Computable 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. A set is noncomputable (or undecidable) if it is not computable.

Definition

A subset $S$ of the natural numbers is computable if there exists a total computable function $f$ such that:

$f(x)=1$ if $x\in S$

$f(x)=0$ if $x\notin S$.

In other words, the set $S$ is computable if and only if the indicator function ${\mathrm{𝟙}}_S$ 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.

Non-examples

Main article: List of undecidable problems

• The set of Turing machines that halt is not computable.
• The set of pairs of homeomorphic finite simplicial complexes is not computable.
• The set of busy beaver champions is not computable.
• Hilbert's tenth problem is not computable.

Properties

Both A, B are sets in this section.

• If A is computable then the complement of A is computable.
• If A and B are computable then:
  • A ∩ B is computable.
  • A ∪ B is computable.
  • The image of A × B under the Cantor pairing function is computable.

In general, the image of a computable set under a computable function is computably enumerable, but possibly not computable.

• A is computable if and only if A and the complement of A are both computably enumerable(c.e.).
• The preimage of a computable set under a total computable function is computable.
• The image of a computable set under a total computable bijection is computable.

A is computable if and only if it is at level ${\Delta }_1^0$ of the arithmetical hierarchy.

A is computable if and only if it is either the image (or range) of a nondecreasing total computable function, or the empty set.

See also

• Computably enumerable
• Decidability (logic)
• Recursively enumerable language
• Recursive language
• Recursion

Bibliography

• Cutland, N. Computability. Cambridge University Press, Cambridge-New York, 1980. ;
• Rogers, H. The Theory of Recursive Functions and Effective Computability, MIT Press. ;
• Soare, R. Recursively enumerable sets and degrees. Perspectives in Mathematical Logic. Springer-Verlag, Berlin, 1987.