DocumentCode :
1709197
Title :
A Unified Framework for Testing Linear-Invariant Properties
Author :
Bhattacharyya, Arnab ; Grigorescu, Eduard ; Shapira, Asaf
fYear :
2010
Firstpage :
478
Lastpage :
487
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. 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.
Keywords :
Boolean functions; linear algebra; set theory; Boolean function; F2-linear transformation; linear equation; linear subspace; linear-invariant property testing; one-sided error; Boolean functions; Hypercubes; Linearity; Mathematical model; Polynomials; Testing; linear invariance; property testing;
fLanguage :
English
Publisher :
ieee
Conference_Titel :
Foundations of Computer Science (FOCS), 2010 51st Annual IEEE Symposium on
Conference_Location :
Las Vegas, NV
ISSN :
0272-5428
Print_ISBN :
978-1-4244-8525-3
Type :
conf
DOI :
10.1109/FOCS.2010.53
Filename :
5671239
Link To Document :
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