Cheap Non-standard Analysis and Computability

Cheap Non-standard Analysis and Computability

Olivier Bournez    Sabrina Ouazzani

Cheap Non-standard Analysis and Computability

Olivier Bournez    Sabrina Ouazzani
Abstract

Non standard Analysis is an area of Mathematics dealing with notions of infinitesimal and infinitely large numbers, in which many statements from classical Analysis can be expressed very naturally. Cheap non-standard analysis introduced by Terence Tao in 2012 is based on the idea that considering that a property holds eventually is sufficient to give the essence of many of its statements. This provides constructivity but at some (acceptable) price.

We consider computability in cheap non-standard analysis. We prove that many concepts from computable analysis as well as several concepts from computability can be very elegantly and alternatively presented in this framework. It provides a dual view and dual proofs to several statements already known in these fields.

Computability Theory, Computable Analysis, Non-Standard Analysis

Ecole Polytechnique, LIX, 91128 Palaiseau Cedex, Francebournez@lix.polytechnique.frANR PROJECT RACAF LACL, Université Paris-Est Créteil, 61 avenue du Général de Gaulle, 94010 Créteil, Francesabrina.ouazzani@lacl.fr \subjclass Theory of computation/Computability, Mathematics of computing/Numerical analysis, Models of Computation/Continuous functions.

1 Introduction

While historically reasonings in mathematics were often based on the used of infinitesimals, in order to avoid paradoxes and debates about the validity of some of the arguments, this was later abandoned in favor of epsilon-delta based definitions such as today’s classical definition of continuity for functions over the reals.

Non standard analysis (NSA) originated from the work of Abraham Robinson in the 1960’s who came with a formal construction of non-standard models of the reals and of the integers [16]. Many statements from classical Analysis can be expressed very elegantly, using concepts such as infinitesimals or infinitely large numbers in NSA: See e.g. [16, 6, 11, 12]. It not only have interests for understanding historical arguments and the way we came to some of today’s notions, but also clear interests for pedagogy and providing results that have not been obtained before in Mathematics. See e.g. [11] for nice presentations of NSA, or [12] for an undergraduate level book presenting in a very natural way the whole mathematical calculus, based on Abraham Robinson’s infinitesimals approach.

However, the construction and understanding of concepts from NSA is sometimes hard to grasp. Its models are built using concepts such as ultrafilter that are obtained using non-constructive arguments through the existence of a free ultrafilters whose existence requires to use the axiom of choice. Moreover, the dependance on the choice of this ultrafilter is sometimes not easy to understand (at least for non model-theory experts).

Terence Tao came in 2012 in a post in his blog [18] with a very elegant explanation of the spirit of many of the statements of non-standard analysis but using only very simple and elegant arguments. He called this cheap non-standard analysis in opposition to classical non-standard analysis. This theory is based on the idea that the asymptotic limit of a sequence given by its value after some finite rank is enough to define non standard object. Cheap non-standard analysis provides constructivity but this of course comes with some price (e.g. a non-total order on cheap non-standard integers, i.e. some indeterminacy).

Computability theory, classically dealing with finite or discrete structures such as a finite alphabet or the integers, has been far extended in many directions at this date. Various approaches have been considered for formalizing its issues in analysis, but at this stage the most standard approaches for dealing with computations over the reals are from computable analysis [20] and computability for real functions [13]. For other approaches and how they relate, see e.g. [3] or the appendices of [20].

In this paper, we explore how computability mixes with cheap non-standard analysis. We prove that many concepts from computable analysis as well as several concepts from computability can be very elegantly and alternatively presented in this framework. In particular, we prove that computable analysis concepts have very nice and simple formulations in this framework. We also obtain alternative, equivalent and nice formulations of many of its concepts in this framework.

