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.