Bhattacharyya–Grigorescu–Shapira Conjecture (testability of linear-invariant properties)

Show that for any (possibly infinite) collection of affine linear constraints {(Mi, σi)}, the property Pn of functions f: F2^n → {0,1} being (Mi, σi)-free for all i is testable by a one-sided error, constant-query property tester.

Background

The conjecture is motivated by the program to characterize testability of linear-invariant function properties. Such properties are subspace-hereditary, and many special cases (e.g., rank-1 constraints or certain finite-field regimes with bounded Cauchy–Schwarz complexity) are known to be testable.

Despite progress, the general case remains unresolved. Even the subcase of (M,σ)-freeness for arbitrary σ was conjectured and remains open, highlighting the difficulty beyond rank-1 scenarios.