How unprovable is Rabin's decidability theorem?

We study the strength of set-theoretic axioms needed to prove Rabin's theorem on the decidability of the MSO theory of the infinite binary tree. We first show that the complementation theorem for tree automata, which forms the technical core of typical proofs of Rabin's theorem, is equivalent over the moderately strong second-order arithmetic theory $\mathsf{ACA}_0$ to a determinacy principle implied by the positional determinacy of all parity games and implying the determinacy of all Gale-Stewart games given by boolean combinations of ${\bf \Sigma^0_2}$ sets. It follows that complementation for tree automata is provable from $\Pi^1_3$- but not $\Delta^1_3$-comprehension. We then use results due to MedSalem-Tanaka, M\"ollerfeld and Heinatsch-M\"ollerfeld to prove that over $\Pi^1_2$-comprehension, the complementation theorem for tree automata, decidability of the MSO theory of the infinite binary tree, positional determinacy of parity games and determinacy of $\mathrm{Bool}({\bf \Sigma^0_2})$ Gale-Stewart games are all equivalent. Moreover, these statements are equivalent to the $\Pi^1_3$-reflection principle for $\Pi^1_2$-comprehension. It follows in particular that Rabin's decidability theorem is not provable in $\Delta^1_3$-comprehension.Comment: 21 page

Semilinear order property and infinite games

Falta resumen y palabras claveEn este trabajo se analiza la determinación de juegos de Lipschitz y Wadge, junto con la propiedad de ordenación semilineal, estrechamente relacionada con estos juegos, en el contexto de la Aritmética de segundo orden y el programa de la Matemática inversa (Reverse Mathematics). En primer lugar, se obtienen pruebas directas, formalizables en la Aritmética de segundo orden, de la determinación de los juegos de Lipschitz y Wadge para los primeros niveles de la Jerarquía de diferencias de Hausdorff. A continuación, se determinan los axiomas de existencia suficientes para la formalización de dichas pruebas dentro de los subsistemas clásicos de la Aritmética de segundo orden (fórmula). Finalmente, en algunos casos se muestra que dichos axiomas de existencia son óptimos, probando que resultan ser equivalentes (sobre un subsistema débil adecuado, como RCA0 o ACA0) a las correspondientes formalizaciones de los principios de determinación o de ordenación semilineal. Los principales resultados obtenidos son los siguientes: Teorema A. Sobre RCA0 son equivalentes: (fórmula) (el principio de determinación para juegos de Lipschitz entre subconjuntos del espacio de Cantor que son diferencia de dos cerrados). (fórmula) (la propiedad de ordenación semilineal de la reducibilidadLipschitz entre subconjuntos del espacio de Cantor que son diferencia de dos cerrados). Teorema B. Sobre RCA0 son equivalentes: (fórmula) (el principio de determinación para juegos de Lipschitz entre subconjuntos abiertos o cerrados del espacio de Baire). Teorema C. Sobre ACA0 son equivalentes: (fórmula) (el principio de determinación para juegos de Lipschitz entre subconjuntos del espacio de Baire que son simultáneamente abiertos y cerrados). (fórmula) (la propiedad de ordenación semilineal de la reducibilidadLipschitzentre subconjuntos del espacio de Baire simultáneamente abiertos y cerrados).In this thesis we analyze the determinacy of the Lipschitz and Wadge games, as well as the tightly related semilinear ordering principle, in the setting of second order arithmetic and of the program of Reverse Mathematics. Firstly, we obtain direct proofs, formalizable in second order arithmetic, of the determinacy of Lipschitz and Wadge games for the first levels of the Hausdorff's hierarchy of differences. Then we determine the set existence axioms needed to formalize such proofs within the classical subsystems of second order arithmetic (fórmula). Finally, in some cases we show that these axioms of existence are optimal, proving that they turn out to be equivalent (over a suitable weak subsystem asRCA0 orACA0) to the corresponding formalization of the principles of determinacy or semilinear ordering. The main results are: Theorem A.The following assertions are pairwise equivalent over RCA0: (fórmula) (determinacy of Lipschitz games for subsets of the Cantor space which are differences of closed sets). (fórmula) (Lipschitz semilinear ordering for subsets of the Cantor space which are differences of closed sets). Theorem B.The following assertions are pairwise equivalent over RCA0: (fórmula) (determinacy of Lipschitz games for open or closed subsets of the Baire space). Theorem C.The following assertions are pairwise equivalent over ACA0: (fórmula) (determinacy of Lipschitz games for clopen subsets of the Baire space). (fórmula) (Lipschitz semilinear ordering for clopen subsets of the Baire space)

無限ゲーム,帰納的定義,超限的再帰

要約のみTohoku University田中 一之課

μ計算と算術の階層構造の探索---ゲール・スチュワートゲームの視点より

Tohoku University横山啓太課

Evolving Computability

We consider the degrees of non-computability (Weihrauch degrees) of finding winning strategies (or more generally, Nash equilibria) in infinite sequential games with certain winning sets (or more generally, outcome sets). In particular, we show that as the complexity of the winning sets increases in the difference hierarchy, the complexity of constructing winning strategies increases in the effective Borel hierarchy.Comment: An extended abstract of this work has appeared in the Proceedings of CiE 201