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Computability and Complexity in Constructive Mathematics

Research Project

Project/Area Number 15500005
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Fundamental theory of informatics
Research InstitutionJapan Advanced Institute of Science and Technology

Principal Investigator

ISHIHARA Hajime  Japan Advanced Institute of Science and Technology, School of Information Science, Associate Professor, 情報科学研究科, 助教授 (10211046)

Project Period (FY) 2003 – 2006
Project Status Completed (Fiscal Year 2006)
Budget Amount *help
¥3,600,000 (Direct Cost: ¥3,600,000)
Fiscal Year 2006: ¥900,000 (Direct Cost: ¥900,000)
Fiscal Year 2005: ¥900,000 (Direct Cost: ¥900,000)
Fiscal Year 2004: ¥900,000 (Direct Cost: ¥900,000)
Fiscal Year 2003: ¥900,000 (Direct Cost: ¥900,000)
KeywordsConstructive Mathematics / Computability / Computational Complexity / Reverse Mathematics / 国際研究者交流 / ドイツ:アメリカ:ニュージー
Research Abstract

From 2003 to 2006, for 4 years, we have done a research on computability and complexity in constructive mathematics. During the research, we have got some important and better understandings on the subject. One of the most important understanding is that computability and complexity in constructive mathematics can be dealt with better within a more general framework of constructive reverse mathematics. Moreover, progress in constructive mathematics, such as constructive set theory and topology in constructive mathematics, has produced new problems in computability and complexity in constructive mathematics, and in constructive reverse mathematics.
In this research project, we have proposed a new framework of constructive reverse mathematics. We have investigated relationship between Brouwer's fan theorem and weak Koenig's lemma, with Dr. Josef Berger, and computability in these theorems. With Professor Peter Aczel, Dr. Laura Crosilla, Professor Erik Palmgren, and Associate Professor Peter Schuster, we have dealt with a problem of constructive reverse mathematics in the constructive Zermelo-Fraenkel set theory. Concerning topology in constructive mathematics, we have worked on constructions of quotient topologies in constructive set theory and type theory, with Professor Erik Palmgren, and on quasi-apartness and neighbourhood spaces, with Professor Ray Mines, Associate Professor Peter Schuster and Dr. Luminita Vita. Furthermore, we have treated computational complexity in constructive theory of real numbers and the constructive intermediate value theorem, and a constructive version of Banach's inverse mapping theorem in F-spaces as an application of Baire's theorem.
Further research project is putting research in constructive reverse mathematics forward with progress in constructive mathematics such as constructive set theory and constructive topology.

Report

(5 results)
  • 2006 Annual Research Report   Final Research Report Summary
  • 2005 Annual Research Report
  • 2004 Annual Research Report
  • 2003 Annual Research Report
  • Research Products

    (46 results)

All 2007 2006 2005 2004 Other

All Journal Article (46 results)

  • [Journal Article] A Piano Duo Support System for Parents to Lead Children to Practice Musical Performances2007

    • Author(s)
      Chika Oshima, Kazushi Nishimoto, Norihiro Hagita
    • Journal Title

      ACM Transactions on Multimedia Computing, Communications and Applications (ACM TOMCCAP) Vol.3, No.2

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] A Voice-to-MIDI System for Singing Melodies with Lyrics2007

    • Author(s)
      Naoki Ito, Kazushi Nishimoto
    • Journal Title

      Proc. International Conference on Advances in Computer Entertainment Technology 2007

      Pages: 13-15

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] A Method of Voice-to-MIDI for Singing with Lyrics2007

    • Author(s)
      Naoki Itou, Kazushi Nishimoto
    • Journal Title

      Proc. Interaction 2007, IPSJ Vol.2007, No.4

      Pages: 71-72

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Quotient topologies in constructive set theory and type theory2006

    • Author(s)
      Hajime Ishihara
    • Journal Title

      Ann. Pure Appl. Logic 141

      Pages: 257-265

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Quasi-apartness and neighbourhood spaces2006

    • Author(s)
      Hajime Ishihara
    • Journal Title

      Ann. Pure Appl. Logic 141

      Pages: 296-306

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Weak Koenig's lemma implies Brouwer's fan theorem : a direct proof2006

    • Author(s)
      Hajime Ishihara
    • Journal Title

      Notre Dame J. Formal Logic 47

      Pages: 249-252

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Reverse mathematics in Bishop's constructive mathematics2006

    • Author(s)
      Hajime Ishiahra
    • Journal Title

      Philosophia Scientiae, Cahier special 6

      Pages: 43-59

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Binary refinement implies discrete exponentiation2006

    • Author(s)
      Peter Aczel
    • Journal Title

      Studia Logica 84

      Pages: 361-368

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] A constructive Banach's inverse mapping theorem in F-spaces2006

    • Author(s)
      Hajime Ishihara
    • Journal Title

      New Zealand J. Math. 35

      Pages: 183-188

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Quotient topologies in constructive set theory and type theory2006

