逻辑和计算 I
This is an advanced undergraduate and graduate-level course in mathematical logic and theory of computation. Topics to be presented in the first semester include: computable functions, undecidability, propositional logic, NP-completeness, first-order logic, Goedel's completeness theorem, Ehrenfeucht-Fraisse games, Presburger arithmetic.
In the second semester, we will move on to Goedel's incompleteness theorems, second-order logic, infinite automata, determinacy of infinite games, etc.
In the second semester, we will move on to Goedel's incompleteness theorems, second-order logic, infinite automata, determinacy of infinite games, etc.
讲师
日期
2022年10月27日 至 12月27日
网站
修课要求
Completion of undergraduate course on logic, set theory or automata theory is recommended. But all interested students are welcome.
课程大纲
"Logic and Computation I" consists of the following three parts.
Part 1. Introduction to Computational Theory
Fundamentals on theory of computation and computability theory (recursion theory) of mathematical logic, as well as the connection between them.
This part is the basis for the following lectures.
Part 2. Propositional Logic and Computational Complexity
The basics of propostional logic (Boolean algebra) and complexity theory including some classical results, such as the Cook-Levin theorem.
Part 3. First Order Logic and Decision Problems
The basics of first-order logic, Goedel's completeness theorem, and the decidability of Presburger arithmetic.
We will use Ehrenfeucht-Fraisse game as a basic tool of first-order logic, and apply it to prove Lindstrom's theorem.
In "Logic and Computation II", we will move on to Goedel's incompleteness theorem, second-order logic, infinite automata, determinacy of infinite games, Post's problem, the Kondo-Addison theorem, admissible sets, alpha-recursion theory, etc.
Part 1. Introduction to Computational Theory
Fundamentals on theory of computation and computability theory (recursion theory) of mathematical logic, as well as the connection between them.
This part is the basis for the following lectures.
Part 2. Propositional Logic and Computational Complexity
The basics of propostional logic (Boolean algebra) and complexity theory including some classical results, such as the Cook-Levin theorem.
Part 3. First Order Logic and Decision Problems
The basics of first-order logic, Goedel's completeness theorem, and the decidability of Presburger arithmetic.
We will use Ehrenfeucht-Fraisse game as a basic tool of first-order logic, and apply it to prove Lindstrom's theorem.
In "Logic and Computation II", we will move on to Goedel's incompleteness theorem, second-order logic, infinite automata, determinacy of infinite games, Post's problem, the Kondo-Addison theorem, admissible sets, alpha-recursion theory, etc.
参考资料
[1] H.D. Ebbinghaus, H. Flum and W. Thomas, Mathematical Logic, 3rd ed., Springer 2021.
[2] D.C. Kozen, Theory of Computation, Springer 2006.
[3] K. Tanaka, 計算理論と数理論理学 (Mathematics of Logic and Computation, in Japanese), Kyoritsu 2022.
[2] D.C. Kozen, Theory of Computation, Springer 2006.
[3] K. Tanaka, 計算理論と数理論理学 (Mathematics of Logic and Computation, in Japanese), Kyoritsu 2022.
听众
Graduate
视频公开
公开
笔记公开
公开
语言
英文
讲师介绍
田中一之教授博士毕业于美国加州大学伯克利分校，曾就职于东京工业大学和东北大学，并指导15位博士生和50名硕士生，2022年正式入职BIMSA。他是数理逻辑和计算理论领域的国际知名学者，在反推数学和二阶算术领域开创了新的研究方法，如WKLo的田中嵌入定理和守恒结果的田中公式，取得了一系列奠基性的成果，并将这一研究方向引入日本，将日本的数理逻辑研究推向了世界水平。田中一之教授还致力于模态mu演算，认知逻辑，随机博弈树等交叉领域的研究。