How incomputable is the separable Hahn-Banach theorem?

How incomputable
is the separable Hahn-Banach theorem?

Guido Gherardi Dipartimento di Filosofia, Università di Bologna, via Zamboni 38, 40126 Bologna, Italy guido.gherardi@unibo.it  and  Alberto Marcone Dipartimento di Matematica e Informatica, Università di Udine, viale delle Scienze 206, 33100 Udine, Italy alberto.marcone@dimi.uniud.it
Abstract.

We determine the computational complexity of the Hahn-Banach Extension Theorem. To do so, we investigate some basic connections between reverse mathematics and computable analysis. In particular, we use Weak König’s Lemma within the framework of computable analysis to classify incomputable functions of low complexity. By defining the multi-valued function and a natural notion of reducibility for multi-valued functions, we obtain a computational counterpart of the subsystem of second order arithmetic . We study analogies and differences between and the class of -computable multi-valued functions. Extending work of Brattka, we show that a natural multi-valued function associated with the Hahn-Banach Extension Theorem is -complete.

Key words and phrases:
Computable Analysis, Reverse Mathematics, Weak König’s Lemma, Hahn-Banach Extension Theorem, multi-valued functions
2000 Mathematics Subject Classification:
Primary 03F60; Secondary 03B30; 46A22; 46S30
We would like to thank Vasco Brattka for profitable discussion on the subject.
An extended abstract of this paper appears in the Proceedings of the Fifth International Conference on Computability and Complexity in Analysis (CCA 2008), Electronic Notes in Theoretical Computer Science 221, 85–102, 2008, doi:10.1016/j.entcs.2008.12.009

1. Introduction

In this paper we tackle a problem in computable analysis ([Wei00] is the main reference in the area) borrowing ideas and proof techniques from the research program of reverse mathematics ([Sim99] is the standard reference). The two subjects share the goal of classifying complexity of mathematical practice. Reverse mathematics was started by Harvey Friedman in the 1970’s ([Fri75]). It adopts a proof-theoretic viewpoint (although techniques from computability theory are increasingly important in the subject) and investigates which axioms are needed to prove a given theorem (see Section 3 for details). On the other hand, computable analysis extends to computable separable metric spaces the notions of computability and incomputability by combining concepts of approximation and of computation. To this end the representation approach (Type-2 Theory of Effectivity, TTE), introduced for real functions by Grzegorczyk and Lacombe ([Grz55, Lac55]), is used. This approach provides a realistic and flexible model of computation.

One of the goals of computable analysis is to study and compare degrees of incomputability of (possibly multi-valued) functions between separable metric spaces. Multi-valued functions are the appropriate way of dealing with situations where problems have non-unique solutions, and have been studied in computable analysis since [Wei00]. In this paper we introduce a notion of computable reducibility for multi-valued functions which generalizes at once both notions of reducibility for single-valued functions extensively studied by Brattka in [Bra05]. Let 111The notation means that is a multi-valued function with and . Following [Wei00, §1.4], we view a partial multi-valued function as a subset of . Then and, when , we have . and be two (partial) multi-valued functions, where are separable metric spaces. We say that is computably reducible to , and write , if there are computable multi-valued functions and such that (see Definition 4.1 below for the definition of composition of multi-valued functions) for all . We use and to denote the strict order and the equivalence relation defined in the obvious way.

In [Bra08] Brattka started the study of the separable Hahn-Banach Theorem from the viewpoint of computable analysis. Given a computable separable Banach space consider the multi-valued function mapping a closed linear subspace of and a bounded linear functional with to the set of all bounded linear functionals which extend and are such that . For many computable separable Banach spaces , it turns out that is incomputable. Brattka does not establish precisely the degree of incomputability of these functions, as he shows, in our notation, that for every . Here is a standard function considered in computable analysis, the first in a sequence of increasingly incomputable functions (see Definition 2.10 below).

We generalize Brattka’s approach and consider the following “global separable Hahn-Banach multi-valued function” :  takes as input a separable Banach space , a closed linear subspace and a bounded linear functional of norm , and gives as output the bounded linear functionals which extend and are such that .

