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.

Contents
1 Definition
2 Notes
3 Examples
4 See also

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 1S 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

See also

See also: Recursive set, Algorithm, Alphabet, Complement (set theory), Computability theory, Computable function, Countable, Empty set, Formal language, Iff