• Search Research Projects
  • Search Researchers
  • How to Use
  1. Back to previous page

Categorical and Higher-Categorical Approaches to Duality and Semantics across Mathematics, Physics, and Information

Research Project

Project/Area Number 17K14231
Research Category

Grant-in-Aid for Young Scientists (B)

Allocation TypeMulti-year Fund
Research Field Foundations of mathematics/Applied mathematics
Research InstitutionAdvanced Telecommunications Research Institute International (2020-2021)
Kyoto University (2017-2018)

Principal Investigator

Maruyama Yoshihiro  株式会社国際電気通信基礎技術研究所, 佐藤匠徳特別研究所, 研究員 (20761290)

Project Period (FY) 2021-03-01 – 2022-03-31
Project Status Completed (Fiscal Year 2021)
Budget Amount *help
¥4,160,000 (Direct Cost: ¥3,200,000、Indirect Cost: ¥960,000)
Fiscal Year 2020: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2019: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2018: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2017: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Keywords圏論的双対性 / 圏論的論理 / 圏論的人工知能 / 圏論的機械学習 / 圏論的認知科学 / 圏論的言語学 / 圏論的量子力学 / 圏論的統一科学 / 圏論的意味論 / 圏論的普遍代数 / 圏論的普遍論理 / 圏論的融合AI / 論理推論と統計推論の圏論的融合 / 圏論的ロボティクス / 数学基礎論 / 量子基礎論 / プログラム意味論 / 記号的AIと統計的AIの双対性 / 記号的AIと統計的AIの圏論的統合 / 証明論的意味論 / 情報物理学 / 人工知能の限界 / フレーム問題 / ルーカス・ペンローズの議論 / 科学基礎論としての圏論 / 量子力学の圏論的基礎
Outline of Research at the Start

本研究では層理論や代数幾何のスキーム論的構造を利用することで、数学・物理・情報を横断して現れる非可換代数や部分構造論理に対する圏論的双対性の理論を構築する。同時に、殆ど全ての標準的な論理構造を包含する、一階の部分構造論理と高階の部分構造論理に対する圏論的意味論の研究を行う。その上でゲーデル翻訳・ジラール翻訳・バーズ翻訳などの論理間の翻訳関係の構造を、圏論的論理のハイパードクトリン上に定義されたローヴェア・ティアニー位相として定式化する事で、論理間の翻訳の圏論的位相理論を確立する。これらの研究は、圏論的人工知能・圏論的認知科学・圏論的ロボティクスの数学的基礎の研究に繋がってゆくことが期待される。

Outline of Final Research Achievements

We have developed a universal categorical duality theorem encompassing all Stone dualities for coherent spaces and applied it to computer science, by which we won a paper prize at an international conference on computational intelligence. We have discovered a novel infinitary Stone duality theorem which replaces the known infinitary Stone duality, which was published in a major journal in algebra, and further developed a general theory of categorical dualities which encompasses those infinitary Stone dualities, which was published in a top journal in category theory. To extend topos theory for arbitrary logical systems, we have generalised and applied tripos theory, thus having discovered a general theory of categorical logics, which was published in a top journal in mathematical logic.

Academic Significance and Societal Importance of the Research Achievements

圏論的双対性の普遍理論を構築し、圏論のトップジャーナル等において出版すると共に、計算知能に関する国際会議で論文賞を受賞した。圏論的論理の普遍理論を構築し、数理論理学のトップジャーナル等において出版し、トライポス理論を通じたトポス理論の拡張に成功した。これらは、純粋数学としての革新と同時に、人工知能・機械学習や量子物理・量子情報への応用を企図したものであり、従来の枠組みとは異なる全く新しい角度から、圏論に基づく人工知能と量子情報の新地平を切り拓くものとして非常に大きな可能性を秘めたものである。専門的な国際誌と同時に、成果の社会への普及のため数学セミナーや現代思想など一般向け雑誌にも多数寄稿した。

Report

(4 results)
  • 2021 Annual Research Report   Final Research Report ( PDF )
  • 2018 Research-status Report
  • 2017 Research-status Report
  • Research Products

    (12 results)

