闭选择与一致低基定理
Closed Choice and a Uniform Low Basis Theorem
摘要：
我们研究不同空间的闭选择原理。给定不构成解的信息，封闭选择决定解。我们表明，封闭的选择，可以在一个统一的框架内使用Weihrauch约简表征几种模型的超计算。可归约为单点空间、自然数、康托空间和贝尔空间的闭选择的函数类分别对应于可计算函数类、可计算函数类、弱可计算函数类和有效博雷尔可测函数类。我们还证明了所有这些类对应于类的非确定性可计算的功能与各自的空间作为咨询空间。证明了欧几里得空间上的闭选择可以看作是“局部紧选择”，它是自然数上的闭选择与Cantor空间上的闭选择的乘积.证明了紧选择的商定理，证明了单值函数在一定意义下可以被紧选择整除。另一个结果是独立选择定理，它提供了一个统一的证明，许多选择原则是封闭的组合。最后，我们还研究了与之相关的低可计算函数类，它包括弱可计算函数类和可计算函数类。作为一个主要结果，我们证明了一个统一版本的低基定理，该定理指出，康托空间（和欧几里得空间）上的封闭选择是低可计算的。最后，我们对图灵跳跃操作及其初始拓扑进行了一些相关的观察。
英文摘要：
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.
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