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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.
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).
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 ...
In mathematical logic, Craig's theorem (also known as Craig's trick [1]) states that any recursively enumerable set of well-formed formulas of a first-order language is (primitively) recursively axiomatizable. This result is not related to the well-known Craig interpolation theorem, although both results are named after the same logician ...
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 ...
Though undecidable languages are not recursive languages, they may be subsets of Turing recognizable languages: i.e., such undecidable languages may be recursively enumerable. Many, if not most, undecidable problems in mathematics can be posed as word problems : determining when two distinct strings of symbols (encoding some mathematical ...