The Church Problem for Countable Ordinals

TL;DR

This paper extends the classical Church problem to countable ordinals, proving determinacy and decidability for all countable ordinals and full extension for ordinals less than ^0omega, advancing understanding of infinite game theory.

Contribution

It generalizes the Church problem to arbitrary countable ordinals and establishes determinacy and decidability results for these cases, including a full extension for ordinals below ^0omega.

Findings

Determinacy and decidability hold for all countable ordinals.

Full extension of the theorem holds for ordinals less than ^0omega.

The results advance the understanding of infinite games and their applications.

Abstract

A fundamental theorem of Buchi and Landweber shows that the Church synthesis problem is computable. Buchi and Landweber reduced the Church Problem to problems about ω-games and used the determinacy of such games as one of the main tools to show its computability. We consider a natural generalization of the Church problem to countable ordinals and investigate games of arbitrary countable length. We prove that determinacy and decidability parts of the Bu}chi and Landweber theorem hold for all countable ordinals and that its full extension holds for all ordinals < \omega\^\omega.