aa r X i v : . [ m a t h . L O ] F e b STASHING AND PARALLELIZATION PENTAGONS
VASCO BRATTKAFaculty of Computer Science, Universit¨at der Bundeswehr M¨unchen, Germany and Departmentof Mathematics and Applied Mathematics, University of Cape Town, South Africa e-mail address : [email protected]
Abstract.
Parallelization is an algebraic operation that lifts problems to sequences in anatural way. Given a sequence as an instance of the parallelized problem, another sequenceis a solution of this problem if every component is instance-wise a solution of the originalproblem. In the Weihrauch lattice parallelization is a closure operator that corresponds tothe bang operator in linear logic. Here we introduce a dual operation that we call stashing and that also lifts problems to sequences, but such that only some component has to bean instance-wise solution. In this case the solution is stashed away in the sequence. Thisoperation, if properly defined, induces an interior operator in the Weihrauch lattice, whichcorresponds to the question mark operator known from linear logic. It can also be seen asa countable version of the sum operation. We also study the action of the monoid inducedby stashing and parallelization on the Weihrauch lattice, and we prove that it leads to atmost five distinct degrees, which (in the maximal case) are always organized in pentagons.We also introduce another closely related interior operator in the Weihrauch lattice thatreplaces solutions of problems by upper Turing cones that are strong enough to computesolutions. It turns out that on parallelizable degrees this interior operator corresponds tostashing. This implies that, somewhat surprisingly, all problems which are simultaneouslyparallelizable and stashable have computability-theoretic characterizations. Finally, weapply all these results in order to study the recently introduced discontinuity problem,which appears as the bottom of a number of natural stashing-parallelization pentagons.The discontinuity problem is not only the stashing of several variants of the lesser limitedprinciple of omniscience, but it also parallelizes to the non-computability problem. Thissupports the slogan that “non-computability is the parallelization of discontinuity”. Wealso study the non-majorizability problem as an asymmetric version of the discontinuityproblem and we show that it parallelizes to the hyperimmunity problem. Finally weidentify a phase transition related to the limit avoidance problem that marks a pointwhere pentagons are taking off from the bottom of the Weihrauch lattice.
Contents
1. Introduction 22. Stashing as Interior Operator 73. The Stashing-Parallelization Monoid 94. The Upper Turing Cone Operator 125. The Discontinuity Problem in Pentagons 146. Majorization and Hyperimmunity 207. Retractions, All-or-Unique Choice and G¨odel Numbers 23 © Vasco BrattkaCreative Commons
8. Limit Avoidance and a Phase Transition 269. Conclusions 27References 28Acknowledgments 291.
Introduction
The Weihrauch lattice has been used as a computability theoretic framework to analyzethe uniform computational content of mathematical problems from many different areas ofmathematics, and it can also be seen as a uniform variant of reverse mathematics (a recentsurvey on Weihrauch complexity can be found in [11]).The notion of a mathematical problem has a very general definition in this approach.
Definition 1 (Problems) . A problem is a multi-valued function f : ⊆ X ⇒ Y on representedspaces X, Y that has a realizer.We recall that by a realizer F : ⊆ N N → N N of f , we mean a function F that satisfies δ Y F ( p ) ∈ f δ X ( p ) for all p ∈ dom( f δ X ), where δ X : ⊆ N N → X and δ Y : ⊆ N N → Y arethe representations of X and Y , respectively (i.e., partial surjective maps onto X and Y ,respectively). A problem is called computable if it has a computable realizer and continuous if it has a continuous realizer.By h p, q i we denote the usual pairing function on N N , defined by h p, q i (2 n ) = p ( n ), h p, q i (2 n + 1) = q ( n ) for all p, q ∈ N N , n ∈ N . Weihrauch reducibility can now be defined asfollows. Definition 2 (Weihrauch reducibility) . Let f : ⊆ X ⇒ Y and g : ⊆ W ⇒ Z be problems.Then f is called Weihrauch reducible to g , in symbols f ≤ W g , if there are computable H, K : ⊆ N N → N N such that H h id , GK i is a realizer of f whenever G is a realizer of g . Analogously, one says that f is strongly Weihrauch reducible to g , in symbols f ≤ sW g ,if the expression H h id , GK i can be replaced by HGK . Both versions of the reducibilityhave topological counterparts, where one only requires
H, K to be continuous and thesereducibilities are denoted by ≤ ∗ W and ≤ ∗ sW , respectively.The topological version of Weihrauch reducibility has always been studied alongside thecomputability-theoretic version, and all four reducibilities induce a lattice structure (see [11]for references).Normally, the Weihrauch lattice refers to the lattice induced by ≤ W , but here we willfreely use this term also for the lattice structure induced by ≤ ∗ W . If we want to be moreprecise, we will call the latter the topological Weihrauch lattice . The equivalence classes ofproblems under (strong) Weihrauch reducibility are called (strong) Weihrauch degrees .In [6, Definition 4.1] the operation of parallelization was introduced. For reasons thatwill become clear below, we denote the parallelization b f of a problem f additionally withthe non-standard notation Π f in this article. Definition 3 (Parallelization) . For every problem f : ⊆ X ⇒ Y we define its parallelization Π f : ⊆ X N ⇒ Y N by dom(Π f ) := dom( f ) N andΠ f ( x n ) := { ( y n ) ∈ Y N : ( ∀ n ) y n ∈ f ( x n ) } TASHING AND PARALLELIZATION PENTAGONS 3 for all ( x n ) ∈ X N . We also write b f := Π f and we call a problem parallelizable (or stronglyparallelizable ) if f ≡ W b f (or f ≡ sW b f ) holds.In [6, Proposition 4.2] it was proved that parallelization is a closure operator in theWeihrauch lattice. This holds analogously for the topological versions of Weihrauch re-ducibility. Fact 4 (Parallelization) . f Π f is a closure operator with respect to the following versionsof Weihrauch reducibility: ≤ W , ≤ sW , ≤ ∗ W and ≤ ∗ sW .We recall the definition of a closure operator and an interior operator for a preorderedset. By a preordered set ( P, ≤ ) we mean a set P with a relation ≤ on P that is reflexive andtransitive. The relations ≤ W , ≤ sW , ≤ ∗ W and ≤ ∗ sW are preorders on the set P of problems f : ⊆ N N ⇒ N N on Baire space . Definition 5 (Closure and interior operator) . Let ( P, ≤ ) a preordered set with a function C : P → P . Then C is called a closure operator for ≤ if the following hold for all x, y ∈ P :(1) x ≤ C ( x ) (extensive)(2) x ≤ y = ⇒ C ( x ) ≤ C ( y ) (monotone)(3) CC ( x ) ≤ C ( x ) (idempotent)Analogously, C is called an interior operator for ≤ if the three conditions hold for ≥ inplace of ≤ .Besides f Π f a number of other closure operator appeared in the study of theWeihrauch lattice [11]. However, not so many interior operators have been considered yet.In this article we want to study a dual operation to parallelization that we call stashing andthat can be defined as follows. Definition 6 (Stashing) . For every problem f : ⊆ X ⇒ Y we define its stashing or summa-tion Σ f : ⊆ X N ⇒ Y N by dom(Σ f ) := dom( f ) N andΣ f ( x n ) := { ( y n ) ∈ Y N : ( ∃ n ) y n ∈ f ( x n ) } for all ( x n ) ∈ X N . We also write b f := Σ f and we call a problem stashable (or stronglystashable ) if f ≡ W b f (or f ≡ sW b f ) holds.Essentially, the definition corresponds to parallelization with an existential quantifierin the place of the universal one. This means that given an instance ( x n ) for Σ f , a solutionis a sequence ( y n ) such that y n ∈ f ( x n ) for at least one n ∈ N . This operation can beseen as a countable version of the sum operation + (see [11]), which is the reason why wehave called it summation in earlier presentations of this work. However, stashing bettercorresponds to the intuition of what Σ f does.A subtle technical point in this definition is that we use the completion Y of the space Y on the output side. For a represented space ( Y, δ Y ) the completion ( Y , δ Y ) is defined by Y := Y ∪ {⊥} (with a distinct element ⊥ 6∈ Y ) and δ Y : N N → Y with δ Y ( p ) := (cid:26) δ Y ( p −
1) if p − ∈ dom( δ Y ) ⊥ otherwise , For studying the order structure it is sufficient to consider problems on Baire space as representativesof arbitrary problems. This guarantees that P is actually a set. VASCO BRATTKA
ACC N DISDNC N DNC D N NON
Π ΣΣ ΠFigure 1:
ACC N pentagon in the Weihrauch lattice.where p − ∈ N N ∪ N ∗ is a finite or infinite sequence that is obtained as the concatenationof p (0) − , p (1) − , p (2) − , ... with the understanding that − ε ∈ N ∗ is the emptyword. This operation of completion saw some recent surge of interest after Dzhafarov [15]used it to show that the strong version Weihrauch reducibility ≤ sW actually yields a latticestructure (here completion appeared in the definition of a suitable supremum operation).See [8, 7] for further applications and a more detailed study of completion.One reason that the completion cannot be omitted in Definition 6 is that it allows usto produce dummy outputs with no meaning (without the completion this might not bepossible as, for instance, some represented spaces ( Y, δ Y ) might not even have computablepoints). Another reason is that every partial computable problem with a completion on theoutput side can be extended to a total computable problem in a certain sense. In any casethe completion enables us to prove the following result (see Proposition 18) in Section 2. Proposition 7 (Stashing) . f Σ f is an interior operator with respect to the followingversions of Weihrauch reducibility: ≤ W , ≤ sW , ≤ ∗ W and ≤ ∗ sW .While parallelization Π can be seen as the counterpart of the bang operator “!’ in linearlogic [11], stashing Σ can be seen as the counterpart of the dual why-not operator “?”.Since our lattice is now equipped with a closure operator f Π f and a dual interioroperator f Σ f , it is natural to ask how the monoid generated by { Π , Σ } ∗ acts on thelattice structure? In other words, starting from an arbitrary problem f , what kind ofproblems can we generate by repeated applications of Π and Σ (in any order) to f ? Andhow are these problems related with respect to the lattice structure?In Section 3 we prove that we can generate at most five distinct problems in this way(up to equivalence) and that these five problems (in the maximal case) are always organizedin a pentagon (see Proposition 21, Corollary 22).The maximal case can actually occur and in Section 5 we study a number of specificsuch pentagons, in particular the one shown in the diagram in Figure 1. Here every problemin the diagram allows a ≤ sW –reduction to any problem above it that is connected with aline and no other ≤ ∗ W –reductions are possible (except those that follow by transitivity). Wedefine all the problems that occur in this diagram and some further problems that we aregoing to study in this article. Definition 8 (Some problems) . We consider the following problems for X ⊆ N :(1) LPO : N N → { , } , LPO ( p ) = 1 : ⇐⇒ p = 000 ... , TASHING AND PARALLELIZATION PENTAGONS 5 (2) lim : ⊆ N N → N N , h p , p , p , ... i 7→ lim n →∞ p n ,(3) lim X : ⊆ X N → X, ( x n ) n ∈ N lim n →∞ x n ,(4) J : N N → N N , p p ′ ,(5) EC : N N → N , p range( p − C X : ⊆ N N ⇒ X, p X \ range( p − C X ) = { p ∈ N N : | X \ range( p − | ≥ } ,(7) ACC X : ⊆ N N ⇒ X, p X \ range( p −
