In this post about the difference between the recursive and effective topos, Andrej Bauer said:

If you are looking for a deeper explanation, then perhaps it is fair to say that the Recursive Topos models computability a la Banach-Mazur (a map is computable if it takes computable sequences to computable sequences) and the Effective topos models computability a la Kleene (a map is computable if it is realized by a Turing machine). In many respects Kleene's notion of computability is better, but you'll have to ask another question to find out why :-)

So I'm asking:

1) What is "computability a la Banach-Mazur"? I would guess it has something to do with Baire spaces and computable analysis, but I don't really know.

2) Why is Kleene's notion of computability better?

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    $\begingroup$ Uh oh, now I feel obliged to answer. $\endgroup$ – Andrej Bauer Apr 20 '10 at 13:20

This answer requires a bit of background.

Definition 1: a numbered set $(X,\nu_X)$ is a set $X$ together with a partial surjection $\nu_X : \mathbb{N} \to X$, called a numbering of $X$. When $\nu_X(n) = x$ we say that $n$ is a code for $x$.

Numbered sets are a generalization of Gödel codes. Some typical examples are:

  • $\mathbf{N} = (\mathbb{N}, \mathrm{id}_\mathbb{N})$ is the standard numbering of natural numbers.
  • $\mathbf{P} = (P, \phi)$ where $P$ is the set of partial computable maps and $\phi$ is a standard enumeration of partial computable maps.
  • $\mathbf{R} = (R,\nu_R)$ where $R$ is the set of computable reals and $\nu_R(n) = x$ when, for all $k \in \mathbb{N}$, $\phi_n(k)$ outputs (a code of) a rational number $q$ such that $|x - q| < 2^{-k}$.

Numbered sets can be used to give effective structure to many mathematical structures. What should we take as a morphism between numbered sets? Presumably a map $f : X \to Y$ should be considered a morphism from $(X,\nu_X)$ to $(Y,\nu_Y)$ when it is "computable" in a suitable sense. We understand fairly well what it means to have a computable map $\mathbb{N} \to \mathbb{N}$, namely computed by a Turing machine, so let us take that for granted. It is easy to extend computability of sequences of numbers to computability of arbitrary sequences:

Defintion 2: A map $s : \mathbb{N} \to X$ is a computable sequence in $(X,\nu_X)$ when there exists a computable map $f : \mathbb{N} \to \mathbb{N}$ such that $s(n) = \nu_X(f(n))$ for all $n \in \mathrm{dom}(\nu_X)$.

Now suppose we think a bit like analysts. One way to define a continuous map is to say that it maps convergent sequences to convergent sequences. We could mimick this idea to define general computable maps.

Definition 3: A function $f : X \to Y$ where $(X,\nu_X)$ and $(Y,\nu_Y)$ are numbered sets is Banach-Mazur computable when $f \circ s$ is a computable sequence in $(Y,\nu_Y)$ whenever $s$ is a computable sequence in $(X,\nu_X)$.

How good is this notion? And how does it compare to the following notion, which is taken as the standard one nowadays?

Definition 3: A function $f : X \to Y$ where $(X,\nu_X)$ and $(Y,\nu_Y)$ are numbered sets is Markov computable, or just computable, when there exists a partial computable map $g : \mathbb{N} \to \mathbb{N}$ such that $f(\nu_X(n)) = \nu_Y(g(n))$ for all $n \in \mathrm{dom}(\nu_X)$.

In other words, $f$ is tracked by $g$ in the sense that $g$ does to codes what $f$ does to elements. (Note: in this MO I attributed this notion of computability to Kleene, but I think it's better to attach Markov's name to it, if any.)

Every Markov computable function is Banach-Mazur computable. In some cases the converse holds as well. For example, every Banach-Mazur computable map $\mathbf{N} \to \mathbf{N}$ is Markov computable. However, this is not the case in general:

  1. R. Friedberg demonstrated that there is a Banach-Mazur computable map $\mathbf{N}^\mathbf{N} \to \mathbf{N}$ which is not Markov computable. [R. Friedberg. 4-quantifier completeness: A Banach-Mazur functional not uniformly partial recursive. Bull. Acad. Polon. Sci. Sr. Sci. Math. Astr. Phys., 6:1–5, 1958.]

  2. P. Hertling constructed a Banach-Mazur computable map $\mathbf{R} \to \mathbf{R}$ which is not Markov computable. [P. Hertling. A Banach-Mazur computable but not Markov computable function on the computable real numbers. In Proceedings ICALP 2002, pages 962–972. Springer LNCS 2380, 2002.]

  3. A. Simpson and I showed that there is a Banach-Mazur computable $\mathbf{X} \to \mathbf{R}$ that is not Markov computable when $\mathbf{X}$ is any inhabited computable complete separable metric space computably without isolated points. [A. Bauer and A. Simpson: Two Constructive Embedding-Extension Theorems with Applications to Continuity Principles and to Banach-Mazur Computability, Mathematical Logic Quarterly, 50(4,5):351-369, 2004.]

What this says is that Banach-Mazur computability is too general because it admits functions that cannot be computed in the standard sense of the word, i.e., computed by Turing machine (in terms of codes).

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  • $\begingroup$ B-M is a really pretty idea! It's too bad it doesn't work, but is there some way of salvaging it? $\endgroup$ – Neel Krishnaswami Apr 21 '10 at 7:58
  • $\begingroup$ There are classes of numbered sets for which B-M computability is equivalent to Markov computability. For example, Eršov numbered sets have this property (the conditions that a numbered set must satisfy in order to be an Eršov set do not fit into this comment, or perhaps just barely, but essentially they say that the numbered set behaves like the numbered set of r.e. sets). $\endgroup$ – Andrej Bauer Apr 21 '10 at 9:42
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    $\begingroup$ What is the essence of the counter-examples? Do the 3 counter-examples have some kernel idea in common? [That answer probably doesn't fit in a comment! You should add it either as an edit or even as a 'new' answer]. $\endgroup$ – Jacques Carette Apr 21 '10 at 11:45

Jacques asks about a "kernel" idea for the three counter-examples. I am not sure I can answer the question well. The most general result, namely the third one, is really a consequence of Friedberg's result. Peter Hertling's construction is an adaptation of Friedberg's result (although not an obvious one by any means), too. So the question is, what did Friedberg do? I must admit I cannot explain this succinctly, perhaps it's best if you have a look at Section 9 of

P. Hertling, "A Banach–Mazur computable but not Markov computable function on the computable real numbers", Annals of Pure and Applied Logic 132 (2005) 227–246.

Suffice it to say that the construction involves showing that a certain set of indices of B-M computable functionals is a $\Pi^0_4$-set, but the corresponding set of indices of Markov computable functionals is $\Sigma^0_4$-complete and contained in the first set, therefore the two sets differ. Now if there is anyone around who can explain in simple terms what $\Sigma^0_4$-complete sets are like, I would love to hear it.

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