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Mathematical logic and foundations Definition

Recursively enumerable language

Builds on Turing machineFormal language

Definition

Let Σ\Sigma be an alphabet and let L⊆Σ∗L \subseteq \Sigma^* be a formal language. Let MM be a Turing machine with input alphabet Σ\Sigma and an accepting halting state. The machine MM recognises LL if, for every w∈Σ∗w \in \Sigma^*, the run of MM on ww halts in the accepting state exactly when w∈Lw \in L. On inputs w∉Lw \notin L the machine may halt in another state or run forever. The language LL is recursively enumerable, or Turing-recognizable, if some Turing machine recognises it. The language LL is co-recognizable if its complement Σ∗∖L\Sigma^* \setminus L is recursively enumerable.

Remarks

Equivalently, LL is recursively enumerable when it is empty or there is a machine that prints exactly the members of LL, in some order and possibly with repetitions. Example: the halting language H={⟨M,w⟩:M halts on w}H = \set{\langle M, w \rangle : M \text{ halts on } w} is recursively enumerable. A universal machine simulates MM on ww and accepts if the simulation halts, so it accepts exactly the members of HH. Non-example: the complement of HH is not recursively enumerable. If it were, HH would be decidable by the characterisation of decidable languages, which contradicts the undecidability of the halting problem. Every decidable language is recursively enumerable. Under a computable bijection between Σ∗\Sigma^* and ℕ\N, these languages correspond to the computably enumerable sets.

Reading path

To understand this page from the ground up, read these 28 pages in order. The foundations come first.

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Element, definition, no evidence levelElementdefinitionFunction, concept, no evidence levelFunctionconceptMonoid, concept, no evidence levelMonoidconceptProposition, concept, no evidence levelPropositionconceptSet, concept, no evidence levelSetconceptSet, definition, no evidence levelSetdefinitionTuring machine, definition, no evidence levelTuring machinedefinitionImplication, definition, no evidence levelImplicationdefinitionPredicate, definition, no evidence levelPredicatedefinitionOrdered pair, definition, no evidence levelOrdered pairdefinitionExistential quantifier, definition, no evidence levelExistential quantifierdefinitionUniversal quantifier, definition, no evidence levelUniversal quantifierdefinitionCartesian product, definition, no evidence levelCartesian productdefinitionSubset, definition, no evidence levelSubsetdefinitionUniqueness quantifier, definition, no evidence levelUniqueness quantifierdefinitionBinary relation, definition, no evidence levelBinary relationdefinitionFunction, definition, no evidence levelFunctiondefinitionCodomain, definition, no evidence levelCodomaindefinitionInjective function, definition, no evidence levelInjective functiondefinitionFree monoid, definition, no evidence levelFree monoiddefinitionSurjective function, definition, no evidence levelSurjective functiondefinitionBijective function, definition, no evidence levelBijective functiondefinitionEquinumerous sets, definition, no evidence levelEquinumerous setsdefinitionFinite set, definition, no evidence levelFinite setdefinitionAlphabet, definition, no evidence levelAlphabetdefinitionString, definition, no evidence levelStringdefinitionFormal language, definition, no evidence levelFormal languagedefinitionRecursively enumerable language, definition, no evidence levelRecursively enumerabl…definition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Element definition
  2. Function concept
  3. Monoid concept
  4. Proposition concept
  5. Set concept
  6. Set definition
  7. Turing machine definition
  8. Implication definition
  9. Predicate definition
  10. Ordered pair definition
  11. Existential quantifier definition
  12. Universal quantifier definition
  13. Cartesian product definition
  14. Subset definition
  15. Uniqueness quantifier definition
  16. Binary relation definition
  17. Function definition
  18. Codomain definition
  19. Injective function definition
  20. Free monoid definition
  21. Surjective function definition
  22. Bijective function definition
  23. Equinumerous sets definition
  24. Finite set definition
  25. Alphabet definition
  26. String definition
  27. Formal language definition
  28. Recursively enumerable language definition

Where this leads

1 page builds on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Recursively_enumerable_language
About this page
TagXYZEVGCBXN
URIm2://XYZEVGCBXN
Kinddefinition
WritingW0 stub
MSC03D25, 68Q45, 68Q04
DefinesRecursively enumerable language
UsesTuring machine, Formal language
Changed3 November 2026

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