Math 2.0

Mathematical logic and foundations Proof theory Concept

Proof complexity

Builds on Proof theoryPropositional logicTautology

Summary

Proof complexity is the part of proof theory and of computational complexity that studies the size of formal proofs. Its main objects are proof systems for propositional logic, and the question is how short a proof of a given tautology can be. Cook and Reckhow showed in 1979 that a proof system with short proofs of all tautologies exists if and only if NP equals coNP. So lower bounds on proof size for ever stronger systems are a route toward separating complexity classes.

Reading path

To understand this page from the ground up, read these 14 pages in order. The foundations come first.

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Proposition, concept, no evidence levelPropositionconceptTautology, concept, no evidence levelTautologyconceptConjunction, definition, no evidence levelConjunctiondefinitionDisjunction, definition, no evidence levelDisjunctiondefinitionImplication, definition, no evidence levelImplicationdefinitionNegation, definition, no evidence levelNegationdefinitionPredicate, definition, no evidence levelPredicatedefinitionBiconditional, definition, no evidence levelBiconditionaldefinitionModus ponens, concept, no evidence levelModus ponensconceptMathematical proof, concept, no evidence levelMathematical proofconceptPropositional logic, concept, no evidence levelPropositional logicconceptPredicate logic, concept, no evidence levelPredicate logicconceptProof theory, concept, no evidence levelProof theoryconceptProof complexity, concept, no evidence levelProof complexityconcept
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. Proposition concept
  2. Tautology concept
  3. Conjunction definition
  4. Disjunction definition
  5. Implication definition
  6. Negation definition
  7. Predicate definition
  8. Biconditional definition
  9. Modus ponens concept
  10. Mathematical proof concept
  11. Propositional logic concept
  12. Predicate logic concept
  13. Proof theory concept
  14. Proof complexity concept

Further reading

  1. Web https://en.wikipedia.org/wiki/Proof_complexity
About this page
Tag1Q046J0AW2
URIm2://1Q046J0AW2
Kindconcept
WritingW0 stub
MSC03F20, 68Q17
UsesProof theory, Propositional logic, Tautology
Same as
Changed2 November 2026

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