Recursively enumerable
In computability theory, traditionally called recursion theory, a set S of natural numbers is called recursively enumerable, computably enumerable, semidecidable, provable or Turing-recognizable if:
Or, equivalently,
The first condition suggests why the term semidecidable is sometimes used; the second suggests why computably enumerable is used. The abbreviations r.e. and c.e. are often used, even in print, instead of the full phrase.
In computational complexity theory, the complexity class containing all recursively enumerable sets is RE. In recursion theory, the lattice of r.e. sets under inclusion is denoted .
A set S of natural numbers is called recursively enumerable if there is a partial recursive function whose domain is exactly S, meaning that the function is defined if and only if its input is a member of S.
The following are all equivalent properties of a set S of natural numbers:
The equivalence of semidecidability and enumerability can be obtained by the technique of dovetailing.
The Diophantine characterizations of a recursively enumerable set, while not as straightforward or intuitive as the first definitions, were found by Yuri Matiyasevich as part of the negative solution to Hilbert's Tenth Problem. Diophantine sets predate recursion theory and are therefore historically the first way to describe these sets (although this equivalence was only remarked more than three decades after the introduction of recursively enumerable sets). The number of bound variables in the above definition of the Diophantine set is the best known so far; it might be that a lower number can be used to define all diophantine sets.
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