# Game Characterizations and Lower Cones in the Weihrauch Degrees

## Abstract

We introduce generalized Wadge games and show that each lower cone in the Weihrauch degrees is characterized by such a game. These generalized Wadge games subsume (a variant of) the original Wadge game, the eraser and backtrack games as well as Semmes’s tree games. In particular, we propose that the lower cones in the Weihrauch degrees are the answer to Andretta’s question on which classes of functions admit game characterizations. We then discuss some applications of such generalized Wadge games.

## 1 Introduction

The use of games in set theory has a well-established tradition, going back to work by Banach, Borel, Zermelo, Kőnig, and others (see [16, Sect. 27] for a thorough historical account of the subject). In particular, games which go on for infinitely many rounds have taken a prominent role in the field especially since the work of Gale and Stewart on the determinacy of certain types of such games [12].

In this paper, we will focus on infinite games which have been used to characterize classes of functions in descriptive set theory. Interest in this particular area began with a re-reading of the seminal work of Wadge [40], who introduced a game in order to analyze a notion of reducibility—*Wadge reducibility*—between subsets of Baire space. In the variant—which by a slight abuse we call the *Wadge game*—two players, \(\mathbf {I}\) and \(\mathbf {II}\), are given a partial function \(f:\subseteq {{\mathbb {N}^\mathbb {N}}} \rightarrow {{\mathbb {N}^\mathbb {N}}}\) and play with perfect information for \(\omega \) rounds. In each run of this game, at each round player \(\mathbf {I}\) first picks a natural number and player \(\mathbf {II}\) responds by either picking a natural number or passing, although she must pick natural numbers at infinitely many rounds. Thus, in the long run \(\mathbf {I}\) and \(\mathbf {II}\) build elements \(x \in {\mathbb {N}^\mathbb {N}}\) and \(y\in {\mathbb {N}^\mathbb {N}}\), respectively, and \(\mathbf {II}\) wins the run if and only if \(x \not \in \mathop {\mathrm {dom}}(f)\) or \(f(x) = y\). It can be considered a folklore result, also implicit in Wadge’s work, that this game *characterizes* the continuous functions in the following sense.

### Theorem 1

A partial function \(f:{{\mathbb {N}^\mathbb {N}}} \rightarrow {{\mathbb {N}^\mathbb {N}}}\) is relatively continuous iff player \(\mathbf {II}\) has a winning strategy in the Wadge game for *f*.

By adding new possibilities for player \(\mathbf {II}\) at each round, one can obtain games characterizing larger classes of functions. For example, in the *eraser game* (implicit in [11]) characterizing the Baire class 1 functions, player \(\mathbf {II}\) is allowed to erase past moves, the rule being that she is only allowed to erase each position of her output sequence finitely often. In the *backtrack game* (implicit in [44]) characterizing the functions which preserve the class of \(\mathbf {\Sigma }^0_{2}\) sets under preimages, player \(\mathbf {II}\) is allowed to erase *all* of her past moves at any given round, the rule in this case being that she only do this finitely many times.

In his PhD thesis [39], Semmes introduced the *tree game* characterizing the class of Borel functions in Baire space. Player \(\mathbf {I}\) plays as in the Wadge game and therefore builds some \(x \in {\mathbb {N}^\mathbb {N}}\) in the long run, but at each round *n* player \(\mathbf {II}\) now plays a finite *labeled tree*, i.e., a pair \((T_n,\phi _n)\) consisting of a finite tree \(T_n \subseteq \mathbb {N}^{<\mathbb {N}}\) and a function \(\phi _n: {T_n {\smallsetminus } \{{\langle \rangle }\}} \rightarrow {\mathbb {N}}\), where \({\langle \rangle }\) denotes the empty sequence. The rules are that \(T_n \subseteq T_{n+1}\) and \(\phi _n \subseteq \phi _{n+1}\) must hold for each *n*, and that the *final* labeled tree \((T,\phi ) = (\bigcup _{n \in \mathbb {N}} T_n,\bigcup _{n\in \mathbb {N}}\phi _n)\) must be an infinite tree with a unique infinite branch. Player \(\mathbf {II}\) then wins if the sequence of labels along this infinite branch is exactly *f*(*x*). By providing suitable extra requirements on the structure of the final tree, Semmes was able to obtain a game characterizing the Baire class 2 functions, and although this is not done explicitly in [39], it is not difficult to see that restrictions of the tree game also give his *multitape game* characterizing the classes of functions which preserve \(\mathbf {\Sigma }^0_3\) under preimages and the *multitape eraser game* characterizing the class of functions for which the preimage of any \(\mathbf {\Sigma }^0_2\) set is a \(\mathbf {\Sigma }^0_3\) set.

