Improved Bounds on the Weak
نویسنده
چکیده
We show that the known bounded-depth proofs of the Weak Pigeonhole Principle PHP 2n n in size n O(log(n)) are not optimal in terms of size. More precisely, we give a size-depth trade-oo upper bound: there are proofs of size n O(d(log(n)) 2=d) and depth O(d). This solves an open problem of Maciel, Pitassi and Woods (2000). Our technique requires formalizing the ideas underlying Nepomnja s cij's Theorem which might be of independent interest. Moreover, our result implies a proof of the unboundedness of primes in I0 with a provably weakerìarge number assumption' than previously needed.
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