Higher Type ITTM-recursion and Determinacy of Infinite Games

Abstract

We outline a theory of type 2 recursion for Infinite Time Turing Machines {\em à la Kleene}. We establish a connection between classical descriptive set theory and ittm theory, by calculating the complexity of its halting problem as exactly that of a complete $⅁\hat{I}{\pounds }_3^0$ (or $G_{\hat{I}´\ddot{I}ƒ}$) set. This mirrors exactly what Kleene, Moschovakis {\em et al.} achieved for Kleene's type 2 recursion and $\hat{I}{\pounds }_1^0$ (or Open) Determinacy.} We ascertain the least ordinal which is not generalised recursive in this sense, and its characterisation {\via}a concept of {\em infinite nestings} in Gödel's constructible hierarchy. The results do not require large cardinal axioms, and are all provable within analysis.