Research activity information

• GENERALIZATIONS OF GÖDEL'S INCOMPLETENESS THEOREMS FOR Σn-DEFINABLE THEORIES OF ARITHMETIC

  Makoto Kikuchi, Taishi Kurahashi

  Cambridge University Press, Dec. 2017, Review of Symbolic Logic, 10(4) (4), 603 - 616, English

  [Refereed]

  Scientific journal


• ILLUSORY MODELS OF PEANO ARITHMETIC

  Makoto Kikuchi, Taishi Kurahashi

  Sep. 2016, JOURNAL OF SYMBOLIC LOGIC, 81(3) (3), 1163 - 1175, English

  [Refereed]

  Scientific journal


• Liar-type Paradoxes and the Incompleteness Phenomena

  Makoto Kikuchi, Taishi Kurahashi

  Aug. 2016, JOURNAL OF PHILOSOPHICAL LOGIC, 45(4) (4), 381 - 398, English

  [Refereed]

  Scientific journal


• Preface to Special Section : 60 Years of Mathematics and Logic(60 Years of Mathematics and Logic)

  KIKUCHI Makoto

  Japan Association for Philosophy of Science, 2016, Journal of the Japan Association for Philosophy of Science, 43(1) (1), 1 - 2, Japanese


• Universal Rosser Predicates

  KIKUCHI MAKOTO, KURAHASHI TAISHI

  2016, Journal of Symbolic Logic, 82(1) (1), 292 - 302, English

  [Refereed]

  Scientific journal


• Set-Theoretic Mereology

  HAMKINS, J.D, KIKUCHI MAKOTO

  2016, Logic and Logical Philosophy, 25(3) (3), 285 - 391, English

  [Refereed]

  Scientific journal


• Formalizations of the Concept of Proofs and Interpretations of Quantifiers

  KIKUCHI MAKOTO

  The Philosophy of Science Society, Japan, Dec. 2015, Philosophy of Science: Journal of the Philosophy of Science Society, Japan, 48(2) (2), 33 - 48, Japanese

  [Refereed][Invited]

  Scientific journal


• On proofs of the incompleteness theorems based on Berry's paradox by Vopenka, Chaitin, and Boolos

  Makoto Kikuchi, Taishi Kurahashi, Hiroshi Sakai

  Aug. 2012, MATHEMATICAL LOGIC QUARTERLY, 58(4-5) (4-5), 307 - 316, English

  [Refereed]

  Scientific journal


• Kolmogorov complexity and characteristic constants of formal theories of arithmetic

  Shingo Ibuka, Makoto Kikuchi, Hirotaka Kikyo

  Oct. 2011, MATHEMATICAL LOGIC QUARTERLY, 57(5) (5), 470 - 473, English

  [Refereed]

  Scientific journal


• Three Short Stories around Gödel's Incompleteness Theorems

  Makoto Kikuchi, Taishi Kurahashi

  We describe three short stories around Gödel's incompleteness theorems. Firstly, we show that the second incompleteness theorem is derivable by formalizing Yablo's paradox. Then, we prove a version of the incompleteness theorem which is not formalizable. Finally, we discuss the theorems of arithmetic in nonstandard models of arithmetic and the definability of truth in models of arithmetic within other models.

  Japan Association for Philosophy of Science, May 2011, Journal of the Japan Association for Philosophy of Science, 38巻2号pp.27-32(2) (2), 75 - 80, Japanese

  [Refereed]

  Scientific journal


• On Mathematical Aspects of Varieties of Cognitive Structures of Class-Inclusion Reasoning

  Mizuho Mishima, Makoto Kikuchi

  We use varieties of cognitive structures in cognitive activities. Classes and collections are two kinds of cognitive structures for class-inclusion reasoning introduced by Markman to analyze the Inhelder and Piaget's class-inclusion problem. In this paper, we shall review investigations on classes and collections in developmental psychology, and propose mathematical definitions and models of classes and collections. We shall show also interpretations of these two models.

