2026-07-25T02:25:35Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:16597 2024-03-06T10:57:53Z ddc:004 open_access

Characterizing Derandomization Through Hardness of Levin-Kolmogorov Complexity Liu, Yanyi Pass, Rafael Derandomization Kolmogorov Complexity Hitting Set Generators A central open problem in complexity theory concerns the question of whether all efficient randomized algorithms can be simulated by efficient deterministic algorithms. We consider this problem in the context of promise problems (i.e,. the prBPP v.s. prP problem) and show that for all sufficiently large constants c, the following are equivalent: - prBPP = prP. - For every BPTIME(n^c) algorithm M, and every sufficiently long z ∈ {0,1}ⁿ, there exists some x ∈ {0,1}ⁿ such that M fails to decide whether Kt(x∣z) is "very large" (≥ n-1) or "very small" (≤ O(log n)). where Kt(x∣z) denotes the Levin-Kolmogorov complexity of x conditioned on z. As far as we are aware, this yields the first full characterization of when prBPP = prP through the hardness of some class of problems. Previous hardness assumptions used for derandomization only provide a one-sided implication. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Yanyi Liu and Rafael Pass 2022 Is Part Of LIPIcs, Volume 234, 37th Computational Complexity Conference (CCC 2022) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.CCC.2022.35 urn:nbn:de:0030-drops-165975 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2022.35 eng https://creativecommons.org/licenses/by/4.0/legalcode