\[ \newcommand{\tr}{\Rightarrow} \newcommand{\trs}{\tr^{\!\ast}} \newcommand{\rlnm}[1]{\mathsf{(#1)}} \newcommand{\rred}[1]{\xrightarrow{#1}} \newcommand{\rreds}[1]{\mathrel{\xrightarrow{#1}\!\!^*}} \newcommand{\cl}{\mathsf{Cl}} \newcommand{\pow}{\mathcal{P}} \newcommand{\matches}{\mathrel{\mathsf{matches}}} \newcommand{\kw}[1]{\mathsf{#1}} \]

Reflections: computing under representation

The technology of bijections allow us to define computability for sets $S$ other than $\mathbb{N}$.

Reflecting functions on a single set

First, let us suppose we are able to represent a set $A$ through the natural numbers. In other words, let us suppose we have a bijection $i : A \xrightarrow{\cong} \mathbb{N}$, with inverse $i^{-1} : \mathbb{N} \xrightarrow{\cong} A$.

We may think of $i$ as uniquely representing all the elements of set $A$ as natural numbers.

Let $f : A ⇀ A$ be a partial function. We cannot directly speak to $f$’s computability, as it is a partial function on the set $A$. However, we only know what computability means for functions of type $\mathbb{N} \rightharpoonup \mathbb{N}$ only. Nevertheless, the bijection $i$ allows us to transport the function to the set $\mathbb{N}$!

Definition. The reflection of $f : A \rightharpoonup A$ under $i$ is the function

\[\begin{aligned} & \tilde{f} : \mathbb{N} ⇀ \mathbb{N} \\ & \tilde{f}(n) = i(f(i^{-1}(n))) \end{aligned}\]

Written in point-free style, we define $\tilde{f} = i \circ f \circ i^{-1}$. This definition can also be expressed as a commutative diagram:

\[\require{amscd} \begin{CD} A @>{f}>> A\\ @Ai^{-1}AA @VViV \\ \mathbb{N} @>{\tilde{f}}>> \mathbb{N} \end{CD}\]

(with apologies for not being able to render the partial arrow $⇀$ in the diagram)

The reflection $\tilde{f}$ computes $f$ “under” the encoding $i$. Thus, instead of acting on elements of $A$, it acts on their encoded forms.

In a sense, the function $f : A ⇀ A$ is computable just if its reflection $\tilde{f} : \mathbb{N} ⇀ \mathbb{N}$ is.

Reflecting general functions

In the previous section we dealt with the computability of functions that had a set $A$ as both domain and codomain. How can this be generalised to cover any old function $f : A \to B$ with any domain $A$ and any codomain $B$?

In this case, we will need two bijections, one for each set:

\[\begin{aligned} &\phi : A \xrightarrow{\cong} \mathbb{N} & &\psi : B \xrightarrow{\cong} \mathbb{N} \end{aligned}\]

As before, $\phi$ enables us to encode elements $a \in A$ as numbers $\phi(a) \in \mathbb{N}$, while being invertible.

Similarly, $\psi$ enables us to encode elements $b \in B$ as numbers $\psi(b) \in \mathbb{N}$, while also being invertible.

We can then play the same trick as before, but now using each bijection to appropriately decode and then encode elements of $A$ and $B$ as natural numbers.

Definition. The reflection of $f : A \rightharpoonup B$ under $(\phi, \psi)$ is the function

\[\begin{aligned} & \tilde{f} : \mathbb{N} ⇀ \mathbb{N} \\ & \tilde{f}(n) = \psi(f({\phi}^{-1}(n))) \end{aligned}\]

Written in point-free style, we define $\tilde{f} = \psi \circ f \circ \phi^{-1}$. This definition can also be expressed as a commutative diagram:

\[\require{amscd} \begin{CD} A @>{f}>> B\\ @A{\phi}^{-1}AA @VV{\psi}V \\ \mathbb{N} @>{\tilde{f}}>> \mathbb{N} \end{CD}\]

(with apologies for not being able to render the partial arrow $⇀$ in the diagram)

This enables us to define what it means for a general function $f : A \to B$ between any sets to be computable, as long as these sets can be put in bijection with the natural numbers.