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.
The basic task to be studied in the present paper is the following:
Given information about what does not constitute a solution, find a solution.
The difficulty of this task depends strongly on the structure of the set of potential solutions. In general, each represented space (X, δ) induces a topology, where a set U ⊆ X is open, if its characteristic function
is continuous with respect to the representation δ and a standard representation δ S of Sierpiński space S = {0, 1} (which is equipped with the topology {∅, {1}, {0, 1}}). Such a standard representation of S can be defined by δ S (p) = 1 : ⇐⇒ (∃n) p(n) = 0 for all p ∈ N N . Intuitively, the open sets are those for which membership can be continuously confirmed. Each represented space then comes naturally with a representation δ • of the open sets, defined by
for all p ∈ N N . Here [δ → δ S ] denotes the canonical function space representation (see [36]) of δ and δ S (which is the exponential in the category of represented spaces). The representation δ • in turn induces a representation ψ X -of the closed sets by ψ X -(p) = X \ δ • (p). The restriction to closed sets as solution sets arises from the fact that they are exactly those sets for which one can continuously confirm membership in the complement.
We give some intuitive descriptions of equivalent versions of this very general representation for concrete spaces that we will consider.
• N = {0, 1, 2, …}, the set of natural numbers: the standard representation is defined by δ N (p) := p(0) and an equivalent way of defining ψ N -is by
p is an enumeration of all points that are not in A.
• {0, 1} N , the Cantor space: the standard representation can be obtained by restricting the identity on Baire space to Cantor space δ {0,1} N := id N N | {0,1} N .
In this case one can think that ψ {0,1} N -(p) = A if p is a (potentially empty) enumeration of words w i ∈ {0, 1} * such that A = {0, 1} N \ ∞ i=0 w i {0, 1} N . That is p is a (potentially empty) enumeration of words w i such that the corresponding balls exhaust the exterior of A.
• N N , the Baire space: this case can be handled analogously to Cantor space, except that the representation δ N N is just the identity. • R, the Euclidean real number line (and R n in general): for convenience we assume that we use some standard numbering : N → Q. Then the Cauchy representation ρ :⊆ N N → R can be defined by ρ(p) := lim n→∞ p(n), where the domain dom(ρ) contains only rapidly converging sequences, i.e. p with |p(i)p(j)| < 2 -j for all i > j. Thus, a real number x is represented by a rapidly converging sequence of rational numbers. The representation ψ R -can then be considered as follows: a name p of a set A is a sequence ( a i , b i ) i∈N such that A = R \ ∞ i=0 (a i , b i ). That is, intuitively, p is a list of rational intervals that exhaust the complement of A.
• I := [0, 1], the real unit interval (and I n in general): this can be treated by restricting the case of R n .
For most spaces, closed choice is not computable. Thus, our interest lies on classifying the degree of incomputability, that is the Weihrauch degree of closed choice, depending on the underlying space. Some of the arising Weihrauch degrees are associated with certain models of type-2 hypercomputation, giving an independent justification for our interest in closed choice. Additionally, as already demonstrated in [6], several important mathematical theorems share a Weihrauch degree with an appropriate version of closed choice.
In recursion theory, a question closely related to our notion of closed choice has been studied. Given a Π 0 1 -class of Cantor space (which is a co-c.e. closed set in our terminology), what can we say about its elements? It is known that a co-c.e. closed set may contain no computable points, but always contains a low point [16]. We present a stronger result, which takes the form that closed choice for Cantor space is computable, if we replace the standard representation of the elements with another one, which just renders the low points computable. On the side, we present a few results on the initial topology of the Turing jump operator (called Π-topology by Joseph Miller, see [21]).
This section serves to give a brief introduction into represented spaces, realizers, Weihrauch reducibility and several associated operations. The basic reference for this section is [36]. While the study of (variants of) Weihrauchreducibility has commenced over a decade ago ( [31], [34], [35], [15]), the relevant sources for this section are [7], [6] and [28].
A significant ingredient of the theory of represented spaces is Baire space N N , i.e. the set of natural number sequences, equipped with the topology derived from the metric d N N which is defined by d N N (u, u) = 0 and d N N (u, v) = 2 -min{n|un =vn} for u = v. A useful property of Baire space to be exploited frequently is the existence of an effective and bijective pairing function , : N N × N N → N N . In t
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