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MATHEMATICAL LOGIC AND ITS APPLICATION TO COMPUTATIONAL COMPLEXITY

Research Project

Project/Area Number 12640115
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field General mathematics (including Probability theory/Statistical mathematics)
Research InstitutionUNIVERSITY OF NAGOYA

Principal Investigator

YASUMOTO Masahiro  NAGOYA UNIV.GRADUATE SCHOO OF HUMAN INFORMATICS PROF., 大学院・人間情報学研究科, 教授 (10144114)

Co-Investigator(Kenkyū-buntansha) EDA Katsuya  WASEDA UNIV.GRADUATE SCHOOL OF SCIENCE AND ENGINEERING PROF., 理工学部, 教授 (90015826)
TSUKIJI Tatsuie  NAGOYA UNIV.GRADUATE SCOOL OF HUMAN INFORMATICS RES.ASSO., 大学院・人間情報学研究科, 助手 (70291961)
MATSUBARA Yo  NAGOYA UNIV.GRADUATE SCHOOL OF HUMAN INFORMATICS ASSO.PROF., 大学院・人間情報学研究科, 助教授 (30242788)
OZAWA Masanao  TOHOKU UNIV.GRADUATE SCHOOL OF INFORMATION SCIENCES PROF., 情報科学研究科, 教授 (40126313)
Project Period (FY) 2000 – 2002
Project Status Completed (Fiscal Year 2002)
Budget Amount *help
¥3,300,000 (Direct Cost: ¥3,300,000)
Fiscal Year 2002: ¥1,300,000 (Direct Cost: ¥1,300,000)
Fiscal Year 2001: ¥1,100,000 (Direct Cost: ¥1,100,000)
Fiscal Year 2000: ¥900,000 (Direct Cost: ¥900,000)
Keywordsbounded arithmetic / computational complexity / nonstandard models / ブール値モデル / 多項式時間計算量
Research Abstract

Let $N$ be a nonstandard model of $S_2$. A subset $\a$ of $N$ is called an oracle if $ (N,\a) $ satisfies $S_2 (\a) $. In this reseach we are concerned with bounded oracles $\a$ of $N$ satisfying $P^\a=NP^\a$. We proved that the existence of such oracles implies many interesting results about separations of axioms in bounded arithmetic.
Let $n\in N$ and $M=PTC (n, \a) $ I.e.$M$ be the polynomial time closure of $\ {n\} $ with the oracle $\a$. Then it is known that $M$ is a model of $T_2^0$. Assume that there exists a bounded oracle $\a$ such that $N$ satisfies $P^\a=NP^\a$. Then $M$ satisfies Axioms $S_2$ and we proved that $M$ has no endextension satisfying $R_2^1$. This implies that $U_2^1$ is not a conservative extension of $S_2 (\a) $. In paticular, if there exists a model $N$ of $S_2$ such that $P=NP$ holds in $N$, then there is a first order sentence which is provable in $U^1_2$ but not in $S_2$. It is believed that $P\not=NP$ but there may exist a nonstandard model $N$ of $S_2$ satisfying $P=NP$.

Report

(4 results)
  • 2002 Annual Research Report   Final Research Report Summary
  • 2001 Annual Research Report
  • 2000 Annual Research Report
  • Research Products

    (12 results)

All Other

All Publications (12 results)

  • [Publications] M.Yasumoto: "Endextensions in bounded arithmetic and computational complexity"Preprint Series in Mathematical Sciences Nagoya University. 2001-5. 1-7 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] S.Aida, R.Schuler, T.Tsukiji, O.Watanabe: "The Difference between Polynomial-Time Many-One and Truth-Table Reducibilities on Distributional Problems"Theory of Computing Systems. 35・4. 449-463 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Y.Matsubara: "Stationary preserving ideal over £_κλ"Journal of the Mathematical Society of Japan. (to appear).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] M.Yasumoto: "Endextensions in bounded arithmetic and computational complexity"Preprint Series in Mathmatical Sciences Nagoya University 2001-5. 1-7 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Y.Matsubara: "Stationary preserving ideals over Lκλ"to appear in J.of Mathematical Sosiety of Japan.

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] S.Aida, R.Schuler, T.Tsukiji and O.Watanabe: "The difference between Polynomial-Time Many-One and Truth-Table Reducibilities on Distributational Problems"Theory of Computing Systems 35. 449-463 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Y.Matsubara: "Stationary preserving ideal over Lκλ"Journal of the Mathematical Society of Japan. (to appear).

    • Related Report
      2002 Annual Research Report
  • [Publications] S.Aida: "The Difference between Polynomial-Time Many-One and Truth-Table Reducibilities on Distributional Problems"Theory Comput. Systems(2002). 35・4. 449-463 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] M. Yasumoto: "Endextensions in bounded arithmetic and computational complexity"Preprint Series in Mathmatical Sciences Nagoya University. 2001-5. 1-7 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] T. Tsukiji: "A limit law for outputs in random recursive circuits"Algorithmica. 31. 403-412 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] M.Ozawa: "Entanglement Measures and Hilbert-Schmidt Distance"Phys.Lett.. A.268. 158-160 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] T.Tsukiji: "On the difference between polynomial-time many-one and truth-table reducibility on distributional problems"Electronic Collq. on Computational Complexity. 81. 1-14 (2000)

    • Related Report
      2000 Annual Research Report

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Published: 2000-04-01   Modified: 2016-04-21  

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