Computational Aspects of Randomness and Their Structural Analysis via Nonstandard Methods

• Project/Area Number: 23340020
• Research Category: Grant-in-Aid for Scientific Research (B)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: General mathematics (including Probability theory/Statistical mathematics)
• Research Institution: Tohoku University
• Principal Investigator: TANAKA Kazuyuki 東北大学, 理学(系)研究科(研究院), 教授 (70188291)
• Co-Investigator(Kenkyū-buntansha): YAMAZAKI Takeshi 東北大学, 大学院理学研究科, 准教授 (30336812) ; SUZUKI Toshio 首都大学東京, 大学院理学研究院, 准教授 (30235973) ; TADAKI Kohtaro 中央大学, 研究開発機構, 機構准教授 (70407881)
• Co-Investigator(Renkei-kenkyūsha): KURODA Satoru 群馬県立女子大学, 文学部, 准教授 (30300586) ; YOKOYAMA Keita 北陸先端科学技術大学院大学, 情報科学研究科, 助教 (10534430)
• Project Period (FY): 2011-04-01 – 2015-03-31
• Project Status: Completed (Fiscal Year 2014)
• Budget Amount: ¥17,940,000 (Direct Cost: ¥13,800,000、Indirect Cost: ¥4,140,000); Fiscal Year 2014: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000); Fiscal Year 2013: ¥7,410,000 (Direct Cost: ¥5,700,000、Indirect Cost: ¥1,710,000); Fiscal Year 2012: ¥5,720,000 (Direct Cost: ¥4,400,000、Indirect Cost: ¥1,320,000); Fiscal Year 2011: ¥3,380,000 (Direct Cost: ¥2,600,000、Indirect Cost: ¥780,000)
• Keywords: 計算可能性理論 / 超準モデル / ランダム性 / 数学基礎論

Outline of Final Research Achievements

The main aim of this research is to cultivate systematic understanding of randomness via various logical methods such as non-standard analysis so that computational aspects of basic concepts about probability and games deeply correlated to randomness should be clarified. We focus on the following five topics. (1) Setting a logical framework of nonstandard arguments for constructive measure theory. (2) Investigating computational structures by randomness notions. (3) Determining the equilibrium points of game trees. (4) Elucidating physical meaning of Chaitin’s Omega. (5) Logical treatments for determinacy of games.

