe
Continuity, Computability, Constructivity From Logic to Algorithms 2023.9
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Solovay reducibility and signed-digit representation Invited
Kenshi Miyabe
NUS workshop, From omega to Omega 2023.6
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Subclasses of weakly computable reals International conference
International conference on computability, complexity and randomness 2022 2022.6
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Generality of computable measures International conference
Leeds Computability Days 2022 2022.6
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計算可能な予測の収束速度
日本数学会2022年度年会 2022.3
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計算可能な予測の収束速度
RIMS共同研究 (公開型)「証明と計算の理論と応用」 2021.12
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The rate of convergence of computable predictions
Kenshi Miyabe
German Mathematical Society 2020.9 ドイツ数学会
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Computable prediction International conference
Kenshi Miyabe
Artificial General Intelligence 2019.8 AGI
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Uniform relativization Invited International conference
Kenshi Miyabe
Computability in Europe 2019.7 CiE
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Schnorr triviality via decidable machines International conference
Kenshi Miyabe
Computability Theory and Foundations of Mathematics 2019.3 CTFM
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Muchnik degrees and Medvedev degrees of the randomness notions
Kenshi Miyabe
American Mathematical Society 2019.3 AMS
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Coherence between reducibility and randomness notions
Kenshi Miyabe
Mathematical Society of Japan 2019.3 MSJ
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The chance that the sun will rise tomorrow revisited
Kenshi Miyabe
SIG-AGI 2019.3 SIG-AGI
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Erdos-Feller-Kolmogorov-Petrowsky law of the iterated logarithm International conference
Kenshi Miyabe
Workshop on Computability Theory and Foundations of Mathematics 2018 2018.9 CTFM
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A tutorial in game-theoretic probability and algorithmic randomness International conference
Kenshi Miyabe
Workshop on Computability Theory and Foundations of Mathematics 2018 2018.9 CTFM
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A hierarchy of functions corresponding randomness hierarchy
Kenshi Miyabe
RIMS Symposia (open) 2017.12 RIMS
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Computation at random points
Kenshi Miyabe
IPA Math 2017 2017.12 JST
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Face randomness by mathematics Invited
Kenshi Miyabe
MATH POWER 2017.10
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Solomonoff’s universal induction, or algorithmic probability Invited
Kenshi Miyabe
SIG-AGI 2017.9 SIG-AGI
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Can one compute a more random set uniformly?
Kenshi Miyabe
JMO 2017.3 JMO
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Randomness notions in Muchnik and Medvedev degrees
Kenshi Miyabe
Dagstuhl seminar on “Computability Theory” 2017.2 Schloss Dagstuhl
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Randomness notions in Muchnik and Medvedev degrees
Kenshi Miyabe
Computability Theory and Foundations of Mathematics 2016 2016.9 CTFM
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Computability of reals on the space where the triangle inequality does not hold
Kenshi Miyabe
JMO 2016.9 JMO
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Variants of layerwise computability International conference
Kenshi Miyabe
Thirteenth International Conference on Computability and Complexity in Analysis 2016.6 CCA
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Mass problems for randomness notions
Kenshi Miyabe
JMO 2016.3 JMO
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Characterizations of 3-randomness via complexity
Kenshi Miyabe
JMO 2015.9 JMO
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Reducibilities as refinements of the randomness hierarchy
Kenshi Miyabe
Computability Theory and Foundations of Mathematics 2015 2015.9 CTFM
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Total-machine reducibility and randomness notions International conference
Kenshi Miyabe
Asian Logic Conference 2015 2015.1 ALC
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Derandomization in GameTheoretic Probability
Kenshi Miyabe
Fifth Workshop on Game-Theoretic Probability and Related Topics 2014.11
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Schnorr randomness versions of K, C, LR, vL-reducibilities International conference
