Document Open Access Logo

Instance-Wise Hardness Versus Randomness Tradeoffs for Arthur-Merlin Protocols

Authors Dieter van Melkebeek, Nicollas Mocelin Sdroievski

PDF
Thumbnail PDF

File

LIPIcs.CCC.2023.17.pdf
  • Filesize: 0.89 MB
  • 36 pages

Document Identifiers

Author Details

Dieter van Melkebeek
  • University of Wisconsin-Madison, WI, USA
Nicollas Mocelin Sdroievski
  • University of Wisconsin-Madison, WI, USA

Funding

Partial support for this research was provided by the University of Wisconsin-Madison Office of the Vice Chancellor for Research and Graduate Education with funding from the Wisconsin Alumni Research Foundation, and by the National Science Foundation under Grant No. 2312540.

Acknowledgements

We thank Ronen Shaltiel and Chris Umans for answering questions about their work, Oded Goldreich for helpful feedback on the write-up, and Lijie Chen for suggesting the potential use of PCPs during a presentation of our preliminary results.

Cite AsGet BibTex

Dieter van Melkebeek and Nicollas Mocelin Sdroievski. Instance-Wise Hardness Versus Randomness Tradeoffs for Arthur-Merlin Protocols. In 38th Computational Complexity Conference (CCC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 264, pp. 17:1-17:36, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)
https://doi.org/10.4230/LIPIcs.CCC.2023.17

Abstract

A fundamental question in computational complexity asks whether probabilistic polynomial-time algorithms can be simulated deterministically with a small overhead in time (the BPP vs. P problem). A corresponding question in the realm of interactive proofs asks whether Arthur-Merlin protocols can be simulated nondeterministically with a small overhead in time (the AM vs. NP problem). Both questions are intricately tied to lower bounds. Prominently, in both settings blackbox derandomization, i.e., derandomization through pseudo-random generators, has been shown equivalent to lower bounds for decision problems against circuits. Recently, Chen and Tell (FOCS'21) established near-equivalences in the BPP setting between whitebox derandomization and lower bounds for multi-bit functions against algorithms on almost-all inputs. The key ingredient is a technique to translate hardness into targeted hitting sets in an instance-wise fashion based on a layered arithmetization of the evaluation of a uniform circuit computing the hard function f on the given instance. In this paper we develop a corresponding technique for Arthur-Merlin protocols and establish similar near-equivalences in the AM setting. As an example of our results in the hardness to derandomization direction, consider a length-preserving function f computable by a nondeterministic algorithm that runs in time n^a. We show that if every Arthur-Merlin protocol that runs in time n^c for c = O(log² a) can only compute f correctly on finitely many inputs, then AM is in NP. Our main technical contribution is the construction of suitable targeted hitting-set generators based on probabilistically checkable proofs for nondeterministic computations. As a byproduct of our constructions, we obtain the first result indicating that whitebox derandomization of AM may be equivalent to the existence of targeted hitting-set generators for AM, an issue raised by Goldreich (LNCS, 2011). Byproducts in the average-case setting include the first uniform hardness vs. randomness tradeoffs for AM, as well as an unconditional mild derandomization result for AM.

Subject Classification

ACM Subject Classification
  • Theory of computation → Pseudorandomness and derandomization
Keywords
  • Hardness versus randomness tradeoff
  • Arthur-Merlin protocol
  • targeted hitting set generator

