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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.
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).
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.
In computer science, in particular in the field of formal language theory, an abstract family of languages is an abstract mathematical notion generalizing characteristics common to the regular languages, the context-free languages and the recursively enumerable languages, and other families of formal languages studied in the scientific literature.
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 ...
In formal language theory, a cone is a set of formal languages that has some desirable closure properties enjoyed by some well-known sets of languages, in particular by the families of regular languages, context-free languages and the recursively enumerable languages. [1]
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.