A comparison of concepts from computable analysis and effective descriptive set theory
Abstract
Computable analysis and effective descriptive set theory are both concerned with complete metric spaces, functions between them and subsets thereof in an effective setting. The precise relationship of the various definitions used in the two disciplines has so far been neglected, a situation this paper is meant to remedy.
As the role of the Cauchy completion is relevant for both effective approaches to Polish spaces, we consider the interplay of effectivity and completion in some more detail.
1 Introduction
Both computable analysis (Weihrauch [35, 38]) and effective descriptive set theory (Moschovakis [18]) have a notion of computability on (complete, separable) metric spaces as a core concept. Nevertheless, the definitions are prima facie different, and the precise relationship has received little attention so far (contrast e.g. the wellestablished connections between Weihrauch’s and PourEl & Richard’s approach [29] to computable analysis).
The lack of exchange between the two approaches becomes even more regrettable in the light of recent developments that draw on both computable analysis and descriptive set theory:

The study of Weihrauch reducibility often draws on concepts from descriptive set theory via results that identify various classes of measurable functions as lower cones for Weihrauch reducibility [2, 3, 27]. The Weihrauch lattice is used as the setting for a metamathematical investigation of the computable content of mathematical theorems [4, 10, 22].

In fact, Weihrauch reducibility was introduced partly as an analogue to Wadge reducibility for functions (see the original papers by Weihrauch [36, 37, 38] and subsequent work by Hertling [12]), and as such, can itself be seen as a subfield of (effective) descriptive set (or rather function) theory.

The QuasiPolish spaces [6] introduced by de Brecht allow the generalization of many results from descriptive set theory to a much larger class of spaces (e.g. [7, 20]), and admit a very natural characterization in terms of computable analysis as those countably based spaces with a total admissible Bairespace representation.

Even more so, the suggested synthetic descriptive set theory [26, 28] by the third author and de Brecht would extend some fundamental results from descriptive set theory even further, to general represented spaces [23]. This could pave the way to apply some very strong results by Kihara [15] to the longoutstanding questions regarding generalizations of the JayneRogers theorem [13, 21, 14, 34].
Our goal with the present paper is to facilitate the transfer of results between the two frameworks by pointing out both similarities and differences between definitions. For example, it turns out that the requirements of an effective metric space (as used by Moschovakis) are strictly stronger than those Weihrauch imposes on a computable metric space – however, this is only true for specific metrics, by moving to an equivalent metric, the stronger requirements can always be satisfied. Hence, effective Polish spaces and computable Polish spaces are the same concept.
Besides the fundamental layer of metric spaces, we shall also consider the computability structure on hyperspaces such as all measurable subsets of some given Polish spaces. While these spaces do not carry a meaningful topology, they can nevertheless be studied as represented spaces. This was done implicitly in [18], and more explicitly in [2, 33, 27] and [26, 28].
As a digression, we will consider a more abstract view point on the Cauchy completion to illuminate the different approaches to metric spaces.
2 Effective Polish Spaces and Computable Polish Spaces
We begin by contrasting the definitions of the fundamental structure on metric spaces used to derive computability notions; Moschovakis defines a recursively presented metric space (RPMS) and Weihrauch a computable metric space (CMS). Throughout the text, by we denote some standard bijection.
Definition(Moschovakis [18]) 1 (3b).
Suppose is a separable, complete metric space with distance function . A recursive presentation of is any function whose image is dense in and such that the relations
are recursive.
A recursively presented metric space is a triple as above. To every recursively presented metric space we assign the nbhd system , where
and is the empty set if does not have the form .
When referring to recursively presented metric space, we usually omit the metric and the recursive presentation, and simply write .
Definition(Weihrauch [38]) 2 (cf 8.1.2).
We define a computable metric space with its Cauchy representation as follows:

An effective metric space is a tuple such that is a metric space and is a dense sequence in .

