A unified framework for testing linear-invariant properties

Arnab Bhattacharyya, Elena Grigorescu, Asaf Shapira

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

Abstract

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.

Original languageEnglish
Title of host publicationProceedings - 2010 IEEE 51st Annual Symposium on Foundations of Computer Science, FOCS 2010
PublisherIEEE Computer Society
Pages478-487
Number of pages10
ISBN (Print)9780769542447
DOIs
StatePublished - 2010
Externally publishedYes
Event2010 IEEE 51st Annual Symposium on Foundations of Computer Science, FOCS 2010 - Las Vegas, NV, United States
Duration: 23 Oct 201026 Oct 2010

Publication series

NameProceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS
ISSN (Print)0272-5428

Conference

Conference2010 IEEE 51st Annual Symposium on Foundations of Computer Science, FOCS 2010
Country/TerritoryUnited States
CityLas Vegas, NV
Period23/10/1026/10/10

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