Infinite games and transfinite recursion of multiple inductive definitions

Keisuke Yoshii, Kazuyuki Tanaka

Research output: Chapter in Book/Report/Conference proceedingConference contribution

1 Citation (Scopus)

Abstract

The purpose of this research is to investigate the logical strength of weak determinacy of Gale-Stewart games from the standpoint of reverse mathematics. It is known that the determinacy of sets (open sets) is equivalent to system ATR 0 and that of Σ 2 0 corresponds to the axiom of Σ 1 1 inductive definitions. Recently, much effort has been made to characterize the determinacy of game classes above Σ 2 0 within second order arithmetic. In this paper, we show that for any k ε ω, the determinacy of Δ((Σ 2 0) k+1) sets is equivalent to the axiom of transfinite recursion of Σ 1 1 inductive definitions with k operators, denote [Σ 1 1] k -IDTR. Here, (Σ 2 0) k+1 is the difference class of k + 1 Σ 2 0 sets and Δ((Σ 2 0) k+1) is the conjunction of (Σ 2 0) k+1 and co-(Σ 2 0) k+1.

Original languageEnglish
Title of host publicationHow the World Computes - Turing Centenary Conference and 8th Conference on Computability in Europe, CiE 2012, Proceedings
Pages374-383
Number of pages10
DOIs
Publication statusPublished - 2012
EventTuring Centenary Conference and 8th Conference on Computability in Europe, CiE 2012 - Cambridge, United Kingdom
Duration: 2012 Jun 182012 Jun 23

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume7318 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349

Other

OtherTuring Centenary Conference and 8th Conference on Computability in Europe, CiE 2012
CountryUnited Kingdom
CityCambridge
Period12/6/1812/6/23

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Computer Science(all)

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