Characterizing Derandomization Through Hardness of Levin-Kolmogorov Complexity

Yanyi Liu, Rafael Pass

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

Abstract

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.

Original languageEnglish
Title of host publication37th Computational Complexity Conference, CCC 2022
EditorsShachar Lovett
ISBN (Electronic)9783959772419
DOIs
StatePublished - 1 Jul 2022
Event37th Computational Complexity Conference, CCC 2022 - Philadelphia, United States
Duration: 20 Jul 202223 Jul 2022

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
Volume234

Conference

Conference37th Computational Complexity Conference, CCC 2022
Country/TerritoryUnited States
CityPhiladelphia
Period20/07/2223/07/22

Keywords

  • Derandomization
  • Hitting Set Generators
  • Kolmogorov Complexity

All Science Journal Classification (ASJC) codes

  • Software

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