As examples of applications of these games, Semmes found a new proof of a theorem of Jayne and Rogers characterizing the class of functions which preserve \(\mathbf {\Sigma }^0_2\) under preimages and extended this theorem to the classes characterized by the multitape and multitape eraser games, by performing a detailed analysis of the corresponding game in each case.

Given the success of such game characterizations, in [1] Andretta raised the question of which classes of functions admit a characterization by a suitable game. Significant progress towards an answer was made by Motto Ros in [23]: Starting from a general definition of a reduction game, he shows how to construct new games from existing ones in ways that mirror the typical constructions of classes of functions (e.g., piecewise definitions, composition, pointwise limits). In particular, Motto Ros’s results show that all the usual subclasses of the Borel functions studied in descriptive set theory admit game characterizations.

In order to arrive at a full characterization of the classes of functions characterizable by a game, we need to find the appropriate language to formulate such a result. Weihrauch reducibility (in its modern form) was introduced by Gherardi and Marcone [13] and Brattka and Gherardi [2, 3] based on earlier work by Weihrauch on a reducibility between sets of functions analogous to Wadge reducibility [41, 42].

We will show that game characterizations and Weihrauch degrees correspond closely to each other. We can thus employ the results and techniques developed for Weihrauch reducibility to study function classes in descriptive set theory, and vice versa. In particular, we can use the algebraic structure available for Weihrauch degrees [6, 15] to obtain game characterizations for derived classes of functions from game characterizations for the original classes, similar to the constructions found by Motto Ros [23]. As a further feature of our work, we should point out that our results apply to the effective setting firsthand, and are then lifted to the continuous setting via relativization. They thus follow the recipe laid out by Moschovakis in [21].

While the traditional scope of descriptive set theory is restricted to Polish spaces, their subsets, and functions between them, these restrictions are immaterial for the approach presented here. Our results naturally hold for multivalued functions between represented spaces. As such, this work is part of a larger development to extend descriptive set theory to a more general setting, cf., e.g., [7, 18, 29, 33, 35].

We shall freely use standard concepts and notation from descriptive set theory, referring to [17] for an introduction.

## 2 Preliminaries on Represented Spaces and Weihrauch Reducibility

Represented spaces and continuous/computable maps between them form the setting for computable analysis [43]. For a comprehensive modern introduction we refer to [31].

A *represented* space \(\mathbf {X} = (X, \delta _\mathbf {X})\) is given by a set *X* and a partial surjection \(\delta _\mathbf {X} : \subseteq {{\mathbb {N}^\mathbb {N}}} \rightarrow {X}\). A (multivalued) function between represented spaces is just a (multivalued) function on the underlying sets. We say that a partial function \(F : \subseteq {{\mathbb {N}^\mathbb {N}}} \rightarrow {{\mathbb {N}^\mathbb {N}}}\) is a *realizer* for a multivalued function \(f : \subseteq {\mathbf {X}} \rightrightarrows {\mathbf {Y}}\) (in symbols: \(F \vdash f\)) if \(\delta _\mathbf {Y}(F(p)) \in f(\delta _\mathbf {X}(p))\) for all \(p \in \mathop {\mathrm {dom}}(f\delta _\mathbf {X})\). We call *f**computable* (*continuous*), if it admits some computable (continuous) realizer.

Represented spaces and continuous functions (in the sense just defined) do indeed generalize Polish spaces and continuous functions (in the usual sense). Indeed, let \((X,\tau )\) be some Polish space, and fix a countable dense sequence \((a_i)_{i \in \mathbb {N}}\) and a compatible metric *d*. Now define \(\delta _\mathbf {X}\) by \(\delta _\mathbf {X}(p) = x\) iff \(d(a_{p(i)},x) < 2^{-i}\) holds for all \(i \in \mathbb {N}\). In other words, we represent a point by a sequence of basic points converging to it with a prescribed speed. It is a foundational result in computable analysis that the notion of continuity for the represented space \((X,\delta _\mathbf {X})\) coincides with that for the Polish space \((X,\tau )\).