Our approach provides an alternative to usual presentation of computable analysis. In particular, nowadays, a popular approach to formalize computable analysis is based on Type-2 computability, i.e. Turing machines working over representations of objects by infinite sequences of integers: See [20] for a monograph. Other presentations include original ones from [19], [7], and [15]. More recently, links have been established between type-2 computability and transfinite computations (see [5] for example) using surreal numbers. NSA has also been used in the context of various applications such as for modeling of systems: See e.g. [14] or [2].

The paper is organized as follows. In Section 2, we recall cheap non-standard analysis. In Section 3, we present the very basics of constructions from NSA, and we state some relations to cheap non-standard analysis. In Section 4, we start to discuss computability issues, and we consider computability of cheap non-standard integers and rational numbers. In Section 5, we discuss some computability issues related to infinitesimals and infinitely large numbers. In Section 6, we discuss computability for real numbers. In Section 7, we go to computability for functions over the reals and we discuss continuity and uniform continuity. In Section 8, we discuss computability of functions over the reals. In Section 9, we discuss some applications illustrating the interest of using our framework. Finally, in Section 10, we discuss our constructions and we discuss some interesting perspectives.

In all what follows, , and are respectively the set of integers, rational numbers and real numbers. We will sometimes also write for a synonym for : we will use preferably when talking about indices. In what follows, is either or , and we assume . By number, we mean either a natural or a rational number. In the current version, we use a color coding to help our reader to visualize the type of each variable. However, the paper can be read without this color coding.

2 Cheap Non-Standard Analysis

We start by presenting/recalling cheap non-standard analysis [18]. It makes the distinction between two types of mathematical objects: standard objects and (strictly) non-standard objects . Cheap non-standard objects are allowed to depend on an asymptotic parameter , contrary to standard objects that come from classical analysis. A cheap non-standard “object” is then defined by a sequence , which is studied in the asymptotic limit , that is to say for sufficiently large . Every standard “object” is also considered as a non-standard “object” identified by a constant sequence having value . The underlying idea is similar to what is for example done in probability theory where an element of is also implicitly considered as a probabilistic element of , depending if it has a measure associated. This is done with “object” referring to any natural mathematical concept such as “natural number”, “rational number”, “real number” “set”, “function” (but this could also be “point”, “graph”, etc. [18]).

The idea is then to consider that all reasonings are done in the asymptotic limit , that is to say for sufficiently large . In particular, two cheap non-standard elements , are considered as equal if after some finite rank. More generally, any standard true relation or predicate is considered to be true for some cheap non-standard object if it is true for all sufficiently large values of the asymptotic parameter, that is to say after some finite rank. Any operation which can be applied to standard object can also be applied to cheap non-standard object by applying the standard operation for each choice of rank parameter . For example, given two cheap non-standard integers and , we say that if one has after some finite rank. Similarly, we say that the relation is false if it is false after some finite rank. As another example, the sum of two cheap non-standard integers and is given by A cheap non-standard set is given by , where each is a set, and if we write , we have as expected if after some finite rank . Similarly, if is a cheap non-standard function from a cheap non-standard set to another cheap non-standard set , then is the cheap nonstandard element defined by . Every standard function is also a nonstandard function using all these conventions, as expected.

One key point is that introducing a dependence to a rank parameter leads to the definition of fully new concepts: infinitely small and large numbers. A cheap non-standard rational is infinitesimal if for all standard rational number . For example, is an infinitesimal. A cheap non-standard number can be infinitely large too: as an example, consider or , greater than any standard number. Note that the inverse of an infinitely large cheap non-standard number is a cheap non-standard infinitesimal.

From the fact that applying a standard operation to a cheap non-standard number is basically applying it to each possible value of , separately, most of the classical analysis properties on operations can hence be transferred from the standard framework to the cheap non-standard one: for example, commutativity and associativity for addition and multiplication operations.