Kenshi Miyabe
Conference on Computability, Complexity and Randomness CCR 2014 2014.6 CCR
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Characterization of Lebesgue points via integral tests
Kenshi Miyabe
JMO 2014.3 JMO
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A Gap Phenomenon for Schnorr Randomness International conference
Kenshi Miyabe
Computability Theory and Foundations of Mathematics 2014.2 CTFM
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Unpredictability of initial points
Kenshi Miyabe
RIMS研究集会「力学系と計算」 2014.1 RIMS
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Variants of Layerwise Computability
MIYABE Kenshi
ARGENTINA-JAPAN-NEW ZEALAND WORKSHOP, UNIVERSITY OF AUCKLAND 2013.12
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Almost uniform weak n-randomness International conference
Kenshi Miyabe
Eighth International Conference on Computability, Complexity and Randomness 2013.9 CCR
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Being a Lebesgue point for integral tests International conference
Kenshi Miyabe
Asian Logic Conference 2013 2013.9 ALC
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The emergence of probability from randomness and games International conference
Kenshi Miyabe
Modeling Market Dynamics and Equilibrium – New Challenges, New Horizons 2013.8
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L^1-computability and the computability of conditional probability
Kenshi Miyabe
Analysis, Randomness and Applications 2013 2013.6 ARA
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The preordering related to uniform Schnorr randomness
Kenshi Miyabe
ELC Workshop on Randomness and Probability Through Computability 2013.5 RPTC
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Uniform relativization and almost uniform relativization International conference
Kenshi Miyabe
ELC Seminar on Algorithmic Randomness 2013.5 ELC
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Van Lambalgen’s Theorem for uniform Kurtz randomness
Kenshi Miyabe
JMO 2013.3 JMO
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The other history of probability theory
Kenshi Miyabe
JMO 2013.3 JMO
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Schnorr and Kurtz randomness versions of Merkle’s criterion
Kenshi Miyabe
COMP 2013.3 COMP
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Things to do in and with algorithmic randomness Invited
Kenshi Miyabe
Sendai Logic School 2013 2013.2 SLS
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Computably measurable sets and computably measurable functions in terms of algorithmic randomness International conference
Kenshi Miyabe
Computability Theory and Foundations of Mathematics 2013.2 CTFM
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Computability of conditional probability
Kenshi Miyabe
LA Symposium 2013.1
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An introduction to algorithmic randomness International conference
Kenshi Miyabe
Fourth Workshop on Game-Theoretic Probability and Related Topics 2012.11 GTP
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The law of the iterated logarithm in game-theoretic probability International conference
Kenshi Miyabe
Fourth Workshop on Game-Theoretic Probability and Related Topics 2012.11 GTP
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Natural properties that a randomness notion should have
Kenshi Miyabe
JMO 2012.9 JMO
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An introduction to game-theoretic probability
Kenshi Miyabe
SLACS 2012 2012.9 SLACS
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Schnorr triviality and a base for uniform Schnorr randomness
Kenshi Miyabe
RIMS Symposia (open) 2012.9 RIMS
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On the effectivization of Lusin’s theorem
Kenshi Miyabe
LA Symposium 2012.7
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Schnorr triviality is equivalent to being a basis for tt-Schnorr randomness International conference
Kenshi Miyabe
Seventh International Conference on Computability, Complexity and Randomness 2012.7 CCR
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Weak L^1-computability and Limit L^1-computability International conference
Kenshi Miyabe
Ninth International Conference on Computability and Complexity in Analysis 2012.6 CCA
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An integral test for Schnorr randomness and its applications International conference
Kenshi Miyabe
Computability in Europe 2012 2012.6 CiE
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The convergence rate of SLLN in the case of non-i.i.d.
Kenshi Miyabe
JMO 2012.3 JMO
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At which points are functions differentiable?
