Title of the Report: Inspecting Hierarchy Collapse inμ-Calculus and Borel Games

Presenter:Kazuyuki Tanaka

Affiliation:Beijing Institute of Mathematical Science and Applications (BIMSA)

Date of the Report: July 13, 2024 (Saturday), 9:30-10:15

Location of the Report:Lecture Hall on the Second Floor, Block A, Feicui Science and Education Building

Abstract:In this talk, I will introduce some of our recent results in interdisciplinary areas of logic, games, and computation. First, I would like to discuss the collapse of the alternation hierarchy ofμ-calculus. Theμ-calculus is obtained from modal logic by adding the least and greatest fixed-point operatorsμandν. The alternation hierarchy ofμ-formulas is defined by measuring the entanglement of operatorsμandνin 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, everyμ-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 from 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 Fσsets, which is in contrast to the above results onμ-calculus.

Biography of the Presenter:Kazuyuki Tanaka is a professor at the 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 Institute of Technology 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 onμ-calculus, epistemic logic, random game trees, etc.