The Cauchy representation associated with the effective metric space is defined by

Finally, a computable metric space is an effective metric space such that the following relation (involving a standard numbering )
Both definitions can only ever apply to separable metric spaces, however, a noticeable difference is Moschovakis’ requirement of completeness, which is not demanded by Weihrauch. This is only a superfluous distinction, though:
Observation 3.
If is a computable metric space (CMS) with a Cauchy representation then its completion (where is the expanded distance function for the completion, specifically ) is also a CMS.
A more substantial difference lies in the decidabilityrequirement of distances between basic points and rational numbers. For Weihrauch’s definition, being able to semidecide and is enough, whereas Moschovakis demands these to be decidable. By identifying^{1}^{1}1That this identification actually makes sense follows from the investigation of the class of computable functions between spaces in Section 3. and , we immediately find:
Observation 4.
Every recursively presented metric space is a computable metric space.
The converse fails in general:
Example 5.
Consider the following CMS: Let the base set be , the dense set also (with a standard bijection) and the distance function be defined as follows (assuming is the th element of the first copy of , from the second):
Where is the step count of the th Turing machine started with no arguments if it halts
if it does not. Then, to ensure the validity of the triangle inequality, we set
This space is a CMS but not an RPMS.
Proof.
To output the upper bound one only has to simulate , the th program for steps, if it did not halt yet, output , if it did halt it will be a lower bound. We can avoid outputting the exact term for the exact step count in case it halts. Similarly we semidecide the other types of distances.
This will form a CMS (with the representation of eventually constant sequences of points).
Suppose towards a contradiction that admits a recursive presentation . Since the set is dense in and the latter space is discrete we have that is surjective. It follows easily that there exists a recursive function such that and .
The decidabilityrequirements now imply in particular that is a decidable property – but by the construction of , this would mean that the Halting problem is decidable, providing the desired contradiction. ∎
We will proceed to find a weaker counterpart to Observation 4. First, note that in a recursively presented metric space we can decide whether , whereas we cannot decide in a computable metric space. It is possible, however, to avoid having duplicate points in the dense sequence even in the latter case. First we shall provide a general criterion for when two dense sequences give rise to homeomorphic computable metric spaces (which presumably is a folklore result):
Lemma 6.
For two CMSs if is uniformly computable in then the
identity function is computable.
Proof.
The assumption means that given , we can compute some such that . Now we are given some via some sequence such that . Consider the sequence . We find that ; thus this sequence constitutes a name for . ∎
In general, we shall write iff there is a bijection such that and are computable. In this paper, will generally be the identity on the underlying sets.
Corollary 7.
For two CMSs if is uniformly computable in and is uniformly computable in then
As a slight detour, we will prove a more general, but ultimately too weak result. We recall from [23] that a represented space is called computably Hausdorff, if inequality is recognizable. Inequality is (computably) recognizable in a represented space X iff the function is computable, where is the Sierpiński space (with underlying set and open sets , ) and and otherwise. Note that every computable metric space is computably Hausdorff. We define a multivalued map by and iff and . In words, RemoveDuplicates takes a sequence with infinite range, and produces a sequence with the same range but without duplicates.
Proposition 8.
Let be computably Hausdorff. Then is computable.
Proof.
Given a sequence in a computable Hausdorff space, we can compute , i.e. as a recursively enumerable set (relative to the sequence). By assumption on the range of the sequence, this set is infinite. It is a basic result from recursion theory that any infinite recursively enumerable set is the range of an injective computable function, and this holds uniformly. Let be such a function. Then satisfies the criteria for the output. ∎
The combination of Lemma 6 and Proposition 8 allows us to conclude that for any infinite computable metric space , there is a computable metric space with the same underlying set and metric, and a repetitionfree dense sequence such that is computable – but we cannot guarantee computability of thus. Consequently, we shall employ a more complicated construction:
Theorem 9.
For any infinite CMS , there is a repetitionfree sequence such that is a CMS with .
Proof.
We will first describe an algorithm obtaining the sequence from the original sequence .

At any stage, let be the finite prefix sequence of the emitted so far. We also keep track of a precision parameter , starting with .

In the first stage, we emit into (i.e. we set )

Do the following iteration:

Take the next element from and place it in an auxiliary set , increment

For all elements , we can compute the number where is the finite sequence of s emitted so far.

For each , check (nondeterministically) in parallel: if skip , if emit , remove from , emit as an (thus also suffix it on ), repeat.

If all elements in were skipped, repeat.

