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We should therefore choose [MATH] as the eventual output at some stage [MATH] , when only a finite initial segment of [MATH] has been read. However, this output is incorrect if the sequence encoded by [MATH] never mentions a specific element |
[MATH] , which is possible on the basis of the finite initial segment of [MATH] read by the computation at sage [MATH] The next result is also given in Proposition 24 of NP18 |
Proposition 3.3 [MATH] Proof. It is easy to see that [MATH] . To prove that the opposite reduction does not hold, apply the Invariance Principle: [MATH] has only computable outputs, whereas [MATH] maps some computable input to an incomputable output. |
For the following results, we need the Bolzano Weierstraß operators [MATH] , with [MATH] for [MATH] , and the operators [MATH] , which are the restrictions of the operators |
[MATH] to the sequences with compact range for which the accumulation point is unique. Proposition 3.4 For every [MATH] [MATH] Proof. |
[MATH] would imply [MATH] by Proposition 3.2 . Since [MATH] by BGM12 , Corollary 11.24] and obviously [MATH] , this would in turn imply [MATH] , which is impossible by |
BGM12 , Proposition 13.9] We now show that [MATH] is not non deterministically computable with finitely many mind changes: Proposition 3.5 |
[MATH] Proof. Recall that an operation [MATH] is non-uniformly computable , if [MATH] contains a computable solution for all computable [MATH] . A Weihrauch degree [MATH] |
is called low , if [MATH] . Both properties are preserved downwards under Weihrauch reduction. Notice that [MATH] is non-uniformly computable as all solutions are computable. Moreover [MATH] computes the characteristic function of the [MATH] -complete set |
[EQUATION] Therefore [MATH] . Since [MATH] and [MATH] is monotone, it follows that [MATH] is not low. On the other hand, [MATH] is low by BdBP12 , Theorem 8.7] , but not non-uniformly computable because |
[MATH] and there exist computable infinite binary trees without computable infinite branches. The incomparability of [MATH] and [MATH] |
then follows immediately. 4. Exact projections operators We start with the formal definition of the exact projection operators. Definition 4.1 |
Given a metric space [MATH] , a point [MATH] and a nonempty set [MATH] we say that [MATH] is a projection point of [MATH] onto [MATH] if |
[MATH] (where, as usual, [MATH] ). In other words, the projection points of [MATH] onto [MATH] are the points of [MATH] with minimal distance from [MATH] |
Notice that if [MATH] then [MATH] itself is the unique projection point of [MATH] onto [MATH] . Obviously, projection points of [MATH] onto [MATH] exist if and only if the infimum in the definition of [MATH] is actually a minimum. We will be mostly interested in the case where [MATH] is a Euclidean space and |
[MATH] is closed; in this situation projection points of any [MATH] onto [MATH] do exist. If [MATH] is a computable metric space, projections points give rise to several multi-valued functions, depending on the representation of [MATH] which we will always assume to be at least closed. |
Definition 4.2 Given a computable metric space [MATH] the (exact) negative, positive and total closed projection operators on [MATH] are the partial multi-valued functions [MATH] [MATH] and [MATH] which associate to every |
[MATH] (with Cauchy representation) and every closed [MATH] (with negative, positive and total representation, respectively) the set of the projection points of [MATH] onto [MATH] |
Thus [MATH] [MATH] , and [MATH] The (exact) negative, positive and total projections operators for compact sets are defined by replacing [MATH] [MATH] , and [MATH] |
with [MATH] [MATH] , and [MATH] respectively. These are denoted [MATH] [MATH] , and [MATH] The first obvious observation is that the negative projection operators compute the corresponding choice operators. |
Fact 4.3 [MATH] and [MATH] for all computable metric spaces [MATH] Projections on compact sets are special cases of projections on closed sets. |
