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We work in the Weihrauch lattice, which has become a widespread tool to classify the level of incomputability of mathematical problems from several branches of classical mathematics since GM09 . Intuitively, given two (multi-valued) functions [MATH] and [MATH] on represented spaces, [MATH] is |
Weihrauch reducible to [MATH] if [MATH] can be computed by [MATH] , with computable translations from [MATH] to [MATH] , and, viceversa, from |
[MATH] to [MATH] , allowed. (More details are in § 2.2 below.) Recall that in this approach mathematical objects are encoded by sequences of infinite length whose information is based on the topological properties of the underlying spaces. For example, [MATH] (for a fixed [MATH] ) is naturally represented by an effecti... |
positive representation is based on the upper Fell topology [MATH] and consists, for nonempty closed sets, in enumerating dense sequences of points in them. Finally, the total representation, corresponding to the Fell topology, is obtained by combining both kinds of information ( Sch03 ). More details about these repre... |
In the literature ( BD10 Neu15 ) it has been proved that the projection operators, for some metric spaces and closed sets with optimal conditions (such as convexity, boundedness of [MATH] , uniqueness of the solution) are computable. But what happens when such optimal conditions fail? It is not surprising that the prob... |
In many concrete applications one may be content already with approximations of arbitrarily accurate precision. In other words, we investigate the computational complexity of operators selecting points [MATH] such that |
[MATH] for some fixed [MATH] Intuitively, we expect that the loss of accuracy results in a simpler computational complexity. We indeed prove that these operators are simpler than their exact counterparts, but still not computable for negative and positive representations of closed sets. In contrast, the approximate pro... |
Table summarizes our main results. Quite surprisingly, it turns out that in most cases the (approximate) projection operators are Weihrauch complete with respect to some fundamental computational class which is represented in the last column by its emblematic representative, already studied in the literature, and defin... |
It is also remarkable that, as far as exact projections are concerned, negative or positive information for closed sets can be used interchangeably, as this has no effect in the classification obtained with respect to any given dimension [MATH] . The difference between negative and positive information only arises when... |
To see that the approximate projections are of practical importance we suggest a concrete application, which is actually the original motivation for our research. The Whitney Extension Theorem was originally proved in |
Whi34 and, roughly speaking, generalizes the well-known Urysohn-Tietze Extension Lemma to the case of differentiable functions. An expository paper on modern developments concerning the Whitney Extension Theorem and its generalizations is Fef09 . By using approximate projections in place of the exact ones originally us... |
We now explain the organization of the paper. In Section we give a brief introduction to computable analysis, introduce the representations we will be using throughout the paper, and recall Weihrauch reducibility and some milestones in the Weihrauch lattice. Section |
provides a new characterization of the Weihrauch degree of the function [MATH] , introduced by Neumann and Pauly ( NP18 ). Sections |
and are the core of the paper and are devoted respectively to the exact and approximated projection operators: after defining them we prove the results summarized in Table . In Section we briefly sketch the application of approximate projections to the Whitney Extension Theorem. |
2. Computable analysis: notation and terminology This Section recalls basic definitions and terminology of computable analysis and of Weihrauch reducibilities (see BGP for a self-contained introduction to the subject). The reader familiar with the topics can safely skip it and refer back to this section as needed. |
We work in the framework is the so called Type-2 Theory of Effectivity (TTE), which finds a systematic foundation in Wei00 and provides a realistic and flexible model of computation. The salient features of TTE Turing machines are that they work on infinite sequences of bits and that no correction is allowed on the out... |