### Definition 2

Let *f* and *g* be partial multivalued functions between represented spaces. We say that *f* is *Weihrauch reducible* to *g*, in symbols \(f\le _{\mathrm {W}}g\), if there are computable functions \(K:\subseteq {{\mathbb {N}^\mathbb {N}}} \rightarrow {{\mathbb {N}^\mathbb {N}}}\) and \(H:\subseteq {{\mathbb {N}^\mathbb {N}}\times {\mathbb {N}^\mathbb {N}}} \rightarrow {{\mathbb {N}^\mathbb {N}}}\) such that whenever \(G \vdash g\), the function \(F := \left( p \mapsto H(p,G(K(p)))\right) \) is a realizer for *f*.

If there are computable functions \(K,H:\subseteq {{\mathbb {N}^\mathbb {N}}} \rightarrow {{\mathbb {N}^\mathbb {N}}}\) such that whenever \(G \vdash g\) then \(HGK \vdash f\), then we say that *f* is *strongly Weihrauch reducible* to *g* (\(f \le _{\mathrm {sW}}f\)). We write \(f \le _{\mathrm {W}}^\mathrm {c}g\) and \(f \le _{\mathrm {sW}}^\mathrm {c}g\) for the variations where computable is replaced with continuous.

A multivalued function *f**tightens**g*, denoted by \(f \preceq g\), if \(\mathop {\mathrm {dom}}(g) \subseteq \mathop {\mathrm {dom}}(f)\) and \(f(x) \subseteq g(x)\) whenever \(x \in \mathop {\mathrm {dom}}(g)\), cf. [30, 34].

### Proposition 3

**(cf., e.g.,**[28,

**Chapter 4**]

**).**Let \(f:\subseteq {\mathbf {A}} \rightrightarrows {\mathbf {B}}\) and \(g:\subseteq {\mathbf {C}} \rightrightarrows {\mathbf {D}}\). We have

- 1.
\(f \le _{\mathrm {sW}}g\) (\(f \le _{\mathrm {sW}}^\mathrm {c}g\)) iff there exist computable (continuous) \(k:\subseteq {\mathbf {A}} \rightrightarrows {\mathbf {C}}\) and \(h:\subseteq {\mathbf {D}} \rightrightarrows {\mathbf {B}}\) such that \(hgk \preceq f\); and

- 2.
\(f \le _{\mathrm {W}}g\) (\(f \le _{\mathrm {W}}^\mathrm {c}g\)) iff there exist computable (continuous) \(k:\subseteq {\mathbf {A}} \rightrightarrows {\mathbf {{\mathbb {N}^\mathbb {N}}\times C}}\) and \(h:\subseteq {\mathbf {{\mathbb {N}^\mathbb {N}}\times D}} \rightrightarrows {\mathbf {B}}\) such that \(h(\mathrm {id}_{{\mathbb {N}^\mathbb {N}}} \times g)k \preceq f\).

Although there are plenty of interesting operations defined on Weihrauch degrees (cf., e.g., the introduction of [4] for a recent overview), here we only require the sequential composition operator \(\star \) from [5, 6]. Rather than defining it explicitly as in [6], we will make use of the following characterization:

### Theorem 4

**(Brattka and Pauly** [6]**).**\(\displaystyle f \star g \equiv _{\mathrm {W}}\max _{\le _{\mathrm {W}}} \{f' \circ g' \,;\, f' \le _{\mathrm {W}}f \ \wedge g' \le _{\mathrm {W}}g\}\)

## 3 Transparent Cylinders

We call \(f : \subseteq {\mathbf {X}} \rightrightarrows {\mathbf {Y}}\) a *cylinder* if \(\mathrm {id}_{\mathbb {N}^\mathbb {N}}\times f \le _{\mathrm {sW}}f\). Note that *f* is a cylinder iff \(g \le _{\mathrm {W}}f\) and \(g \le _{\mathrm {sW}}f\) are equivalent for all *g*. This notion is from [3].

### Definition 5

Call \(T : \subseteq {\mathbf {X}} \rightrightarrows {\mathbf {Y}}\)* transparent* iff for any computable or continuous \(g : \subseteq {\mathbf {Y}} \rightrightarrows {\mathbf {Y}}\) there is a computable or continuous, respectively, \(f : \subseteq {\mathbf {X}} \rightrightarrows {\mathbf {X}}\) such that \(T \circ f \preceq g \circ T\).