1) withdom(
ACC X ) = { p ∈ N N : | range( p − | ≤ } ,(8) AoUC X : ⊆ N N ⇒ X, p X \ range( p −
1) withdom(
AoUC X ) = { p ∈ N N : | X \ range( p − | = 1 or range( p −
1) = ∅} ,(9) DNC X : N N ⇒ X N , p
7→ { q ∈ X N : ( ∀ i ∈ N ) ϕ pi ( i ) = q ( i ) } ,(10) PA : N N ⇒ N N , p
7→ { q ∈ N N : p ≪ q } ,(11) WKL : Tr ⇒ N , T [ T ] with dom( WKL ) = { T : T infinite } ,(12) NON : N N ⇒ N N , p
7→ { q ∈ N N : q T p } ,(13) DIS : N N ⇒ N N , p
7→ { q ∈ N N : U ( p ) = q } ,(14) NRNG : N N ⇒ N , p
7→ { A ∈ N : A = range( p − } .Here LPO is also known as limited principle of omniscience and it is nothing but thecharacteristic function of the zero sequence. By lim we just denote the ordinary limit mapwith respect to the Baire space topology, where for convenience, the input sequence isencoded by h p , p , p , ... ih n, k i := p n ( k ) where h n, k i := ( n + k )( n + k + 1) + k is the usualCantor pairing function for n, k ∈ N . By p ′ we denote the Turing jump of p ∈ N N . Weidentify n ∈ N with the set n = { , ..., n − } . The problem EC (this name was introduced in[32, Exercise 8.2]) was originally studied under the name C [30, 24, 1, 2, 25, 6]. Intuitively, EC translates enumerations into characteristic functions. The problem C N is known as choiceon the natural numbers and was introduced and studied in [5, 4]. The problems ACC X arealso known under the name LLPO X and have been studied in [31, 17, 12]. The acronym ACC stands for all-or-co-unique choice and
LLPO := C = ACC = AoUC is known as lesser limited principle of omniscience . We recall that p − | A | denotes the cardinality of the set A . The acronym AoUC stands for all-or-unique choice .This problem was studied mostly for the unit interval X = [0 ,
1] [27, 9, 21] and not forspaces X ⊆ N that we are interested in here. The acronym DNC stands for diagonally non-computable and by ϕ p we denote a standard G¨odel numbering of the partial computablefunctions ϕ pi : ⊆ N → N relative to some oracle p ∈ N N . The acronym PA stands for Peanoarithmetic and by p ≪ q we express the fact that q is of PA–degree relative to p , whichmeans that q computes a path through every infinite binary tree that is computable relativeto p . The relation ≪ was introduced by Simpson [28]. By Tr we denote the set of binarytrees T ⊆ { , } ∗ and [ T ] denotes the set of infinite paths of such a tree and WKL standsfor
Weak K˝onig’s Lemma . By ≤ T we denote Turing reducibility and by U : ⊆ N N → N N wedenote some universal computable function. Such a function can be defined, for instance,by U hh i, r i , p i := ϕ h r,p i i whenever ϕ h r,p i i is total (and undefined otherwise). Here
NON and
DIS are called the non-computability problem and the discontinuity problem , respectively.The problem
NRNG is called range non-equality problem and is introduced here. The universal function U has been defined differently in [3], but our definition is equivalent, as thefunction U defined here also satisfies a utm- and an smn-theorem [32]. In particular, the discontinuityproblem DIS defined with one version of U is strongly Weihrauch equivalent to the one defined with theother version of U . VASCO BRATTKA
Nobrega and Pauly used Wadge games to characterize certain lower cones in theWeihrauch lattice [26]. In [3] we have characterized the upper cone of the discontinuityproblem by Wadge games on problems. The characterization goes as follows [3, Theo-rem 17, Corollary 28].
Theorem 9 (Wadge games and the discontinuity problem) . Let f : ⊆ X ⇒ Y be a problem.Then DIS ≤ W f ⇐⇒ Player I has a computable winning strategy in the Wadge game f .An analogous result holds for ≤ ∗ W and (not necessarily computable) winning strate-gies [3, Theorem 27]. We are going to use Theorem 9 in the proof of Proposition 32 thatestablishes the pentagon of ACC N shown in Figure 1.Several facts are known about the parallelization of the problems summarized in Defi-nition 8. These results were proved in [6, Lemma 6.3, Theorem 8.2], [4, Lemma 8.9] and theresult \ ACC X ≡ sW DNC X has first been proved by Higuchi and Kihara [17, Proposition 81]and independently in [12, Theorem 5.2]. See also the survey [11]. Fact 10 (Parallelization of problems) . d LPO ≡ sW c C N ≡ sW [ lim X ≡ sW lim ≡ sW J ≡ sW EC , \ LLPO ≡ sW c C n ≡ sW WKL , and \ ACC X ≡ sW DNC X for X = N or X ≥ n ≥ ACC N and NON were the two weakest unsolvable (andincomparable) natural problems in the Weihrauch lattice, and, in fact, they are the twoweakest problems discussed in [12]. Hence, it is a somewhat surprising coincidence thatthese problems appear together in the diagram in Figure 1. In fact, they are related through
NON ≡ sW ΠΣ(
ACC N ). The discontinuity problem DIS was introduced in [3] and it wasproved that (under the axiom of determinacy)
DIS is actually the weakest discontinuousproblem with respect to the topological version of Weihrauch reducibility ≤ ∗ W . Part of theabove relation between ACC N and NON is that we are going to prove that
DIS parallelizesto
NON (see Theorem 34).
Theorem 11 (Non-computability is parallelized discontinuity) . NON ≡ sW d DIS .This result supports the slogan that “non-computability is parallelized discontinuity”.In Section 5 with study a number of further pentagons with the discontinuity problem atthe bottom and we show that the discontinuity problem can be obtain by stashing of severaldifferent problems.
Theorem 12 (Discontinuity as stashing) . DIS ≡ sW d LPO ≡ sW \ LLPO ≡ sW \ ACC N ≡ sW \ AoUC n for n ≥ Proposition 13 (Range-non-equality problem) . DIS ≡ sW NRNG .In Section 4 we introduce and study another interior operator in the Weihrauch latticethat replaces a problem by its upper Turing cone version.
Definition 14 (Upper Turing cone version) . Let f : ⊆ X ⇒ Y be a problem. We definethe upper Turing cone version f D : ⊆ X ⇒ D by dom( f D ) := dom( f ) and f D ( x ) := { deg T ( q ) ∈ D : ( ∃ y ≤ T q ) y ∈ f ( x ) } . TASHING AND PARALLELIZATION PENTAGONS 7
Here D denotes the set of Turing degrees deg T ( p ) = { q ∈ N N : q ≡ T p } , represented by δ D : N N → D , p deg T ( p ). If ( Y, δ Y ) is a represented space, then we define y ≤ T q : ⇐⇒ ( ∃ p ∈ δ − Y { y } ) p ≤ T q ⇐⇒ ( ∃ computable F : ⊆ N N → Y ) F ( q ) = y. Hence, y ≤ T q means that y has a name that can be computed from q and if Y = N N thenthis is the usual version of Turing reducibility. This version of Turing reducibility has alsobeen called representation reducibility [22].Besides the fact that f f D is an interior operator in the Weihrauch lattice, ourmain result in this direction shows that on parallelizable problems the two interior operator f Σ f and f f D coincide (up to equivalence). Proposition 15 (Closure under upper Turing cones) . Σ b f ≡ sW b f D for all problems f .As a corollary of this result we obtain the following surprising consequence (see Corol-lary 28). Corollary 16 (Closure under upper Turing cones) . If f is a problem that is parallelizableand stashable, then it is also closed under upper Turing cones, i.e., f ≡ W f D .This means that all such problems are essentially of computability theoretic nature.This includes all problems that occur on the right-hand side of stashing-parallelization pen-tagons. The remarkable situation here is that parallelization and stashing are in some sensepurely set-theoretic operations (with no mention of any computability theoretic property ornotion) and yet a combination of both generates computability-theoretic problems (startingfrom any problem f whatsoever).In Section 6 we study the non-marjorizability problem NMAJ as an asymmetric versionof the discontinuity problem and we investigate its pentagon. This is of interest as
NMAJ parallelizes to the hyperimmunity problem
HYP . In Section 7 we investigate the retractionproblem
RET X that characterizes the complexity of multi-valued retractions R : X ⇒ X and turns out to be equivalent to AoUC X for X ≥
2. We also study the correspondingpentagons. Finally, in Section 8 we identify a phase transition related to the limit avoidanceproblem
NLIM N . This is the weakest problem known to us that does not stash away to thediscontinuity problem. 2. Stashing as Interior Operator
The main purpose of this section is to show that stashing f Σ f is an interior operator forvarious versions of the Weihrauch lattice (see Proposition 18). For this result it is essentialthat the completion Y of Y is used on the output side of Σ f : ⊆ X N ⇒ Y N . As a technicalpreparation we need the following lemma. In [7, Corollary 2.7] we proved that there is acomputable retraction r : Y → Y , which is a computable map such that r | Y = id Y . Herewe will have to use a similar property of the product space Y N . Lemma 17 (Retractions for product spaces) . For every represented space Y there is acomputable retraction r : Y N → Y N , i.e., a computable r with r | Y N = id Y N . Proof.
The space Z := Y is represented by a total representation δ Z = δ Y and by [7, Corol-lary 2.7] there is a computable retraction R : Z → Z . We assume that Z N is representedby the completion δ Z N of the usual product representation δ Z N . We claim that there is a VASCO BRATTKA computable map T : Z N → Z N with T | Z N = id Z N . We assume that Z N = Z N ∪ {⊥ N } and Z = Z ∪ {⊥} . The names of ⊥ N with respect to δ Z N are exactly those names that containonly finitely many digits different from 0 (since δ Z and hence δ Z N are total). Now T canbe realized in a computable way by interpreting the non-zero content of a given name p as a name h p , p , p , ... i of a point in Z N with respect to δ Z N . As long as no non-zerocontent is available in p , the names p i are filled up by zeros. Altogether this shows that T is computable and it acts as the identity with the exception that T ( ⊥ N ) = ( ⊥ , ⊥ , ⊥ , ... ).Now r : Z N → Z N with r := b R ◦ T is the desired computable retraction for Z N .Now we are prepared to prove that stashing is an interior operator on the (strong)Weihrauch lattice. Properties (1) and (2) in the following result are only made possi-ble by the usage of the completion Y , whereas the existence of a retraction according toLemma 17 guarantees that the completion is not an obstacle for property (3). For a problem f : ⊆ X ⇒ Y the completion is defined by f : X ⇒ Y , x (cid:26) f ( x ) if x ∈ dom( f ) ⊥ otherwiseThis completion was introduced and studied in [8, 7] and we will use it in part (2) of thefollowing proof. Proposition 18 (Stashing) . The stashing operation f Σ f is an interior operator for ≤ sW , ≤ W , ≤ ∗ W and ≤ ∗ sW . That is, for all problems f, g :(1) Σ f ≤ sW f ,(2) f ≤ sW g = ⇒ Σ f ≤ sW Σ g ,(3) Σ f ≤ sW ΣΣ f .Analogous statements hold for ≤ W , ≤ ∗ W and ≤ ∗ sW . Proof.