    • Author(s)
      H.Ishihara, E.Palmgren
    • Journal Title

      Ann. PureAppl. Logic 141

      Pages: 257-265

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Quasi-apartness and neighbourhood spaces2006

    • Author(s)
      H.Ishihara, R.Mines, P.Schuster, L.Vita
    • Journal Title

      Ann. PureAppl. Logic 141

      Pages: 296-306

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Weak Koenig's lemma implies Brouwer's fan theorem2006

    • Author(s)
      H.Ishihara
    • Journal Title

      Notre Dame J. Formal Logic 47

      Pages: 249-252

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Reverse mathematics in Bishop's constructive mathematics2006

    • Author(s)
      H.Ishihara
    • Journal Title

      Philosophia Scientiae, Cahier special 6

      Pages: 43-59

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Binary refinement implies discrete exponentiation2006

    • Author(s)
      P.Aczel, L.Crosilla, H.Ishihara, E.Palmgren, P.Schuster
    • Journal Title

      Studia Logica 84

      Pages: 361-368

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Annual Research Report 2006 Final Research Report Summary
  • [Journal Article] A constructive Banach's inverse mapping theorem in F-spaces2006

    • Author(s)
      H.Ishihara, L.Vita
    • Journal Title

      New Zealand J. Math. 35

      Pages: 183-188

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Analyses on Effects and Characteristics of Key-release Velocity in a Piano Performance2006

    • Author(s)
      Chika Oshima, Kauzshi Nishimoto, Akinori Abe
    • Journal Title

      Journal of Information Processing Society Japan Vol. 47, No. 5

      Pages: 1546-1557

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] The Phantom of The Piano : A Piano Education Support System that Encourages to Study for Learner-self2006

    • Author(s)
      Naoto Hikawa, Chika Oshima, Kazushi Nishimoto, Masahide Naemura
    • Journal Title

      Proc. Interaction 2006, IPSJ Vol.2006, No.4

      Pages: 69-70

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] A discussion on characteristics of piano performance that appear in "Key-release" actions2006

    • Author(s)
      Chika Oshima, Akinori Abe, Kazushi Nishimoto
    • Journal Title

      Proc. SIG-LSE-A503, JSAI

      Pages: 135-142

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Toward Developing of Gradual Musical Scores that Allow People to Challenge Difficult Pieces2006

    • Author(s)
      Chika Oshima, Naoki Itou, Kazushi Nishimoto, Masahide Naemura
    • Journal Title

      IPSJ SIG Technical Report 2006-EC-3 Vol.2006, No.24

      Pages: 57-64

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] How a Piano Practice Support System Should Present Model Performances : A Pilot Study2006

    • Author(s)
      Chika Oshima, Naoto Hikawa, Kazushi Nishimoto, Naomi Inoue
    • Journal Title

      IPSJ SIG Technical Report, 2006-HI-119 Vol.2006, No.72

      Pages: 71-78

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] A Voice-to-MIDI Pitch Input Method with Concurrently Using Rhythm-Tapping2006

    • Author(s)
      Naoki Itou, Kazushi Nishimoto
    • Journal Title

      IPSJ SIG Technical Report, 2006-EC-5 Vol. 2006, No. 134

      Pages: 43-48

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Quotient topologies in constructive set theory and type theory2006

    • Author(s)
      H.Ishihara, E.Palmgren
    • Journal Title

      Ann. Pure Appl. Logic 141

      Pages: 257-265

    • Related Report
      2006 Annual Research Report
  • [Journal Article] Quasi-apartness and neighbourhood spaces2006

    • Author(s)
      H.Ishihara, R.Mines, P.Schuster, L.Vita
    • Journal Title

      Ann. Pure Appl. Logic 141

      Pages: 296-306

    • Related Report
      2006 Annual Research Report
  • [Journal Article] Weak Koenig lemma implies Brouwer' s fan theorem : a direct proof2006

    • Author(s)
      H.Ishihara
    • Journal Title

      Notre Dame J. Formal Logic 47

      Pages: 249-252

    • Related Report
      2006 Annual Research Report
  • [Journal Article] Reverse mathematics in Bishop' s constructive mathematics2006

    • Author(s)
      H.Ishihara
    • Journal Title

      Philosophia Scientiae, Cahier special 6

      Pages: 43-59

    • Related Report
      2006 Annual Research Report
  • [Journal Article] A constructive Banach' s inverse mapping theorem in F-spaces2006

    • Author(s)
      H.Ishihara, L.Vita
    • Journal Title

      New Zealand J. Math. 35

      Pages: 183-188

    • Related Report
      2006 Annual Research Report
  • [Journal Article] Constructive reverse mathematics : compactness properties2005

    • Author(s)
      Hajime Ishihara
    • Journal Title

      Oxford Logic Guides 48

      Pages: 245-267

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Brouwer's fan theorem and unique existence in constructive analysis2005

