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Excluding PH Pessiland

Authors Shuichi Hirahara, Rahul Santhanam

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Subject Classification

ACM Subject Classification
  • Theory of computation → Complexity classes
  • Theory of computation → Problems, reductions and completeness
Keywords
  • average-case complexity
  • pseudorandomness
  • meta-complexity

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Abstract

Heuristica and Pessiland are "worlds" of average-case complexity [Impagliazzo95] that are considered unlikely but that current techniques are unable to rule out. Recently, [Hirahara20] considered a PH (Polynomial Hierarchy) analogue of Heuristica, and showed that to rule it out, it would be sufficient to prove the NP-completeness of the problem GapMINKT^PH of estimating the PH-oracle time-bounded Kolmogorov complexity of a string.
In this work, we analogously define "PH Pessiland" to be a world where PH is hard on average but PH-computable pseudo-random generators do not exist. We unconditionally rule out PH-Pessiland in both non-uniform and uniform settings, by showing that the distributional problem of computing PH-oracle time-bounded Kolmogorov complexity of a string over the uniform distribution is complete for an (error-prone) average-case analogue of PH. Moreover, we show the equivalence between error-prone average-case hardness of PH and the existence of PH-computable pseudorandom generators.

Cite As Get BibTex

Shuichi Hirahara and Rahul Santhanam. Excluding PH Pessiland. In 13th Innovations in Theoretical Computer Science Conference (ITCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 215, pp. 85:1-85:25, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022) https://doi.org/10.4230/LIPIcs.ITCS.2022.85

Author Details

Shuichi Hirahara
  • Principle of Informatics Research Division, National Institute of Informatics, Tokyo, Japan
Rahul Santhanam
  • Department of Computer Science, University of Oxford, UK

Funding

  • Hirahara, Shuichi: Supported by JST, PRESTO Grant Number JPMJPR2024, Japan.
  • Santhanam, Rahul: Supported by ESPRC New Horizons Grant No. EP/V048201/1 on "Structure vs Randomness in Algorithms and Computation", UK.

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