If we prove (1) and (3) for ≤ sW , then this implies the corresponding statements for ≤ ∗ sW , ≤ W and ≤ ∗ W . Only in the case of (2) we explicitly need to consider the different typesof reducibilities.(1) We consider the computable projection K : X N → X, ( x n ) x and the computablefunction H : Y → Y N , y ( y, ⊥ , ⊥ , ... ). Here H is computable, since the embedding ι : Y → Y is computable according to [8, Corollary 3.10] and the element ⊥ ∈ Y is com-putable too. These two functions K, H witness the reduction Σ f ≤ sW f , i.e., Hf K ( x n ) ∈ Σ f ( x n ) for all ( x n ) ∈ dom(Σ f ). We note that the usage of the completion Y guaranteesthe existence of computable default outputs ⊥ ∈ Y .(2) We consider problems f : ⊆ X ⇒ Y and g : ⊆ W ⇒ Z . If f ≤ sW g , then there arecomputable K : ⊆ X ⇒ W and H : ⊆ Z ⇒ Y such that ∅ 6 = HgK ( x ) ⊆ f ( x ) for all x ∈ dom( f ) [11, Proposition 3.2]. Then the completion H : Z ⇒ Y is computable by [8,Proposition 4.9] and so are the parallelizations b H : Z N ⇒ Y N and b K : ⊆ X N ⇒ W N . Weobtain ∅ 6 = b H ◦ Σ g ◦ b K (( x n ) n )= b H { ( z n ) n ∈ Z N : ( ∃ n ) z n ∈ gK ( x n ) } = { ( y n ) ∈ Y N : ( ∃ n ) y n ∈ HgK (( x n ) n ) } ⊆ Σ f (( x n ) n )for all ( x n ) n ∈ dom(Σ f ). We note that the totality of the completion H guarantees thatalso those components of ( z n ) n ∈ Σ g ◦ b K (( x n ) n ) ⊆ Z N are processed, which are not in the TASHING AND PARALLELIZATION PENTAGONS 9 domain of H . Altogether, this proves Σ f ≤ sW Σ g .Let now f ≤ W g hold with computable K : ⊆ X ⇒ N N × W and H : ⊆ N N × Z ⇒ Y accordingto [14, Lemma 2.5], i.e., ∅ 6 = H ◦ (id N N × g ) ◦ K ( x ) ⊆ f ( x ) for all x ∈ dom( f ). Then, again,the completion H : N N × Z ⇒ Y is computable. By [8, Proposition 3.8, Corollary 3.10]there is a computable ι : N N × Z → N N × Z with ι ( p, z ) = ( p, z ) for all ( p, z ) ∈ N N × Z . Then \ ( H ◦ ι ) : ( N N × Z ) N ⇒ Y N can also be considered as a problem of type H ′ : ( N N ) N × Z N ⇒ Y N and b K can be seen as a problem of type b K : ⊆ X N ⇒ ( N N ) N × W N . As above we obtain ∅ 6 = H ′ ◦ (id ( N N ) N × Σ g ) ◦ b K (( x n ) n ) ⊆ Σ f (( x n ) n ) , for all ( x n ) n ∈ dom(Σ f ), i.e., Σ f ≤ W Σ g . The statements for ≤ ∗ sW and ≤ ∗ W can be provedanalogously with continuous K, H . We note that by [8, Proposition 4.9] H is continuous, if H is so.(3) We consider ΣΣ f : ⊆ ( X N ) N ⇒ Y NN . By Lemma 17 there is a computable retraction r : Y N → Y N . For every represented space X the map s X : ( X N ) N → X N , (( x n,k ) k ) n ( x n,k ) h n,k i that interleaves a double sequence in a single sequence, is a computable isomorphism, i.e.,it is bijective and computable and its inverse is computable too. We now consider thecomputable functions K := s − X and H := s Y ◦ b r : Y NN → Y N . Then we obtain ∅ 6 = H ◦ ΣΣ f ◦ K (( x n,k ) h n,k i )= s Y ◦ b r { ( z n ) n ∈ Y NN : ( ∃ n ) z n ∈ Σ f (( x n,k ) k ) } = s Y { (( y n,k ) k ) n ∈ ( Y N ) N : ( ∃ n )( ∃ k ) y n,k ∈ f ( x n,k ) } = { ( y n,k ) h n,k i ∈ Y N : ( ∃h n, k i ) y n,k ∈ f ( x n,k ) } = Σ f (( x n,k ) h n,k i ) . This proves Σ f ≤ sW ΣΣ f .We can conclude from property (2) in Proposition 18 that stashing is, in particular,invariant under (strong) Weihrauch reducibility and can hence be seen as an operation on(strong) Weihrauch degrees. Corollary 19.
Stashing can be extended to an operation on (strong) Weihrauch degrees.3.
The Stashing-Parallelization Monoid
We adopt the convention that we denote the parallelization of a problem f by b f and thestashing by b f when we deal with single applications of these operators. However, for iteratedapplications it is useful to use the notation Π f and Σ f instead.The closure and interior operators Π and Σ generate a monoid { Π , Σ } ∗ under composi-tion and we want to study the action of this monoid on the Weihrauch lattice. To this end,it is worth spelling out the problems ΣΠ f and ΠΣ f explicitly:(1) ΣΠ f : ⊆ X N × N ⇒ Y NN , ( x n,k )
7→ { ( y n,k ) : ( ∃ n )( ∀ k ) y n,k ∈ f ( x n,k ) } ,(2) ΠΣ f : ⊆ X N × N ⇒ Y N × N , ( x n,k )
7→ { ( y n,k ) : ( ∀ n )( ∃ k ) y n,k ∈ f ( x n,k ) } . We can see that stashing corresponds to a usage of an existential quantifier whereasparallelization corresponds to a usage of a universal quantifier in a certain sense. Hence ΣΠand ΠΣ correspond to applications of these quantifiers in different order.There is a subtle technical point here: we have to deal with the spaces Y N and Y N , whichare not computably isomorphic. We recall that Y N = ( Y ∪{⊥} ) N , whereas Y N = Y N ∪{⊥ N } .Hence, formally there is no subset relation between these two sets. In order to make thelatter a subset of the former, we can choose ⊥ N := ( ⊥ , ⊥ , ... ), as implicitly done in the proofof Lemma 17. In this sense the double sequence notation ( y n,k ) in the description of ΣΠ f should be understood.Besides the retraction r from Lemma 17 we also need the maps s, t that exist accordingto the following lemma. Intuitively speaking, s maps every sequence that contains a ⊥ to ⊥ N and t maps ⊥ N to some sequence that contains a ⊥ (which one it is, might depend onthe given name of ⊥ N ). Lemma 20 (Completion of product spaces) . For every represented space Y there arecomputable s : Y N → Y N and t : Y N ⇒ Y N such that s | Y N = t | Y N = id Y N . Proof.
A suitable computable map s : Y N → Y N with s | Y N = id | Y N is realized by a com-putable F : N N → N N with the property that F h p , p , p , ... i − h q , q , q , ... i with q i = p i − p i with p i − ∈ N N . This can be achieved by copying the non-zerocontent of p i subtracted by 1 into the q i , where the resulting sequence h q , q , q , ... i is filledup by zeros, whenever necessary (i.e., whenever no non-zero content is available for some p i then the entire output is filled up only with zeros as long as no non-zero content appears).That is, if one of the p i is a name of ⊥ ∈ Y (either because it has only finitely many digitsdifferent from zero or because p i − dom( δ Y )), then F h p , p , ... i is a name of ⊥ N ∈ Y N .In this way, F realizes the identity on Y N .For the second part of the statement, we note that Y N has a precomplete and totalrepresentation by [8, Proposition 3.8] and hence we can extend the parallelization of thecomputable embedding Y ֒ → Y to a computable problem t : Y N ⇒ Y N with t | Y N = id Y N by [7, Proposition 2.6].Our core observation on the action of the monoid { Π , Σ } ∗ on the (strong) Weihrauchlattice is captured by the following result. Proposition 21 (Action of the stashing-parallelization monoid) . For every problem f weobtain:(1) ΠΣ f ≤ sW ΣΠ f ,(2) ΠΣΠ f ≡ sW ΣΠ f ,(3) ΣΠΣ f ≡ sW ΠΣ f . Proof. (1) Given an instance ( x n,k ) of ΠΣ f , we just swap n – with k –positions in ( x n,k ), thenwe apply ΣΠ f to the result, then we use the parallelization of the problem t : Y N ⇒ Y N from Lemma 20 in order to convert the output of ΣΠ f from ( Y N ) N into a double sequencein ( Y N ) N and then we swap the n – and k –positions again to obtain a result in ΠΣ f ( x n,k ),which is correct because( ∃ k )( ∀ n ) y n,k ∈ f ( x n,k ) = ⇒ ( ∀ n )( ∃ k ) y n,k ∈ f ( x n,k ) . TASHING AND PARALLELIZATION PENTAGONS 11 f Σ f Π f ΣΠ f ΠΣ f Π ΣΣ ΠFigure 2: Parallelization-stashing pentagon in the Weihrauch lattice.(2) Since Π is a closure operator we have ΣΠ f ≤ sW ΠΣΠ f and ΠΠ f ≤ sW Π f , and by (1)and monotonicity of Σ we obtain ΠΣΠ f ≤ sW ΣΠΠ f ≤ sW ΣΠ f .(3) Since Σ is an interior operator we have ΣΠΣ f ≤ sW ΠΣ f and Σ f ≤ ΣΣ f , and by (1) andmonotonicity of Π we obtain ΠΣ f ≤ sW ΠΣΣ f ≤ sW ΣΠΣ f .That is the action of the stashing-parallelization monoid { Π , Σ } ∗ on a problem f inthe (strong) Weihrauch lattice leads to at most five distinct degrees { f, Π f, Σ f, ΣΠ f, ΠΣ f } ,which are arranged in a pentagon, see the diagram in Figure 2. Every line in the diagramindicates a ≤ sW –reduction in the upwards direction.Of course, if a problem f is parallelizable and stashable (such as any problem of the form f = ΣΠ g is), then the pentagon reduces to a single degree. Other smaller sizes than fivecan be realized too and, as we will see, also the maximal size of five can be realized. In anycase, the stashing-parallelization pentagon can be seen as the trace of f under { Σ , Π } ∗ thatreveals some information about the underlying problem f . It follows from Proposition 21and the fact that Π and Σ are closure and interior operators, respectively, that for a problem f with a full pentagon of size five, f is incomparable with the opposite problems ΣΠ f andΠΣ f . Corollary 22 (Full pentagons) . For every problem f we obtain:(1) f ≤ sW ΣΠ f ⇐⇒ Π f ≡ sW ΣΠ f ,(2) ΣΠ f ≤ sW f ⇐⇒ Σ f ≡ sW ΣΠ f ,(3) f ≤ sW ΠΣ f ⇐⇒ Π f ≡ sW ΠΣ f ,(4) ΠΣ f ≤ sW f ⇐⇒ Σ f ≡ sW ΠΣ f .Analogous statements hold for the reductions ≤ W , ≤ ∗ sW , ≤ ∗ W .There are many interesting questions regarding the interaction of parallelization andstashing. For instance, we will see later in Corollary 58 that the map f (Σ f, Π f ) isnot injective on Weihrauch degrees, i.e., problems are not characterized by their respectivepentagons. However, these pentagons still reveal some interesting information in manycases, as we will see. The Upper Turing Cone Operator
The main purpose of this section is to prove that the upper Turing cone operator f f D is an interior operator on the (strong) Weihrauch lattice that coincides with the interioroperator f Σ f , restricted to (strongly) parallelizable problems.In the following it is useful to have a simplified version of ΣΠ f , where we replacedouble sequences on the input side by ordinary sequences. For this purpose we consider theinjection I X : X ֒ → X N , x ( x, x, x, ... ) for every represented space X . Lemma 23. ΣΠ f ≡ sW (ΣΠ f ) ◦ I X N for every problem f : ⊆ X ⇒ Y . Proof.
