極限学習を用いた実効的閉集合の次数構造の解明

• Project/Area Number: 12J08528
• Research Category: Grant-in-Aid for JSPS Fellows
• Allocation Type: Single-year Grants
• Section: 国内
• Research Field: General mathematics (including Probability theory/Statistical mathematics)
• Research Institution: Japan Advanced Institute of Science and Technology
• Principal Investigator: 木原 貴行 北陸先端科学技術大学院大学, 情報科学研究科, 特別研究員(PD)
• Project Period (FY): 2012-04-01 – 2015-03-31
• Project Status: Completed (Fiscal Year 2014)
• Budget Amount: ¥3,630,000 (Direct Cost: ¥3,300,000、Indirect Cost: ¥330,000); Fiscal Year 2014: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000); Fiscal Year 2013: ¥1,100,000 (Direct Cost: ¥1,100,000); Fiscal Year 2012: ¥1,100,000 (Direct Cost: ¥1,100,000)
• Keywords: 記述集合論 / 無限次元トポロジー / チューリング次数 / 強制法 / 関数解析 / バナッハ空間 / 計算理論 / 弱無限次元空間 / ランダム性 / コルモゴロフ複雑性 / 実解析学 / ボレル階層 / マルチンゲール / フラクタル / 実数上の集合論 / ベール函数

Outline of Annual Research Achievements

H25年度に引き続き，H26年度も，ボレル可測函数の構造解析および無限次元ポーランド空間の分類研究を推し進めた．本研究の主要成果は，記述集合論，無限次元トポロジー，そしてベール函数環の構造解析に跨る，計算可能性理論の新たな応用の道を切り開いたことである．
1. 近年，記述集合論で中心的問題とされているボレル可測函数の分解への応用を目的として，チューリング次数に関するShore-Slaman和定理を無限次元空間上に拡張することに成功した．この定理を応用することにより，連続函数環の理論におけるゲルファント-コルモゴロフの定理の類似物として，ポーランド空間上の神託付チューリング次数構造が，その上の有限級ベール函数環の構造によって決定されることを示した．
2. 第2級ボレル同型の不変量として，チューリング次数の理論を用いた余次数スペクトルの概念を導入し，Haverの性質Cを持つ無限次元カントール多様体の連続体濃度の族で，それぞれの各有限級ボレル構造がいずれも非同型であるものを構成した．これは有限級ベール函数のなすバナッハ環の線型等長（環同型）に関するMotto Rosの問題を解決するのみならず，Roman PolによるAlexandov問題の解決を初めとする，無限次元トポロジーの既存の様々な定理を拡張するものである．
3. その他，Sacks強制法から得られる逆系の逆極限による空間の構成手法を導入し，Gregoriadesの問題を解決し，ボレル同型の理論においては非可算解析空間が弱余ススリン-F-同型の下で2種類しか存在しないことと，任意の実数に対するシャープが存在することがZFC上同値であることを示した．他には，KP集合論の(Σ1-projectumがωであるような)ω-モデル上の強制法について，クリーチャー強制法およびイデアル商強制法の理論の観点に基づいた再整備を行った．