  Japan Association for Philosophy of Science, Mar. 2009, Journal of the Japan Association for Philosophy of Science, 111, 9-13(2) (2), 53 - 57, Japanese

  [Refereed]

  Scientific journal


• A Mathematical Model of Deductive and Non-Deductive Inferences

  Makoto Kikuchi

  Mar. 2009, Annals of the Japan Association for Philosophy of Science, 17, 1-11, English

  [Refereed]

  Scientific journal


• Two Phases of Class-Inclusion Learning

  Mizuho Mishima, Makoto Kikuchi

  Jul. 2008, Proceedings of the International Conference of Cognitive Science 2008, 206-211, English

  [Refereed]

  Scientific journal


• On characteristic constants of theories defined by Kolmogorov complexity

  Shingo Ibuka, Makoto Kikuchi, Hirotaka Kikyo

  2008, LOGIC, LANGUAGE, INFORMATION AND COMPUTATION, 5110, 218 - 225, English

  [Refereed]

  International conference proceedings


• On the Hierarchical Structure of Abstract Concepts in General Design Theory

  Makoto Kikuchi

  2006, Proceedings of the 6th International Workshop of Emergent Synthesis, 261-266, English

  [Refereed]

  Scientific journal


• Mathematical models of category-based induction

  Mizuho Mishima, Makoto Kikuchi

  2006, DISCOVERY SCIENCE, PROCEEDINGS, 4265, 327 - 331, English

  [Refereed]

  Scientific journal


• Study on Applications of Linguistics to Design Theory (2nd Report) - Functions as Meanings and Simon's Definition of Artifacts -

  Makoto Kikuchi, Ichiro Nagasaka

  The aim of design theory is to investigate the domain-independent natures of artifacts and design activities. The notion of function is important in design theory since design specifications are described in terms of functions, but it is not easy to define functions because the notion is closely related to our subjective intentions. In this report, we shall analyze the paradox of General Design Theory in in order to show the importance of the concept of function in a design theory, and we show that the concept has a correspondence with the notion of meanings of natural language expressions. Then, we give a formulation of the notion of function by using Situation Semantics which corresponds to Simion's definition of an artifact as an interface between its inner and outer environments. These arguments show the relationships between design theory and natural language semantics.

  The Japan Society for Precision Engineering, Jan. 2006, Journal of the Japan Society for Precision Engineering, 72(1),142-146(1) (1), 142 - 146, Japanese

  [Refereed]

  Scientific journal


• General Design Theory and Genetic Epistemology

  Mizuho Mishima, Makoto Kikuchi

  Aug. 2005, Proceedings of 15th International Conference on Engeineering Design, 260.81, English

  [Refereed]

  Scientific journal


• 言語学の設計学への応用に関する研究（第１報）-一般設計学の逆理と名詞句としての実体概念の形式化－

  菊地 誠, 長坂 一郎

  2004, 精密工学会誌, 70(3),354-358, English

  [Refereed]

  Scientific journal


• 教員方法論の二つの系譜に対応する2種類の設計論について

  三木 大史, 角田 譲, 菊池 誠

  2004, Design Symposium 講演論文集, 169-172, Japanese

  [Refereed]

  Scientific journal


• Mathematical analysis of information flows in design and education

  H Miki, Y Kakuda, M Kikuchi

  2004, Design 2004: Proceedings of the 8th International Design Conference, Vols 1-3, 未記入, 409 - 414, English

  [Refereed]

  International conference proceedings


• A Relational Theory of Interaction and Learning Based on Classifier System

  NAGASAKA Ichiro, KIKUCHI Makoto, KITAMURA M

  2004, Proceedings of 4th International ICSC Symposium on Engineering of Intelligent Systems, File No.18, English

  [Refereed]

  International conference proceedings


• Situation Theoretic Analysis of Functions for a Formal Theory of Design

  KIKUCHI Makoto, NAGASAKA Ichirou

  2003, Proceedings of 14th International Conference on Engeineering Design, File No.1448, English

  [Refereed]

  International conference proceedings


• On the Three Forms of Non-Deductive Inferences: Induction, Abduction, and Design.

