Computational Aspects of Randomness and Their Structural Analysis via Nonstandard Methods

2014 Fiscal Year Final Research Report

• Project/Area Number: 23340020
• Research Category: Grant-in-Aid for Scientific Research (B)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: General mathematics (including Probability theory/Statistical mathematics)
• Research Institution: Tohoku University
• Principal Investigator: TANAKA Kazuyuki 東北大学, 理学(系)研究科(研究院), 教授 (70188291)
• Co-Investigator(Kenkyū-buntansha): YAMAZAKI Takeshi 東北大学, 大学院理学研究科, 准教授 (30336812) ; SUZUKI Toshio 首都大学東京, 大学院理学研究院, 准教授 (30235973) ; TADAKI Kohtaro 中央大学, 研究開発機構, 機構准教授 (70407881)
• Co-Investigator(Renkei-kenkyūsha): KURODA Satoru 群馬県立女子大学, 文学部, 准教授 (30300586) ; YOKOYAMA Keita 北陸先端科学技術大学院大学, 情報科学研究科, 助教 (10534430)
• Project Period (FY): 2011-04-01 – 2015-03-31
• Keywords: 計算可能性理論 / 超準モデル / ランダム性

Outline of Final Research Achievements

The main aim of this research is to cultivate systematic understanding of randomness via various logical methods such as non-standard analysis so that computational aspects of basic concepts about probability and games deeply correlated to randomness should be clarified. We focus on the following five topics. (1) Setting a logical framework of nonstandard arguments for constructive measure theory. (2) Investigating computational structures by randomness notions. (3) Determining the equilibrium points of game trees. (4) Elucidating physical meaning of Chaitin’s Omega. (5) Logical treatments for determinacy of games.

Free Research Field

数学基礎論