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

References

  1. Barış Aydınlıoğlu and Dieter van Melkebeek. Nondeterministic circuit lower bounds from mildly derandomizing Arthur-Merlin games. Computational Complexity, 26(1):79-118, 2017. URL: https://doi.org/10.1007/s00037-014-0095-y.
  2. László Babai, Lance Fortnow, Noam Nisan, and Avi Wigderson. BPP has subexponential time simulations unless EXPTIME has publishable proofs. Computational Complexity, 3(4):307-318, 1993. URL: https://doi.org/10.1007/BF01275486.
  3. László Babai and Shlomo Moran. Arthur-Merlin games: A randomized proof system, and a hierarchy of complexity classes. Journal of Computer and System Sciences, 36(2):254-276, 1988. URL: https://doi.org/10.1016/0022-0000(88)90028-1.
  4. Eli Ben‐Sasson, Oded Goldreich, Prahladh Harsha, Madhu Sudan, and Salil Vadhan. Robust PCPs of proximity, shorter PCPs, and applications to coding. SIAM Journal on Computing, 36(4):889-974, 2006. URL: https://doi.org/10.1137/S0097539705446810.
  5. Venkatesan T. Chakaravarthy and Sambuddha Roy. Arthur and Merlin as oracles. Computational Complexity, 20(3):505-558, 2011. URL: https://doi.org/10.1007/s00037-011-0015-3.
  6. L. Chen, R. D. Rothblum, and R. Tell. Unstructured hardness to average-case randomness. In Symposium on Foundations of Computer Science (FOCS), pages 429-437, 2022. URL: https://doi.org/10.1109/FOCS54457.2022.00048.
  7. Lijie Chen, Ron D. Rothblum, Roei Tell, and Eylon Yogev. On exponential-time hypotheses, derandomization, and circuit lower bounds: Extended abstract. In Symposium on Foundations of Computer Science (FOCS), pages 13-23, 2020. URL: https://doi.org/10.1109/FOCS46700.2020.00010.
  8. Lijie Chen and Roei Tell. Hardness vs randomness, revised: Uniform, non-black-box, and instance-wise. In Symposium on Foundations of Computer Science (FOCS), 2021. URL: https://doi.org/10.1109/FOCS52979.2021.00021.
  9. Oded Goldreich. In a world of P=BPP. In Studies in Complexity and Cryptography. Miscellanea on the Interplay between Randomness and Computation, pages 191-232. Springer, 2011. Part of the Lecture Notes in Computer Science book series (LNCS, volume 6650). URL: https://doi.org/10.1007/978-3-642-22670-0_20.
  10. Oded Goldreich. On doubly-efficient interactive proof systems. Foundations and Trends in Theoretical Computer Science, 13:157-246, 2018. URL: https://doi.org/10.1561/0400000084.
  11. Oded Goldreich. Two comments on targeted canonical derandomizers. In Computational Complexity and Property Testing: On the Interplay Between Randomness and Computation, pages 24-35. Springer, 2020. URL: https://doi.org/10.1007/978-3-030-43662-9_4.
  12. Shafi Goldwasser, Yael Tauman Kalai, and Guy N. Rothblum. Delegating computation: Interactive proofs for muggles. Journal of the ACM, 62(4), 2015. URL: https://doi.org/10.1145/2699436.
  13. Dan Gutfreund, Ronen Shaltiel, and Amnon Ta-Shma. Uniform hardness versus randomness tradeoffs for Arthur-Merlin games. Computational Complexity, 12(3):85-130, 2003. URL: https://doi.org/10.1007/s00037-003-0178-7.
  14. Russell Impagliazzo, Valentine Kabanets, and Avi Wigderson. In search of an easy witness: Exponential time vs. probabilistic polynomial time. Journal of Computer and System Sciences, 65(4):672-694, 2002. URL: https://doi.org/10.1016/S0022-0000(02)00024-7.
  15. Russell Impagliazzo and Avi Wigderson. P = BPP if E requires exponential circuits: Derandomizing the XOR lemma. In Symposium on Theory of Computing (STOC), page 220–229, 1997. URL: https://doi.org/10.1145/258533.258590.
  16. Russell Impagliazzo and Avi Wigderson. Randomness vs time: Derandomization under a uniform assumption. Journal of Computer and System Sciences, 63(4):672-688, 2001. URL: https://doi.org/10.1006/jcss.2001.1780.
  17. Adam R. Klivans and Dieter van Melkebeek. Graph nonisomorphism has subexponential size proofs unless the polynomial-time hierarchy collapses. SIAM Journal on Computing, 31(5):1501-1526, 2002. URL: https://doi.org/10.1137/S0097539700389652.
  18. Yanyi Liu. Personal communication, October 2022. Google Scholar
  19. Yanyi Liu and Rafael Pass. Characterizing derandomization through hardness of Levin-Kolmogorov complexity. In Computational Complexity Conference (CCC), volume 234, pages 35:1-35:17, 2022. URL: https://doi.org/10.4230/LIPIcs.CCC.2022.35.
  20. Yanyi Liu and Rafael Pass. Leakage-resilient hardness v.s. randomness. In Computational Complexity Conference (CCC), 2023. URL: https://eccc.weizmann.ac.il/report/2022/113/.
  21. Chi-Jen Lu. Derandomizing Arthur-Merlin games under uniform assumptions. Computational Complexity, 10(3):247-259, 2001. URL: https://doi.org/10.1007/s00037-001-8196-9.
  22. Peter Bro Miltersen and N. V. Vinodchandran. Derandomizing Arthur-Merlin games using hitting sets. Computational Complexity, 14(3):256-279, 2005. URL: https://doi.org/10.1007/s00037-005-0197-7.
  23. Noam Nisan and Avi Wigderson. Hardness vs randomness. Journal of Computer and System Sciences, 49(2):149-167, 1994. URL: https://doi.org/10.1016/S0022-0000(05)80043-1.
  24. Ronen Shaltiel and Christopher Umans. Simple extractors for all min-entropies and a new pseudorandom generator. Journal of the ACM, 52(2):172-216, 2005. URL: https://doi.org/10.1145/1059513f.1059516.
  25. Ronen Shaltiel and Christopher Umans. Low-end uniform hardness versus randomness tradeoffs for AM. SIAM Journal on Computing, 39(3):1006-1037, 2009. URL: https://doi.org/10.1137/070698348.
  26. Luca Trevisan and Salil Vadhan. Pseudorandomness and average-case complexity via uniform reductions. Computational Complexity, 16(4):331-364, 2007. URL: https://doi.org/10.1007/s00037-007-0233-x.
  27. Christopher Umans. Pseudo-random generators for all hardnesses. Journal of Computer and System Sciences, 67(2):419-440, 2003. URL: https://doi.org/10.1016/S0022-0000(03)00046-1.
  28. R. Ryan Williams. Natural proofs versus derandomization. SIAM Journal on Computing, 45(2):497-529, 2016. URL: https://doi.org/10.1137/130938219.
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2024 Schloss Dagstuhl – LZI GmbH Imprint Privacy Contact