Kenshi Miyabe
JMO 2012.3 JMO
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L^1-computability and weak L^1-computability
Kenshi Miyabe
京都計算可能解析学シンポジウム2012 2012.2
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Schnorr Layerwise Computability
Kenshi Miyabe
証明論及び計算可能性理論に関するワークショップ 2012.2
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Characterization of Kurtz randomness by a differentiation theorem International conference
Kenshi Miyabe
Twelfth Asian Logic Conference 2011.12 ALC
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Randomness and separation axioms
Kenshi Miyabe
ANALYSIS AND RANDOMNESS IN AUCKLAND 2011.12
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An optimal superfarthingale and its convergence over a computable topological space International conference
Kenshi Miyabe
Solomonoff 85th Memorial Conference 2011.12
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Randomness and differentiability
Kenshi Miyabe
証明論と計算論研究集会 2011.9
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An optimal semimeasure and its convergence
Kenshi Miyabe
ALGi22 2011.8 ALGi
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Chara terizing randomness by integral tests International conference
Kenshi Miyabe
CiE 2011 2011.6 CiE
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Degree of non-randomness and uniform Solovay reducibility
Kenshi Miyabe
RIMS研究集会 2011.2 RIMS
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Towards prediction via randomness
Kenshi Miyabe
Statistics for Stochastic Processes: Inference, Asymptotic Methods, Finance and Data Analysis 2011.2
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Algorithmic randomness over general spaces International conference
Kenshi Miyabe
CCA-CCR 2011 2011.1 CCA-CCR
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Frequency and belief are probability
Kenshi Miyabe
SLACS 2010 2010.11 SLACS
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Algorithmic randomness over general spaces
Kenshi Miyabe
数理解析研究所短期共同研究ー形式体系と計算理論ー 2010.9 RIMS
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Limit Theorems on a Computable Topological Space
Kenshi Miyabe
ALGI21 2010.9 ALGI
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Computable Topology and Randomness
Kenshi Miyabe
Computable Analysis in Kyoto 2010.7
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An extension of van Lambalgen’s Theorem to infinitely many relative 1-random reals
Kenshi Miyabe
GTP 2010.3 GTP
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Another relativization of Schnorr randomness and computably randomness
Kenshi Miyabe
ALGI20 2009.9 ALGI
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An extension of van Lambalgen’s Theorem to infinitely many relative 1-random reals
Kenshi Miyabe
SLACS 2009 2009.9 SLACS
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Classification of Index Sets of Functions
Kenshi Miyabe
CSCAT 2007 2007.3 CSCAT
Works
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アルゴリズム的ランダムネスと計算可能測度論
宮部賢志
2021.2
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アルゴリズム情報理論とAI
2019.12
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ゲームによって予測不可能性を捉える
2014.10
Research Projects
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Solovay reducibility in Banach space
Grant number：25K07105 2025.4 - 2029.3
Japan Society for the Promotion of Science Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (C)
Grant amount：\4680000 （ Direct Cost: \3600000 、 Indirect Cost：\1080000 ）
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Study on convergence rate of weakly computable reals
Grant number：22K03408 2022.4 - 2027.3
Japan Society for the Promotion of Science Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (C)
Grant amount：\3640000 （ Direct Cost: \2800000 、 Indirect Cost：\840000 ）
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Model-free theory of finance based on game-theoretic probability
Grant number：21K18585 2021.7 - 2024.3
Japan Society for the Promotion of Science Grants-in-Aid for Scientific Research Grant-in-Aid for Challenging Research (Exploratory)
Grant amount：\5850000 （ Direct Cost: \4500000 、 Indirect Cost：\1350000 ）
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様相ミュー計算と確率ゲームへの逆数学的アプローチ：ハイブリッド検査法の創出と分析
Grant number：21H03392 2021.4 - 2025.3
日本学術振興会 科学研究費助成事業 基盤研究(B) 基盤研究(B)
田中 一之, 鹿島 亮, 宮部 賢志, 木原 貴行
Grant amount：\17290000 （ Direct Cost: \13300000 、 Indirect Cost：\3990000 ）
