aa r X i v : . [ m a t h . L O ] A ug WEIHRAUCH GOES BROUWERIAN
VASCO BRATTKA AND GUIDO GHERARDI
Abstract.
We prove that the Weihrauch lattice can be transformed into aBrouwer algebra by the consecutive application of two closure operators inthe appropriate order: first completion and then parallelization. The closureoperator of completion is a new closure operator that we introduce. It trans-forms any problem into a total problem on the completion of the respectivetypes, where we allow any value outside of the original domain of the problem.This closure operator is of interest by itself, as it generates a total versionof Weihrauch reducibility that is defined like the usual version of Weihrauchreducibility, but in terms of total realizers. From a logical perspective com-pletion can be seen as a way to make problems independent of their premises.Alongside with the completion operator and total Weihrauch reducibility weneed to study precomplete representations that are required to describe theseconcepts. In order to show that the parallelized total Weihrauch lattice formsa Brouwer algebra, we introduce a new multiplicative version of an implication.While the parallelized total Weihrauch lattice forms a Brouwer algebra withthis implication, the total Weihrauch lattice fails to be a model of intuitionisticlinear logic in two different ways. In order to pinpoint the algebraic reasonsfor this failure, we introduce the concept of a Weihrauch algebra that allowsus to formulate the failure in precise and neat terms. Finally, we show thatthe Medvedev Brouwer algebra can be embedded into our Brouwer algebra,which also implies that the theory of our Brouwer algebra is Jankov logic.
Keywords:
Weihrauch complexity, computable analysis, Brouwer algebra,intuitionistic and linear logic.
MSC classifications:
Contents
1. Introduction 22. Closure Operators and Weihrauch Algebras 53. Precomplete Representations 84. Total Weihrauch Reducibility 115. Completion 156. Algebraic Operations 187. Co-Residual Operations 258. The Brouwer Algebra of Parallelizable Total Degrees 319. Conclusion 34Acknowledgments 34References 35
Date : August 7, 2020. Introduction
Over the previous ten years Weihrauch complexity has been developed as acomputability theoretic approach to classify the uniform computational content oftheorems. A survey article that summarizes some of the current research directionsin Weihrauch complexity can be found in [6]. The advantage of this approach isthat it provides a direct computability theoretic way to classify problems, whileheuristic observation shows that the approach can be seen as a uniform version ofreverse mathematics in the sense of Friedman and Simpson [21].Weihrauch complexity is based on Weihrauch reducibility ≤ W that induces alattice structure. Beyond the lattice operations the Weihrauch lattice is equippedwith a number of additional algebraic operations. Early on it was noticed thatthe semantics of these operations has the flavor of linear logic. Table 1 providesa dictionary that shows how the usual symbols for operations on problems in theWeihrauch lattice are translated into operations of linear logic.logical operation in linear logic algebraic operation on problems ⊗ multiplicative conjunction × product& additive conjunction ⊔ coproduct ⊕ additive disjunction ⊓ infimum& multiplicative disjunction + sum! bang b parallelization Table 1.
Linear logic versus the algebra of problemsHowever, so far no satisfactory interpretation of the Weihrauch lattice as a modelof (intuitionistic) linear logic has been found. This is partially due to the lack of aninternal implication operation that corresponds to the linear implication ⊸ . Suchan implication would have to fulfill( g ⊸ f ) ≤ W h ⇐⇒ f ≤ W g × h and it can be proved that such an implication does not exist, given ≤ W and × [8,Proposition 37]. However, Weihrauch reducibility f ≤ W g can be seen at least asan external implication operation f ⇐ = g .The Weihrauch lattice has also additional algebraic operations such as the com-positional product ⋆ , which can be seen as a non-commutative version of conjunc-tion. Here f ⋆ g captures what can be computed by first using the problem g andthen the problem f , possibly with some intermediate computation. There is animplication operation g → f in the Weihrauch lattice that is a right co-residualoperation of ⋆ [8], i.e., we have( g → f ) ≤ W h ⇐⇒ f ≤ W g ⋆ h. However, this setting does not provide a model for classical linear logic, since theoperation ⋆ is not commutative. While the connections to linear logic might not be as tight as one wishes, thereis still hope that there is a close connection to intuitionistic logic. In linear logicintuitionistic implication is represented by ! A ⊸ B . Hence, it is to be expectedthat the parallelized Weihrauch reducibility f ≤ W b g gives us an external form ofintuitionistic implication. This could theoretically be substantiated by showing that A comprehensive up-to-date bibliography is maintained at the following web page: http://cca-net.de/publications/weibib.php Girard also proposed a less known non-commutative version of linear logic, but also this logicdoes not seem to fit to our model [28].
EIHRAUCH GOES BROUWERIAN 3 the resulting structure is a Brouwer algebra, since Brouwer algebras are modelsfor intermediate propositional logics in between classical and intuitionistic logic.However, also this hope did not materialize as Higuchi and Pauly proved that theparallelized Weihrauch lattice is not a Brouwer algebra [13].In this article we prove that one does obtain a Brouwer algebra if one combinestwo closure operators in the Weihrauch lattice in the appropriate order: first com-pletion f f and then parallelization f b f . While parallelization is a well un-derstood operation [3] that corresponds somewhat to the usage of countable choicein constructive mathematics, completion is a new operation that we introduce inthis article. Formally, the completion f : X ⇒ Y of a problem f : ⊆ X ⇒ Y isdefined by f ( x ) := (cid:26) f ( x ) if x ∈ dom( f ) Y otherwise , i.e., by a totalization of f on the completions X, Y of the corresponding types. Logically, completion can be seen as a way to make problems independent oftheir premises. In general, a logical statement of the form( ∀ x ∈ X )( x ∈ D = ⇒ ( ∃ y ∈ Y ) P ( x, y ))is translated into a problem f : ⊆ X ⇒ Y in the Weihrauch lattice by settingdom( f ) = D and f ( x ) := { y ∈ Y : P ( x, y ) } for all x ∈ dom( f ). Now the transitionto the completion f corresponds to the statement( ∀ x ∈ X )( ∃ y ∈ Y )( x ∈ D = ⇒ P ( x, y )) , where the existence is required independent of the premise x ∈ D . The completionof the data types is relevant here, as it guarantees the existence of total represen-tations of the underlying types.The completion operation f f is of interest by itself as it is a closure operatorthat yields a total version of Weihrauch reducibility ≤ tW by f ≤ tW g ⇐⇒ f ≤ W g .Total Weihrauch reducibility ≤ tW can also be defined directly almost as the usualreducibility ≤ W , but in terms of total realizers instead of partial realizers. In thiscase the completion of the types features again, since one needs to consider so-calledprecomplete representations for the underlying types.Among other things we prove that total Weihrauch reducibility induces a latticestructure with operations induced by the original operations of the Weihrauch lat-tice. The lattice structure of the total Weihrauch lattice is somewhat different fromthe original Weihrauch lattice, but it does not change all too dramatically as manyproblems are actually complete, i.e., Weihrauch equivalent to their own completion.We list some examples of complete and incomplete problems: • Complete problems:
LPO , LLPO , lim , J , WKL , SORT , IVT , PA , MLR , DNC n . • Incomplete problems: C N , C N N , WWKL .The reader who does not know these problems will find relevant definitions ofsome of them later. The topic of completion of choice problems is subject of anentirely separate article [4].When we move to the total Weihrauch lattice W tW of total Weihrauch reducibil-ity ≤ tW , then we can introduce a new implication f ։ g that can almost be seenas a multiplicative co-residual of × . However, also in this case we fail to obtain amodel for intuitionistic linear logic.In order to make the spectacular twofold failure of obtaining a model of intu-itionistic linear logic more understandable, we introduce the concept of a Weihrauch We were inspired to continue the study of completions by recent work of Dzhafarov who usedthem to show that strong Weihrauch reducibility induces a lattice structure [10].
V. BRATTKA AND G. GHERARDI
Weihrauch algebrascommutative deductiveBrouweralgebrasTroelstraalgebras × , ։ ⋆, → models ofinuitionisticlinear logicsmodels ofinuitionisticlogics Figure 1.
Different types of algebras as models of logicalgebra in the following section 2. These are lattice-ordered monoids with some ad-ditional implication operation. The total Weihrauch lattice W tW is a commutativeWeihrauch algebra with respect to × , ։ and a deductive Weihrauch algebra withrespect to ⋆, → . However, none of these Weihrauch algebras is commutative anddeductive simultaneously, which is what is required in order to obtain, in our terms,a Troelstra algebra , i.e., a model of some form of intuitionistic linear logic. Seethe diagram in Figure 1 for an illustration of the situation.When we apply parallelization after completion, then we obtain the parallelizedtotal Weihrauch lattice W ptW which then leads to a Brouwer algebra, i.e., a Troel-stra algebra where the monoid structure is merged with the lattice structure (inour terms × and ⊔ are merged). In section 8 we prove that one can embed theMedvedev Brouwer algebra [24] into our Brouwer algebra. Like in the case of theMedvedev Brouwer algebra we obtain Jankov logic as the theory of our algebra.In the following section 2 we provide some very basic lattice theoretic resultsregarding closure operators that are helpful for our study, and we define Weihrauchand Troelstra algebras alongside with Brouwer algebras. In section 3 we studyprecomplete representations and the data type of completion that is needed to in-troduce the closure operator of completion and total Weihrauch reducibility. Insection 4 we introduce total Weihrauch reducibility and we prove some basic prop-erties of it. In section 5 we introduce and study the closure operator of completion.Section 6 provides results that show how the algebraic operations of the Weihrauchlattice interact with completion. In particular, we prove that total Weihrauch re-ducibility actually yields a lattice structure. In section 7 we review the operations ⋆ and → and study their interaction with completion and we also introduce the newimplication operation ։ . Finally, in section 8 we prove that the parallelized totalWeihrauch lattice W ptW is a Brouwer algebra with the implication derived from ։ .We also discuss the embedding of the Medvedev lattice. We close this article witha brief survey on the classification of concrete problems in the parallelized totalWeihrauch lattice. This is the dual structure of what Troelstra called an intuitionistic linear algebra [26].
EIHRAUCH GOES BROUWERIAN 5 Closure Operators and Weihrauch Algebras
In this section we prepare some basic order theoretic concepts that we are goingto use frequently. We recall that a preorder ≤ on a set X is a binary relation on X that is reflexive and transitive. We also speak of a preordered space ( X, ≤ ) inthis context. An equivalence relation ≡ on a set X is a binary relation on X thatis reflexive, symmetric and transitive. In the following we will have to deal withseveral closure operators. Definition 2.1 (Closure operator) . Let ( X, ≤ ) be a preordered space together witha map c : X → X . Then c is called a closure operator , if(1) x ≤ c ( x ),(2) cc ( x ) ≤ c ( x ) and(3) x ≤ y = ⇒ c ( x ) ≤ c ( y )hold for all x, y ∈ X . We say that x ∈ X is closed if c ( x ) ≤ x .We call a map c : X → X monotone , if x ≤ y = ⇒ c ( x ) ≤ c ( y ) holds and antitone , if x ≤ y = ⇒ c ( y ) ≤ c ( x ) holds. We use the same terminology for binarymaps (cid:3) : X × X → X with respect to individual arguments. We use the usualconcepts of a suprema (also called a least upper bound) and an infima (also calleda greatest lower bound) for preordered sets in the usual way, and we note that on apreordered space they are only uniquely determined up to equivalence in the caseof existence. If one has a preordered space ( X, ≤ ) and one identifies all equivalentelements with each other, then one obtains a quotient structure ( X/ ≡ , ≤ ), which isa partially ordered space , i.e., the resulting order is a preorder that is additionallyanti-symmetric. A lattice ( X, ≤ , ∧ , ∨ ) is a partially ordered set together with asupremum operation ∨ and an infimum operation ∧ . If ≤ c is a preorder on X and c : X → X a map, then we say that c generates ≤ c on ( X, ≤ ) if x ≤ c y ⇐⇒ x ≤ c ( y ) holds for all x, y ∈ X . The following result is straightforward to prove. Itshows how closure operators act on lattices and preordered spaces. Proposition 2.2 (Closure operators) . Let ( X, ≤ ) be a preordered space with twoclosure operators c, c ′ : X → X and binary operations (cid:3) , ∨ , ∧ : X × X → X . Then (1) x ≤ c y : ⇐⇒ x ≤ c ( y ) ⇐⇒ c ( x ) ≤ c ( y ) defines a preorder that satisfies x ≤ y = ⇒ x ≤ c y for all x, y ∈ X , (2) x ≡ c y : ⇐⇒ ( x ≤ c y and y ≤ c x ) defines an equivalence relation, (3) (cid:3) c : X × X → X, ( x, y ) c ( x ) (cid:3) c ( y ) shares corresponding monotonicityproperties as (cid:3) , more precisely: (a) if (cid:3) is monotone (antitone) in one argument with respect to ≤ , thenso is (cid:3) c in the same argument with respect to ≤ c , (b) if ∧ is an infimum with respect to ≤ , then so is ∧ c with respect to ≤ c , (c) if ∨ is a supremum with respect to ≤ , then so is ∨ c with respect to ≤ c . (4) If ( X, ≤ , ∧ , ∨ ) is a lattice, then so is ( X/ ≡ c , ≤ c , ∧ c , ∨ c ) . (5) c ′ ◦ c : X → X is monotone with respect to ≤ and ≤ c .Proof. (1) Reflexivity of ≤ c follows from x ≤ c ( x ), transitivity from monotonicityof c together with cc ( x ) ≤ c ( x ). It is also clear that x ≤ c ( y ) ⇐⇒ c ( x ) ≤ c ( y )holds. Finally, x ≤ y = ⇒ x ≤ c y holds as c is monotone.(2) Is obvious.(3) (a) Suppose (cid:3) is antitone in the first argument and x , x , y ∈ X with x ≤ c x .Then c ( x ) ≤ c ( x ) and hence c ( x ) (cid:3) c ( y ) ≤ c ( x ) (cid:3) c ( y ), since (cid:3) is antitone in thefirst argument. Hence x (cid:3) c y ≤ x (cid:3) c y ≤ c ( x (cid:3) c y ), which means x (cid:3) c y ≤ c x (cid:3) c y ,i.e., (cid:3) c is antitone in the first argument with respect to ≤ c . The other cases aretreated analogously.(b), (c) If ∧ is an infimum with respect to ≤ and x, y ∈ X , then x ∧ y ≤ x and V. BRATTKA AND G. GHERARDI x ∧ y ≤ y and hence x ∧ y ≤ c x and x ∧ y ≤ c y . Hence x ∧ y is a lower boundof x and y with respect ≤ c . Let now z ∈ X be such that z ≤ c x and z ≤ c y .Then z ≤ c ( x ) and z ≤ c ( y ), which implies z ≤ c ( x ) ∧ c ( y ) and hence z ≤ c x ∧ c y .This means that x ∧ c y is above every lower bound of x and y with respect to ≤ c and hence it is an infimum with respect to ≤ c . The statement for suprema can beproved analogously.(4) This follows from (1)–(3).(5) If x ≤ y , then c ′ c ( x ) ≤ c ′ c ( y ) follows. If x ≤ c y , then cc ′ c ( x ) ≤ cc ′ c ( y ) followsand hence c ′ c ( x ) ≤ c c ′ c ( y ). (cid:3) We also need to deal with situations where a closure operator respects certainunderlying algebraic operations or other closure operators. Hence, we use the fol-lowing terminology.
