Numbering (computability theory)

In computability theory a numbering is the 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.

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

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.

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 recursively enumerable sets, the numbering η is computable if the set of pairs (x,y) where yη(x) is recursively enumerable. Similarly, a numbering g of partial functions is computable if the relation R(x,y,z) = "[g(x)](y) = z" is partial recursive (Ershov 1999:487).