Numbering (computability theory)

In computability theory a numbering is an assignment of natural numbers to a set of objects such as functions, rational numbers, graphs, or words in some formal language. A numbering can be used to transfer the idea of computability^[1] and related concepts, which are originally defined on the natural numbers using computable functions, to these different types of objects.

Common examples of numberings include Gödel numberings in first-order logic, the description numbers that arise from universal Turing machines and admissible numberings of the set of partial computable functions.

Definition and examples

A numbering of a set ${\displaystyle S}$ is a surjective partial function from ${\displaystyle \mathbb {N} }$ to S (Ershov 1999:477). The value of a numbering ${\displaystyle \nu }$ at a number i (if defined) is often written ${\displaystyle \nu _{i}}$ instead of the usual ${\displaystyle \nu (i)}$.

Examples of numberings include:

• The set of all finite subsets of ${\displaystyle \mathbb {N} }$ has a numbering ${\displaystyle \gamma }$ , defined so that ${\displaystyle \gamma (0)=\emptyset }$ and so that, for each finite nonempty set ${\displaystyle A=\{a_{0},\ldots ,a_{k}\}}$, ${\displaystyle \gamma (n_{A})=A}$ where ${\displaystyle n_{A}=\sum _{i\leq k}2^{a_{i}}}$ (Ershov 1999:477). This numbering is an injection.
• A fixed Gödel numbering ${\displaystyle \varphi _{i}}$ of the computable partial functions can be used to define a numbering W of the computably enumerable sets, by letting by W(i) be the domain of ${\displaystyle \varphi _{i}}$. This numbering will be surjective (like all numberings) but not injective: there will be distinct numbers that map to the same computably enumerable set under W.

Types of numberings

A numbering is total if it is a total function. If the domain of a partial numbering is computably enumerable then there always exists an equivalent total numbering (equivalence of numberings is defined below).

A numbering η is decidable if the set ${\displaystyle \{(x,y):\eta (x)=\eta (y)\}}$ is a decidable set.

A numbering η is single-valued if η(x) = η(y) if and only if x=y; in other words if η is an injective function. A single-valued numbering of the set of partial computable functions is called a Friedberg numbering.

Comparison of numberings

There is a preorder on the set of all numberings. Let ${\displaystyle \nu _{1}:\mathbb {N} \rightharpoonup S}$ and ${\displaystyle \nu _{2}:\mathbb {N} \rightharpoonup S}$ be two numberings. Then ${\displaystyle \nu _{1}}$ is reducible to ${\displaystyle \nu _{2}}$, written ${\displaystyle \nu _{1}\leq \nu _{2}}$, if

${\displaystyle \exists f\in \mathbf {P} ^{(1)}\,\forall i\in \mathrm {Domain} (\nu _{1}):\nu _{1}(i)=\nu _{2}\circ f(i).}$

If ${\displaystyle \nu _{1}\leq \nu _{2}}$ and ${\displaystyle \nu _{1}\geq \nu _{2}}$ then ${\displaystyle \nu _{1}}$ is equivalent to ${\displaystyle \nu _{2}}$; this is written ${\displaystyle \nu _{1}\equiv \nu _{2}}$.

Computable numberings

When the objects of the set S being numbered are sufficiently "constructive", it is common to look at numberings that can be effectively decoded (Ershov 1999:486). For example, if S consists of computably enumerable sets, the numbering η is computable if the set of pairs (x,y) where y ∈ η(x) is computably enumerable. Similarly, a numbering g of partial functions is computable if the relation R(x,y,z) = "[g(x)](y) = z" is computably enumerable (Ershov 1999:487).

A computable numbering is called principal if every computable numbering of the same set is reducible to it. Both the set of all computably enumerable subsets of ${\displaystyle \mathbb {N} }$ and the set of all partial computable functions have principle numberings (Ershov 1999:487). A principle numbering of the set of partial computably functions is known as an admissible numbering in the literature.

See also

• Complete numbering
• Cylindrification
• Gödel numbering
• Description number