Lower bounds for DNF-refutations of a relativized weak pigeonhole principle
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The relativized weak pigeonhole principle states that if at least 2n out of n(2) pigeons fly into n holes, then some hole must be doubly occupied. We prove that every DNF-refutation of the CNF encoding of this principle requires size 2((log n)3/2-is an element of) for every is an element of > 0 and every sufficiently large n. By reducing it to the standard weak pigeonhole principle with 2n pigeons and n holes, we also show that this lower bound is essentially tight in that there exist DNF-refutations of size 2((log n)O(1)) even in R(log). For the lower bound proof we need to discuss the existence of unbalanced low-degree bipartite expanders satisfying a certain robustness condition.
CitacióAtserias, A.; Müller, M.; Oliva, S. Lower bounds for DNF-refutations of a relativized weak pigeonhole principle. "Journal of symbolic logic", 01 Juny 2015, vol. 80, núm. 2, p. 450-476.