Math 2.0

Mathematical logic and foundations Definition

Decidable language

Builds on Turing machineFormal language

Definition

Let Σ\Sigma be an alphabet and let L⊆Σ∗L \subseteq \Sigma^* be a formal language. A Turing machine MM with input alphabet Σ\Sigma, an accepting halting state and a rejecting halting state decides LL if both conditions hold.

  1. MM halts on every input w∈Σ∗w \in \Sigma^*.
  2. MM halts in the accepting state exactly when w∈Lw \in L.

The language LL is decidable, or recursive, if some Turing machine decides it.

Remarks

Example: L={anbn:n≥0}L = \set{a^n b^n : n \ge 0} over {a,b}\set{a, b} is decidable. A machine first rejects strings not of the form aibja^i b^j. Then it repeatedly crosses out the first aa and the last bb, and rejects if one kind runs out before the other. Each round shortens the remaining string, so the machine halts on every input, and it accepts exactly the strings in LL. Non-example: the halting language H={⟨M,w⟩:M halts on w}H = \set{\langle M, w \rangle : M \text{ halts on } w} is not decidable, by the undecidability of the halting problem. A machine that simulates MM on ww satisfies clause 2 but not clause 1, since it runs forever when MM does. Decidable languages are closed under complement, union, intersection, concatenation and Kleene star. Under a computable bijection between Σ∗\Sigma^* and ℕ\N, they correspond to the computable 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 languagedefinitionDecidable language, definition, no evidence levelDecidable languagedefinition
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. Decidable language definition

Where this leads

3 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Recursive_language
About this page
Tag3J6YG01N22
URIm2://3J6YG01N22
Kinddefinition
WritingW0 stub
MSC03D25, 68Q45, 68Q04
DefinesDecidable language
UsesTuring machine, Formal language
Changed3 November 2026

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