Reverse mathematics suggests a plausible representative for the degree of incomputability of . To see this, recall that reverse mathematics singled out five subsystems of second order arithmetic: in order of increasing strength these are RCA, WKL, ACA, ATR and -CA. Most theorems of ordinary mathematics are either provable in the weak base system RCA or are equivalent, over RCA, to one of the other systems. Computable functions naturally correspond to RCA and is easy to see that (and indeed any with ) corresponds to ACA (the correspondence between a system and a function will be made precise in Section 5 below). Brown and Simpson ([BS86]) showed that, over the base theory RCA, the Hahn-Banach Theorem for separable Banach spaces is equivalent to WKL. Thus to define a representative for the incomputability degree of  we could look for a function in computable analysis corresponding to WKL.

We consider the multi-valued function defined on by

In other words, the domain of is the collection of pair of functions from the natural numbers into themselves (i.e. of elements of Baire space) with disjoint ranges, and, for any such pair , is the set of the characteristic functions of sets of natural numbers (i.e. elements of Cantor space) separating the range of and the range of . Thus  corresponds to a statement (strictly connected with -separation) which is well-known to be equivalent to WKL (see [Sim99, Lemma IV.4.4]).  is not computable and, using our definition of computable reducibility between multi-valued functions, we obtain, as expected, . We also show that , where  is the multi-valued function associating to an infinite subtree of the set of its infinite paths. Moreover we prove that , establishing the degree of incomputability of the separable Hahn-Banach Theorem.

The “reversal” in Brown and Simpson’s result (i.e. the proof that the separable Hahn-Banach Theorem implies WKL) is based on a construction due to Bishop, Metakides, Nerode and Shore ([MNS85]) and appears also in [Sim99, Theorem IV.9.4]. We exploit the ideas of this proof to show . The original proof by Brown and Simpson of the “forward direction” (showing that WKL proves the separable Hahn-Banach Theorem) has been simplified first by Shioji and Tanaka ([ST90], this is essentially the proof contained in [Sim99, §IV.9]) and then by Humphreys and Simpson ([HS99]). No details of these or other proofs of the Hahn-Banach Theorem are needed for showing . Brattka noticed the possibility of avoiding these details in [Bra08] and wrote: Surprisingly, the proof of this theorem does not require a constructivization of the classical proof but just an “external analysis”. We explain this fact by observing that the computable analyst is allowed to conduct an unbounded search for an object that is guaranteed to exist by (nonconstructive) mathematical knowledge, whereas the reverse mathematician has the burden of an existence proof with limited means. We give another instance of this phenomenon in Example 5.9 below.

Of course, each of the mathematical objects mentioned above needs some “coding” (in reverse mathematics jargon) or “representation” (using computable analysis terminology). In this respect the computable analysis and the reverse mathematics traditions have developed slightly different approaches to separable Banach spaces.

The plan of the paper is as follows. Sections 2 and 3 are brief introductions to computable analysis and reverse mathematics, respectively. The reader with some basic knowledge in one of these fields can safely skip the corresponding section and refer back to it when needed. Section 4 deals with multi-valued functions and computable reductions among them. In Section 5 we compare reverse mathematics and computable analysis. We show the similarities of the two approaches, but also note that results cannot be translated automatically in either direction. The multi-valued function  is studied in Section 6. Section 7 sets up the study of Banach spaces in computable analysis, while Section 8 contains the proof of .

1.1. Notation for sequences

We finish this introduction by establishing our notation for finite and infinite sequences of natural numbers.

Let , resp. , be the sets of all finite, resp. infinite, sequences of natural numbers. When we use to denote its length and, for , to denote the -th element in the sequence. Similarly, is defined for every when . Let be set of all with . We use to denote the empty sequence, i.e. the only element of , and to denote the infinite sequence which always takes value . When we write to mean that is an initial segment of . is the sequence obtained by concatenating after , and when , abbreviates , and abbreviates . When we write also and , which are the obvious elements of . If and we write for the sequence . If we let be such that and .

