TY - GEN

T1 - A unified framework for testing linear-invariant properties

AU - Bhattacharyya, Arnab

AU - Grigorescu, Elena

AU - Shapira, Asaf

PY - 2010

Y1 - 2010

N2 - There has been a sequence of recent papers devoted to understanding the relation between the testability of properties of Boolean functions and the invariance of the properties with respect to transformations of the domain. Invariance with respect to F2-linear transformations is arguably the most common such symmetry for natural properties of Boolean functions on the hypercube. Hence, it is an important goal to find necessary and sufficient conditions for testability of linear-invariant properties. This is explicitly posed as an open problem in a recent survey of Sudan [1]. 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.

AB - There has been a sequence of recent papers devoted to understanding the relation between the testability of properties of Boolean functions and the invariance of the properties with respect to transformations of the domain. Invariance with respect to F2-linear transformations is arguably the most common such symmetry for natural properties of Boolean functions on the hypercube. Hence, it is an important goal to find necessary and sufficient conditions for testability of linear-invariant properties. This is explicitly posed as an open problem in a recent survey of Sudan [1]. 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.

UR - http://www.scopus.com/inward/record.url?scp=78751556852&partnerID=8YFLogxK

U2 - 10.1109/FOCS.2010.53

DO - 10.1109/FOCS.2010.53

M3 - פרסום בספר כנס

AN - SCOPUS:78751556852

SN - 9780769542447

T3 - Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS

SP - 478

EP - 487

BT - Proceedings - 2010 IEEE 51st Annual Symposium on Foundations of Computer Science, FOCS 2010

PB - IEEE Computer Society

Y2 - 23 October 2010 through 26 October 2010

ER -