    • Author(s)
      Josef Berger
    • Journal Title

      MLQ Math. Logic Q. 51

      Pages: 360-364

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] On constructive completions2005

    • Author(s)
      Laura Crosilla
    • Journal Title

      J. Symbolic Logic 70

      Pages: 969-978

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Constructive reverse mathematics : compactness properties2005

    • Author(s)
      H.Ishihara
    • Journal Title

      In : L.Crosilla and P. Schuster eds., From Sets and Types to Analysis and Topology, Oxford Univ. Press

      Pages: 245-267

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Brouwer's fan theorem and unique existence in constructive analysis2005

    • Author(s)
      J.Berger, H.Ishihara
    • Journal Title

      MLQ Math. Log. Q. 51

      Pages: 360-364

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] On constructing completions2005

    • Author(s)
      L.Crosilla, H.Ishihara, P.Schuster
    • Journal Title

      J. Symbolic Logic 70

      Pages: 969-978

    • NAID

      120001746752

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] A Piano Duo Performance Support System to Motivate Children's Practice at Home2005

    • Author(s)
      Chika Oshima, Kazushi Nishimoto, Masami Suzuki
    • Journal Title

      Journal of Information Processing Society Japan Vol.46, No.1

      Pages: 157-171

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Lesson Support Toward Creative Musical Performance Based on Analysis of Pupil's Musical Comprehension and Acquirement of Skills for Piano Performance2005

    • Author(s)
      Chika Oshima, Kazushi Nishimoto, Masami Suzuki
    • Journal Title

      Journal of Japan Creativity Society Vol.8

      Pages: 21-35

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] A Tangible Composition System for Promoting Children to Create Music by Reflectively Manipulating Notes2005

    • Author(s)
      Mao Makino, Chika Oshima, Rodney Berry, Naoto Hikawa, Kazushi Nishimoto, Masami Suzuki, Norihiro Hagita
    • Journal Title

      Journal of Japan Society for Fuzzy Theory and Intelligent Informatics Vol.17, No.2

      Pages: 28-38

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] What are "the Essences of Music" ?2005

    • Author(s)
      Homei Miyashita, Kazushi Nishimoto
    • Journal Title

      Transactions of Virtual Reality Society Japan Vol. 10, No. 1

      Pages: 11-20

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] The expansion of "score" as an inspiring interface for musical performers2005

    • Author(s)
      Homei Miyashita, Kazushi Nishimoto
    • Journal Title

      Journal of Human Interface Society Vol. 7, No. 2

      Pages: 37-42

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] What are the "Essences of Music"?--A Case Study on Japanese Audience2005

    • Author(s)
      Homei Miyashita, Kazushi Nishimoto
    • Journal Title

      Proc. of Int'l Conf. on Sound and Music Computing '05 (SMC05), CD-ROM

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Constructive reverse mathematics : compactness properties2005

    • Author(s)
      Hajime Ishihara
    • Journal Title

      From Sets and Types to Analysis and Topology (Oxford Univ.Press)

      Pages: 245-267

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Brouwer's fan theorem and unique existence in constructive analysis2005

    • Author(s)
      Josef Bergen
    • Journal Title

      MLQ Math.Log.Q. 51

      Pages: 360-364

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Family Ensemble : A Collaborative Musical Edutainment System for Children and Parents2004

    • Author(s)
      Chika Oshima, Kazushi Nishimoto, Masami Suzuki
    • Journal Title

      Proceedings of the ACM Multimedia 2004 International Conference

      Pages: 556-563

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Improvisation Support using Thermoscore-Display2004

    • Author(s)
      Homei Miyashita, Kazushi Nishimoto
    • Journal Title

      Proceedings of the International Symposium on "INTERACTION : Systems, Practice and Theory"

      Pages: 193-208

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Relativization of real numbers to a uniferse

    • Author(s)
      Hajime Ishihara
    • Journal Title

      Synthese Library (印刷中)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Unique existence and computability in constructive reverse mathematics

    • Author(s)
      Hajime Ishihara
    • Journal Title

      Lect. Notes Comput. Sci. (印刷中)

    • NAID

      120001063235

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Relativization of real numbers to a universe'

    • Author(s)
      H.Ishihara
    • Journal Title

      In : S. Lindestroem, E. Palmgren, K. Segerberg and V. Stoltenerg-Hansen eds., Logicism, Intuitionism, and Formalism-What has become of them?, Synthese Library, to appear

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Unique existence and computability in constructive reverse mathematics

    • Author(s)
      H.Ishihara
    • Journal Title

      S.B. Cooper, B. Loewe and A. Sorbi eds., Computation and Logic in the Real World, Lect. Notes Comput. Sci., to appear

    • NAID

      120001063235

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary

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Published: 2003-04-01   Modified: 2016-04-21  

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