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From algorithmic randomness to continuous real functions and real closed fields
Grant number：21K03340 2021.4 - 2025.3
Japan Society for the Promotion of Science Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (C) Grant-in-Aid for Scientific Research (C)
Grant amount：\4160000 （ Direct Cost: \3200000 、 Indirect Cost：\960000 ）
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様相ミュー計算と確率ゲームへの逆数学的アプローチ：ハイブリッド検査法の創出と分析
Grant number：23K21641 2021.4 - 2023.3
日本学術振興会 科学研究費助成事業 基盤研究(B)
田中 一之, 宮部 賢志, 鹿島 亮, 木原 貴行
Grant amount：\17290000 （ Direct Cost: \13300000 、 Indirect Cost：\3990000 ）
本研究の目的は，モデル検査など論理学の応用技術において仮定される論理や集合の公理が何かを探査し，異なる技法の間の類似性や相異性を顕在化することで，ハイブリッド技術の創出を目指すものである．初年度(令和３年度)においては５つの領域に焦点を絞り，それぞれの課題を探求し，得られた知見を統合して多角的な検討を加えて，より高度な視点を得る計画であった．しかし，コロナ禍の行動制限により研究交流がほとんど行えず，とくに本研究で重要な役割を担う中国とシンガポールの協力者たちとの交流には長い空白ができてしまったので，代表者は北京の応用数学研究所に滞在して彼らとの交流基盤の回復に努め，体制が整うまで本研究を中断することにした．従って，本年度の研究も未完結であるが，個別課題ごとには次のような顕著な成果が得られた．
1. 様相μ計算の論理式は，演算子μ,νの入れ替わりが何回生じるかで分類される．μν型の論理式でもνμ型の論理式でも表現されるクラスΔ2はμ型とν型の論理式の超限的な組合せによって表せることを示す論文を出版した．さらに，様相μ計算の階層の崩壊を認識論理学の視点で分析し，無知の形式化を行った．2. 様相μ計算の階層が様相の種類によってどう変わるかを，真理値評価ゲームを用いて調べた．3. 確率的モデル検査は，検査法や評価アルゴリムがランダム化されたもので，ここではBlackwellゲームの一種として扱う．代表者が開発したWKLoの超準的手法を用いて，確率アルゴリズムを逆数学的に分析する研究を進めた．4. 決定木において，クエリの最小コストを最大化する入力分布（固有分布）について，より一般な木構造に対して固有分布を調べて論文にまとめた．さらにはグラフ構造への一般化を検討している．５. 確率アルゴリズムの視点から確率的推論を捕捉する独自の研究を目指しながら，先行研究の成果を整理した． -
strategy and control in continuous-time games
Grant number：16K12399 2015.4 - 2019.3
Japan Society for the Promotion of Science Grants-in-Aid for Scientific Research Grant-in-Aid for Challenging Exploratory Research
TAKEMURA AKIMICHI
Authorship：Coinvestigator(s)
Grant amount：\3120000 （ Direct Cost: \2400000 、 Indirect Cost：\720000 ）
The game-theoretic probability established by Shafer and Vovk in their book "Probability and Finance: It's Only a Game!" constructs the whole framework of probability theory based on a betting game between two players without using the measure theory. In this study, we mainly considered continuous time games and obtained results on refinements of the law of the iterated logarithm and the efficiency of Bayesian betting strategies.
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Construction of computable measure theory
Grant number：26870143 2014.4 - 2018.3
Japan Society for the Promotion of Science Grants-in-Aid for Scientific Research Grant-in-Aid for Young Scientists (B)
Miyabe Kenshi
Authorship：Principal investigator
Grant amount：\4800000 （ Direct Cost: \3900000 、 Indirect Cost：\900000 ）
In this research, we aimed to develop the basis of computable measure theory. As products, we proposed some new measures of randomness and proved its nice properties. In concrete, we prove some nice properties of Schnorr randomness version of LR-reducibility, that naturally comes from uniform relativization of Schnorr randomness as LR-reducibility with Martin-Lof randomness does.
We also proposed density randomness as a new randomness notions, which appears in the study of computability of Lebesgue density theorem. Density randomness is closely related with the convergence of martingales and has many equivalent characterizations. We hope to have many and wide applications of this notion. -
Theory of prediction based on game-theoretic probability and randomness
Grant number：25540009 2014.1 - 2016.3
Japan Society for the Promotion of Science Grants-in-Aid for Scientific Research Grant-in-Aid for Challenging Exploratory Research
Takemura Akimichi, MIYABE Kenshi
Authorship：Coinvestigator(s)
Grant amount：\480000 （ Direct Cost: \390000 、 Indirect Cost：\90000 ）
In game-theoretic probability stochastic behavior of the players results from a game between players. However the stochastic behavior is often a consequence of a deterministic strategy of the players. In particular, the strategy of the player "Reality" does not have a counterpart in the usual measure-theoretic probability and it is a new field of research. In this project we succeeded in establishing a general method for constructing Reality's deterministic strategy.
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アルゴリズム的観点での確率及び予測概念の研究
Grant number：13J08139 2013.4 - 2014.3
日本学術振興会 特別研究員奨励費 特別研究員奨励費
宮部 賢志
Authorship：Principal investigator
Grant amount：\3600000 （ Direct Cost: \3600000 ）
本研究の目的は, アルゴリズム的確率の確率および予測としての性質を明らかにすることにあった. 平成25年度は特に「A, アルゴリズム的確率の一般化」に重点を置いた. これまでアルゴリズム的確率は, 文字列上でのみ定義されていた. これをもっと一般的な空間で定義できるようにするためには, 計算可能性解析の手法を導入する必要がある.
