报告题目：Inspecting hierarchy collapse in μ-calculus and Borel games

报告人：Kazuyuki Tanaka

单位：Beijing Institute of Mathematical Science and Applications

报告时间：2024年7月13日(周六)上午09:30-10:15

报告地点：翡翠科教楼A座二楼报告厅

报告摘要：In this talk, I will introduce some of our recent results in interdisciplinary areas of logic, games and computation.First, I would like to talk about collapse of the alternation hierarchy of μ-calculus. The μ-calculus is obtained from modal logic by adding least and greatest fixed-point operators μ and ν. The alternation hierarchy of μ-formulas is defined by measuring the entanglement of operators μ and ν in a formula. Bradfield showed that, for all n∈N, there is a formula W_n with alternation depth n which is, over all Kripke frames, equivalent to no formulas with depth smaller than n. However, it is also known that, over frames of modal logic S5, every μ-formula is equivalent to a formula without fixed point operators. We have been investigating such collapse phenomena in more depth over different classes of frames.Secondly, we have been studying the determinacy strength of infinite games in the standpoint of Reverse Mathematics, that is, a foundational program aimed at pinpointing which axioms are needed to prove a theorem. We consider an infinite game where two players I and II alternately choose an element from N and player I wins the game if the resulting sequence f belongs to a given set G. A set G is said to be determinate if one of the two players has a winning strategy. Recently, we found collapse phenomena in the determinacy hierarchy over boolean combinations of Fσ sets, which is in contrast to the above results on μ-calculus.

报告人简介：：Kazuyuki Tanaka is a professor at Beijing Institute of Mathematical Science and Applications (BIMSA). He 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 WKL_0 and the Tanaka conservation (STY theorem). Also, he has been working on μ-calculus, epistemic logic, random game trees, etc.