We define , , and as the subsets of , , and whose elements take values in .

We fix a bijection between  and and, as usual in the literature, we identify an element of  with the corresponding natural number. We assume that the maps , , , and are all computable.

Of course,  has a natural topology, namely the product topology starting from the discrete topology on . When we view  as a topological space we call it the Baire space. Similarly,  with the relative topology is the Cantor space.

2. Computable analysis

2.1. TTE computability

In contrast with the case of natural numbers, several nonequivalent approaches to computability theory for the reals have been proposed in the literature. We work in the framework of the so called Type-2 Theory of Effectivity (TTE), which finds a systematic foundation in [Wei00]. TTE extends the ordinary notion of Turing computability to second countable -topological spaces, and therefore deals with computability over the reals as a particular case within a more general theory.

The basic idea of TTE is that concrete computing machines do not manipulate directly abstract mathematical objects, but they perform computations on sequences of digits which are codings for such objects. In general, mathematical objects require an infinite amount of information to be completely described, and it is therefore natural to extend the ordinary theory of computation to infinite sequences. This extension does not compromise the concreteness of the model, since computations on infinite sequences have a very natural translation in terms of ordinary Turing computations on finite sequences (see [Wei00, Lemma 2.1.11]). The most important feature that differentiates TTE Turing machines from ordinary Turing machines is the fact that they are conceived to work on infinite strings of ’s and ’s, and they do that according to the following specifications. TTE Turing machines have one input tape, one working tape and one output tape. Each tape is equipped with a head. All ordinary instructions for Turing machines are allowed for the working tape, while the head of the input tape can only read and move rightward, and the head of the output tape can only write and move rightward. These limitations (in particular, those for the output tape) imply the impossibility of correcting the output; once a digit is written, it cannot be canceled or changed. Hence at each stage of the computation the partial output is reliable (this is the most we can ask, since in finitely many steps we never obtain a complete output).

It is straightforward to enumerate all TTE Turing machines and let be the -th such machine. Let be the partial function computed by as follows. Given let consist of ’s followed by a single , ’s, and so on; write on the input tape and start ; if the computation is infinite and the output tape eventually contains an infinite sequence of ’s and ’s with infinitely many ’s we translate back to an element of  which is ; otherwise .

Notice that is a subset of  for every .

Definition 2.1 (Computable functions on ).

We say that a function is computable if there exists such that and for every .

As noticed by Weihrauch ([Wei00, p. 38]), TTE Turing machines can be viewed as ordinary oracle Turing machines; the oracle supplies the information about the input and the -th bit of the output is computed when we give as input to the oracle Turing machine. Therefore the computable partial functions from  to  coincide with the computable (or recursive) functionals222Beware that in some literature “functional” means function from  to , rather than function from  to  as here. of classical computability (or recursion) theory, also known as Lachlan functionals.

The restrictions on the instructions allowed in TTE Turing machines imply the following fact ([Wei00, Theorem 2.2.3]).

Lemma 2.2.

Every computable function is continuous.

We transfer the notion of computability for the Baire space to spaces with cardinality less than or equal to the continuum using the notion of representation.

Definition 2.3 (Representations and represented spaces).

A representation of a set is a surjective function . The pair is a represented space.

If a -name for is any such that .

We say that is -computable when it has a computable -name (i.e.  is a computable set).

2.2. Effective metric spaces

The definition of representation is too general for practical purposes, as it allows an object in to be coded by arbitrary sequences. However, there are important cases in which we can find meaningful representations, for example when is a separable metric space.

Definition 2.4 (Effective metric space).

An effective metric space is a triple where

  • is a separable metric space;

  • is a dense sequence in .

If there is no danger of confusion, we often write in place of .

We equip every effective metric space with the Cauchy representation , such that if and only if for all and all , , and if and only if . In other words, is a name for when encodes a Cauchy sequence of elements in the fixed dense subset of which converges effectively to .

A rational open ball in is an open ball of the form with , and .

In particular, we have the effective metric space , where and is a standard computable enumeration of the set of the rational numbers (it is convenient to assume and ).

