recursively enumerable

For a languagePlanetmathPlanetmath L, TFAE:

  • There exists a Turing machineMathworldPlanetmath f such that x.(xL)the computation f(x) terminates.

  • There exists a total recursive function f:L which is onto.

  • There exists a total recursive function f:L which is one-to-one and onto.

A language L fulfilling any (and therefore all) of the above conditions is called recursively enumerablePlanetmathPlanetmath.


  1. 1.
  2. 2.

    The set of encodings of Turing machines which halt when given no input.

  3. 3.

    The set of encodings of theorems of Peano arithmeticMathworldPlanetmathPlanetmath.

  4. 4.

    The set of integers n for which the hailstone sequence starting at n reaches 1. (We don’t know if this set is recursive, or even if it is N; but a trivial program shows it is recursively enumerable.)

Title recursively enumerable
Canonical name RecursivelyEnumerable
Date of creation 2013-03-22 12:44:32
Last modified on 2013-03-22 12:44:32
Owner ariels (338)
Last modified by ariels (338)
Numerical id 6
Author ariels (338)
Entry type Definition
Classification msc 03D25
Synonym semi-recursive
Related topic HaltingProblem
Related topic TuringComputable
Defines semi-recursive
Defines recursively enumerable function