  The study of the interplay between the testability of properties of Boolean
functions and the invariances acting on their domain which preserve the
property was initiated by Kaufman and Sudan (STOC 2008). Invariance with
respect to F_2-linear transformations is arguably the most common symmetry
exhibited by natural properties of Boolean functions on the hypercube. Hence,
an important goal in Property Testing is to describe necessary and sufficient
conditions for the testability of linear-invariant properties. This direction
was explicitly proposed for investigation in a recent survey of Sudan.
  We obtain the following results:
  1. We show that every linear-invariant property that can be characterized by
forbidding induced solutions to a (possibly infinite) set of linear equations
can be tested with one-sided error.
  2. We show that every linear-invariant property that can be tested with
one-sided error can be characterized by forbidding induced solutions to a
(possibly infinite) set of systems of linear equations.
  We conjecture that our result from item (1) can be extended to cover systems
of linear equations. We further show that the validity of this conjecture would
have the following implications:
  1. It would imply that every linear-invariant property that is closed under
restrictions to linear subspaces is testable with one-sided error. Such a
result would unify several previous results on testing Boolean functions, such
as the testability of low-degree polynomials and of Fourier dimensionality.
  2. It would imply that a linear-invariant property P is testable with
one-sided error if and only if P is closed under restrictions to linear
subspaces, thus resolving Sudan's problem.
