Completion of choice

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Abstract

We systematically study the completion of choice problems in the Weihrauch lattice. Choice problems play a pivotal rôle in Weihrauch complexity. For one, they can be used as landmarks that characterize important equivalences classes in the Weihrauch lattice. On the other hand, choice problems also characterize several natural classes of computable problems, such as finite mind change computable problems, non-deterministically computable problems, Las Vegas computable problems and effectively Borel measurable functions. The closure operator of completion generates the concept of total Weihrauch reducibility, which is a variant of Weihrauch reducibility with total realizers. Logically speaking, the completion of a problem is a version of the problem that is independent of its premise. Hence, studying the completion of choice problems allows us to study simultaneously choice problems in the total Weihrauch lattice, as well as the question which choice problems can be made independent of their premises in the usual Weihrauch lattice. The outcome shows that many important choice problems that are related to compact spaces are complete, whereas choice problems for unbounded spaces or closed sets of positive measure are typically not complete.

Introduction

Choice problems play a crucial rôle in Weihrauch complexity. A recent survey on the field can be found in [9]. A choice problem is a problem of the logical form( closed AX)(AD(xX)xA). Here X is typically a computable metric space, the closed set AX is typically given by negative information in order to make the statement non-trivial, and the premise D could be a property such as non-emptiness, sometimes combined with additional properties, such as positive measure, connectedness, etc. The multi-valued Skolem function of such a choice problem is a function of the formCX:A(X)X,AA, where A(X) denotes the space of closed subsets of X with respect to negative information and dom(CX)={A:A} is a particular D. If the domain is further restricted to sets of positive measure or connected sets, then we denote the problem by PCX and CCX, respectively. By KX we denote compact choice that considers compact sets with respect to negative information. Many basic systems from reverse mathematics [22] have certain choice problems as uniform counterparts. Also some classes of problems that are computable in a certain sense can be characterized as cones below certain choice problems in the Weihrauch lattice. The Table 1 gives a survey on such correspondences (see [9] for further details).

We assume that the reader is familiar with Weihrauch reducibility W. The statement fWg roughly speaking expresses that the problem f can be computably reduced to the problem g in the sense that each realizer of g computes a realizer of f in a uniform way (see [9]). In [6] we have introduced the closure operator of completion ff that induces total Weihrauch reducibility tW byftWg:fWg. Total Weihrauch reducibility tW is a variant of the usual concept of Weihrauch reducibility W and it can be directly defined using total realizers [6]. Our main motivation for studying this total variant of Weihrauch reducibility and the completion operator ff is that one can obtain a Brouwer algebra in this way. More precisely, if the completion operator is combined with the closure operator ffˆ of parallelization, then the resulting lattice structure is a Brouwer algebra, i.e., a model of some intermediate logic that, like in the case of the Medvedev lattice, turns out to be Jankov logic [6].

Formally, the completion f:XY of a problem f:XY is defined byf(x):={f(x)if xdom(f)Yotherwise, i.e., by a totalization of f on the completions X,Y of the corresponding types.3 Logically, the completion f of a problem f can be seen as a way to make f independent of its premise. For choice problems this means to consider them in the form(AA(X))(xX)(ADxA), where the existence of x is now independent of the premise AD. If we use intuitionistic logic, then we cannot just export the quantifier without changing the meaning of the formula. Likewise, the computational content of the formula with the exported quantifier is different from the original one.

The main question that we study in this article is: which choice problems CX and their variants are complete, i.e., when do we obtain CXWCX?

Some examples of complete and incomplete choice problems are the following:

  • Complete choice problems: Cn for n1, KN, C2N, CC[0,1].

  • Incomplete choice problems: C0,CN,CR,CNN,PC2N,PCC[0,1].

From the perspective of a complete problem its lower cones in the Weihrauch lattice and in the total Weihrauch lattice coincide. Together with the notion of completeness we also study the notion of co-completeness. For co-complete problems the upper cones in the two lattices coincide. Since many important problems are either complete or co-complete or even both, we obtain very similar reducibility relations between important choice problems in the usual and the total Weihrauch lattice. The incomplete problems in Fig. 1 are all shown in dashed boxes together with their completions. If we disregard the completions, then all relations between any two problems shown in Fig. 1 are the same for ordinary Weihrauch reducibility W and its total variant tW, except those that involve the weak Bolzano-Weierstraß theorem WBWT2 on the space {0,1}.

