Closed choice and a Uniform Low Basis Theorem

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. The class of limit computable functions can be characterized with parallelized choice on natural numbers. On top of these results we provide further insights into algebraic properties of closed choice. In particular, 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.

How constructive is constructing measures?

“…For well-behaved spaces, using closed choice iteratively does not increase its power by [4,Theorem 7.3]…”

“…For any perfect computably compact compact computable metric space X we find that C X ≡ W C [0,1] by [4,Corollary 4.5]. For well-behaved spaces, using closed choice iteratively does not increase its power by [4,Theorem 7.3]…”

“…The degree C [0,1] is closely linked to WKL in reverse mathematics (discussed in [5]), while C N is Weihrauch-complete for functions computable with finitely many mindchanges (cf [4]). …”

“…For the reverse direction 4 we use Proposition 4 to express the given overt set as the closure of a sequence (x i ) i∈N . Now we can compute a sum of point measures via µ(U) = {i∈N|xi∈U} 2 −i .…”

“…An important source for examples of Weihrauch degrees relevant in order to classify theorems are the closed choice principles studied in, eg, Brattka and Gherardi [5] and Brattka, Le Roux and Pauly [4]: Definition 12 Given a represented space X, the associated closed choice principle C X is the partial multivalued function C X :⊆ A(X) ⇒ X mapping a non-empty closed set to an arbitrary point in it.…”