Resolution proofs of generalized pigeonhole principles. (Note)
Theoretical Computer Science
Exponential lower bounds for the pigeonhole principle
Computational Complexity
Bounded arithmetic, propositional logic, and complexity theory
Bounded arithmetic, propositional logic, and complexity theory
An exponential lower bound to the size of bounded depth Frege proofs of the Pigeonhole Principle
Random Structures & Algorithms
A new proof of the weak pigeonhole principle
STOC '00 Proceedings of the thirty-second annual ACM symposium on Theory of computing
Time—space tradeoffs for satisfiability
Journal of Computer and System Sciences - Eleventh annual conference on computational learning theory&slash;Twelfth Annual IEEE conference on computational complexity
Resolution lower bounds for the weak pigeonhole principle
STOC '02 Proceedings of the thiry-fourth annual ACM symposium on Theory of computing
Time-Space Tradeoffs for Nondeterministic Computation
COCO '00 Proceedings of the 15th Annual IEEE Conference on Computational Complexity
FOCS '99 Proceedings of the 40th Annual Symposium on Foundations of Computer Science
Resolution Lower Bounds for Perfect Matching Principles
CCC '02 Proceedings of the 17th IEEE Annual Conference on Computational Complexity
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We show that the known bounded-depth proofs of the Weak Pigeonhole Principle PHPn2n in size nO(log(n)) are not optimal in terms of size. More precisely, we give a size-depth trade-off upper bound: there are proofs of size nO(d(log(n))2/d) and depth O(d). This solves an open problem of Maciel et al. (Proceedings of the 32nd Annual ACM Symposium on the Theory of Computing, 2000). Our technique requires formalizing the ideas underlying Nepomnjascij's Theorem which might be of independent interest. Moreover, our result implies a proof of the unboundedness of primes in Iδ0 with a provably weaker 'large number assumption' than previously needed.