2.1. Representations To extend the notion of computability to functions between spaces different from [MATH] we need the notion of representation. Recall that a |
representation [MATH] of a set [MATH] is a surjective function [MATH] , and in this case we say that the pair [MATH] is a represented space . If [MATH] |
[MATH] -name for [MATH] is any [MATH] such that [MATH] . By routine syntactic pairing techniques it is straightforward to obtain representations for finite and countably infinite product of represented spaces. |
Given represented spaces [MATH] and [MATH] and a partial multi-valued function [MATH] , we say that [MATH] is a [MATH] -realizer of [MATH] (and write [MATH] ) if [MATH] , for all [MATH] . We can now say that a function between represented spaces is |
computable if it has a computable realizer. For representations [MATH] and [MATH] of the same set [MATH] , we say that [MATH] is computably reducible to [MATH] (we write |
[MATH] ) if there is a computable [MATH] such that for every [MATH] we have [MATH] . If [MATH] and [MATH] , the two representations are computably equivalent |
[MATH] ). The general notion of representation is too broad for practical purposes. Concretely, representations are associated to the final topologies they induce on the represented space, and usually admissible representations for [MATH] -spaces are considered. Such representations are those that make the use of reali... |
An important example is the Cauchy representation which is admissible with respect to the topology of a separable (computable) metric space. |
Definition 2.1 (Computable metric spaces) computable metric space is a triple [MATH] , where [MATH] is a metric on [MATH] [MATH] is a dense sequence in [MATH] , and [MATH] is a computable double sequence in [MATH] . We then represent [MATH] by the Cauchy representation |
[MATH] , defined by [EQUATION] When [MATH] we say that [MATH] is an effective Cauchy sequence , and that it converges effectively to [MATH] |
Notice that with this representation, the metric [MATH] is computable. A particularly important example is provided by the Euclidean spaces [MATH] which are computable metric spaces when we fix a function |
[MATH] enumerating in an effective way [MATH] . Here [MATH] is the usual Euclidean metric. By using the same effective numbering [MATH] , there are other ways to represent real numbers, by changing the underlying topology over |
[MATH] . The representation [MATH] is given by [MATH] iff [MATH] , and analogously [MATH] is given by [MATH] iff [MATH] . These two representations are admissible with respect to the topologies [MATH] and |
[MATH] whose open sets are of the form [MATH] and [MATH] respectively (see Wei00 for more details). Given a computable metric space [MATH] we can effectively enumerate the open balls with center in [MATH] and rational radius in an obvious way using a computable pairing function: to [MATH] we associate the open ball [MA... |
[MATH] is a standard enumeration of the nonnegative rational numbers (notice that [MATH] when [MATH] ). We call these sets open basic balls . We denote the closed ball [MATH] by [MATH] or [MATH] (notice that in general this is not the same as the closure |
[MATH] of [MATH] , although in [MATH] they coincide). Definition 2.2 (Closed set representations) Let [MATH] be a computable metric space. |
By [MATH] we denote the hyperspace of closed subsets of [MATH] equipped with the negative information representation [MATH] such that |
[EQUATION] By [MATH] we denote the hyperspace of closed subsets of [MATH] equipped with the positive information representation [MATH] such that |
[EQUATION] Finally, by [MATH] we denote the hyperspace of closed subsets of [MATH] equipped with the total information representation |
[MATH] , that is [EQUATION] It is clear from Definitions 2.1 and 2.2 that we can view [MATH] as an element of [MATH] if and only if we can semi-decide whether |
[MATH] for every [MATH] . This means that to show that (a name for) some [MATH] can be computed from some input [MATH] it suffices to give a definition of [MATH] by a [MATH] formula with parameter [MATH] |
It is well known that the operations [MATH] are computable, as well as [MATH] Closed sets with positive information are also known in the literature as |