Since I X N is computable, it is clear that (ΣΠ f ) ◦ I X N ≤ sW ΣΠ f holds. On the otherhand, given an instance (( x n,k ) k ) n ∈ X N of ΣΠ f , we can compute the interleaved se-quence ( x n,k ) h n,k i ∈ X N and then apply I X N followed by ΣΠ f . This yields a sequence(( y m, h n,k i ) h n,k i ) m ∈ Y NN that satisfies the property ( ∃ m )( ∀h n, k i ) y m, h n,k i ∈ f ( x n,k ). Usingthe parallelization of the problem t : Y N ⇒ Y N from Lemma 20 we can convert this into asequence in Y N and then extract a diagonal sequence (( y n, h n,k i ) h n,k i ) n ∈ Y N from it. Thissequence can be converted back to Y NN by the parallelization of the function s : Y N → Y N from Lemma 20 and we claim that the result is a solution of ΣΠ f ((( x n,k ) k ) n ). This isbecause ( ∃ m )( ∀h n, k i ) y m, h n,k i ∈ f ( x n,k ) = ⇒ ( ∃ n )( ∀ k ) y n, h n,k i ∈ f ( x n,k ) . The exact relation between the upper Turing cone operator and the stashing is capturedin the following result, again with the help of the injection I X . Proposition 24 (Upper Turing cone operator) . f D ≡ sW Σ f ◦ I X holds for every problem f : ⊆ X ⇒ Y . In particular, f D ≤ sW Σ f . Proof.
We consider the represented spaces (
X, δ X ) and ( Y, δ Y ). We claim that every realizer F : ⊆ N N → N N of Σ f ◦ I X is also a realizer of f D , which proves f D ≤ sW Σ f ◦ I X . To thisend, let F be such a realizer and x := δ X ( p ) ∈ dom( f D ) = dom( f ). Let ( y n ) := δ Y N F ( p ) ∈ Σ f ( x, x, ... ). Let h q , q , ... i := F ( p ) and let G : ⊆ N N → N N be a computable realizer of thepartial inverse ι − : ⊆ Y → Y of the embedding ι : Y → Y , which is computable according to[8, Corollary 3.10]. Then there is some n ∈ N with y n = δ Y G ( q n ) = δ Y ( q n ) ∈ f δ X ( p ) = f ( x )and y n ≤ T G ( q n ) ≤ T F ( p ). This proves the claim.For the reverse reduction Σ f ◦ I X ≤ sW f D we first note that δ Y is a precomplete representa-tion by [8, Proposition 3.8]. Hence there is a total computable r : N → N such that for all q ∈ N N , n ∈ N the function ϕ qr ( n ) is always total and if ϕ qn is total and ϕ qn ∈ dom( δ Y ), then δ Y ( ϕ qr ( n ) ) = δ Y ( ϕ qn ). Intuitively, the programme with code r ( n ) works as the programme n , but it adds 1 to all output results and fills up the output with dummy symbols 0 inappropriate positions as long as no other better information becomes available. Now weconsider the computable function H : N N → N N with H ( q ) := h ϕ qr (0) , ϕ qr (1) , ... i . We claimthat HF is a realizer of Σ f ◦ I X for every realizer F of f D . To this end, let F be such arealizer and x := δ X ( p ) ∈ dom(Σ f ◦ I X ) = dom( f ). Then there exists some y ≤ T F ( p ) with y ∈ f ( x ). Hence, there is some n ∈ N such that y = δ Y ( ϕ F ( p ) n ) = δ Y ( ϕ F ( p ) r ( n ) ). This impliesthat δ Y N HF ( p ) ∈ Σ f ◦ I X ( x ), which completes the proof. TASHING AND PARALLELIZATION PENTAGONS 13
We note that the main idea of the proof, namely to compute on all G¨odel numbers inparallel, can only be realized because the stashing uses a completion Y of the space Y onthe output side.By a combination of Propositions 24 and 21 with Lemma 23 we obtain the followingcorollary. Corollary 25 (Upper Turing cone operator) . (Π f ) D ≡ sW ΣΠ f and (ΠΣ f ) D ≡ sW ΠΣ f forevery problem f .This means that both problems on the right-hand side of the diagram in Figure 2 canbe seen as upper Turing cone versions and hence as computability-theoretic problems. Weemphasize that Turing cones appear here out of a purely topological context without anycomputability theory being involved. This is because Corollary 25 is also correct when thecomputability-theoretic Weihrauch reducibility is replaced by its topological counterpart.We mention in passing that the upper Turing cone operator is an interior operator onthe Weihrauch lattice. Proposition 26 (Upper Turing cone operator as interior operator) . The operation f f D is an interior operator on the (strong) Weihrauch lattice. That is, for all problems f, g wehave:(1) f D ≤ sW f ,(2) f ≤ sW g = ⇒ f D ≤ sW g D ,(3) f D ≤ sW f DD .Analogous statements hold for ≤ W , ≤ ∗ W and ≤ ∗ sW . Proof.
We consider problems f : ⊆ X ⇒ Y and g : ⊆ W ⇒ Z .(1) This follows from f D ≤ sW Σ f ≤ sW f , which holds by Propositions 24 and 18.(2) Let f ≤ W g hold via computable H, K : ⊆ N N → N N , i.e., H h id , GK i is a realizer of f whenever G is a realizer of g . We claim that f D ≤ W g D holds via id , K . Let p be a nameof some input x ∈ dom( f ). Then K ( p ) is a name of a point w ∈ dom( g ) and any name q of a point in g ( w ) yields a name H h p, q i of a point in f ( x ), since there is a realizer G of g with GK ( p ) = q . Let now G be a realizer of g D . Then there is a name q of a point in g ( w )such that q ≤ T GK ( p ). Hence H h p, q i ≤ T h p, GK ( p ) i . This shows that h id , GK i is a realizerof f D whenever G is a realizer of g D and hence f D ≤ W g D . The statement for ≤ sW can beproved analogously. In the topological cases we have to work with continuous H, K . Then H is computable relative to some r ∈ N N and we obtain as above H h p, q i ≤ T h r, p, GK ( p ) i .Hence f D ≤ ∗ W g D holds via continuous H ′ , K , where H ′ h p, q i := h r, p, q i .(3) We have even f DD = f D by transitivity of Turing reducibility.By Proposition 21 stashing extends to an interior operator on parallelizable Weihrauchdegrees and by Corollary 25 the upper Turing cone operator coincides on those degrees withstashing. Corollary 27. f Σ f and f f D are identical interior operators restricted to (strongly)parallelizable (strong) Weihrauch degrees.We note that f Σ f and f f D are not identical on arbitrary Weihrauch degrees.The problem f D is always computable when f has only computable solutions. For instance, LPO D is computable, while this is not the case for Σ( LPO ) (see Proposition 40).We can also formulate this result such that problems which are simultaneously stashableand parallelizable are automatically closed under applying the upper Turing cone operator.
Corollary 28.
For every problem f the following conditions are equivalent to each other:(1) Σ f ≡ sW f and Π f ≡ sW f ,(2) f D ≡ sW f and Π f ≡ sW f .An analogous property holds with ≡ W instead of ≡ sW .It follows from Corollary 25 that problems g that are simultaneously parallelizableand stashable can only occur in certain regions of the Weihrauch lattice. For one, everyproblem with the set of Turing degrees as target set is densely realized by [12, Corollary 4.9],which means that a realizer of such a problem can produce outputs that start with arbitraryprefixes. This in turn implies by [14, Proposition 6.3] that any problem with discrete outputbelow it has to be computable. We formulate this as a corollary. Corollary 29 (Parallelizable and stashable problems) . Let f : ⊆ X ⇒ N be a problemand let g be a problem that is parallelizable and stashable. If f ≤ W g holds, then f iscomputable.One of the weakest problems with discrete output that is discontinuous is ACC N , theall-or-co-unique choice problem on N . This problem was studied in [12] and an equivalentproblem was investigated earlier under the name LLPO X [31, 16] (and under the name LLPO ∞ [25, Definition 16] in the case of ACC N ). Intuitively speaking, ACC N is the problemthat given a list of natural numbers which is either empty or contains exactly one number,one has to produce a number which is not in the list. For f = ACC N we can also phraseCorollary 29 as follows. Corollary 30 (The cone of all-or-co-unique choice) . If ACC N ≤ W g holds for some problem g , then g cannot be simultaneously parallelizable and stashable.Hence, in a certain sense, problems that are parallelizable and stashable at the sametime are rare, even rarer than this result suggests. Namely, ACC N is not the weakest discon-tinuous problem with discrete output. Mylatz has proved that there are also discontinuousproblems of type f : ⊆ N N ⇒ N with f < W ACC N [25, Satz 14].5. The Discontinuity Problem in Pentagons
In this section we investigate the discontinuity problem
DIS by studying a number ofstashing-parallelization pentagons in which it appears as the bottom problem. Along theline we will formulate some problems that are equivalent to
DIS . In the following we use thenotation b f = Σ f for the stashing of specific problems f . We start with defining a numberof problems related to ACC N . Definition 31 (Problems related to all-or-co-unique choice) . We consider the followingproblems:(1) A : N N ⇒ N , hh i, n i , p i 7→ { k ∈ N : ϕ pi ( n ) = k } ,(2) B : N N ⇒ N N , h i, p i 7→ { q ∈ N N : ( ∃ n ) ϕ pi ( n ) = q ( n ) ∈ N } ,(3) C : N N ⇒ N N , h i, p i 7→ { q ∈ N N : ( ∃ n ) ϕ pi ( n ) = q ( n ) } .As a first result we prove that the discontinuity problem is the stashing of ACC N . Proposition 32 (All-or-co-unique choice) . We obtain \ ACC N ≡ sW b A ≡ sW B ≡ sW C ≡ sW DIS and
ACC N ≡ sW A . TASHING AND PARALLELIZATION PENTAGONS 15
Proof.
It is straightforward to see that
ACC N ≡ sW A , which implies \ ACC N ≡ sW b A by Propo-sition 18.We prove b A ≡ sW B . We note that b A is of type b A : ( N N ) N ⇒ N N . In order to show b A ≤ sW B , we consider instances p = ( hh i , n i , p i , hh i , n i , p i , hh i , n i , p i , ... ) ∈ ( N N ) N of b A . There is a j ∈ N such that ϕ h p i j ( k ) = ϕ p k i k ( n k ) for all k ∈ N and all p of theabove form. Hence, a solution to B h j, h p ii is a solution to b A ( p ). Thus b A ≤ sW B . For theinverse reduction we consider the computable function K : N N → ( N N ) N with K h i, p i :=( hh i, i , p i , hh i, i , p i , hh i, i , p i , ... ). This function reduces B to b A , i.e., we obtain b A ≡ sW B .It is easy to see that B ≤ sW C holds, since the parallelization b ι : N N → N N of theembedding ι : N → N is computable, where ι is computable according to [8, Corollary 3.10].We now prove C ≡ sW DIS . The reduction C ≤ sW DIS follows with help of the computablefunction K : N N → N N with K h i, h r, p ii := hh i, r i , p i . The computable inverse of K yields DIS ≤ sW C . We note that these reductions are obvious when the corresponding ϕ h r,p i i istotal, but otherwise any q ∈ N N is allowed as a solution in both cases.Finally, we prove DIS ≤ sW B . By Theorem 9 it suffices to show that player I has acomputable winning strategy in the Wadge game B . Therefore we consider a G¨odel number i ∈ N such that ϕ pi ( n ) = k if and only if h n, k i + 1 is the first number of the form h n, m i + 1listed in p . If there is no number of this form, then ϕ pi ( n ) is undefined. In other words, theprogram i upon input n and oracle p searches for the first number of the form h n, k i +1 listedin p and outputs k if such a number is found. Now player I in the Wadge game B startsplaying h i, p i with p = 000 ... . This corresponds to the nowhere defined function ϕ p i andhence player II must play a name of some q ∈ N N with some n ∈ N such that q ( n ) ∈ N ;otherwise player II looses. However, when the first candidate n ∈ N with q ( n ) ∈ N appears, then player I modifies the p in its play to a p by appending h n , q ( n ) i in thecurrent position to it. Hence ϕ p i ( n ) = q ( n ). This forces player II to modify its play toa name of some q with another n ∈ N with q ( n ) ∈ N and n = n ; otherwise playerII looses. Now player I modifies p by appending h n , q ( n ) i to it. This strategy can becontinued inductively and describes a computable winning strategy for player I.We can conclude from Proposition 32 that the discontinuity problem is stashable. Corollary 33.
