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

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.