Definition 2.3 (Preservation) . Let ( X, ≤ ) be a preordered space with closureoperators c, c ′ : X → X and a binary operation (cid:3) : X × X → X .(1) We say that c is preserved by (cid:3) if c ( x (cid:3) y ) ≤ c ( x ) (cid:3) c ( y ) for all x, y ∈ X .(2) We say that c is co-preserved by (cid:3) if c ( x ) (cid:3) c ( y ) ≤ c ( x (cid:3) y ) for all x, y ∈ X .(3) We say that c is preserved by c ′ if c ◦ c ′ ( x ) ≤ c ′ ◦ c ( x ) for all x ∈ X .Whenever a closure operator is preserved by a certain operation, then we candraw certain conclusions. The proof of the following result is straightforward. Proposition 2.4 (Preservation) . Let ( X, ≤ ) be a preordered space with closureoperators c, c ′ : X → X and a binary monotone operation (cid:3) : X × X → X . (1) If c is preserved by (cid:3) , then for all x, y ∈ Xc ( x (cid:3) y ) ≤ c ( x ) (cid:3) c ( y ) ≡ c ( c ( x ) (cid:3) c ( y )) . In particular, x (cid:3) y is closed if x and y are. (2) If c is co-preserved by (cid:3) , then for all x, y ∈ Xc ( x ) (cid:3) c ( y ) ≤ c ( x (cid:3) y ) ≤ c ( c ( x ) (cid:3) c ( y )) . (3) If c is preserved by c ′ , then for all x ∈ Xcc ′ ( x ) ≤ c ′ c ( x ) ≡ cc ′ c ( x ) . In particular, c ′ c is a closure operator with respect to ≤ c and ≤ , and c ′ ( x ) is closed with respect to c if x is so.Proof. The equivalences in (1) and (3) are consequences of the respective first re-lations and the fact that c is a closure operator. For the second relation in (2)we just use that c is a closure operator. It is clear that c ′ ( x ) is closed if x isclosed. That c ′ c is monotone with respect to ≤ c follows from Proposition 2.2.Clearly, also x ≤ c c ′ c ( x ) holds. Finally, c ′ cc ′ c ( x ) ≤ c ′ c ′ c ( x ) ≤ c ′ c ( x ) and hence c ′ cc ′ c ( x ) ≤ c c ′ c ( x ). (cid:3) If the binary operation is a supremum or an infimum operation, then it is alwayspreserved in certain ways.
Proposition 2.5 (Preservation of suprema and infima) . Let ( X, ≤ ) be a preorderedspace with a closure operator c : X → X , and binary operations ∨ , ∧ : X × X → X . (1) If ∨ is a supremum operation, then it co-preserves c . (2) If ∧ is an infimum operation, then it preserves c .In particular, x ∨ y ≡ c x ∨ c y , if ∨ is a supremum. EIHRAUCH GOES BROUWERIAN 7
Proof.
Since x ∨ y is a supremum, we obtain c ( x ) ≤ c ( x ∨ y ) and c ( y ) ≤ c ( x ∨ y ) dueto monotonicity of c . Hence c ( x ) ∨ c ( y ) ≤ c ( x ∨ y ), which means that ∨ co-preserves c . The statement for ∧ can be proved analogously. That ∨ co-preserves c means x ∨ c y ≤ c x ∨ y . We also have x ∨ y ≤ c ( x ) ∨ c ( y ), i.e., x ∨ y ≤ c x ∨ c y . (cid:3) We note that this result implies that the we can replace ∨ c by ∨ in Proposi-tion 2.2.In the following we will have to deal with lattices that have some additional alge-braic operations and we propose the following concept that encapsulates a structurethat we will see in different variations. Definition 2.6 (Weihrauch algebra) . We call ( X, ≤ , ∧ , ∨ , · , → , , ⊥ , ⊤ ) a Weihrauchalgebra if the following hold:(1) ( X, ≤ , ∧ , ∨ ) is a bounded lattice with bottom ⊥ and top ⊤ . (Lattice)(2) ( X, · ,
1) is a monoid with neutral element 1. (Monoid)(3) · : X × X → X is monotone in both components. (Monotonicity)(4) → : X × X → X is monotone in the second component, antitone in the firstcomponent. (Monotonicity)(5) x ≤ y · z = ⇒ ( y → x ) ≤ z holds for all x, y, z ∈ X . (Implication)A Weihrauch algebra is called commutative , if · is commutative, and it is called deductive , if “ ⇐⇒ ” holds instead of “= ⇒ ” in (5).One could add additional distributivity requirements to this definition. Struc-tures that satisfy (1), (2) and (3) have also been called lattice-ordered monoids .Using these building blocks, we can define structures that have been already con-sidered for other purposes. Definition 2.7 (Algebras) . Let X = ( X, ≤ , ∧ , ∨ , · , → , , ⊥ , ⊤ ) be a Weihrauchalgebra. We call X a Troelstra algebra if it is commutative and deductive. If,additionally, · = ∨ and 1 = ⊥ , then X is called a Brouwer algebra .If we denote a Brouwer algebra as a tuple, then we omit the double occurrenceof · = ∨ and 1 = ⊥ , respectively. What we call a Troelstra algebra is exactlywhat Troelstra [25] called an intuitionistic linear algebra , except that the order isreversed. A bottom element in our sense is not required in Troelstra’s axioms, butit always exists by [25, Lemma 8.3]. The relevance of Troelstra algebras is that theyform sound and complete models of intuitionistic linear logic [25, Theorem 8.15].In an analogous sense Brouwer algebras (that are just defined dually to Heytingalgebras ) are known as models of intermediate logics, i.e., predicate logics betweenclassical logic and intuitionistic logic [12].A Brouwer algebra embedding is an injective map from one Brouwer algebrato another one that is monotone in both directions, preserves suprema, infima,implications and the bottom and top elements.In the case of a deductive Weihrauch algebra the condition (5) can be seen as alaw of (co-)residuation. We need to add the prefix “co-” as residuation is normallyconsidered in the opposite order [12].
Definition 2.8 (Co-residuation) . Let ( X, ≤ ) be a preordered set with a binaryoperation · : X × X → X . Then we call · right co-residuated , if there is a binaryoperation → : X × X → X such that x ≤ y · z ⇐⇒ ( y → x ) ≤ z The term Brouwer algebra is used in different versions in different references, we mean by a
Brouwer algebra just the dual concept of a Heyting algebra, as usual in computability theory [24].
V. BRATTKA AND G. GHERARDI holds for all x, y, z ∈ X . Analogously, we call · left co-residuated , if an analogouscondition holds with z · y in place of y · z . Finally, · is called co-residuated if andonly if it is left and right co-residuated.Hence, a deductive Weihrauch algebra is a right co-residuated lattice-orderedmonoid and a Troelstra algebra is a co-residuated lattice-ordered monoid.3. Precomplete Representations
We will need some pairing functions in the following. Firstly, we define a pairingfunction π : N N × N N → N N , ( p, q )
7→ h p, q i by h p, q i (2 n ) := p ( n ) and h p, q i (2 n +1) := q ( n ) for p, q ∈ N N and n ∈ N . We define a pairing function of type h , i : ( N N ) N → N N by h p , p , p , ... ih n, k i := p n ( k ) for all p i ∈ N N and n, k ∈ N , where h n, k i is thestandard Cantor pairing defined by h n, k i := ( n + k + 1)( n + k ) + k . Finally, wenote that by np we denote the concatenation of a number n ∈ N with a sequence p ∈ N N . By π i : N N → N N , h p , p , p , ... i 7→ p i we denote the projection on the i –th component of a tuple and we also use the binary tupling functions π h p, q i = p and π h p, q i = q . It will always be clear from the context whether we apply thesefunctions in a countable or binary setting.We recall that a represented space ( X, δ ) is a set X together with a surjective(partial) map δ : ⊆ N N → X , called the representation of X . For the purposes of ourtopic so-called precomplete representations are important. They were introducedby Kreitz and Weihrauch [15] following the concept of a precomplete numbering,that was originally introduced by Erˇsov [11]. Definition 3.1 (Precompleteness) . A representation δ : ⊆ N N → X is called pre-complete , if for any computable function F : ⊆ N N → N N there exists a total com-putable function G : N N → N N such that δF ( p ) = δG ( p ) for all p ∈ dom( F ).In this situation we also say that the represented space ( X, δ ) is precomplete .We point out that we demand that the equation in the definition holds for all p ∈ dom( F ), not only for p ∈ dom( δF ). The precomplete representations areexactly those that satisfy a certain version of the recursion theorem [15]. For usthey are relevant since we are going to work with total functions. It is clear thatnot all represented spaces are precomplete. By id : N N → N N we denote the identity of Baire space. For other sets X we usually add an index X and write the identity as id X : X → X . By b n := nnn... ∈ N N we denote the constant sequence with value n ∈ N . Example 3.2.
There are partial computable functions F : ⊆ N N → N N withouttotal computable extension, such as the function defined by F ( p ) = b n : ⇐⇒ p starts with exactly n digits 0, where dom( F ) = { n p : n ∈ N , p ∈ N N , p (0) = 0 } .This shows that the represented space ( N N , id) is not precomplete.However, it is not too hard to see that in every equivalence class of representa-tions there is a precomplete representation. We recall that for two representations δ , δ of the same set X we say that δ is computably reducible to δ , in symbols δ ≤ δ , if and only if there is a computable F : ⊆ N N → N N such that δ = δ F . Wedenote the corresponding equivalence by ≡ . For p ∈ N N we denote by p − ∈ N N ∪ N ∗ the sequence or word that is formed as concatenation of p (0) − p (1) − p (2) − − ε is the empty word. Definition 3.3 (Precompletion) . Let (
X, δ X ) be a represented space. Then the precompletion δ ℘X of δ X is defined by δ ℘X ( p ) := δ X ( p −
1) for all p ∈ N N such that p − ∈ dom( δ X ). This result is due to Matthias Schr¨oder (personal communication 2009), see the constructionin [20, Lemmas 4.2.10, 4.2.11, Section 4.2.5].
EIHRAUCH GOES BROUWERIAN 9
We note that the identity id : N N → N N , considered as a representation of N N , has the precompletion id ℘ : ⊆ N N → N N with id ℘ ( p ) := p − δ ℘X = δ X ◦ id ℘ . Now we can prove the following result. Proposition 3.4 (Precompleteness) . Let ( X, δ X ) be a represented space. The pre-completion δ ℘X of δ X is precomplete and satisfies δ ℘X ≡ δ X .Proof. The computable function F : N N → N N , p p + 1 satisfies δ X ( p ) = δ ℘X F ( p )and hence it witnesses δ X ≤ δ ℘X . The computable function G : ⊆ N N → N N , p p − δ ℘X ( p ) = δ X G ( p ) and hence it witnesses δ ℘X ≤ δ X . Altogether δ ℘X ≡ δ X .We need to prove that δ ℘X is precomplete. Let F : ⊆ N N → N N be computable andlet M be a Turing machine that computes F . We modify this machine such thatit never halts and after every n steps for some suitable fixed number n ∈ N themachine writes a 0 on the output tape, irrespective of the input. Otherwise themachine is left unchanged. Then the modified machine computes a total function G : N N → N N with F ( p ) − G ( p ) − δ ℘X F ( p ) = δ ℘X G ( p ) for all p ∈ dom( F ). (cid:3) We will also need the fact that other classes of functions can be extended to totalones under precomplete representations. Hence we introduce the following concept.
