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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.
Thus the halting problem is an example of a computably enumerable (c.e.) set, which is a set that can be enumerated by a Turing machine (other terms for computably enumerable include recursively enumerable and semidecidable). Equivalently, a set is c.e. if and only if it is the range of some computable function.
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.
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).
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 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 ...
In the context of formal language theory, a Turing machine is capable of enumerating some arbitrary subset of valid strings of an alphabet. A set of strings which can be enumerated in this manner is called a recursively enumerable language. The Turing machine can equivalently be defined as a model that recognises valid input strings, rather ...