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Computer science Definition

Descriptive complexity theory

Builds on StructureSignatureComplexity classSentence

Definition

Let ℒ\mathcal L be a logic and let 𝒞\mathcal C be a complexity class. Fix a finite relational signature σ\sigma. A finite σ\sigma-structure is encoded as a binary string by listing the tuples of each relation for some ordering of its universe. The logic ℒ\mathcal L captures 𝒞\mathcal C if the following holds for every σ\sigma and every class KK of finite σ\sigma-structures that is closed under isomorphism.

The class KK belongs to 𝒞\mathcal C if and only if some ℒ\mathcal L-sentence φ\varphi defines KK. This means that a finite σ\sigma-structure is in KK exactly when it satisfies φ\varphi.

Descriptive complexity theory is the study of which logics capture which complexity classes.

Remarks

Example: existential second-order logic captures NP, by Fagin's theorem. For instance, the class of 33-colorable graphs is in NP, and it is defined by an existential second-order sentence that guesses three color classes.

Non-example: first-order logic does not capture the class P. Over the empty signature, the class of finite sets with an even number of elements can be decided in polynomial time. Yet no first-order sentence defines it, as one shows with Ehrenfeucht–Fraïssé games.

On ordered structures, where σ\sigma contains a linear order, first-order logic with a least fixed point operator captures P (Immerman and Vardi). Whether some logic captures P on all finite structures is a major open question.

Reading path

To understand this page from the ground up, read these 15 pages in order. The foundations come first.

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Complexity class, concept, no evidence levelComplexity classconceptFunction, concept, no evidence levelFunctionconceptProposition, concept, no evidence levelPropositionconceptSentence, concept, no evidence levelSentenceconceptStructure, concept, no evidence levelStructureconceptConjunction, definition, no evidence levelConjunctiondefinitionDisjunction, definition, no evidence levelDisjunctiondefinitionImplication, definition, no evidence levelImplicationdefinitionNegation, definition, no evidence levelNegationdefinitionPredicate, definition, no evidence levelPredicatedefinitionBiconditional, definition, no evidence levelBiconditionaldefinitionPropositional logic, concept, no evidence levelPropositional logicconceptPredicate logic, concept, no evidence levelPredicate logicconceptSignature, definition, no evidence levelSignaturedefinitionDescriptive complexity theory, definition, no evidence levelDescriptive complexit…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. Complexity class concept
  2. Function concept
  3. Proposition concept
  4. Sentence concept
  5. Structure concept
  6. Conjunction definition
  7. Disjunction definition
  8. Implication definition
  9. Negation definition
  10. Predicate definition
  11. Biconditional definition
  12. Propositional logic concept
  13. Predicate logic concept
  14. Signature definition
  15. Descriptive complexity theory definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Descriptive_complexity_theory
About this page
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MSC68Q19, 03C13, 68Q15
DefinesDescriptive complexity theory
UsesStructure, Signature, Complexity class, Sentence
Changed3 November 2026

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