The transparent (singlevalued) functions on Baire space were studied by de Brecht under the name *jump operator* in [8]. One of the reasons for their relevance is that they induce endofunctors on the category of represented spaces, which in turn can characterize function classes in DST [32]. The term *transparent* was coined in [5]. Our extension of the concept to multivalued functions between represented spaces is rather straightforward, but requires the use of the notion of tightening. Note that if \(T : \subseteq {\mathbf {X}} \rightrightarrows {\mathbf {Y}}\) is transparent, then for every \(y \in \mathbf {Y}\) there is some \(x \in \mathop {\mathrm {dom}}(T)\) with \(T(x) = \{y\}\), i.e., *T* is *slim* in the terminology of [5, Definition 2.7].

Given \(p \in {\mathbb {N}^\mathbb {N}}\), recall that for each \(n \in \mathbb {N}\) we can define \((p)_n \in {\mathbb {N}^\mathbb {N}}\) by \((p)_n(k) = p(\ulcorner n,k \urcorner )\), where \(\ulcorner \cdot ,\cdot \urcorner \) is some standard pairing function on natural numbers. Two examples of transparent cylinders which will be relevant in what follows are the functions \(\lim \), \(\lim _\varDelta : \subseteq {{\mathbb {N}^\mathbb {N}}} \rightarrow {{\mathbb {N}^\mathbb {N}}}\) defined by letting \(\lim (p) = \lim _{n \in \mathbb {N}} (p)_n\) with \(\mathop {\mathrm {dom}}(\lim ) = \{p \in {\mathbb {N}^\mathbb {N}} \,;\, \lim _{n \in \mathbb {N}} (p)_n \text { exists}\}\) and letting \(\lim _\varDelta (p)\) be the restriction of \(\lim \) to the domain \(\{p \in {\mathbb {N}^\mathbb {N}} \,;\, \exists n \in \mathbb {N}\forall k \ge n ((p)_k = (p)_n)\}\).

To see an example relating to Semmes’s tree game characterizing the Borel functions, one first needs to define the appropriate represented space of labeled trees. For this, it is best to work in a quotient space of labeled trees under the equivalence relation of bisimilarity (see [27] for the details, which we omit). The resulting quotient space can be seen as the space of labeled trees in which the order of the subtrees rooted at the children of a node, and possible repetitions among these subtrees, are abstracted away. Then the function \(\mathrm {Prune}\), which removes from (any representative of the equivalence class of) a labeled tree which has one infinite branch all of the nodes which are not part of that infinite branch, is a transparent cylinder.

### Theorem 6

**(Brattka and Pauly** [6]**).** For every multivalued function *g* there is a multivalued function \(g^t \equiv _{\mathrm {W}}g\) which is a transparent cylinder.

### Proposition 7

Let \(T :\subseteq {\mathbf {X}} \rightrightarrows {\mathbf {Y}}\) and \(S : \subseteq {\mathbf {Y}} \rightrightarrows {\mathbf {Z}}\) be cylinders. If *T* is transparent then \(S \circ T\) is a cylinder and \(S \circ T \equiv _{\mathrm {W}}S \star T\). Furthermore, if *S* is also transparent then so is \(S \circ T\).

## 4 Generalized Wadge Games

In order to define our generalization of the Wadge game, first we need the following notion, which is just the dual notion to being an admissible representation as in the approach taken by Schröder in [36].

### Definition 8

A *probe* for \(\mathbf {Y}\) is a computable partial function \(\zeta : \subseteq {\mathbf {Y}} \rightarrow {{\mathbb {N}^\mathbb {N}}}\) such that for every computable or continuous \(f : \subseteq {\mathbf {Y}} \rightrightarrows {{\mathbb {N}^\mathbb {N}}}\) there is a computable or continuous, respectively, \(e : \subseteq {\mathbf {Y}} \rightrightarrows {\mathbf {Y}}\) such that \(\zeta e \preceq f\).

Note that a probe is always transparent.

The following definition generalizes the definition of a reduction game from [23, Subsect. 3.1], which is recovered as the special case in which all involved spaces are \({\mathbb {N}^\mathbb {N}}\), the map \(\zeta \) is the identity on \({\mathbb {N}^\mathbb {N}}\), and *T* is a singlevalued function.