Research Progress Status

26年度が最終年度であるため、記入しない。

Strategy for Future Research Activity

26年度が最終年度であるため、記入しない。

Research Products

• [Journal Article] Decomposing Borel functions using the Shore-Slaman join theorem (2015)
  • Author(s): Takayuki Kihara
  • Journal Title: Fundamenta Mathematicae Volume: 230 Issue: 1 Pages: 1-13
  • DOI: 10.4064/fm230-1-1
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] Unified characterizations of lowness properties via Kolmogorov complexity (2015)
  • Author(s): Takayuki Kihara and Kenshi Miyabe
  • Journal Title: Archive for Mathematical Logic Volume: 54 Issue: 3-4 Pages: 329-358
  • DOI: 10.1007/s00153-014-0413-8
  • Peer Reviewed
• [Journal Article] On effectively closed sets of effective strong measure zero (2014)
  • Author(s): Kojiro Higuchi and Takayuki Kihara
  • Journal Title: Annals of Pure and Applied Logic Volume: 165 Issue: 9 Pages: 1445-1469
  • DOI: 10.1016/j.apal.2014.04.013
  • Peer Reviewed
• [Journal Article] Uniform Kurtz randomness (2014)
  • Author(s): Takayuki Kihara and Kenshi Miyabe
  • Journal Title: Journal of Logic and Computation Volume: (印刷中) Issue: 4 Pages: 863-882
  • DOI: 10.1093/logcom/ext054
  • Peer Reviewed
• [Journal Article] Inside the Muchnik degrees, part I : Discontinuity, learnability, and cons tructivism (2014)
  • Author(s): Kojiro Higuchi and Takayuki Kihara
  • Journal Title: Annals of Pure and Applied Logic Volume: 165 Issue: 5 Pages: 1058-1114
  • DOI: 10.1016/j.apal.2014.01.003
  • Peer Reviewed
• [Journal Article] Inside the Muchnik degrees, Part II : The degree structures induced by tha arithmatiral hiararchy of countably continuous functions (2014)
  • Author(s): Kojiro Higuchi and Takayuki Kihara
  • Journal Title: Annals of Pure and Applied Logic Volume: 165 Issue: 6 Pages: 1201-1241
  • DOI: 10.1016/j.apal.2014.03.001
  • Peer Reviewed
• [Journal Article] 実数の集合論とランダムネス : 概説 (2013)
  • Author(s): 木原貴行
  • Journal Title: 京都大学数理解析研究所講究録 Volume: 1832 Pages: 97-113
• [Journal Article] Incomputability of simply connected planar continua (2012)
  • Author(s): Takayuki Kihara
  • Journal Title: Computability Volume: 1 Pages: 131-152
  • Peer Reviewed
• [Journal Article] ボレル関数の分解問題への計算論の応用
  • Author(s): 木原貴行
  • Journal Title: 京都大学数理解析研究所講究録 Volume: (掲載確定)
• [Presentation] The second-level Borel isomorphism problem: An encounter of recursion theory and infinite dimensional topology (2015)
  • Author(s): Takayuki Kihara
  • Organizer: JAIST Logic workshop series 2015: Constructivism and Computability
  • Place of Presentation: Shiinoki Cultural Complex, Ishikawa, Kanazawa, Japan
  • Year and Date: 2015-03-03
  • Invited
• [Presentation] 弱無限次元空間の余次数スペクトルと第2級ボレル同型問題 (2015)
  • Author(s): 木原貴行
  • Organizer: 第２回 山陰 基礎論・解析学 研究集会
  • Place of Presentation: 国際ファミリープラザ，鳥取県 米子市
  • Year and Date: 2015-01-24
• [Presentation] 計算論的ランダムネスの手法と応用について (2015)
  • Author(s): 木原貴行
  • Organizer: 大阪大学確率論セミナー
  • Place of Presentation: 大阪大学理学部，大阪府 豊中市
  • Year and Date: 2015-01-13
  • Invited
• [Presentation] Degree spectra and the second-level Borel isomorphism problem on C-compacta (2014)
  • Author(s): 木原貴行
  • Organizer: RIMS研究集会：証明論・計算論とその周辺 2014
  • Place of Presentation: 京都大学数理解析研究所，京都府 京都市
  • Year and Date: 2014-12-25
• [Presentation] Recursion theoretic methods in topological dimension theory (2014)
  • Author(s): Takayuki Kihara
  • Organizer: The Seventh Desctiptive Set Theory in Paris
  • Place of Presentation: Institut de Mathematique de Jussieu, Paris, France
  • Year and Date: 2014-12-09
  • Invited
• [Presentation] Recursion theoretic methods in topological dimension theory (2014)
  • Author(s): Takayuki Kihara
  • Organizer: Logic Seminar in National University of Singapore
  • Place of Presentation: National University of Singapore, Singapore
  • Year and Date: 2014-11-26
• [Presentation] 無限次元ポーランド空間のσ-同相型と次数スペク トル (2014)
  • Author(s): 木原貴行
  • Organizer: 日本数学会2014年度秋季総合分科会
  • Place of Presentation: 広島大学 東広島キャンパス，広島県 東広島市