アルゴリズム的確率において, 2007年のHutter-MuchnikによるMLランダムな点での非収束という重要な結果がある. 一方, ランダムネスの理論においてランダムの概念の解析的な特徴付けが知られるようになった. やがて, これらの問題が本質的に同じ問題であることが明らかになってきた.
当該年度に得た大きな結果として, 密度ランダムネスの微分定理による特徴付けがある. これはランダムネスの理論の文脈での貢献であると同時に, Hutter-Muchnikの結果の特殊な場合の拡張と見ることもできるもので, 計算可能解析の手法がアルゴリズム的磯率の理解に有用であることを示す大きな一歩となった. 本結果は現在執筆中の論文に含まれる予定である.
次に「B, ゲーム論的確率論での確率との関係」について述べる. アルゴリズム的確率の最も大きな特徴は「普遍性」(universal property)を持つということである. 一方, ゲーム論的確率論ではこのような性質は存在せず, そのため脱乱択化ができる. 当該年度にはその具体的な手法を与えることができた. このような性質の違いを明らかにすることで, 確率の概念に追ることができると考えている.
アルゴリズム的確率の研究を進める上で, 計算可能測度論構築の必要性を強く感じたため, そちらにも力を注いだ. -
ランダムネスを用いた確率概念の拡張
Grant number：23740072 2011.4 - 2013.3
科学研究費補助金 若手研究(B) 若手研究(B)
宮部 賢志
Authorship：Principal investigator
Grant amount：\3000000 （ Direct Cost: \2600000 、 Indirect Cost：\400000 ）
本研究の目的はランダムと確率の概念の関係を計算可能性の観点から明らかにすることであった．現在広く使われているKolmogorovによる公理的確率論は，確率の意味を問わず，計算可能性との関係も明らかではない．「そもそも確率とは何か」という科学哲学的動機と共に，予測限界の理論的解明という実際的な意味を持つテーマである．このテーマはvon Mises, Solomonoff, 晩年のKolmogorovなどの研究の流れにあり，現在海外では注目を厚め始めているトピックである．
本研究では，具体的には，計算可能解析の理論を使って，ランダムネスの理論を拡張し，統計的予測理論における考え方とSolomonoffの予測との関係を明らかにすることを目標としている．これまでの研究により，計算可能距離空間上でのランダムネスの諸概念の取り扱い方や計算可能可測関数との関係などを明らかにすることができた．また測度論でよく知られた結果の計算可能性を考えると，Schnorrランダムネスが自然に現れることも明らかにした．さらに予測理論の関係も少しずつ明らかになりつつある．
現在は技術的な困難が1つ1つ解決されている段階であり，実際に応用されるようになるにはもう少し時間がかなりそうであるが，現在基礎論の研究者を中心に研究者が増え始めており，今後ますます研究が進むと予想される．特に計算量との関係の重要性が明らかになってきており，その分野の研究者との交流も進みつつある． -
計算論的ランダムネスによる確率解釈
2010.4 - 2011.3
Teaching Experience
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計算理論
2026.4 Institution：明治大学
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基礎微分積分2
2018.4 Institution：明治大学
Level：Undergraduate (liberal arts)
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確率論2
2018.4 Institution：明治大学
Level：Undergraduate (specialized)
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確率論1
2018.4 - 2023.3 Institution：明治大学
Level：Undergraduate (specialized)
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関数論1
2017.4 Institution：明治大学
Level：Undergraduate (specialized)
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関数論1演習
2017.4 Institution：明治大学
Level：Undergraduate (specialized)
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理工学研究科総合講義A
2017.4 - 2025.3 Institution：明治大学
Level：Graduate (liberal arts)
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数値計算の基礎2
2016.4 - 2018.3 Institution：明治大学
Level：Undergraduate (specialized)
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数理コンピュータ科学2
2015.4 - 2019.3 Institution：明治大学
Level：Undergraduate (specialized)
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数値計算の基礎1
2014.4 - 2018.3 Institution：明治大学
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数学B2(微分積分)
Institution：早稲田大学
Level：Undergraduate (liberal arts)
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