Lesson plan
Proof complexity
In logic and theoretical computer science, and specifically proof theory and computational complexity theory, proof complexity is the field aiming to understand and analyse the computational resources that are required to prove or refute statements. Research in proof complexity is predominantly concerned with proving proof-length lower and upper bounds in various propositional proof systems. For example, among the major challenges of proof complexity is showing that the Frege system, the usual propositional calculus, does not admit polynomial-size proofs of all tautologies. Here the size of the proof is simply the number of symbols in it, and a proof is said to be of polynomial size if it is polynomial in the size of the tautology it proves. Systematic study of proof complexity began with the work of Stephen Cook and Robert Reckhow (1979), who provided the basic definition of a propositional proof system from the perspective of computational complexity. Specifically, Cook and Reckhow observed that proving proof size lower bounds on stronger and stronger propositional proof systems can be viewed as a step towards separating NP from co-NP (and thus P from NP), since the existence of a propositional proof system that admits polynomial size proofs for all tautologies is equivalent to NP = co-NP.…
Article summary
In logic and theoretical computer science, and specifically proof theory and computational complexity theory, proof complexity is the field aiming to understand and analyse the computational resources that are required to prove or refute statements. Research in proof complexity is predominantly concerned with proving proof-length lower and upper bounds in various propositional proof systems. For example, among the major challenges of proof complexity is showing that the Frege system, the usual propositional calculus, does not admit polynomial-size proofs of all tautologies. Here the size of the proof is simply the number of symbols in it, and a proof is said to be of polynomial size if it is polynomial in the size of the tautology it proves. Systematic study of proof complexity began with the work of Stephen Cook and Robert Reckhow (1979), who provided the basic definition of a propositional proof system from the perspective of computational complexity. Specifically, Cook and Reckhow observed that proving proof size lower bounds on stronger and stronger propositional proof systems can be viewed as a step towards separating NP from co-NP (and thus P from NP), since the existence of a propositional proof system that admits polynomial size proofs for all tautologies is equivalent to NP = co-NP.…
Key claims
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Discussion questions
- Summarise the article in your own words. Which single claim would you verify first, and why?
- Which claim in this article is most likely to be revised as the sources update? How would you check its current status?
- If two sources disagreed about one of these facts, what would you need in order to trust one over the other?
Bibliography
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Key claims and bibliography are generated from the same tracked claims and sources as the article page.