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. Formally, a subset ${\displaystyle S}$ of the natural numbers is computable if there exists a total computable function ${\displaystyle f}$ such that

${\displaystyle f(x)={\begin{cases}1&{\text{if }}\,x\in S,\\0&{\text{if }}\,x\not \in S.\end{cases}}}$

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.

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 ${\displaystyle \Delta _{1}^{0}}$ 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

Notes

References