However this is not always the case when one considers statement on cheap non-standard objects. One typical illustration that transfer from standard predicates to cheap non-standard ones is not automatic is the failure of the law of excluded middle. Repeating [18]: For instance, the nonstandard real number is neither positive, negative, nor zero, because none of the three statements , , or are true for all sufficiently large . Nevertheless, despite some peculiarities in the manipulation of statements, most of the standard first-order logic statements remains the same when quantified over cheap non-standard objects. We refer to [18] for a very pedagogical and more complete discussions about cheap non-standard concepts and some of its properties.

3 More on NSA: filters and ultrafilters

The classical constructions for non-standard analysis are done using free ultrafilters.

We recall the definition of an ultrafilter over an infinite set , called the index set. Typically, for us .

{definition}

[Filter] A filter over is a non-emptyset of subsets of such that:

  1. is closed under superset: if , and , then .

  2. is closed under finite intersections: If and , then .

  3. , but .

In particular, since , and , one cannot have both and its complement in .

{lemma}

[Fréchet filter] The set of all cofinite (i.e. complements of finite) subsets of is a filter. It is called the Fréchet filter.

{definition}

[Ultrafilter] An ultrafilter over is a filter over with the additional property that for each , exactly one of the sets et belongs to . A free ultrafilter is an ultrafilter such that no finite set belongs to . In the literature, a free ultrafilter is sometimes called a non-principal ultrafilter (in opposition to principal or fixed ones that thus contain a smallest element, called the principal element).

We just comment in the remaining lines how this relates to NSA. In NSA, one fixes a free ultrafilter . One also considers sequences indexed by . Sequences and are considered equal iff the set of indices such that is in the fixed free ultrafilter . Consequently, basically, cheap non-standard analysis corresponds to the case where is not a free ultrafilter, but the Fréchet filter. One deep interest of the above construction (also called ultraproduct) is that taking as a ultrafilter provides a transfer theorem (Łós’s theorem) that guarantees that any first order formula is true in the ultraproduct iff the set of indices when the formula is true belongs to the ultrafilter .

To some extend, cheap non-standard analysis constructions allow to reason on objects independantly from the ultrafilter, in the following sense (missing proofs are in appendix or in arXiv.org version of this article111Current reference submit/2240185.).

{theorem}

Two cheap non-standard numbers and , respectively corresponding to the sequences and , are equal iff for all free ultrafilter over we have .

The following lemma is based on a statement from [11]. For selfcontentness, and completeness we provide its proof, mostly repeating [11, Theorem 1.42] but proving also the required extension, as we need a variation of it.

{lemma}

[Folklore] For every infinite set , there exists a free ultrafilter over . Fix some infinite set . There exists a free ultrafilter over that contains .

Proof.

The set of all cofinite (complements of finite) subsets of is a filter over (called the Fréchet filter).

The set of all cofinite subsets of and of such that is finite is also a filter.

Let be the set of all filters over such that contains all cofinite subsets of and all such that is finite.

Then is nonempty and is closed under unions of chains. By Zorn’s Lemma, has a maximal element (in fact, infinitely many maximal elements).

is a filter and contains no finite set, because contains all cofinite sets but .

(resp. Furthermore, for , is infinite, because contains all such that is finite: otherwise but )

To show that an ultrafilter, we consider an arbitrary set and prove that there is a filter which contains either or , so by maximality, or .

Case : For all , is infinite. and each belong to the set

is a filter over I, because is obviously closed under supersets and finite intersections, and the hypothesis of Case 1 guarantees that each is infinite.

Case 2: For some , is finite. Then for every , is infinite, for otherwise would be finite. Case 1 applies to , so the set is a filter over such that , .

We see that belongs to iff for all , is infinite. In particular, belongs to iff for all , is infinite. Hence, . ∎

We now go to the proof of Theorem 3.

Proof.

Fix a free ultrafilter . Suppose that and represent the same cheap non-standard number. After some finite rank . Then is in (as its complement is finite, and hence not inside). So for that free ultrafilter.