The notion of effective metric space can be generalized.

Definition 2.5 (Effective topological space).

An effective topological space is a triple , where is a second countable -topology on and is an enumeration of a sub-base of .

Each effective topological space has a standard representation such that if and only if .

It is immediate that effective metric spaces are particular examples of effective -topological spaces. In fact, if is an effective metric space we let enumerate the rational open balls of . We will always assume that there exist computable functions and such that has center and radius . In this context we usually write in place of .

The Cauchy representation of an effective metric space is equivalent to the representation of considered as an effective topological space. This equivalence means that each representation is reducible to the other, where a representation of a set is reducible to a representation of the same set when there is a continuous function such that for all . A representation of which is equivalent to the standard representation is said to be admissible for .

Definition 2.6 (Realizers).

Given represented spaces and and a partial function , we say that is a -realizer of when , for all .

The function is said to be -computable if it has a computable -realizer. In practice we often omit explicit mention of the representations and write just computable.

Using the notion of realizer we thus extend the notion of computable from the Baire space to the effective topological spaces. This extension is particularly successful when we use admissible representations, as the following results (due to Kreitz and Weihrauch) show.

Theorem 2.7.

Let and be effective topological spaces with admissible representations and . A function is continuous if and only if it has a continuous -realizer.

Corollary 2.8.

Let and be effective topological spaces with admissible representations and . Then every function which is -computable is continuous.

Corollary 2.8 is an extension of Lemma 2.2. We point out that Theorem 2.7 and Corollary 2.8 hold in particular for effective metric spaces and Cauchy representations.

The notions of effective metric and effective topological spaces in their complete generality have no computational content. In fact, notwithstanding the established terminology ([Wei00]), we are not requiring any “effectivity” property (even the computable enumeration of the rational open balls of an effective metric space is nothing but an enumeration of pairs of natural numbers). In the case of effective metric spaces, the natural “effective” requirement is the computability of the distance between points.

Definition 2.9 (Computable metric space).

A computable metric space is an effective metric space such that the function is computable.

If is a computable metric space it is straightforward that the distance function is -computable. Typical examples of computable metric spaces are and the Baire space (recall that for such that we let for the least such that ).

In the case of effective -topological spaces, the “effective” requirement is the computability of the operation of intersection of open sets (see [GW05]).

2.3. Representations of continuous functions

Notice that the set of all continuous partial functions on the Baire space is too large to have a representation. However, every partial continuous function has a continuous extension to a set ([Wei00, Theorem 2.3.8]; this is an instance of a classical result due to Kuratowski, see e.g. [Kec95, Theorem 3.8]). Thus it suffices to represent

Lemma 2.2 implies that each computable has an extension in . Define by , for and . is a representation of .

Given effective metric spaces and , we define a representation of the set of total continuous functions from into by if and only if is a -realizer of . This representation satisfies the following fundamental properties:

Evaluation:

the map is -computable;

Type conversion:

let be a represented space; every function is -computable if and only if , defined by , is -computable.

The evaluation and type conversion properties witness the reliability of the simulation of continuous functions on separable metric spaces via realizers.

2.4. Borel complexity

Computable analysis provides a method to classify incomputable functions between separable metric spaces in complexity hierarchies, analogously to the classification of functions from to pursued in classical computability theory. In particular, [Bra05] studied the following functions of strictly increasing complexity.

Definition 2.10 (The ’s).

For every let be defined by

where is when is odd and when is even.

Using natural representations for Borel sets of each given finite level, Brattka ([Bra05]) says that a function , for and computable metric spaces, is -computable (for ) if there exists a computable function that maps every name of an open set to a name of a set such that . It follows immediately that every -computable function is -measurable (equivalently, of Baire class ). Brattka shows that is -computable if and only if is computably reducible to 333We refer the reader to Section 4 below for the definition of reducibility..

