Propositional proof complexity studies the lengths of propositional proofs or equivalently the time complexity of non-deterministic algorithms accepting some coNP-complete set. The main problem is the NP versus coNP problem, a questionwhether the computational complexity class NP is closed under complementation. Central objects studied are propositional proof systems (non-deterministic algorithms accepting the set of propositional tautologies). Time lower bounds then correspond to lengths-of-proofs lower bounds.
Bounded arithmetic is a generic name for a collection of first-order and second-order theories of arithmetic linked to propositional proof systems (and to a variety of other computational complexity topics). The qualification bounded refers to the fact that the induction axiom is typically restricted to a subclass of bounded formulas.
The links between propositional proof systems and bounded arithmetic theories have many facets but informally one can view them as two sides of the same thing: the former is a non-uniform version of the latter. In particular, it is known that proving lengths-of-proofs lower bounds for propositional proof systems is very much related to proving independence results for bounded arithmetic theories. In fact, proving such lower bounds is equivalent to constructing non-elementary extensions of particular models of bounded arithmetic theories.
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