Definition 3.5 (Respect for precompleteness) . We say that a set P of functions F : ⊆ N N → N N respects precompleteness , if for every precomplete representation δ and any function F ∈ P there exists a total function G ∈ P such that δF ( p ) = δG ( p )for all p ∈ dom( F ).It is clear that the set of computable functions respects precompleteness by def-inition. However, also other classes of functions do. Some of them, simply becausethey can already be extended to total functions in the same class irrespectively ofthe representation. We provide a number of examples. We call a function non-uniformly computable if it maps all computable inputs in its domain to computableoutputs. By J : N N → N N , p p ′ we denote the Turing jump operator and by U : ⊆ N N → N N a universal computable function such that for every continuous func-tion F : ⊆ N N → N N there is a q ∈ N N with F ( p ) = U h q, p i for all p ∈ dom( F ) [27,Theorem 2.3.8]. Proposition 3.6 (Respect for precompleteness) . The following classes of partialfunctions F : ⊆ N N → N N respect precompleteness: computable, continuous, limitcomputable, Borel measurable and non-uniformly computable.Proof. The statement for computable functions is a consequence of the definitionof precompleteness. Let δ be a precomplete representation and let F : ⊆ N N → N N be a continuous function. Then there is a q ∈ N N such that F ( p ) = U h q, p i forall p ∈ dom( F ). By precompleteness of δ , there is a total function V : N N → N N with δV ( p ) = δU ( p ) for all p ∈ dom( U ). Hence G ( p ) := V h q, p i defines a totalcontinuous function with δG ( p ) = δF ( p ) for all p ∈ dom( F ). This shows that theclass of continuous functions respects precompleteness. For every limit computablefunction F : ⊆ N N → N N there exists a computable function H : ⊆ N N → N N suchthat F = H ◦ J [1, Theorem 14]. By precompleteness of δ there exists a totalcomputable function I : N N → N N such that δI ( p ) = δH ( p ) for all p ∈ dom( H ).Hence G := I ◦ J is a total function that is limit computable and satisfies δF ( p ) = δG ( p ) for all p ∈ dom( F ). Hence the class of limit computable functions respectsprecompleteness. Every partial Borel measurable function F : ⊆ N N → N N canbe extended to a total Borel measurable function by a theorem of Kuratowski(see [14, Theorem 2.2]). The class of non-uniformly computable functions respectsprecompleteness since every non-uniformly computable function F : ⊆ N N → N N can simply be extended to a total non-uniformly computable function G : N N → N N bydefining G ( p ) = b ... for all p ∈ N N \ dom( F ). (cid:3) The proof for limit computable functions (which are exactly the effectively Σ –computable functions) can easily be extended to any finite level of the Borel hier-archy. We prove in [4, Corollaries 8.4, 9.3] that functions that are computable withfinitely many mind changes and low functions do not respect precompleteness.We also need to study how certain algebraic constructions on represented spacesbehave with respect to precompleteness. For any sets X and Y we denote by X × Y and X N the usual products , by X ⊔ Y := ( { } × X ) ∪ ( { } × Y ) the disjoint union of X and Y , by X ∗ := S ∞ i =0 ( { i } × X i ) the set of words over X , where X i denotesthe i –fold product of X with itself, and X := { } . By X := X ∪ {⊥} we denotethe completion X , where we assume that ⊥ 6∈ X . Definition 3.7 (Constructions on representations) . Let (
X, δ X ) and ( Y, δ Y ) berepresented spaces. We define(1) δ X × Y : ⊆ N N → X × Y , δ X × Y h p, q i := ( δ X ( p ) , δ Y ( q ))(2) δ X ⊔ Y : ⊆ N N → X ⊔ Y , δ X ⊔ Y (0 p ) := (0 , δ X ( p )) and δ X ⊔ Y (1 p ) := (1 , δ Y ( p ))(3) δ X ∗ : ⊆ N N → X ∗ , δ X ∗ ( n h p , p , ..., p n i ) := ( n, ( δ X ( p ) , δ X ( p ) , ..., δ X ( p n )))(4) δ X N : ⊆ N N → X N , δ X N h p , p , p , ... i := ( δ X ( p n )) n ∈ N (5) δ X : N N → X , δ X ( p ) := δ ℘X ( p ) if p ∈ dom( δ ℘X ) and δ X ( p ) := ⊥ otherwise.We warn the reader that all these constructions on represented spaces preserveequivalence of representations, except the last one for the completion. In otherwords, the equivalence class of δ X does not only depend on the equivalence class of δ X , but on the concrete representative δ X itself. For our applications this does notcause any problems (see the remark after Corollary 5.3; the problem could also becircumvented by moving to multi-valued representations [20, Lemma 4.2.11]).The next observation is that finite and countable products preserve precomplete-ness. Proposition 3.8 (Products and precompleteness) . Let ( X, δ X ) and ( Y, δ Y ) be pre-complete represented spaces. Then so are ( X × Y, δ X × Y ) and ( X N , δ X N ) .Proof. If F : ⊆ N N → N N is computable, then so are the projections F i = π i ◦ F for i ∈ { , } with π h p, q i = p and π h p, q i = q . Hence, by precompletenessthere are total computable functions G i : N N → N N with δ X F ( p ) = δ X G ( p )for all p ∈ dom( F ) and with an analogous statement for δ Y , F and G . Let G ( p ) = h G ( p ) , G ( p ) i for all p ∈ N N . Then G : N N → N N is computable andtotal, and we obtain δ X × Y F ( p ) = δ X × Y G ( p ) for all p ∈ dom( F ). Hence δ X × Y is precomplete. If F : ⊆ N N → N N is computable, then so is the function H : ⊆ N N → N N , h i, p i 7→ π i ◦ F ( p ), where π i : N N → N N , h p , p , p , ... i 7→ p i denotes the i –th projection. Due to precompleteness of δ X there is a total computable function I : N N → N N with δ X H ( p ) = δ X I ( p ) for all p ∈ dom( H ). Then also the function G : N N → N N , p
7→ h I h , p i , I h , p i , I h , p i , ... i is computable and total and satisfies δ X N F ( p ) = δ X N G ( p ) for all p ∈ dom( F ). This shows that δ X N is precomplete. (cid:3) The coproduct constructions for X ⊔ Y and X ∗ are less nicely behaved with re-spect to precompleteness. One problem is that also the natural number componentthat selects the argument has to be handled in a precomplete manner. One canmodify the definition of δ X ⊔ Y and δ X ∗ to take this into account. However, eventhen it is not clear why the construction should preserve precompleteness. We justobtain that if δ X and δ Y are the precompletions according to Proposition 3.4, then δ X ⊔ Y and δ X ∗ are precomplete in the modified definition. We formulate this moreformally. We use the total representation δ N of N given by δ N ( p ) := p (0). EIHRAUCH GOES BROUWERIAN 11
Proposition 3.9 (Coproducts and precompleteness) . Let ( X, δ X ) and ( X i , δ X i ) berepresented spaces for i ∈ { , } . (1) We define a representation δ of X ⊔ X by δ h q, p i := ( δ ℘ N ( q ) , δ ℘X i ( p )) for all q, p ∈ N N such that δ ℘ N ( q ) = i ∈ { , } and p ∈ dom( δ ℘X i ) . Then δ isprecomplete and δ ≡ δ X ⊔ X . (2) We define a representation δ of X ∗ by δ h q, h p , ..., p n ii := ( δ ℘ N ( q ) , ( δ ℘X ( p ) , ..., δ ℘X ( p n ))) for all q, p , ..., p n ∈ N N such that δ ℘ N ( q ) = n and p i ∈ dom( δ ℘X ) for i =1 , ..., n . Then δ is precomplete and δ ≡ δ X ∗ .Proof. The proof is similar to the proof of Proposition 3.4. We only considerthe case of X ∗ and leave the case X ⊔ X to the reader. Given a δ X ∗ –name h n, h p , ..., p n ii of x ∈ X ∗ we compute q := b n = nnn... and then h q + 1 , h p +1 , ..., p n + 1 ii is a δ –name of the same point x . Since r r + 1 is computable, weobtain δ X ∗ ≤ δ . Given a δ –name h q, h p , ..., p n ii of a point x ∈ X ∗ , we can searchfor the first non-zero value k ∈ N in q , in which case we know that n = k −
1, andthen we can compute h n, h p − , ..., p n − ii , which is a δ X ∗ –name of the same point x . Since r r − r − ∈ N N , we obtain δ ≤ δ X ∗ . Any machine that computes a function F : ⊆ N N → N N can be modified asin the proof of Proposition 3.4 such that it computes a total function G : N N → N N ,potentially with extra zeros on the output side and such that δF ( p ) = δG ( p ) for all p ∈ dom( F ). (cid:3) We mention that the completion (
X, δ X ) of a represented space is always pre-complete. This follows like in the proof of Proposition 3.4. The only additionalobservation required in the proof is that if δ X F ( p ) = ⊥ , then also δ X G ( p ) = ⊥ .We recall that a computable embedding f : X → Y is a computable function thatis injective and whose partial inverse f − : ⊆ Y → X is computable too. Corollary 3.10 (Completion) . ( X, δ X ) is a precomplete represented space for everyrepresented space ( X, δ X ) and ι : X → X, x x is a computable embedding. Total Weihrauch Reducibility
In this section we are going to introduce a total variant of Weihrauch reducibilitythat behaves very similarly to the usual reducibility from a practical perspective,but that has different algebraic properties.By a problem f : ⊆ X ⇒ Y we mean a partial multi-valued map f : ⊆ X ⇒ Y on represented spaces ( X, δ X ) and ( Y, δ Y ). We recall that composition of problems f : ⊆ X ⇒ Y and g : ⊆ Y ⇒ Z is defined by g ◦ f ( x ) := { z ∈ Z : ( ∃ y ∈ f ( x )) z ∈ g ( y ) } for all x ∈ dom( g ◦ f ) := { x ∈ dom( f ) : f ( x ) ⊆ dom( g ) } . For two problems f : ⊆ X ⇒ Y and g : ⊆ X ⇒ Z with identical source space X we define the juxtaposition ( f, g ) : ⊆ X ⇒ Y × Z by ( f, g )( x ) := f ( x ) × g ( x ) and dom( f, g ) := dom( f ) ∩ dom( g ).If f, g : ⊆ N N ⇒ N N are problems on Baire space, then we also call h f, g i := h i◦ ( f, g )the juxtaposition of f and g and h f × g i defined by h f × g ih p, q i := h f ( p ) , g ( q ) i forall p, q ∈ N N the product of f and g .We say that a function F : ⊆ N N → N N is a realizer of f , if δ Y F ( p ) ∈ f δ X ( p )for all p ∈ dom( f δ X ). We denote this by F ⊢ f . We say that f is computable if ithas a computable realizer. Other notions, such as continuity, Borel measurability and so forth that are well-defined for functions F : ⊆ N N → N N are transferred inan analogous manner to problems f : ⊆ X ⇒ Y .We write F ⊢ t f , if F is a total realizer of f . We now recall the definitionof ordinary and strong Weihrauch reducibility on problems f, g , which is denotedby f ≤ W g and f ≤ sW g , respectively, and we introduce two new concepts of totalWeihrauch reducibility and strong total Weihrauch reducibility , which are denotedby f ≤ tW g and f ≤ stW g , respectively. Definition 4.1 (Weihrauch reducibility) . Let f : ⊆ X ⇒ Y and g : ⊆ U ⇒ V beproblems. We define:(1) f ≤ W g : ⇐⇒ ( ∃ computable H, K : ⊆ N N → N N )( ∀ G ⊢ g ) H h id , GK i ⊢ f .(2) f ≤ sW g : ⇐⇒ ( ∃ computable H, K : ⊆ N N → N N )( ∀ G ⊢ g ) HGK ⊢ f .(3) f ≤ tW g : ⇐⇒ ( ∃ computable H, K : ⊆ N N → N N )( ∀ G ⊢ t g ) H h id , GK i ⊢ t f .(4) f ≤ stW g : ⇐⇒ ( ∃ computable H, K : ⊆ N N → N N )( ∀ G ⊢ t g ) HGK ⊢ t f .For (3) and (4) we assume that we replace each of the given representations of X, Y, U and V by a computably equivalent precomplete representation of the cor-responding set.We call the reducibilities ≤ W and ≤ sW partial in order to distinguish themfrom their total counterparts ≤ tW and ≤ stW . We note that precompleteness is notrequired or relevant in the partial case, but it can be assumed without loss of gener-ality since the concept of partial (strong) Weihrauch reducibility is invariant undercomputably equivalent representations [3, Lemma 2.11]. In the total cases (3) and(4), however, precompleteness is essential, since otherwise these definitions wouldnot be invariant under computably equivalent representations. By Proposition 3.4we can always choose precomplete representations that are computably equivalentto the given representations of the spaces X, Y, U and V . But we still need to showthat the definition of ≤ tW and ≤ stW does not depend on this choice.We will prove a slightly more general result that highlights the places whereprecompleteness is actually needed. For this purpose we introduce the followingterminology: we say that f ≤ tW g holds with respect to ( δ X , δ Y , δ U , δ V ), if Defini-tion 4.1 (3) holds as it stands but exactly for the given representations of X, Y, U and V , respectively, and these representations are not required to be precomplete.Hence the statement defined here is weaker than f ≤ tW g in the sense defined above.We use a corresponding terminology for ≤ stW . Now we obtain the following result. Lemma 4.2 (Invariance under representations) . Let f : ⊆ X ⇒ Y , g : ⊆ U ⇒ V beproblems on represented spaces ( X, δ X ) , ( Y, δ Y ) , ( U, δ U ) and ( V, δ V ) . Let δ ′ X , δ ′ Y , δ ′ U and δ ′ V be further representations of the given sets, respectively, such that (1) δ ′ X ≤ δ X , δ Y ≤ δ ′ Y , δ U ≤ δ ′ U and δ ′ V ≤ δ V , (2) δ V , δ ′ Y and δ ′ U are precomplete.If f ≤ tW g holds with respect to ( δ X , δ Y , δ U , δ V ) , then it also holds with respect to ( δ ′ X , δ ′ Y , δ ′ U , δ ′ V ) . An analogous statement holds for f ≤ stW g .Proof. We follow the construction as outlined in the proof of [3, Lemma 2.11]. Since δ ′ U and δ ′ V are precomplete according to (2), we can additionally assume that thecomputable functions S, T : N N → N N in that proof are total. In that proof it isshown that whenever G ′ ⊢ g holds with respect to ( δ ′ U , δ ′ V ), then G := T G ′ S ⊢ g holds with respect to ( δ U , δ V ). Due to totality of T, S , the same holds true if wereplace ⊢ by ⊢ t in both occurrences. If we assume that H h id , GK i ⊢ t f holdswith respect to ( δ X , δ Y ), then we obtain as in the proof mentioned above that H ′ h id , G ′ K ′ i ⊢ f holds with respect to ( δ ′ X , δ ′ Y ). Due to precompleteness of δ ′ U and This is not related to the preorders being partial or total in an order theoretic sense; they areboth partial in that sense.
EIHRAUCH GOES BROUWERIAN 13 δ ′ Y according to (2), we can always assume that H ′ , K ′ are even total computablefunctions. Hence, we even obtain H ′ h id , G ′ K ′ i ⊢ t f , which completes the proof.The proof for ≤ stW is analogous. (cid:3) If f ≤ tW g holds with respect to ( δ X , δ Y , δ U , δ V ) and at least δ V is precompleteamong these representations, then according to Lemma 4.2 we can always replacethe non-precomplete representations by equivalent precomplete ones and f ≤ tW g holds with respect to these precomplete representations and hence f ≤ tW g holdsin terms of Definition 4.1.For the moment Lemma 4.2 is useful as it implies that ≤ tW and ≤ stW are well-defined and invariant under computably equivalent representations. Corollary 4.3 (Invariance under equivalent representations) . Let f : ⊆ X ⇒ Y and g : ⊆ U ⇒ V be problems. The relations f ≤ W g , f ≤ sW g , f ≤ tW g and f ≤ stW g remain unchanged if we replace the representations of X, Y, U and V by computablyequivalent ones. We note that the statement for ≤ W and ≤ sW was proved in [3, Lemma 2.11]. Thefollowing example shows that precompleteness in Definition 4.1 cannot be omittedif one wants to achieve invariance under equivalent representations. Example 4.4.
Every computable function F : ⊆ N N → N N without total com-putable extension (see Example 3.2) has a total computable realizer with respectto (id , id ℘ ), but not with respect to (id , id). Hence F ≤ tW F does not hold withrespect to (id , id , id , id ℘ ). Clearly, F ≤ tW F holds with respect to (id ℘ , id ℘ , id ℘ , id ℘ )and hence F ≤ tW F holds in terms of Definition 4.1.The argument used in the proof of Lemma 4.2 concerning H ′ and K ′ also allowsus to slightly rephrase Definition 4.1. Due to precompleteness we can demand total H, K (and replace ⊢ t by ⊢ on the right-hand side.) Lemma 4.5 (Weihrauch reducibility) . Let f : ⊆ X ⇒ Y and g : ⊆ U ⇒ V beproblems. We choose precomplete representations that are computably equivalent tothe given representations of X, Y, U and V . Then: (1) f ≤ W g ⇐⇒ ( ∃ computable H, K : N N → N N )( ∀ G ⊢ g ) H h id , GK i ⊢ f . (2) f ≤ sW g ⇐⇒ ( ∃ computable H, K : N N → N N )( ∀ G ⊢ g ) HGK ⊢ f . (3) f ≤ tW g ⇐⇒ ( ∃ computable H, K : N N → N N )( ∀ G ⊢ t g ) H h id , GK i ⊢ f . (4) f ≤ stW g ⇐⇒ ( ∃ computable H, K : N N → N N )( ∀ G ⊢ t g ) HGK ⊢ f . The proof of the backward direction is immediate and the forward directionfollows from precompleteness of the representations of U and Y , respectively.In [3, Lemma 2.4] we have proved that ≤ W and ≤ sW are preorders , i.e., they arereflexive and transitive. The associated equivalences are denoted by ≡ W and ≡ sW ,respectively. Using Lemma 4.5 we can now easily transfer these proofs to the caseof the total reducibilities. Proposition 4.6 (Preorders) . The relations ≤ tW and ≤ stW are preorders on theclass of problems.Proof. We follow the proof of [3, Lemma 2.4] and the notations used therein. Re-flexivity is obvious as the corresponding functions
H, K are total. For the tran-sitivity proof, we assume that the reductions f ≤ tW g and g ≤ tW h are given bytotal H, K, H ′ , K ′ . Then the corresponding functions H ′′ and K ′′ constructedin the proof of [3, Lemma 2.4] are also total and hence the claim follows fromLemma 4.5. (cid:3) By ≡ tW and ≡ stW we denote the equivalence relations that are associated with ≤ tW and ≤ stW , respectively. If the different versions of Weihrauch reducibility are expressed as in Lemma 4.5, then it is immediately clear that a partial reductionimplies the corresponding total reduction. Using Lemma 4.5, Corollary 4.3 andProposition 3.4 obtain the following corollary. Corollary 4.7 (Partial and total Weihrauch reducibility) . Let f and g be problems.Then f ≤ W g = ⇒ f ≤ tW g and f ≤ sW g = ⇒ f ≤ stW g . This means that all positive results that hold for a partial version of Weihrauchreducibility can be transferred to the corresponding total variant. Together with theobvious other implications we obtain the diagram for the logical relations betweendifferent versions of Weihrauch reducibility that is displayed in Figure 2. Thediagram is complete up to transitivity (see Example 4.8). The diagram also showsthe generating closure operators of cylindrification and completion that we discusslater. f ≤ sW g f ≤ W gf ≤ tW gf ≤ stW g partial strongtotal weakcylindrification completion Figure 2.
Implications between notions of reducibility
Example 4.8.
