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.

• 03B45 - Modal logic (including the logic of norms)
• 03C90 - Nonclassical models (Boolean-valued, sheaf, etc.)
• 03G30 - Categorical logic, topoi
• 18C10 - Theories (e.g., algebraic theories), structure, and semantics
• 18C50 - Categorical semantics of formal languages

• 1 zbMATH Open