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TR23-029 | 18th March 2023 04:33

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


Authors: Nicollas Sdroievski, Dieter van Melkebeek
Publication: 18th March 2023 04:34
Downloads: 442


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 $c=O(\log^2 a)$ 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.

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