Conversely, assume that for all rank , there is a rank with . Then is infinite. By Lemma 3, one can build a free ultrafilter with . Hence for that free ultrafilter . ∎

4 Computability for Integers or Rational Numbers

4.1 Very Basic Notions From Computability

We assume basic familiarity with computability theory. In computability theory, any integer is computable: there exists some Turing machine that writes in binary. However, not all total functions are computable (to avoid ambiguities we will say total recursive for “computable” in this context): there does not always exist some Turing machine that takes as input in binary and outputs in binary. An example of total recursive function is . An example of a non total recursive function is function which maps to , where is the output of the th Turing machine on input , for a given (non-assumed computable) enumeration of terminating Turing machines. will denote in the rest such a non total recursive function.

We used the wording “Turing machines”, but it is well known that the set of total recursive functions can be defined abstractly without referring to Turing machines: this is the smallest set that contains the constant function , the successor function , projection functions, and closed by composition, primitive recursion, and safe minimization. Safe minimization is minimization over safe predicates, that is to say predicates where for all there is a with .

We will several times use the following easy remark: If a function is computable for all arguments above a certain rank, then this function is computable. More formally: {theorem}[Computability for all indices] For any total function , for any finite , if there is some total recursive function such that for all , then there is a total recursive function such that for all .

4.2 Computable Cheap Non-Standard Numbers

In the literature, no discussions exist about the computability of numbers: any standard number is computable. But here cheap non-standard integer or cheap non-standard rational numbers may not be computable:

{definition}

[Computable cheap non-standard number] A cheap non-standard number is computable if seen as a sequence from to is total recursive .

For example, is not a computable cheap non-standard integer. Computable cheap non-standard integers include .

Our purpose is first to understand to what corresponds the subset of the computable cheap non-standard numbers among all cheap non-standard numbers: can it be defined abstractly, i.e. taking cheap non-standard analysis as a basis (i.e. in the spirit of [12] that presents mathematical calculus taking NSA as a basis)?

The following facts are easy: As usual denotes .

{theorem}

[Stability by total recursive functions] For any cheap non-standard computable numbers , , …, , for any standard total recursive function , we have that is a computable cheap non-standard number.

{theorem}

[Basic properties] The set of cheap non-standard computable natural numbers is a semiring: In particular, it is stable by , , . The set of cheap non-standard computable rational numbers is a ring: In particular, it is stable by , , inverse, .

{theorem}

[First characterization]

  • The set of cheap non-standard computable numbers is the smallest set that contains and that is stable by standard total recursive .

  • The set of cheap non-standard computable numbers is also the set of for total recursive standard .

Proof.

The Cheap non-standard integer is computable. When is computable and is a standard total recursive function, then, is computable. Now, from definitions is computable, iff for some standard total recursive , hence . First item follows.

Second item is a direct corollary of above reasoning. ∎

4.3 Shift Operation and Preservation Property

The previous properties can also be stated in another alternative way: Consider the following operation that maps cheap non-standard numbers to cheap non-standard numbers:

{definition}

[Shift operation] Whenever , is defined by .

Notice that for all standard integer . However, is not necessarily for a cheap non-standard . In particular, . In other words, using non-standard analysis vocabulary, is not an internal operation.

{theorem}

The set of computable cheap non-standard numbers is the smallest set that contains all solutions of for standard total recursive, and that is stable by standard total recursive .

Proof.

Cheap non-standard integer can be obtained as a solution of . Hence this class contains all computable cheap non-standard numbers from above statements.

We only need to state that cheap non-standard numbers are stable by such a equation: Assume that is solution of . Then after some finite rank , we must have , where denotes th iteration of (computability of follows from the Theorem 4.1). This yields computability for indices . And hence, this yields computability for all by the Theorem 4.1. ∎

A key remark is that the unary operation can actually be extended to a binary operation. A cheap non-standard element of is called a cheap non-standard index.

