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Numbering (computability theory)

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In computability theory, the assignment of natural numbers to a set of objects

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

Examples of numberings include:

  • The set of all finite subsets of N {\displaystyle \mathbb {N} } has a numbering γ {\displaystyle \gamma } , defined so that γ ( 0 ) = ∅ {\displaystyle \gamma (0)=\emptyset } and so that, for each finite nonempty set A = { a 0 , … , a k } {\displaystyle A=\{a_{0},\ldots ,a_{k}\}} , γ ( n A ) = A {\displaystyle \gamma (n_{A})=A} where n A = ∑ i ≤ k 2 a i {\displaystyle n_{A}=\sum _{i\leq k}2^{a_{i}}} (Ershov 1999:477). This numbering is a (partial) bijection.
  • A fixed Gödel numbering φ i {\displaystyle \varphi _{i}} of the computable partial functions can be used to define a numbering W of the recursively enumerable sets, by letting by W(i) be the domain of φi. This numbering will be surjective (like all numberings) but not injective: there will be distinct numbers that map to the same recursively 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 recursively enumerable then there always exists an equivalent total numbering (equivalence of numberings is defined below).

A numbering η is decidable if the set { ( x , y ) : η ( x ) = η ( y ) } {\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 ν 1 : N ⇀ S {\displaystyle \nu _{1}:\mathbb {N} \rightharpoonup S} and ν 2 : N ⇀ S {\displaystyle \nu _{2}:\mathbb {N} \rightharpoonup S} be two numberings. Then ν 1 {\displaystyle \nu _{1}} is reducible to ν 2 {\displaystyle \nu _{2}} , written ν 1 ≤ ν 2 {\displaystyle \nu _{1}\leq \nu _{2}} , if

∃ f ∈ P ( 1 ) ∀ i ∈ D o m a i n ( ν 1 ) : ν 1 ( i ) = ν 2 ∘ f ( i ) . {\displaystyle \exists f\in \mathbf {P} ^{(1)}\,\forall i\in \mathrm {Domain} (\nu _{1}):\nu _{1}(i)=\nu _{2}\circ f(i).}

If ν 1 ≤ ν 2 {\displaystyle \nu _{1}\leq \nu _{2}} and ν 1 ≥ ν 2 {\displaystyle \nu _{1}\geq \nu _{2}} then ν 1 {\displaystyle \nu _{1}} is equivalent to ν 2 {\displaystyle \nu _{2}} ; this is written ν 1 ≡ ν 2 {\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 recursively enumerable sets, the numbering η is computable if the set of pairs (x,y) where y ∈ η(x) is recursively enumerable. Similarly, a numbering g of partial functions is computable if the relation R(x,y,z) = "(y) = z" is partial recursive (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 recursively enumerable subsets of N {\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 recursive functions is known as an admissible numbering in the literature.

See also

References

  1. "Computability Theory - an overview | ScienceDirect Topics". www.sciencedirect.com. Retrieved 2021-01-19.
  • Y.L. Ershov (1999), "Theory of numberings", Handbook of Computability Theory, Elsevier, pp. 473–506.
  • V.A. Uspenskiĭ, A.L. Semenov (1993), Algorithms: Main Ideas and Applications, Springer.
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