Let f : N N → N N denote a constant function with computablevalue. Then id ≡ W f , but id stW f . Let 0 : ⊆ N N → N N denote the nowheredefined function. Then id ≡ tW
0, but id W
0. Let id | { p } : ⊆ N N → N N be theidentity restricted to a non-computable p ∈ N N . Then id ≡ stW id | { p } × id, butid W id | { p } × id.We note that the reducibilities ≤ tW and ≤ stW share similar properties as ≤ W and ≤ sW when it comes to the preservation of computability or other properties.We say that a class C of problems is preserved downwards by a reducibility ≤ forproblems if f ≤ g and g ∈ C imply f ∈ C . Proposition 4.9 (Downwards preservation) . Computability, continuity, limit com-putability, Borel measurability and non-uniform computability are preserved down-wards by ≤ tW .Proof. Let C be the class of computable, continuous, limit computable, Borel mea-surable or non-uniformly computable problems. We choose precomplete represen-tations and total computable H, K that witness f ≤ tW g according to Lemma 4.5.If g ∈ C , then it has a realizer G : ⊆ N N → N N that is in C . Since the target spaceof g is represented with a precomplete representation, we can assume without lossof generality that G is total by Proposition 3.6. Hence H h id , GK i is a (even total)realizer of f that is also in the class C . This proves that f ∈ C . (cid:3) Any class C of functions F : ⊆ N N → N N constitutes a property of problemsthat is preserved downwards by total Weihrauch reducibility if the following condi-tions are satisfied: C contains the identity, is closed under composition with com-putable functions, is closed under juxtaposition with the identity and C respectsprecompleteness. In [4, Corollaries 6.2, 7.4, 8.3] we prove that finite mind changecomputability and Las Vegas computability is not preserved downwards by ≤ tW ,whereas non-deterministic computability is preserved. EIHRAUCH GOES BROUWERIAN 15
It is known that the class of the nowhere defined problems (often denoted by ) forms the bottom element of the Weihrauch lattice [3, Lemma 2.7], while theWeihrauch equivalence class of id (often denoted by ) is the class of all computableproblems with at least one computable point in the domain [3, Lemma 2.8]. More-over, a problem f is computable if and only if f ≤ W id. The statement about thenowhere defined function 0 : ⊆ N N → N N in Example 4.8, namely that id ≡ tW Corollary 4.10 (Minimal total degree) . The equivalence class of all computableproblems forms the minimal element with respect to total Weihrauch reducibility.
This already shows that the algebraic structure induced by total Weihrauch re-ducibility is significantly different from the structure induced by partial Weihrauchreducibility. In between and one obtains a complicated structure for partialWeihrauch reducibility, and among other results one can show that one can embedthe entire Medvedev lattice (and hence the Turing semi-lattice) in an order-reversingway into the Weihrauch lattice between and [13, Lemma 5.6]. In contrast to thisthe two degrees and fall together with respect to total Weihrauch reducibility.Strictly speaking, the class of problems is not a set, but we can always considerrepresentatives of problems on Baire space to obtain a set as underlying structure.This is known for ≤ W and ≤ sW (see [6, Lemma 3.8]) and holds correspondingly for ≤ tW and ≤ stW . Corollary 4.11 (Realizer version) . Let f : ⊆ X ⇒ Y be a problem on representedspaces ( X, δ X ) and ( Y, δ Y ) . Then f r := δ − Y ◦ f ◦ δ X : ⊆ N N ⇒ N N satisfies f r ≡ stW f .Proof. By f r ≡ sW f holds according to [6, Lemma 3.8] (and is easy to see, since f r and f share exactly the same realizers). Hence f r ≡ stW f follows by Corollary 4.7. (cid:3) We note that we do not need to assume that δ X and δ Y are precomplete in thisresult. However, for f r : ⊆ N N ⇒ N N we need to use precomplete representations of N N for the total versions of Weihrauch reducibility.5. Completion
In this section we discuss the closure operation of completion f f that gener-ates ≤ tW on ≤ W and ≤ stW on ≤ sW . For the definition of the completion f we usethe completion X of a represented space according to Definition 3.7. Definition 5.1 (Completion) . Let f : ⊆ X ⇒ Y be a problem. We define the completion of f by f : X ⇒ Y , x (cid:26) f ( x ) if x ∈ dom( f ) Y otherwiseWe note that the completion f is always pointed , i.e., it has a computable pointin its domain. This is because ⊥ ∈ X is always computable (as it has the constantzero sequence as a name).Sometimes it is useful to think of f in terms of its realizer version f r : N N ⇒ N N ,which is given by f r ( p ) = δ − Y ◦ f ◦ δ X ( p ) = (cid:26) ( δ ℘Y ) − ◦ f ◦ δ ℘X ( p ) if p ∈ dom( f ◦ δ ℘X ) N N otherwise . Since f has exactly the same realizers as f r , one can deduce from this formula thatthe realizers of f are exactly the total realizers of f with respect to δ ℘X and δ ℘Y ,which immediately yields the following conclusion with the help of Lemma 4.5. Lemma 5.2 (Completion and total Weihrauch reducibility) . For all problems f, g : f ≤ W g ⇐⇒ f ≤ W g ⇐⇒ f ≤ tW g and f ≤ sW g ⇐⇒ f ≤ sW g ⇐⇒ f ≤ stW g . Thus, we could define total Weihrauch reducibility also using the completionoperation and partial Weihrauch reducibility. Lemma 5.2 also shows that the to-tal Weihrauch degrees can be order theoretically embedded into the pointed partialWeihrauch degrees. Together with Corollary 4.7 we obtain that completion is mono-tone.
Corollary 5.3 (Monotonicity of completion) . Let f and g be problems. Then (1) f ≤ W g = ⇒ f ≤ W g , (2) f ≤ sW g = ⇒ f ≤ sW g . We note that this result also implies that completion is a well-defined opera-tion on (strong) Weihrauch degrees: if f , f are identical problems with possiblydifferent but computably equivalent representations on the input and output side,respectively, then f ≡ sW f and hence f ≡ sW f follows. This is so, even so therepresentations of the corresponding completions of the spaces on the input andoutput side are not necessarily computably equivalent (see the remark after Defi-nition 3.7). Now we can see that completion is a closure operator. Proposition 5.4 (Completion as closure operator) . Completion f f is a closureoperator on ≤ W and ≤ sW .Proof. By Lemma 5.2 f ≤ sW f is equivalent to f ≤ stW f , which holds since ≤ stW is reflexive by Proposition 4.6. By Lemma 5.2 f ≤ sW f is equivalent to f ≤ sW f ,which holds since ≤ sW is reflexive. Completion is monotone with respect to ≤ W and ≤ sW according to Corollary 5.3. Altogether completion is a closure operatorwith respect to ≤ W and ≤ sW . (cid:3) We have used properties of ≤ tW and ≤ stW in order to obtain properties of com-pletion. Vice versa Proposition 5.4 and Lemma 5.2 also imply Proposition 4.6and Corollary 4.7. Hence, these concepts yield different perspectives on the sameproperties.It is clear that every f is strongly totally equivalent to its completion by Lemma 5.2and Proposition 5.4. Corollary 5.5. f ≡ stW f for every problem f . In the study of total Weihrauch reducibility the degrees that have identical coneswith respect to partial and total Weihrauch reducibility play an important role.Hence, we introduce a name for such degrees.
Definition 5.6 (Complete problems) . A problem f is called complete if f ≡ W f and strongly complete if f ≡ sW f .Now we obtain the following straightforward characterization of completeness. Theorem 5.7 (Completeness) . Let g be a problem. Then the following hold: (1) g complete ⇐⇒ ( ∀ problems f )( f ≤ W g ⇐⇒ f ≤ tW g ) . (2) g strongly complete ⇐⇒ ( ∀ problems f )( f ≤ sW g ⇐⇒ f ≤ stW g ) .Proof. If g is (strongly) complete, then the respective given equivalence holds byLemma 5.2. On the other hand, if f ≤ W g ⇐⇒ f ≤ tW g holds for all f , then g ≡ W g follows from Corollary 5.5. The case of strong completeness can be handledanalogously. (cid:3) EIHRAUCH GOES BROUWERIAN 17
Examples of complete problems are abundant. We study a number of landmarksin the Weihrauch lattice, among them the Turing jump operator J and and thebinary sorting problem SORT that was introduced and studied by Neumann andPauly [18]. The problems
WBWT , ACC X , PA and MLR were studied for instancein [7]. We identify X ∈ N with the set X = { , , ..., X − } . Many furthercompleteness questions regarding choice are studied in [4]. Proposition 5.8 (Complete problems) . The following problems are all stronglycomplete: (1) id : N N → N N , p p , (2) J : N N → N N , p p ′ , (3) lim : ⊆ N N → N N , h p , p , p , ... i 7→ lim n →∞ p n , (4) LPO : N N → { , } , LPO ( p ) = 0 : ⇐⇒ ( ∃ n ∈ N ) p ( n ) = 0 , (5) SORT : 2 N → N with SORT ( p ) := (cid:26) k b if p contains exactly k ∈ N zeros b if p contains infinitely many zeros . (6) WBWT : 2 N ⇒ N , p
7→ { q ∈ N : lim n →∞ q ( n ) is a cluster point of p } . (7) ACC X : ⊆ N N ⇒ N , p
7→ { n ∈ X : n + 1 range( p ) } , where X ≥ or X = N and dom( ACC X ) := { p ∈ N N : range( p ) ⊆ { , n + 1 } for some n ∈ X } . (8) PA : 2 N ⇒ N , p
7→ { q ∈ N : q is a PA-degree relative to p } . (9) MLR : 2 N ⇒ N , p
7→ { q ∈ N : q Martin-L¨of random relative to p } .Proof. (1) Follows since id ≤ sW id N N ≤ sW id.(2) There is a total computable function H : N N → N N such that H ◦ J ( p ) = J ( p −
1) + 1 for all p with p − ∈ N N . This can be proved using the smn-Theorem.Together with the identity K this function H witnesses the reduction J ≤ sW J .(3) Follows by Corollary 5.3 since lim ≡ sW J holds (see [6, Theorem 6.7]).(4) Given a name p ∈ N N of a point in N N with respect to δ N N , we can compute K ( p ) as follows: K ( p )( n ) = 0 : ⇐⇒ p ( n ) = 1 and K ( p )( n ) := 1 otherwise. If δ N N ( p ) = q ∈ N N , then LPO ◦ K ( p ) = LPO ( q ). Hence, together with H ( r ) := r + 1the functions H, K witness
LPO ≤ sW LPO .(5) As always we assume that 2 N is represented by δ N : ⊆ N N → N , p p withdom( δ N ) = 2 N . Given a name p ∈ N N of some q ∈ N , i.e., δ N ( p ) = q we can com-pute K ( p ) as follows: K ( p )( n ) = 0 : ⇐⇒ p ( n ) = 1 and K ( p )( n ) := 1 otherwise.Then SORT ◦ K ( p ) = SORT ( q ) if q ∈ N . Hence, H ( r ) := r + 1 and K witness SORT ≤ sW SORT .(6) We represent 2 N as above. Given a name p ∈ N N of some q ∈ N we can compute K ( p ) as follows, we let K ( p )( n ) := p ( n ) − p ( n ) = 0 and we let K ( p )( n ) = i for the number i ∈ { , } such that i + 1 appears a maximal number of timeswithin p (0) , ..., p ( n ) (and we choose i = 0 if 1 and 2 appear equally often). Thisconstruction guarantees that we do not generate any additional cluster points, i.e., WBWT K ( p ) = WBWT ( q ) for q ∈ N . Similarly as in the other cases above, thisproves WBWT ≤ sW WBWT .(7) Given some name p ∈ N N of a point q ∈ N N we compute K ( p ) as follows:we let K ( p )( n ) := k + 1 if k + 2 = p ( n ) is the first number larger than 1 among p (0) , ..., p ( n ) and k ∈ X . Otherwise, we let K ( p )( n ) := 0. This guarantees that ACC X K ( p ) = ACC X ( q ), if q ∈ dom( ACC X ). Similarly as in the other cases above,this proves ACC X ≤ sW ACC X .(8), (9) We use K : N N → N , p p (0)+1 p (1)+1 p (2)+1 ... , which is total com-putable. It is straightforward to see that every problem F : 2 N ⇒ N that is antitonein the sense that p ≤ T q implies F ( q ) ⊆ F ( p ) is strongly complete. This is because p − ≤ T p ≡ T K ( p ) if p ∈ N N is such that p − ∈ N , and hence F K ( p ) ⊆ F ( p − F ≤ sW F . This applies in particular to PA and MLR . (cid:3) These results show that the cones below the given problems are identical inthe total and partial Weihrauch lattices. It is known, for instance, that f is limitcomputable if and only if f ≤ W lim [6]. Hence, an analogous statement holds for ≤ tW . 6. Algebraic Operations
In this section we want to discuss properties of certain algebraic operations andwe want to prove that the total versions of Weihrauch reducibility yield latticestructures. We start recalling the usual algebraic operations on the Weihrauchlattice [6].
Definition 6.1 (Algebraic operations) . Let f : ⊆ X ⇒ Y and g : ⊆ U ⇒ V bemulti-valued functions. We define the following operations:(1) f × g : ⊆ X × U ⇒ Y × V, ( f × g )( x, u ) := f ( x ) × g ( u ) anddom( f × g ) := dom( f ) × dom( g ) (product)(2) f ⊔ g : ⊆ X ⊔ U ⇒ Y ⊔ V , ( f ⊔ g )(0 , x ) := { }× f ( x ), ( f ⊔ g )(1 , u ) := { }× g ( u )and dom( f ⊔ g ) := dom( f ) ⊔ dom( g ) (coproduct)(3) f ⊞ g : ⊆ X ⊔ U ⇒ Y × V , ( f ⊞ g )(0 , x ) := f ( x ) × V , ( f ⊞ g )(1 , u ) := Y × g ( u )and dom( f ⊞ g ) := dom( f ) ⊔ dom( g ) (box sum)(4) f ⊓ g : ⊆ X × U ⇒ Y ⊔ V, ( f ⊓ g )( x, u ) := f ( x ) ⊔ g ( u ) anddom( f ⊓ g ) := dom( f ) × dom( g ) (meet)(5) f + g : ⊆ X × U ⇒ Y × V , ( f + g )( x, u ) := ( f ( x ) × V ) ∪ ( Y × g ( u )) anddom( f + g ) := dom( f ) × dom( g ) (sum)(6) f ∗ : ⊆ X ∗ ⇒ Y ∗ , f ∗ ( i, x ) := { i } × f i ( x ) anddom( f ∗ ) := dom( f ) ∗ (finite parallelization)(7) b f : ⊆ X N ⇒ Y N , b f ( x n ) n ∈ N := X i ∈ N f ( x i ) anddom( b f ) := dom( f ) N (parallelization)For every operation (cid:3) ∈ {× , ⊔ , ⊞ , ⊓ , + } we define its completion (cid:3) by f (cid:3) g := f (cid:3) g . It follows from Lemma 5.2 that these operations are monotone with respectto total Weihrauch reducibility, since the underlying operations (cid:3) are monotonewith respect to partial Weihrauch reducibility by [6, Proposition 3.6]. Corollary 6.2 (Monotonicity) . ( f, g ) f (cid:3) g for (cid:3) ∈ {× , ⊔ , ⊞ , ⊓ , + } , f b f and f f ∗ are monotone with respect to ≤ tW and ≤ stW .Proof. By Lemma 5.2 completion generates ≤ tW on ≤ W (and ≤ stW on ≤ sW ). By[6, Proposition 3.6] the given operations (cid:3) are monotone with respect to ≤ W and ≤ sW , respectively. The claim now follows with Proposition 2.2. (cid:3) Now we prove that the algebraic operations preserve completeness in the senseof Definition 2.3. It is clear by Proposition 2.5 that we also get co-preservation forsuprema (see Proposition 6.5). Later we will show that this also holds for + (seeProposition 6.8).
