Published: March 29, 2011

**Keywords:**property testing, Fourier analysis

**ACM Classification:**F.2.2

**AMS Classification:**68W20, 68Q25

**Abstract:**
[Plain Text Version]

We consider the task of testing properties of Boolean functions that are invariant under linear transformations of the Boolean cube. Previous work in property testing, including the linearity test and the test for Reed-Muller codes, has mostly focused on such tasks for linear properties. The one exception is a test due to Green for “triangle freeness:” a function $f:\mathbb{F}_{2}^{n}\to \{0,1\}$ has this property if $f(x),f(y),f(x+y)$ do not all equal $1$, for any pair $x,y\in \mathbb{F}_{2}^ {n}$.

Here we extend this test to a more systematic study of testing for
linear-invariant non-linear properties. We consider properties that
are described by a single forbidden pattern (and its linear transformations),
i.e., a property is given by $k$ points
$v_{1},\ldots,v_{k}\in\mathbb{F}_{2}^{k}$
and $f:\mathbb{F}_{2}^{n}\to \{0,1\}$ has the property that if for all
linear maps $L:\mathbb{F}_{2}^{k}\to\mathbb{F}_{2}^{n}$
it is the case that $f(L(v_{1})),\ldots,f(L(v_{k}))$
do not all equal $1$. We show that this property is testable if the
underlying matroid specified by $v_{1},\ldots,v_{k}$ is a graphic
matroid. This extends Green's result to an infinite class of new properties.
Part of our main results was obtained independently by
Král', Serra, and Venna
[*Journal of Combinatorial Theory Series A*, 116 (2009), pp 971--978].

Our techniques extend those of Green and in particular we establish a link between the notion of “$1$-complexity linear systems” of Green and Tao, and graphic matroids, to derive the results.