G_{\delta \sigma}-games and generalized computation

TL;DR

This paper establishes a deep connection between winning strategies in certain complex infinite games and higher-type recursive functions, revealing their computational and ordinal-theoretic properties.

Contribution

It introduces a novel equivalence between $G_{\delta \sigma}$-game strategies and higher-type recursive functions, characterizing their computational complexity and ordinal length.

Findings

The set of convergent recursions is a complete $ ext{Game} \Sigma_3^0$ set.

Strategies for the first player are recursive in the higher-type functional.

Identifies the ordinal length of monotone $ ext{Game} \Sigma_3^0$-inductive operators.

Abstract

We show the equivalence between the existence of winning strategies for $G_{\delta \sigma}$ (also called $\Sigma^{0}_{3}$) games in Cantor or Baire space, and the existence of functions generalized-recursive in a higher type-2 functional. (Such recursions are associated with certain transfinite computational models.) We show, inter alia, that the set of indices of convergent recursions in this sense is a complete $\Game \Sigma_{3}^{0}$ set: as paraphrase, the listing of those games at this level that are won by player I, essentially has the same information as the `halting problem' for this notion of recursion. Moreover the strategies for the first player in such games are recursive in this sense. We thereby establish the ordinal length of monotone $\Game \Sigma^{0}_{3}$-inductive operators, and characterise the first ordinal where such strategies are to be found in the constructible hierarchy. In summary: Theorem (a) The following sets are recursively isomorphic. (i) The complete ittm-semi-recursive-in-${eJ}$ set, $H^{eJ}$; (ii) the $\Sigma_{1}$-theory of $( L_{\eta_{0}} , \in ) $, where $\eta_{0}$ is the closure ordinal of $\Game \Sigma_{3}^{0}$-monotone induction; (iii) the complete $\Game \Sigma_{3}^{0}$ set of integers. (b) The ittm-recursive-in-${eJ}$ sets of integers are precisely those of $L_{\eta_{0}}$.