Fact 4.4 For all computable metric spaces [MATH] (1) [MATH] (2) [MATH] (3) [MATH] Proof. The proof follows immediately by the definition of the representations. |
In some important cases, the inverse reduction holds as well. In the next result, a computable metric space [MATH] is computably compact when it is computable as a member of [MATH] , that is, it has some computable |
[MATH] -name (or, equivalently, of [MATH] ). Fact 4.5 For all computably compact metric spaces [MATH] (1) [MATH] (2) [MATH] (3) [MATH] |
Proof. The inverse reductions of Fact 4.4 can be obtained by fixing a finite cover of [MATH] by basic balls and use it to show that [MATH] |
[MATH] , and [MATH] are computable. Theorem 4.6 For [MATH] (1) [MATH] (2) [MATH] (3) [MATH] Proof. The inverse reductions of Fact 4.4 can be obtained as follows. |
We first deal with the positive representation. According to Remark 2.3 , let [MATH] and [MATH] . By Wei00 , Lemma 5.1.7] we can compute [MATH] as an element of |
[MATH] , hence we can determine a (natural) [MATH] . Given [MATH] we can also determine an upper bound [MATH] for [MATH] . Let [MATH] . Using Remark 2.5 , and since clearly [MATH] , it suffices to compute a dense sequence in [MATH] . This is not difficult, starting from the positive information for [MATH] : we list all... |
[MATH] . Notice that all projection points of [MATH] onto [MATH] belong to [MATH] Obviously, these points also are projections points of [MATH] onto [MATH] , so that an application of [MATH] to this new set releases a correct result. |
We now deal with the total representation. Let then [MATH] be given. We want to compute, as a suitable input for [MATH] , some compact [MATH] with total information such that the projections points of [MATH] |
onto [MATH] should be also projection points of [MATH] onto [MATH] . However, we cannot set [MATH] (with [MATH] the same of the previous case). This is because the possible elements in [MATH] that are not accumulation points of the dense set enumerated in [MATH] do not belong to [MATH] , but they are inevitably preserv... |
[MATH] . Thus, the two descriptions needed to provide the total information of [MATH] can fail to be coherent. To obtain consistent information for both types of information, we add to [MATH] the whole set [MATH] . Therefore we define [MATH] to be |
[EQUATION] The left hand term of the equation guarantees that a [MATH] -name of [MATH] can be effectively obtained, while the right hand side guarantees the same with respect to a [MATH] - name. Finally, use Remark |
2.5 and the fact that [MATH] to obtain a [MATH] -name of [MATH] as a member of [MATH] We now consider the negative representation with the goal of showing that |
[MATH] . We make use of the homeomorphism [MATH] between [MATH] and the open ball [MATH] defined by [EQUATION] We claim that [MATH] is computable. The critical points are the vectors [MATH] |
close to 0, but we can handle them as follows: until the test [MATH] fails and the parallel test [MATH] succeeds, we let [MATH] . Notice in fact, that for all [MATH] |
(including 0), [MATH] . Analogously, one can prove that [MATH] is also computable. Now suppose we are given [MATH] . We compute a compact set [MATH] as follows. The main idea is to use [MATH] to rescale [MATH] within the compact [MATH] . However, the function [MATH] produces unavoidable metric distortions, as [MATH] -i... |
[EQUATION] The second line provides a [MATH] definition of [MATH] with [MATH] and [MATH] as parameters, and hence (a name for) [MATH] is computed from (a name for) [MATH] and [MATH] . Since [MATH] by Remark 2.5 we have [MATH] |
Since [MATH] and [MATH] , the monotonicity of [MATH] implies that [MATH] if and only if [MATH] for all [MATH] . Thus the members of |
[MATH] are exactly those of the form [MATH] for some [MATH] . Therefore from [MATH] we can compute [MATH] . Notice that we are using |
[MATH] (which is part of the original input) in this final computation, so that we do not prove [MATH] Since we are interested mainly in projections in Euclidean spaces, Theorem |
4.6 allows us to concentrate on operators for closed sets. The proof of the next theorem shows that we can obtain upper bounds for all three exact projection operators by using essentially the same argument. |
