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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.
This means that every formal language that can be decided by such a machine is a context-sensitive language, and every context-sensitive language can be decided by such a machine. This set of languages is also known as NLINSPACE or NSPACE(O(n)), because they can be accepted using linear space on a non-deterministic Turing machine. [1]
With regard to what actions the machine actually does, Turing (1936) [2] states the following: "This [example] table (and all succeeding tables of the same kind) is to be understood to mean that for a configuration described in the first two columns the operations in the third column are carried out successively, and the machine then goes over into the m-configuration in the final column."
An oracle machine or o-machine is a Turing a-machine that pauses its computation at state "o" while, to complete its calculation, it "awaits the decision" of "the oracle"—an entity unspecified by Turing "apart from saying that it cannot be a machine" (Turing (1939), The Undecidable, p. 166–168).
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 machine always halts: it is known as a decider and is said to decide the recursive language.
There are no constraints on the productions rules. They generate exactly all languages that can be recognized by a Turing machine, thus any language that is possible to be generated can be generated by a Type-0 grammar. [8] These languages are also known as the recursively enumerable or Turing-recognizable languages. [8]
Deterministic context-free languages can be recognized by a deterministic Turing machine in polynomial time and O(log 2 n) space; as a corollary, DCFL is a subset of the complexity class SC. [3] The set of deterministic context-free languages is closed under the following operations: [4] complement; inverse homomorphism; right quotient with a ...
The equivalence of unrestricted grammars to Turing machines implies the existence of a universal unrestricted grammar, a grammar capable of accepting any other unrestricted grammar's language given a description of the language. For this reason, it is theoretically possible to build a programming language based on unrestricted grammars (e.g. Thue).