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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.
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).
Examples of RE-complete problems: Halting problem: Whether a program given a finite input finishes running or will run forever. By Rice's theorem, deciding membership in any nontrivial subset of the set of recursive functions is RE-hard. It will be complete whenever the set is recursively enumerable.
A recursive formal language is a recursive subset in the set of all possible words over the alphabet of the language. A recursive language is a formal language for which there exists a Turing machine that, when presented with any finite input string, halts and accepts if the string is in the language, and halts and rejects otherwise. The Turing ...
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.
A recursive language is a computable subset of a formal language. The set of Gödel numbers of arithmetic proofs described in Kurt Gödel's paper "On formally undecidable propositions of Principia Mathematica and related systems I" is computable; see Gödel's incompleteness theorems. Non-examples:
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 set being enumerated is then called recursively enumerable (or computably enumerable in more contemporary language), referring to the use of recursion theory in formalizations of what it means for the map to be computable. In this sense, a subset of the natural numbers is computably enumerable if it is the range of a computable function. In ...