### Definition 9

Let \(\zeta :\subseteq {\mathbf {Y}} \rightarrow {{\mathbb {N}^\mathbb {N}}}\) be a probe, \(T: \subseteq {\mathbf {X}} \rightrightarrows {\mathbf {Y}}\) and \(f : \subseteq {\mathbf {A}} \rightrightarrows {\mathbf {B}}\). The \((\zeta ,T)\)*-Wadge game for**f* is played by two players, \(\mathbf {I}\) and \(\mathbf {II}\), who take turns in infinitely many rounds. At each round of a run of the game, player \(\mathbf {I}\) first plays a natural number and player \(\mathbf {II}\) then either plays a natural number or passes, as long as she plays natural numbers infinitely often. Therefore, in the long run player \(\mathbf {I}\) builds \(x \in {\mathbb {N}^\mathbb {N}}\) and \(\mathbf {II}\) builds \(y \in {\mathbb {N}^\mathbb {N}}\), and player \(\mathbf {II}\)*wins* the run of the game if \(x \not \in \mathop {\mathrm {dom}}(f\delta _\mathbf {A})\), or \(y \in \mathop {\mathrm {dom}}(\delta _\mathbf {B}\zeta T\delta _\mathbf {X}) \) and \(\delta _\mathbf {B}\zeta T\delta _\mathbf {X}(y) \subseteq f\delta _\mathbf {A}(x)\).

It is easy to see that the Wadge game is the \((\mathrm {id},\mathrm {id})\)-Wadge game, the eraser game is the \((\mathrm {id},\lim )\)-Wadge game, and the backtrack game is the \((\mathrm {id},\lim _\varDelta )\)-Wadge game. Semmes’s tree game for the Borel functions is the \((\mathrm {Label},\mathrm {Prune})\)-Wadge game, where \(\mathrm {Label}\) is the probe extracting the infinite sequence of labels from (any representative of the equivalence class of) a pruned labeled tree consisting of exactly one infinite branch.

### Theorem 10

Let *T* be a transparent cylinder. Then player \(\mathbf {II}\) has a (computable) winning strategy in the \((\zeta ,T)\)-Wadge game for *f* iff \(f \le _{\mathrm {W}}^\mathrm {c}T\) (\(f \le _{\mathrm {W}}T\)).

### Proof

(\(\Rightarrow \)) Any (computable) strategy for player \(\mathbf {II}\) gives rise to a continuous (computable) function \(k : \subseteq {{\mathbb {N}^\mathbb {N}}} \rightarrow {{\mathbb {N}^\mathbb {N}}}\). If the strategy is winning, then \(\delta _{\mathbf {B}} \zeta T \delta _{\mathbf {X}} k \preceq f \delta _{\mathbf {A}}\), which implies \(\delta _{\mathbf {B}} \zeta T \delta _{\mathbf {X}} k \delta ^{-1}_{\mathbf {A}} \preceq f \delta _{\mathbf {A}} \delta ^{-1}_{\mathbf {A}} = f\). Thus the continuous (computable) maps \(\delta _\mathbf {B} \circ \zeta \) and \(\delta _\mathbf {X}k\delta _\mathbf {A}^{-1}\) witness that \(f \le _{\mathrm {sW}}^\mathrm {c}T\) (\(f \le _{\mathrm {sW}}T\)).

(\(\Leftarrow \)) As *T* is a cylinder, if \(f \le _{\mathrm {W}}^\mathrm {c}T\) (\(f \le _{\mathrm {W}}T\)), then already \(f \le _{\mathrm {sW}}^\mathrm {c}T\) (\(f \le _{\mathrm {sW}}T\)). Thus, there are continuous (computable) *h*, *k* with \(h \circ T \circ k \preceq f\). As \(\delta _\mathbf {B} \circ \delta _\mathbf {B}^{-1} = \mathrm {id}_\mathbf {B}\), we find that \(\delta _\mathbf {B} \circ \delta _\mathbf {B}^{-1} \circ h \circ T \circ k \preceq f\). Now \(\delta _\mathbf {B}^{-1} \circ h : \subseteq {\mathbf {Y}} \rightrightarrows {{\mathbb {N}^\mathbb {N}}}\) is continuous (computable), so by definition of a probe, there is some continuous (computable) \(e : \subseteq {\mathbf {Y}} \rightrightarrows {\mathbf {Y}}\) with \(\delta _\mathbf {B} \circ \zeta \circ e \circ T \circ k \preceq f\). As *T* is transparent, there is some continuous (computable) *g* with \(e \circ T \succeq T \circ g\), thus \(\delta _\mathbf {B} \circ \zeta \circ T \circ g \circ k \preceq f\).