All 2022 2021 2019 2018

All Journal Article (9 results) (of which Peer Reviewed: 9 results,  Open Access: 1 results) Presentation (3 results) (of which Int'l Joint Research: 3 results,  Invited: 2 results)

  • [Journal Article] Chu Duality Theory and Coalgebraic Representation of Quantum Symmetries2022

    • Author(s)
      Y. Maruyama
    • Journal Title

      Journal of Pure and Applied Algebra

      Volume: 226 Pages: 106960-106960

    • Related Report
      2021 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Fibred Algebraic Semantics for a Variety of Non-Classical First-Order Logics and Topological Logical Translation2021

    • Author(s)
      Y. Maruyama
    • Journal Title

      The Journal of Symbolic Logic

      Volume: 86 Pages: 1189-1213

    • Related Report
      2021 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Higher-Order Fuzzy Logics and their Categorical Semantics: Higher-Order Linear Completeness and Baaz Translation via Substructural Tripos Theory2021

    • Author(s)
      Y. Maruyama
    • Journal Title

      Proceedings of IEEE-FUZZ

      Volume: -

    • Related Report
      2021 Annual Research Report
    • Peer Reviewed
  • [Journal Article] A Reasoning System for Fuzzy Distributed Knowledge Representation in Multi-Agent Systems2021

    • Author(s)
      Y. Maruyama
    • Journal Title

      Proceedings of IEEE-FUZZ

      Volume: -

    • Related Report
      2021 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Harmony, Higher-Order Rules, and the Curry-Howard-Lambek Correspondence2019

    • Author(s)
      Y. Maruyama
    • Journal Title

      Proceedings of Proof-Theoretic Semantics Conference

      Volume: vol. 3

    • Related Report
      2018 Research-status Report
    • Peer Reviewed / Open Access
  • [Journal Article] Pancomputationalism and Data Science: From Symbolic to Statistical AI, and to Quantum AI2018

    • Author(s)
      Y. Maruyama
    • Journal Title

      Springer SAPERE

      Volume: 44 Pages: 207-211

    • Related Report
      2018 Research-status Report
    • Peer Reviewed
  • [Journal Article] The Frame Problem, Gödelian Incompleteness, and the Lucas-Penrose Argument: A Structural Analysis of Arguments about Limits of AI, and its Physical Consequence2018

    • Author(s)
      Y. Maruyama
    • Journal Title

      Springer SAPERE

      Volume: 44 Pages: 194-206

    • Related Report
      2018 Research-status Report
    • Peer Reviewed
  • [Journal Article] The Frame Problem, Goedelian Incompleteness, and the Lucas-Penrose Argument: A Structural Analysis of Arguments about Limits of AI, and its Physical and Metaphysical Consequences2018

    • Author(s)
      Y. Maruyama
    • Journal Title

      Springer SAPERE Series

      Volume: 印刷中

    • Related Report
      2017 Research-status Report
    • Peer Reviewed
  • [Journal Article] Quantum Pancomputationalism and Statistical Data Science: From Symbolic to Statistical AI, and to Quantum AI2018

    • Author(s)
      Y. Maruyama
    • Journal Title

      Springer SAPERE Series

      Volume: 印刷中

    • Related Report
      2017 Research-status Report
    • Peer Reviewed
  • [Presentation] Harmony, Higher-Order Rules, and the Curry-Howard-Lambek Correspondence2019

    • Author(s)
      Y. Maruyama
    • Organizer
      Proof-Theoretic Semantics Conference
    • Related Report
      2018 Research-status Report
    • Int'l Joint Research
  • [Presentation] A structural perspective on quantum cognition: From Penrose's argument to No-Go theorems in cognitive science2018

    • Author(s)
      Y. Maruyama
    • Organizer
      Quantum Contextuality in Quantum Mechanics and Beyond
    • Related Report
      2018 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Contextuality of Reason, Contextuality of Reality, and the Nature of Probabilities Involved2018

    • Author(s)
      Y. Maruyama
    • Organizer
      Purdue Wiener Memorial Lectures
    • Related Report
      2018 Research-status Report
    • Int'l Joint Research / Invited

URL: 

Published: 2017-04-28   Modified: 2023-01-30  

Information User Guide FAQ News Terms of Use Attribution of KAKENHI

Powered by NII kakenhi