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| In [[computability theory]] and [[computational complexity theory]], '''RE''' ([[Recursively enumerable set|recursively enumerable]]) is the [[complexity class|class]] of [[decision problem]]s for which a 'yes' answer can be verified by a [[Turing machine]] in a finite amount of time.<ref>{{CZoo|Class RE|R#re}}</ref> Informally, it means that if the answer is 'yes', then there is some procedure which takes finite time to determine this. On the other hand, if the answer is 'no', the machine might never halt. Equivalently, '''RE''' is the class of decision problems for which a Turing machine can list all the 'yes' instances, one by one (this is what 'enumerable' means).
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| Similarly, '''co-RE''' is the set of all languages that are complements of a language in '''RE'''. In a sense, '''co-RE''' contains languages of which membership can be disproved in a finite amount of time, but proving membership might take forever.
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| Each member of '''RE''' is a [[recursively enumerable set]] and therefore a [[Diophantine set]].
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| ==Relations to other classes==
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| The set of [[recursive language]]s ('''[[R (complexity)|R]]''') is a subset of both '''RE''' and '''co-RE'''.<ref>{{CZoo|Class co-RE|C#core}}</ref> 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. Therefore:
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| :<math>\mbox{R} = \mbox{RE}\cap\mbox{co-RE}.</math>
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| ==RE-complete==
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| '''RE-complete''' is the set of decision problems that are complete for '''RE'''. In a sense, these are the "hardest" recursively enumerable problems. All such problems are non[[recursive language|recursive]]. Generally, no constraint is placed on the reductions used except that they must be [[many-one reduction]]s.
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| Examples of RE-complete problems:
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| #[[Halting problem]]: Whether a program given a finite input finishes running or will run forever.
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| #By [[Rice's Theorem]], deciding membership in any nontrivial subset of the set of [[Computable function|recursive functions]] is '''RE'''-hard. It will be complete whenever the set is recursively enumerable.
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| #{{harvs|first=John|last=Myhill|authorlink=John Myhill|year=1955|txt}}<ref>{{citation
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| | last = Myhill | first = John | authorlink = John Myhill
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| | doi = 10.1002/malq.19550010205
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| | journal = Zeitschrift für Mathematische Logik und Grundlagen der Mathematik
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| | mr = 0071379
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| | pages = 97–108
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| | title = Creative sets
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| | volume = 1
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| | year = 1955}}.</ref> has proven that all [[creative set]]s are '''RE'''-complete.
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| #The uniform [[word problem (mathematics)|word problem]] for [[group (mathematics)|groups]] or [[semigroup]]s. [Indeed, the [[word problem for groups|word problem for some individual groups]] is '''RE'''-complete.]
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| #Deciding membership in a general [[unrestricted grammar|unrestricted]] [[formal grammar]]. [Again, certain individual grammars have '''RE'''-complete membership problem.]
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| #The [[validity]] problem for [[first-order logic]].
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| #[[Post correspondence problem]]: Given a finite set of strings, determine if there is a string that can be factored into a composition of the strings (allowing repeats) in two different ways.
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| #Determining if a [[Diophantine equation]] has any integer solutions.
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| ==co-RE-complete==
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| '''co-RE-complete''' is the set of decision problems that are complete for '''co-RE'''. In a sense, these are the complements of the hardest recursively enumerable problems.
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| Examples of co-RE-complete problems:
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| #The [[Domino Problem]] for [[Wang tile]]s.
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| #The [[satisfiability]] problem for [[first-order logic]]
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| ==See also==
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| [[List of undecidable problems]]
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| ==References==
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| <references/>
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| {{ComplexityClasses}}
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| [[Category:Complexity classes]]
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