A pseudorandom generator G n : f0; 1g is hard for a propositional proof system P if (roughly speaking) P can not ef- ciently prove the statement G n (x 1 ; : : : ; x n ) 6= b for any string b 2 . We present a function (m 2 ) generator which is hard for Res( log n); here Res(k) is the propositional proof system that extends Resolution by allowing k-DNFs instead of clauses.

We prove a quasi-polynomial lower bound on the size of bounded-depth Frege proofs of the pigeonhole principle PHP m n where m = (1 + 1/polylog n) n. This lower bound qualitatively matches the known quasipolynomial-size bounded-depth Frege proofs for these principles. Our technique, which uses a switching lemma argument like other lower bounds for bounded-depth Frege proofs, is novel in that the tautology to which this switching lemma is applied remains random throughout the argument.

Recent results established exponential lower bounds for the length of any Resolution proof for the weak pigeonhole principle. More formally, it was proved that any Resolution proof for the weak pigeonhole principle, with n holes and any number of pigeons, is of length ), (for a constant = 1=3). One corollary is that certain propositional formulations of the statement P 6= NP do not have short Resolution proofs. After a short introduction to the problem of P 6= NP and to the research area of propositional proof complexity, I will discuss the above mentioned lower bounds for the weak pigeonhole principle and the connections to the hardness of proving P 6= NP .

We initiate the study of the proof complexity of propositional encoding of (weak cases of) concrete independence results. In particular we study the proof complexity of Paris-Harrington’s Large Ramsey Theorem. We prove a conditional lower bound in Resolution and a quasipolynomial upper bound in bounded-depth Frege.