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 - 10.4230/LIPIcs.CCC.2022.35
DO - 10.4230/LIPIcs.CCC.2022.35
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AN - SCOPUS:85134402737
T3 - Leibniz International Proceedings in Informatics, LIPIcs
BT - 37th Computational Complexity Conference, CCC 2022
A2 - Lovett, Shachar
PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
T2 - 37th Computational Complexity Conference, CCC 2022
Y2 - 20 July 2022 through 23 July 2022
ER -