Lower bounds on the query complexity of non-uniform and adaptive reductions showing hardness amplification

• Authors: Sergei Artemenko;Ronen Shaltiel
• Venue: APPROX'11/RANDOM'11 Proceedings of the 14th international workshop and 15th international conference on Approximation, randomization, and combinatorial optimization: algorithms and techniques
• Year: 2011

Hardness amplification results show that for every function f there exists a function Amp(f) such that the following holds: if every circuit of size s computes f correctly on at most a 1 - δ fraction of inputs, then every circuit of size s′ computes Amp(f) correctly on at most a 1/2 + ε fraction of inputs. All hardness amplification results in the literature suffer from "size loss" meaning that s′ ≤ ε ċ s. In this paper we show that proofs using "non-uniform reductions" must suffer from size loss. To the best of our knowledge, all proofs in the literature are by non-uniform reductions. Our result is the first lower bound that applies to non-uniform reductions that are adaptive. A reduction is an oracle circuit R(ċ) such that when given oracle access to any function D that computes Amp(f) correctly on a 1/2 + ε fraction of inputs, RD computes f correctly on a 1 - δ fraction of inputs. A non-uniform reduction is allowed to also receive a short advice string α that may depend on both f and D in an arbitrary way. The well known connection between hardness amplification and list-decodable error-correcting codes implies that reductions showing hardness amplification cannot be uniform for ε We also prove the same lower bounds on the number of queries of nonuniform and adaptive reductions that are allowed to rely on arbitrary specific properties of the function f. Previous limitations on reductions were proven for "function-generic" hardness amplification, in which the non-uniform reduction needs to work for every function f and therefore cannot rely on specific properties of the function.