However, a certain amount of expressiveness is lost by the transition from the ordinary Weihrauch lattice to its total variant:

  • (1)

    Finite mind change computable problems and Las Vegas computable problems can be characterized as lower cones with Weihrauch reducibility W, but not with total Weihrauch reducibility tW.

  • (2)

    Low problems can be characterized as lower cones with strong Weihrauch reducibility sW, but not with strong total Weihrauch reducibility stW.

  • (3)

    Limit computable problems and non-deterministically computable problems can be characterized as lower cones for all mentioned reducibilities.

We provide a list of some references for some crucial reductions and separations given in Fig. 1. Several further references can be found in the survey [9].

  • (a)

    In Corollary 5.3 we prove that in general CXsWCXsWTCXsWTCX holds. In Theorem 11.6 we provide the necessary separations for X=NN and in Corollary 8.3 the corresponding separations for X=N. The reduction limWCNN follows, for instance, from [2, Theorem 7.7]. The reduction TCNWKNKN follows from Corollary 8.10 and Proposition 8.13.

  • (b)

    Neumann and Pauly introduced SORT and proved CN<WSORT<Wlim [21, Proposition 24]. This is improved by Corollary 8.16, which yields CNWSORT. The reduction SORTWKNKN follows from Corollary 8.10 and Proposition 8.13. The reduction CC[0,1]WSORT was proved in [12, Proposition 16].

  • (c)

    The reduction CRWL was proved in [2, Corollary 4.9, Theorem 8.7]. The separation of L and L and, in fact, several other separations in the diagram follow, since WBWT2WL by Proposition 9.5 and WBWT2WCN by Proposition 8.8. The problem WBWT2 was introduced in [11]. By [8, Corollary 11.11] we have BWT2sWLLPO and hence WBWT2WBWT2WLPO.

In the following section 2 we continue the study of precomplete representations that was started in [6]. We characterize represented spaces that admit total precomplete representations as spaces that allow computable multi-valued retractions from their completions onto themselves. We call such spaces multi-retraceable. In section 3 we briefly recall some basic facts about total Weihrauch reducibility that were provided in [6]. In section 4 we continue the study of completion of problems that was started in [6] and we introduce the notion of co-completeness and co-totality. In particular, we introduce a criterion that is sufficient to guarantee co-completeness and co-totality for jumps of non-constant discrete functions. In section 5 we start to study the main theme of this article, namely the completion of choice problems. We formulate a number of results that hold for general choice problems and in section 6 we focus on choice on compact spaces. While choice on Cantor space, on non-empty finite spaces and connected choice on the unit interval are complete, most other choice principles that we study are incomplete. In section 7 we establish incompleteness of choice problems for sets of positive measure and in section 8 we establish incompleteness of choice for natural numbers. In section 9 we briefly discuss lowness and we show that the low problem L:=J1lim is not complete. Finally, in section 10 we discuss variants of choice on Euclidean space and in section 11 choice on Baire space.

Section snippets

Precompleteness, completeness and retraceability

We recall that a represented space (X,δ) is a set X together with a surjective (partial) map δ:NNX, called the representation of X. For the purposes of our topic so-called precomplete representations are important. They were introduced by Kreitz and Weihrauch [19] following the concept of a precomplete numbering introduced by Eršov [17]. We recall some results on precomplete representations from [6].

Definition 2.1 Precompleteness

A representation δ:NNX is called precomplete, if for any computable F:NNNN there exists a

Total Weihrauch reducibility

In this section we are going to recall the definition of Weihrauch reducibility and of total Weihrauch reducibility, which was introduced in [6]. We write Ftf, if F is a total realizer of f. We now recall the definition of ordinary and strong Weihrauch reducibility on problems f,g, which is denoted by fWg and fsWg, respectively, and we recall the two new concepts of total Weihrauch reducibility and strong total Weihrauch reducibility, which are denoted by ftWg and fstWg, respectively.

Definition 3.1 Weihrauch reducibility

Let f:

Completion, totalization and co-completion

In this section we recall the definition of the closure operation ff on (strong) Weihrauch reducibility that was introduced in [6] and we prove some further properties of it. For the definition of the completion f we use the completion X of a represented space.