Research Products

• [Journal Article] Infinite games recognized by 2-stack visibly pushdown automata (2015)
  • Author(s): W. Li, S. Okisaka and K. Tanaka
  • Journal Title: RIMS Kokyuroku Volume: 印刷中
• [Journal Article] Hardness of Classically Simulating Quantum Circuits with Unbounded Toffoli and Fan-Out Gates (2014)
  • Author(s): Y.Takahashi, T.Yamazaki and K.Tanaka
  • Journal Title: Quantum Information and Computation Volume: 14 Pages: 1149-1164
  • NAID: 110009820596
  • Peer Reviewed
• [Journal Article] On the Ramseyan factorization theorem (2014)
  • Author(s): S.Murakami, T.Yamazaki and K.Yokoyama
  • Journal Title: Lecture Notes in Computer Science Volume: 8493 Pages: 324-332
  • DOI: 10.1007/978-3-319-08019-2_33
  • NAID: 120005617819
  • ISBN: 9783319080185, 9783319080192
  • Peer Reviewed
• [Journal Article] Hardness of Classically Simulating Quantum Circuits with Unbounded Toffoli and Fan-Out Gates (2013)
  • Author(s): Y. Takahashi, T. Yamazaki and K. Tanaka
  • Journal Title: Lecture Notes in Computer Science Volume: 8087 Pages: 801-812
  • DOI: 10.1007/978-3-642-40313-2_70
  • NAID: 110009820596
  • ISBN: 9783642403125, 9783642403132
  • Peer Reviewed
• [Journal Article] Spectral representation of some computably enumerable sets with an application to quantum provability (2013)
  • Author(s): C. S. Calude and K. Tadaki
  • Journal Title: Lecture Notes in Computer Science Volume: 7956 Pages: 43-54
  • DOI: 10.1007/978-3-642-39074-6_6
  • ISBN: 9783642390739, 9783642390746
  • Peer Reviewed
• [Journal Article] Relative randomness for Mart in-Loef random sets (2012)
  • Author(s): NingNing Peng, Kojiro Higuchi, Takeshi Ymazaki and Kazuyuki Tanaka
  • Journal Title: Lecture Notes in Computer Science Volume: 7318 Pages: 303-312
  • DOI: 10.1007/978-3-642-30870-3_58
  • ISBN: 9783642308697, 9783642308703
  • Peer Reviewed
• [Journal Article] Infinite Games and Transfinite Recursion of Multiple Inductive Definitions (2012)
  • Author(s): K. Yoshii and K. Tanaka
  • Journal Title: Lecture Notes in Computer Science Volume: 7318 Pages: 374-383
  • DOI: 10.1007/978-3-642-30870-3_38
  • ISBN: 9783642308697, 9783642308703
  • Peer Reviewed
• [Journal Article] A statistical mechanical interpretation of algorithmic information theory III: Composite systems and fixed points (2012)
  • Author(s): K. Tadaki
  • Journal Title: Mathematical Structures in Computer Science Volume: 22 Issue: 5 Pages: 752-770
  • DOI: 10.1017/s096012951100051x
  • Peer Reviewed
• [Journal Article] Fixed point theorems on partial randomness (2012)
  • Author(s): K. Tadaki
  • Journal Title: Annals of Pure and Applied Logic Volume: 163 Issue: 7 Pages: 763-774
  • DOI: 10.1016/j.apal.2011.09.018
  • Peer Reviewed
• [Journal Article] Phase transition between unidirectionality and bidirectionality (2012)
  • Author(s): K. Tadaki
  • Journal Title: Lecture Notes in Computer Science Volume: 7060 Pages: 203-223
  • DOI: 10.1007/978-3-642-27654-5_16
  • ISBN: 9783642276538, 9783642276545
  • Peer Reviewed
• [Journal Article] A Chaitin Ω number based on compressible strings (2012)
  • Author(s): K. Tadaki
  • Journal Title: Natural Computing Volume: 11 Issue: 1 Pages: 117-128
  • DOI: 10.1007/s11047-011-9272-y
  • Peer Reviewed
• [Journal Article] Newwork Games with and without Synchroneity (2011)
  • Author(s): Ahmad Termimi Ab Ghani and Kazuyuki Tanaka
  • Journal Title: Lecture Notes in Computer Science Volume: 7037 Pages: 87-103
  • DOI: 10.1007/978-3-642-25280-8_9
  • ISBN: 9783642252792, 9783642252808
  • Peer Reviewed
• [Presentation] Finitisations of second order principles (2015)
  • Author(s): K.Tanaka and F. Pelupessy
  • Organizer: 日本数学会2015年会
  • Place of Presentation: 明治大学
  • Year and Date: 2015-03-22
• [Presentation] 決定問題の相転移 (2015)
  • Author(s): 田中一之
  • Organizer: 情報処理学会第７７回全国大会
  • Place of Presentation: 京都大学
  • Year and Date: 2015-03-17
  • Invited
• [Presentation] Phase transitions and Reverse Mathematics (2014)
  • Author(s): K.Tanaka and F. Pelupessy
  • Organizer: 日本数学会秋季総合分科会
  • Place of Presentation: 広島大学
  • Year and Date: 2014-09-27
• [Presentation] Variants of Infinite Games and Their Strenth (2014)
  • Author(s): K. Tanaka
  • Organizer: IMS-JSPS Joint Workshop in Mathematical Logic and Foundations of Mathmematics
  • Place of Presentation: シンガポール国立大学
  • Year and Date: 2014-09-01
• [Presentation] Probability distributions achieving the equilibrium of an AND-OR tree under directional algorithms (2012)
  • Author(s): Toshio Suzuki and Ryota Nakamura
  • Organizer: The International MultiConference of Engineers and Computer Scientists 2012
  • Place of Presentation: 香港（中国）
• [Presentation] Infinite Games and Reverse Mathematics (2012)
  • Author(s): Kazuyuki Tanaka and Keisuke Yoshii
  • Organizer: WORKSHOP ON PROOF THEORY AND COMPUTABILITY THEORY 2012
  • Place of Presentation: 東京
• [Presentation] Robustness of statistical mechanical interpretation of algorithmic information theory (2011)
  • Author(s): K. Tadaki
  • Organizer: The Twelfth Asian Logic Conference (ALC 2011)
  • Place of Presentation: ウェリントン（ニュージーランド）
• [Presentation] Newwork Games with and without Synchroneity (2011)
  • Author(s): Ahmad Termimi Ab Ghani and Kazuyuki Tanaka
  • Organizer: GameSec 2011
  • Place of Presentation: メリーランド（アメリカ合衆国）
• [Presentation] Reverse Mathematics on Measure-Theoretic Probability (Plenary talk)
  • Author(s): K. Tanaka
  • Organizer: Asian Logic Conference
  • Place of Presentation: Guangzhou, China
  • Invited
• [Presentation] Infinite Games and Reverse Mathematics
  • Author(s): K. Tanaka
  • Organizer: Fourth Workshop on Game-Theoretic Probability and Related Topics
  • Place of Presentation: 東京大学情報理工
  • Invited
• [Presentation] Reverse Mathematics and Nonstandard Proof Mathods
  • Author(s): K. Tanaka
  • Organizer: Workshop on Logic
  • Place of Presentation: Ghent Univ., Belgium
  • Invited
• [Book] ロジックの世界 (2015)
  • Author(s): D. クライアン、S. シュアテル、B. メイブリン 原著 田中一之訳
  • Total Pages: 192
  • Publisher: 講談社
• [Book] チューリングと超パズル (2013)
  • Author(s): 田中一之
  • Total Pages: 266
  • Publisher: 東京大学出版会
• [Book] ゲーデルに挑む (2012)
  • Author(s): 田中一之
  • Total Pages: 188
  • Publisher: 東京大学出版会
• [Remarks] CTFM2014
  • URL: http://www.jaist.ac.jp/CTFM/CTFM2014/
• [Remarks] CTFM2013
  • URL: http://sendailogic.math.tohoku.ac.jp/CTFM/