2002 Fiscal Year Final Research Report Summary
MATHEMATICAL LOGIC AND ITS APPLICATION TO COMPUTATIONAL COMPLEXITY
| Project/Area Number |
12640115
|
|---|
| Research Category |
Grant-in-Aid for Scientific Research (C)
|
| Allocation Type | Single-year Grants |
|---|
| Section | 一般 |
|---|
| Research Field |
General mathematics (including Probability theory/Statistical mathematics)
|
|---|
| Research Institution | UNIVERSITY 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
|
| Keywords | bounded arithmetic / computational complexity / nonstandard models / ブール値モデル |
|---|
| 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 $P=NP$.
|
