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Notes on Computable Analysis

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Abstract

Computable analysis has been part of computability theory since Turing’s original paper on the subject (Turing, Proc. London Math. Sc. 42:230–265, 1936). Nevertheless, it is difficult to locate basic results in this area. A first goal of this paper is to give some new simple proofs of fundamental classical results (highlighting the role of \({{\Pi }_{1}^{0}}\) classes). Naturally this paper cannot cover all aspects of computable analysis, but we hope that this gives the reader a completely self-contained ingress into this area. A second goal is to use tools from effective topology to analyse the Darboux property, particularly a result by Sierpiński, and the Blaschke Selection Theorem.

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Notes

  1. In this year Richard Dedekind developed Dedekind cuts which gave an equivalent definition of the real numbers.

  2. This list is a sample, and by no means exhaustive. For further contributions, see reference list.

  3. Although it is fair to say that the interpretation of what a computable function should be differs in these texts, as we will soon see.

  4. Both equivalences follow by well-known isomorphisms.

  5. In the case of sets, A is a \({{\Sigma }_{1}^{0}}\) subset of \(\mathbb {N}\) if A can be expressed as x ∈ A iff ∃n R(x, n) where R is a computable predicate on \(\mathbb {N \times N}\).

  6. For English translation of [22], see [23]. For an English translation of [36], see [37]. For an English translation of [46], see [47]. For an English translation of [70], see [71].

  7. Quotes and comments from Borel’s paper [11] are based on a translation (French to English) by Avigad and Brattka [3].

  8. This definition is not really attributed to Borel, however, as these types of functions are classically referred to as ‘Borel computable’ we will stick with this notation to avoid confusion.

  9. Define a function ϕ(m, n) = f(x) n , where f(x) n is the n th term in f(x)′ s Cauchy name. By the S-m-n Theorem we can find a total computable function g such that ϕ(m, n) = φ g(m)(n) for all \(n \in \mathbb {N}\). By the Recursion Theorem g has a fixed point. That is, there exists an e ′ such that \(\varphi _{g(e^{\prime })}(n)=\varphi _{e^{\prime }}(n)\). Then \(\phi (m,n)=\phi (e^{\prime },n)= \varphi _{g(e^{\prime })}(n)=\varphi _{e}^{\prime }(n)\), and so e ′ is the index of the Cauchy name of f(x), and is computable from g.

  10. In detail; we now take the partial function with index e and input it into the index function ν of f. Then ν(n) is the index of the partial function that outputs a Cauchy name for f(b). We take the (n+1)th approximation of f(b) and are done.

  11. The use of a converging oracle computation ΦA(n) is z+1 for the largest z such that A(z) is queried during the computation. Let the use function be Use \(:\mathbb {N} \to \mathbb {N}\). That is, Use (ΦA(n)) = z+1 from above. The Use Principle is as follows; let ΦA be a converging oracle computation and B a set such that \(B \upharpoonright \textit {Use}({\Phi }^{A}(n))=A \upharpoonright \textit {Use}({\Phi }^{A}(n))\). Then ΦB(n) = ΦA(n). For more details see, for example, [25] Section 2.

  12. Hertling has also written other papers about Banach-Mazur computability, notably ‘Banach-Mazur computable functions on metric Spaces’ [38].

  13. An English translation of this paper can be found in [10].

  14. It may be that this result appeared earlier that in the text given.

  15. The function f is Type II computable, so there exists an oracle machine \({{\Phi }_{e}^{x}}\) that outputs a Cauchy name for f(x). If \({{\Phi }_{e}^{x}}\) reads the first m terms in a Cauchy name of x, and outputs a n th approximation of f(x), then by the Use Principle, this is also a sufficient n th approximation of f(y) for any real y ∈ (x m −2−m,x m +2−m). That is, the interval (x m −2−m,x m +2−m) maps into the interval (f(x) n −2−n,f(x) n +2−n).

  16. As with the computable real and computable real-valued function, there exist other classifications of a computable subset of \(\mathbb {R}^{n}\). See, for example, the Braverman and Yampolsky book [14].

  17. We emphasise that our result is for Baire rather than Cantor space. This does not have any effect in this paper but is worth bearing in mind as our \({{\Pi }_{1}^{0}}\) classes are not computably bounded. As a result, classical theorems (for example the Low Basis Theorem) would not apply here.

  18. We can not just take the open balls B((a n , b n ),2−(n)) and B((a n , f(a) n ),2−(n)) here because the point (a, b) and (a, f(a)) may actually fall outside of these balls.

  19. Note that \(d_{{\Gamma }_{f}, n}((a,b))\leq |f(x)_{n+2} - y_{n+2}|+2^{-(n+1)} =g(n,0)\).

  20. If we were asked to evaluate u at this stage, defer to Stage 2.

  21. An English translation of this paper can be found in [10].

  22. He formed this example while preparing for lectures [20].

  23. Folklore or any early text in recursive analysis.

  24. ‘Computable’ is intended again in the classical sense here.

  25. This example is similar to an example of a real-valued canonical Darboux function given in [7].

  26. The paper cited here, ‘Kreis und Kugel’, is written in German and has not been translated into English. A number of secondary sources confirmed this reference, for example [40] and [34].

  27. An English translation of this paper can be found in [10].

  28. Weierstrass published relatively little in his lifetime, but some of his original works can be found in [72]. See [29] for historical information.

  29. Metric sourced from [28].

  30. Notice we are using two different metrics here. Recall that d was defined earlier; d X (x i )= infx ∈ X|x − x i |.

  31. The jump is considered as a set here again.

  32. Notice that x n is thought of as a real here, rather than a term in a Cauchy name for a real x. This is one of the only notational exceptions of this type, and was made for convenience.

  33. This is just a more convenient reformulation of Definition 6.2.2.

  34. If a closed set X has a computable distance function then it can be expressed as a \({{\Pi }_{1}^{0}}\) class in Baire space. Check d X (x). If the n th approximation of d X (x) is ever greater than 2−n, kill the branch that extends however much of the Cauchy name of x we have seen.

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Acknowledgments

This work was supported by the Marsden Fund of New Zealand, and a MSc scholarship to Porter. Much of the material is based around Porter’s MSc Thesis supervised by Day and Downey, and for this reason Porter is the first author on the paper.

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  1. School of Mathematics and Statistics, Victoria University, PO Box 600, Wellington, New Zealand

    Michelle Porter, Adam Day & Rodney Downey

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  1. Michelle Porter
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Correspondence to Michelle Porter.

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This article is part of the Topical Collection on 50th Anniversary

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Porter, M., Day, A. & Downey, R. Notes on Computable Analysis. Theory Comput Syst 60, 53–111 (2017). https://doi.org/10.1007/s00224-016-9732-y

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