The parallel test in is a common trick in computable analysis. The relations by themselves are not decidable, but as at least one of them has to be true, we can wait until we recognize a true proposition. If there are multiple such that lies between and , then the choice is nondeterministic in the high level view of real numbers as inputs. If all codings and implementations are fixed, then the choice here is determined, too, though.
First, we shall argue that is dense and repetition free. If were not dense, then there would be some such that . However, then once has been placed into and incremented beyond , would have been chosen for – contradiction. The sequence cannot have repetitions, because a duplicate element could never satisfy the test in .
In remains to show that is computable in and vice versa. From Corollary 7 we would then know that .
By construction is computable in : Given , just follow the construction above in order to identify which is the th element to be put into , then we have .
Now to prove that is computable in ; i.e. that given some we can compute a sequence such that . For this, we inspect the algorithm above beginning from the point when is put into . If is moved into as the th element to enter , then , and we can continue the sequence as the constant sequence . If is not moved into in the th round, then this is due to , and there must be some such that witnesses this distance, i.e. . Thus, continuing the sequence with works. ∎
In [11] it is proved that for every recursively presented metric space there exists a recursive real such that the metric takes values in on the dense sequence. This idea combined with Theorem 9 gives the following result.
Theorem 10.
For every CMS there is a CMS with a computable real such that , and satisfies the criteria for a recursively presented metric space.
Proof.
If is finite, the result is straightforward. If is infinite, we may assume by Theorem 9 that is repetitionfree. Let the following be computable bijections:
Then consider the computable sequence defined via . We can diagonalize to find a computable real number not in with . By choice of , we find for , hence, the problematic case in the requirements for a recursively presented metric space becomes irrelevant. To compute the identity , one just needs to map a fast Cauchy sequence to (as ), the identity in the other direction does not require any changes at all. ∎
3 The induced computability structures
Having proved that computable and recursively presented metric spaces are interconnected through a computable rescaling of the metric, it is natural to compare some of the basic objects derived from them. There are three fundamental types of objects in recursively presented metric spaces: recursive sets, functions and points, all of which have the corresponding analogue in computable metric spaces. We will see that these concepts do coincide.
A comparison of the computability structure induced by recursive presentations and computable metric spaces respectively is more illuminating in the framework of represented spaces. We recall some notions from [23], and then prove some basic facts about them – these results are known and included here for completeness. A represented space is a pair of a set and a partial surjection . A multivalued function between represented spaces is a multivalued function between the underlying sets. For and , we call a realizer of (notation ), iff for all . A map between represented spaces is called computable (continuous), iff it has a computable (continuous) realizer. Similarly, we call a point computable, iff there is some computable with . Any computable metric space induces a represented space via its Cauchy representation, and a function between computable metric spaces is called computable, iff it is computable between the induced representations. Note that the realizerinduced notion of continuity coincides with ordinary metric continuity is a basic fact about admissible representation [38].
The category of represented spaces is cartesian closed, meaning we have access to a general function space construction as follows: Given two represented spaces , we obtain a third represented space of functions from to by letting be a name for , if the th Turing machine equipped with the oracle computes a realizer for . As a consequence of the UTM theorem, is the exponential in the category of continuous maps between represented spaces, and the evaluation map is even computable.
Still drawing from [23], we consider the Sierpiński space , which allows us to formalize semidecidability. An explicit representation for this space is where and for . The computable functions are exactly those where is recursively enumerable (and thus corecursively enumerable). In general, for any represented space we obtain two spaces of subsets of ; the space of open sets by identifying with , and the space of closed sets by identifying with . In particular, the computable elements of are precisely the recursively enumerable sets. An explicit representation for is found in defined via iff .
The focus of computable analysis has traditionally been on the computable admissible spaces. Following Schröder [32] we call a space computably admissible, iff the canonic map has a computable inverse. This essentially means that a point can be effectively recovered from its neighborhood filter. The computably admissible spaces are those represented spaces that correspond to topological spaces.
We proceed to introduce the notion of an effective countable base. The effectively countably based computable admissible spaces are exactly the computable topological spaces studied by Weihrauch (e.g. [30, Definition 3.1]). The countably based admissible spaces that admit a total Baire space representation are the QuasiPolish spaces introduced by de Brecht [6].
Definition 11.
An effective countable base for is a computable sequence such that the multivalued partial map is computable. Here and iff . Note that the requirement on Base also gives that forms a basis of .
Proposition 12 ((^{2}^{2}2As mentioned in the introduction to this section, this result is folklore. It has appeared e.g. as [16, Theorem 2.3].)).
Let be a CMS. Then provides an effective countable base for .
Proof.
We start to prove that this is a computable sequence. By the definition of , it suffices to show that given , , we can recognize . Let , i.e. . Now iff . By the conditions on a CMS, the property is r.e., and existential quantification over an r.e. property still produces an r.e. property.
Next, we need to argue that Base is computable. Given some and some open set with , we do know by definition of that will be recognized at some finite stage. Moreover, we can simulate the computation until this happens. At this point, only some finite prefix of the name of has been read, say of length . But then we must have . It is easy to verify that we can identify a particular ball inside the intersection still containing . ∎
We now have the ingredients to give a more specific characterization of both and for countably based spaces , and computably admissible .
Proposition 13.
Let have an effective countable base and let be a computable sequence that is dense in . Then the map defined via is computable and has a computable multivalued inverse.
Proof.
That the map is computable follows from [23, Proposition 4.2(4), Proposition 3.3(4)]. For the inverse, fix some computable realizer of Base. Given some , test for any if . If this is confirmed, compute and list it in .
We will now argue that with the algorithm described above. If , then by construction , hence . On the other hand, let . The realizer for Base will choose some on input , . As this happens after some finite time, there is some so close to that the realizer works in exactly the same way^{3}^{3}3For this, it is important to fix one realizer of Base and to use the same name of for all calls.. This ensures that is listed in , thus . ∎
Proposition 14.
Let be computably admissible and have an effective countable base , and let be a computable sequence that is dense in . Then is a computable embedding.
Proof.
That the map is computable is straightforward. For the inverse, we shall first argue that is computable. By typeconversion, this is equivalent to . Here we understand if and if . By employing Proposition 13, this follows from being computable.
Finally, we can compute from as is computably admissible. ∎
As a consequence of Proposition 13, we see that for countably based spaces , we may conceive of open sets being given by enumeration of basic open sets exhausting them. For computable metric spaces in particular, an open set is given by an enumeration of open balls with basic points as centers and radii of the form (or equivalently, rational radii):
Definition(Weihrauch [38]) 15 (cf 4.1.2).
Given a computable metric space , we define a numbering I for the open balls with basic centers and radii of the form via . For convenience, we shall assume that .
Definition(Weihrauch [38]) 16 (5.1.15.4).
Given a computable metric space , we define the representation by
which is intuitively a name consisting of the descriptions of open balls that exhaust the particular set (but not necessarily all of them).
An open is computably open if for some recursive .
The analogous notion in effective descriptive set theory is the following.
Definition(Moschovakis [18]) 17 (1b.1).
Given a recursively presented metric space , a pointset is semirecursive (or else ) if
with some recursive
The following lemma is simple but useful tool.
Lemma 18.
Suppose that is a recursively presented metric space, which by Obesrvation 4 is a computable metric space. Then there exist computable functions
such that
for all .
Proof.
The existence of such a function follows easily since in the definitions we use computable encoding. Regarding , we first claim that
where .
To see the latter, assume first that , where . Then . We choose large enough such that , from which it follows that . Now we consider some . Clearly belongs to , and
hence . This also implies that and so . Hence is a suitable pair of naturals such that . Conversely if are such that , and is a member of , then we have that
and we have proved the preceding equality. Now we consider some recursive function such that and we define
We also let be if the naturals do not have the form above. ∎
We are now ready to compare the notions of computablyopen and semirecursive set.
Theorem 19.
Suppose that is a recursively presented metric space, is a computable metric space, and is as in Theorem 10. Then:

For every ,
(recall that is recursively presented).
In particular, the family of all semirecursive subsets of a recursively presented metric space is also the family of all computably open subsets of the latter space; and the family of all computably open subsets of a computable metric space is the family of all semirecursive subsets of a recursive presented metric space, which is equivalent to the original one.
Proof.
The second assertion follows from the first one and the fact that the metric space is recursively presented, so let us prove the first assertion. Let and assume that is semirecursive. Then for some recursive . From Lemma 18 we have that
and is computably open from the closure properties of the latter class of sets, cf. [23, Proposition 6 (4)]. The converse follows again from Lemma 18 by using the function and the closure properties of semirecursive sets, cf. [18] 3C.1 (closure under ). ∎
We now shift our attention to computable/recursive functions.
Proposition 20.
For two represented spaces , the map is computable. If is computably admissible, then this map admits a computable inverse.
Proof.
That is computable follows by combining computability of and computability of and using type conversion.
For the inverse direction, recall that for computably admissible the map is computable. By computability of defined via and composition, we find that is computable. Currying produces the claim. ∎
Corollary 21.
Let be computably admissible. Then is computable iff is computable.
Lemma 22.
is computable.
Proof.
We start with . That this map is computable follows from open sets being effectively closed under products and intersection.
A similar characterization of computability of functions is used in effective descriptive set theory:
Definition(Moschovakis [18]) 23 (3d).
A function is recursive if and only if the neighborhood diagram of defined by
is semirecursive.
Theorem 24.
Suppose that , are recursively presented metric spaces, , are computable metric spaces with being admissible, and , are as in Theorem 10. Then:

For every , is recursive exactly when is computable.

