Games for functions: Baire classes, Weihrauch degrees, transfinite computations, and ranks
Hugo de Holanda Cunha Nobrega
Abstract:
Game characterizations of classes of functions in descriptive set theory have their origins in the seminal work of Wadge, with further developments by Van Wesep, Andretta, Duparc, Motto Ros, and Semmes, among others. In this thesis we study such characterizations from several perspectives.
We define modifications of Semmes's game characterization of the Borel functions, obtaining game characterizations of the Baire class $\alpha$ functions for each fixed $\alpha < \omega_1$. Some of our results were independently proved by Louveau and Semmes in unpublished work. We also define a construction of games which transforms a game characterizing a class $\Lambda$ of functions into a game characterizing the class of functions which are piecewise $\Lambda$ on a countable partition of their domains by $\Pi^0_\alpha$ sets, for each $0 < \alpha < \omega_1$.
We then define a framework of parametrized Wadge games by using tools from computable analysis, and show how the choice of parameters can be used to fine-tune what class of functions is characterized by the resulting game. As an application, we recast our games characterizing the Baire classes into this framework.
Furthermore, we generalize our game characterizations of function classes to generalized Baire spaces, i.e., the spaces of functions from an uncountable cardinal to itself. We also show how the notion of computability on Baire space can be generalized to the setting of generalized Baire spaces, and show that this is indeed appropriate for developing a generalized version of computable analysis by defining a representation of Galeotti's generalized real line and analyzing the Weihrauch degree of the intermediate value theorem for that space.
In the final part of the thesis, we show how the game characterizations of function classes discussed lead in a natural way to a stratification of each class into a hierarchy, intuitively measuring the complexity of functions in that class. This idea and the results presented open new paths for further research.