Proposition 6.3 (Completion and algebraic operations) . Let f and g be problems.We obtain (1) f (cid:3) g ≤ sW f (cid:3) g ≡ sW f (cid:3) g for (cid:3) ∈ {× , ⊔ , ⊞ , ⊓ , + } , (2) b f ≤ sW b f ≡ sW b f , (3) f ∗ ≤ sW f ∗ ≡ sW f ∗ . EIHRAUCH GOES BROUWERIAN 19
In particular, if f and g are (strongly) complete, then so are f × g , f ⊔ g , f ⊞ g , f ⊓ g , f + g , b f and f ∗ .Proof. We consider problems f : ⊆ X ⇒ Y and g : ⊆ U ⇒ V and (cid:3) ∈ {× , ⊔ , ⊞ , ⊓ , + } .Since X ⊔ U ⊆ X ⊔ U and X × U ⊆ X × U , it follows that dom( f (cid:3) g ) ⊆ dom( f (cid:3) g ),and restricted to x ∈ dom( f (cid:3) g ) we have f (cid:3) g ( x ) = ( f (cid:3) g )( x ) ⊆ ( f (cid:3) g )( x ). The“ ⊆ ” is even an equality in the cases (cid:3) ∈ {× , ⊔ , ⊓} . In the other cases it is notan equality simply because V $ V and Y $ Y . We can also assume that therepresentations of X × U and X ⊔ U are total (since the representations of X and U are so). Hence every realizer of f (cid:3) g is total. By Corollary 3.10 ι : Z → Z, z z is computable for every represented space Z , hence it follows that f (cid:3) g ≤ sW f (cid:3) g ,since a realizer for f (cid:3) g can choose any value outside of dom( f (cid:3) g ). This also holdsin the cases where we only have “ ⊆ ” above, since the representation of V is total,every name of a point in V is also a name of some point in V and an analogousstatement holds for Y . The proofs for the unary operations are analogous. We have X N ⊆ X N and X ∗ ⊆ X ∗ and hence b f ≤ sW b f and f ∗ ≤ sW f ∗ . The remaining claimsfollow by Proposition 2.4 as completion is a closure operator by Proposition 5.4. (cid:3) The closure properties of complete problems are very useful. For instance, it isknown that lim ≡ sW d LPO [6] and hence the statement on lim in Proposition 5.8 couldalso be derived from the statement on
LPO . Likewise, we obtain a number of furthercomplete problems in this way. We refrain from giving exact definitions of the listedproblems, but we rather point the reader to [7] were all stated equivalences havebeen proved [7, Theorem 5.2, Corollary 5.3, Proposition 14.10]. For the purpose ofthis article, the equivalences can be read as definitions.
Corollary 6.4 (Complete problems) . WKL ≡ sW C N ≡ sW \ ACC , DNC X ≡ sW \ ACC X for X ∈ N with X ≥ or X = N are strongly complete, and COH ≡ W \ WBWT iscomplete. In [3, Proposition 3.11] we proved that ⊓ is the infimum operation with respectto ≤ sW and ≤ W . That ⊔ is the supremum operation with respect to ≤ W was firstproved by Pauly [19, Theorem 4.5] (see also [6, Theorem 3.9]). Dzhafarov provedthat ⊞ is a supremum operation for ≤ sW [10] and he also showed f ⊞ g ≡ W f ⊔ g .Using Propositions 2.2 and 2.5 we can transfer these results to the total versions ofWeihrauch reducibility. Proposition 6.5 (Infima and suprema) . Let f, g be problems. Then (1) f ⊓ g is an infimum of f and g with respect to ≤ tW and ≤ stW . (2) f ⊔ g is a supremum of f and g with respect to ≤ tW . (3) f ⊞ g is a supremum of f and g with respect to ≤ stW . (4) f ⊔ g ≡ W f ⊔ g and hence f ⊔ g ≡ tW f ⊔ g ≡ tW f ⊞ g . (5) f ⊞ g ≡ sW f ⊞ g and hence f ⊞ g ≡ stW f ⊞ g . In Lemma 6.9 we will see that the equivalences in (4) cannot be strengthened tostrong equivalences.By a (strong) total Weihrauch degree we mean an equivalence class with respectto ≤ tW (or with respect to ≤ stW in the strong case). We denote the correspondingclasses by W tW and W stW . Strictly speaking, these are not sets, but every equiva-lence class has a representative on Baire space according to Corollary 4.11, and ifdesired, we can turn the classes W tW and W stW into sets of such representatives.The same applies to further classes of degrees that we consider in the following.We can extend the reducibilities ≤ tW and ≤ stW to the corresponding degrees and any monotone algebraic operation too. By Proposition 6.5 ( W tW , ≤ tW , ⊓ , ⊔ ) yieldsa lattice structure.It was first proved by Pauly [19, Theorem 4.22] that the Weihrauch lattice isdistributive. In fact, he proved that it is a distributive join semi-lattice, whichimplies distributivity as a lattice. That is, we have f ⊔ ( g ⊓ h ) ≡ W ( f ⊔ g ) ⊓ ( f ⊔ h )and f ⊓ ( g ⊔ h ) ≡ W ( f ⊓ g ) ⊔ ( f ⊓ h ) [8, Theorem 31]. Also the total Weihrauchdegrees form a distributive lattice. Theorem 6.6 (Total Weihrauch lattice) . ( W tW , ≤ tW , ⊓ , ⊔ ) is a distributive lattice.Proof. By Proposition 6.5 we obtain f ⊓ ( g ⊔ h ) = f ⊓ ( g ⊔ h ) ≡ W f ⊓ ( g ⊔ h ) ≡ W ( f ⊓ g ) ⊔ ( f ⊓ h ) = ( f ⊓ g ) ⊔ ( f ⊓ h )and hence f ⊓ ( g ⊔ h ) ≡ tW ( f ⊓ g ) ⊔ ( f ⊓ h ) by Corollary 4.7. With Proposition 6.5 andCorollary 5.5 we obtain similarly as above f ⊔ ( g ⊓ h ) ≡ tW f ⊔ ( g ⊓ h ) = f ⊔ ( g ⊓ h ) ≡ W ( f ⊔ g ) ⊓ ( f ⊔ h ) ≡ W ( f ⊔ g ) ⊓ ( f ⊔ h ) = ( f ⊔ g ) ⊓ ( f ⊔ h )and hence f ⊔ ( g ⊓ h ) ≡ tW ( f ⊔ g ) ⊓ ( f ⊔ h ). Altogether, this shows that the totalWeihrauch lattice is distributive. (cid:3) Proposition 6.5 implies that W stW is a lattice. Dzhafarov proved that the lattice W sW is not distributive [10, Theorem 4.4]. We can transfer his proof to W stW . Theorem 6.7 (Strong total Weihrauch lattice) . ( W stW , ≤ stW , ⊓ , ⊞ ) is a lattice,which is not distributive.Proof. Proposition 6.5 implies that W stW is a lattice. Suppose that this lattice isdistributive. Then, in particular again by Proposition 6.5( f ⊞ g ) ⊓ h ≡ sW ( f ⊞ g ) ⊓ h = ( f ⊞ g ) ⊓ h ≤ stW ( f ⊓ h ) ⊞ ( g ⊓ h ) = ( f ⊓ h ) ⊞ ( g ⊓ h ) , i.e., ( f ⊞ g ) ⊓ h ≤ stW ( f ⊓ h ) ⊞ ( g ⊓ h ), which by Lemma 5.2, Propositions 6.5 and6.3 is equivalent to( f ⊞ g ) ⊓ h ≤ sW ( f ⊓ h ) ⊞ ( g ⊓ h ) ≡ sW ( f ⊓ h ) ⊞ ( g ⊓ h ) ≡ sW ( f ⊓ h ) ⊞ ( g ⊓ h ) . Hence, it suffices to provide a counterexample for ( f ⊞ g ) ⊓ h ≤ sW ( f ⊓ h ) ⊞ ( g ⊓ h ).We use the proof idea of [10, Theorem 4.4] and we consider the constant problems c p,q : ⊆ N N → N N , p q with dom( c p,q ) = { p } for p, q ∈ N N . Let p i , q i ∈ N N for i ∈ { , , } be mutually Turing incomparable and such that none of thesepoints can be computed from the supremum of the others (this is possible, see forinstance [22, Exercise 2.2 in Chapter VII]). We choose f := c p ,q , g := c p ,q and h := c p ,q . We recall that N N = N N ∪ {⊥} is represented with a precompleterepresentation δ , defined by δ ( p ) = id ℘ ( p ) = p − p − ∈ N N and δ ( p ) = ⊥ otherwise. Now assume that ( f ⊞ g ) ⊓ h ≤ sW ( f ⊓ h ) ⊞ ( g ⊓ h ) via computable H, K .We claim that K hh i, p i + 1 i , p + 1 i = h i, h p ′ i , p ′ ii for i ∈ { , } with names p ′ k of p k for k ∈ { , , } . Firstly, if K hh i, p i +1 i , p +1 i = h j, h r, s ii such that r is not a nameof p j or s is not a name of p , then a realizer of e := ( f ⊓ h ) ⊞ ( g ⊓ h ) on h j, h r, s ii could return any value, for instance a computable one, and in this case H couldneither compute q i nor q from this result. Hence K hh i, p i + 1 i , p + 1 i = h j, h p ′ j , p ′ ii with j ∈ { , } and p ′ k a name for p k for k ∈ { , , } . Secondly, if j = i , then arealizer of e upon input of h j, h p ′ j , p ′ ii could return a name of q j together withsome computable values, from which H can neither compute q i nor q . This provesthe claim above. Now on input h , h p ′ , p ′ ii as above, a realizer of e can produce r := hh , q ′ i + 1 , b i with a name q ′ of q . Suppose H ( r ) = h , s i with some s ∈ N N .Since H is continuous, a certain prefix of r is sufficient to produce the output 0 EIHRAUCH GOES BROUWERIAN 21 in the first component. Now on input h , h p ′ , p ′ ii a realizer of e can produce theoutput t := hh , c i + 1 , n ( h , q ′ i + 1) i with a computable c that shares a sufficientlylong prefix with q ′ and a sufficiently large n ∈ N and a name q ′ of q . Then H ( t ) = h , s ′ i with some s ′ ∈ N N . However, s ′ is computable from q and hence itcan neither compute q nor q , which is a contradiction. Hence H ( r ) = h , s i withsome s ∈ N N . Again, due to continuity of H , some prefix of the input is sufficientto produce the component 1 on the output side. On input h , h p ′ , p ′ ii a realizer of e can now produce the output t := hh , q ′ i + 1 , n ( h , q ′ i + 1) i for sufficiently large n ∈ N and H ( t ) = h , s ′ i with s ′ ∈ N N . However, since s ′ is computable from q and q , it cannot compute q , which is a contradiction. (cid:3) We are going to prove that + also co-preserves completion with respect to ≤ tW and ≤ stW . Proposition 6.8 (Sums) . f + g ≡ sW f + g and hence f + g ≡ stW f + g for allproblems f, g .Proof. We consider problems f : ⊆ X ⇒ Y and g : ⊆ U ⇒ V . We obtain theproblems f + g : X × U ⇒ Y × V with( f + g )( x, u ) = ( f ( x ) × V ) ∪ ( Y × g ( u )) if ( x, u ) ∈ dom( f ) × dom( g )( f ( x ) × V ) ∪ ( Y × V ) if x ∈ dom( f ) and u dom( g )( Y × V ) ∪ ( Y × g ( u )) if x dom( f ) and u ∈ dom( g )( Y × V ) ∪ ( Y × V ) otherwiseand f + g : X × U ⇒ Y × V with( f + g )( z ) = ( ( f ( x ) × V ) ∪ ( Y × g ( u )) if z = ( x, u ) ∈ dom( f ) × dom( g ) Y × V otherwise . And we also consider h : X × U ⇒ Y × V with h ( x, u ) := (cid:26) ( f ( x ) × V ) ∪ ( Y × g ( u )) if ( x, u ) ∈ dom( f ) × dom( g ) Y × V otherwise . Then we have h ( x, u ) ⊆ ( f + g )( x, u ) for all ( x, u ) ∈ X × U and hence together withProposition 6.3 f + g ≤ sW f + g ≤ sW h . On the other hand, there is a computablefunction s : Y × V → Y × V with s ( y, v ) = ( y, v ) for all ( y, v ) ∈ Y × V . Namely, onecan just consider S : ⊆ N N → N N , p p − s under the representation of Y × V , which is possible, since this spacehas a precomplete representation by Proposition 3.8. Analogously to s , there is alsoa computable function ι : X × U → X × U with ι ( x, u ) = ( x, u ) for ( x, u ) ∈ X × U .Then h = s ◦ ( f + g ) ◦ ι and hence h ≤ sW f + g . (cid:3) The following example shows that × and ⊓ do not co-preserve completion withrespect to ≤ W and that ⊔ does not co-preserve completion with respect to ≤ sW . Lemma 6.9.