{definition}

[] Given some cheap non-standard number and some cheap non-standard index , Let be defined by

It can be checked that this definition is valid: its value is independant of the representative. It can also be checked that it satisfies , , , for any cheap non-standard number and cheap indices and .

{theorem}

Assume that and are computable. Then is computable.

From previous definitions, we derive easily the following preservation property.

{theorem}

[Preservation property] Let be some standard property over numbers. If holds for non standard numbers , then holds.

More generally, holds for all cheap non-standard index .

In some axiomatic view, computability of cheap non-standard numbers can be summarized as follows:

{theorem}
  • Not all cheap non-standard numbers are computable.

  • Computable cheap non-standard numbers include all standard numbers. The image of a computable cheap non-standard number by a standard total recursive function is computable.

  • The infinitely large cheap non-standard number satisfying is among computable numbers.

  • Computable cheap non-standard numbers are exactly those that can be obtained by above rules.

  • There exists some operation over cheap non-standard numbers, that preserves standard numbers, and that satisfies preservation property (Theorem 4.3).

5 Infinitesimals and Infinitely Large Numbers

Any cheap non-standard rational number is of the form for some cheap non-standard integers and . It is computable iff it is of the form with and computable.

Cheap non-standard integers as well as cheap non-standard rational numbers can be infinitely large. Cheap non-standard rational numbers can also be infinitesimals. For example, and are computable infinitesimals. Cheap non-standard rationals as well as are non-computable infinitesimals.

{definition}

[Infinitely large and infinitesimal numbers] A cheap non-standard number is infinitely large iff for all standard number , one has . A cheap non-standard rational number is infinitesimal iff for all standard rational number , one has .

One key point in the above concept is that this involves a quantification over all standard number , which is weaker than over all cheap non-standard . Actually, we however have the following phenomenon:

{theorem}

Let (respectively: ) be some cheap non-standard number that is infinitely large (resp. infinitesimal). For any cheap non-standard number (resp. ) there exists some cheap non-standard number , of the form for some cheap non-standard finite index , with (resp. ).

Proof.

Let some cheap non-standard number. Write .

Consider some . As is infinitely large (respectively: infinitesimal), there must exists some finite rank such that for all , we have (resp. ).

Let be the function that maps to the corresponding for all . Consider cheap non-standard index defined by .

From definitions is such that and hence satisfy (resp. ) for all . The conclusion follows. ∎

Fix some computable infinitesimal . We have that for any computable infinitesimal , there always exists some cheap non-standard finite index with However, this can be non-computable. Therefore, it is natural to consider the following notion.

{definition}

[Effectiveness] We say that some computable infinitesimal is effective with respect to computable iff there exists some cheap non-standard computable index with

This is clearly a reflexive relation as . It is also transitive:

{theorem}

[Transitivity of computably bounded relation]

Let be non zero positive computable infinitesimals. If is effective with respect to and effective with respect to , then is effective with respect to .

Proof.

If is effective with respect to and effective with respect to then there exists some cheap non-standard computable finite index with , and some cheap non-standard computable finite index with . But then : Indeed, apply to members of to get

from previously stated properties of . ∎

As a consequence, the following notion is natural and provides an equivalence relation:

{definition}

We say that two computable infinitesimals and are computably equivalent iff is effective with respect to and conversely.

{theorem}

is effective with respect to any computable infinitesimal .

Proof.

Consider computable cheap non-standard index given by where standard function for , and say (as , its components are non-zero after some rank). Then .

A computable infinitesimal is said to be monotone if with for all . Monotone computable infinitesimals include and .

{theorem}

Any monotone computable infinitesimal is effective with respect to . All monotone computable infinitesimals are computably equivalent.

Proof.

Consider be some computable monotone infinitesimal, i.e. with total recursive and decreasing. From Theorem 5, there exists for some cheap non-standard finite index , with . We get that predicate given by is safe. It follows that is computable. Consider , hence computable. We have .

First statement follows.

