Completion of choice
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 Here X is typically a computable metric space, the closed set 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 form where denotes the space of closed subsets of X with respect to negative information and is a particular D. If the domain is further restricted to sets of positive measure or connected sets, then we denote the problem by and , respectively. By 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 . The statement 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 that induces total Weihrauch reducibility by Total Weihrauch reducibility is a variant of the usual concept of Weihrauch reducibility 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 is that one can obtain a Brouwer algebra in this way. More precisely, if the completion operator is combined with the closure operator 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 of a problem is defined by i.e., by a totalization of f on the completions of the corresponding types.3 Logically, the completion 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 where the existence of x is now independent of the premise . 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 and their variants are complete, i.e., when do we obtain ?
Some examples of complete and incomplete choice problems are the following:
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Complete choice problems: for , , , .
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Incomplete choice problems: .
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 and its total variant , except those that involve the weak Bolzano-Weierstraß theorem on the space .
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 , but not with total Weihrauch reducibility .
- (2)
Low problems can be characterized as lower cones with strong Weihrauch reducibility , but not with strong total Weihrauch reducibility .
- (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 holds. In Theorem 11.6 we provide the necessary separations for and in Corollary 8.3 the corresponding separations for . The reduction follows, for instance, from [2, Theorem 7.7]. The reduction follows from Corollary 8.10 and Proposition 8.13.
- (b)
Neumann and Pauly introduced and proved [21, Proposition 24]. This is improved by Corollary 8.16, which yields . The reduction follows from Corollary 8.10 and Proposition 8.13. The reduction was proved in [12, Proposition 16].
- (c)
The reduction was proved in [2, Corollary 4.9, Theorem 8.7]. The separation of and and, in fact, several other separations in the diagram follow, since by Proposition 9.5 and by Proposition 8.8. The problem was introduced in [11]. By [8, Corollary 11.11] we have and hence .
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 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 is a set X together with a surjective (partial) map , 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 is called precomplete, if for any computable 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 , if F is a total realizer of f. We now recall the definition of ordinary and strong Weihrauch reducibility on problems , which is denoted by and , respectively, and we recall the two new concepts of total Weihrauch reducibility and strong total Weihrauch reducibility, which are denoted by and , respectively.
Definition 3.1 Weihrauch reducibility Let
Completion, totalization and co-completion
In this section we recall the definition of the closure operation on (strong) Weihrauch reducibility that was introduced in [6] and we prove some further properties of it. For the definition of the completion we use the completion of a represented space.
Definition 4.1 Completion Let be a problem. We define the completion of f by
We note that the completion is always pointed, i.e., it has a computable point in its domain. This is because 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 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 using a computable multi-valued retraction . We illustrate this with choice on Cantor space .
Proposition 6.1 Choice on Cantor space . Proof We consider . By Corollary 5.3 it suffices to prove . Firstly, we note that the set is computable, as we can easily check whether there is a point that is not covered by . Hence, given a list
Positive choice
In this section we want to study , which is 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 , and . 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 (see [7, Proposition 8.2] for these results and the definition of ). By we denote the restriction of to connected sets. The following
Choice on the natural numbers
In this section we study choice on natural numbers. Since , we get the following conclusion from Corollary 4.22.
Corollary 8.1 is co-complete and co-total.
On the other hand, is not complete. Since it is known by [8, Theorem 7.12] that holds if and only if f is computable with finitely many mind changes, it suffices to show that is not computable with finitely many mind changes in order to conclude that holds.
Proposition 8.2 Choice on natural numbers is limit computable and not computable with finitely many mind
Lowness
Proposition 2.10 can also be used to prove that is not low. We recall that a problem is called low, if it has a realizer of the form with some computable and . Lowness was studied, for instance, in [2], [1]. By [2, Theorem 8.10] f is low if and only if . Likewise, f is called low2, if , where .
Corollary 9.1 is low2 but not low. Proof We first prove that is not low. By Proposition 2.10 there is a retraction that is computable with
Choice on Euclidean space
By Lemma 8.7 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 and . Proof Let us assume that . Then we obtain where the first reduction holds since 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 , which is known to be false [4, Corollary 4.2]. can be proved
Choice on Baire space
Next we want to study the choice problem on Baire space . For this purpose we consider the wellfounded tree problem, i.e., the characteristic function of the singleton with the empty set as its member:
By [3, Theorem 5.2] the set is equivalent to the set of wellfounded trees that is known to be –complete. By we denote the wellfounded tree problem with target space .
We start with proving that for every closed set that is
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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).