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LIPIcs - Leibniz International Proceedings in Informatics. 26th International Symposium on Theoretical Aspects of Computer Science
Doi
Abstract
We consider the task of testing properties of Boolean functions that are invariant under linear transformations of the Boolean cube. Previous work in property testing, including the linearity test and the test for Reed-Muller codes, has mostly focused on such tasks for linear properties. The one exception is a test due to Green for {}``triangle freeness\u27\u27: A function f:mathbbF2nβtomathbbF2β satisfies this property if f(x),f(y),f(x+y) do not all equal 1, for any pair x,yinmathbbF2nβ.
Here we extend this test to a more systematic study of testing for linear-invariant non-linear properties. We consider properties that are described by a single forbidden pattern (and its linear transformations), i.e., a property is given by k points v1β,ldots,vkβinmathbbF2kβ and f:mathbbF2nβtomathbbF2β satisfies the property that if for all linear maps L:mathbbF2kβtomathbbF2nβ it is the case that f(L(v1β)),ldots,f(L(vkβ)) do not all equal 1. We show that this property is testable if the underlying matroid specified by v1β,ldots,vkβ is a graphic matroid. This extends Green\u27s result to an infinite class of new properties.
Our techniques extend those of Green and in particular we establish a link between the notion of {}``1-complexity linear systems\u27\u27 of Green and Tao, and graphic matroids, to derive the results
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