DIS is strongly stashable.The next result supports the slogan that “non-computability is the parallelization of(effective) discontinuity”.
Theorem 34 (Discontinuity and non-computability) . d DIS ≡ sW NON . Proof.
By Proposition 32 it suffices to show b C ≡ sW NON . For this purpose it is helpful toreformulate
NON as follows:
NON : N N ⇒ N N , p
7→ { q ∈ N N : ( ∀ i )( ∃ n ) ϕ pi ( n ) = q ( n ) } . We note that this formulation of
NON is equivalent to the usual one, as for non-totalfunctions ϕ pi there exists always an n ∈ N \ dom( ϕ pi ), which implies ϕ pi ( n ) = q ( n ), since q is total. With NON written in this form it is clear that C ≤ sW NON . Moreover,
NON isstrongly parallelizable, since given p := h p , p , ... i it is clear that q T p implies q T p i for every i ∈ N . Hence, b C ≤ sW NON . ACC n DISDNC n PANON
Π ΣΣ ΠFigure 3:
ACC n pentagon in the Weihrauch lattice for n ≥ NON ≤ sW b C , we assume that we have given some p ∈ N N .Then we can evaluate b C on the instance ( p i ) i ∈ N with p i := h i, p i in order to get someoutput ( q i ) i ∈ N ∈ b C ( p i ) i ∈ N with the property that ( ∀ i )( ∃ n ) ϕ pi ( n ) = q i ( n ). We claim that q := h q , q , q , ... i 6≤ T p . If we assume the contrary, then there is some total computable r : N → N such that ϕ pr ( i ) ( n ) = q i ( n ) for all i, n ∈ N . Hence, by the relativized version ofKleene’s fixed point theorem [29, Theorem 2.2.1] there is some i ∈ N with ϕ pi ( n ) = ϕ pr ( i ) ( n ) = q i ( n ) for all n ∈ N , which contradicts the assumption that q i ∈ C h i, p i . Altogether, thisproves d DIS ≡ sW b C ≡ sW NON .We note that
DNC D N < W DNC N follows since ACC N ≤ W DNC N , but ACC N W DNC D N byCorollary 30. Together with Fact 10, we have established the pentagon of ACC N given inFigure 1. Perhaps the pentagon in Figure 1 is the most natural pentagon in which thediscontinuity problem DIS appears, but it is by far not the only one. It was observed byJockusch [20, Theorem 6] and Weihrauch [31, Theorem 4.3] that the problems
DNC n and ACC n , respectively, form strictly decreasing chains, i.e., we have the following fact (see also[17, Corollary 82], [12, Corollary 3.8]). Fact 35 (Jockusch 1989, Weihrauch 1992) . For all n ≥ DNC N < sW DNC n +1 < sW DNC n ,(2) ACC N < sW ACC n +1 < sW ACC n .On the other hand, it turns out that the stashing of the problem ACC n is stronglyequivalent to DIS for all n ≥
2. In order to express this result, it is useful to consider auniversal function of type U n : ⊆ N N → { , ..., n − } N . Such a function can be defined bytruncating U accordingly: U n hh i, r i , p i := max( n − , ϕ h r,p i i ) = max( n − , U hh i, r i , p i )whenever ϕ h r,p i i is total (where the maximum is understood pointwise). Using this definitionwe can also modify the problem DIS accordingly and in this way we obtain
DIS n : N N ⇒ { , ..., n − } N , p
7→ { q ∈ { , ..., n − } N : U n ( p ) = q } for all n ≥
2. In these terms we obtain the following result.
Proposition 36 (All-or-co-unique choice) . \ ACC n ≡ sW DIS n ≡ sW DIS for all n ≥ TASHING AND PARALLELIZATION PENTAGONS 17
LLPO DISWKL PANON
Π ΣΣ ΠFigure 4:
LLPO pentagon in the Weihrauch lattice.
Proof.
By Fact 35 it suffices to consider the case n = 2. The remaining cases follow byProposition 32 since stashing is an interior operator by Proposition 18. The reduction DIS ≡ sW \ ACC N ≤ sW \ ACC also follows. The reduction \ ACC ≤ sW DIS can be proved almostliterally following the lines of the proof of Proposition 32 with some obvious modifications.For instance, one needs to replace all terms ϕ pi ( n ) in the definitions of A, B and C bymax(1 , ϕ pi ( n )); one has to replace the output types of A, B and C by { , } , { , } N and { , } N , respectively; and one has to work with the embedding ι : { , } → { , } . It remainsto prove the reduction DIS ≤ sW DIS . For this direction we use the computable embedding ι : N N → N , p p (0) p (1) p (2) ... . There is a total computable s : N → N such that ι ( ϕ ts ( i ) ) = max(1 , ϕ ti ) for all i ∈ N and t ∈ N N such that ϕ ti is total and max(1 , ϕ ti ) containsinfinitely many ones. Now, given in instance hh i, r i , p i of DIS we compute the instance hh s ( i ) , r i , p i of DIS . If q ∈ N N satisfies q = U hh s ( i ) , r i , p i , then ι ( q ) = U hh i, r i , p i follows.This is clear if ϕ h r,p i i is total and max(1 , ϕ h r,p i i ) contains infinitely many ones. But otherwiseevery q ∈ N N satisfies the conclusion. Altogether, this completes the proof.The upper Turing cone version of DNC n for n ≥ PA of finding aPA degree relative to the input. By a result of Jockusch and Friedberg [20, Theorem 5] theTuring degrees of q ≫ p are exactly the degrees that compute a diagonally non-computablefunction f : N → { , ..., n − } relative to p for every n ≥ Fact 37. PA ≡ sW DNC D n for every n ≥ PA < W DNC n follows since ACC n ≤ W DNC n , but ACC n W PA by Corol-lary 29. Altogether, we have thus established the pentagon of ACC n for n ≥ n = 2. Since LLPO ≡ sW ACC , Fact 10 yields the stashing-parallelization pentagon of LLPO given in Fig-ure 4.We have a number of basic discrete problems ordered in the following way [12, Fact 3.4],[5, Theorem 3.10], [10, Section 13].
Fact 38.
ACC N ≤ W LLPO ≤ W LPO ≤ W lim ≤ W C N .Now the question appears how far up in this chain of discrete problems we can go suchthat we still obtain the discontinuity problem DIS as stashing of the corresponding discrete problem? We will see in Proposition 60 that a phase transition in this respect happensbetween
LPO and lim .We now study the pentagon of LPO . We use the notation W pi := dom( ϕ pi ) and by χ A : N → { , } we denote the characteristic function of A ⊆ N with A = χ − A { } . We firstdefine some problems related to LPO . Definition 39 (Problems related to
LPO ) . We consider:(1) L : N N → { , } , hh i, n i , p i 7→ − χ W pi ( n ) = (cid:26) n ∈ dom( ϕ pi )1 otherwise ,(2) D : N N ⇒ { , } N , h i, p i 7→ { q ∈ { , } N : ( ∃ n ) χ W pi ( n ) = q ( n ) ∈ { , }} ,(3) E : N N ⇒ { , } N , h i, p i 7→ { q ∈ { , } N : ( ∃ n ) χ W pi ( n ) = q ( n ) } .Now we can prove the following result. Proposition 40 (Stashing of
LPO ) . LPO ≡ sW L and d LPO ≡ sW b L ≡ sW D ≡ sW E ≡ sW DIS . Proof.
We proceed as in the proof of Proposition 32. It is easy to see that
LPO ≡ sW L ,which implies d LPO ≡ sW b L . The same reductions that prove b A ≡ sW B in Proposition 32also show b L ≡ sW D . The reduction D ≤ sW E follows using the computable embedding ι : { , } → { , } . By Proposition 32 and Fact 38 and since stashing is an interior op-erator by Proposition 18, we obtain DIS ≡ sW \ ACC N ≤ sW d LPO .It only remains to show E ≤ sW DIS . By the proof of Proposition 36 it suffices to show E ≤ sW C , where C : N N ⇒ { , } N , h i, p i 7→ { q ∈ { , } N : ( ∃ n ) min(1 , ϕ pi ( n )) = q ( n ) } is the modification of the function from Proposition 32 that was used in the proof of Propo-sition 36 in order to show DIS ≡ sW DIS ≡ sW C . For the reduction E ≤ sW C we proceed asfollows. Given an instance h i, p i of E , we try to find out for each n ∈ N , which of thetwo consecutive values 2 n, n + 1 appears in W pi first, if any. More precisely, there is acomputable function r : N → N such that ϕ pr ( i ) ( n ) = n ∈ W pi is found first1 if 2 n + 1 ∈ W pi is found first ↑ if { n, n + 1 } ∩ W pi = ∅ holds for all i, n ∈ N and p ∈ N N . We use the computable function K : N N → N N with K h i, p i = h r ( i ) , p i to translate instances of E into instances of C and the computablefunction H : { , } N → { , } N with H ( q )(2 n ) := 1 − q ( n ) and H ( q )(2 n + 1) := q ( n ) inorder to translate solutions of C into solutions of E . That this reduction is correct can beseen as follows. Given an instance h i, p i of E and q ∈ C h r ( i ) , p i = CK h i, p i , there is some n ∈ N with min(1 , ϕ pr ( i ) ( n )) = q ( n ). We are now in exactly one of the following three cases:(1) ϕ pr ( i ) ( n ) = 0 = ⇒ ( q ( n ) = 1 and 2 n ∈ W pi ) = ⇒ H ( q )(2 n ) = 0 = χ W pi (2 n ),(2) ϕ pr ( i ) ( n ) = 1 = ⇒ ( q ( n ) = 0 and 2 n + 1 ∈ W pi )= ⇒ H ( q )(2 n + 1) = 0 = χ W pi (2 n + 1),(3) ϕ pr ( i ) ( n ) = ↑ = ⇒ ( q ( n ) ∈ { , } and { n, n + 1 } ∩ W pi = ∅ )= ⇒ ( H ( q )(2 n ) = 1 = χ W pi (2 n ) or H ( q )(2 n + 1) = 1 = χ W pi (2 n + 1)).In any case we obtain H ( q ) ∈ E h i, p i . Altogether we obtain E ≤ sW C ≡ sW DIS . TASHING AND PARALLELIZATION PENTAGONS 19
LPO DIS lim J D NON
Π ΣΣ ΠFigure 5:
LPO pentagon in the Weihrauch lattice.By Fact 10 we have d LPO ≡ sW lim ≡ sW J . It is clear that J D is strongly Weihrauchequivalent to J D : D ⇒ D , a
7→ { b ∈ D : a ′ ≤ T b } . We note that J D < W lim follows since LPO ≤ W lim, but LPO W J D by Corollary 29. Alto-gether, we have established the pentagon of LPO given in Figure 5.Proposition 40 also leads to another characterization of the discontinuity problem interms of ranges. This characterization is unique among all the characterizations that we haveprovided because it is purely set-theoretic (i.e., no G¨odel numberings or other computability-theoretic concepts are used) and because it only involves standard data types (i.e., nocompletions are mentioned).
