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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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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.

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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) 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

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