  KIKUCHI Makoto, NAGASAKA Ichiro

  2003, Proc.of the 21th IASTED International Conference Applied Informatics., 357-362, English

  [Refereed]

  International conference proceedings


• On a set theory with uncertain membership relations

  S Yatabe, Y Kakuda, M Kikuchi

  2003, DESIGN AND APPLICATION OF HYBRID INTELLIGENT SYSTEMS, 104, 458 - 467, English

  [Refereed]

  International conference proceedings


• Channel Theory for User-Interactions in Constraint Satisfaction and Design

  KIKUCHI Makoto, NAGASAKA Ichiro, Banbara Mutsunori

  2003, Proceedings of the Third International Workshop on User-Interaction in Constraint Satisfaction, 56-69, English

  [Refereed]

  International conference proceedings


• A mathematical model of interactions in artifact environments

  A Kikuchi, Nagasaka, I, S Toyoda, S Kitamura

  2003, SICE 2003 ANNUAL CONFERENCE, VOLS 1-3, 2085-2090, 1614 - 1619, English

  [Refereed]

  International conference proceedings


• A Formal Analysis of Classifier System and Interface between Learning System Environment.

  NAGASAKA Ichiro, KIKUCHI Makoto, KITAMURA Shinzo

  2003, Proc.of the 21th IASTED International Conference Applied Informatics., 20-25, English

  [Refereed]

  International conference proceedings

• 1936年の奇跡：チューリング機械の誕生

  KIKUCHI MAKOTO

  Oct. 2018, 数学セミナー, (2018年11月号) (2018年11月号), 30 - 34, Japanese

  [Invited]

  Introduction commerce magazine


• ヒルベルトの第10問題

  KIKUCHI MAKOTO

  日本評論社, Sep. 2017, 数学セミナー, 2017年(10月) (10月), 19 - 23, Japanese

  [Invited]

  Introduction commerce magazine


• ルベーグ非可測集合の存在とバナッハ-タルスキの逆理

  KIKUCHI MAKOTO

  Oct. 2016, 数学セミナー, Japanese

  [Invited]

  Introduction commerce magazine


• 実数の連続性がわからない

  KIKUCHI MAKOTO

  日本評論社, May 2015, 数学セミナー, Japanese

  [Invited]

  Introduction commerce magazine


• Academic information distribution in mathematics and philosophy

  KIKUCHI Makoto

  Academic information distribution is important for researchers from the viewpoints of evaluation and promotion of research activities. In old days, researchers in our country had disadvantages of isolation about information caused by the location which is far from the center of research, but now a days, the research environments in our country have been improved significantly by means of the development of the information communication technologies and the grows of the international interchange of researchers. However, these improvements also have risk of causing problems of excessive homogeneity and competition in researches, and isolation of research disciplines. In this article, we discuss the current state and further possibility of alternations of academic information distributions in mathematics and philosophy.

  Information Science and Technology Association (INFOSTA), 2013, The journal of Information Science and Technology Association, 63(11) (11), 458 - 463, Japanese


• A Note on General Design Theory and Abstract Design Theory (Algebra, Logic and Geometry in Informatics)

  Kikuchi Makoto

  Kyoto University, May 2003, RIMS Kokyuroku, 1318, 136 - 148, Japanese


• Kolmogorov complexity and the second incompleteness theorem(Mathematical Incompleteness in Arithmetic)

