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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.
The set of recursive languages is a subset of both RE and co-RE. [3] In fact, it is the intersection of those two classes, because we can decide any problem for which there exists a recogniser and also a co-recogniser by simply interleaving them until one obtains a result.
Note that the set of grammars corresponding to recursive languages is not a member of this hierarchy; these would be properly between Type-0 and Type-1. Every regular language is context-free, every context-free language is context-sensitive, every context-sensitive language is recursive and every recursive language is recursively enumerable.
The halting language is therefore recursively enumerable. It is possible to construct languages which are not even recursively enumerable, however. A simple example of such a language is the complement of the halting language; that is the language consisting of all Turing machines paired with input strings where the Turing machines do not halt ...
An Enumerable Language is Turing Recognizable. It's very easy to construct a Turing Machine that recognizes the enumerable language . We can have two tapes. On one tape we take the input string and on the other tape, we run the enumerator to enumerate the strings in the language one after another.
A recursively enumerable language is a computably enumerable subset of a formal language. The set of all provable sentences in an effectively presented axiomatic system is a computably enumerable set. Matiyasevich's theorem states that every computably enumerable set is a Diophantine set (the converse is trivially true).
A universal Turing machine can calculate any recursive function, decide any recursive language, and accept any recursively enumerable language. According to the Church–Turing thesis , the problems solvable by a universal Turing machine are exactly those problems solvable by an algorithm or an effective method of computation , for any ...
Thus a language is computable just in case there is a procedure that is able to correctly tell whether arbitrary words are in the language. A language is computably enumerable (synonyms: recursively enumerable, semidecidable) if there is a computable function f such that f(w) is defined if and only if the word w is in the language.