Second statement is a clear corollary. ∎

We say that some computable infinitesimal is effective if it belongs to the above class: it is monotone or computably equivalent to some monotone computable infinitesimal.

{corollary}

A computable infinitesimal is effective iff it is effective with respect to . Any effective computable infinitesimal is effective with respect to any computable .

6 Computability for Real Numbers

After having studied computability for cheap non-standard integer and rational numbers, we now go to computability for cheap non-standard real numbers.

{definition}

[Computability for real numbers] Fix some effective computable infinitesimal . A standard real is said to be computable if there exists some cheap non-standard computable rational , such that

{theorem}

The previous definition is not depending on : if this holds for an effective infinitesimal , then it holds for any other effective computable infinitesimal .

Proof.

Infinitesimal is effective with respect to for . Let the corresponding infinitesimal: , that is . Let and . We know (using preservation Theorem 4.3) that Consider then , . This guarantees Indeed, implies using definition of what integer part is, and then

Two cheap non-standard reals are said to be infinitely close (respectively: effectively infinitely close) if the absolute value of their difference is less than some (resp. effective) computable infinitesimal .

{definition}

[Left and Right-Computability for real numbers] A standard real is said to be left-computable (respectively: right-computable) if it is infinitely close to some cheap non-standard computable rational with (resp. ).

{theorem}

A standard real is computable iff it is effectively infinitely close to some cheap non-standard computable rational . A standard real is computable iff it is right-computable and left-computable.

Proof.

First statement is just a restatement of the definition. Concerning second statement: Direction from left to right of second item is trivial. Direction from right to left is the following. Assume that is right and left-computable. There exists some , , , such that and and both infinitesimal. This must hold componentwise for for some . Replacing if needed the values of functions for indices less than , we can assume without loss of generality that .

Given , consider , and . (respectively: ) is an increasing (resp. decreasing) function converging to . We have

The predicate given by is safe. Consequently, is computable, and is a computable sequence of rational numbers proving that is computable, since considering , we get

Let : these are the rationals with finite binary representation. They are sometimes also called dyadic rationals.

{theorem}

We can always assume in previous statements, i.e. to be of the form for some cheap non-standard integer .

Proof.

This is the case in the proof of Theorem 6. The case of left and right-computability is similar. ∎

One important theorem is that this corresponds to the classical definition of computability for reals (in the sense of computable analysis): Formally, according to classical definitions and statements from [20, 13, 1], this is equivalent to say that the following holds:

{theorem}

A standard real is computable iff there exist some total recursive functions and such that for all integer . A standard real is left-computable (resp. right-computable) iff there exist some total recursive functions and such that (resp. ).

Proof.

Consider monotone computable infinitesimal . There must exist some computable cheap non-standard integers and such that . The total recursive functions and such that and satisfies the above property after some finite rank . They can be fixed to and on the finitely many before so that this holds for all .

Conversely, if this holds, is a monotone computable infinitesimal, and and such that and are computable cheap non-standard integers such that .

The statements for right and left-computability are obtained in a similar fashion.

7 Continuity and Effective Uniform Continuity for Real Functions

The following theorem is left as an exercice in [18].

{theorem}

A function is continuous iff for all standard element of , and for all cheap non-standard element infinitely close to , then is infinitesimally close to .

We provide here the proof for completeness:

Proof.

Function is continuous in iff for all there exists some such that whenever we have .

For the right to the left direction: Assume is continuous. Consider some standard of , and some cheap non-standard element infinitely close to . Consider some standard , and the corresponding standard . As is infinitesimal, writing , there is some such that for all , we have . Consequently, , that is to say we have . As this holds for all standard , is infinitely close to .

For the left to the right direction: Assume is not continuous. That means that there exists some such that for all , say , there exists some with and . That means that is some cheap non-standard element infinitely close to but with not infinitely close to .

Similarly, the following can be established:

{theorem}

A function is uniformly continuous if for all cheap non-standard element of , and for all cheap non-standard element infinitely close to , then is infinitesimally close to .