  KIKUCHI MAKOTO

  Kyoto University, May 1995, RIMS Kokyuroku, 912, 33 - 42, English


• Berry's paradox and the incompleteness theorem

  KIKUCHI MAKOTO

  Kyoto University, Aug. 1993, RIMS Kokyuroku, 847, 56 - 60, Japanese

• Proof and Truth in Mathematics: Modal Logic and the Foundations of Mathematics

  SANO KATSUHIKO, KURAHASHI TAISHI, USUBA TOMOMICHI, KUROKAWA HIDENORI, KIKUCHI MAKOTO

  Joint work, Kyoritsu Shuppan, Mar. 2016, Japanese, ISBN: 4320111486

  Scholarly book


• The Incompleteness Theorems

  KIKUCHI MAKOTO

  Single work, Kyoritsu Shuppan, Oct. 2014, Japanese, This book is a comprehensive introduction to the incompleteness theorems. Basic notions and facts in mathematical logic are explained thoroughly, and the whole accounts of the proofs of the incompleteness theorems and their philosophical consequences are explicated. A brief introduction to the historical context of the theorems and further topics such as Rosser's theorem, Hilbe, ISBN: 9784320110960

  Scholarly book

• 量子論理と二つの中間層を持つオーソモジュラー束について

  KIKUCHI MAKOTO, HIYOSHI RYOTA

  日本数学会2019年度年会, Mar. 2019, Japanese, 東京工業大学, Domestic conference

  Oral presentation


• 計算の哲学：推論および物理的現象との関係の再考に向けて

  KIKUCHI MAKOTO

  日本哲学会第51回大会, Oct. 2018, Japanese, 立命館大学, Domestic conference

  Public symposium


• 幾何学の基礎に関するフレーゲとヒルベルトの論争について

  KIKUCHI MAKOTO

  日本数学会2018年度秋季総合分科会, Sep. 2018, Japanese, 岡山大学, Domestic conference

  Oral presentation


• ZFC の可算モデル上の包含関係について

  KIKUCHI MAKOTO, JOEL DAVID, HAMKINS

  日本数学会2017年度秋季総合分科会, Sep. 2017, Japanese, 山形大学, Domestic conference

  Oral presentation


• 人工知能の哲学：知能の理解と実現に挑む工学と哲学の対話

  KIKUCHI MAKOTO

  日本科学哲学会第49回大会, Nov. 2016, Japanese, Domestic conference

  [Invited]

  Nominated symposium


• 2階論理の意味論

  KIKUCHI MAKOTO

  科学基礎論学会秋の研究例会, Nov. 2016, Japanese, Domestic conference

  Public symposium


• 数学における真理の絶対性と相対性

  KIKUCHI MAKOTO

  科学基礎論学会2016年度総会と講演会, Jun. 2016, Japanese, Domestic conference

  Public symposium


• On the element-of relation and the inclusion relation in set theory

  KIKUCHI MAKOTO, JOEL DAVID, HAMKINS

  日本数学会2016年度年会, Mar. 2016, Japanese, 筑波大学, Domestic conference

  Oral presentation


• 論理の基本概念の再検討

  KIKUCHI MAKOTO

  日本科学哲学会第48回（2015年）大会, Nov. 2015, Japanese, 首都大学東京, Domestic conference

  Public symposium


• New Developments of Barwise and Seligman’s Channel Theory

  KIKUCHI MAKOTO

  科学基礎論学会秋の研究例会, Nov. 2015, Japanese, 東京大学, Domestic conference

  Public symposium


• The Incompleteness Theorems: The Bounds and Possibilities of Formalized Mathematics

  KIKUCHI MAKOTO

  日本数学会2015年度秋季総合分科会, Sep. 2015, Japanese, 京都産業大学, Domestic conference

  [Invited]

