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Oblivious Complexity Classes Revisited: Lower Bounds and Hierarchies

Authors Karthik Gajulapalli , Zeyong Li, Ilya Volkovich

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Author Details

Karthik Gajulapalli
  • Georgetown University, Washington, DC, USA
Zeyong Li
  • National University of Singapore, Singapore
Ilya Volkovich
  • Boston College, MA, USA

Acknowledgements

The authors would like to thank Alexander Golovnev, William Gasarch and Edward Hirsch for many helpful discussions and feedback on an earlier draft of this manuscript. The authors would also like to thank the anonymous referees for useful comments.

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Karthik Gajulapalli, Zeyong Li, and Ilya Volkovich. Oblivious Complexity Classes Revisited: Lower Bounds and Hierarchies. In 44th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 323, pp. 23:1-23:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.FSTTCS.2024.23

Abstract

In this work we study oblivious complexity classes. These classes capture the power of interactive proofs where the prover(s) are only given the input size rather than the actual input. In particular, we study the connections between the symmetric polynomial time - S₂P and its oblivious counterpart - O₂P. Among our results:  
- For each k ∈ ℕ, we construct an explicit language L_k ∈ O₂P that cannot be computed by circuits of size n^k. 
- We prove a hierarchy theorem for O₂TIME. In particular, for any time constructible function t:ℕ → ℕ and any ε > 0 we show that: O₂TIME[t(n)] ⊊ O₂TIME[t(n)^{1 + ε}].
- We prove new structural results connecting O₂P and S₂P. 
- We make partial progress towards the resolution of an open question posed by Goldreich and Meir (TOCT 2015) that relates the complexity of NP to its oblivious counterpart - ONP. 
- We identify a natural class of problems in O₂P from computational Ramsey theory, that are not expected to be in 𝖯 or even BPP. 
To the best of our knowledge, these results constitute the first explicit fixed-polynomial lower bound and hierarchy theorem for O₂P. The smallest uniform complexity class for which such lower bounds were previously known was S₂P due to Cai (JCSS 2007). In addition, this is the first uniform hierarchy theorem for a semantic class. All previous results required some non-uniformity. In order to obtain some of the results in the paper, we introduce the notion of uniformly-sparse extensions which could be of independent interest.
Our techniques build upon the de-randomization framework of the powerful Range Avoidance problem that has yielded many new interesting explicit circuit lower bounds.

Subject Classification

ACM Subject Classification
  • Theory of computation → Complexity classes
  • Theory of computation → Problems, reductions and completeness
  • Theory of computation → Circuit complexity
  • Theory of computation → Interactive proof systems
  • Theory of computation → Pseudorandomness and derandomization
Keywords
  • fixed circuit lower bounds
  • semantic time hierarchy
  • oblivious complexity
  • range avoidance

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