Theorem 4.7 (1) For [MATH] [MATH] and [MATH] are non deterministically limit computable, that is [MATH] (2) For [MATH] [MATH] is non deterministically computable, that is |
[EQUATION] Proof. We first show (2). Given [MATH] and [MATH] we can compute [MATH] by Wei00 , Lemma 5.1.7] . We use this distance to compute first [MATH] as an element in [MATH] , and then [MATH] . This set obviously consists precisely of all projection points of [MATH] onto [MATH] |
We use then an upper bound [MATH] of [MATH] and an upper bound [MATH] of [MATH] to translate [MATH] into an element of [MATH] : it holds in fact that |
[MATH] Finally, to determine a projection point of [MATH] onto [MATH] , it suffices to select a point from this compact set. This is the only non computable step in the construction, but it is non deterministically computable by BG11a , Theorem 2.10] . This shows [MATH] , and [MATH] follows because [MATH] is a cylinder... |
When [MATH] is not provided with total information, we can initially use [MATH] to obtain the total information about [MATH] . For the input [MATH] , this follows by BG09 , Proposition 4.2] . For the input [MATH] this follows by BG09 , Proposition 4.5] (since [MATH] is effectively locally compact). The remainder of the... |
By BGM12 , Corollary 11.19] this means that [MATH] and again we obtain [MATH] because [MATH] is a cylinder (see BGM12 , Corollary 11.13] ). |
4.1. Exact negative projection operators In the previous section we have seen that [MATH] But is this reduction in fact an equivalence? This is indeed the case for [MATH] as the following result shows: |
Theorem 4.8 [MATH] for [MATH] Proof. Recall that by BGM12 , Corollary 11.7] [MATH] . Hence we can substitute in the proof [MATH] by [MATH] . Moreover, it suffices to work with [MATH] because the results for [MATH] follow by transitivity of |
[MATH] as [MATH] Fix the usual (and computable, by BGH15b , Lemma 7.1] ) embedding [MATH] . Moreover, let [MATH] be computable and such that [MATH] iff [MATH] for every [MATH] |
Throughout this proof it is convenient to represent points of [MATH] in polar coordinates (which we will write [MATH] ). This does not cause any problem, because we use only points with radial coordinate not smaller than [MATH] and angular coordinate in the interval [MATH] : for such points both directions of the conve... |
Without loss of generality, we are given as input a sequence of trees [MATH] converging to an infinite binary tree [MATH] and we want to find, using [MATH] , an infinite path in [MATH] . To achieve this goal we compute a closed subset [MATH] of [MATH] such that if [MATH] , then [MATH] is an infinite path in [MATH] |
[MATH] is defined as the intersection of closed sets [MATH] To describe the [MATH] , for each [MATH] and [MATH] , let us denote by [MATH] the closed set [MATH] (here, as usual, [MATH] denotes that the finite binary string [MATH] is an initial segment of [MATH] ). In words, [MATH] is obtained by removing from [MATH] the... |
At stage [MATH] , for every [MATH] we let [MATH] be the cardinality of the set [EQUATION] We then define, for every [MATH] [EQUATION] |
Eventually, we let [EQUATION] In this way we compute [MATH] and we need to show that if [MATH] then [MATH] is a path in [MATH] Since [MATH] converges, for every [MATH] the sequence [MATH] is non-decreasing and eventually takes a constant value [MATH] . Therefore the sequences [MATH] and [MATH] stabilize at [MATH] and [... |
If [MATH] then the ray starting at [MATH] and moving in direction [MATH] meets [MATH] at distance [MATH] from [MATH] . To see this notice first that if [MATH] then [MATH] . Thus, even when for some [MATH] with [MATH] we deleted the ray in direction |
[MATH] up to [MATH] , at some later stage [MATH] (such that [MATH] and so [MATH] ) the deletion up to [MATH] superseded it. If instead [MATH] and [MATH] is least such that [MATH] then the ray starting at [MATH] and moving in direction [MATH] |
meets [MATH] at distance [MATH] from [MATH] (because at a stage [MATH] such that [MATH] we delete the ray up to distance [MATH] ). |
