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

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 $L_{\eta_{0}}$.