  • Year and Date: 2014-09-25
• [Presentation] Unfolded forcing constructions via Kreisel compactness (2014)
  • Author(s): Takayuki Kihara
  • Organizer: Analysis, Randomness and Applications (ARA Japan 2014)
  • Place of Presentation: 国立中央青少年交流の家，静岡県 御殿場市
  • Year and Date: 2014-09-05
• [Presentation] On the effectively Gδ-decomposable functions (2014)
  • Author(s): Takayuki Kihara
  • Organizer: Eleventh International Conference on Computability and Complexity in Analysis (CCA2014)
  • Place of Presentation: Darmstadt, Germany
  • Year and Date: 2014-07-21
• [Presentation] Scott ideals in infinite dimensional topology (2014)
  • Author(s): Takayuki Kihara
  • Organizer: Logic Seminar in Technische Universitat Darmstadt
  • Place of Presentation: Technische Universitat Darmstadt, Germany
  • Year and Date: 2014-07-11
• [Presentation] Decomposition, dimension, and degrees (2014)
  • Author(s): Takayuki Kihara
  • Organizer: Oberseminar Mathematische Logik Sommersemester 2014
  • Place of Presentation: Mathematical Institute of the University of Freiburg, Germany
  • Year and Date: 2014-07-02
• [Presentation] Triviality within and beyond hyperarithmetic (2014)
  • Author(s): Takayuki Kihara
  • Organizer: Ninth International Conference on Computability, Complexity and Randomness (CCR 2014)
  • Place of Presentation: National University of Singapore, Singapore
  • Year and Date: 2014-06-11
• [Presentation] Infinite dimensional topology for computability theorists (2014)
  • Author(s): Takayuki Kihara
  • Organizer: Matsuyama Seminar on Topology, Geometry, Set Theory and their Applications
  • Place of Presentation: Ehime University, Ehime, Matsuyama
  • Year and Date: 2014-04-15
• [Presentation] Arboreal Forcings over Admissible Sets (2014)
  • Author(s): Takayuki Kihara
  • Organizer: Correctness by Construction, CORCON 2014 Workshop
  • Place of Presentation: Genoa, Italy
  • Year and Date: 2014-03-24
• [Presentation] ランダム性の保存とnull-additivity (2014)
  • Author(s): 木原貴行
  • Organizer: 2014年度日本数学会年会
  • Place of Presentation: 学習院大学, 東京
  • Year and Date: 2014-03-15
• [Presentation] Effective methods in descriptive set theory (2014)
  • Author(s): Takayuki Kihara
  • Organizer: Computability Theory and Foundations of Mathematics 2014
  • Place of Presentation: Tokyo Institute of Technology, Tokyo, Japan
  • Year and Date: 2014-02-17
  • Invited
• [Presentation] ボレル可測関数の連続関数への分解に関するルージンの予想の周辺 (2014)
  • Author(s): 木原貴行
  • Organizer: 山陰 基礎論・解析学セミナー2014
  • Place of Presentation: 米子高専, 米子
  • Year and Date: 2014-02-01
• [Presentation] A hierarchy of the countably computable functions (2013)
  • Author(s): Takayuki Kihara
  • Organizer: Argentina-Japan-New Zealand workshop 2013
  • Place of Presentation: University of Auckland, New Zealand
  • Year and Date: 2013-12-02
• [Presentation] On a question about Martin-Löf null-additivity (2013)
  • Author(s): Takayuki Kihara
  • Organizer: Kerikeri Workshop
  • Place of Presentation: Tokerau beach, New Zealand
  • Year and Date: 2013-11-27
• [Presentation] An application of the Shore-Slaman join theorem in descriptive set theory (2013)
  • Author(s): Takayuki Kihara
  • Organizer: The 13th Asian Logic Conference
  • Place of Presentation: Guangzhou, China
  • Year and Date: 2013-09-20
  • Invited
• [Presentation] Thinly-sliced Borel analysis (2013)
  • Author(s): 木原貴行
  • Organizer: 証明論研究集会2013
  • Place of Presentation: 慶応義塾大学
  • Year and Date: 2013-08-08
• [Presentation] Lightface methods in boldface mathematics (2013)
  • Author(s): Takayuki Kihara
  • Organizer: Logic@JAIST
  • Place of Presentation: Hotel Arrowle, Ishikawa, Japan
  • Year and Date: 2013-07-23
• [Presentation] An application of classical recursion theory to descriptive set theory via computable analysis (2013)
  • Author(s): Takayuki Kihara
  • Organizer: Tenth International Conference on Computability and Complexity in Analysis
  • Place of Presentation: LORIA, Nancy, France
  • Year and Date: 2013-07-08