It thus suffices to check that [MATH] for every [MATH] . Indeed we have [EQUATION] Thus every point in [MATH] is in direction [MATH] for some [MATH] , as required. |
Corollary 4.9 [MATH] for [MATH] Proof. By Theorem 4.7 .(1) and Theorem 4.8 For case [MATH] we do not obtain the full power of [MATH] . We can prove however that a characterization for the one dimensional case can be found in terms of [MATH] . As a preliminary result, we prove: |
Proposition 4.10 [MATH] Proof. Analogously to the treatment of negative information in the proof of Theorem 4.6 , given [MATH] and [MATH] we can compute the set |
[EQUATION] Notice that for such a set both [MATH] and [MATH] always exist, which would not hold true in general for our original [MATH] Moreover [MATH] . Since [MATH] |
[MATH] for all [MATH] . More precisely, as in the proof of Theorem 4.6 , the members of [MATH] are exactly those of the form [MATH] for some [MATH] Recalling that [MATH] we can determine [MATH] as an element of |
[MATH] and [MATH] as an element of [MATH] . We then use [MATH] to obtain [MATH] . Let now denote as [MATH] the function mapping any given [MATH] for which the elements [MATH] defined as above exist to the pair [MATH] . We have then just proved that |
[MATH] Consider now the function [MATH] such that [MATH] and [MATH] . It is easy to see that [MATH] (an application of [MATH] finds [MATH] such that |
[MATH] , then we use the input [MATH] of [MATH] , which is still available by definition of [MATH] , to recover the value of [MATH] ). |
Let now [MATH] . Then, in virtue of what observed above, [MATH] This shows that [MATH] (the transformation [MATH] was indeed computable uniformly in [MATH] and notice also that, by definition of [MATH] , the original [MATH] is still available after the application of [MATH] , hence it can be used to compute [MATH] ). |
By definition of compositional product, [MATH] for every [MATH] and [MATH] . Therefore [MATH] For the next result we need to use the Sierpinski space and its ordinary admissible representation: |
Definition 4.11 (Sierpinski space) The Sierpinski space is given by the topology [MATH] on the set [MATH] As a represented space, the Sierpinski space is equipped with the representation [MATH] and [MATH] for |
[MATH] In other words, [MATH] can be seen as the identity function [MATH] , where the codomain is equipped with the discrete topology. |
Lemma 4.12 [MATH] Proof. Consider the space [MATH] . As [MATH] , for the identity function [MATH] we find that [MATH] , as obviously |
[MATH] and moreover [MATH] BG11b , Lemma 6.3] BGP , Theorem 6.7] ). In the statement we can thus replace [MATH] with [MATH] Notice now that the computable embedding [MATH] we already used in the proof of Theorem 4.8 gives naturally rise to a corresponding computable embedding [MATH] . Recall that |
[MATH] preserves the order on binary sequences, hence [MATH] does the same. Finally, observe that [MATH] , the lifted version of |
[MATH] such that [MATH] coincides with [MATH] , is computable. In the following, by notational abuse, we identify [MATH] with [MATH] , that is, we will not distinguish a binary sequence [MATH] from any [MATH] such that [MATH] . This produces no ambiguity for |
[MATH] and [MATH] that still remain single-valued, whereas the single-valuedness of [MATH] can be preserved by its replacement with a computable realizer. For instance, we will see |
[MATH] as defined by [MATH] , and [MATH] as being of the form [MATH] with [MATH] Now, given inputs [MATH] , we compute [MATH] and |
[MATH] , where [EQUATION] for [MATH] and [MATH] . Since [MATH] coincides with [MATH] for all [MATH] , we notice that [MATH] if and only if only if [MATH] contains infinitely many [MATH] (in this case indeed [MATH] ), and [MATH] if and only if [MATH] |
contains infinitely many [MATH] (in this case indeed [MATH] ). Given now [MATH] and [MATH] , we can compute [EQUATION] From [MATH] we can use |
[MATH] to compute [MATH] and then [MATH] . Moreover, the sign of [MATH] yields a valid answer to [MATH] (notice that the sign of [MATH] is always decidable, as they are necessarily different from [MATH] , which is not necessarily the case for [MATH] and [MATH] ). |