  Invited oral presentation


• 自然数概念の無際限拡張可能性

  KIKUCHI MAKOTO, KUROKAWA HIDENORI

  科学基礎論学会2015年度総会と講演会, Jun. 2015, Japanese, 北海道教育大学札幌校, Domestic conference

  Oral presentation


• Σn 定義可能な算術の不完全性定理

  KIKUCHI MAKOTO, KURAHASHI TAISHI

  日本数学会2015年度年会, Mar. 2015, Japanese, 東京都千代田区, Domestic conference

  Oral presentation


• ゲーデルの完全性定理と不完全性定理

  KIKUCHI MAKOTO

  共立出版第169回数学文献を読む会, Dec. 2014, Japanese, Domestic conference

  Public discourse


• 状況理論としてのチャンネル理論

  KIKUCHI MAKOTO

  科学基礎論学会秋の研究例会, Nov. 2014, Japanese, 東京都目黒区, Domestic conference

  Public symposium


• 形式的数学と非形式的数学

  KIKUCHI MAKOTO

  日本科学哲学会第47回大会, Nov. 2014, Japanese, Domestic conference

  [Invited]

  Nominated symposium


• 不完全性定理の構成的性質について

  KIKUCHI MAKOTO, FUCHINO SAKAE

  日本数学会2014年度秋季総合分科会, Sep. 2014, Japanese, 東広島市, Domestic conference

  Oral presentation


• 算術の超準モデルにおける定理と証明について

  KIKUCHI MAKOTO, KURAHASHI TAISHI

  日本数学会2014年度秋季総合分科会, Sep. 2014, Japanese, 東広島市, Domestic conference

  Oral presentation


• On proofs of the incompleteness theorems based on liar-type paradoxes

  KIKUCHI MAKOTO, KURAHASHI TAISHI

  2013年度日本数学会秋季総合分科会, Sep. 2013, Japanese, 愛媛大学, Domestic conference

  Oral presentation


• Notes on the incompleteness theorems

  Makoto Kikuchi, Taishi Kurahashi

  Logic Colloquium '11, Jul. 2011, English, Barcelona, Catalonia, Spain, International conference

  Oral presentation


• Berry の逆理に基づく Boolos の不完全性定理の証明について

  倉橋 大志, 酒井 拓史, 菊池 誠

  2010 日本数学会年会, Mar. 2010, Japanese, 慶應義塾大学矢上キャンパス, Domestic conference

  Oral presentation


• On Characteristic Constants of Theories Defined by Kolmogorov Complexity

  Shingo Ibuka, Hirotaka Kikyo, Makoto Kikuchi

  Fall Meeting of the Mathematical Society of Japan, Sep. 2008, Japanese, 日本数学会, Tokyo Institute of Technology, Domestic conference

  Oral presentation


• On Characteristic Constants of Theories Defined by Kolmogorov Complexity

  Shingo Ibuka, Makoto Kikuchi, Hirotaka Kikyo

  15th International Workshop, WoLLIC 2008, Jul. 2008, English, Helliot-Watt University, Edinburgh, UK, International conference

  Oral presentation


• An Introduction to General Design Theory

  Makoto Kikuchi

  Hayashibara Forum 2007: Science of Design (Workshop), Sep. 2007, English, 東京大学人工物工学研究センター・社団法人林原共済会・毎日新聞社, Kyoto, International conference

  Invited oral presentation


• 一般設計学における設計知識の数学的表現について

  菊池 誠

  スキルの科学コロキュアム, 2005, Japanese, 国際高等研究所, 国際高等研究所, Domestic conference

  Oral presentation


• 長坂一郎, 菊池誠 状況理論による機能概念の分析

  長坂 一郎, 菊池 誠

  科学基礎論学会2003年度年会, 2003, Japanese, 化学基礎論学会, 未記入, Domestic conference

  Oral presentation


• 状況理論に基づく機能概念の数学的形式化について

  菊池 誠, 長坂 一郎

  日本数学会年会, 2003, Japanese, 日本数学会, 未記入, Domestic conference

  Oral presentation


• On functions of artifacts and organisms.