For example standard function with domain is not-uniformly continuous as is not infinitely close to when is infinitely large. However, it is uniformly-continuous (and hence continuous) on as for infinitesimal, is always infinitely close to when (hence is bounded).

Notice that above theorems are defining concepts of continuity and uniform continuity very elegantly: there is no alternance of quantifiers compared to classical - definition. This sounds more natural and easier to grasp than usual definition.

8 Computability For Real Functions

We now go to computability issues:

{definition}

Fix some effective computable infinitesimal . A function has an effective modulus of continuity iff there exists some computable nonstandard such that for all standard and ,

Obviously, such a function is uniformly continuous, and hence continuous. More fundamentally:

{theorem}

The previous concept is not depending on : if this holds for a effective computable infinitesimal , then it holds for any other effective computable infinitesimal .

Proof.

Infinitesimal is effective with respect to . Let the corresponding infinitesimal: that is to say . But then provides the property for as we know using Property 4.3 that

{theorem}

We can always assume in previous statements, i.e. to be of the form for some cheap non-standard integer .

Proof.

This follows from the proof of previous theorem as applying the sense from left to right, and then left to right provides a of that form. ∎

This still corresponds to the classical notion from computable analysis. Formally, according to classical definitions and statements from [20, 13, 1], this is equivalent to say that the following holds:

{theorem}

A function has an effective modulus of continuity iff there exists some total recursive such that if then for all standard .

Proof.

Assume there is such a recursive . Consider monotone computable cheap non-standard infinitesimal given by . Consider computable cheap non-standard rational given by . Then

holds.

Conversely, assume that function has an effective modulus of continuity. Consider monotone computable infinitesimal . There must exists some computable cheap non-standard such that

That means that there exists some finite rank such that the properties holds componentwise for . Write for some total recursive . Consider for and for . This provides the expected property for all . ∎

Consider an indexed family of cheap non-standard numbers: to some parameter , is associated some cheap non-standard number . We say that the family is uniformly computable in if there exists some standard total recursive function such that .

{definition}

[Computability for functions over the reals] Fix some effective computable infinitesimal. We say that (standard domain) is computable iff

  1. [discretization property]: there exists some computable such that

  2. [it has some uniform approximation function over the rationals]: There exists some indexed family of cheap non-standard rationals , uniformly computable in , such that

    for all

Before going to the statement and proof that this corresponds to classical notion of computability for functions over the reals, notice that one main interest of above definition is that it sounds more natural and easier to grasp than classical ones222See statement of Theorem 8 for example of a classical definition.: in particular, item . is a very natural discretization property333It follows from the proofs that the discretization property is equivalent to the existence of an effective modulus of continuity. However, we believe the latter concept is harder to grasp, as basically talking about the dependence of a from , whose meanings is not so natural..

{theorem}

The previous definition is not depending on .

Proof.

This follows from Theorems 8 for item 1. Item 2. holds for some effective iff it holds for any effective using a reasoning similar to Theorem 8. ∎

{theorem}

Without loss of generality, we can always assume and in above definiton to be in and , i.e. to be of the form for some cheap non-standard integer or instead of being rational numbers.

Proof.

The fact that this is true for item 1. is Theorem 8. Now, if 1. holds, then can be replaced by where is approximating cheap non-standard rational at precision . The error would then be at most instead of . But as this is equivalent to hold for by above Theorem, we get the statement. ∎

It turns out that our definition is equivalent to the classical notion of computability in computable analysis. Formally, according to classical definitions and statements from [20, 13, 1], this is equivalent to say that the following holds:

{theorem}

A real function is computable iff it is computable in the sense of computable analysis.

Proof.

It is proved for example in [13, Corollary 2.14] that as above is computable in the sense of computable analysis iff

  1. [it has an effective modulus of continuity] there exists some total recursive such that if then for all standard .