3. Reverse mathematics

In the 1970’s Harvey Friedman started the research program of reverse mathematics, which was pursued in the two next decades by Steve Simpson and his students and increasingly by other researchers. Nowadays reverse mathematics is an important area of mathematical logic, crossing the boundary between computability theory and proof theory, but employing ideas and techniques also from model theory and set theory. We refer the reader to [Sim99] for details about the topics we will sketch in this section (the collection [Sim05] documents more recent advances).

Reverse mathematics searches in a systematic way for equivalences between different statements with respect to some base theory (which does not prove any of them) in the context of subsystems of second order arithmetic. Recall that the language  of second order arithmetic has variables for natural numbers and variables for sets of natural numbers, constant symbols and , binary function symbols for addition and product of natural numbers, symbols for equality and the order relation on the natural numbers and for membership between a natural number and a set. Second order arithmetic is the -theory with classical logic consisting of the axioms stating that is a commutative ordered semiring with identity, the induction scheme for arbitrary formulas, and the comprehension scheme for sets of natural numbers defined by arbitrary formulas. Hermann Weyl and Hilbert and Bernays already noticed that  was rich enough to express, using appropriate codings, significant parts of mathematical practice, and that many mathematical theorems were provable in (fragments of) second order arithmetic.

Formulas of  are classified in the usual hierarchies: those with no set quantifiers and only bounded number quantifiers are , while counting the number of alternating unbounded number quantifiers we obtain the classification of all arithmetical (= without set quantifiers) formulas as and formulas (one uses or depending on the type of the first quantifier in the formula, existential in the former, universal in the latter). Formulas with set quantifiers in front of an arithmetical formula are classified by counting their alternations as and . A formula is a certain theory if it is equivalent in that theory both to a formula and to a formula.

Reverse mathematics starts with the fairly weak base theory RCA, where the induction scheme and the comprehension scheme are restricted respectively to and formulas. RCA is strong enough to prove some basic results about many mathematical structures, but too weak for many others.

If a theorem is expressible in  but unprovable in RCA, reverse mathematics asks the question: what is the weakest axiom we can add to RCA to obtain a theory that proves ? In principle, we could expect that this question has a different answer for each . The “discovery” of reverse mathematics is that this is not the case. In fact, most theorems of ordinary mathematics expressible in  are either provable in RCA or equivalent over RCA to one of the following four subsystems of second order arithmetic, listed in order of increasing strength: WKL, ACA, ATR, and -CA. This leads to a neat picture where theorems belonging to quite different areas of mathematics are classified in five levels, roughly corresponding to the mathematical principles used in their proofs. RCA corresponds to “computable mathematics”, WKL embodies a compactness principle, ACA is linked to sequential compactness, ATR allows for transfinite arguments, -CA includes impredicative principles.

In this paper we will refer extensively to WKL and, in passing, to ACA. Therefore we describe these two theories in a little more detail.

ACA is obtained from RCA by extending the comprehension scheme to all arithmetical formulas. The statements without set variables provable in ACA coincide exactly with the theorems of Peano arithmetic, so that in particular the consistency strength of the two theories is the same. Within ACA one can develop a fairly extensive theory of continuous functions, using the completeness of the real line as an important tool. ACA proves (and often turns out to be equivalent to) also many basic theorems about countable fields, rings, and vector spaces.

To obtain WKL we add to RCA the statement of Weak König’s Lemma, i.e. every infinite binary tree has a path, which is essentially the compactness of Cantor space. An equivalent statement, clearly showing that WKL is stronger than RCA and weaker than ACA, is -separation: if and are -formulas such that there exists a set such that and for all . WKL and RCA have the same consistency strength of Primitive Recursive Arithmetic, and are thus proof-theoretically fairly weak. Nevertheless, WKL proves (and often turns out to be equivalent to) a substantial amount of classical mathematical theorems, including many results about real-valued functions, basic Banach space facts, etc. For example, WKL is equivalent, over RCA, to the Peano-Cauchy existence theorem for solutions of ordinary differential equations.

4. Multi-valued functions in computable analysis

The main goal of this section is to give the definition of reducibility of multi-valued functions.

Since we will often compose multi-valued functions, we spell out Weihrauch’s definition for this operation.