There are problems f, g : ⊆ N N → N N such that (1) f × g W f × g , and hence f × g tW f × g , (2) f ⊓ g W f ⊓ g , and hence f ⊓ g tW f ⊓ g , (3) f ⊔ g sW f ⊔ g , and hence f ⊔ g stW f ⊔ g .Proof. We consider the constant problems c p,q : ⊆ N N → N N , p q with dom( c p,q ) = { p } for p, q ∈ N . Let p, q, r, s ∈ N N be mutually Turing incomparable. We choose f := c p,q and g := c r,s . We recall that N N = N N ∪ {⊥} is represented with aprecomplete representation δ , defined by δ ( p ) = id ℘ ( p ) = p − p − ∈ N N and δ ( p ) = ⊥ otherwise. We only need to prove the former statements regarding ≤ W , since the latter statements regarding ≤ tW follow in each case with Lemma 5.2.(1) holds since a name for the input pair ( p, p ) ∈ dom( c p,q × c r,s ) can only bemapped computably to a name of an input outside of dom( c p,q × c r,s ) = { ( p, r ) } since r is not computable from p , and a realizer for c p,q × c r,s can map such a nameto any name, for instance a computable name. From a computable name and aname for ( p, p ) one cannot compute q .(2) Let us assume that c p,q ⊓ c r,s ≤ W c p,q ⊓ c r,s is witnessed by computable H, K .We consider the name p +1 of p and the name b ⊥ . Since ( p, ⊥ ) ∈ dom( c p,q ⊓ c r,s ), K h p +1 , b i has to be defined, but it cannot be a name of a point in dom( c p,q ⊓ c r,s ) = { p } × { r } . Let G be a realizer of c p,q ⊓ c r,s that maps every name of a point outsideof dom( c p,q ⊓ c r,s ) to b
0. Then H hh p + 1 , b i , GK h p + 1 , b ii = H hh p + 1 , b i , b i = h , t i for some t ∈ N N , since it cannot be equal to h , u i for some u ∈ N N because q cannot be computed from p and H h id , GK i has to be a realizer of c p,q ⊓ c r,s . Dueto continuity of H the output 1 in the first component is determined already bya prefix of the input, say by w ⊑ p + 1 and 0 n ⊑ b
0. Hence, on the names w b n ( r + 1) of ⊥ and r , respectively, the function H will also produce 1 in thefirst component. Moreover K h w b , n ( r + 1) i is also a name of a point outside ofdom( c p,q ⊓ c r,s ) = { p } × { r } and hence GK h w b , n ( r + 1) i = b
0. In this case wemust have H hh w b , n ( r + 1) i , GK h w b , n ( r + 1) ii = H hh w b , n ( r + 1) i , b i = h , t i with a name t of s , which is impossible, since s cannot be computed from r .(3) Let us assume that c p,q ⊔ c r,s ≤ sW c p,q ⊔ c r,s is witnessed by computable H, K .Upon input of the name h i, b i of ( i, ⊥ ) ∈ dom( c p,q ⊔ c r,s ) with i ∈ { , } the function K cannot produce a name of a point in dom( c p,q ⊔ c r,s ) = { (0 , p ) , (1 , r ) } . There isa realizer G of f ⊔ g that produces the name b ⊥ on any input outside of thedomain of dom( c p,q ⊔ c r,s ) and hence HGK h i, b i = h j, t i for some fixed j ∈ { , } and t ∈ N N and both values i ∈ { , } . The fixed j can only be correct for one of thevalues i , since we need i = j for the correctness of H, K , which is impossible. (cid:3)
With the help of Corollary 5.5 it follows that × and ⊓ are not monotone withrespect to the total versions of Weihrauch reducibility. Corollary 6.10. × , ⊓ are neither monotone with respect to ≤ tW nor with respectto ≤ stW , and ⊔ is not monotone with respect to ≤ stW . Many further algebraic properties of the Weihrauch lattice have been studiedin [8]. Some of these results can be transferred to the total case by Corollary 4.7.In some cases we can also transfer results for pointed problems, since the completion f of any problem is always pointed. For instance, the completions of the algebraicoperations are ordered in the following way, as the corresponding reductions holdmore generally for pointed problems (by [6, Proposition 5.7], and that f ∗ ≤ W b f holds for pointed f , is easy to see). Corollary 6.11 (Order of operations) . For all problems f and g we obtain: f + g ≤ sW f ⊓ g ≤ sW f ⊞ g ≤ sW f ⊔ g ≤ W f × g , f ⊞ g ≤ sW f × g and f ∗ ≤ W b f . Now we study the completions of parallelization f b f and finite parallelization f f ∗ . In [3, Proposition 4.2] we proved that f b f is a closure operator for ≤ W and ≤ sW and Pauly proved in [19, Theorem 6.2] that f f ∗ is a closureoperator for (the topological version of) ≤ W . We note that the latter one is nota closure operator for ≤ sW . Nevertheless, the completions of both operators areclosure operators for ≤ tW and ≤ stW . In order to prove this, we need the followingadditional lemma. EIHRAUCH GOES BROUWERIAN 23
Lemma 6.12 (Arno Pauly ) . f ∗∗ ≡ sW f ∗ for all pointed problems f .Proof. It is easy to see that f ≤ sW f ∗ holds for all problems f , in particular, weobtain f ∗ ≤ sW f ∗∗ . For the inverse reduction we assume that f is pointed. Let p be a computable name of a point in dom( f ). We use the computable functions K, H with K h n, hh i , h p , , ..., p ,i ii , h i , h p , , ..., p ,i ii , ..., h i n , h p n, , ..., p n,i n iiii := * k, h p , , ..., p ,i , p , , ..., p ,i , ......, p n, , ..., p n,i n , p , ..., p | {z } m times i + where k := h n, h i , ..., i n ii ≥ i + ... + i n and m := k − ( i + ... + i n ), and for arbitrary k = h n, h i , ..., i n ii ∈ N and j := i + ... + i n ≤ k we define H h k, h q , ..., q k ii := h n, hh i , h q , ..., q i ii , h i , h q i +1 , ..., q i + i ii , ..., h i n , h q i + ... + i n − +1 , ..., q j iiii . Then
H, K are computable and witness f ∗∗ ≤ sW f ∗ . (cid:3) Now we are prepared to prove the following result.
Proposition 6.13 (Parallelization) . f b f and f f ∗ are closure operators for ≤ tW and ≤ stW (and also for ≤ W and ≤ sW ).Proof. Since parallelization f b f and completion f f are both closure operatorsfor ≤ sW and ≤ W by [3, Proposition 4.2] and Proposition 5.4, and parallelizationpreserves completion by Proposition 6.3, the claim follows from Propositions 2.2and 2.4. The claim for f f ∗ with respect to ≤ tW follows analogously. In order toprove the claim for ≤ stW , we note that f f ∗ is a closure operator with respect to ≤ sW restricted to pointed problems. This follows from Corollary 6.2, Lemma 6.12and since f ≤ sW f ∗ obviously holds true. Hence, we also obtain that f f ∗ is a closure operator with respect to ≤ stW , since all problems of the form f arepointed. (cid:3) With the following counterexamples we show that (finite) parallelization doesnot co-preserve completion. Some of the statements can be seen as a strengtheningof the first statement in Lemma 6.9.
Lemma 6.14.
There is a problem f with f × f W b f and f × f W f ∗ . This implies (1) f × f W f × f , and hence f × f tW f × f , (2) f ∗ W f ∗ , and hence f ∗ tW f ∗ , (3) b f W b f , and hence b f tW b f .Proof. We consider the function f : ⊆ N N → N N with f ( p ) = q , f ( r ) = s , dom( f ) = { p, r } and pairwise Turing incomparable p, q, r, s ∈ N N such that none of these iscomputable from the supremum of the others (this is possible, see for instance [22,Exercise 2.2 in Chapter VII]). We recall that N N = N N ∪ {⊥} is represented with aprecomplete representation δ , defined by δ ( p ) = id ℘ ( p ) = p − p − ∈ N N and δ ( p ) = ⊥ otherwise. Let us assume that the reduction f × f ≤ W b f holds, witnessedby computable H, K . The names p + 1 , r + 1 of p, q are mapped by K to a name K h p + 1 , r + 1 i of a point ( q n ) n ∈ N in dom( b f ), since a realizer of b f can choose acomputable output outside of dom( b f ) and the result ( q, s ) cannot be computedfrom p, r alone. For the same reason q n = r for at least one n ∈ N and hence π n K h p + 1 , r + 1 i is a name r of r . Due to continuity of K there are prefixes By personal communication 2018. w ⊑ p + 1 and v ⊑ r + 1 that are sufficient for K to produce a prefix u ⊑ r thatis long enough so that it cannot be extended to a name of p . We can now replace r + 1 by t = v b
0, which is a name of ⊥ ∈ dom( f ). Now π n K h p + 1 , t i cannot be aname of r , since r cannot be computed from p and t and it cannot be a name of p either, since u ⊑ π n K h p + 1 , t i . Hence K h p + 1 , t i is a name for a point outsideof dom( b f ) and a realizer of b f can choose a computable result c on this name. But H hh p + 1 , t i , c i cannot compute q , which is required by the assumption. This proves f × f W b f . The second statement can be proved analogously, one has to choose w, v such that also the natural number component of the name of an output in( N N ) ∗ is fixed.All other statements that involve ≤ W are consequences since f × f ≤ W f ∗ ≤ W b f , f × f ≤ W f ∗ and f × f ≤ W b f . These reductions follow since obviously g × g ≤ W g ∗ and g × g ≤ W b g for any problem g , completion is a closure operator by Proposi-tion 5.4, and by Corollary 6.11, since f is pointed. The statements that involve ≤ tW follow from Lemma 5.2. (cid:3) As an immediate consequence of these counterexamples we can conclude thatparallelization and finite parallelization are not monotone operations for the totalvariants of Weihrauch reducibility. Since f ≤ stW f holds by Corollary 5.5, we obtainthe following conclusion using Lemma 6.14. Corollary 6.15. f b f and f f ∗ are neither monotone with respect to ≤ tW nor with respect to ≤ stW . Another consequence of Lemma 6.14 is that completion does neither preserveidempotency nor parallelizability. We recall that a problem f is called idempotent ,if f ≡ W f × f and it is called parallelizable , if b f ≡ W f . If we consider the problem f from Lemma 6.14, then we can take f ∗ and b f as examples to obtain the followingresult. Corollary 6.16 (Idempotency and parallelizability) . (1) There is an idempotent problem f such that f is not idempotent. (2) There is a parallelizable problem f such that f is not parallelizable. In the next step we want to clarify the relation between ≤ tW and ≤ stW andfor this purpose we need to study cylinders. We recall that a problem f is called cylinder if id × f ≤ sW f holds, and id × f is called the cylindrification of f [3].It follows from [4, Proposition 4.16] that “total cylinders” are exactly the usualcylinders. Corollary 6.17 (Total cylinders) . id × f ≤ sW f ⇐⇒ id × f ≤ stW f holds for allproblems f . It is known that g is a cylinder if and only if f ≤ W g ⇐⇒ f ≤ sW g holds for allproblems f [3, Proposition 3.5, Corollary 3.6]. We provide a similar result for thetotal variant of Weihrauch reducibility. Proposition 6.18 (Cylinder) . A problem g is a cylinder if and only if for everyproblem f one has f ≤ tW g ⇐⇒ f ≤ stW g .Proof. Let us assume that f ≤ tW g ⇐⇒ f ≤ stW g holds for every problem f . It isclear that id × g ≡ W g and hence id × g ≡ tW g by Corollary 4.7. By the assumptionthis implies id × g ≤ stW g and hence id × g ≤ sW g by Corollary 6.17. This showsthat g is a cylinder.For the other direction, let us now assume that g is a cylinder, i.e., id × g ≤ sW g and hence id × g ≤ stW g by Corollary 4.7. We only need to prove that f ≤ tW g EIHRAUCH GOES BROUWERIAN 25 implies f ≤ stW g . Let us assume that f ≤ tW g holds. Since f ≤ sW id × f , we obtain f ≤ stW id × f by Lemma 5.2. Now it suffices to show id × f ≤ stW id × g . Butthis can be done by using the construction of the proof of [3, Proposition 3.5]. ByLemma 4.5 it suffices to note that if H, K from the proof of [3, Proposition 3.5] aretotal, then also the H ′ , K ′ constructed in the first half of that proof are total. (cid:3) Hence, the relations between strong and weak versions of the reducibility can beexpressed in the same way in the partial and the total case, respectively.We can also say something on the interaction between cylindrification and com-pletion. While the completion of a cylinder f is only a cylinder in the trivial casethat the original problem f is already strongly complete, the cylindrification of acomplete problem is always complete. Proposition 6.19 (Completion and cylindrification) . Let f be a problem. Then (1) f is a cylinder ⇐⇒ f is strongly complete and a cylinder, (2) id × f is complete ⇐⇒ f is complete.The implication “ ⇐ = ” in (2) also holds for strongly complete instead of complete.Proof. (1) If f ≡ sW f and f is a cylinder, then clearly id × f ≤ sW id × f ≤ sW f ≤ sW f and hence f is a cylinder. If, on the other hand, f is a cylinder, then id × f ≤ sW f .Hence id × f ≤ stW f and since id × f is diverse, we obtain by [4, Proposition 4.16]that id × f ≤ sW f . This implies id × f ≤ sW id × f ≤ sW f , which means that f is acylinder and f ≤ sW id × f ≤ sW f , which means that f is strongly complete.(2) If f is complete, then id × f ≤ W id × f ≤ W id × f by Propositions 6.3 and 5.8,which means that id × f is complete. The proof in the strong case is analogous. If,on the other hand, id × f is complete, then f ≤ W id × f ≤ W id × f ≤ W f , where thefirst reduction holds since f ≤ W id × f and completion is a closure operator. (cid:3) Co-Residual Operations
In this section we will discuss certain algebraic operations that are co-residualoperations. In this context we have to deal with a top element in the Weihrauchlattice. The Weihrauch lattice has no natural top element, but we can just attacha top element ∞ to it. The algebraic operations are then naturally extended to thetop element, so that the lattice structure and the order among the operations ispreserved. We are led to the following choice of values for all problems f including ∞ (see also the discussion in [8]):(1) f ⊓ ∞ = ∞ ⊓ f = f ,(2) f ⊔ ∞ = ∞ ⊔ f = ∞ ,(3) f × ∞ = ∞ × f = ∞ ,(4) f + ∞ = ∞ + f = f ,(5) ∞ = c ∞ = ∞ ∗ = ∞ .One arguable alternative could be to choose 0 × ∞ = 0, given that 0 × f ≡ W f = ∞ . However this seems to be less natural for our purposes. It is consistentwith our usage of the term to say that a problem f is pointed , if 1 ≤ W f holds.According to this definition ∞ is pointed too.Using our universal function U : ⊆ N N → N N , we can define a representation Φof certain continuous functions by Φ q ( p ) := U h q, p i for all p, q ∈ N N . Then anycontinuous F : ⊆ N N → N N has an extension of the form Φ q : ⊆ N N → N N and for acomputable F we can choose a computable q (see [27]). From this representationwe can derive a G¨odel numbering ϕ of the computable F : ⊆ N N → N N , i.e., forevery computable F there is some n ∈ N such that ϕ n : ⊆ N N → N N extends F . Wealso assume that ϕ satisfies suitable utm- and smn-Theorems (see [27] for details).We use Φ and ϕ to define the compositional product and two implications. The compositional product f ⋆ g was originally defined in [5] using the property(1) stated in Fact 7.2 below. It expresses a problem that can be obtained by firstapplying g and then f with some possible intermediate computation. A correspond-ing compositional implication operation g → f was introduced and studied in [8].It characterizes the minimal problem h such that f ≤ W g ⋆ h (see Fact 7.2). Herewe phrase these operations in a type free version on Baire space (as in [6]). Wealso introduce a multiplicative implication g ։ f , which is supposed to capture aproblem simpler than every h such that f ≤ W g × h (see Proposition 7.9). Definition 7.1 (Compositional product and implications) . Let f, g be problems.We define problems f ⋆ g , ( g → f ), ( g ։ f ) : ⊆ N N ⇒ N N by(1) ( f ⋆ g ) h q, p i := h id × f r i ◦ Φ q ◦ g r ( p ),(2) ( g → f )( p ) := {h t, q i : ∅ 6 = Φ t ◦ g r ( q ) ⊆ f r ( p ) } ,(3) ( g ։ f )( p ) := {h n, k, q i : ∅ 6 = ϕ n h q, g r ◦ ϕ k ( p ) i ⊆ f r ( p ) } ,where we assume for (2) and (3) that dom( g ) = ∅ or dom( f ) = ∅ . In the case ofspecial constants we define:(1) f ⋆ ∞ := ∞ ⋆ f := ∞ ,(2) ( g →