  2. [it has some computable approximation function]: there exists some total recursive such that for all standard rational , standard integer

Now, Item 1. is equivalent to our Item 1. by Theorem 8. Concerning Item 2. Using Theorem 8, and Theorem 8, considering infinitesimal , suppose there exists some such that

for all cheap non-standard number . Write . Above inequality yields item 2 above.

Conversely, assume we have for some . Consider . This yields item 2. of our definition.

9 Examples of Applications

Hence, as expected, results known about computable functions are true in this framework, and conversely. However, our framework can present alternative ways to establish proofs.

Notice that cheap non-standard analysis is however distinct from NSA, and some of NSA statements and concepts needs to be adapted. As an example, a cheap non-standard number is said to be limited if there exists some standard real such that . In NSA, to every limited non-standard number is associated some unique standard real number , called its standard-part, such that is infinitesimal. This is not possible in cheap non-standard analysis since for example is clearly non infinitely close to any standard real number. We can however talk about the following:

{definition}

[Standard part and ] Assume is limited. We write (respectively: ) for the limit inf (resp. limit inf) of .

We write (respectively: ) for the inf (resp. sup) of . The following is easy to establish: {lemma} Assume is limited, where is some standard function.

Some standard is some accumulation point of iff there exists some infinitely large cheap non-standard index , monotone (i.e. ), such that is infinitesimal. Consequently, and are respectively the least and largest such , i.e. infinitely close to some and .

The following two statements have very elegant classical proofs in NSA: See e.g. [16, 12] This can be adapted to a proof using cheap non-standard arguments: see Appendix 9.

{theorem}

[Intermediate Value Theorem] Every continuous standard function with has a zero at some standard .

We need the following whose proof is easy.

{lemma}

Assume that is some continuous function. Assume that is limited. If then . If then . Similarly for . If then . If then . Similarly for .

We now prove Theorem 9.

Proof.

Assume w.l.o.g that and that and . Take infinitely large cheap non-standard integer . The idea is to consider the of the form for cheap non-standard integer .

To do so, consider where and

This latter set is non-empty as and does not contain since .

Function is continuous, hence uniformly continuous on its domain. Since and are infinitely close, necessarily and must be infinitely close by Theorem 7.

We have and by definition of . Consequently, using Lemma 9, necessarily and , hence for standard .

Notice that we could have considered , or the defined symmetrically, and this could provide possibly other zeros.

{theorem}

[Extreme Value Theorem] Every continuous standard function attains its maximum at some standard .

We need the following whose proof is easy.

{lemma}

Assume that is some continuous function. Assume that is limited and is some standard value. If then .

We can now go to the proof of Theorem 9:

Proof.

Assume w.l.o.g that . Take infinitely large cheap non-standard integer . The idea is to consider the of the form for cheap non-standard integer .

To do so, consider , where and

This latter set is non-empty as a finite set always has a maximum.

Function is continuous, hence uniformly continuous on its domain.

Consider , and . Then we claim that for all standard . Indeed, any contain at least one , infinitely close to it: Consider .

Hence is infinitely close to . The latter, is less than by definition of , and hence less than by Lemma 9. ∎

Notice that we could have considered , or the defined symmetrically, and this could provide possibly other maximums.

We now adapt these proofs to go to computability issues:

{lemma}

Assume that is limited and computable. Then is right-computable and is left-computable.

{theorem}

Every computable function with has a right-computable zero and a left-computable zero.

Proof.

Assume w.l.o.g that and that and .

Consider effective infinitesimal . There must exists some computable such that There must also exists some indexed family of cheap non-standard rationals , uniformly computable in , such that for all .

Consider infinitely large computable cheap non-standard integer . The idea is to consider the of the form for cheap non-standard integer .

To do so, consider where and

Function is continuous, hence uniformly continuous on its domain. Since and are infinitely close, necessarily and must be infinitely close by Theorem 7.

We have and