As \((g \circ k) : \subseteq {\mathbf {A}} \rightrightarrows {\mathbf {X}}\) is continuous (computable), it has some (continuous) computable realizer \(K : \subseteq {{\mathbb {N}^\mathbb {N}}} \rightarrow {{\mathbb {N}^\mathbb {N}}}\). By Theorem 1, player \(\mathbf {II}\) has a winning strategy in the Wadge game for *K*, and it is easy to see that this strategy also wins the \((\zeta , T)\)-Wadge game for *f* for her.

### Corollary 11

Let *T* and *S* be transparent cylinders. If the \((\zeta , T)\)-Wadge game characterizes the class Open image in new window and the \((\zeta ', S)\)-Wadge game characterizes the class Open image in new window, then the \((\zeta ', S \circ T)\)-Wadge game characterizes the class Open image in new window.

We thus get game characterizations of many classes of functions, including, e.g., ones not covered by Motto Ros’s constructions in [23]. For example, consider the function \(\mathrm {Sort}:{{\{0, 1\}^\mathbb {N}}} \rightarrow {{\{0, 1\}^\mathbb {N}}}\) given by \(\mathrm {Sort}(p) = 0^{n}1^\mathbb {N}\) if *p* contains exactly *n* occurrences of 0 and \(\mathrm {Sort}(p) = 0^\mathbb {N}\) otherwise. This map was introduced and studied in [26]. From their results it follows that the class Open image in new window of total functions on \({\mathbb {N}^\mathbb {N}}\) which are Weihrauch-reducible to \(\mathrm {Sort}\) is neither the class of pointwise limits of functions in some other class, nor the class of \(\mathbf {X}\)-measurable functions for any boldface pointclass \(\mathbf {X}\) of subsets of \({\mathbb {N}^\mathbb {N}}\) closed under countable unions and finite intersections. By Theorem 6, \(\mathrm {Sort}\) is Weihrauch-equivalent to a a transparent cylinder \(\mathrm {Sort}^\mathrm {t}_{\mathbb {N}^\mathbb {N}}\). Thus, by Theorem 10, Open image in new window is characterized by the \((\mathrm {id},\mathrm {Sort}^\mathrm {t}_{\mathbb {N}^\mathbb {N}})\)-Wadge game.

The converse of Theorem 10 is almost true, as well:

### Proposition 12

If the \((\zeta ,T)\)-Wadge game characterizes a lower cone in the Weihrauch degrees, then it is the lower cone of \(\zeta \circ T\), and \(\zeta \circ T\) is a transparent cylinder.

## 5 Using Game Characterizations

One main advantage of having game characterizations of some properties is realized together with determinacy: either by choosing our set-theoretic axioms accordingly, or by restricting to simple cases and invoking, e.g., Borel determinacy [20], we can conclude that if the property is false, i.e., player \(\mathbf {II}\) has no winning strategy, then player \(\mathbf {I}\) has a winning strategy. Thus, player \(\mathbf {I}\)’s winning strategies serve as explicit witnesses of the failure of a property. Applying this line of reasoning to our generalized Wadge games, we obtain the following corollaries of Theorem 10:

### Corollary 13

**(ZFC).** Let *T* be a transparent cylinder and \(\zeta \) a probe such that \(\zeta \circ T\) is single-valued and \(\mathop {\mathrm {dom}}(\zeta \circ T)\) is Borel. Then for any \(f : {\mathbf {A}} \rightrightarrows {\mathbf {B}}\) such that \(\mathop {\mathrm {dom}}(\delta _\mathbf {A})\) and *f*(*x*) are Borel for any \(x \in \mathbf {A}\), we find that \(f \nleq _\text {W}^\mathrm {c}T\) iff player \(\mathbf {I}\) has a winning strategy in the \((\zeta ,T)\)-Wadge game for *f*.

### Corollary 14

**ZF + DC+ AD).** Let *T* be a transparent cylinder and \(\zeta \) a probe. Then \(f \nleq _\text {W}^\mathrm {c}T\) iff player \(\mathbf {I}\) has a winning strategy in the \((\zeta ,T)\)-Wadge game for *f*.