0) := ( g ։
0) := 0, (0 → f ) := (0 ։ f ) := ∞ for f W ∞ → f ) := ( ∞ ։ f ) := 0, ( g → ∞ ) := ( g ։ ∞ ) := ∞ for g = ∞ .We call f ⋆ g the compositional product , ( g → f ) the compositional implication and( g ։ f ) the multiplicative implication .The definition of ( g →
0) := ( g ։
0) := 0 is consistent with what is defined in thefirst two items (2) and (3) above. The domains in the first items (1)–(3) are alwaysmeant to be maximal. For instance dom( g → f ) = dom( f r ) if g is somewheredefined. The fact that we use G¨odel numbers n, k ∈ N for ( g ։ f ) actually hassome reason: the crucial properties of this implication are computability theoreticones (see Proposition 7.11) and do not relativize to a topological version in anobvious way. However, the fact that we use G¨odel numbers makes the domain of( g ։ f ) relatively complicated. If g is somewhere defined, thendom( g ։ f ) = { p ∈ dom( f r ) : ( ∃ q ∈ dom( g r )) q ≤ T p } . For pointed g (that have a computable point in the domain) the domain is morenatural and we obtain dom( g ։ f ) = dom( f r ). The following facts were proved in[8, Corollaries 18 and 25, Theorem 24, Proposition 31]. Fact 7.2 (Compositional product and implication) . For all problems f and g in-cluding ∞ : (1) f ⋆ g ≡ W max ≤ W { f ◦ g : f ≤ W f, g ≤ W g } , (2) ( g → f ) ≡ W min ≤ W { h : f ≤ W g ⋆ h } , (3) ( g → f ) ≤ W h ⇐⇒ f ≤ W g ⋆ h , (4) ⋆ is monotone with respect to ≤ W in both components, (5) → is monotone with respect to ≤ W in the second component and antitonein the first component. We note that for (3) to be correct in the case of dom( g ) = ∅ and dom( f ) = ∅ ,we actually use ( g → f ) = ∞ and f ⋆ ∞ = ∞ ⋆ f = ∞ .By W we denote the class of Weihrauch degrees including ∞ . We extend allthe algebraic operations to degrees in the usual way without introducing a newnotation. It is known that the underlying structure is a lattice [8] and togetherwith Fact 7.2 (3) we obtain the following conclusion. Corollary 7.3 (Weihrauch algebra) . ( W , ≤ W , ⊓ , ⊔ , ⋆, → , , , ∞ ) is a deductiveWeihrauch algebra that is not commutative. EIHRAUCH GOES BROUWERIAN 27
For instance lim ⋆ WKL ≡ W lim < W WKL ⋆ lim and hence ⋆ is clearly not commu-tative.We can interpret ( f ։ ∞ ) = ( f → ∞ ) as negation operation in the Weihrauchlattice and we formally define negation correspondingly. Definition 7.4 (Negation) . For every problem f we define its negation ¬ f by ¬ f := ∞ for f = ∞ and ¬∞ := 0 (the nowhere defined problem 0 : ⊆ N N → N N ).It is then obvious that our negation behaves as in Jankov logic. Corollary 7.5 (Jankov rule) . ¬¬ f ⊓ ¬ f ≡ W is computable. We note that ¬ f ≡ W ¬ f ≤ W ¬ f , but equivalence does not hold as we obtain ¬∞ = 0 < W ≡ W ¬∞ . Here we are in particular interested in how the composi-tional product and the implications interact with completion in general. We showthat ⋆ co-preserves completion with respect to ≤ sW and → preserves completionwith respect to ≤ W . Proposition 7.6 (Completion and compositional products and implication) . Forall problems f, g including ∞ : (1) f ⋆ g ≤ sW f ⋆ g ≡ sW f ⋆ g . (2) ( g → f ) ≤ W ( g → f ) ≤ W ( g → f ) .In particular f ⋆ g is (strongly) complete, if f and g are so.Proof. (1) It is routine to check the claim for the special cases where the problem ∞ is involved. Otherwise, it suffices to consider f, g : ⊆ N N ⇒ N N and for such problemswe have f ⋆ g = h id × f i ◦ U ◦ h id × g i . Hence, f ⋆ g = h id × f i ◦ U ◦ h id × g i . Since U is computable and id is complete, this implies by Proposition 6.3 f ⋆ g ≤ W (id × f ) ⋆ (id × g ) ≤ W (id × f ) ⋆ (id × g ) ≤ W f ⋆ g. Since every compositional product is a cylinder by [8, Lemma 17], we even obtainthe strong Weihrauch reduction. The equivalence follows as in Proposition 6.3.(2) Since f ≤ W g ⋆ ( g → f ) by Fact 7.2 and completion is a closure operator, weobtain with (1) f ≤ W g ⋆ ( g → f ) ≤ W g ⋆ ( g → f ) . Hence Fact 7.2 implies ( g → f ) ≤ W ( g → f ), which in turn implies the statement,as completion is a closure operator. (cid:3) We note that neither of the reductions in (2) are equivalences in general, as thefollowing examples show:(1) ( ∞ → ∞ ) ≡ W < W ≡ W ( ∞ → ∞ ),(2) (0 → ≡ W < W ∞ ≡ W (0 → g → f ) does not need to be complete, even though g and f are.We now want to study the multiplicative implication ( g ։ f ) somewhat further.We first study its monotonicity properties. Proposition 7.7 (Monotonicity of multiplicative implication) . Let f i , g i be prob-lems for i ∈ { , } including ∞ . If f ≤ W f , g ≤ W g and g is pointed, then ( g ։ f ) ≤ W ( g ։ f ) .Proof. It is routine to check that the claim holds in those cases where the implica-tion takes the values 0 or ∞ . This includes the cases where ∞ is among f i , g i . Webreak the proof for the other cases into two manageable pieces, where we either fix f = f = f or g = g = g . It suffices to consider problems g i , f i , g, f : ⊆ N N ⇒ N N for i ∈ { , } .(1) Let g ≤ W g hold via computable H, K . We prove ( g ։ f ) ≤ W ( g ։ f ). Let us assume that g is pointed. This implies that g is also pointed and we alsoobtain dom( g ։ f ) = dom( g ։ f ) = dom( f ). By the smn-Theorem there arecomputable functions r, h : N → N such that • ϕ h h n,k i hh p, q i , t i = ϕ n h q, H h π ϕ k ( p ) , t ii , • ϕ r ( k ) ( p ) = K ◦ π ϕ k ( p )for all n, k ∈ N and p, q, t ∈ N N . Let p ∈ dom( g ։ f ) = dom( g ։ f ) = dom( f ).Let h n, k, q i ∈ ( g ։ f )( p ). Then we obtain ϕ h h n,k i hh p, q i , g ◦ ϕ r ( k ) ( p ) i = ϕ n h q, H h π ϕ k ( p ) , g ◦ ϕ r ( k ) ( p ) ii = ϕ n h q, H h π ϕ k ( p ) , g ◦ K ◦ π ϕ k ( p ) ii = ϕ n h q, H h id , g ◦ K i ◦ ϕ k ( p ) i⊆ ϕ n h q, g ◦ ϕ k ( p ) i ⊆ f ( p ) . This means h h h n, k i , r ( k ) , h p, q ii ∈ ( g ։ f )( p ). Since the function H ′ with H ′ h p, h n, k, q ii := h h h n, k i , r ( k ) , h p, q ii is computable, we obtain the desired con-clusion ( g ։ f ) ≤ W ( g ։ f ).(2) Let now f ≤ W f hold via computable functions H, K . We prove that we ob-tain ( g ։ f ) ≤ W ( g ։ f ). By the smn-Theorem there are computable functions r, h : N → N such that • ϕ h h n,k i hh p, q i , t i = H h p, ϕ n h q, t ii , • ϕ r ( k ) ( p ) = ϕ k ◦ K ( p )for all n, k ∈ N and p, q, t ∈ N N . Since g is pointed, we have dom( g ։ f i ) = dom( f i )for i ∈ { , } . Let p ∈ dom( g ։ f ). Then K ( p ) ∈ dom( g ։ f ). Let h n, k, q i ∈ ( g ։ f ) K ( p ). This means that we have ∅ 6 = ϕ n h q, gϕ k K ( p ) i ⊆ f K ( p ). Then weobtain ∅ 6 = ϕ h h n,k i hh p, q i , gϕ r ( k ) ( p ) i = H h p, ϕ n h q, gϕ k K ( p ) ii ⊆ H h p, f K ( p ) i ⊆ f ( p ) , i.e., h h h n, k i , r ( k ) , h p, q ii ∈ ( g ։ f )( p ). This proves ( g ։ f ) ≤ W ( g ։ f ). (cid:3) The pointedness assumption is not necessary when we deal with total Weihrauchreducibility. Hence, analogously to the proof of Proposition 2.2 we can obtain thefollowing conclusion.
Corollary 7.8 (Monotonicity of multiplicative implication) . ։ is monotone in thesecond argument and antitone in the first argument with respect to ≤ tW . Now we would like to have an analog of Fact 7.2 (3) for ։ . Unfortunately, thisis not possible, but we can say at least the following. Proposition 7.9 (Multiplicative implication) . For all problems f, g including ∞ : (1) f ≤ W g × h = ⇒ ( g ։ f ) ≤ W h , (2) ( g ։ f ) ≤ W h = ⇒ f ≤ W g ⋆ h , provided that g is pointed, (3) ( g → f ) ≤ W ( g ։ f ) , provided that g is pointed.Proof. It is routine to check that the claim holds in those cases where the impli-cation takes the values 0 or ∞ . This includes the cases where ∞ is among f, g, h .Otherwise, it suffices to consider problems f, g, h : ⊆ N N ⇒ N N .(1) Let f ≤ W g × h be witnessed by computable functions H and K . Then thereare n, k ∈ N with ϕ n hh p, r i , s i = H h p, h s, r ii and ϕ k = π K . We need to prove( g ։ f ) ≤ W h . We define K ′ , H ′ : ⊆ N N → N N by K ′ := π K and H ′ h p, r i := h n, k, h p, r ii for all p, r ∈ N N and n, k ∈ N . Given an input p ∈ dom( g ։ f )we claim that H ′ h p, hK ′ ( p ) i ⊆ ( g ։ f )( p ), i.e., H ′ , K ′ witness ( g ։ f ) ≤ W h : if h n, k, h p, r ii ∈ H ′ h p, hK ′ ( p ) i , then r ∈ hπ K ( p ) and hence ϕ n hh p, r i , gϕ k ( p ) i ⊆ H h p, h gπ K ( p ) , hπ K ( p ) ii = H h p, h g × h i ◦ K ( p ) i ⊆ f ( p ) . EIHRAUCH GOES BROUWERIAN 29
This means h n, k, h p, r ii ∈ ( g ։ f )( p ), which proves the claim.(2) This follows from (3) together with Fact 7.2.(3) Given a p ∈ dom( g → f ) we can use ( g ։ f ) in order to determine a h n, k, q i ∈ ( g ։ f )( p ). Here we use that g is pointed and hence dom( f ։ g ) = dom( f → g ).We can then compute a t ∈ N N with Φ t ( r ) = ϕ n h q, r i for all h q, r i ∈ dom( ϕ n ). Weclaim that h t, ϕ k ( p ) i ∈ ( g → f )( p ):Φ t ◦ g ◦ ϕ k ( p ) = ϕ n h q, g ◦ ϕ k ( p ) i ⊆ f ( p ) . This proves the claim. (cid:3)
Again the pointedness assumptions can be removed when we deal with totalWeihrauch reducibility and the corresponding completions of operations. In thisway Proposition 7.9 shows that we have an instance of a commutative Weihrauchalgebra. We formulate this result together with the deductive Weihrauch algebrawhose existence follows from Fact 7.2 (3).
Corollary 7.10 (Weihrauch algebra of total Weihrauch degrees) . The total Weih-rauch degrees give rise to the following Weihrauch algebras: (1) ( W tW , ≤ tW , ⊓ , ⊔ , × , ։ , , , ∞ ) is a commutative Weihrauch algebra. (2) ( W tW , ≤ tW , ⊓ , ⊔ , ⋆, → , , , ∞ ) is a deductive Weihrauch algebra. It would be desirable to have an equivalence in Proposition 7.9 (1) instead ofjust an implication, which would mean that ։ is a co-residual operation of × inthe same way as → is a co-residual of ⋆ . However, in [8, Proposition 37] it wasproved that there is no such co-residual operation to × . The following result showsthat ։ has such a co-residual property at least restricted to special problems. Proposition 7.11 (Multiplicative deduction) . ( g ։ f ) ≤ W h = ⇒ f ≤ W b g × h forall problems f, g, h including ∞ , such that g is pointed.Proof. It is routine to check the claim for the special cases where the problem ∞ is involved. Otherwise, it suffices to consider problems f, g, h : ⊆ N N ⇒ N N .Let g be pointed and let ( g ։ f ) ≤ W h hold via computable H, K . Then givena point p ∈ dom( f ) = dom( g ։ f ) any h n, k, q i ∈ H h p, hK ( p ) i satisfies ∅ 6 = ϕ n h q, g ◦ ϕ k ( p ) i ⊆ f ( p ). Since N N has a precomplete representation δ N N , it followsthat there is a total computable universal function u : N N → N N with δ N N ◦ u h k, p i = δ N N ( ϕ k ( p )+1) = ϕ k ( p ) for all k ∈ N and p ∈ dom( ϕ k ). We define a total computablefunction K ′ ( p ) := hh u h , p i , u h , p i , u h , p i , ... i , K ( p ) i and a computable function H ′ h p, hh q , q , q , ... i , r ii := ϕ n h q, q k − i where h n, k, q i = H h p, r i . Whenever G isa realizer of b g , with respect to δ N NN , then we obtain H ′ h p, h G × h i ◦ K ′ ( p ) i = H ′ h p, h G h u h , p i , u h , p i , u h , p i , ... i , hK ( p ) ii⊆ [ { ϕ n h q, g ◦ ϕ k ( p ) i : h n, k, q i ∈ H h p, hK ( p ) i}⊆ f ( p ) , i.e., f ≤ W b g × h . (cid:3) The basic idea of the proof is that using the parallelization we can evaluate g on all possible inputs ϕ k ( p ) with G¨odel numbers k ∈ N and only after we learnthe result of h we know which of these values is actually needed. The completionguarantees that all these values actually exist.A similar idea as in the proof of Proposition 7.11 has been independently usedby Neumann and Pauly [18, Proposition 31] to prove the following result, which werephrase in terms of our terminology. The notion of precompleteness used by Neumann and Pauly is not the usual one; what isrequired is rather a uniform version of completeness, which is satisfied by our completion g . Proposition 7.12 (Neumann and Pauly 2018) . g ⋆ h ≤ W b g × h for all problems g and h : ⊆ X ⇒ N . This result yields a similar transition from g ⋆ h to b g × h as the one that happensfrom Proposition 7.9 to 7.11, except that we do not need problems h with naturalnumber output for the latter transition. We obtain the following obvious corollaryof Proposition 7.11. Corollary 7.13 (Multiplicative deduction) . ( g ։ f ) ≤ W h ⇐⇒ f ≤ W g × h forall problems f, g, h including ∞ and such that g is parallelizable and complete. This is the key observation that is used in the next section in order to showthat the parallelized total Weihrauch degrees form a Brouwer algebra. We notethat by [8, Proposition 37] it is known that there is no way to define ։ such thatthe statement in Corollary 7.13 holds for all problems g . This remains so, evenif we replace Weihrauch reductions ≤ W by total Weihrauch reductions ≤ tW andthe product × by its completion × , as a refined version of the argument from [8,Proposition 37] shows. Proposition 7.14.
The operation × is not co-residuated and ⋆ is not left co-residuated with respect to ≤ tW .Proof. We have(1) C N × C N ≤ W C N × ( C N ⊔ C N ),(2) C N × C N ≤ W C N × ( C N ⊔ C N ),(3) ( C N ⊓ C N ) ⋆ ( C N ⊔ C N ) ≤ W C N ⊔ ( C N ⋆ C N ),(4) C N × C N W C N ⊔ ( C N ⋆ C N ).While (1) and (2) are clear, it remains to justify (3) and (4). We obtain (3) since C N ⋆ C N ≡ W C N and by distributivity properties of ⋆ [8, Proposition 39]( C N ⊓ C N ) ⋆ ( C N ⊔ C N ) ≡ W (( C N ⊓ C N ) ⋆ C N ) ⊔ (( C N ⊓ C N ) ⋆ C N ) ≤ W ( C N ⊓ ( C N ⋆ C N )) ⊔ (( C N ⋆ C N ) ⊓ ( C N ⋆ C N )) ≤ W C N ⊔ ( C N ⋆ C N ) . Now we need to justify why (4) holds. Since C N is a fractal by [2, Corollary 5.6],[5, Fact 3.2] and C N is a fractal as proved in [4, Lemma 8.7], it follows that C N × C N is a fractal and hence join irreducible by [5, Proposition 2.6] . This means that C N × C N ≤ W C N ⊔ ( C N ⋆ C N ) would imply that C N × C N ≤ W C N or C N × C N ≤ W C N ⋆ C N holds. The latter is impossible, as C N has computable inputs without computablesolutions, while C N ⋆ C N has computable solutions for all inputs. The former isimpossible as even C N W C N .By Propositions 6.3 and 7.6 all degrees that appear in (1)–(4) are complete,as C N is complete by Corollary 6.4. Hence, all the statements (1)–(4) hold trueif we replace ≤ W by ≤ tW . Suppose now a binary operation (cid:3) would exist suchthat ( g (cid:3) f ) ≤ tW h ⇐⇒ f ≤ tW h × g holds for all problems f, g, h . We consider g :=( C N ⊔ C N ) and h := C N , h := C N and f := C N × C N . Then by (1)–(4) f ≤ tW h × g and f ≤ tW h × g , but f tW ( h ⊓ h ) ⋆g , which also implies f tW ( h ⊓ h ) × g Thissimultaneously shows that (cid:3) does not exist and also a corresponding operation for ⋆ does not exist. (cid:3) The Weihrauch algebra of total Weihrauch degrees fails in two different waysbeing a model of some intuitionistic linear logic. The multiplicative and composi-tional versions of the algebra both fail to be Troelstra algebras, the former is notdeductive, the latter is not commutative.
Corollary 7.15.
The Weihrauch algebras from Corollary 7.10 are not Troelstraalgebras, i.e.,
EIHRAUCH GOES BROUWERIAN 31 (1) ( W tW , ≤ tW , ⊓ , ⊔ , × , ։ , , , ∞ ) is not deductive, (2) ( W tW , ≤ tW , ⊓ , ⊔ , ⋆, → , , , ∞ ) is not commutative. The Brouwer Algebra of Parallelizable Total Degrees
In [3] we have already studied parallelized Weihrauch reducibility ≤ pW , which isthe reducibility that is generated by the closure operator of parallelization on ≤ W .Likewise we want to study parallelized total Weihrauch reducibility ≤ ptW . Definition 8.1 (Parallelized Weihrauch reducibility) . For problems f, g we write(1) f ≤ pW g : ⇐⇒ f ≤ W b g (parallelized Weihrauch reducibility)(2) f ≤ ptW g : ⇐⇒ f ≤ W b g (parallelized total Weihrauch reducibility)Analogously, we write ≡ pW and ≡ ptW for the corresponding equivalences.It is clear that ≤ pW and ≤ ptW are actually preorders by Propositions 2.2, ascompletion and parallelized completion are closure operators (the latter by Propo-sition 6.13). We note that we also have f ≤ ptW g ⇐⇒ f ≤ tW b g by Proposition 6.3.It is important to mention that the order in which we apply the closure operatorsmatters. While b g is always complete and parallelizable, b g is always complete, butnot necessarily parallelizable (see Lemma 6.14).For each operation (cid:3) ∈ {× , ⊔ , ⊞ , ⊓ , + , ⋆, → , ։ } we define its parallelized comple-tion b (cid:3) by f b (cid:3) g := b f (cid:3) b g . Since parallelized completion is a closure operator for ≤ W by Proposition 6.13, we straightforwardly obtain the following by Proposition 2.2. Corollary 8.2 (Monotonicity) . (1) ( f, g ) f b (cid:3) g for (cid:3) ∈ {× , ⊔ , ⊞ , ⊓ , + , ⋆ } is monotone with respect to ≤ ptW . (2) ( f, g ) f b (cid:3) g for (cid:3) ∈ {→ , ։ } is monotone with respect to ≤ ptW in thesecond argument and antitone in the first argument.Proof. The corresponding monotonicity properties with respect to ≤ W are knownby [6, Proposition 3.6], except for ։ : ։ is monotone with respect to ≤ tW byCorollary 7.8. Hence the claims follow from Proposition 2.2. (cid:3) An interesting property of parallelized (total) Weihrauch reducibility is thatsuprema and products are merged in a certain sense. We summarize some factsregarding preservation and co-preservation of parallelization that were proved in [3,Propositions 4.5, 4.8, 4.9] and [8, Propositions 41, 44].
Fact 8.3 (Parallelization and algebraic operations) . For all problems f, g including ∞ : (1) [ f × g ≡ sW b f × b g ≤ W [ f ⊔ g , (2) b f ⊔ b g ≤ W [ f ⊔ g ≤ W [ b f ⊔ b g , (3) [ f ⊓ g ≤ sW b f ⊓ b g ≡ sW [ b f ⊓ b g , (4) [ f ⋆ g ≤ sW b f ⋆ b g ≡ sW [ b f ⋆ b g , (5) [ f × g ≡ sW b f × b g ≡ W [ f ⊔ g ≡ W [ f ⊔ g . Hence, × and ⊔ are equivalent operations under parallelized total Weihrauchreducibility. This follows from Fact 8.3 and Proposition 2.5. Corollary 8.4 (Products and coproducts) . f × g ≡ ptW b f × b g ≡ ptW b f ⊔ b g ≡ ptW f ⊔ g for all problems f, g . By W ptW we denote the class of parallelized total Weihrauch degrees including ∞ . We use the same notation ≤ ptW for the order on degrees and we consider the operations to be extended to these degrees. In order to avoid too clumsy notationwe use the abbreviation ⇛ for c ։ in the following. We prove that the parallelizedtotal Weihrauch degrees form a Brouwer algebra. Theorem 8.5 (Brouwer algebra) . ( W ptW , ≤ ptW , b ⊓ , ⊔ , ⇛ , , ∞ ) is a Brouwer alge-bra.Proof. ( W ptW , ≤ ptW , b ⊓ , ⊔ ) is a lattice by Proposition 2.2 as parallelized completionis a closure operator. We obtain by Corollary 7.13 and Fact 8.3( g ⇛ f ) ≤ ptW h ⇐⇒ ( b g ։ b f ) ≤ W b h ⇐⇒ b f ≤ W b g × b h ⇐⇒ b f ≤ W [ g ⊔ h ⇐⇒ f ≤ ptW g ⊔ h. This proves the claim. (cid:3)
In [3] we have proved that the Medvedev lattice can be embedded into theparallelized Weihrauch lattice. This embedding can actually be extended to aBrouwer algebra embedding into the parallelized total Weihrauch lattice. We recallsome basic definitions for the Medvedev lattice [24]. Let
A, B ⊆ N N . Then A issaid to be Medvedev reducible to B , in symbols A ≤ M B , if there is a computablefunction F : ⊆ N N → N N such that B ⊆ dom( F ) and F ( B ) ⊆ A . We recall thedefinition of the algebraic operations of the Medvedev lattice:(1) A ⊗ B := 0 A ∪ B = h A ⊔ B i ,(2) A ⊕ B := h A × B i ,(3) B → A := {h n, q i ∈ N N : ( ∀ p ∈ B ) ϕ n h q, p i ∈ A } .By M we denote the set of Medvedev degrees. We identify degrees with their mem-bers and use the same notation for the algebraic operations on degrees. Medvedev [17]proved that ( M , ⊗ , ⊕ , → , N N , ∅ ) is a Brouwer algebra (see [24, Theorem 9.1]). In[3] we have considered the constant problems c A : N N ⇒ N N , p A for every non-empty A ⊆ N N and c ∅ = ∞ . The following facts were proved in [3,Theorem 5.1]. Fact 8.6 (Medvedev embedding) . For all
A, B ⊆ N N : (1) A ≤ M B ⇐⇒ c A ≤ W c B , (2) c A ⊗ B ≡ sW c A ⊓ c B , (3) c A ⊕ B ≡ sW c A × c B ≡ c A ⋆ c B . The equivalence c A × c B ≡ W c A ⋆ c B , was not proved in the references, but itis easy to see. For one, f × g ≤ W f ⋆ g holds in general and on the other hand, c A ⋆ c B = h id × c A i ◦ U ◦ h id × c B i ≤ W c A × c B , as the output of c A does not dependon the input. Here we add the observation that also the implication is preserved.In fact, since the product and the compositional product for problems of the form c A coincide, also the multiplicative and compositional implications coincide. Lemma 8.7 (Medvedev implication) . c B → A ≡ W ( c B ։ c A ) ≡ W ( c B → c A ) for all A, B ⊆ N N .Proof. It is routine to check the special cases of problems that involve
A, B ∈{∅ , N N } . Since the Medvedev lattice is a Brouwer algebra by [24, Theorem 9.1], wehave A ≤ M B ⊕ ( B → A ). With the help of Proposition 7.9 and Fact 8.6 we obtain A ≤ M B ⊕ ( B → A ) = ⇒ c A ≤ W c B ⊕ ( B → A ) ≡ W c B × c B → A = ⇒ ( c B ։ c A ) ≤ W c B → A . EIHRAUCH GOES BROUWERIAN 33
We can also prove c B → A ≤ W ( c B ։ c A ). Given a p ∈ N N we obtain h n, k, q i ∈ ( c B ։ c A )( p ) ⇐⇒ ∅ 6 = ϕ n h q, c B ◦ ϕ k ( p ) i ⊆ c A ( p ) ⇐⇒ h n, q i ∈ c B → A ( p ) . Hence, c B → A ≤ W ( c B ։ c A ) follows. We have ( c B → c A ) ≤ W ( c B ։ c A ) by Propo-sition 7.9. We also obtain ( c B ։ c A ) ≤ W ( c B → c A ). To this end, let h be aproblem such that c A ≤ W c B ⋆ h . Like above we obtain c B ⋆ h ≤ W c B × h , sincethe output of c B does not depend on its input. That means c A ≤ W c B × h andhence ( c B ։ c A ) ≤ W h by Proposition 7.9. However, if g is a problem such that c A ≤ W c B ⋆ h implies g ≤ W h for every h , then g ≤ W ( c B → c A ) follows. Hence,( c B ։ c A ) ≤ W ( c B → c A ). (cid:3) Hence the map A c A is a lattice embedding from the Medvedev lattice into theWeihrauch lattice that also preserves the corresponding implications (even thoughthe Weihrauch lattice itself is not a Brouwer algebra). It is easy to see that everyWeihrauch degree of the form c A with A ⊆ N N is parallelizable and complete, i.e., c c A ≡ W c A . Hence the above embedding is also an embedding into the parallelizedtotal Weihrauch degrees. We note that c ∅ = ∞ and c N N ≡ W
1. Hence, we obtain a
Brouwer algebra embedding , i.e., a lattice embedding that preserves the implicationand the lower and upper bound.
Theorem 8.8 (Embedding of the Medvedev lattice) . c : M → W ptW , A c A isa Brouwer algebra embedding. The fact that the parallelized total Weihrauch lattice is a Brouwer algebra im-plies that it is a model for some intermediate logic (i.e., some propositional logicintermediate between intuitionistic logic and classical logic). The existence of anembedding from the Medvedev lattice into the parallelized total Weihrauch lat-tice allows us to conclude that the logic of the parallelized total Weihrauch latticeis
Jankov logic , i.e., the deductive closure of intuitionistic logic together with the weak principle of excluded middle ¬¬ A ∨ ¬ A . We follow Sorbi [23, 24] for a formaldefinition of the theory of a Brouwer algebra. Let Form denote the set of wellformed propositional formulas. Then we call a map v : Form → W ptW valuation ifit satisfies the following for all A, B ∈ Form:(1) v ( A ∨ B ) = v ( A ) b ⊓ v ( B ),(2) v ( A ∧ B ) = v ( A ) ⊔ v ( B ),(3) v ( A → B ) = ( v ( A ) ⇛ v ( B )),(4) v ( ¬ A ) = ( v ( A ) ⇛ ∞ ).We write W ptW (cid:15) A if v ( A ) = 1 for all valuations v . Then the set of formulasTh( W ptW ) := { A ∈ Form : W ptW (cid:15) A } is called the theory of W ptW . It was provedby Medvedev [16] (see [24, Corollary 6.4]) that the theory of the Brouwer algebra M is Jankov logic. We obtain the same result for our Brouwer algebra W ptW .For one, it contains Jankov logic by Corollary 7.5. On the other hand, it cannotvalidate any additional propositional formulas as the Medvedev Brouwer algebra isembeddable by Theorem 8.8. Corollary 8.9 (Theory of the parallelized complete Weihrauch degrees) . The the-ory of the Brouwer algebra W ptW is Jankov logic. We note that Higuchi and Pauly proved [13, Theorems 4.1, 4.2] that neither theWeihrauch lattice by itself nor the parallelized Weihrauch lattice (restricted to thepointed problems) is a Brouwer algebra. Hence, the closure operator of completionseems to be essential in order to obtain a Brouwer algebra.In view of Corollary 7.10 one could obtain a way to transform the total Weihrauchlattice into a Troelstra algebra by restricting it to a linear fragment. We call
L ⊆ W ptW linear if f × g ≡ tW f ∗ g holds for all f, g ∈ L . If there would be anylinear sublattice of interest that also preserves the monoid structure, then thatwould be a potential candidate for a Troelstra algebra. We note that the constantmulti-valued problems c A used for the embedding of the Medvedev lattice forma linear subset of the total Weihrauch degrees by Fact 8.6, however, this is nota sublattice and leads directly to a Brouwer algebra, i.e., a trivial example of aTroelstra algebra. WKL ′′ ≡ ptW RT k +2 ≡ ptW LLPO ′′ lim ′ ≡ ptW LPO ′ WKL ′ ≡ ptW KL ≡ ptW BWT R ≡ ptW RT k +2 ≡ ptW LLPO ′ lim ≡ ptW SORT ≡ ptW C R ≡ ptW C N ≡ ptW LPODNC DNC DNC N PA MLR COH GENNONWKL ≡ ptW C N ≡ ptW WWKL ≡ ptW IVT ≡ ptW K N ≡ ptW LLPO
Figure 3.
Problems in the parallelized total Weihrauch lattice W ptW Conclusion
We have proved that the Weihrauch lattice can be transformed into a Brouweralgebra by completion followed by parallelization. It would be desirable to under-stand the structure of this Brouwer algebra somewhat better. Is it isomorphic tothe Medvedev Brouwer algebra? Presumably not, as the Medvedev algebra con-siders only problems that are independent of the input. However, we need morestructural information on the lattices and algebras in order to prove such properties.The Medvedev lattice has, for instance, a second smallest degree, called 0 ′ , whichconsists of all non-computable p ∈ N N . Is there such a second smallest degree in theparallelized total Weihrauch lattice? Or is the structure dense? We do not evenknow the answer to this question for the ordinary Weihrauch lattice or its totalvariant. What we can say, though, is that the parallelized total Weihrauch latticeis still inhabited by a variety of interesting problems. The diagram in Figure 3shows a number of problems (that are taken without further explanation from [7]and [9]), and that inhabit W ptW . Even though a lot of problems that are normallyseparated in the Weihrauch lattice are identified in W ptW , the structure is still richand non-linear. Acknowledgments
We would like to thank Paulo Oliva for discussions of models of intuitionisticlinear logic at the Logic Colloquium 2018 in Udine that have helped us to identifythe relevance of Troelstra and Weihrauch algebras.
EIHRAUCH GOES BROUWERIAN 35
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Faculty of Computer Science, Universit¨at der Bundeswehr M¨unchen, Germany andDepartment of Mathematics & Applied Mathematics, University of Cape Town, SouthAfrica E-mail address : [email protected]
Dipartimento di Filosofia e Comunicazione, Universit`a di Bologna, Italy
E-mail address : [email protected]