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

Abstract

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·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(•) 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 < 1/4. A reduction is non-adaptive if it makes non-adaptive queries to its oracle. Shaltiel and Viola (STOC 2008) showed lower bounds on the number of queries made by non-uniform reductions that are non-adaptive. We show that every non-uniform reduction must make at least Ω(1/ε) queries to its oracle (even if the reduction is adaptive). This implies that proofs by non-uniform reductions must suffer from size loss. We also prove the same lower bounds on the number of queries of non-uniform 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. © 2011 Springer-Verlag.