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Excluding PH Pessiland

Authors Shuichi Hirahara, Rahul Santhanam

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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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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

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.

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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