Through the notion of jump that we recalled in Section 2.3 we are now able to characterize [MATH] in terms of [MATH] Corollary 4.13 |
[MATH] Proof. This follows by Proposition 4.10 and Lemma 4.12 since [MATH] BGM12 , Corollary 11.11] ) and since for generic multi-valued functions [MATH] it holds [MATH] |
4.2. Exact positive projection operators Quite surprisingly, for the projections with positive information for closed sets we obtain the same characterizations obtained for the case of negative information. We start with the dimensions [MATH] for which we are still able to prove the equivalence with [MATH] |
Proposition 4.14 [MATH] for [MATH] Proof. We prove the statement for [MATH] . As before, the results for [MATH] follow by transitivity of [MATH] as [MATH] . As in the proof of Theorem 4.8 , also here it is convenient to represent points of [MATH] in polar coordinates. In this case we use only points with radial coordin... |
[MATH] , so that again for our purposes both directions of the conversion between Cartesian and polar coordinates are computable. |
Let [MATH] be given as input. We want to find a cluster point of this sequence. We consider the points [MATH] . Let now [MATH] . Notice that [MATH] because [MATH] is bounded, while [MATH] if and only if |
[MATH] is a cluster point of [MATH] . Thus if [MATH] then [MATH] and [MATH] Corollary 4.15 [MATH] for [MATH] Proof. By Theorem 4.7 .(1) and Proposition 4.14 |
For [MATH] , by reasoning analogously to the case of negative information, we obtain the same characterization in terms of [MATH] . We start with the following result, which is an analoguous of Proposition 4.10 |
Proposition 4.16 [MATH] Proof. Let [MATH] and [MATH] be given. We can now compute [EQUATION] Notice that both [MATH] and [MATH] always exist, which would not hold true in general for our original [MATH] . Moreover [MATH] . Recalling that |
[MATH] we can determine [MATH] as an element of [MATH] and [MATH] as an element of [MATH] . Analogously to the proof of Proposition |
4.10 , we can then use [MATH] to obtain [MATH] and consequently [MATH] to determine [MATH] such that [MATH] . As observed, this gives a member of |
[MATH] Lemma 4.17 [MATH] Proof. The proof is almost the same of that of Lemma 4.12 . But the replacement of the negative representation for closed sets with its dual requires to switch the positions of [MATH] and [MATH] with respect to [MATH] . We compute then the new set [MATH] . From [MATH] we can then extract [MATH]... |
Corollary 4.18 [MATH] Proof. This follows by Proposition 4.16 and Lemma 4.17 , analogously to the proof of Corollary 4.13 4.3. Exact total projection operators |
For the case of total information we can fully characterize the Weihrauch degree of [MATH] already for [MATH] . We start by determining the following upper bound: |
Proposition 4.19 [MATH] Proof. Let the input [MATH] be given with [MATH] . It holds obviously that [MATH] , and in fact [MATH] for |
[MATH] . By using the total information on [MATH] we can compute the exact value of [MATH] via approximations that become at every stage more reliable. We produce then a valid input [MATH] for [MATH] in the following way. At stage [MATH] , by considering the initial segment of the negative information on [MATH] that we... |
still are plausible candidates as members of [MATH] we let [MATH] Otherwise, suppose that we realize that one of the two points, say [MATH] , is not in [MATH] . Then we put [MATH] . We then let [MATH] |
for all [MATH] . If instead we realize that [MATH] then we switch the roles of [MATH] and [MATH] Given now [MATH] we compute (again) |
[MATH] or [MATH] , depending on whether [MATH] or [MATH] , finding an element of [MATH] Notice that in the above proof the use of the original input after the application of [MATH] is essential: [MATH] cannot hold for mere cardinality reasons. But the opposite reduction even holds for the strong version of Weihrauch re... |
Proposition 4.20 [MATH] Proof. Let [MATH] . We construct then a valid input [MATH] for [MATH] according to the following idea: if a point of |
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