Closed Choice and a Uniform Low Basis Theorem

TL;DR

This paper explores closed choice principles across various spaces, linking them to models of hypercomputation and establishing their properties, including a uniform Low Basis Theorem, within a Weihrauch reducibility framework.

Contribution

It introduces a uniform framework connecting closed choice principles to hypercomputation models and proves a uniform Low Basis Theorem for Cantor and Euclidean spaces.

Findings

Characterization of function classes via closed choice in different spaces

Proof that closed choice on Cantor and Euclidean spaces is low computable

Establishment of product and quotient theorems for closed choice principles

Abstract

We study closed choice principles for different spaces. Given information about what does not constitute a solution, closed choice determines a solution. We show that with closed choice one can characterize several models of hypercomputation in a uniform framework using Weihrauch reducibility. The classes of functions which are reducible to closed choice of the singleton space, of the natural numbers, of Cantor space and of Baire space correspond to the class of computable functions, of functions computable with finitely many mind changes, of weakly computable functions and of effectively Borel measurable functions, respectively. We also prove that all these classes correspond to classes of non-deterministically computable functions with the respective spaces as advice spaces. Moreover, we prove that closed choice on Euclidean space can be considered as "locally compact choice" and it is obtained as product of closed choice on the natural numbers and on Cantor space. We also prove a Quotient Theorem for compact choice which shows that single-valued functions can be "divided" by compact choice in a certain sense. Another result is the Independent Choice Theorem, which provides a uniform proof that many choice principles are closed under composition. Finally, we also study the related class of low computable functions, which contains the class of weakly computable functions as well as the class of functions computable with finitely many mind changes. As one main result we prove a uniform version of the Low Basis Theorem that states that closed choice on Cantor space (and the Euclidean space) is low computable. We close with some related observations on the Turing jump operation and its initial topology.