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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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