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Electronic Colloquium on Computational Complexity

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TR20-153 | 6th October 2020 02:11

Total Functions in the Polynomial Hierarchy



We identify several genres of search problems beyond NP for which existence of solutions is guaranteed. One class that seems especially rich in such problems is PEPP (for "polynomial empty pigeonhole principle"), which includes problems related to existence theorems proved through the union bound, such as finding a bit string that is far from all codewords, finding an explicit rigid matrix, as well as a problem we call Complexity, capturing Complexity Theory's quest. When the union bound is generous, in that solutions constitute at least a polynomial fraction of the domain, we have a family of seemingly weaker classes $\alpha$-PEPP, which are inside FP}$^{\text{NP}}|$poly. Higher in the hierarchy, we identify the constructive version of the Sauer-Shelah lemma and the appropriate generalization of PPP that contains it. The resulting total function hierarchy turns out to be more stable than the polynomial hierarchy: it is known that, under oracles, total functions within FNP may be easy, but total functions a level higher may still be harder than FP$^{\text{NP}}$.

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