overt sets (see Pau16 for a discussion of nomenclature). Remark 2.3 By BP03 , Theorems 3.7 and 3.8] , in every complete computable metric space [MATH] , the positive information representation for |
nonempty closed sets is equivalent to the representation which assigns to a name [MATH] the set [MATH] . See also Wei00 , Lemma 5.1.10] for the case [MATH] |
As for the negative information, in the Euclidean space this is equivalent to the representation encoding a closed [MATH] by enumerating all [MATH] |
such that [MATH] Wei00 , Lemma 5.1.10] ). We are also interested in representing the space of the compact subsets of a fixed computable metric space. |
Definition 2.4 (Compact set representations) Let [MATH] be a computable metric space. By [MATH] we denote the hyperspace of compact subsets of [MATH] equipped with the negative information representation |
[MATH] such that: [EQUATION] Analogously, one defines the hyperspace [MATH] of compact subsets of [MATH] equipped with the positive information representation |
[MATH] and the hyperspace [MATH] of compact subsets of [MATH] equipped with the total information representation [MATH] , by replacing [MATH] |
with [MATH] and [MATH] , respectively. Remark 2.5 In the case of the Euclidean space [MATH] , the balls [MATH] can be more simply replaced by a single ball |
[MATH] , for [MATH] , satisfying [MATH] (in agreement with Wei00 , Definition 5.2.1] ). Abstracting from the purely syntactic elements, representations often denote objects by enumerating sequences of objects. Therefore one is often allowed to skip the annoying linguistic aspects by describing the represented element d... |
[MATH] directly as [MATH] , by meaning that [MATH] (which really should be [MATH] ) is the [MATH] -th rational open ball enumerated by some [MATH] -name of [MATH] |
2.2. Weihrauch reducibility The original definition of Weihrauch reducibility between functions over represented spaces is due to Weihrauch in an unpublished report from 1992, and in the next decade the notion was explored in several thesis by some of Weihrauch’s students. The authors GM09 extended Weihrauch reducibili... |
[MATH] is Weihrauch reducible to [MATH] , and write [MATH] , if there are computable [MATH] and [MATH] such that [MATH] whenever [MATH] (here [MATH] is the identity function on Baire space). |
The intuition behind the definition is that [MATH] means that the problem of computing [MATH] can be computably and uniformly solved by using in each instance a single computation of [MATH] [MATH] modifies (each name for) the input of [MATH] to feed it to [MATH] , while [MATH] , using also the original input, transform... |
A direct consequence of the definition of Weihrauch reducibility is the following Invariance Principle [MATH] implies that for any given |
[MATH] -name [MATH] of some [MATH] there is some [MATH] with a [MATH] -name [MATH] such that [MATH] (here [MATH] denotes the usual Turing reducibility). In other words, [MATH] provides an upper bound for the computational complexity of (some element in) [MATH] |
The relation [MATH] is reflexive and transitive and induces an equivalence relation denoted by [MATH] . The partial order on the sets of |
[MATH] -equivalence classes (called Weihrauch degrees ) is a distributive bounded lattice BG11b Pau10b with several natural and useful algebraic operations BGM12 . As usual, we use [MATH] to denote |
[MATH] and [MATH] , and [MATH] to denote [MATH] and [MATH] The Weihrauch lattice can be used as a tool for comparing multi-valued functions arising from theorems from different areas of mathematics, once the theorems are translated into mathematical problems on represented spaces. This line of research has blossomed in... |
In some cases, one can prove the reducibility of [MATH] to [MATH] by using a computable [MATH] that does not access to the original input, that is, we have |
[MATH] whenever [MATH] . In this case we say that [MATH] is strongly Weihrauch reducible to [MATH] and write [MATH] . We then use |
[MATH] for the induced equivalence relation. We notice that [MATH] always hold, whereas [MATH] holds when [MATH] is a cylinder , that is [MATH] , where id is the identity function on the Baire space. |
2.3. Some milestones in the Weihrauch lattice Multi-valued functions [MATH] can be seen as problems: given [MATH] , find a [MATH] . The algebraic structure of the Weihrauch lattice can provide a useful tool to determine the computational complexity of fundamental mathematical problems which are not computable, at least... |
[MATH] , the computational version of the limited principle of omniscience of constructive mathematics defined by [MATH] if [MATH] and [MATH] if [MATH] |
[MATH] , the computational version of the lesser limited principle of omniscience of constructive mathematics defined by [MATH] iff [MATH] , where for at most one [MATH] and one [MATH] it holds [MATH] |
[MATH] , the Weak König’s Lemma operator , mapping each infinite binary tree to its infinite paths; here a tree [MATH] is represented by its characteristic function [MATH] , that is, [MATH] iff [MATH] |
for a recursive enumeration [MATH] of all finite binary words; [MATH] , the closed choice operators , selecting members from any given non-empty closed set (encoded by negative information) in a computable metric space [MATH] |
[MATH] , the compact choice operators , selecting members from any given non-empty compact set (encoded by negative information) in a computable metric space [MATH] |
[MATH] , for every convergent sequence [MATH] in the Baire space; [MATH] , the Bolzano-Weierstraß Theorem operators, that maps every sequence with compact range in a computable metric space [MATH] to its accumulation points. |
For instance, the problem of finding the derivative [MATH] of [MATH] is Weihrauch equivalent to [MATH] . As for [MATH] [MATH] , and [MATH] , we obtain very important cases when we set [MATH] or [MATH] . For example, the (contrapositive of) the Baire Category Theorem is Weihrauch equivalent to |
[MATH] . See BGP for a general overview of this program of classification of mathematical problems and for further references. It is well known that [MATH] (where [MATH] ) and |
[MATH] A multi-valued function [MATH] is called non deterministically computable if [MATH] computable with finitely many mind changes if |
[MATH] , and limit computable if [MATH] . This terminology arises from the non standard models of computation that make such [MATH] |
computable. For instance, [MATH] is computable with finitely many mind changes if it can be computed by a non standard TTE-machine that is allowed to revise the output with the restraint that only finitely many corrections can occur. See BGP for more details and further references. |
Some degrees can be seen as the parallelization or composition of other degrees. The parallelization of [MATH] of [MATH] is defined as [MATH] . We have for example [MATH] and [MATH] . It is known that parallelization is a closure operator, that is [MATH] |
[MATH] , and [MATH] The composition of multi-valued functions is defined so that the range of the first function not necessarily has to be included in the domain of the second function. Intuitively, some computational transformation is allowed so that the two spaces can match. It is easier to define such compositional ... |
[EQUATION] Here the leftmost occurrences of [MATH] and [MATH] must be understood as denoting the corresponding degrees, and the maximum as a degree defined by the partial order induced on the Weihrauch degrees by [MATH] . (Notice that the Weihrauch lattice is not complete, but the [MATH] above always exists by |
BP , Corollary 18] BGP , Theorem 5.2] .) It holds then [MATH] and [MATH] These equivalences justify the following terminology: [MATH] is said to be non deterministically limit computable and [MATH] is said to be non deterministically computable with finitely many mind changes |
Finally, some degrees can be seen as jumps of others. Given a multi-valued function [MATH] on represented spaces [MATH] , the jump [MATH] of [MATH] coincides with [MATH] but the representation of [MATH] is weakened into the representation |
[MATH] , where [MATH] and [MATH] iff [MATH] . It holds then [MATH] and [MATH] Notice that [MATH] is a cylinder, hence [MATH] , and the same holds for [MATH] and [MATH] |
3. The functions [MATH] and [MATH] In NP18 Neumann and Pauly introduced the function [MATH] defined as [EQUATION] Our results support the importance of this function, so that one might see it as a candidate for a new possible milestone in the Weihrauch lattice. To this end we first show that [MATH] is strongly Weihrauc... |
Consider the space [EQUATION] This space is seen as a subspace of the represented space [MATH] , hence its members are represented as real numbers via Cauchy sequences of elements of the dense set of rationals in [MATH] , i.e., [MATH] itself. |