Definition 41 (Range non-equality problem) . We call
NRNG : N N ⇒ N , p
7→ { A ∈ N : A = range( p − } the range non-equality problem .We now obtain the following characterization. Corollary 42 (Range non-equality problem) . DIS ≡ sW NRNG . Proof.
By Proposition 40 it suffices to show
NRNG ≡ sW E . For one, there is a j ∈ N suchthat W pj = range( p − NRNG ≤ sW E . On the other hand,there is a computable f : N N → N N such that range( f h i, p i −
1) = W pi for all p ∈ N N and i ∈ N , which shows E ≤ sW NRNG .The reader might have noticed that a lot of problems that occur in the lower parts of ourpentagons can actually be seen as complementary problems of other well-known problems.We briefly make this more precise.
Definition 43 (Complementary problem) . For every problem f : ⊆ X ⇒ Y we define the complementary problem f c : ⊆ X ⇒ Y by graph( f c ) := graph( f ) c = ( X × Y ) \ graph( f ).That is dom( f c ) = { x ∈ X : f ( x ) = Y } and f ( x ) := Y \ f ( x ) for all x ∈ dom( f c ).Using this concept we see that DIS = U c , NRNG = EC c , and NON = ( ≥ T ) c , where ≥ T : N N ⇒ N N , p
7→ { q ∈ N N : q ≤ T p } . Even though complementation yields a neat way of expressing these problems, f f c is not an operation on the Weihrauch lattice. For instance J c is obviously computable,whereas EC c ≡ sW DIS is not, although J ≡ sW EC by Fact 10.6. Majorization and Hyperimmunity
In this section we study the stashing-parallelization pentagons of the non-majorizationproblem
NMAJ that can be seen as an asymmetric version of the discontinuity problem.The non-majorization problem
NMAJ is introduced in the following definition and it isrelated to the well-known hyperimmunity problem.
Definition 44 (Problems related to hyperimmunity) . We consider the following problems:(1)
NGEQ : N N ⇒ N , hh i, n i , p i 7→ { k ∈ N : ϕ pi ( n ) k } ,(2) NMAJ : N N ⇒ N N , h i, p i 7→ { q ∈ N N : ( ∃ n ) ϕ pi ( n ) q ( n ) } ,(3) HYP : N N ⇒ N N , p
7→ { q ∈ N N : ( ∀ r ≤ T p )( ∃ n ) r ( n ) < q ( n ) } ,(4) MEET : N N ⇒ N N , p
7→ { q ∈ N N : ( ∀ r ≤ T p )( ∃ n ) r ( n ) = q ( n ) } ,(5) 1- WGEN : N N ⇒ N N , p
7→ { q ∈ N N : q is weakly 1–generic relative to p } .The non-majorization problem NMAJ has been defined here ad hoc, whereas the Weih-rauch complexity of the hyperimmunity problem
HYP and the weak –genericity problem WGEN have already been studied in [12, 13]. The principle
MEET was introduced in areverse mathematics context in [18]. We note that the existential quantifier “ ∃ n ” in HYP and
MEET could equivalently be replaced by “ ∃ ∞ n ”. We recall that a point p ∈ N N is called weakly –generic relative to q ∈ N N if p ∈ U for every dense open set U ⊆ N N that is c.e. openrelative to q . A set U ⊆ N N is c.e. open relative to q if U = U qi := { p ∈ N N : 0 ∈ dom( ϕ qi ) } for some i ∈ N . By a theorem of Kurtz the hyperimmune degrees coincide with the weakly1–generic degrees and this also holds uniformly in the following sense [12, Corollary 9.5]. Fact 6.1 (Uniform theorem of Kurtz) . HYP ≡ W WGEN .On the first sight, the non-majorization problem
NMAJ looks similar to the discontinuityproblem
DIS in the form of C , as defined in Definition 31. In fact, NMAJ can be seen asan asymmetric version of C , since the inequality = is simply replaced by (we note that is not the same as < here, as ϕ pi might be partial and ϕ pi ( n ) q ( n ) is supposed to meanthat either ϕ pi ( n ) does not exist or ϕ pi ( n ) exists and ϕ pi ( n ) < q ( n ).) Despite the similaritybetween NMAJ and
DIS , it turns out that
NMAJ is neither equivalent to
DIS nor stashable.Among all the problems that we have studied here, it is perhaps the one that comes closestto
DIS without being equivalent to it. The following result clarifies the relation of theseproblems to each other.
Proposition 45 (The non-majorization problem) . We obtain \ NMAJ ≡ W \ NGEQ ≡ W DIS , \ NMAJ ≡ sW HYP , DIS < W NMAJ < W HYP , and
NON < W HYP D . In particular, NMAJ is notstashable.
Proof.
With C from Definition 31 we obtain DIS ≡ sW C ≤ sW NMAJ ≤ sW NGEQ ≤ W LPO .The latter reduction holds since
LPO can be used to determine whether ϕ pi ( n ) is definedand if it is defined then one can use the original input to find a larger value; otherwise 0 isa suitable output. This implies DIS ≡ W \ NGEQ ≡ W \ NMAJ by Proposition 40 since stashingis an interior operator by Proposition 18. Moreover, we have
HYP ( p ) = { q ∈ N N : ( ∀ r ≤ T p )( ∃ n ) r ( n ) < q ( n ) } = { q ∈ N N : ( ∀ i )( ∃ n ) ϕ pi ( n ) q ( n ) } . TASHING AND PARALLELIZATION PENTAGONS 21
NMAJ DISHYP HYP D NON
Π ΣΣ ΠFigure 6: Non-marjorization pentagon in the Weihrauch lattice.Here clearly “ ⊇ ” holds regarding the second equality since the ϕ pi include all the total r ≤ T p and the other inclusion “ ⊆ ” holds as for ϕ pi that are not total the condition ϕ pi ( n ) q ( n )is satisfied by definition for all n dom( ϕ pi ). We claim that \ NMAJ ≡ sW HYP . For one, it isclear that
NMAJ ≤ sW HYP holds. Moreover,
HYP is strongly parallelizable, as there is somecomputable r : N → N with ϕ h p ,p ,p ,... i r h i,k i ( n ) = ϕ p k i ( n ) for all i, n, k ∈ N and p , p , ... ∈ N N .Together, this implies \ NMAJ ≤ sW HYP . On the other hand, there is a computable s : N → N such that ϕ ps ( i ) ( n ) = ϕ pi h i, n i for all i, n ∈ N and p ∈ N N . Hence, the function K : N N → N N with K ( p ) := hh s (0) , p i , h s (1) , p i , h s (2) , p i , ... i is computable and with q = h q , q , q , ... i ∈ \ NMAJ ◦ K ( p ) we obtain( ∀ i )( ∃ n ) ϕ pi h i, n i = ϕ ps ( i ) ( n ) q i ( n ) = q h i, n i , so in particular q ∈ HYP ( p ). This proves HYP ≤ sW \ NMAJ .Suppose
DIS ≡ W NMAJ , then
NON ≡ W HYP would follow by Theorem 34, since paral-lelization is a closure operator. However, it is well-known that there are hyperimmune-freenon-computable degrees [23, Section 2], i.e., there is non-computable q which is not of hy-perimmune degree. The problem NON has a realizer that, on computable inputs, producessuch non-computable q , which is not of hyperimmune degree. Since hyperimmune degreesare upwards closed by [23, Theorem 1.1], we obtain that NON < W HYP D ≤ W HYP . Thisimplies
DIS < W NMAJ . Finally, we clearly have
NMAJ < W HYP as NMAJ has computablesolutions on all instances, while
HYP does not.We emphasize that we have only proved \ NMAJ ≡ W DIS with an ordinary Weihrauchequivalence, unlike in all previous cases, where we have established a strong Weihrauchequivalence. Hence we are left with the following open question.
Question 46.
Does \ NMAJ ≡ sW DIS hold?We note that
HYP D is exactly the problem of finding a hyperimmune degree relativeto the input, i.e., it can equivalently be described as HYP D : D ⇒ D , a
7→ { b ∈ D : b is of hyperimmune degree relative to a } , since hyperimmune degrees are upwards closed by [23, Theorem 1.1]. This establishes thepentagon in Figure 6, except that we did not yet prove HYP D < W HYP . In the case of theearlier pentagons discussed here, we have used Corollary 29 for the corresponding separation.
In the case of the hyperimmunity problem a more tailor-made argument is required, since
NMAJ has no natural number output and is densely realized itself. We combine ideas fromthe proof of [14, Proposition 6.3] and the proof of Proposition 32.
Proposition 47.
HYP D < W HYP (even restricted to computable instances).
Proof.
Here we consider
HYP D to be defined as HYP D : N N ⇒ N N , p
7→ { s ∈ N N : ( ∃ q ≤ T s )( ∀ r ≤ T p )( ∃ n ) r ( n ) < q ( n ) } . It suffices to prove
NMAJ W HYP D . Let us assume the contrary, i.e., let H, K : ⊆ N N → N N be computable functions such that H h id , GK i is a realizer of NMAJ whenever G is a realizerof HYP D . Then there is a computable monotone function h : N ∗ → N ∗ that approximates H in the sense that H ( p ) = sup w ⊑ p h ( w ) for all p ∈ dom( H ). As in the proof of Proposition 32we consider a fixed G¨odel number i ∈ N such that ϕ pi ( n ) = k if and only if h n, k i + 1 isthe first number of the form h n, m i + 1 listed in p . Now we use h and the finite extensionmethod to construct an input p ∈ N N of NMAJ on which the above reduction fails. Westart with p := 000 ... , which yields the nowhere defined function ϕ p i . For this p there isa lexicographically first w ∈ { , , , ... } ∗ such that | h h p | | w | +1 , d w i| > d ∈ { , } . This is because there is some s ∈ { , , , ... } N with d s ∈ HYP D ( p ) for bothvalues d ∈ { , } . We choose a := max { h h p | | w | +1 , d w i (0) : d ∈ { , }} and b := h , a i + 1and we continue with p := 0 | w | +1 b ... , which yields a function that satisfies ϕ p i (0) = a and is undefined otherwise. Again there is a lexicographically first w ∈ { , , , ... } ∗ oflength | w | > | h h p | | w | + | w | +2 , d w d w i| > d , d ∈ { , } andnow we choose a := max { h h p | | w | + | w | +2 , d w d w i (1) : d , d ∈ { , }} and b := h , a i + 1 . The next input is p := 0 | w | +1 b | w | b ... , which represents a function ϕ p i that satisfies ϕ p i ( j ) = a j for j ∈ { , } and that is undefined otherwise. We continue the constructioninductively and obtain computable sequences ( p n ) n ∈ N in N N and ( w n ) n ∈ N in N ∗ in this way.The sequence ( p n ) n ∈ N converges to a computable p ∈ N N . This is because the constructionabove only depends on the computable function h and yields longer and longer portionsof p . For every d ∈ { , } N we denote by s d the sequence s d := d w d w d w ... with d j := d ( j ). The construction ensures that H h p, s d i ( n ) ≤ ϕ pi ( n ) for every n ∈ N and d ∈ { , } N and hence H h p, s d i 6∈ NMAJ ( p ) for every d ∈ { , } N . On the other hand, thereis some d ∈ { , } N of hyperimmune degree, which implies s d ∈ HYP D ( p ), since d ≤ T s d .This yields a contradiction to the assumption and hence HYP D < W HYP .Hence, there is no uniform computable method to find a hyperimmune q ∈ N N from anarbitrary member of a hyperimmune degree. This also yields a second proof of NMAJ W DIS .Finally, we mention that the separation in Proposition 47 also yields a separation of thecorresponding problems (i.e., the sets given by the respective solutions on computable in-stances) in the Medvedev lattice (see [11, Theorem 9.1]).We now want to show that
MEET is equivalent to
HYP . The corresponding proof of[18, Theorem 38] can be transferred into our setting. On the first sight it might be abit surprising that replacing = by makes a difference, while replacing < by = does not.However, this comparison does not take the aspect of totality into account. For completenesswe include the proof, which is interesting by itself. TASHING AND PARALLELIZATION PENTAGONS 23
Proposition 48.
