Recursively enumerable set |
In the theory of computability (often less suggestively called recursion theory), a set S of natural numbers or tuples of natural numbers, or of literal strings, is recursively enumerable or
computably enumerable or semi-decidable if it satisfies either (and therefore both) of the
following equivalent conditions.
- There is an algorithm that, when given a natural number n (or tuple
of natural numbers, or word, as the case may be) eventually halts if n is a member of S and otherwise runs
forever.
- There is an algorithm that "generates" the members of S. That means that its output is simply a list of the members
of S: s1, s2, s3, ... If necessary it runs forever.
Common-programming-sense should suggest how to convert either of these algorithms to the other, thus showing the equivalence
of the existence of either with the existence of the other. The first condition suggests why the term semi-decidable is
sometimes used; the second suggests why computably enumerable is used. The word recursive is in this context
taken to be synonymous with computable; see recursive function.
It may be fairly readily seen that any set S is recursive
(i.e., decidable) if and only if both S and the complement of S are recursively enumerable.
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