Math 2.0

Computer science Definition

Decision problem

Builds on Formal languageEncoding of data

Definition

Let Σ\Sigma be an alphabet. A decision problem over Σ\Sigma is a formal language L⊆Σ∗L \subseteq \Sigma^*. Its instances are the strings w∈Σ∗w \in \Sigma^*, the yes-instances are the strings in LL, and the no-instances are the strings not in LL.

More generally, a decision problem is given by a set II of instances and a subset Y⊆IY \subseteq I of yes-instances. With an encoding e ⁣:I→Σ∗e \colon I \to \Sigma^*, it corresponds to the language e(Y)⊆Σ∗e(Y) \subseteq \Sigma^*.

Remarks

Example: PRIMES is the set of binary numerals of prime numbers, a language over {0,1}\set{0, 1}. The string 111111 is a yes-instance, since 77 is prime, and 110110 is a no-instance, since 66 is not. Non-example: computing the greatest common divisor of two numbers is not a decision problem. Its answers are numbers, not yes or no. It is a function problem; a related decision problem asks whether the greatest common divisor of aa and bb equals 11. For computability, the choice of a reasonable encoding does not matter. For complexity, the encoding should not inflate the input. Writing nn in unary instead of binary can change a running time from exponential to polynomial in the input length.

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To understand this page from the ground up, read these 27 pages in order. The foundations come first.

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Element, definition, no evidence levelElementdefinitionInjective function, concept, no evidence levelInjective functionconceptMonoid, concept, no evidence levelMonoidconceptProposition, concept, no evidence levelPropositionconceptSet, definition, no evidence levelSetdefinitionImplication, 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 levelStringdefinitionEncoding of data, definition, no evidence levelEncoding of datadefinitionFormal language, definition, no evidence levelFormal languagedefinitionDecision problem, definition, no evidence levelDecision problemdefinition
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. Injective function concept
  3. Monoid concept
  4. Proposition concept
  5. Set definition
  6. Implication definition
  7. Predicate definition
  8. Ordered pair definition
  9. Existential quantifier definition
  10. Universal quantifier definition
  11. Cartesian product definition
  12. Subset definition
  13. Uniqueness quantifier definition
  14. Binary relation definition
  15. Function definition
  16. Codomain definition
  17. Injective function definition
  18. Free monoid definition
  19. Surjective function definition
  20. Bijective function definition
  21. Equinumerous sets definition
  22. Finite set definition
  23. Alphabet definition
  24. String definition
  25. Encoding of data definition
  26. Formal language definition
  27. Decision problem definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Decision_problem
About this page
TagB3DM1AYBVE
URIm2://B3DM1AYBVE
Kinddefinition
WritingW0 stub
MSC68Q01, 03D35
DefinesDecision problem
UsesFormal language, Encoding of data
Changed3 November 2026

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