recursive set

A subset S of the natural numbersMathworldPlanetmath is said to be recursive if its characteristic functionMathworldPlanetmathPlanetmathPlanetmath

χS(x) := {1 if xS0 if x-S

is recursive (computable). In other words, there is an algorithmMathworldPlanetmath (via Turing machineMathworldPlanetmath for example) that determines whether an element is in S or not in S.

More generally, a subset Sn is recursive if its characteristic function fS is recursive.

A recursive setMathworldPlanetmath is also known as a decidable set or a computable set.

Examples of recursive sets are finite subset of , the set itself, the set of even integers, the set of Fibonacci numbersMathworldPlanetmath, the set of pairs (a,b) where a divides b, and the set of prime numbersMathworldPlanetmath. In the last example, one may use the Sieve of EratosthenesMathworldPlanetmathPlanetmath as an algorithm to determine the primality of an integer.

A set S is recursively enumerable if the partial functionMathworldPlanetmath

f(x) := {1 if xSundefined if x-S

is computable. In other words, there is an algorithm that halts (and returns 1) only when an element in S is used as an input.


  • A special case of a recursive set is that of a primitive recursive set. A set is primitive recursive if its characteristic function is primitive recursive ( All of the examples cited above are primitive recursive.

  • On the other hand, one can broaden the scope of recursiveness to sets which are not necessarily subsets of n. Below are two examples:

    • Since can be effectively embedded in , so the notion of recursive sets be extended to subsets of .

    • Since every finite setMathworldPlanetmath Σ can be encoded by the natural numbers, we can define a recursive language over Σ to be a subset LΣ* such that L, when encoded by the natural numbers, is a recursive set. Equivalently, L is recursive iff there is a Turing machine that decides L (accepts L and rejects Σ*-L).

  • Similarly, recursive enumerability can be defined on languagesPlanetmathPlanetmath: a language L over Σ is recursively enumerable if its encoding by the natural numbers is a recursively enumerable set. This is equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath to saying that L is accepted by a Turing machine.

  • Using the above definition, one can define a recursive predicate or a recursively enumerable predicate φ(x) according to whether {xφ(x)} is a recursive or recursively enumerable set respectively.

Title recursive set
Canonical name RecursiveSet
Date of creation 2013-03-22 17:34:52
Last modified on 2013-03-22 17:34:52
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 16
Author CWoo (3771)
Entry type Definition
Classification msc 03B25
Classification msc 03D20
Synonym decidable set
Synonym computable set
Synonym decidable predicate
Synonym computable predicate
Defines recursively enumerable set
Defines recursive predicate
Defines recursively enumerable predicate
Defines recursive language