HYP ≡ W MEET . Proof.
For p ∈ N N we have MEET ( p ) = { q ∈ N N : ( ∀ i )( ϕ pi total = ⇒ ( ∃ n ) ϕ pi ( n ) = q ( n )) } . We note that in the case of
HYP we obtain a corresponding formulation with insteadof =. In this case totality does not need to be mentioned as the negative condition isautomatically satisfied by partial ϕ pi . It is obvious that HYP ≤ sW MEET , as we just have touse q q + 1 to translate the solution on the same input p . For the opposite direction wenote that by the smn-theorem there is a computable s : N → N such that ϕ ps ( i ) ( n ) = Φ pi h i, n i for all i, n ∈ N . Here Φ pi ( n ) denotes the time complexity, i.e., the number of computationsteps required to compute ϕ pi ( n ) (if it exists and undefined otherwise). Then given an input p ∈ N N and q ∈ HYP ( p ) we can compute r ∈ N N with r h i, n i := (cid:26) ϕ pi h i, n i if Φ pi h i, n i < q ( n )0 otherwisefor all i, n ∈ N . If i ∈ N is such that ϕ pi is total, then Φ pi and hence ϕ ps ( i ) are totaltoo and hence there is some n ∈ N with Φ pi h i, n i = ϕ ps ( i ) ( n ) < q ( n ). This implies that r h i, n i = ϕ pi h i, n i , i.e., r ∈ MEET ( p ). Since r can be computed, given p, q , we obtain MEET ≤ W HYP .We note that the backwards reduction is not a strong Weihrauch reduction.7.
Retractions, All-or-Unique Choice and G¨odel Numbers
In the previous sections we have discussed the stashing of a number of problems of thetype f : ⊆ X ⇒ N with natural number output. In this particular situation we can alsodescribe stashing in an alternative way. This is because the space N N is related to the space N ⊆ N of partial functions f : ⊆ N → N that we can represent by δ N ⊆ N h i, p i := ϕ pi for all i ∈ N , p ∈ N N . The exact relation between these two spaces is captured in the followinglemma. The function ι essentially identifies ↑ (i.e., undefined) with ⊥ . Lemma 49 (Space of partial functions) . The function ι : N ⊆ N → N N with ι ( p )( n ) := (cid:26) p ( n ) if n ∈ dom( p ) ⊥ otherwiseis computable and there is a computable function σ : N N → N ⊆ N such that σ ◦ ι ( p ) is anextension of p for all p ∈ N ⊆ N .The proof is straightforward. We just note that a prefix of a name of ⊥ can start like aname of a natural number n ∈ N and continue with dummy symbols 0, which means that itis actually a name of ⊥ . Hence, we do not get that the spaces N N and N ⊆ N are computablyisomorphic. However, Lemma 49 roughly speaking states that they are “isomorphic up toextensions”. Hence, for every property that is invariant under extensions, it does not matterwhether we work with N N or N ⊆ N . This does, in particular, apply to stashing. Hence, forproblems of type f : ⊆ X ⇒ N we can also describe stashing by the following definition. Definition 50.
For every problem f : ⊆ X ⇒ N we define ϕ f : ⊆ X N ⇒ N N by ϕ f ( x n ) n ∈ N := {h i, p i ∈ N N : ( ∃ n ∈ dom( ϕ pi )) ϕ pi ( n ) ∈ f ( x n ) } for all ( x n ) n ∈ N ∈ dom( f ) N .Hence, as an immediate corollary of Lemma 49 we obtain the following corollary. Corollary 51 (Stashing for discrete outputs) . Σ f ≡ sW ϕ f for all problems f : ⊆ X ⇒ N .This again sheds light on the fact that stashing of parallelizable problems gives uscomputability-theoretic problems.We note that the usage of the G¨odel numbering and partial functions ϕ pi in the previoussections was mostly related to the input side of problems. Hence, implicitly, we have workedwith the input space N ⊆ N . Now, we want to get some better understanding of the effect ofthe space N N on the output side. Corollary 51 can be seen as a way of replacing N N on theoutput side by N N .For some problems, as those discussed in Propositions 32 and 40 it appeared thatdirectly replacing N N by N N on the output side was also possible without changing thedegree. However, the problem NGEQ from Definition 44 is an example that shows that thisis not always possible. If we just replace N N by N N in Σ( NGEQ ), then we obtain
NMAJ ,which is not equivalent to Σ(
NGEQ ).Hence, it is useful to have upper bounds on the price that such a direct replacement of N N by N N costs. This is exactly captured by the following retraction problem RET X thatwe define together with the closely related extension problem EXT X . Definition 52 (Retraction and extension problems) . For every represented space X and Y ⊆ N we define the following problems:(1) RET X : X ⇒ X, x
7→ { y ∈ X : x ∈ X = ⇒ x = y } ,(2) EXT Y : ⊆ N N ⇒ Y N , h i, p i 7→ { q ∈ N N : q is a total extension of ϕ pi } ,where dom( EXT Y ) := {h i, p i : range( ϕ pi ) ⊆ Y } .The parallelization \ RET X captures the complexity of translating X N into X N . ByLemma 49 we obtain the following. Corollary 53. \ RET X ≡ sW EXT X for all X ⊆ N .The problem RET X : X ⇒ X is a multi-valued retraction, i.e., a problem that satisfies RET X | X = id X . And it is the simplest such retraction in terms of strong Weihrauchreducibility. Hence, we obtain the following result. Proposition 54 (Multi-retraceability) . Let X be a represented space. Then there exists acomputable multi-valued retraction R : X ⇒ X if and only if RET X is computable. Proof.
It is clear that
RET X is a multi-valued retraction, which yields the “if”–directionof the proof. On the other hand, if there is a computable multi-valued retraction R , then RET X ≤ sW R and hence RET X is computable too. TASHING AND PARALLELIZATION PENTAGONS 25
Spaces that allow for computable multi-valued retractions were called multi-retraceable in [8]. This condition was further studied by Hoyrup in [19] and related to fixed-pointproperties. Here we rather have to deal with spaces that are not multi-retraceable. In thesecases we can consider the complexity of
RET X as a measure of how far the space X is awayfrom multi-retraceability or, in other words, how difficult it is to determine total extensions.We gain some upper bounds directly from [7, Proposition 2.10]. Fact 7.1.
RET X ≤ sW lim for all totally represented X and RET N ≤ sW C N .For finite X ⊆ N we can classify RET X somewhat more precisely. It is quite easy tosee that the retraction problem RET n is just equivalent to all-or-unique choice AoUC n andhence located in between LLPO = AoUC and C n , as well as LPO . Proposition 55 (Retraction problem) . LLPO ≤ sW RET n ≡ sW AoUC n ≤ sW C n , and RET n ≤ sW RET N ≡ W LPO for all n ≥ Proof.
We start with proving
RET n ≤ sW AoUC n . Given a name p of x ∈ N we generate aname q of the set { , , ..., n − } , as long as only dummy information appears in p . If someprefix of p starts to look like a name of some k < n , then we modify the output q to a nameof the set { k } . Since k ∈ RET n ( x ), this yields the desired reduction.Now we consider the inverse reduction AoUC n ≤ sW RET n . Given a name p of a set A ⊆ { , , ..., n − } with A = { , , ..., n − } or A = { k } for some k < n , we generate aname of ⊥ as long as the name looks like a name of { , , ..., n − } . In the moment wherethe represented set is clearly smaller, we wait until the information suffices to identify thesingleton { k } , and then we modify the output to an output of k ∈ { , , ..., n − } . If weapply RET n to the generated point in { , , ..., n − } it yields a point in A . This establishesthe desired reduction.The reductions LLPO = AoUC ≤ sW AoUC n ≤ sW C n are clear for all n ≥
2. The reduc-tion
RET n ≤ sW RET N is easy to see, as well as RET N ≤ W LPO . For the latter reduction, weconsider a name p of an input x ∈ N . We use LPO in order to decide whether p = b
0. If p = b
0, then we can produce any output k ∈ N ; otherwise we search for the first non-zerocomponent k + 1 in p and produce the corresponding k as output. This yields the reduction RET N ≤ W LPO . Finally, the reduction
LPO ≤ W RET N can be seen as follows. Given aninput p ∈ N N , we seek the first non-zero component. If this component appears in position n , then we generate the output x = n ∈ N . If there is no non-zero component, then wegenerate the output x = ⊥ ∈ N . Now given some k ∈ RET N ( x ) and p , we search a non-zerocomponent in p up to position k . If such a non-zero component appears, then LPO ( p ) = 0;otherwise LPO ( p ) = 1. This describes the reduction LPO ≤ W RET N .Using this result, we obtain the pentagon of RET n in the Weihrauch lattice. By Fact 10we have \ LLPO ≡ sW c C n ≡ sW WKL , which implies \ RET n ≡ sW WKL . On the other hand, byProposition 40 we have \ LLPO ≡ sW d LPO ≡ sW DIS and this yields
RET n ≡ W DIS . Corollary 56 (Pentagon of retractions) . \ RET n ≡ W \ RET N ≡ W DIS , \ RET n ≡ sW WKL , and \ RET N ≡ W lim for all n ≥ RET n for n ≥ LLPO in Figure 4, while the pentagon of
RET N looks like the pentagonof LPO in Figure 5 (except that one cannot use strong Weihrauch reducibility). The firstmentioned fact is also interesting, as the problems
RET n form a strictly increasing chain,as we prove next. Proposition 57.
AoUC n < W AoUC n +1 < W AoUC N for all n ≥ Proof.
It suffices to prove
AoUC n +1 W AoUC n for all n ≥
2. Let us assume for a contra-diction that
AoUC n +1 ≤ W AoUC n for some n ≥ H, K . Now consider thename p = 000 ... of the empty set ∅ = range( p − a k := H h p, k i ∈ { , ..., n } for k < n satisfy |{ a , ..., a n − }| ≤ n . For each k < n there is some prefix of p of length l k thatsuffices to produce the output a k . For the prefix l := max { l , ..., l n } we have | H h p | l N N × { , ..., n }i| = |{ a , ..., a n − }| ≤ n, despite the fact that p | l N N contains names of all singleton sets A ⊆ { , , ..., n } . Since thereare n + 1 such sets, this is a contradiction!That means that with the problems RET n ≡ sW AoUC n we have a strictly increasingsequence of problems in between LLPO and
LPO that all parallelize and stash to the sameproblems, respectively. This is a strong refutation of injectivity of the following operation.
Corollary 58.