Definition 4.1 (Composition of multi-valued functions).

Given two (partial) multi-valued functions and , the composition is the multi-valued function defined by

To define the notion of computable multi-valued function we look at realizers.

Definition 4.2 (Realizers of multi-valued functions).

Let and be represented spaces and . A -realizer for is a (single-valued) function such that for every .

Notice that in Definition 4.2 we do not require that implies . In other words a realizer does not, in general, lift to a single-valued selector for the multi-valued function.

Definition 4.3 (Computability of multi-valued functions).

Let and be represented spaces. A multi-valued function is -computable if it has a computable -realizer. In practice we often omit explicit mention of the representations and write just computable.

Our definition of computable multi-valued function agrees with [Wei00, Definition 3.1.3.4] and [Bra05, p.21]. Notice however that Brattka’s paper includes also the definition of -computable multi-valued function; for single-valued functions the notions of computable and -computable coincide, but for arbitrary multi-valued functions the latter is stronger.

4.1. Reducibility of multi-valued functions

We now define the notion of computable reducibility for multi-valued functions. The intuitive idea is that one problem is reducible to another, provided that whenever we have a method to compute a solution for the second problem, we can uniformly find a way to compute a solution for the first one. This generalizes the notion of reducibility between single-valued functions investigated in [Bra05] and extensively used in recent work in computable analysis. Actually, in [Bra05] there are two distinct notions, introduced in Definitions 5.1 and 7.1, of computable reducibility between single-valued functions. Our definition generalizes the former, and Lemma 4.5 below shows that the generalization of the latter (realizer reducibility) leads to an equivalent concept444Brattka’s notion of realizer reducibility, as well its generalization to the case of multi-valued functions (Lemma 4.5.(ii)), are particular cases of Wadge’s reducibility for sets of functions as defined in [Wei00, Def. 8.2.5].. Thus the notion of computable reducibility appears to be more robust in the multi-valued setting.

Definition 4.4 (Reducibility of multi-valued functions).

Let , , , be represented spaces. Let and be multi-valued functions. We say that is computably reducible to , and write , if there exist computable multi-valued functions and such that for all .

Notice that when and are single-valued is single-valued on , but it may be the case that is not single-valued. Therefore the restriction of our notion of computable reducibility to single-valued functions is weaker than Brattka’s notion of computable reducibility for single-valued functions. However when dealing with multi-valued functions it is natural to allow and to be multi-valued as well. As we have pointed out, the following Lemma gives further support to our definition, by showing that it coincides with the natural generalization of Brattka’s notion of realizer reducibility.

Lemma 4.5.

Let , , , be represented spaces. Let and be multi-valued functions. The following are equivalent

  1. ;

  2. there exist computable functions and such that is a realizer for whenever is a realizer for .

Proof.

First assume and let the computable multi-valued functions and witness this. Let and , respectively, be computable realizers for and . Suppose is a realizer for ; we claim that is a realizer for . In fact if then and hence , so that .

Now suppose (ii) holds and let and witness this. Define and by

Since and are computable realizers for and respectively, the latter are computable multi-valued functions.

To check that and witness let and suppose . There exist and such that . By definition of let be such that , and . Let be a realizer for such that . Then

where membership follows from the fact that is a realizer for . We have thus shown , as needed. ∎

Since transitivity of for multi-valued functions is not immediately obvious, we state it explicitly.

Lemma 4.6.

is transitive.

Proof.

Let , and be multi-valued functions. Let and witness , while and witness . It is easy to check that and the map witness . ∎

Thus is a preorder (reflexivity is obvious) and we can give the usual definitions.

Definition 4.7.

As usual we use and for the strict relation and the equivalence relation arising from .

We now prove two simple Lemmas about .

Lemma 4.8.

Let be multi-valued functions such that and for every . Then .

Proof.

It is straightforward to check that the identity on and projection on the second coordinate from witness this. ∎

Lemma 4.9.

Let and be computable multi-valued functions. For any multi-valued function we have .

Proof.