  • Invited
• [Presentation] Lowness for uniform Kurtz randomness (2013)
  • Author(s): Takayuki Kihara
  • Organizer: Computability in Europe 2013, The Nature of Computation : Logic, Algorithms, Applications
  • Place of Presentation: University of Milano-Bicocca, Italy
  • Year and Date: 2013-07-01
• [Presentation] A theorem on computable martingales (2013)
  • Author(s): Takayuki Kihara
  • Organizer: Continuity, Computability, Constructivity-From Logic to Algorithms (CCC 2013)
  • Place of Presentation: Swansea University, Gregynog, UK
  • Year and Date: 2013-06-26
• [Presentation] Bitwise addition and algorithmic randomness (2013)
  • Author(s): Takayuki Kihara
  • Organizer: ELC Workshop on Randomness and Probability Through Computability
  • Place of Presentation: University of Tokyo
  • Year and Date: 2013-05-15
• [Presentation] Lowness for randomness and set theory of the real line (2013)
  • Author(s): Takavuki Kihara
  • Organizer: ELC Seminar on Algorithmic Randomness
  • Place of Presentation: University of Tokyo
  • Year and Date: 2013-05-13
• [Presentation] WKL for infinite trees with few infinite paths (2013)
  • Author(s): Takayuki Kihara
  • Organizer: Workshop on Reverse Mathematics and Type Theory
  • Place of Presentation: Hoam Faculty House, Seoul, Korea
  • Year and Date: 2013-03-26
• [Presentation] 隈部-Slaman強制法によるボレル関数のω-分解問題の部分的解決 (2013)
  • Author(s): 木原貴行
  • Organizer: 日本数学会2012年度年会
  • Place of Presentation: 京都大学
  • Year and Date: 2013-03-21
• [Presentation] An application of Turing Degree Theory to the ω-decomposability problem on Borel functions (2013)
  • Author(s): 木原貴行
  • Organizer: Computability Theory and Foundations of Mathe matics
  • Place of Presentation: 東京工業大学
  • Year and Date: 2013-02-18
• [Presentation] ボレル関数の分解問題への計算論の応用 (2013)
  • Author(s): 木原貴行
  • Organizer: 2012年度冬のLAシンポジウム
  • Place of Presentation: 京都大学数理解析研究所
  • Year and Date: 2013-01-30
• [Presentation] 次数の理論の実解析学への応用 (2012)
  • Author(s): 木原貴行
  • Organizer: 〓012年度数学基礎論若手の会
  • Place of Presentation: 奈良県青少年会館
  • Year and Date: 2012-11-24
• [Presentation] Strong nullness and lowness for randomness (2012)
  • Author(s): 木原貴行
  • Organizer: Fourth Workshop on Game-Theoretic Probability and Related Topics
  • Place of Presentation: 東京大学
  • Year and Date: 2012-11-13
• [Presentation] 計算可能性理論における強零集合I…細字H01集合と完全集合性… (2012)
  • Author(s): 木原貴行
  • Organizer: 日本数学会2012年度秋季分科会
  • Place of Presentation: 九州大学, 福岡
  • Year and Date: 2012-09-20
• [Presentation] 計算可能性理論における強零集合II --コルモゴロフ複雑性とトリビアル性-- (2012)
  • Author(s): 木原貴行
  • Organizer: 日本数学会2012年度秋季分科会
  • Place of Presentation: 九州大学, 福岡
  • Year and Date: 2012-09-20
• [Presentation] On some variants of the Jayne-Rogers Theorem via the Shore-Slaman Join Theorem (2012)
  • Author(s): 木原貴行
  • Organizer: SLACS 2012
  • Place of Presentation: 京都産業大学
  • Year and Date: 2012-09-16
• [Presentation] Low for random=strong measure zero (2012)
  • Author(s): 木原貴行
  • Organizer: R IMS研究集会, 証明論と複雑性
  • Place of Presentation: 京都大学数理解析研究
  • Year and Date: 2012-09-13
• [Presentation] 不完全性定理から零点の幾何学へ (2012)
  • Author(s): 木原貴行
  • Organizer: 夏のLAシンポジウム2012
  • Place of Presentation: 天橋立宮津ロイヤルホテル
  • Year and Date: 2012-07-19
• [Presentation] A hierarchy of immunity and density for sets of reals (2012)
  • Author(s): Takayuki Kihara
  • Organizer: Computability in Europe 2012, "How the World Computes
  • Place of Presentation: University of Cambridge, UK
  • Year and Date: 2012-06-19
• [Presentation] On the computability relative to non-constructive principles (2012)
  • Author(s): Takayuki Kihara
  • Organizer: 2nd Workshop on Continuity, Computability, Constructivity -- From Logic to Algorithms --
  • Place of Presentation: Universitaet Trier, Germany
  • Year and Date: 2012-05-29
• [Remarks] 木原貴行 - 研究者 - researchmap
  • URL: http://researchmap.jp/kihara/
• [Remarks]
  • URL: http://researchmap.jp/kihara/
• [Remarks]
  • URL: http://researchmap.jp/kihara/