  KIKUCHI Makoto

  LoRwi, 2003, English, 未記入, 未記入, Domestic conference

  Oral presentation

• 精密工学会


• 日本認知科学会


• 日本科学哲学会


• 科学基礎論学会


• Association for Symbolic Logic


• 日本数学会

• Mathematics and Philosophy of Set Theory of the Continuum

  藤田 博司, 黒川 英徳, 菊池 誠, 池田 真治

  Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research, Grant-in-Aid for Scientific Research (C), Ehime University, 01 Apr. 2020 - 31 Mar. 2024

  本研究では数学基礎論の従来の理解を問いなおす作業を進めている。2021年度もひきつづき研究代表者および3名の研究分担者(池田・菊池・黒川)がそれぞれに文献の検討をおこなった。具体的には以下のテーマを中心に研究が行なわれた。(1)ニュートンとライプニッツの無限小解析、および、その哲学的批判者たちの議論の検討（池田）(2)集合論の公理化の沿革の哲学的評価（藤田）(3)現代の数学基礎論において連続体の論理的代替物として研究されている２階の自然数論および実閉順序体の理論の現状の分析（菊池）(4)証明や計算、構成といった概念の数学的定式化についての論理学的および哲学的考察（菊池・黒川） 今後重要な意味をもつであろう思潮として、量子計算の分野から「量子力学は情報の物理学である」「計算とは量子計算のことである」といった考え方が興ってきていることに注目し、意識や連続体の研究とこうした考え方との関連性を検討を始めている。 これまでの藤田の実績をまとめたものとして、連続体の数理に密接に関連する位相数学の基礎概念を概説した書籍を藤田が出版準備中である。この書籍においては、位相数学の諸概念をあえてイデア的なものと位置づけ、図形的・直観的なイメージの世界と抽象的な論理の世界の往還をテーマとして解説している。しかしながら本研究の目的である「連続体の哲理の歴史的展望の上に立って最先端の集合論研究との対話を重ね、集合と連続体の数学と哲学の新たな可能性を見出すこと」については、なお端緒についたばかりと言わねばならない。


• 数学の哲学の新たな展開

  菊池 誠

  科学研究費補助金／基盤研究(B), Apr. 2017 - Mar. 2022, Principal investigator

  Competitive research funding


• 算術の超準モデルと不完全性定理

  菊池 誠

  科学研究費補助金／基盤研究(C), Apr. 2012 - Mar. 2015, Principal investigator

  Competitive research funding


• Emergent Design Theories in Interactive Information Fields

  KITAMURA Shinzo, TAMAKI Hisashi, NAGSAKA Ichiro, KIKUCHI Makoto, MURAO Hajime

  Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research, Grant-in-Aid for Scientific Research (B), Kobe University, 2002 - 2003

  The project aimed at clarifying and also interpreting the creativity and the comprehensive ability of human which were revealed in the process of designing artifacts, and included the following sub-themes : (1) analysis of the mathematical structure of "interactive information fields" which describe the informational flow between human and computers, (2) emergent design theories based on learning (application of reinforcement learning to such problems that the information on an environment is incomplete), and (3) emergent design theories based on interaction (application of interactive evolutionary computation to such problems that the specification is also incomplete). In the course of the sub-theme (1), first by taking classifier systems for the example, their mathematical modeling based on the situation theory and the channel theory was investigated. As for the abstract design theory, by focusing on abduction, the mathematical formulation of non-deductive reasoning was given, which clarified the relationship among induction, abduction and system design. In the sub-theme (2), the frameworks based on reinforcement learning and genetics-based machine learning were reconstructed as the mechanism of acquiring the internal model of an environment as well as seeking for a solution. Through some computational experiments for the problems of real-time scheduling, generating a walking pattern of biped robots, navigating a robot, and so on, the effectiveness of the frameworks was examined. Furthermore, in the sub-theme (3), methods of interactive evolutionary computation, which utilized the ambiguous decisions of human, were mainly considered. The computational models of a co-evolutionary genetic algorithm as well as a genetic algorithm with sexual selection were investigated, and the potential of these models was confirmed through computational experiments.


