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Diagonalization in proof complexity

Jan Krajíček — 2004

Fundamenta Mathematicae

We study diagonalization in the context of implicit proofs of [10]. We prove that at least one of the following three conjectures is true: ∙ There is a function f: 0,1* → 0,1 computable in that has circuit complexity 2 Ω ( n ) . ∙ ≠ co . ∙ There is no p-optimal propositional proof system. We note that a variant of the statement (either ≠ co or ∩ co contains a function 2 Ω ( n ) hard on average) seems to have a bearing on the existence of good proof complexity generators. In particular, we prove that if a minor variant...

On the weak pigeonhole principle

Jan Krajíček — 2001

Fundamenta Mathematicae

We investigate the proof complexity, in (extensions of) resolution and in bounded arithmetic, of the weak pigeonhole principle and of the Ramsey theorem. In particular, we link the proof complexities of these two principles. Further we give lower bounds to the width of resolution proofs and to the size of (extensions of) tree-like resolution proofs of the Ramsey theorem. We establish a connection between provability of WPHP in fragments of bounded arithmetic and cryptographic assumptions (the existence...

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