Unfortunately, as determinacy fails in a computable setting (cf., e.g., [10, 19]), we do not retain the computable counterparts. More generally, we are lacking a clear understanding of how these winning strategies of player \(\mathbf {I}\) might look like. As pointed out to the authors by Carroy and Louveau, this holds even for the original Wadge game, i.e., the \((\mathrm {id},\mathrm {id})\)-Wadge game. Here, we already have a notion of explicit witnesses for discontinuity: points of discontinuity. We can thus inquire about their relation:

### Question 15

Let a point of discontinuity of a function \(f : {{\mathbb {N}^\mathbb {N}}} \rightarrow {{\mathbb {N}^\mathbb {N}}}\) be given as a sequence \((a_n)_{n \in \mathbb {N}}\), a point \(a \in {\mathbb {N}^\mathbb {N}}\), and a word \(w \in \mathbb {N}^{<\mathbb {N}}\) with \(w \sqsubseteq f(a)\) such that \(\forall n \ d(a_n, a) < 2^{-n} \wedge w \not \sqsubseteq f(a_n)\). Let \(\text {Point}\) be the multivalued map that takes as input a winning strategy for player \(\mathbf {I}\) in the \((\mathrm {id},\mathrm {id})\)-Wadge game for some function \(f : {{\mathbb {N}^\mathbb {N}}} \rightarrow {{\mathbb {N}^\mathbb {N}}}\), and outputs a point of discontinuity for that function. Is \(\text {Point}\) computable? More generally, what is the Weihrauch degree of \(\text {Point}\)?

We can somewhat restrict the range of potential answers for the preceding question:

### Theorem 16

**(**^{1}**).** Let player \(\mathbf {I}\) have a computable winning strategy in the \((\mathrm {id},\mathrm {id})\)-Wadge game for \(f : {{\mathbb {N}^\mathbb {N}}} \rightarrow {{\mathbb {N}^\mathbb {N}}}\). Then *f* has a computable point of discontinuity.

A more convenient way of exploiting determinacy of the \((\zeta ,T)\)-Wadge games could perhaps be achieved if a more symmetric version were found. In this, we could hope for a dual principle *S*, where for any *f* either \(f \le _{\mathrm {W}}^c T\) or \(S \le _{\mathrm {W}}^c f\) holds. More generally, we hope that a better understanding of the \((\zeta ,T)\)-Wadge games would lead to structural results about the Weihrauch lattice, similar to the results obtained by Carroy on the strong Weihrauch reducibility [9].

## 6 Generalized Wadge Reductions

As mentioned in the introduction, the Wadge game was introduced not to characterize continuous functions, but in order to reason about a reducibility between sets. Given \(A, B \subseteq {\mathbb {N}^\mathbb {N}}\), we say that *A* is *Wadge-reducible* to *B*, in symbols \(A \le _w B\), if there exists a continuous \(F : {{\mathbb {N}^\mathbb {N}}} \rightarrow {{\mathbb {N}^\mathbb {N}}}\) such that \(F^{-1}[B] = A\). Equivalently, we could consider the multivalued function \(\frac{B}{A} : {{\mathbb {N}^\mathbb {N}}} \rightrightarrows {{\mathbb {N}^\mathbb {N}}}\) defined by \(\frac{B}{A}(x) = B\) if \(x \in A\) and \(\frac{B}{A}(x) = ({\mathbb {N}^\mathbb {N}}{\smallsetminus } B)\) if \(x \notin A\). Now, \(A \le _w B\) iff \(\frac{B}{A}\) is continuous. A famous structural result following from the determinacy of the corresponding Wadge game is that for any Borel \(A, B \subseteq {\mathbb {N}^\mathbb {N}}\), either \(A \le _w B\) or \({\mathbb {N}^\mathbb {N}}{\smallsetminus } B \le _w A\). In particular, the Wadge hierarchy on the Borel sets is a strict weak order of width 2.

Both definitions immediately generalize to the case where \(A \subseteq \mathbf {X}\) and \(B \subseteq \mathbf {Y}\) for represented spaces \(\mathbf {X}\), \(\mathbf {Y}\). However, they yield different notions, for not every continuous multivalued function has a continuous choice function. As noted, e.g., by Hertling [14], extending the former definition to the reals already introduces infinite antichains in the resulting degree structure. The second generalization was proposed by Pequignot [35] as an alternative.^{2}

It is a natural variation to replace *continuous* in the definition of Wadge reducibility by some other class of functions (ideally one closed under composition). Motto Ros has shown that for the typical candidates of more restrictive classes of functions, the resulting degree structures will not share the nice properties of the standard Wadge degrees (they are *bad*) [24]. Larger classes of functions as reduction witnesses have been explored by Motto Ros, Schlicht, and Selivanov [25] in the setting of quasi-Polish spaces—using the generalization of the first definition of the reduction.

