2003 Fiscal Year Final Research Report Summary
Logic of Limit Computing and its Applications
| Project/Area Number |
13480084
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| Research Category |
Grant-in-Aid for Scientific Research (B)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
計算機科学
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| Research Institution | Kobe University |
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Principal Investigator |
HAYASHI Susumu Kobe University, Computers and Systems Engineering, Professor, 工学部, 教授 (40156443)
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| Co-Investigator(Kenkyū-buntansha) |
YASUGI Mariko Kyoto Sangyo University, Computer Science, Professor, 理学部, 教授 (90022277)
TAMURA Naoyuki Kobe University, Information center, Professor, 学術情報基盤センター, 教授 (60207248)
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| Project Period (FY) |
2001 – 2003
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| Keywords | Inductive inference / mathematical logic / classical proof execution |
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| Research Abstract |
The fundamental theory of Limit Computable Mathematics (LCM) were founded and many new results have been obtained in the project. The followings are the main results among them :
1. Improved realizability interpretations for LCM
2. sublearning hierarchy
3. LCM game
4. the arithmetical hierarchy
5. LCM categories
6. Computability in analysis in the light of LCM. Computability theory of discontinuous functions
7. Models of Delta-0-2 maps by means of concurrent computation
Among them, the discovery of sublearning hierarchy is quite important. Below the hierarchy of LEM (the laws of excluded middle), principles corresponding to LLPO of constructive mathematics exist and they are related to WKL (Weak Koenig Lemma). This fact has been conceived by some researchers intuitively. By establishing computational and learning theoretic meanings of this fact, we finally explored the position of WKL in the hierarchy of LCM. We found that the principle represents the limit computational model of Popperian game of non-deterministic computation of refutability by finite numbers of processes.
The full arithmetical hierarchy of LCM including this sublearning hierarchy was fully explored.
This was done through various techniques of mathematical logic, and it is important also from mathematical point of view. The calibration theory according to this hierarchy has been started by a researcher out of our project.
We also found that Coquand's game semantics could be restricted in a very natural way so that it coincides with LCM.
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