recursive set

In computability theory a countable set is called recursive, computable or decidable if we can construct an algorithm which terminates after a finite amount of time and decides whether a given element belongs to the set or not. A more general class of sets are called recursively enumerable sets. For those sets the algorithm only terminates if the element belongs to the set otherwise it does not halt.

Definition

A subset S of the natural numbers is called recursive if there exists a total computable function

f:\mathbb{N} \to \mathbb{N}

with

f(x) =

\left\{\begin{matrix} 0 &\mbox{if}\ x \in S \\ \neq 0 &\mbox{if}\ x \notin S \end{matrix}\right. In other words the set S is recursive iff the indicator function

1_{S}

is computable

Notes

If A is a recursive set then the complement of A is a recursive set. If A and B are recursive sets then A ∩ B and A ∪ B are recursive sets. A set A is a recursive set iff A and the complement of A are recursively enumerable sets. The preimage of a recursive set under a total computable function is a recursive set.

Examples

the empty set
the natural numbers
every finite subset of the natural numbers
the set of prime numbers
a recursive language is a recursive set in the set of all possible words over the alphabet of the language.

Recursive Set

Definition

Notes

Examples

See also