### Definition 17

Let *T* be a Weihrauch degree. We define a relation \(\preceq ^T\) on subsets of represented spaces as follows: For \(A \subseteq \mathbf {X}\), \(B \subseteq \mathbf {Y}\) let \(A \preceq ^T B\) hold iff \(\frac{B}{A} \le _{\mathrm {W}}^\mathrm {c}T\).

### Observation 18

If \(T \star T \equiv _{\mathrm {W}}T\), then \(\preceq ^T\) is a quasiorder.

The following partially generalizes [23, Theorem 6.10]:

### Theorem 19

Let \(A \subseteq \mathbf {X}\) and \(B \subseteq \mathbf {Y}\), let \(T : {\mathbf {U}} \rightrightarrows {\mathbf {V}}\) be a transparent cylinder, and let \(\zeta : \subseteq {\mathbf {Y}} \rightarrow {{\mathbb {N}^\mathbb {N}}}\) be a probe such that the \((\zeta , T)\)-Wadge game for \(\frac{B}{A}\) is determined. Then either \(A \preceq ^T B\) or \(B \le _w {\mathbb {N}^\mathbb {N}}{\smallsetminus } A\).

### Proof

If player \(\mathbf {II}\) has a winning strategy in the \((\zeta , T)\)-Wadge game for \(\frac{B}{A}\), then by Theorem 10, we find that \(\frac{B}{A} \le _{\mathrm {W}}^\mathrm {c}T\), hence \(A \preceq ^T B\).

Otherwise, player \(\mathbf {I}\) has a winning strategy. This winning strategy induces a continuous function \(s : {{\mathbb {N}^\mathbb {N}}} \rightarrow {{\mathbb {N}^\mathbb {N}}}\), such that if player \(\mathbf {II}\) plays \(y \in {\mathbb {N}^\mathbb {N}}\), then player \(\mathbf {I}\) plays \(s(y) \in {\mathbb {N}^\mathbb {N}}\). As *T* is a transparent cylinder and \(\zeta \) a probe, there is a continuous function \(t : {{\mathbb {N}^\mathbb {N}}} \rightarrow {{\mathbb {N}^\mathbb {N}}}\) such that \((\zeta \circ T \circ \delta _\mathbf {U} \circ t) = \mathrm {id}_{\mathbb {N}^\mathbb {N}}\). Now we consider \(s \circ t : {{\mathbb {N}^\mathbb {N}}} \rightarrow {{\mathbb {N}^\mathbb {N}}}\). If \(\delta _\mathbf {X}(x) \in A\), then if player \(\mathbf {II}\) plays *t*(*x*), player \(\mathbf {I}\) needs to play some *s*(*t*(*x*)) such that \(\delta _\mathbf {Y}(s(t(x))) \notin B\). Likewise, if \(\delta _\mathbf {X}(x) \notin A\), then for player \(\mathbf {I}\) to win, it needs to be the case that \(\delta _\mathbf {Y}(s(t(x))) \in B\). Thus, \(s \circ t\) is a continuous realizer of \(\frac{B}{{\mathbb {N}^\mathbb {N}}{\smallsetminus } A}\), and \(B \le _w {\mathbb {N}^\mathbb {N}}{\smallsetminus } A\) follows.

### Corollary 20

**(ZF + DC + AD).** Suppose \(T \star T \equiv _{\mathrm {W}}T\). Then \(\prec ^T\) is strict weak order of width at most 2.

In [22], Motto Ros has identified sufficient conditions on a generalized reduction (although in a different formalism) to ensure that its degree structure is equivalent to the Wadge degrees. We leave for future work the task of determining precisely for which *T* the degree structure of \(\prec ^T\) (restricted to subsets of \({\mathbb {N}^\mathbb {N}}\)) is equivalent to the Wadge degrees, and which other structure types are realizable.

## Footnotes

- 1.
A key lemma for the proof of this theorem goes back to helpful comments by Takayuki Kihara.

- 2.
While Pequignot only introduces the notion for second countable \(T_0\) spaces, the extension to all represented spaces is immediate. Note that one needs to take into account that for general represented spaces, the Borel sets can show unfamiliar properties, e.g., even singletons can fail to be Borel (cf. also [37, 38]).

## Notes

### Acknowledgments

We are grateful to Benedikt Löwe, Luca Motto Ros, Takayuki Kihara and Raphaël Carroy for helpful and inspiring discussions. We would also like to thank the anonymous referees for the many corrections which have significantly improved the paper.

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