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| In [[mathematics]], [[logic]] and [[computer science]], a [[formal language]] is called '''recursively enumerable''' (also '''recognizable''', '''partially decidable''', '''semidecidable''' or '''Turing-acceptable''') if it is a [[recursively enumerable set|recursively enumerable subset]] in the [[set (mathematics)|set]] of all possible words over the [[alphabet (computer science)|alphabet]] of the language, i.e., if there exists a [[Turing machine]] which will enumerate all valid strings of the language.
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| Recursively enumerable languages are known as '''type-0''' languages in the [[Chomsky hierarchy]] of formal languages. All [[regular language|regular]], [[context-free grammar|context-free]], [[context-sensitive language|context-sensitive]] and [[recursive language|recursive]] languages are recursively enumerable.
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| The class of all recursively enumerable languages is called '''[[RE (complexity)|RE]]'''. | |
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| ==Definitions==
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| There exist three equivalent major definitions for the concept of a recursively enumerable language.
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| # A recursively enumerable language is a [[recursively enumerable set|recursively enumerable]] [[subset]] in the [[set (mathematics)|set]] of all possible words over the [[alphabet (computer science)|alphabet]] of the [[formal language|language]].
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| # A recursively enumerable language is a formal language for which there exists a [[Turing machine]] (or other [[computable function]]) which will enumerate all valid strings of the language. Note that if the language is [[Infinity|infinite]], the enumerating algorithm provided can be chosen so that it avoids repetitions, since we can test whether the string produced for number ''n'' is "already" produced for a number which is less than ''n''. If it already is produced, use the output for input ''n+1'' instead (recursively), but again, test whether it is "new".
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| # A recursively enumerable language is a formal language for which there exists a Turing machine (or other computable function) that will halt and accept when presented with any [[literal string|string]] in the language as input but may either halt and reject or loop forever when presented with a string not in the language. Contrast this to [[recursive language]]s, which require that the Turing machine halts in all cases.
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| All [[regular language|regular]], [[context-free language|context-free]], [[context-sensitive language|context-sensitive]] and [[recursive language|recursive]] languages are recursively enumerable.
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| [[Post's theorem]] shows that '''[[RE (complexity)|RE]]''', together with its [[complement (complexity)|complement]] [[co-RE]], correspond to the first level of the [[arithmetical hierarchy]].
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| ==Example==
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| The [[Halting problem]] is recursively enumerable but not recursive. Indeed one can run the Turing Machine and accept if the machine halts, hence it is r.e. On the other hand the problem is undecidable.
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| Some other r.e. languages that are not recursive:
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| * [[Post correspondence problem]]
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| * [[Mortality (computability theory)]]
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| * [[Entscheidungsproblem]]
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| == Closure properties ==
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| Recursively enumerable languages are [[closure (mathematics)|closed]] under the following operations. That is, if ''L'' and ''P'' are two recursively enumerable languages, then the following languages are recursively enumerable as well:
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| * the [[Kleene star]] <math>L^*</math> of ''L''
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| * the [[Concatenation#Concatenation_of_sets_of_strings|concatenation]] <math>L \circ P</math> of ''L'' and ''P''
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| * the [[Union (set theory)|union]] <math>L \cup P</math>
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| * the [[Intersection (set theory)|intersection]] <math>L \cap P</math>.
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| Note that recursively enumerable languages are not closed under set difference or complementation. The set difference ''L'' - ''P'' may or may not be recursively enumerable. If ''L'' is recursively enumerable, then the complement of ''L'' is recursively enumerable if and only if ''L'' is also recursive.
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| == References ==
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| * Sipser, M. (1996), ''Introduction to the Theory of Computation'', PWS Publishing Co.
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| * Kozen, D.C. (1997), ''Automata and Computability'', Springer.
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| == External links ==
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| * {{CZoo|Class RE|R#re}}
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| * [http://www.cs.colostate.edu/~massey/Teaching/cs301/RestrictedAccess/Slides/301lecture23.pdf Lecture slides]
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| {{Formal languages and grammars}}
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| [[Category:Formal languages]]
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| [[Category:Theory of computation]]
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| CNF
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