It is easy to see that this Cauchy representation [MATH] is computably equivalent to the representation [MATH] with [MATH] and [EQUATION] |
To see that [MATH] , take any [MATH] -name [MATH] of [MATH] and consider the Cauchy sequence [MATH] such that [MATH] if [MATH] for all [MATH] , and |
[MATH] if [MATH] for a (unique) [MATH] . For the opposite reduction, let [MATH] converge effectively to [MATH] . To obtain a [MATH] -name [MATH] of [MATH] just put [MATH] if [MATH] |
and [MATH] otherwise. Intuitively, according to the representation [MATH] a name of [MATH] is a (computable!) oracle that for every [MATH] replies “yes” or “no” to the question “is |
[MATH] ?”; if the answer is always “no” then [MATH] It is often more convenient to represent [MATH] by [MATH] However, when representing the space of closed subsets of [MATH] we will view [MATH] as a computable metric space and use the standard enumeration of the basic open balls [MATH] , for [MATH] and [MATH] a nonneg... |
[MATH] As a subset of [MATH] [MATH] is also well-ordered by the usual order [MATH] The single valued function [MATH] mapping [MATH] to [MATH] is then defined for every [MATH] |
Proposition 3.1 [MATH] Proof. We first show that [MATH] . Given [MATH] we construct a set [MATH] that will provide us with the necessary information to compute [MATH] . More precisely, we define [MATH] , where [MATH] if [MATH] contains exactly [MATH] occurrences of [MATH] . Let now [MATH] be a [MATH] -name of [MATH] . ... |
[MATH] , so that [MATH] if [MATH] for all [MATH] . As soon as we find an [MATH] such that [MATH] , then we let [MATH] for every [MATH] , so that in the end [MATH] |
To prove [MATH] argue as follows. Let [MATH] be given as input to [MATH] . Our strategy consists simply in choosing at any stage [MATH] the smallest element of [MATH] not contained in [MATH] |
and we want to write an input [MATH] for Sort that reflects our choice. At stage 0 let then [MATH] be the least element of [MATH] not contained in |
[MATH] . If this is 0, then we write [MATH] . Otherwise, let it be [MATH] for some [MATH] . Then we put [MATH] as initial segment of the input [MATH] of Sort. At stage [MATH] we consider the sets [MATH] . Let |
[MATH] be the least element not contained in [MATH] , and let [MATH] be the initial segment of [MATH] obtained at stage [MATH] . If [MATH] then we let [MATH] . Otherwise, if [MATH] for some |
[MATH] we extend [MATH] so to obtain a finite prefix [MATH] with [MATH] containing exactly [MATH] occurrences of [MATH] (possibly [MATH] ). In the end, by construction, [MATH] contains exactly [MATH] occurrences of 0 if |
[MATH] for some [MATH] , and [MATH] contains infinitely many [MATH] if [MATH] . We now inspect [MATH] to compute a [MATH] -name [MATH] of [MATH] . Recall that [MATH] if [MATH] contains infinitely many occurrences of [MATH] , that is, [MATH] , otherwise |
[MATH] if [MATH] contains exactly [MATH] occurrences of [MATH] , that is, [MATH] . Therefore, to obtain a correct [MATH] -name [MATH] , we let [MATH] as long as [MATH] . If suddenly |
[MATH] , then we let [MATH] and [MATH] for all [MATH] . In this way we obtain exactly the [MATH] -name of [MATH] Using Proposition 3.1 , we now study the degree of [MATH] in more detail. The following result is given already in Proposition 24 of NP18 |
but we give here a more direct proof of the same result in terms of computability with finitely many mind changes using [MATH] Proposition 3.2 |
[MATH] Proof. By Proposition 3.1 we can substitute [MATH] with [MATH] To prove [MATH] , consider the operator [MATH] , for [MATH] , which is known to be Weihrauch equivalent to [MATH] by PFD18 , Lemma 2.3] . We obtain [MATH] (for the rightmost reduction observe that the map [MATH] from [MATH] to [MATH] is clearly compu... |
To prove that [MATH] , we will show that [MATH] is not computable with finitely many mind changes. Let [MATH] indeed be a [MATH] -name of a nonempty closed [MATH] . The task is to output the minimal element in [MATH] . Suppose that [MATH] |
lists only open balls of the type [MATH] for various [MATH] . If the sequence encoded by [MATH] will in the end contain every open ball of the form |
[MATH] , the temporary choice of any element of the form [MATH] will sooner or later force us to select a larger candidate. In this case we obtain the name of the correct output [MATH] only after infinitely many mind changes. |
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