TY - GEN

T1 - Characterizing Derandomization Through Hardness of Levin-Kolmogorov Complexity

AU - Liu, Yanyi

AU - Pass, Rafael

N1 - Publisher Copyright: © Yanyi Liu and Rafael Pass

PY - 2022/7/1

Y1 - 2022/7/1

N2 - 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(nc) 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.

AB - 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(nc) 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.

KW - Derandomization

KW - Hitting Set Generators

KW - Kolmogorov Complexity

UR - http://www.scopus.com/inward/record.url?scp=85134402737&partnerID=8YFLogxK

U2 - https://doi.org/10.4230/LIPIcs.CCC.2022.35

DO - https://doi.org/10.4230/LIPIcs.CCC.2022.35

M3 - منشور من مؤتمر

T3 - Leibniz International Proceedings in Informatics, LIPIcs

BT - 37th Computational Complexity Conference, CCC 2022

A2 - Lovett, Shachar

T2 - 37th Computational Complexity Conference, CCC 2022

Y2 - 20 July 2022 through 23 July 2022

ER -