Definition 4.1 Completion

Let f:XY be a problem. We define the completion of f byf:XY,x{f(x)if xdom(f)Yotherwise

We note that the completion f is always pointed, i.e., it has a computable point in its domain. This is because X is always computable

Choice problems

Choice principles form the backbone of the Weihrauch lattice, and many other problems can be classified by proving their equivalence to a suitable choice problem [9]. Hence, it is important to understand which choice principles are complete in order to see how the picture for the total Weihrauch lattice changes compared to the partial version.

In order to recall the definition of choice we need to introduce the set A(X) of closed subsets of a topological space X. Typically, we will consider

Choice on compact spaces

Even though the assumptions of Lemma 4.10 (2) are not satisfied in many cases, we can often even prove TCXsWCX using a computable multi-valued retraction r:A(X)dom(CX). We illustrate this with choice on Cantor space 2N.

Proposition 6.1 Choice on Cantor space

C2NsWC2NsWTC2N.

Proof

We consider X=2N. By Corollary 5.3 it suffices to prove TCXsWCX. Firstly, we note that the setB:={k,n0,...,nkN:Bn0...Bnk=X} is computable, as we can easily check whether there is a point xX that is not covered by Bn0...Bnk. Hence, given a list pNN

Positive choice

In this section we want to study PCX, which is CX restricted to sets of positive measures. This requires that we have a fixed Borel measure on X and we are mostly interested in the cases X=2N, X=[0,1] and X=R. In the first case we use the uniform measure μ and in the second and third case the Lebesgue measure λ. It is known that PC2NsWPC[0,1]sWWWKL (see [7, Proposition 8.2] for these results and the definition of WWKL). By PCCX we denote the restriction of PCX to connected sets. The following

Choice on the natural numbers

In this section we study choice on natural numbers. Since limNsWCN, we get the following conclusion from Corollary 4.22.

Corollary 8.1

CN is co-complete and co-total.

On the other hand, CN is not complete. Since it is known by [8, Theorem 7.12] that fWCN holds if and only if f is computable with finitely many mind changes, it suffices to show that CN is not computable with finitely many mind changes in order to conclude that CN<WCN holds.

Proposition 8.2 Choice on natural numbers

CN is limit computable and not computable with finitely many mind

Lowness

Proposition 2.10 can also be used to prove that CN is not low. We recall that a problem f:XY is called low, if it has a realizer of the form F=LG with some computable G:NNNN and L:=J1lim. Lowness was studied, for instance, in [2], [1]. By [2, Theorem 8.10] f is low if and only if fsWL. Likewise, f is called low2, if fsWL2, where L2:=J1J1limlim.

Corollary 9.1

CN is low2 but not low.

Proof

We first prove that CN is not low. By Proposition 2.10 there is a retraction r:NN that is computable with

Choice on Euclidean space

By Lemma 8.7 CN is a total fractal. This fact allows us to give a very simple proof of the following result.

Proposition 10.1 Choice on Euclidean space

CR<WCR and PCR<WPCR.

Proof

Let us assume that CRWCR. Then we obtainCNWCRWCRWC2NCN, where the first reduction holds since CNWCR and completion is a closure operator and the last mentioned equivalence is known [8, Example 4.4 (2)]. By the choice elimination principle [20, Theorem 2.4] it follows that CNWCNWC2N, which is known to be false [4, Corollary 4.2]. PCR<WPCR can be proved

Choice on Baire space

Next we want to study the choice problem on Baire space CNN. For this purpose we consider the wellfounded tree problem, i.e., the characteristic function of the singleton with the empty set as its member:WFT:A(NN){0,1},A{1if A=0otherwise.

By [3, Theorem 5.2] the set {}A(NN) is equivalent to the set of wellfounded trees that is known to be Π11–complete. By WFTS:A(NN)S we denote the wellfounded tree problem with target space S.

We start with proving that for every closed set ANN that is

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    1

    Vasco Brattka has been supported by the National Research Foundation of South Africa (Grant Number 115269).

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    Guido Gherardi has been supported by MIUR-PRIN project Analysis of Program Analyses (ASPRA, ID 201784YSZ5_004).

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