It is straightforward to check that and witness this. ∎

Our definition of -computability for multi-valued functions is motivated by the characterization of this notion for singled-valued functions of Theorems 5.5 and 7.6 (one for each notion of reducibility) in [Bra05]. The reader should however be aware that Brattka defined a notion of -computability for multi-valued functions which is properly stronger than ours ([Bra05, Definition 3.5]).

Definition 4.10 (-computable and -complete).

Let and , be represented spaces. A multi-valued function is -computable if , and -complete if .

Lemma 4.5 above and Theorem 7.6 in [Bra05] imply that a multi-valued function is -computable if and only if it has a -computable realizer.

5. Reverse mathematics and computable analysis

5.1. Correspondence between statements of second order arithmetic and functions

Many mathematical statements expressed in  have the form

where and range over sets of natural numbers. Here are a few examples (we use the standard coding techniques for expressing in RCA functions, real numbers, sequences, etc.).

  1. The statement of Weak König’s Lemma (the main axiom of WKL) is

  2. The existence of the range of any function is

  3. The existence of the least upper bound for any sequence in is

  4. Separation of disjoint ranges is

  5. The statement of the Heine-Borel compactness of the interval is

  6. The statement of the separable Hahn-Banach Theorem is

If holds (this is the case in (2) and (3) above) it is natural to consider the partial function with such that for every . When the uniqueness condition fails we could consider all possible functions with the properties above. However it seems more useful to study the multi-valued function defined by for all such that .

Remark 5.1.

In many cases, including some of the examples given above, it is best to view the domain and the range of as represented spaces different from , thus unraveling the coding used in the reverse mathematics approach. E.g. the functions arising from examples (1) and (3) are best viewed respectively as a partial multi-valued function from to  and a total single-valued function from to .

We have thus associated to the mathematical statement expressed in  a function between represented spaces which can be studied within the framework of computable analysis. Notice that the lack of restrictions on the complexity of corresponds to the principle of computable analysis stating that “the user is responsible for the correctness of the input” (see [GSW07, §6] for a discussion).

We can also reverse the procedure. If we want to study from the viewpoint of computable analysis a multi-valued function , we can look at the reverse mathematics of the statement

with the hope of gaining some useful insight. E.g. if , from we obtain the statement

which is easily seen to be equivalent (over RCA) to -comprehension.

In any case, we expect some connection between the proof-theoretic strength of the statement and the computability strength of the function.

Notice that statements corresponding to functions belonging to different degrees of incomputability may collapse into a single system of reverse mathematics. Indeed, for any , the statement obtained above in correspondence with is equivalent to arithmetic comprehension. This means that each with corresponds to ACA, while it is well-known that . In other words, at the level of ACA computable analysis is finer than reverse mathematics.

The correspondence between proof-theoretic and computable equivalence is more useful when we are at the level of RCA or WKL. First, the computable sets are the intended -model of RCA, which is therefore a formal version of computable mathematics. Hence we expect that a statement provable in RCA gives rise to a computable function. Second, we expect most statements equivalent to WKL to give rise to computably equivalent uncomputable functions.

Sometimes these expectations are fulfilled, and some reverse mathematics proofs even translate naturally into a computable analysis proof. This is the case with Theorems 6.7 and 8.12 below. However the existence of this translation cannot be taken for granted, and for each direction of the correspondence we will give examples of failures. In other words, no automatic translation from the reverse mathematics literature into computable analysis, or vice versa, is possible. This phenomenon is a consequence of the different methods and goals of the two approaches. On one hand, the subsystems of second order arithmetic studied in reverse mathematics uses freely classical principles with no algorithmic content, such as excluded middle and proofs by contraposition. On the other hand, the algorithms of computable analysis assume the existence of the objects they have to compute, without the need of proving it. The examples of failure of the correspondence below highlight these differences.

5.2. Success of the correspondence

An often-used equivalent of ACA is the statement that the range of every one-to-one function from to is a set. Using the approach described above this translates into the following function.

Definition 5.2 ().

Let be the function that maps any one-to-one function to the characteristic function of its range, i.e.

for every injective and every .

