Recursive set


In computability theory, a set of natural numbers is called recursive, computable or decidable if there is an algorithm which takes a number as input, terminates after a finite amount of time and correctly decides whether the number belongs to the set or not.
A set which is not computable is called noncomputable or undecidable.
A more general class of sets than the decidable ones consists of the recursively enumerable sets, also called semidecidable sets. For these sets, it is only required that there is an algorithm that correctly decides when a number is in the set; the algorithm may give no answer for numbers not in the set.

Formal definition

A subset S of the natural numbers is called recursive if there exists a total computable function f such that f = 1 if x∈S and f = 0 if x∉S. In other words, the set S is recursive if and only if the indicator function 1S is computable.

Examples and non-examples

Examples:
Non-examples:
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, A ∪ B and the image of A × B under the Cantor pairing function are recursive sets.
A set A is a recursive set if and only if A and the complement of A are both recursively enumerable sets. The preimage of a recursive set under a total computable function is a recursive set. The image of a computable set under a total computable bijection is computable.
A set is recursive if and only if it is at level of the arithmetical hierarchy.
A set is recursive if and only if it is either the range of a nondecreasing total computable function or the empty set. The image of a computable set under a nondecreasing total computable function is computable.