In mathematics, logic and computer science, a formal language is called recursively enumerable (also recognizable, partially decidable, semidecidable, Turing-acceptable or Turing-recognizable) if it is a recursively enumerable subset in the set of all possible words over the alphabet of the language, i.e., if there exists a Turing machine which will enumerate all valid strings of the language. Recursively enumerable languages are known as type-0 languages in the Chomsky hierarchy of formal languages. All regular, context-free, context-sensitive and recursive languages are recursively enumerable. The class of all recursively enumerable languages is called RE.

Definitions

There are three equivalent definitions of a recursively enumerable language: All regular, context-free, context-sensitive and recursive languages are recursively enumerable. Post's theorem shows that RE, together with its complement co-RE, correspond to the first level of the arithmetical hierarchy.

Example

The set of halting Turing machines is recursively enumerable but not recursive. Indeed, one can run the Turing machine and accept if the machine halts, hence it is recursively enumerable. On the other hand, the problem is undecidable. Some other recursively enumerable languages that are not recursive include:

Closure properties

Recursively enumerable languages (REL) are closed under the following operations. That is, if L and P are two recursively enumerable languages, then the following languages are recursively enumerable as well: Recursively enumerable languages are not closed under set difference or complementation. The set difference L - P is recursively enumerable if P is recursive. If L is recursively enumerable, then the complement of L is recursively enumerable if and only if L is also recursive.

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