Computable set
Set where an algorithm can take a number as an input and can decide whether the number belongs to the set
In computability theory, a set of natural numbers is computable (or decidable or recursive) if there is an algorithm that computes the membership of every natural number in a finite number of steps.
Definition
A subset
S
{\displaystyle S}
of the natural numbers is computable if there exists a total computable function
f
{\displaystyle f}
such that:
f
(
x
)
=
1
{\displaystyle f(x)=1}
if
x
∈
S
{\displaystyle x\in S}
f
(
x
)
=
0
{\displaystyle f(x)=0}
if
x
∉
S
{\displaystyle x\notin S}
.
In other words, the set
S
{\displaystyle S}
is computable if and only if the indicator function
1
S
{\displaystyle \mathbb {1} _{S}}
is computable.
Examples
Every recursive language is computable.
Every finite or cofinite subset of the natural numbers is computable.
The empty set is computable.
The entire set of natural numbers is computable.
Every natural number is computable.
The subset of prime numbers is computable.
The set of Gödel numbers is computable.
Begin with the source’s own compact description: “Computable set” is set where an algorithm can take a number as an input and can decide whether the number belongs to the set. The dossier treats that line as a proposition to test through Computable, where and algorithm, not as a finished interpretation.
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This entry incorporates text from “Computable set” on English Wikipedia. Contributors are listed in the page history. Text is available under the Creative Commons Attribution-ShareAlike 4.0 License. Selected authority identifiers and statements are retrieved from Wikidata under CC0; their references and qualifiers remain part of the verification path.