For every , is computable (equivalently computable) exactly when is recursive.
Proof.
As before the second assertion follows from the first one and the fact that the metric spaces , , are recursively presented. Regarding the first one, if is semirecursive then it is computably open by Theorem 19. By [23, Proposition 6(7)], the map is computable. Thus, is computable. Lemma 18 together with Proposition 14 show that we can compute from . As the composition of computable functions is computable, we conclude that is computable.
Conversely if is computable then from Corollary 21 it follows that is computable. Using the characterization of the open sets in Proposition 13 together with Lemma 18 shows that is computable. Then Lemma 22 implies that is computably open, and so from Theorem 19 we have that is semirecursive, i.e., is a recursive function. ∎
Finally we deal with points. A point in a computable metric space is defined to be computable if it has a computable name, i.e. if it is the limit of a computable fast Cauchy sequence. On the other hand, point in a recursively presented metric space is recursive if the set is semirecursive, cf. the comments preceding 3D.7 [18].
Theorem 25.
Suppose that is a recursively presented metric space, is a computable metric space, and is as in Theorem 10. Then:

For every , is recursive exactly when it is computable.

For every , is computable (equivalently )computable exactly when it is recursive.
4 On Cauchycompletions
As a digression, we shall explore the role Cauchycompletion plays in obtained effective versions of metric spaces. An effective version of Cauchycompletion underlies both the definition of computable metric spaces and recursively presented metric spaces. A crucial distinction, though classically vacuous, lies in the question whether spaces embed into their Cauchycompletion. Our goal in this section is to explore the variations upon effective Cauchycompletion, and to subsequently understand the origin of the discrepancy exhibited in Section 2.
Given a represented space and some metric on , we define the space of fast Cauchy sequences by iff . If is complete, the map is of natural interest (if is not complete, we can still study as a partial map). In fact, it can characterize admissibility as follows:
Proposition 26.
Let admit a computable dense sequence. Let be a computable metric, and let be computable. Then is computably admissible.
Proof.
To show that is computably admissible, we need to show that is computable. We search for some point such that . Then we search for with etc. These points form a fast Cauchy sequence converging to . ∎
Proposition 27.
Let be computably admissible and let contain some computable dense sequence. Let be a computable metric, and let be an effective countable basis for . Then is computable.
Proof.
We are given some fast Cauchy sequence converging to some with as input. As and , we can compute . Then we can invoke Proposition 14 to extract . ∎
This characterization of computable metric spaces in terms of fast Cauchy limits of course presupposes the represented space with its canonical structure. In the beginnings of computable analysis, various nonstandard representations of have been investigated. We will investigate what happens to Cauchy completions, if some other represented space (with again the reals as underlying set) is used in place of .
Definition 28.
Let be a represented space, such that the metric is computable. We obtain its Cauchyclosure by taking the usual quotient of .
Observation 29.
Any computable metric space embeds^{4}^{4}4A computable embedding is a computable injection such that the partial inverse is computable, too. into its Cauchyclosure , and can be extended canonically to . Definition 2 reveals that a complete computable metric space is the Cauchyclosure of a countable metric space with continuous metric into .
In order to find a contrasting picture of the recursively presented metric spaces, we first introduce the represented space . Informally, any real number is encoded by its decimal expansion, with infinite repetitions clearly marked^{5}^{5}5For example, the unique name of is . The number has the names and .. This just ensures that and become both decidable for and .
Observation 30.
The space does not embed into . Let be a computable metric. In general, may fail to be computable. Definition 1 reveals that a recursively presented metric space is (essentially) the Cauchyclosure of a countable metric space with continuous metric into .
Proof.
The claims all follow from the observation that . ∎
5 Representations of point classes
With a correspondence of the spaces, the continuous/computable functions and the open sets in place, we shall conclude this paper by considering higherorder classes of sets (typically called pointclasses), such as sets (), Borel sets or analytic sets. These have traditionally received little attention in the computable analysis community, with the exception of [2] by Brattka and [33] by Selivanov. One reason for this presumably was the focus on admissible representations, i.e. spaces carrying a topology – and the natural representations of these classes of sets generally fail to be admissible. The ongoing development of synthetic descriptive set theory does provide representations of all the natural pointclasses.
In descriptive set theory the usual representation of pointclasses is through universal sets and good universal systems. Let be a pointclass, and , two spaces^{7}^{7}7Usually the spaces involved would be restricted to Polish spaces. However, the formalism is useful for us in a more general setting.. For any and , we write . We write for all the subsets of . Now we call a universal set for and iff .
If is a represented space and a universal set for and , then we obtain a representation of via