The map f ( b f, b f ) is not injective on Weihrauch degrees.This means that pentagons do not characterize the Weihrauch degrees that generatethem, even though they reveal some useful information about them.8. Limit Avoidance and a Phase Transition
The pentagons determined so far are all related to the discontinuity problem. This couldlead to the false conclusion that
DIS is the bottom vertex of typical pentagons. However,this is rather based on the fact that our study of pentagons was motivated by an analysisof the discontinuity problem. In this closing section we study the weakest problem knownto us whose stashing is not
DIS . This is the limit avoidance problem
NLIM N , defined below.We define it together with the related non-lowness problem NLOW . Definition 59 (Problems related to limits) . We consider the following problems for X ⊆ N :(1) NLIM X : ⊆ X N ⇒ X, p
7→ { k ∈ X : lim n →∞ p ( n ) = k } withdom( NLIM X ) = { p : ( p ( n )) n ∈ N converges } ,(2) NLOW : D ⇒ D , a
7→ { b ∈ D : b ′ a ′ } .The following fact is easy to see: Fact 8.1.
ACC N ≤ sW NLIM N ≤ sW NLIM ≡ sW lim ≤ sW lim N ≡ sW C N .If we can prove that DIS < W \ NLIM N holds, then this implies that the stashings of allproblems above NLIM N also lie above DIS . We prove the following stronger result.
Proposition 60. \ NLIM N W WKL . Proof.
We recall that we are working with \ NLIM N : ⊆ N N ⇒ N N , h p , p , p , ... i 7→ { q ∈ N N : ( ∃ n ) lim i →∞ p i ( n ) = q ( n ) ∈ N } . Let us assume for a contradiction that \ NLIM N ≤ W WKL holds. As
WKL is a cylinder (i.e.,satisfies id × WKL ≡ sW WKL ), it follows that \ NLIM N ≤ sW WKL . Since stashing is an interioroperator, this yields \ NLIM N ≤ sW WKL D ≡ sW PA by Fact 37. But this means that every fixed r ∈ N of PA–degree has the property that it computes some q ∈ \ NLIM N h p , p , p , ... i for TASHING AND PARALLELIZATION PENTAGONS 27 every computable h p , p , p , ... i ∈ dom( \ NLIM N ). There are low r ∈ N of PA–degree, i.e.,such that r ′ ≤ T ∅ ′ . This is because the set of diagonally non-computable binary functionsis co-c.e. closed, every such function is of PA-degree (see [29, Theorem 10.3.3]) and by thelow basis theorem there is a low function among those (see [29, Theorem 3.7.2]). Hence forevery computable h p , p , p , ... i ∈ dom( \ NLIM N ) there is some low q ∈ \ NLIM N h p , p , p , ... i .By Corollary 56 this implies that there is some t ∈ \ RET N ( q ), which is limit computable, as \ RET N ≤ sW lim and limit computable operations yield some limit computable outputs on lowinputs. This means that for a fixed limit computable t : N → N we have that lim i →∞ p i = t for all computable h p , p , p , ... i ∈ dom( \ NLIM N ), which is impossible, as lim i →∞ p i is limitcomputable.We mention that Fact 8.1 implies that DNC N ≤ sW \ NLIM N ≤ sW lim. We leave it to afuture study to determine the exact pentagons of the problems listed in Fact 8.1 other than ACC N . Here we just mention one upper bound on ΠΣ( NLIM N ).Since DIS < W NON < W WKL it is clear that Proposition 60 implies \ NLIM N W DIS andalso ΠΣ(
NLIM N ) W NON . It is not difficult to show that the non-lowness problem is anupper bound for ΠΣ(
NLIM N ). Proposition 61.
ΠΣ(
NLIM N ) ≤ sW NLOW . Proof.
For simplicity we work with the equivalent definition
NLOW : N N ⇒ N N , p
7→ { q ∈ N N : q ′ T p ′ } . We first note that
NLOW is parallelizable: this follows since q ′ T h p , p , p , ... i ′ implies q ′ T p ′ i for all i ∈ N . Hence, it suffices to show \ NLIM N ≤ sW NLOW in order to obtainΠΣ(
NLIM N ) ≤ sW NLOW . Now given p := h p , p , p , ... i such that r = lim i →∞ p i exists in N N , we have r ≤ T p ′ . Now let q ∈ N N be such that q ′ T p ′ . Then using the time complexityfunction Φ qi we can define by s h i, n i := Φ qi ( n ) a partial function s : ⊆ N → N that iscomputable from q . This function has the property that every total extension t : N → N ofit computes q ′ . This is because we can simulate the computation of ϕ qi ( n ) for given i, n for t h i, n i time steps. And either the computation halts within this time bound, which implies q ′ h i, n i = 1 or it does not halt, which implies that t h i, n i is not a correct time bound, henceΦ qi ( n ) is undefined and hence q ′ h i, n i = 0. By Lemma 49 we can consider s as a functionof type s : N → N . Since every total extension t : N → N of it computes q ′ , we obtain t T p ′ , which implies r = t for every such extension t . Hence there is some n ∈ N with r ( n ) = s ( n ) ∈ N , i.e., s ∈ \ NLIM N ( p ). This establishes the reduction ΠΣ( NLIM N ) ≤ sW NLOW .9.
Conclusions
We have introduced the stashing operation as a dual of parallelization and we have provedthat it is an interior operator. The action of parallelization and stashing on Weihrauchdegrees naturally leads to the study of stashing-parallelization pentagons, which can beused to describe a number of natural Weihrauch degrees. In many cases of the studiedpentagons the discontinuity problem featured as the bottom problem of the respective pen-tagon. However, we were also able to identify a phase transition point, where this no longerhappens to be the case. The duality inherent in pentagons needs to be studied further in order to simplify the calculation of pentagons, which tends to be difficult in more advancedexamples.
References [1] Vasco Brattka. Computable invariance.
Theoretical Computer Science , 210:3–20, 1999.[2] Vasco Brattka. Effective Borel measurability and reducibility of functions.
Mathematical Logic Quarterly ,51(1):19–44, 2005.[3] Vasco Brattka. The discontinuity problem. arXiv 2012.02143, 2020.[4] Vasco Brattka, Matthew de Brecht, and Arno Pauly. Closed choice and a uniform low basis theorem.
Annals of Pure and Applied Logic , 163:986–1008, 2012.[5] Vasco Brattka and Guido Gherardi. Effective choice and boundedness principles in computable analysis.
The Bulletin of Symbolic Logic , 17(1):73–117, 2011.[6] Vasco Brattka and Guido Gherardi. Weihrauch degrees, omniscience principles and weak computability.
The Journal of Symbolic Logic , 76(1):143–176, 2011.[7] Vasco Brattka and Guido Gherardi. Completion of choice.
Annals of Pure and Applied Logic ,172(3):102914, 2021.[8] Vasco Brattka and Guido Gherardi. Weihrauch goes Brouwerian.
The Journal of Symbolic Logic , elec-tronically published 2020.[9] Vasco Brattka, Guido Gherardi, and Rupert H¨olzl. Probabilistic computability and choice.
Informationand Computation , 242:249–286, 2015.[10] Vasco Brattka, Guido Gherardi, and Alberto Marcone. The Bolzano-Weierstrass theorem is the jumpof weak K˝onig’s lemma.
Annals of Pure and Applied Logic , 163:623–655, 2012.[11] Vasco Brattka, Guido Gherardi, and Arno Pauly. Weihrauch complexity in computable analysis. InVasco Brattka and Peter Hertling, editors,
Handbook of Computability and Complexity in Analysis .Springer, 2021. (to appear).[12] Vasco Brattka, Matthew Hendtlass, and Alexander P. Kreuzer. On the uniform computational contentof computability theory.
Theory of Computing Systems , 61(4):1376–1426, 2017.[13] Vasco Brattka, Matthew Hendtlass, and Alexander P. Kreuzer. On the uniform computational contentof the Baire category theorem.
Notre Dame Journal of Formal Logic , 59(4):605–636, 2018.[14] Vasco Brattka and Arno Pauly. On the algebraic structure of Weihrauch degrees.
Logical Methods inComputer Science , 14(4:4):1–36, 2018.[15] Damir D. Dzhafarov. Joins in the strong Weihrauch degrees.
Mathematical Research Letters , 26(3):749–767, 2019.[16] Kojiro Higuchi and Takayuki Kihara. Inside the Muchnik degrees I: Discontinuity, learnability andconstructivism.
Annals of Pure and Applied Logic , 165(5):1058–1114, 2014.[17] Kojiro Higuchi and Takayuki Kihara. Inside the Muchnik degrees II: The degree structures inducedby the arithmetical hierarchy of countably continuous functions.
Annals of Pure and Applied Logic ,165(6):1201–1241, 2014.[18] Rupert H¨olzl, Dilip Raghavan, Frank Stephan, and Jing Zhang. Weakly represented families in reversemathematics. In Adam Day, Michael Fellows, Noam Greenberg, Bakhadyr Khoussainov, AlexanderMelnikov, and Frances Rosamond, editors,
Computability and Complexity: Essays Dedicated to RodneyG. Downey on the Occasion of His 60th Birthday , volume 10010 of
Lecture Notes in Computer Science ,pages 160–187. Springer, Cham, 2017.[19] Mathieu Hoyrup. The fixed-point property for represented spaces. hal-03117745, 2021.[20] Carl G. Jockusch, Jr. Degrees of functions with no fixed points. In
Logic, methodology and philosophyof science, VIII (Moscow, 1987) , volume 126 of
Stud. Logic Found. Math. , pages 191–201, Amsterdam,1989. North-Holland.[21] Takayuki Kihara and Arno Pauly. Dividing by zero - how bad is it, really? arXiv 1606.04126, May 2016.[22] Joseph S. Miller. Degrees of unsolvability of continuous functions.
The Journal of Symbolic Logic ,69(2):555–584, 2004.[23] Webb Miller and D. A. Martin. The degrees of hyperimmune sets.
Zeitschrift f¨ur Mathematische Logikund Grundlagen der Mathematik , 14:159–166, 1968.[24] Uwe Mylatz.
Vergleich unstetiger Funktionen in der Analysis . PhD thesis, Fachbereich Informatik,FernUniversit¨at Hagen, 1992. Diplomarbeit.
TASHING AND PARALLELIZATION PENTAGONS 29 [25] Uwe Mylatz.
Vergleich unstetiger Funktionen: “Principle of Omniscience” und Vollst¨andigkeit in der C –Hierarchie . PhD thesis, Faculty for Mathematics and Computer Science, University Hagen, Hagen,Germany, 2006. Ph.D. thesis.[26] Hugo Nobrega and Arno Pauly. Game characterizations and lower cones in the Weihrauch degrees. Logical Methods in Computer Science , 15(3):Paper No. 11, 29, 2019.[27] Arno Pauly.
Computable Metamathematics and its Application to Game Theory . PhD thesis, Universityof Cambridge, Computer Laboratory, Clare College, Cambridge, 2011. Ph.D. thesis.[28] Stephen G. Simpson. Degrees of unsolvability: A survey of results. In Jon Barwise, editor,
Handbook ofMathematical Logic , volume 90 of
Studies in Logic and the Foundations of Mathematics , pages 631–652.North-Holland, Amsterdam, 1977.[29] Robert I. Soare.
Turing Computability . Theory and Applications of Computability. Springer, Berlin,Heidelberg, 2016.[30] Thorsten von Stein.
Vergleich nicht konstruktiv l¨osbarer Probleme in der Analysis . PhD thesis, Fach-bereich Informatik, FernUniversit¨at Hagen, 1989. Diplomarbeit.[31] Klaus Weihrauch. The TTE-interpretation of three hierarchies of omniscience principles. InformatikBerichte 130, FernUniversit¨at Hagen, Hagen, September 1992.[32] Klaus Weihrauch.
Computable Analysis . Springer, Berlin, 2000.
Acknowledgments
This work has been supported by the