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Lower Bounds on the Query Complexity of Non-uniform and Adaptive Reductions Showing Hardness Amplification

Abstract

Hardness amplification results show that for every Boolean function f, there exists a Boolean function Amp(f) such that if every size s circuit computes f correctly on at most a 1 − δ fraction of inputs, then every size s′ circuit computes Amp(f) correctly on at most a \({1/2+\epsilon}\) fraction of inputs. All hardness amplification results in the literature suffer from “size loss” meaning that \({s' \leq \epsilon \cdot s}\). We show that proofs using “non-uniform reductions” must suffer from such size loss.

A reduction is an oracle circuit \({R^{(\cdot)}}\) which given oracle access to any function D that computes Amp(f) correctly on a \({1/2+\epsilon}\) fraction of inputs, computes f correctly on a 1 − δ fraction of inputs. A non-uniform reduction is allowed to also receive a short advice string that may depend on both f and D. The well-known connection between hardness amplification and list-decodable error-correcting codes implies that reductions showing hardness amplification cannot be uniform for \({\epsilon < 1/4}\). We show that every non-uniform reduction must make at least \({\Omega(1/\epsilon)}\) queries to its oracle, which implies size loss. Our result is the first lower bound that applies to non-uniform reductions that are adaptive, whereas previous bounds by Shaltiel & Viola (SICOMP 2010) applied only to non-adaptive reductions. We also prove similar bounds for a stronger notion of “function-specific” reductions in which the reduction is only required to work for a specific function f.

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Correspondence to Ronen Shaltiel.

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Artemenko, S., Shaltiel, R. Lower Bounds on the Query Complexity of Non-uniform and Adaptive Reductions Showing Hardness Amplification. comput. complex. 23, 43–83 (2014). https://doi.org/10.1007/s00037-012-0056-2

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Keywords

  • Hardness amplification
  • black-box reductions

Subject classification

  • 68Q17
  • 68Q25