As expected, we have the following Lemma.

Lemma 5.3.

.

Proof.

First we show . Given let be defined by if and only if . is computable and it is immediate that for every injective and .

We now show that . Given let be defined by

The function is computable with (i.e. each is one-to-one). Moreover . ∎

A basic example of reverse mathematics deals with the existence of least upper bounds of bounded sequences of real numbers. Indeed, this mathematical principle turns out to be equivalent to ACA ([Sim99, Theorem III.2.2]). We now show how this equivalence translates into computable analysis.

Definition 5.4 ().

Let be the function that maps any sequence in to its least upper bound.

Theorem 5.5.

.

Proof.

We start by showing that . Given observe that it is easy to use to compute the (characteristic function of the) set . Now we can computably define a sequence of rationals , where is such that and . Clearly is a Cauchy representation of the real number .

By Lemma 5.3, to prove it suffices to show that . Given , define by setting . Given we can define by letting to be the least satisfying . Then for every we have

and hence

This shows that, after using  to obtain , we can establish whether by first computing by search, and then checking finitely many values of . ∎

5.3. Failure of the correspondence

We now exhibit some examples where the correspondence outlined above fails. We first show that sometimes functions arising from statements provable in RCA are incomputable.

Example 5.6.

The following function, known as the “Allwissenheitsprinzip” (Principle of Omniscience), has been studied in detail from the viewpoint of computable analysis ([vS89, Myl89]).

Let be defined by

The incomputability of follows immediately from Lemma 2.2. On the other hand, the statement corresponding to is

which is obviously provable in RCA (and indeed just from the excluded middle, except for the coding of functions in the language of second order arithmetic).

We now give another example, which is more mathematical, but again has its roots in the use of classical logic in reverse mathematics.

Example 5.7.

Let be the hyperspace of closed subsets of  represented by negative information (see Definition 7.3 below) and be the multi-valued function which selects a point from nonempty closed subsets of . In other words , but on the left-hand side of this equality is a closed set (and hence a single element in the hyperspace), while on the right-hand side it is a set of points in the space .

The statement corresponding to  is , which is a tautology, since is an abbreviation for , and hence provable in RCA. On the other hand it follows from Theorem 8.3 below that, if we represent closed sets with respect to negative information (coherently with the reverse mathematics definition of closed set), and hence  is incomputable.

Example 5.8.

It is well-known that the intermediate value theorem is not constructive, and it can be shown that the corresponding multi-valued function is not computable (Brattka and Gherardi have forthcoming results about the incomputability strength of this function). On the other hand, a standard proof of the intermediate value theorem which uses the excluded middle can be carried out in RCA ([Sim99, Theorem II.6.6]).

We now give an example of the opposite phenomena, i.e. a theorem which is not provable in RCA but corresponds to a computable function.

Example 5.9.

The Heine-Borel compactness of the interval is Example (5) at the beginning of this Section. In reverse mathematics it is well-known that this statement is equivalent to WKL ([Sim99, Theorem IV.1.2]). On the other hand in computable analysis it is well-known that the function which maps each countable open covering of consisting of intervals with rational endpoints to a finite subcovering is computable ([Wei00]). We sketch the proof, to emphasize the difference between the reverse mathematics and the computable analysis approaches in this case.

There exists a computable enumeration of all finite open coverings of consisting of intervals with rational endpoints (in RCA we can even prove, e.g. using the ideas of the last part of the proof of Lemma 8.8 below, that the set of all these finite open coverings does exist). If we are given an (infinite) open covering of , where each is an interval with rational endpoints, it suffices to search for such that any interval in is for some . Then is the desired finite subcovering.

In this proof our knowledge of the compactness of insures that the search will sooner or later succeed. From the reverse mathematics viewpoint, the algorithm can be defined in RCA, but the proof of its termination requires WKL.

6. The multi-valued function

For the reader’s convenience, we repeat here the definition of  given in the introduction.

Definition 6.1 ().

Let be defined by ,

Thus is the set of the characteristic functions of the sets separating and