3 research outputs found
A Characterization of Locally Testable Affine-Invariant Properties via Decomposition Theorems
Let be a property of function for
a fixed prime . An algorithm is called a tester for if, given
a query access to the input function , with high probability, it accepts
when satisfies and rejects when is "far" from satisfying
. In this paper, we give a characterization of affine-invariant
properties that are (two-sided error) testable with a constant number of
queries. The characterization is stated in terms of decomposition theorems,
which roughly claim that any function can be decomposed into a structured part
that is a function of a constant number of polynomials, and a pseudo-random
part whose Gowers norm is small. We first give an algorithm that tests whether
the structured part of the input function has a specific form. Then we show
that an affine-invariant property is testable with a constant number of queries
if and only if it can be reduced to the problem of testing whether the
structured part of the input function is close to one of a constant number of
candidates.Comment: 27 pages, appearing in STOC 2014. arXiv admin note: text overlap with
arXiv:1306.0649, arXiv:1212.3849 by other author