• The construction of Abstract Design Theory and its development

  KAKUDA Yuzuru, NAGASAKA Ichiro, KIKUCHI Makoto

  Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research, Grant-in-Aid for Scientific Research (C), KOBE UNIVERSITY, 2001 - 2002

  We defined directly the concept of information flow using the notion of conditional morphism which is different from the notion of infomorphism in Barwise-Seligman's Channel Theory, and showed that they are equivalent. (The proof of the completeness and soundness) The above result allows us to define a concept of an entity, an image of an entity, a function of an entity, and an attribute of an entity, and a scheme of their relations by regarding an information flow between a classification of a theory of desires or requirements and a classification of behaviors of entities and a channel covering the information flow as an idealized production and a design respectively. We introduced the concept of an uncertain classification, and developed design theory based on it, and showed results which is similar to the above, since we can't handle the incompleteness of our knowledge and hence we can't describe an design theory for a realistic case when we use only standard classifications. We formalized a set theory ZF^± which contains a notion "containing uncertainly" to consider the uncertainty fundamentally, and proved its consistency from the consistency of ZF. We describe Bloom's discussion on education with taxonomy in mathematical way, by applying Abstract Design Theory to education. We introduced the concept of a multiple classification to develop a concept of synthesis in design theory in a mathematical and rigorous way, and develop a theory of information flow about it.


• 命題論理の証明図の長さに関する研究

  菊池 誠

  日本学術振興会, 科学研究費助成事業, 奨励研究(A), 神戸大学, 1998 - 1999

  1.概要: 本研究は命題論理の証明図の長さの下限について,計算量理論と情報の理論との関係から研究をおこなうものである.命題論理の証明図の長さの研究においては不完全性定理の分析が重要であるという昨年度の研究結果に基づき,本年度は命題論理の証明図の長さと算術の公理系の不完全性の関係,および命題論理の体系と情報の理論の関係について,特にBarwiseのチャンネル理論との関連を視野に入れながら研究をおこなった. 2.不完全性定理関連: 昨年度に引き続き算術のモデルを用いたAdamowiczの新しい不完全性定理の証明の分析をおこない,その証明の弱い算術の体系への一般化を試みた.この考察により,算術のモデル論とBarwiseのチャンネル理論との関係の糸口が見い出され,この問題への新たなアプローチの可能性が明らかになった. 3.情報の理論との関連: 命題論理の体系と情報の理論の関係については,工学における設計の公理的な理論に関連して,証明図を用いた形式的な設計のプロセスの定式化が可能であることを示した.このことは田浦敏春との共著論文「A General Model of Design Sysnthesis」に発表された.この研究から,数学基礎論と設計論,および表現の理論との関係が示され,そうした理論とチャンネル理論の関係についての新たな研究の可能性が明らかになった.


• Studies on logical structures of arithmetic and computation

  TANAKA Kazuyuki, KASHIMA Ryo, HORAI Masako, AKAMA Yohji, HASEGAWA Ryu, TATSUTA Makoto

  Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research, Grant-in-Aid for Scientific Research (B), TOHOKU UNIVERSITY, 1997 - 1999

  K. Tanaka studied along the program of reverse mathematics how much of set theory is needed to prove a theorem of ordinary mathematics. By observing properties of countable models of WKL_0, he proved (jointly with T. Yamazaki and S. Simpson) new conservation results of WKL_0 over RCA_0. Y. Akama introduced an intersection typing system for combinatory logic, and showed that it is sound and complete for the class of partial combinatory algebras. M. Horai has shown that the Curry-Howard isomorphism between higher-order intuitionistic logic and calculus of constructions can be better formulated when a modified version of higher-order type theory is introduced. She also studied the lambda-representability of functions over free structures with respect to the simple type system, as well as the notion of recursive functions over free-structures. M. Tatsuta investigated realizability interpretations of monotone coinductive definition. He proved, among others, that full monotone coinductive definitions are not sound while restricted ones are. R. Hasegawa studied applications of analytic functions to theoretical computer science. R. Kashima obtained several important results on relevant logic.