Property Testing Lower Bounds via a Generalization of Randomized Parity Decision Trees

A few years ago, Blais, Brody, and Matulef: Comput. Complex. 21(2), 311-358 (2012) presented a methodology for proving lower bounds for property testing problems by reducing them from problems in communication complexity. Recently, Bhrushundi, Chakraborty, and Kulkarni (2014) showed that some reductions of this type can be deconstructed to two separate reductions, from communication complexity to randomized parity decision trees and from the latter to property testing. This work follows up on these ideas. We introduce a model called linear-access algorithms, which is a generalization of randomized parity decision trees, and show several methods to reduce communication complexity problems to problems for linear-access algorithms and problems for linear-access algorithms to property testing problems. This approach yields a new interpretation for several well-known reductions, since we present these reductions as a composition of two steps with fundamentally different functionalities. Furthermore, we demonstrate the potential of proving lower bounds on property testing problems by reducing them directly from problems for linear-access algorithms. In particular, we provide an alternative and simple proof for a known lower bound of (k) queries on testing k-linearity; that is, the property of k-sparse linear functions over F2. This alternative proof relies on a theorem by Linial and Samorodnitsky: Combinatorica 22(4), 497-522 (2002). We then extend this result to a new lower bound of (s) queries for testing s-sparse degree-d polynomials over F2, for any dN. In addition we provide a simple proof for the hardness of testing some families of linear subcodes.