Game characterizations of function classes and Weihrauch degrees
Hugo de Holanda Cunha Nobrega
Abstract:
Games are an important tool in mathematics and logic, providing a
clear and intuitive understanding of the notions they define or
characterize. In particular, since the seminal work of Wadge in the
1970s, game characterizations of classes of functions in Baire space
have been a rich area of research, having had significant and
far-reaching development by van Wesep, Andretta, Duparc, Motto Ros,
and Semmes, among others.
In this thesis we study the connections between these games and the
notion of Weihrauch reducibility, introduced in the context of
computable analysis to express a particular type of continuous
reducibility of functions. Especially through the work of Brattka and
his collaborators, Gherardi, de Brecht and Pauly, among others,
several functions — commonly referred to as choice principles — have
been isolated that capture the complexity of classes of functions with
respect to Weihrauch reducibility.
In particular, we use the games for the corresponding classes to
provide new proofs of the Weihrauch-completeness of countable choice
for the Baire class 1 functions and of discrete choice for the
functions preserving ∆0_2 under preimages, and to introduce new
Weihrauch-complete choice principles for the class of functions
preserving ∆0_3 under preimages and for a particular class Λ_2,3
characterized by a games of Semmes. In the process, we recast some of
these games in a different style, which also allows for a uniform
intuitive view of the way each game presented is related to the class
of functions it characterizes.