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

Authors Dieter van Melkebeek, Nicollas Mocelin Sdroievski

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


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.

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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)


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
  • Hardness versus randomness tradeoff
  • Arthur-Merlin protocol
  • targeted hitting set generator


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  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:
  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:
  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:
  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:
  5. Venkatesan T. Chakaravarthy and Sambuddha Roy. Arthur and Merlin as oracles. Computational Complexity, 20(3):505-558, 2011. URL:
  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:
  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:
  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:
  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:
  10. Oded Goldreich. On doubly-efficient interactive proof systems. Foundations and Trends in Theoretical Computer Science, 13:157-246, 2018. URL:
  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:
  12. Shafi Goldwasser, Yael Tauman Kalai, and Guy N. Rothblum. Delegating computation: Interactive proofs for muggles. Journal of the ACM, 62(4), 2015. URL:
  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:
  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:
  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:
  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:
  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:
  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:
  20. Yanyi Liu and Rafael Pass. Leakage-resilient hardness v.s. randomness. In Computational Complexity Conference (CCC), 2023. URL:
  21. Chi-Jen Lu. Derandomizing Arthur-Merlin games under uniform assumptions. Computational Complexity, 10(3):247-259, 2001. URL:
  22. Peter Bro Miltersen and N. V. Vinodchandran. Derandomizing Arthur-Merlin games using hitting sets. Computational Complexity, 14(3):256-279, 2005. URL:
  23. Noam Nisan and Avi Wigderson. Hardness vs randomness. Journal of Computer and System Sciences, 49(2):149-167, 1994. URL:
  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:
  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:
  26. Luca Trevisan and Salil Vadhan. Pseudorandomness and average-case complexity via uniform reductions. Computational Complexity, 16(4):331-364, 2007. URL:
  27. Christopher Umans. Pseudo-random generators for all hardnesses. Journal of Computer and System Sciences, 67(2):419-440, 2003. URL:
  28. R. Ryan Williams. Natural proofs versus derandomization. SIAM Journal on Computing, 45(2):497-529, 2016. URL: