Logic and Computation 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.
Lecturer
Date
27th October ~ 27th December, 2022
Website
Prerequisite
Completion of undergraduate course on logic, set theory or automata theory is recommended. But all interested students are welcome.
Syllabus
"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.
Reference
[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.
Audience
Graduate
Video Public
Yes
Notes Public
Yes
Language
English
Lecturer Intro
Kazuyuki Tanaka received his Ph.D. from U.C. Berkeley. Before joining BIMSA in 2022, he taught at Tokyo Inst. Tech and Tohoku University, and supervised fifteen Ph.D. students. He is most known for his works on second-order arithmetic and reverse mathematics, e.g., Tanaka's embedding theorem for WKLo and the Tanaka formulas for conservation results. For more details: https://sendailogic.com/tanaka.html