A Unified Framework for Testing Linear-Invariant Properties

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 l...

• Type: Journal
• Year: 2010
• Where: ECCC
• Authors: Arnab Bhattacharyya, Elena Grigorescu, Asaf Shapira
• Keywords: Boolean Functions, ECCC 2010, Linear-invariant Property, One-sided Error