Finding Errorless Pessiland in Error-Prone Heuristica

Authors Shuichi Hirahara, Mikito Nanashima

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Shuichi Hirahara
  • National Institute of Informatics, Tokyo, Japan
Mikito Nanashima
  • Tokyo Institute of Technology, Japan


The authors would like to thank the anonymous reviewers for many helpful comments.

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Shuichi Hirahara and Mikito Nanashima. Finding Errorless Pessiland in Error-Prone Heuristica. In 37th Computational Complexity Conference (CCC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 234, pp. 25:1-25:28, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


Average-case complexity has two standard formulations, i.e., errorless complexity and error-prone complexity. In average-case complexity, a critical topic of research is to show the equivalence between these formulations, especially on the average-case complexity of NP. In this study, we present a relativization barrier for such an equivalence. Specifically, we construct an oracle relative to which NP is easy on average in the error-prone setting (i.e., DistNP ⊆ HeurP) but hard on average in the errorless setting even by 2^o(n/log n)-size circuits (i.e., DistNP ⊈ AvgSIZE[2^o(n/log n)]), which provides an answer to the open question posed by Impagliazzo (CCC 2011). Additionally, we show the following in the same relativized world: - Lower bound of meta-complexity: GapMINKT^𝒪 ∉ prSIZE^𝒪[2^o(n/log n)] and GapMCSP^𝒪 ∉ prSIZE^𝒪[2^(n^ε)] for some ε > 0. - Worst-case hardness of learning on uniform distributions: P/poly is not weakly PAC learnable with membership queries on the uniform distribution by nonuniform 2ⁿ/n^ω(1)-time algorithms. - Average-case hardness of distribution-free learning: P/poly is not weakly PAC learnable on average by nonuniform 2^o(n/log n)-time algorithms. - Weak cryptographic primitives: There exist a hitting set generator, an auxiliary-input one-way function, an auxiliary-input pseudorandom generator, and an auxiliary-input pseudorandom function against SIZE^𝒪[2^o(n/log n)]. This provides considerable insights into Pessiland (i.e., the world in which no one-way function exists, and NP is hard on average), such as the relativized separation of the error-prone average-case hardness of NP and auxiliary-input cryptography. At the core of our oracle construction is a new notion of random restriction with masks.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational complexity and cryptography
  • average-case complexity
  • oracle separation
  • relativization barrier
  • meta-complexity
  • learning
  • auxiliary-input cryptography


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